Spectral spaces 9781107146723, 9781316543870

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Spectral Spaces Spectral spaces are a class of topological spaces. They are a tool linking algebraic structures, in a very wide sense, with geometry. They were invented to give a functional representation of Boolean algebras and distributive lattices and subsequently gained great prominence as a consequence of Grothendieck’s invention of schemes. There are more than 1000 research articles about spectral spaces, but this is the first monograph. It provides an introduction to the subject and is a unified treatment of results scattered across the literature, filling in gaps and showing the connections between different results. The book includes new research going beyond the existing literature, answering questions that naturally arise from the comprehensive approach. The authors serve graduates by starting gently with the basics. For experts, they lead them to the frontiers of current research, making this book a valuable reference source. M a x D i c k m a n n has been a researcher at the CNRS, France, since 1974; Directeur de Recherche since 1988; emeritus since 2007. His research interests include the applications of spectral spaces to real algebraic geometry, quadratic forms, and related topics. N i e l s S c h wa r t z is Professor of Mathematics at the Universit¨at Passau, retired since 2016. Many of his publications are concerned with, or use, spectral spaces in essential ways. In particular, he has used spectral spaces to introduce the notion of real closed rings, an important topic in real algebra and geometry. M a r c u s Tr e s s l is a mathematician working in the School of Mathematics at the University of Manchester. His research interests include model theory, ordered algebraic structures, ring theory, differential algebra, and non-Hausdorff topology.

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M. Cabanes and M. Enguehard Representation Theory of Finite Reductive Groups J. B. Garnett and D. E. Marshall Harmonic Measure P. Cohn Free Ideal Rings and Localization in General Rings E. Bombieri and W. Gubler Heights in Diophantine Geometry Y. J. Ionin and M. S. Shrikhande Combinatorics of Symmetric Designs S. Berhanu, P. D. Cordaro and J. Hounie An Introduction to Involutive Structures A. Shlapentokh Hilbert’s Tenth Problem G. Michler Theory of Finite Simple Groups I A. Baker and G. W¨ustholz Logarithmic Forms and Diophantine Geometry P. Kronheimer and T. Mrowka Monopoles and Three-Manifolds B. Bekka, P. de la Harpe and A. Valette Kazhdan’s Property (T) J. Neisendorfer Algebraic Methods in Unstable Homotopy Theory M. Grandis Directed Algebraic Topology G. Michler Theory of Finite Simple Groups II R. Schertz Complex Multiplication S. Bloch Lectures on Algebraic Cycles (2nd Edition) B. Conrad, O. Gabber and G. Prasad Pseudo-reductive Groups T. Downarowicz Entropy in Dynamical Systems C. Simpson Homotopy Theory of Higher Categories E. Fricain and J. Mashreghi The Theory of H(b) Spaces I E. Fricain and J. Mashreghi The Theory of H(b) Spaces II J. Goubault-Larrecq Non-Hausdorff Topology and Domain Theory ´ J. Sniatycki Differential Geometry of Singular Spaces and Reduction of Symmetry E. Riehl Categorical Homotopy Theory B. A. Munson and I. Voli´c Cubical Homotopy Theory B. Conrad, O. Gabber and G. Prasad Pseudo-reductive Groups (2nd Edition) J. Heinonen, P. Koskela, N. Shanmugalingam and J. T. Tyson Sobolev Spaces on Metric Measure Spaces Y.-G. Oh Symplectic Topology and Floer Homology I Y.-G. Oh Symplectic Topology and Floer Homology II A. Bobrowski Convergence of One-Parameter Operator Semigroups K. Costello and O. Gwilliam Factorization Algebras in Quantum Field Theory I J.-H. Evertse and K. Gy˝ory Discriminant Equations in Diophantine Number Theory G. Friedman Singular Intersection Homology S. Schwede Global Homotopy Theory M. Dickmann, N. Schwartz and M. Tressl Spectral Spaces

Spectral Spaces MAX DICKMANN Centre National de la Recherche Scientifique (CNRS), Paris, France N I E L S S C H WA RT Z Universit¨at Passau, Germany MARCUS TRESSL University of Manchester, UK

University Printing House, Cambridge CB2 8BS, United Kingdom One Liberty Plaza, 20th Floor, New York, NY 10006, USA 477 Williamstown Road, Port Melbourne, VIC 3207, Australia 314–321, 3rd Floor, Plot 3, Splendor Forum, Jasola District Centre, New Delhi – 110025, India 79 Anson Road, #06–04/06, Singapore 079906 Cambridge University Press is part of the University of Cambridge. It furthers the University’s mission by disseminating knowledge in the pursuit of education, learning, and research at the highest international levels of excellence. www.cambridge.org Information on this title: www.cambridge.org/9781107146723 DOI: 10.1017/9781316543870 © Max Dickmann, Niels Schwartz, and Marcus Tressl 2019 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 2019 Printed and bound in Great Britain by Clays Ltd, Elcograf S.p.A. A catalogue record for this publication is available from the British Library. Library of Congress Cataloging-in-Publication Data Names: Dickmann, M. A., 1940– author. | Schwartz, Niels, 1950– author. | Tressl, Marcus, 1964– author. Title: Spectral spaces / Max Dickmann, Centre National de la Recherche Scientifique (CNRS), Niels Schwartz, Universit¨at Passau, Germany, Marcus Tressl, University of Manchester. Description: First edition. | New York : Cambridge University Press, [2019] | Series: New mathematical monographs ; 35 | Includes bibliographical references and index. Identifiers: LCCN 2018045567 | ISBN 9781107146723 (hardback : alk. paper) Subjects: LCSH: Topological spaces. | Geometry, Algebraic. | Commutative rings. Classification: LCC QA611.25 .D53 2019 | DDC 514/.32–dc23 LC record available at https://lccn.loc.gov/2018045567 ISBN 978-1-107-14672-3 Hardback Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this publication and does not guarantee that any content on such websites is, or will remain, accurate or appropriate.

Contents

Preface

page ix

An Outline of the History of Spectral Spaces

xiii

1

Spectral Spaces and Spectral Maps 1.1 The Definition of Spectral Spaces 1.2 Spectral Maps and the Category of Spectral Spaces 1.3 Boolean Spaces and the Constructible Topology 1.4 The Inverse Topology 1.5 Specialization and Priestley Spaces 1.6 Examples 1.7 Further Reading

1 2 10 13 23 27 36 46

2

Basic Constructions 2.1 Spectral Subspaces 2.2 Products of Spectral Spaces 2.3 Spectral Subspaces of Products 2.4 Finite Coproducts 2.5 Zariski, Real, and Other Spectra

48 49 52 57 64 66

3

Stone Duality 3.1 The Spectrum of a Bounded Distributive Lattice 3.2 Stone Duality 3.3 Spectral Spaces via Prime Ideals and Prime Filters 3.4 The Boolean Envelope of a Bounded Distributive Lattice 3.5 Inverse Spaces and Inverse Lattices 3.6 The Spectrum of a Totally Ordered Set 3.7 Further Reading

78 79 82 88 92 94 96 98

v

vi

Contents

4

Subsets of Spectral Spaces 4.1 Quasi-Compact Subsets, Closure, and Generalization 4.2 Directed Subsets and Specialization Chains 4.3 Rank and Dimension 4.4 Minimal Points and Maximal Points 4.5 Convexity and Locally Closed Sets and Points

102 103 107 113 118 131

5

Properties of Spectral Maps 5.1 Images of Proconstructible Sets under Spectral Maps 5.2 Monomorphisms and Epimorphisms 5.3 Closed and Open Spectral Maps 5.4 Embeddings 5.5 Irreducible Maps and Dominant Maps 5.6 Extending Spectral Maps

141 142 145 151 156 163 165

6

Quotient Constructions 6.1 Spectral Quotients Modulo Relations 6.2 Saturated Relations 6.3 Spectral Orders and Spectral Relations 6.4 Quotients Modulo Equivalence Relations and Identifying Maps 6.5 Spectral Quotients and Lattices 6.6 The Space of Connected Components

168 169 174 179

7

Scott Topology and Coarse Lower Topology 7.1 When Scott is Spectral 7.2 Fine Coherent Posets and Complete Lattices 7.3 The Coarse Lower Topology on Root Systems and Forests 7.4 Finite and Infinite Words

205 206 221 230 239

8

Special Classes of Spectral Spaces 8.1 Noetherian Spaces 8.2 Spectral Spaces with Scattered Patch Space 8.3 Heyting Spaces 8.4 Normal Spectral Spaces 8.5 Spectral Root Systems and Forests

246 247 261 267 279 290

9

Localic Spaces 9.1 Frames and Completeness 9.2 Localic Spaces – Spectra of Frames 9.3 Localic Maps 9.4 Localic Subspaces 9.5 Localic Points

299 300 306 311 315 322

186 197 199

Contents

vii

10

Colimits in Spec 10.1 Coproducts 10.2 Fiber Sums 10.3 Colimits 10.4 Constructions with Fiber Sums

328 329 340 354 360

11

Relations of Spec with Other Categories 11.1 The Spectral Reflection of a Topological Space 11.2 The Sobrification 11.3 Spectral Reflections of Continuous Maps 11.4 Properties of Topological Spaces and their Spectral Reflections 11.5 How Localic Spaces are Located in the Category of Spectral Spaces 11.6 The Categories Spec and PoSets 11.7 The Subcategory BoolSp of Spec

369 371 384 390

The Zariski Spectrum 12.1 The Zariski Spectrum – Topology on the Set of Prime Ideals of a Ring 12.2 Functoriality 12.3 Locally Closed Points and the Nullstellensatz 12.4 The Spectrum of a Noetherian Ring 12.5 Zariski Spectra under Ring-Theoretic Constructions 12.6 Hochster’s Results

416 419 434 442 450 458 469

13

The Real Spectrum 13.1 Motivation and Elementary Examples 13.2 Specialization and Maximal Points 13.3 Spectral Morphisms Induced by Ring Homomorphisms 13.4 Real Spectra under Ring-Theoretic Constructions 13.5 The Real Spectrum in Real Algebraic Geometry 13.6 Further Results and Reading

485 487 501 504 515 518 531

14

Spectral Spaces via Model Theory 14.1 The Model-Theoretic Setup 14.2 Spectral Spaces of Types 14.3 Spectra of Structures and their Elementary Description

540 541 543 560

12

393 402 408 412

viii

Contents

Appendix: The Poset Zoo

579

References

590

Index of Categories and Functors

607

Index of Examples

608

Symbol Index

613

Subject Index

618

Preface

Spectral spaces constitute a class of topological spaces used in various branches of mathematics. They were introduced in the 1930s by M. H. Stone and have been used extensively ever since. There was a marked growth of interest following A. Grothendieck’s revolution of algebraic geometry. It was realized that spectral spaces can be associated with many mathematical structures. Numerous publications are devoted to various properties of spectral spaces and to a growing number of diverse applications. The area is extremely active and is growing at a fast pace. With this book we provide the first comprehensive and coherent treatment of the basic topological theory of spectral spaces. It is possible to study spectral spaces largely with algebraic tools, namely using bounded distributive lattices, or, in more abstract form, using category theory, model theory, or topos theory. However, our focus is clearly on the topology, which provides geometric tools and intuition for applications that, a priori, do not have geometric meaning. Also, in our experience, the topological techniques are very flexible towards possible extensions of techniques and results to wider classes of spaces, where a corresponding algebraic framework does not exist. We start with a careful analysis of the definition of spectral spaces, describe fundamental structural features, and discuss elementary properties. Numerous examples, counterexamples, and constructions, listed in an index of examples, show how one can work with spectral spaces in concrete situations or illustrate results. We exhibit methods illustrating how spectral spaces can be associated with different classes of structures and describe some of the most important applications.

ix

x

Preface

It was our original intention to assemble basic material about spectral spaces in one place to make it more easily accessible. Collecting the material and preparing a coherent presentation proved to be more laborious than anticipated: the terminology and notation differ from publication to publication. The Zariski spectrum of commutative unital rings is undoubtedly the most widely used construction of spectral spaces. Therefore, many results on spectral spaces are found in publications about rings and are expressed in the corresponding language. These needed to be translated into topological language to make them compatible with our intentions and presentation. There are numerous points, some small, some more substantial, about basic topological properties of spectral spaces that had not been considered before, but have to be studied in a comprehensive treatment. So, to round off the picture, we had to fill various gaps in the existing literature. The construction of new spaces from a collection of given spaces is an important topic to which we added new facets. The category of spectral spaces (i.e., spectral spaces with spectral maps as morphisms) is naturally related to several other categories – for example, the categories of topological spaces and of partially ordered sets. The precise relationship with these other categories had previously been studied only in a fragmentary way. In our book we do this systematically, which requires the development of suitable tools. Taking all of this into account, after starting off writing a book in the style of a graduate text, our project has turned, to a considerable extent, into a research monograph. We think that our text is suitable for graduate students who study spectral spaces and for researchers who find the subject relevant and need a solid basis for their own work. This may be in algebra, algebraic geometry, ordered algebraic structures, partially ordered sets, universal algebra, logic, or theoretical computer science. We hope that we have succeeded in writing a book that can serve these different interests. Writing a book requires decisions about what to include and what not to include. So far we have described what is included. Now a word about what is not included. There are various classes of topological spaces that are not spectral, but are related to them and serve purposes that are similar to those of spectral spaces (e.g., representation spaces in algebra or the Ziegler spectrum of module categories). We focus our attention (almost) entirely on spectral spaces. But we point out that our topological approach to spectral spaces may sometimes provide substantial clues on how to deal with questions about related, or broader, classes of spaces. Spectral spaces have a richer theory and offer a more coherent panorama than related larger classes of spaces, which we address

Preface

xi

sporadically at a few points in the book. The applications of spectral spaces are so extensive that they clearly merit an in-depth investigation. We try to keep the presentation as elementary and concrete as possible. We start with basic definitions and give complete proofs of all results. Concreteness means, first of all, that we include a large number of examples illustrating notions and results, as well as showing how spectral spaces can be produced and handled. Concreteness also means that we are generally not satisfied with just showing that a particular construction with spectral spaces exists. Rather, we always strive to describe the points and the topology of a new space that is produced by a construction. For example, it is not difficult to show that spectral spaces have arbitrary colimits. This statement is a pure existence result. We want to reach a deeper level of understanding and give a detailed description of a colimit (at least in special cases) in terms of the points and topologies of the spaces that are used to produce it. Despite all efforts to keep the book elementary, concrete, and accessible, the text requires a certain amount of mathematical maturity, as well as basic knowledge in several areas. We name some general references where everything we need and use can be found. • General topology, [Eng89], [Kel75]. • Algebraic structures such as rings and fields, [AtMa69], [Jac85], [Jac89], [Mat80], [Lan02]. • Partially ordered sets, lattices, and Boolean algebras, [Bly05], [Hal63], [Har05], [Kop89]. • The language of category theory, [HeSt79], [Mac71]. • Notions of domain theory, [GHK+ 03]. • Model theory, [Hod93].

Acknowledgments We want to express our gratitude to the people and organizations that helped and supported us during the writing of the book. We thank the Centre International de Rencontres Mathématiques (CIRM), Luminy, France, for its hospitality and material support for many of our working meetings, especially in the period 2007–2011. Our thanks go to the SCI Tuileries d’Affiac, whose house in the southwest of France hosted many of our working sessions, mainly during the years 2011–2017. Similarly, the Université Paris 7 (Diderot) and the MIMS at the University of Manchester hosted us several times. The Universität Passau provided technical support. Thanks are also due to the Nesin Mathematical Village in Sirince, Turkey, for supporting our meeting

xii

Preface

in spring 2015, as well as to the University of Camerino, Italy, for the invitation in summer 2017. Big thanks go to Anne Klingenberg for having us in Saint-Léger en Yvelines in fall 2011. Finally, we wish to thank Lorna Gregory (Marcus Tressl) and Luise Mayer (Niels Schwartz) for their support and patience during the many years of our work for this book.

Max Dickmann Niels Schwartz Marcus Tressl

An Outline of the History of Spectral Spaces

A review of some principal landmarks in the development of the theory of spectral spaces sheds some light on this special part of general topology. It explains the growing general interest in the subject and, in parts, our motivation for writing this book. Spectral spaces first appeared through the work of Marshall H. Stone in [Sto37b]; one identifiable motivation was the quest for a topological representation of Brouwerian logic. Related to ideas of MacNeille’s dissertation, cf. [Mac36] and Birkhoff’s [Bir33], Stone extended his celebrated work on the duality between Boolean algebras and Boolean spaces ([Sto36], [Sto37a]) to the realm of distributive lattices. 1 He introduced a topology on the set of prime ideals of a distributive lattice; the resulting space is nowadays called the spectrum of the lattice. Stone then singled out certain properties of the spectrum and proved in [Sto37b, Theorems 15,16] that any topological space with these properties is the spectrum of a distributive lattice; this is the birth of spectral spaces and the essence of Stone duality between distributive lattices and spectral spaces, cf. Chapter 3. A spectral space is not Hausdorff, unless it is Boolean. As in Stone’s earlier duality between Boolean algebras and Boolean spaces, the distributive lattice associated with a spectral space can be conceived of as an algebra 2 of continuous functions. Explicitly, the Boolean algebra 2 = {0, 1} is topologized such that the singleton {0} is the only nontrivial open set. Then a distributive lattice is the lattice of continuous functions from its spectrum to 2. 1 2

To simplify the terminology we assume in this outline of the history that distributive lattices are bounded (i.e., have a smallest and a largest element). By an algebraic structure, or an algebra, we mean a universal algebra enriched by a set of relations, which is also known (in model theory) as a first-order structure.

xiii

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An Outline of the History of Spectral Spaces

It was quickly realized that arbitrary rings (not necessarily commutative) can be represented in a related way. Jacobson introduced a topology on the set of primitive ideals of a not necessarily commutative ring, [Jac45, p. 234], and called this the structure space. Jacobson refers to [Sto37a] and Gelfand– Silov, cf. [GeŠi41], for the topology he used. Even if he does not use Stone’s work in a technical way, he seems to have taken inspiration from Stone. His way of defining the topology follows Stone’s lead. Before Jacobson, roughly simultaneously with Stone, the study of rings of continuous functions was under way. This research direction picked up speed in the 1940s, see the bibliography of [GiJe60] (which includes references to some of Stone’s work). The idea of representing algebraic structures via something like continuous functions was an active research topic. Possibly starting with Jacobson, this idea took hold in abstract algebra, where rings are represented via structure spaces. In the case of a commutative unital ring, the primitive ideals are exactly the maximal ideals. Going one step further, one can use the set of all prime ideals to represent a commutative ring. In general, this leads to a more faithful representation. The topologies introduced by Stone and by Jacobson are built on the same principle. These type of topologies were later named hull–kernel topologies. By the same method, the Zariski spectrum of a commutative unital ring is the set of prime ideals equipped with a natural hull–kernel topology. It is difficult to pinpoint who was first to define this topology (i.e., who actually invented the Zariski spectrum and who first named it after Zariski). 3 For early sources about the spectrum we refer to [Gro60, p. 80] and [Bou61, p. 124], where the word “spectrum” is used and the topology with its main properties is described, in particular showing that the axioms of a spectral space are satisfied. Another important development, dating from the end of the 1940s, was the introduction (by Zariski, [Zar52, p. 79 f]) of a topology on any classical algebraic variety over a field. (A variety is the common zero set of a set of polynomials.) For each variety, the subvarieties are a basis of closed subsets for a topology, which was later called the Zariski topology. In general, this is not a Hausdorff topology. A variety over the real field or the complex field also carries the Euclidean topology. The Zariski topology is coarser than the Euclidean topology, but is defined in much greater generality (e.g., in the case of varieties over fields with positive characteristic). With each classical variety one associates its coordinate ring (i.e., its ring 3

Also it seems to be unknown who first called this space a “spectrum.” Stone, who was a functional analyst, saw connections with spectra of operators, as evidenced by his papers [Sto40], [Sto41]. For a simple formal connection between spectra of rings and spectra of operators, consider a finite-dimensional k-vector space V with an operator T : V → V . One can view V as a module over the k-algebra k[T ]. Then the spectrum of the operator is essentially the same as the Zariski spectrum of k[T ].

An Outline of the History of Spectral Spaces

xv

of polynomial functions). This is akin to the ring of continuous functions on topological spaces. The coordinate ring has a Zariski spectrum. Sending a point of the variety to the maximal ideal of all polynomial functions vanishing at the point, defines a map from the variety to the Zariski spectrum of the coordinate ring. It is injective, and its image is dense in the Zariski spectrum in a very strong sense. The topology of the Zariski spectrum restricts to the Zariski topology on the variety. Thus, the Zariski spectrum of the coordinate ring is essentially the same thing as the variety with its Zariski topology. More explanations can be found in the introduction to Chapter 12 and in 12.3.15. Considering algebraic geometry from this point of view, a variety consists of a ring of polynomial functions together with its Zariski spectrum. Going one step further in this direction, one can consider arbitrary commutative unital rings with their Zariski spectrum as a generalization of varieties. This leads us to the starting point of Grothendieck’s new algebraic geometry. Its basic building blocks are affine schemes, namely: each commutative unital ring A, defines a corresponding affine scheme. The underlying space is the Zariski spectrum of the ring, and the rings of sections are locally defined in terms of rings of fractions of A. The given ring A is the ring of global sections of the sheaf. Thus, informally speaking, in this way every commutative ring can be viewed as a ring of functions on its Zariski spectrum. This brings us back to Jacobson and the original idea of representing a ring as a ring of functions. Apparently Grothendieck never referred to Stone’s work on spectral spaces (cf. [Joh86, p. xix]). But, even if the details are not well documented, one can perceive an evolution of ideas from Stone to Jacobson and, eventually, to Grothendieck. The next landmark came, at the end of the 1960s, with Hochster’s dissertation, [Hoc67], [Hoc69]. The functor from rings to spectral spaces sending a ring A to its Zariski spectrum is not a duality, as in the case of distributive lattices. However, Hochster showed that every spectral space is homeomorphic to the Zariski spectrum of some ring, although in a provably non-canonical way. Hochster also coined the name “spectral spaces” for the topological spaces satisfying Stone’s axioms, cf. [Hoc69, Introduction]. Hochster’s results received a great deal of attention. In view of Grothendieck’s work in algebraic geometry it was clear that Zariski spectra, hence spectral spaces, are an important class of topological spaces. There was a growing realization that one can associate spectral spaces with many other kinds of mathematical structures. Often this does not require much more than an adaptation of the construction of spectra of distributive lattices or of Zariski spectra, mutatis mutandis. Other constructions include (the list is far from exhaustive):

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An Outline of the History of Spectral Spaces

• Spectra of Abelian -groups (= lattice-ordered groups) and of f -rings, [Kei71], [BKW77, Chapitre 10] and very much related: spectra of MValgebras, [BDNS94], [DuPo10]. • The real spectrum of a ring, [CoRo82], [BCR87, Chapitre 7], also treated in Chapter 13 of the present book. • Differential spectra, for example [Kei77]. • Spectra in logic, for example [Èsa74], [Èsa04], [McTa44], [McTa46], [Lou83]. • Spectra used in sheaf representations of structures, [Die84], [Joh86, Chapter V]. • Valuation spectra of all sorts, for example [HuKn94], [Rob86]. Shortly after Hochster’s work, Priestley in [Pri70], gave an alternative description of the representation spaces of distributive lattices. Referring to Stone’s characterization, she intended to give “a much simpler characterization in terms of ordered topological space,” loc. cit., Introduction. Her representation spaces are Boolean spaces together with a partial order (which we call a spectral order) satisfying a compatibility condition. These partially ordered Boolean spaces are now called Priestley spaces. Priestley spaces are the same as the representation spaces of Stone (i.e., spectral spaces), but under a somewhat different clothing, cf. [Pri74] and [Fle00]. The achievements of Hochster and Priestley were picked up by mathematicians right away. However, the development progressed in almost disjoint communities. This has not changed much over the years and is a further motivation for writing this book. Priestley spaces and their connection with spectral spaces are presented in Section 1.5. We use Priestley’s version of spectral spaces primarily for the construction of examples. Starting in the late 1970s, spectral spaces played a key role in real algebraic geometry (i.e., in the study of algebraic varieties defined over the field of real numbers). Real algebraic geometry had existed for a long time, but very much in the shadow of complex algebraic geometry. In the 1970s this began to change slowly. The development gathered momentum and became very dynamic with the introduction of the real spectrum of a ring by Coste and Roy (see [CoRo82], as well as references therein, and Chapter 13 of the present book). Similar to the map from a classical variety to the Zariski spectrum of its coordinate ring (see above), every real variety can be mapped injectively and densely into the real spectrum of its coordinate ring. It is a major difference compared with arbitrary varieties and Zariski spectra that the topology of the real spectrum restricts to the Euclidean topology on the variety. Nowadays the real spectrum is an indispensable tool in real algebra and real algebraic geometry.

An Outline of the History of Spectral Spaces

xvii

Over the past two decades we have seen about 400 publications directly studying, or significantly using, spectral spaces. This was triggered by applications in an enormous range of mathematical subject areas like theoretical computer science, category theory, functional analysis, representation theory, to name just a few. In addition to these there are the classical core areas of topology, lattices, ring theory, algebraic geometry, and logic. We hope that the present book will give a common ground to the subject and help readers to navigate its wider ramifications.

1 Spectral Spaces and Spectral Maps

Spectral spaces are a class of topological spaces. They were first described in terms of three topological conditions (but not yet named) in [Sto37b, Theorems 15, 16]. A slightly different, but equivalent, axiomatization was given in [Hoc67], [Hoc69], where the name spectral space was also introduced. Hochster used four axioms, which we present and discuss in Section 1.1. The axioms contain conditions familiar to every mathematician – the separation axiom T0 and quasi-compactness. The other conditions say that there is a distinguished basis of open sets and that the nonempty closed and irreducible subsets correspond to the points of the space. Spectral spaces can be related to each other via continuous maps, since they are topological spaces. But arbitrary continuous maps between spectral spaces do not connect the distinguished bases of domain and codomain with each other, thus disregarding decisive features of the spaces. Therefore, in Section 1.2, a more suitable class of maps is introduced, the spectral maps. The spectral spaces and the spectral maps together form a category, which provides an excellent framework for the further study of spectral spaces. Spectral spaces carry a great deal of structure – besides the defining topology there are two other topologies, the patch topology, also called the constructible topology, cf. Section 1.3, and the inverse topology, cf. Section 1.4. In every topological space a binary relation, called specialization, is defined by: x  y if and only if the neighborhood filter of y is contained in the neighborhood filter of x. In the case of T0 -spaces, hence of spectral spaces, the relation is a partial order. In Section 1.5 the specialization order is analyzed in the context of spectral spaces. A spectral space is uniquely determined by its patch topology and the specialization order. This fact is the basis for another approach to spectral spaces, which is due to H. Priestley and leads to the notion of a Priestley space, see [Pri70] and numerous other publications. In Section 1.5 we show that spectral spaces and Priestley spaces are the same mathemati1

2

Spectral Spaces and Spectral Maps

cal structures, just viewed differently. The focus of the book is on the topology of spectral spaces, but Priestley spaces are a particularly valuable tool for the construction of examples. Various simple spectral spaces can be produced ad hoc and are presented throughout to illustrate the basic notions. The chapter closes with Section 1.6, where a detailed presentation of several examples and constructions is given, highlighting the different structural features of spectral spaces developed in the preceding sections.

1.1 The Definition of Spectral Spaces Summary Spectral spaces are a class of topological spaces defined by four axioms, see 1.1.5. We analyze each of the axioms on its own and explore the consequences of combinations of different axioms. The definition, as well as the characterization given in 1.1.14, indicate that the quasi-compact open sets play a key role in spectral spaces. In 1.1.15 it is shown that every finite T0 -space is spectral, which gives us a first collection of examples. To start with, it is necessary to explain some terminology and notation that is used throughout. References for basic facts from general topology are [Bou71b], [Eng89] or [Kel75]. 1.1.1 Some Notation and Terminology Let X be a topological space. The set of open subsets (i.e., the topology) is denoted by O(X); the set of closed subsets is denoted by A(X). Both O(X) and A(X) contain ∅ , X, and are closed under finite unions and finite intersections. Thus, O(X) and A(X) are bounded sublattices 1 of the Boolean algebra P(X), the power set of X. It follows that they are distributive lattices. Moreover, O(X) is closed under arbitrary unions and A(X) is closed under arbitrary intersections. Thus, they are even complete lattices – but usually infinite meets in O(X) and infinite joins in A(X) do not coincide with those in P(X). Our topological spaces are typically not Hausdorff. Therefore some care is needed with regard to the terminology we use. In the literature a large part of the terminology for topological spaces is adapted to the needs of analysis and assumes the Hausdorff separation axiom. Whenever we use names that may cause confusion, we shall always explain the way we use them. 1

That is, sublattices containing the smallest and largest elements of P(X).

1.1 The Definition of Spectral Spaces

3

1.1.2 Quasi-Compact Sets A topological space is quasi-compact if every open cover has a finite subcover. A space is compact if it is quasi-compact and Hausdorff. 2 Now let X be any topological space. A subset S of X is quasi-compact if it is quasi-compact in its relative topology. Finite unions of quasi-compact subsets of X are quasi-compact, but finite intersections need not be quasi-compact. The subsets of X that are at the same time open and quasi-compact play a crucial role in this book. We call them quasi-compact open sets and write ◦

K (X) := {O ⊆ X | O is quasi-compact open}. ◦

◦

If τ denotes the topology of X, we also write K (τ) instead of K (X). Note that ◦

∅ ∈ K (X) for any topological space X. There are many spaces, in particular ◦ in classical analysis, having no other quasi-compact open sets. The set K (X) is closed under finite unions, hence is a join-subsemilattice of P(X). 1.1.3 Specialization Every topological space X carries a quasi-order, 3 which is called the specialization order and is defined by: x  y if and only if y ∈ {x}. 4 We say that y is a specialization of x, and x is a generalization of y. If the . topology is denoted by τ then we also write  τ Specialization can be checked using any subbasis S of open sets: x  y if and only if y ∈ O and O ∈ S imply x ∈ O. A subset A ⊆ X is closed under specialization, or specialization-closed, if x ∈ A and x  y implies y ∈ A. The set is closed under generalization, or generically closed, if y ∈ A and x  y implies x ∈ A. Closed subsets are closed under specialization, open subsets are closed under generalization. Any set A ⊆ X is contained in a smallest set closed under specialization,  {x}, Spez(A) = {y ∈ X | ∃x ∈ A: x  y} = x∈A

and in a smallest set closed under generalization, Gen(A) = {x ∈ X | ∃y ∈ A: x  y}. 2 3 4

The reader should be aware that there is no agreement in the literature on whether “compact spaces” are Hausdorff or not. A quasi-order is a reflexive and transitive binary relation and need not be antisymmetric, A.1(i). There is no uniform way in the literature to define specialization, cf. 7.1.7. See 12.1.14 for our motivation.

4

Spectral Spaces and Spectral Maps

If A = {a} is a singleton set then we also write Spez(A) = Spez(a) and Gen(A) = Gen(a). Continuous maps preserve specialization (i.e., if f : X → Y is continuous and x  x  in X then f (x)  f (x ) in Y ). Suppose that X carries two topologies σ and τ. If σ ⊆ τ then the specializais stronger than  (i.e., x  x  implies x  x ). tion relation  σ τ τ σ 1.1.4 Reminder (a) Suppose X is a set and S ⊆ P(X). The set S separates points in X if for all x  y in X there is some S ∈ S that contains exactly one of the points (we do not specify which one!). In particular, a topological space X is a T0 -space if O(X) separates points in X. 5 (b) A subset C of a topological space X is irreducible if for all closed subsets A, B ⊆ X with C ⊆ A ∪ B we have C ⊆ A or C ⊆ B. Notice that a set is irreducible if and only if its closure is irreducible. Thus, C is irreducible if and only if C, as an element of the bounded distributive lattice A(X), is join-irreducible (cf. A.6(vii)). Clearly, (closures of) singletons are irreducible. 1.1.5 Definition A spectral space is a topological space X that satisfies the following four conditions. S1: X is quasi-compact and T0 . ◦

S2: K (X) is a basis of open subsets of X. S3: The intersection of two quasi-compact open subsets of X is again quasicompact. S4: X is sober, that is, for every nonempty closed and irreducible subset C of X, there is a point x ∈ X, necessarily unique (by S1), with C = {x}. The topology of X is called the spectral topology. We start by looking at the conditions in Definition 1.1.5 separately, discuss ◦

some basic facts, and record their impact on the set K (X). 1.1.6 The T0 Property It is easily checked that a topological space X has the T0 -property if and only if specialization is a partial order (i.e., is an antisymmetric quasi-order). Thus, a spectral space is partially ordered by specialization. The specialization relation is irrelevant for many spaces in classical analysis, for these are mostly T1 -spaces, 6 and one characterization of T1 -spaces says that every singleton is a closed set. Thus, the specialization order of a T1 -space is the trivial partial order – every point is comparable only with itself. Usually 5 6

T0 -spaces are also called Kolmogorov spaces. Recall that a topological space is a T1 -space if, given two distinct points, each one has a neighborhood not containing the other one.

1.1 The Definition of Spectral Spaces

5

spectral spaces are not T1 -spaces, and we shall see that the specialization order is one of their essential features. Thus, partially ordered sets (also called posets, A.1(iii)) play an essential role throughout this book. For notation and terminology, we refer the reader to the Poset Zoo in the Appendix. 1.1.7 On Bases of Topological Spaces Suppose X is an arbitrary topological space and L ⊆ O(X) is a basis of the topology closed under finite unions ◦

(e.g., L could be K (X) if S2 is satisfied, cf. 1.1.2, 1.1.5). Thus, L is a joinsubsemilattice of O(X). If O ⊆ X is open, then the set i(O) := {U ∈ L | U ⊆ O} is an ideal of the join-semilattice L (i.e., i(O) is closed under finite unions and, if U ⊆ U  ∈ i(O) and U, U  ∈ L, then U ∈ i(O), cf. A.7(i)). The hypothesis that L is a basis allows us to recover O from i(O). For then, every open set O can be written as  U. (∗) O= U ∈i(O)

Let I(L) be the set of ideals of L. Then the definition of i(O) yields a map i : O(X) → I(L). Now (∗) says that the function j : I(L) → O(X); I →



I =



U,

U ∈I

 satisfies j ◦ i = id O(X) . In fact, i(O) is the largest subset S ⊆ L with O = S. Obviously, i ◦ j(I) ⊇ I for all I ∈ I(L). In general, this inclusion is proper. For example, let X ⊆ R be the closed unit interval, take L = O(X), and let I ⊆ L be the ideal of open subsets of X whose closure does not contain 0. Then j(I) = (0, 1] ∈ i ◦ j(I) \ I. 1.1.8 On Axiom S2 Let X be a topological space and suppose that B is a basis ◦

of the topology, contained in K (X) and closed under finite unions (including the empty union, which is ∅). Then every U ∈ O(X) is a union of sets from B. ◦

By quasi-compactness, every V ∈ K (X) is a finite union of sets from B, hence ◦ is a member of B. We conclude that B = K (X). ◦

Thus, if X has a basis contained in K (X), then there is a unique one that is ◦

closed under finite unions, namely K (X).

6

Spectral Spaces and Spectral Maps

1.1.9 Proposition

Suppose that X is a topological space satisfying S2 (i.e.,

◦

◦

K (X) is a basis of the topology). Then j : I(K (X)) → O(X) and i : O(X) → ◦ I(K (X)) are mutually inverse bijective maps. ◦

Proof As K (X) is closed under finite unions, we can apply 1.1.7. For each ◦ I ∈ I(K (X)) the inclusion I ⊆ i ◦ j(I) holds trivially. It remains to show ◦

that i ◦ j(I) ⊆ I. So, pick an element U ∈ i ◦ j(I) (i.e., U ∈ K (X) and  U ⊆ j(I) = I). As U is quasi-compact there is a finite subset J ⊆ I such   that U ⊆ J. The ideal I is closed under finite unions, hence J ∈ I. Thus U is contained in an element of I, and, since I is an ideal, we get U ∈ I, as claimed.  1.1.10 On Axiom S3 Observe that we have not invoked◦ axiom S3 so far. Now suppose that a topological space X satisfies S3. Thus, K (X) is a sublattice of O(X) since it is always closed under finite unions, 1.1.2, and, assuming S3, is also closed under finite intersections. If, in addition, X is quasi-compact (e.g., ◦

if X is a spectral space), then K (X) ⊆ O(X) is even a bounded sublattice. ◦

Given a spectral space X, the lattice K (X) is an important part of its structure. It builds a bridge between spectral spaces and lattices. In Chapter 3 we shall see that this connection is much closer than the present considerations show. In ◦

fact, the spectral space X can be fully recovered from the lattice K (X) alone. Assuming the axioms S1, S2, and S3, the next result shows that the bijective ◦

correspondence of 1.1.9 between the ideals of K (X) and the open sets of X restricts to a bijection between the complements of nonempty closed and ◦

irreducible subsets of X and the prime ideals of the lattice K (X). 1.1.11 Proposition Suppose that X is a topological space and L ⊆ O(X) is a bounded sublattice and a basis of the topology. Let C ⊆ X be a closed set and set I = i(X \ C) = {U ∈ L | U ∩ C = ∅}. Then I is an ideal of L and (i) C  ∅ if and only if I is a proper ideal (i.e., I  L ). (ii) C is a nonempty and irreducible set if and only if I is a prime ideal (i.e., I  L and for all U,V ∈ L with U ∩ V ∈ I we have U ∈ I or V ∈ I). Proof

By 1.1.7 we know that I is an ideal of L.

(i) Clearly, I = L if C = ∅. Conversely, I = L implies X ∈ I, hence C = X ∩C = ∅.

1.1 The Definition of Spectral Spaces

7

(ii) First suppose that C is irreducible and take U,V ∈ L with U ∩ V ∈ I. Then U ∩ V ∩ C = ∅ and so C ⊆ (X \ U) ∪ (X \ V). As C is irreducible we have C ⊆ X \ U or C ⊆ X \ V (i.e., U ∈ I or V ∈ I). Conversely, suppose C is not irreducible. Take closed sets A1, A2 ⊆ X with C  Ai , but C ⊆ A1 ∪ A2 . There are points ci ∈ C ∩ (X \ Ai ) and neighborhoods Ui ∈ L of ci with Ui ⊆ X \ Ai . We see that Ui  I. On the other hand, the inclusion U1 ∩ U2 ⊆ (X \ A1 ) ∩ (X \ A2 ) shows that (U1 ∩ U2 ) ∩ C ⊆ U1 ∩ U2 ∩ (A1 ∪ A2 ) = ∅. Thus U1 ∩ U2 ∈ I, and I is not prime.



1.1.12 Corollary Let X be a topological space satisfying axioms S1–S3. Then the map ◦

{C ∈ A(X) | C  ∅ irreducible} → {I ∈ I(K (X)) | I prime} C → i(X \ C) 

is bijective.

1.1.13 On Axiom S4 and Soberness Suppose that X is a topological space. Every set {x} is closed, irreducible, and nonempty. If A = {x}, then x is called a generic point of A, cf. A.2. In general, a nonempty closed and irreducible set need not have a generic point and, if it has one, there might be several of them (e.g., if X is an indiscrete space with at least two points). Axiom S4 says that the sets {x} are the only nonempty closed and irreducible sets. A space is T0 if and only if every closed and irreducible set A has at most one generic point. Thus, in this case A has a smallest element for specialization. Hence the points of a sober T0 -space are in bijection with the nonempty closed and irreducible sets. 1.1.14 Conclusion The preceding considerations yield the following alternative characterization of spectral spaces, which highlights the key role played by ◦

the lattice K (X). A topological space X is spectral if and only if ◦

(i) the set K (X) is a bounded sublattice of O(X), separates points of X, is a basis of the topology, and ◦

(ii) for every prime ideal I ⊆ K (X) there is a unique point x ∈ X such that ◦ I = {U ∈ K (X) | x  U}. In 1.1.10 we announced that a spectral space X is completely determined by ◦

the lattice K (X). This will be proved in Chapter 3. The present characterization

8

Spectral Spaces and Spectral Maps

is a first step in this direction. It shows how the points of X can be reconstructed ◦

from the lattice K (X). We exhibit a collection of first examples of spectral spaces. They are all finite spaces. Trivially, if X is a finite topological space then every subset is quasi◦

compact, hence K (X) = O(X). Only the T0 -property and soberness have to be discussed. 1.1.15 Proposition Every finite T0 -space is spectral. Proof It suffices to show soberness (axiom S4). So, let C be a nonempty closed and irreducible set. Since C is finite, the set  {c} c ∈C

is closed, hence equals C. As C is irreducible it cannot be covered by finitely many proper closed subsets. Therefore there is some c ∈ C with C = {c}, as required.  1.1.16 Finite Spectral Spaces vs. Finite Posets According to 1.1.15 the finite T0 -spaces are exactly the finite spectral spaces. Moreover, we emphasize that the finite T0 -spaces essentially coincide with the finite posets. One assigns the underlying specialization poset to a finite T0 -space. Conversely, one notes that a finite poset (P, ≤) carries a unique T0 -topology having specialization order ≤. Namely, the open sets are exactly the down-sets (i.e., the fine lower topology of (P, ≤), which coincides with the coarse lower topology, has this property; see Poset Zoo, A.8). 1.1.17 Example There is a unique topology on the empty set. The topology is trivially T0 , hence ∅ is a spectral space. Every singleton carries a unique topology, which, again, is trivially T0 , hence makes the set a spectral space. We write 1 for the spectral space with underlying set 1 = {0}. 1.1.18 Example The set 2 = {0, 1} carries four different topologies. The only one that is not T0 is the indiscrete topology. The other three topologies are spectral. Explicitly: (i) The discrete topology is spectral. The specialization order is trivial (i.e., every element is comparable only with itself). (ii) The fine lower topology for the total order 0 < 1 has the open sets ∅, {0}, 2. The set 2 with this topology is called the Sierpiński space and is denoted by 2. The Sierpiński space is spectral, the point 0 is isolated and not

1.1 The Definition of Spectral Spaces

9

closed, whereas the point 1 is closed and not isolated. The specialization order is the natural total order, that is, 0  1. (iii) Interchanging the role of 0 and 1 in the Sierpiński space (i.e., starting with the total order 1 < 0), we obtain a spectral space with universe 2 and specialization relation 1  0. 1.1.19 Example Given n ∈ N, the set n = {0, . . . , n − 1} carries many topologies (except for the cases discussed in the previous examples), but we are interested in only three of them. (i) The discrete topology is spectral, again with trivial specialization order. (ii) The natural total order yields a T0 -topology whose open sets are the intervals {0, . . . , k − 1}, 0 ≤ k ≤ n. The set n with this topology is denoted by n. The space n is spectral and the specialization order is the natural total order, that is 0  1  · · ·  n − 1. (iii) Reversing the natural order in n we obtain a spectral space with universe n and specialization relation n − 1  n − 2  · · ·  0. In fact, any topology on n with total specialization order is homeomorphic to n via a suitable permutation of {0, . . . , n − 1}. 1.1.20 Saturated and Coherent Sets We shall encounter topological spaces that have some properties in common with spectral spaces, but are not necessarily (or not a priori) spectral. One important such property is coherence. The definition of coherence requires the following notion. Let X be a T0 -space with topology τ. A subset Q ⊆ X is called saturated if Q is an intersection of open sets. As X is a T0 -space this is the same as saying that Q is generically closed (i.e., x  q ∈ Q implies x ∈ Q, 1.1.3). 7 Note that open sets are generically closed, hence so are their intersections. Conversely, if  Q is generically closed then Q = xQ X \ {x}, hence Q is saturated. A saturated set Q is called quasi-compact saturated if it is also quasicompact. In every topological space there are plenty of quasi-compact saturated sets. Namely, for each x ∈ X, the set Gen(x) of generalizations of x is quasicompact saturated. 7

Equivalently, Q is a down-set of the poset (X,  ), or is open for the fine lower topology, τ A.8(ii).

10

Spectral Spaces and Spectral Maps ◦

Claim Assume that K (X) is a subbasis of open sets for X. Then every quasicompact saturated set is an intersection of quasi-compact open sets. Proof of Claim Let Q ⊆ X be quasi-compact saturated and pick x ∈ X \ Q.   Then X \ {x} = i ∈I k ∈Fi Uik , where the Fi are finite and the Uik belong to ◦  K (X). For each i there is some ki ∈ Fi with x  Uiki . Thus, Q ⊆ i ∈I Uiki ⊆ X \ {x}, and, by quasi-compactness of Q, there is a finite subset J ⊆ I with  Q ⊆ i ∈J Uiki ⊆ X \ {x}. Finite unions of quasi-compact open sets are quasi◦   compact open. Thus, i ∈J Uiki ∈ K (X), proving the claim. Later we shall see that in a spectral space the quasi-compact saturated sets are precisely the subsets of X that are closed for the inverse topology, cf. 1.4.7, also see 1.5.5 and 4.1.6. The saturated sets are closed under arbitrary unions and intersections, whereas, in general, quasi-compact sets are only closed under finite unions and not under intersections. Thus, finite unions of quasi-compact saturated sets are quasi-compact saturated, but intersections need not be quasi-compact saturated. Therefore the following definition is introduced: the space X is coherent if the intersection of two quasi-compact saturated sets is again quasi-compact saturated. Spectral spaces are coherent. This is immediate from the fact (mentioned above) that the quasi-compact saturated sets are exactly the subsets closed for the inverse topology. So, we could have replaced the spectral space axiom S3 by the condition that coherence holds.

1.2 Spectral Maps and the Category of Spectral Spaces Summary Spectral maps are the appropriate tool to describe connections between different spectral spaces. Spectral maps are continuous, but satisfy a stronger requirement, see 1.2.2 and, for motivation, 1.2.1. The spectral spaces together with the spectral maps form the category of spectral spaces, cf. 1.2.3, which is denoted by Spec. The category plays a key role in the development of the theory and in many applications, see Section 2.5 for first examples. Properties of Spec are an important topic throughout the book. The notion of a spectral map is illustrated by examples involving finite spectral spaces. The examples are used to obtain first pieces of information about the category Spec, 1.2.7 and 1.2.8.

1.2 Spectral Maps and the Category of Spectral Spaces

11

1.2.1 From Continuous Maps to Spectral Maps To motivate our definition of spectral maps we recall the notion of a continuous map between topological spaces. Any map f : X → Y between two sets induces a map between their power sets, P( f ) : P(Y ) → P(X), B → f −1 (B), which preserves arbitrary unions and intersections. Now suppose that X and Y are topological spaces. Continuity of the map f is defined by the requirement: • The map P( f ) restricts to a map O( f ) : O(Y ) → O(X). Trivially, O( f ) is a homomorphism of bounded lattices. ◦

The essential role played by the lattice K (X) in the definition of spectral spaces leads to the following notion of a spectral map. 1.2.2 Definition Let X and Y be spectral spaces and f : X → Y be a map between the underlying sets. Then f is called a spectral map if the map ◦

◦

◦

P( f ) : P(Y ) → P(X) restricts to a map K ( f ) : K (Y ) → K (X), that is, inverse images of quasi-compact open sets are quasi-compact open. 1.2.3 The Category Spec The identity map of a spectral space is a spectral map, and compositions of spectral maps are spectral maps. Therefore, the class of spectral spaces together with the spectral maps as morphisms forms a category, which is denoted by Spec and is called the category of spectral spaces. The topological spaces and the continuous maps form the category Top of topological spaces. We note that the construction of O(X), 1.1.1, and O( f ), 1.2.1, yields a functor Top → BDLat (cf. A.6(xv)). Spectral maps f : X → Y ◦

between spectral spaces are continuous, since K (Y ) is a basis of open sets for Y ◦

and f −1 (U) is open for U ∈ K (Y ). In general, a continuous map between spectral spaces is not spectral, since preimages of quasi-compact open sets under a continuous map are always open, but may not be quasi-compact. Examples are given in 1.3.8 and in 2.1.2. Thus, Spec is a subcategory of Top, but not a full subcategory. ◦

The map K ( f ) associated with a spectral map f : X → Y is a homomorphism ◦

of bounded lattices. Thus, the construction K that associates the bounded dis◦ ◦ tributive lattice K (X) with the spectral space X and the map K ( f ) with the ◦

spectral map f is a contravariant functor K : Spec → BDLat, where BDLat is the category of bounded distributive lattices with bounded lattice homomorphisms, A.6(viii). Our first examples of spectral maps involve finite spectral spaces, either as domain or as codomain. Particularly prominent finite spaces are those shown

12

Spectral Spaces and Spectral Maps

in 1.1.17, 1.1.18, and 1.1.19. We start with a general fact about spectral maps with finite domain. 1.2.4 Proposition Suppose that X is a finite spectral space, and let Y be any spectral space. The following conditions on a map f : X → Y are equivalent: (i) f is continuous. (ii) f is spectral. (iii) f is a homomorphism for the posets (X,  ) and (Y,  ). Proof Even without finiteness, (ii) ⇒ (i) is trivial and (i) ⇒ (iii) holds true by 1.1.3. So we only need to show that every poset homomorphism f : (X,  ) → ◦

(Y,  ) is spectral if X is finite. Picking V ∈ K (Y ) we have to show that ◦ f −1 (V) ∈ K (X). Since X is finite, it suffices to show that f −1 (V) is a down-set (cf. 1.1.16). Being open, V is a down-set, and preimages of down-sets under poset homomorphisms are again down-sets.  1.2.5 Example Suppose that X is a spectral space. The unique map ∅ → X is a spectral map. If x ∈ X, then the unique map fx : 1 → X, 0 → x, is spectral. A map f : 2 → X is spectral if and only if f (0)  f (1). Given n ∈ N, a map f : n → X is spectral if and only if f (0)  f (1)  · · ·  f (n − 1). 1.2.6 Example Let X be a spectral space. The unique map f : X → 1 is clearly spectral. A map f : X → 2 is spectral if and only if f −1 (0) is open and quasi-compact. ◦

Thus, for every U ∈ K (X) the characteristic function χX\U : X → 2 (i.e., x → 0 for x ∈ U, x → 1 if x  U) is spectral. Given n ∈ N, a map f : X → n is spectral if and only if the sets Uk−1 = −1 f ({0, ..., k − 1}), 0 ≤ k ≤ n, form a chain of quasi-compact open sets. Conversely, every such sequence ∅ = U−1 ⊆ U0 ⊆ · · · ⊆ Un−1 = X yields a spectral map f : X → 2, which is given by: f (x) = k if x ∈ Uk \ Uk−1 . The maps exhibited in 1.2.5 and 1.2.6 can be used to draw conclusions about the category Spec. 1.2.7 Corollary The category Spec has an initial object, the empty space ∅, and a final object, the one-point space 1. The spectral spaces 1 and 2 can be used to distinguish two distinct spectral maps with the same domain and the same codomain. To recall the categorytheoretic context, we ask whether there exist spectral spaces Z and T with the following properties:

1.3 Boolean Spaces and the Constructible Topology

13

• For all spectral maps f , g : X → Y with f  g there is a spectral map h : Y → Z such that h ◦ f  h ◦ g. If such a space Z exists then it is called a co-separator, [HeSt79, p. 74], [AHS90, 7.16]. • For all spectral maps f , g : X → Y with f  g there is a spectral map e : T → X such that f ◦ e  g ◦ e. If such a space T exists then it is called a separator, [HeSt79, p. 74], [AHS90, 7.10]. 1.2.8 Corollary The one-element space 1 is a separator for the category Spec, and the Sierpińksi space 2 is a co-separator. Proof Let f , g : X → Y be spectral maps, f  g. Then there is an element x ∈ X with f (x)  g(x). Consider the spectral map fx : 1 → X defined in 1.2.5. Then f ◦ fx  g ◦ fx . Thus, 1 is a separator. ◦

To show that 2 is a co-separator we may assume that there is some U ∈ K (Y ) such that f (x) ∈ U and g(x)  U. Consider the spectral map χY \U : Y → 2  defined in 1.2.6. Then χY \U ◦ f  χY \U ◦ g, which proves the assertion. For further reading suggestions we note that spectral spaces are also studied as a full subcategory of Top. Thus, in this category the morphisms between spectral spaces are all continuous maps, not only the spectral maps. We refer to [vG12] and [vG14, Section 2.2] for further information.

1.3 Boolean Spaces and the Constructible Topology Summary Besides its (spectral) topology, every spectral space comes naturally equipped with two other topologies: the constructible topology (also called the patch topology) and the inverse topology. In the present section we introduce and analyze the constructible topology. The inverse topology is studied in Section 1.4. Let X be a spectral space. The underlying set, equipped with the constructible topology, is denoted by Xcon and is called the patch space of X, cf. 1.3.11. The space Xcon is Boolean, 1.3.14. In fact, the Boolean spaces are exactly the patch spaces of spectral spaces, see 1.3.16, or, equivalently, are the spectral spaces that are Hausdorff, 1.3.4. Therefore we start with Boolean spaces, recalling a few basic facts that are needed in the sequel. Boolean spaces are a classical topic; for more information we refer to [Kop89, p. 96 ff]. The patch topology is defined in 1.3.11. It is used to introduce the important classes of constructible subsets and of proconstructible subsets of a spectral

14

Spectral Spaces and Spectral Maps ◦

space. The lattice K (Xcon ) is the Boolean algebra of clopen (= closed and open) subsets of Xcon , see 1.3.15. A map f : X → Y between spectral spaces is a spectral map if and only if it is continuous both for the spectral topologies and for the constructible topologies on X and Y , 1.3.21. The functorial nature of the patch space construction is summarized in 1.3.24. For basic facts about Boolean algebras we refer to [Kop89]. 1.3.1 Definition A topological space is Boolean if it is compact (i.e., quasicompact and Hausdorff) and has a basis of clopen sets. Boolean spaces are also called Stone spaces, cf. [Joh86, II 4.2, bottom of p. 70]. 1.3.2 Total Disconnectedness and Clopen Subsets There is another characterization of Boolean spaces, which is frequently taken as a definition. A space is Boolean if and only if it is compact and totally disconnected (i.e., the only nonempty and connected sets are singletons), see [Bou71b, p. TG I.83], [Kop89, Theorem 7.5], [GiJe60, Theorem 16.17], or [Joh86, Theorem II 4.2, p. 69]. (In [Eng89, p. 360] totally disconnected spaces are called hereditarily disconnected.) Given any topological space X, the set of clopen subsets is denoted by Clop(X), and is a Boolean algebra. If X is a Boolean space, its clopen subsets are a basis of the topology and are (quasi-)compact open. Thus, 1.1.8 shows ◦

that K (X) = Clop(X). For a finite topological space X, the following conditions are equivalent: (i) (ii) (iii) (iv)

X is Boolean. X is Hausdorff. X is discrete. The poset (X,  ) is trivially ordered.

1.3.3 Example The simplest example of an infinite Boolean space is the one-point compactification, S ∗ , of a discrete infinite space S. Recall that S ∗ = S ∪ {∞} (with ∞  S) and the following sets form a basis of opens: • singletons {s} with s ∈ S, and • complements in S ∗ of finite subsets of S. The space S ∗ is clearly Boolean, and ∞ is the unique non-isolated point. The construction of S ∗ is a special instance of the one-point compactification (also called Alexandroff compactification), which is defined for arbitrary locally compact spaces. We refer to [Bou71b, p. 67] and adopt the convention that the one-point compactification of a compact space X is the topological sum of X with a singleton space.

1.3 Boolean Spaces and the Constructible Topology 1.3.4 Theorem equivalent:

15

The following conditions about a topological space X are

(i) X is spectral and Hausdorff. ◦

(ii) X is spectral and K (X) is a Boolean algebra. (iii) X is Boolean. ◦

Proof (i) ⇒ (iii) Axioms S1 and S2 show that X is quasi-compact and K (X) is a basis of the topology. Since X is Hausdorff, every quasi-compact subset of X is closed. In particular, X is compact and X has a basis of closed and open subsets (i.e., X is Boolean). (iii) ⇒ (ii) By definition X is quasi-compact and Hausdorff, in particular, T0 . The Hausdorff property implies that every nonempty closed and irreducible set is a singleton, hence soberness holds (axiom S4). We mentioned in 1.3.2 that ◦

◦

K (X) = Clop(X), thus S2 and S3 are satisfied and K (X) is a Boolean algebra. ◦

(ii) ⇒ (i) Take x  y. As X is spectral, there is some U ∈ K (X) containing ◦ one, but not both, of the points x and y, say x ∈ U and y  U. Since K (X) is closed under complements, U and X \ U are open. Hence U and X \ U are disjoint open neighborhoods of x and y.  Strengthening 1.3.4, we shall prove in 1.3.20 that a spectral space X is already Boolean if it has the T1 -property (i.e., all points are closed). 1.3.5 Example

The simplest nonempty Boolean space is the one-element ◦

space 1, 1.1.17. The Boolean algebra K (1) is isomorphic to the two-element Boolean algebra, which is denoted by 2. Its elements are denoted by 0 < 1 or by ⊥ < . Suppose that f : X → Y is a map between spectral spaces. By definition, the ◦

◦

map is spectral if V ∈ K (Y ) implies f −1 (V) ∈ K (X). Usually continuity alone does not imply spectrality (cf. the comments in 1.2.3). But for maps into a Boolean space, we have: 1.3.6 Proposition Assume X is a spectral space, Y is a Boolean space, and f : X → Y is a map. Then f is continuous if and only if f is spectral. In particular, a map between Boolean spaces is spectral if and only if it is continuous. Proof Spectral maps are always continuous, see 1.1.6. Conversely, suppose f is continuous. Pick a quasi-compact open set V ⊆ Y . Then V is also closed

16

Spectral Spaces and Spectral Maps

since Y is Hausdorff. As f is continuous, f −1 (V) is open and closed in X. By quasi-compactness of X it follows that f −1 (V) is open and quasi-compact.  1.3.7 Continuous Maps to Finite Boolean Spaces Suppose that X is a topological space and Y is a finite Boolean space (i.e., is discrete, 1.3.2). Then a map f : X → Y is continuous if and only if every fiber f −1 (y) is open, if and only if every fiber belongs to Clop(X). 1.3.8 Example We exhibit an example of a continuous map between spectral spaces that is not spectral. Take any infinite Boolean space X. Then X has a non-isolated point p, and the map f : X → 2 with f −1 (0) = X \ {p} is continuous. But it is not spectral, since X \ {p} is not quasi-compact. More generally, if U ⊆ X is any subset that is open and not closed then the map g : X → 2 with g −1 (0) = U is continuous, but not spectral. 1.3.9 The Category BoolSp The Boolean spaces with the continuous maps as morphisms form a full subcategory of Top. It is called the category of Boolean spaces and is denoted by BoolSp. It is clear from 1.3.4 and 1.3.6 that BoolSp is also a full subcategory of Spec. 1.3.10 Definition Let X be a topological space satisfying axioms S1, S2, and S3 (e.g., X could be a spectral space). A subset A ⊆ X is a closed constructible ◦

set if X \ A ∈ K (X). (The terminology will be explained in 1.3.18.) We write ◦

K(X) = {X \ U | U ∈ K (X)}, which, by the de Morgan laws, is a bounded sublattice of P(X). 1.3.11 Definition Let X be a spectral space. The set ◦

K (X) ∪ K(X) is a subbasis of open sets for the patch topology, or constructible topology, of X. The set X with the patch topology is called the patch space of X and is denoted by Xcon . ◦

A subset of X is called constructible if it belongs to K (Xcon ), which is also denoted by K(X). An element of O(Xcon ) is a constructibly open set, or a patch open set; an element of A(Xcon ) is a constructibly closed set, or a proconstructible set. In [Hoc69, p. 45] a proconstructible set is also called a patch. The closure of a subset S ⊆ X for the constructible topology is denoted con by S .

1.3 Boolean Spaces and the Constructible Topology

17

1.3.12 Example The definition of the patch space and 1.3.4 show that a Boolean space coincides with its patch space. The patch space of a finite spectral space X is discrete. For, if x ∈ X then both Spez(x) and Gen(x) are constructibly open, cf. 1.1.16, hence {x} = Spez(x) ∩ Gen(x) is open in Xcon . In particular, the spaces ∅ and 1 are Boolean, hence coincide with their patch spaces. The patch space 2con of the Sierpiński space 2 (cf. 1.1.18) is the discrete space with universe 2 = {0, 1}. 1.3.13 Proposition Let X be a spectral space. (i) The patch topology is Hausdorff and refines the spectral topology. ◦

(ii) The sets V ∩ U, where V ∈ K(X) and U ∈ K (X), are clopen in Xcon and form a basis of open subsets. The sets U ∩V are called basic constructible sets . (iii) A subset C ⊆ X is proconstructible if and only if C is an intersection of ◦

sets of the form V ∪ U, where V ∈ K(X) and U ∈ K (X). Proof

(i) It is clear from the definition and S2 that the patch topology is finer ◦

than the spectral topology. The patch space is Hausdorff since K (X) is a set of clopen subsets of Xcon (by definition) and separates points in X. ◦

(ii) Since both K(X) and K (X) are closed under finite intersections, the sets ◦ V ∩ U, where V ∈ K(X) and U ∈ K (X), are clopen and form a basis of open subsets of Xcon . ◦

(iii) It follows from (ii) that the sets V ∪ U with V ∈ K(X) and U ∈ K (X) are a basis of closed sets.  1.3.14 Theorem For every spectral space X, the patch space Xcon is Boolean. In particular, Xcon is also spectral. Proof By 1.3.1 and 1.3.13 it remains to prove that Xcon is (quasi-)compact. We know from 1.3.13 that ◦

S = {U ∪ V | U ∈ K (X) and V ∈ K(X)} is a basis of closed sets of Xcon . Therefore, it suffices to show that every subset U ⊆ S with the finite intersection property (hereafter abbreviated FIP) 8 has nonempty intersection. By Zorn’s Lemma, we may assume that U is maximal for inclusion among 8

The finite intersection property asserts that the intersection of any finite collection of sets from U is nonempty.

18

Spectral Spaces and Spectral Maps

the subsets of S having the FIP. We set  P= {V | V ∈ U ∩ K(X)} and first show (a) If A, B ∈ S with A ∪ B ∈ U then A ∈ U or B ∈ U . ◦

(b) If U ∈ K (X) then U ∈ U if and only if P ∩ U  ∅. Proof (a) Assume that A, B  U . By maximality of U there are A1, . . . , An, B1, . . . , Bk ∈ U such that A ∩ A1 ∩ · · · ∩ An = B ∩ B1 ∩ · · · ∩ Bk = ∅ . But then (A ∪ B) ∩ A1 ∩ · · · ∩ An ∩ B1 ∩ · · · ∩ Bk = ∅ , and A ∪ B cannot be in U . (b) If U ∈ U , then the set {U} ∪ (U ∩ K(X)) is contained in U , hence it has the FIP, too. As U is quasi-compact, we get P ∩ U  ∅. Conversely, if U  U , then X \ U ∈ U (by (a) using U ∪ (X \ U) ∈ U ). Consequently, X \ U ∈ U ∩ K(X), which implies P ⊆ X \ U, as asserted.  ◦

We now show that P is an irreducible subset of X. Pick U1, U2 ∈ K (X) with P ∩ U1, P ∩ U2  ∅. By (b) we know that U1, U2 ∈ U , and S3 yields ◦

U1 ∩ U2 ∈ K (X) ⊆ S . By maximality of U we obtain U1 ∩ U2 ∈ U . Using (b) again, we conclude that P ∩ U1 ∩ U2  ∅, which proves irreducibility. So P is a closed and irreducible subset of X. By (b), applied to U = X, we see that P  ∅. Soberness (i.e., axiom S4) gives a generic point p ∈ P. We ◦

assert that p belongs to every U ∪ V ∈ U , where U ∈ K (X) and V ∈ K(X). By (a) we know that U ∈ U or V ∈ U . In the second case p ∈ P ⊆ V. In the first case, (b) gives P ∩ U  ∅, hence U contains the generic point p of P.  Altogether it has been shown that p ∈ U .  1.3.15 Corollary Let X be a spectral space. The following conditions about a subset C ⊆ X are equivalent: ◦

(i) C ∈ K (Xcon ) (i.e., C is constructible). (ii) C is clopen in Xcon . (iii) C is a finite union of basic constructible sets (i.e., of sets U ∩ V, where ◦

U ∈ K (X) and V ∈ K(X)).

◦

(iv) C is a finite intersection of sets of the form U ∪ V, where U ∈ K (X) and V ∈ K(X). ◦

In particular, K (Xcon ) is the Boolean algebra of subsets of X generated by ◦ K (X), and ◦

K(Xcon ) = K (Xcon ) = K(Xcon ) = K(X).

1.3 Boolean Spaces and the Constructible Topology

19

(i) is equivalent to (ii) because Xcon is Hausdorff and compact.

Proof

(i) ⇒ (iii) Since C is open in Xcon , we know from 1.3.13 that it is a union of ◦

sets of the form U ∩ V with U ∈ K (X), V ∈ K(X). Being compact in Xcon , C is a finite union of such sets. ◦

(iii) ⇒ (ii) The sets U ∈ K (X) and V ∈ K(X) are clopen in Xcon . Finite unions and intersections of clopen sets are clopen. (iii) ⇔ (iv) follows by taking complements.



1.3.16 Corollary The Boolean spaces are exactly the patch spaces of spectral spaces. Proof The patch space of a spectral space is Boolean by 1.3.14. Conversely, every Boolean space is its own patch space, 1.3.12.  1.3.17 The Boolean Algebra of Constructible Sets ◦

Let X be a spectral

space. Then 1.3.15 shows that K(X) = K (Xcon ) is a Boolean algebra. The Boolean algebras form a full subcategory BoolAlg ⊆ BDLat of the category of bounded distributive lattices. Note that bounded lattice homomorphisms between Boolean algebras necessarily respect complementation. Thus, the functor ◦

◦

K : Spec → BDLat (see 1.2.3) restricts to a functor K : BoolSp → BoolAlg. 1.3.18 Corollary Let S be a subset of a spectral space X. (i) S is quasi-compact open if and only if S is open and constructible. (ii) X \ S is quasi-compact open if and only if S is closed in X and is constructible. Note Item (ii) explains why the complements of quasi-compact open sets are called “closed constructible,” cf. 1.3.10. Proof

(i) The definition of the constructible topology, together with com◦

◦

pactness of Xcon , implies K (X) ⊆ K (Xcon ) = K(X). For the converse it suffices to show that constructible sets are quasi-compact. A constructible subset of X is closed in Xcon , hence is compact in Xcon . It is quasi-compact in X since the spectral topology is coarser than the patch topology. (ii) follows from (i) because the complements of open and constructible subsets of X are precisely the closed and constructible subsets.  1.3.19 Constructible Topology vs. Spectral Topology 1.3.18 can be rephrased in the following way: ◦

◦

(i) K (X) = K (Xcon ) ∩ O(X);

The assertions of

20

Spectral Spaces and Spectral Maps ◦

(ii) K(X) = K (Xcon ) ∩ A(X). A spectral space is determined by its underlying set and the spectral topology, of course. The constructible topology contains not quite as much information, but is easier to handle. Therefore, it is often useful to consider the topologies concurrently. For a case in point, see Theorem 1.3.21 below. Specifically, the constructible topology is often extremely useful for the proof of compactness results. Other characterizations of open quasi-compact sets and closed constructible sets (in terms of specialization) are given in 1.5.4. 1.3.20 Proposition A topological space X is Boolean if and only if X is spectral and T1 (i.e., all points of X are closed). Proof By 1.3.4, we only need to show that a spectral T1 -space is Hausdorff. Take x  y and let ◦

C = {U ∩ V | U,V ∈ K (X), x ∈ U, y ∈ V }. ◦

It is claimed that ∅ ∈ C . Assume this is false. As K (X) is closed under finite intersections (axiom S3), C has the finite intersection property. By definition of the patch topology, every set in C is closed in Xcon and by 1.3.14, Xcon  is compact. Then there is a point z ∈ C . By definition of C this point specializes to x and to y. As x  y, z is not a closed point of X, contradicting the assumption that X is T1 .  We use the constructible topology to give an alternative characterization of spectral maps: 1.3.21 Theorem Let X and Y be spectral spaces and let f : X → Y be a map between the underlying sets. Then f is a spectral map if and only if f is continuous both for the spectral topologies and for the constructible topologies. Proof Suppose that f is continuous for both topologies. Then for every V ∈ ◦

K (Y ), the set f −1 (V) ⊆ X is open (by continuity for the spectral topology) and constructible (by continuity for the patch topology), hence is quasi-compact open by 1.3.18(i). Conversely, assume f is spectral. We only have to prove continuity for the ◦

constructible topology. Since K (Y ) ∪ K(Y ) is a subbasis for the patch topology of Y , it suffices to show that f −1 (U) and f −1 (V) are constructibly open if ◦

◦

◦

U ∈ K (Y ) and V ∈ K(Y ). But this follows from f −1 (U) ∈ K (X) ⊆ K (Xcon ) ◦  and f −1 (V) ∈ K(X) ⊆ K (Xcon ).

1.3 Boolean Spaces and the Constructible Topology

21

1.3.22 Notation Given a spectral map f : X → Y we write fcon for the map f considered as a continuous (hence spectral, 1.3.6) map Xcon → Ycon . The ◦

◦

◦

map f defines the bounded lattice homomorphism K ( f ) : K (Y ) → K (X), cf. 1.2.3, hence, by forming complements, also the bounded lattice homomorphism ◦

K( f ) : K(Y ) → K(X), C →  f −1 (C). Applying the construction of K ( f ) (or K( f )) to the spectral map fcon one obtains the homomorphism ◦

◦

◦

K ( fcon ) = K( fcon ) : K (Ycon ) = K(Ycon ) → K(Xcon ) = K (Xcon ) ◦

◦

of Boolean algebras. Since K (Xcon ) = K(X) and K (Ycon ) = K(Y ), see 1.3.11, we also write K( f ) : K(Y ) → K(X). 1.3.23 Corollary Let f : X → Y be a spectral map. Images of proconstructible subsets of X and preimages of proconstructible subsets of Y under f are again proconstructible. In particular, the image of f is a proconstructible subset of Y and every fiber of f is a proconstructible subset of X. Proof Recall from 1.3.11 that “proconstructible” means “closed for the constructible topology.” By 1.3.21, fcon is a continuous map Xcon → Ycon . Compactness of Xcon and Ycon implies that images and preimages of proconstructible  sets under fcon are proconstructible. 1.3.24 BoolSp as a Subcategory of Spec We look at the category Spec of spectral spaces from the viewpoint of the constructible topology. We know that BoolSp is a full subcategory of Spec, 1.3.9. Let X and Y be spectral spaces and f : X → Y a spectral map. By definition, this means that taking preimages under f induces a bounded lattice ◦

◦

◦

homomorphism K ( f ) : K (Y ) → K (X). It also induces the homomorphism K( f ) : K(Y ) → K(X), C → f −1 (C) of Boolean algebras, cf. 1.3.22. The patch space Xcon is Boolean and the identity map X → X is a spectral map if the domain is equipped with the patch topology and the codomain with ◦

◦

the spectral topology (since K (X) ⊆ K (Xcon )). We denote this map by conX : Xcon → X. ◦

◦

◦

It is clear that K (conX ) : K (X) → K (Xcon ) = K(X) is the inclusion homomorphism. Note that, in general, conX is not the identity map in the category Spec, since the topologies of X and Xcon are usually different. Obviously, conX is a

22

Spectral Spaces and Spectral Maps

homeomorphism (hence the identity in Spec) if and only if X = Xcon , if and only if X is Boolean, 1.3.12 and 1.3.14. The construction that associates the patch space Xcon with the spectral space X and the map fcon : Xcon → Ycon with the spectral map f : X → Y is obviously a functor, which is denoted by con : Spec → BoolSp. We record the main properties of this construction. (i) The following diagram commutes: fcon

Xcon

/ Ycon

con X

conY

 X

 / Y.

f

Clearly, fcon is the only (spectral) map making the diagram commutative. ◦

Applying the functor K we obtain the diagram ◦

K(X) = O K (Xcon ) o

◦

K ( fcon )=K( f )

◦

K (Ycon ) O = K(Y )

⊆

⊆ ◦

◦

◦

K( f )

K (X) o

K (Y ).

(ii) It is obvious from the construction that the functor con is idempotent, that is, conXcon : (Xcon )con → Xcon is always a homeomorphism. (iii) Suppose now that the domain X of the spectral map f is Boolean. Then there is a unique spectral map g such that the following diagram commutes: g

X f

/ Ycon conY

 Y.

In fact, if we identify X = Xcon then g coincides with the map fcon of (i). (iv) We rephrase (i)–(iii) in category-theoretic language: (a) Item (i) says that the family (conX )X is a natural transformation from the composition of the functor con with the inclusion functor BoolSp → Spec to the identity functor of Spec.

1.4 The Inverse Topology

23

(b) The universal mapping property in (iii) says that BoolSp ⊆ Spec is a coreflective subcategory (i.e., the inclusion functor has a right adjoint functor, called the coreflector, namely the functor con). The maps conX : Xcon → X are the coreflection maps; they are the components of the co-unit of the adjunction. Later we shall see that BoolSp is also reflective in Spec, see 6.6.8. (c) Item (ii) says that the coreflector con is idempotent. (Idempotency of the coreflector is obvious from the present considerations, but it also follows from [HeSt79, Proposition 36.5] since BoolSp ⊆ Spec is a full subcategory, see 1.3.6.)

1.4 The Inverse Topology Summary We introduce the inverse topology of a spectral space X, 1.4.1, thus completing the description of the collection of topologies that are naturally present in every spectral space. The underlying set of X together with the inverse topology is a spectral space, 1.4.3, called the inverse space. The construction of the inverse space extends to spectral maps, thus defining a functor from Spec into itself. The inverse of the inverse topology is the original spectral topology. Thus, the inverse space functor is an involution, 1.4.5. We have seen that spectral maps are those maps between spectral spaces that are continuous for the spectral topology and the patch topology, cf. 1.3.21. We extend this characterization using the inverse topology, 1.4.6. The inverse topology was first described by Hochster, see [Hoc69, Proposition 8]. ◦

1.4.1 Definition Let X be a topological space. The set K (X) is a basis of closed sets of a new topology on X, which is called the inverse topology. The set X with the inverse topology is the inverse space of X and is denoted by Xinv . The open sets of Xinv are called inversely open, the closed sets are inversely inv closed. The closure of C ⊆ X for the inverse topology is denoted by C . 1.4.2 Examples (i) The inverse space 2inv of the Sierpiński space 2 (cf. 1.1.18(ii)) is the space with universe 2 = {0, 1} and specialization relation 1  0, see 1.1.18(iii). (ii) The closed unit interval [0, 1] of the real line has only two quasi-compact open sets. The inverse space is the set [0, 1] with the indiscrete topology.

24

Spectral Spaces and Spectral Maps

(iii) Let X be a Boolean space. By 1.3.2 a subset is quasi-compact open if and only if its complement is quasi-compact open. Thus, the inverse topology is the same as the given Boolean topology. 1.4.3 Theorem For every spectral space X, the inverse space Xinv is a spectral space, and: ◦

◦

(i) K (Xinv ) = K(X) and K(Xinv ) = K (X). (ii) X and Xinv have the same patch space, in particular K(Xinv ) = K(X). (iii) For all x, x  ∈ X, x  x  in X ⇐⇒ x   x in Xinv . Proof By definition, Xinv is a coarsening of Xcon . Hence, every constructible subset of X is quasi-compact for the inverse topology. In particular, Xinv is ◦

quasi-compact. Since K (X) separates points in X, the set of complements, ◦ K(X) = {X \ U | U ∈ K (X)}, also separates points. Consequently, Xinv is T0 and axiom S1 holds. It is clear that K(X) is a bounded sublattice of P(X). Every element of K(X) is a constructible set, hence is quasi-compact in Xinv . As K(X) is a basis for the inverse topology it follows from 1.1.8 that ◦

K (Xinv ) = K(X), hence Xinv satisfies axioms S2 and S3, and in (i) the first equality holds. To show that Xinv is spectral it remains to prove soberness (i.e., axiom S4). Let S ⊆ X be nonempty closed and irreducible in Xinv . We must show that S is the closure of a singleton in Xinv . Consider the set C = {S ∩ A | A ∈ K(X) and S ∩ A  ∅}, which has the FIP since S is irreducible in Xinv and all A ∈ K(X) are open in Xinv . Note that S ∈ A(Xinv ) ⊆ A(Xcon ) and K(X) ⊆ A(Xcon ). Since Xcon is inv  compact there is a point p ∈ C. We conclude that {p} = S since K(X) is a basis of open sets of Xinv . The second equality in (i) follows from the first one by complementation. (ii) is immediate from (i). Finally, (iii) also follows from (i) since specialization can be tested using a basis of the topology.  1.4.4 Corollary Let X be a spectral space. Then the patch topology is finer than the interval topology of the poset (X,  ). Proof The sets x ↑ = {x}, y ↓ = {y}

inv

are a subbasis of closed sets for the

1.4 The Inverse Topology

25

interval topology, A.8(iv). Just note that they are closed also for the patch topology.  1.4.5 The Inverse Space Functor Every spectral map f : X → Y defines the bounded lattice homomorphism K( f ) : K(Y ) → K(X), C → f −1 (C), cf. 1.3.22. By 1.4.3, this means that f , considered as a map Xinv → Yinv , is spectral. This map is denoted by finv , and we obtain a functor inv : Spec → Spec. Obviously we have ◦

◦

K ( f ) = K( finv ) and K( f ) = K ( finv ). It is clear from 1.4.3(i) that the functor inv is an involution of Spec (i.e., inv◦inv is the identity functor of Spec). The fixed points of the functor inv are the spectral spaces that satisfy X = Xinv , ◦

in other words, such that K (X) = K(X). By 1.3.4 this is equivalent to X being a Boolean space. We see that the spaces X, Xinv , and Xcon are identical if any two of them coincide. 1.4.6 Theorem Let f : X → Y be a map between spectral spaces. The following conditions are equivalent: (i) f is a spectral map. (ii) f is continuous for the spectral, the inverse, and the constructible topology. (iii) f is continuous for any two of the three topologies. Proof (i) ⇒ (ii) If f is spectral, then finv and fcon are also spectral, see 1.4.5 and 1.3.21. (ii) ⇒ (iii) is obvious. (iii) ⇒ (i) Case 1 Suppose that f is continuous for the spectral topology and for the constructible topology. Then f is spectral by 1.3.21. Case 2 Suppose that f is continuous for the inverse topology and for the constructible topology. Then finv : Xinv → Yinv is spectral by 1.3.21. Now 1.4.5 shows that f = ( finv )inv : X → Y is spectral. Case 3 Suppose that f is continuous for the spectral topology and for the inverse topology. Let V ⊆ Y be open and quasi-compact; we must prove that f −1 (V) is open and quasi-compact in X. The set V is open in Y and Y \ V is open in Yinv . Hence f −1 (V) is open in X and X \ f −1 (V) is open in Xinv . In

26

Spectral Spaces and Spectral Maps

particular, f −1 (V) and X \ f −1 (V) are open in Xcon . Thus f −1 (V) is clopen in Xcon and open in X. By 1.3.18(i), f −1 (V) is open and quasi-compact in X.  1.4.7 The Dual Topology The inverse topology is a very important and useful construction for spectral spaces. However, in the realm of general topology there are many spaces with very few quasi-compact open sets, and then the inverse topology may even be indiscrete, cf. 1.4.2(ii). As a related construction we present the dual topology, which is always finer than the inverse topology. If there are enough quasi-compact open sets, which is the case for spectral spaces, then both topologies coincide. Let X be a T0 -space and denote the topology by τ. The co-compact topology or dual topology of (X, τ) is the topology having the quasi-compact saturated sets as a subbasis of closed sets. In fact, the subbasis is even a basis of closed sets since finite unions of quasi-compact saturated sets are quasi-compact saturated, 1.1.20. The dual topology is denoted by τ ∂ . (In [GL13, Definition 9.1.23, p. 403] the space (X, τ ∂ ) is also called the de Groot dual of X.) If X is coherent and quasi-compact then arbitrary intersections of quasicompact saturated sets are quasi-compact saturated. In this case the closed sets for the dual topology are exactly the quasi-compact saturated sets of τ. In general, the specialization order of τ ∂ is the inverse of the specialization order of τ (since the principal generically closed sets Gen(x) with x ∈ X are quasi-compact saturated for τ, hence are closed in τ ∂ ). In particular, iterating the formation of the dual topology, one sees that τ and τ ∂∂ have the same specialization order. In general, neither of the inclusions τ ⊆ τ ∂∂ and τ ∂∂ ⊆ τ holds. However, if τ is Hausdorff, then τ ∂ ⊆ τ, and equality holds if τ is compact. Assume that X carries two T0 -topologies τ and γ such that τ ⊆ γ and the specialization orders agree. Then every quasi-compact saturated set for γ is also quasi-compact saturated for τ, hence γ ∂ ⊆ τ ∂ . We compare the dual topology with the inverse topology τinv . The dual ◦

topology is always finer than the inverse topology, because K (X) is a basis of closed sets for τinv and is contained in A(τ ∂ ). Moreover, the topologies coincide if and only if every basic closed set of τ ∂ (i.e., every quasi-compact ◦

saturated set) is closed for τinv (i.e., is an intersection of elements of K (X)). ◦ These equivalent conditions hold, for example, if K (X) is a subbasis of open sets, see the claim in 1.1.20. Finally we mention that the open sets of the inverse topology of a spectral space may also be found under the terminology Thomason set in the literature, cf. [Hrb16] with reference to [Tho97].

1.5 Specialization and Priestley Spaces

27

1.5 Specialization and Priestley Spaces Summary Every spectral space carries three topologies (spectral, constructible, and inverse) and a partial order (specialization). After having analyzed the different topologies we now turn to the specialization order. By definition, a spectral space is completely determined by its spectral topology, hence also by its inverse topology, cf. 1.4.5. The patch topology contains less information. We know that every finite T0 -space is spectral, 1.1.15. Every finite set with more than one element carries several different T0 -topologies, corresponding to its different partial orders, hence is the underlying set of various spectral spaces. But the patch space is always the same: the underlying set with the discrete topology. However, we show that the patch topology together with the specialization order determines the spectral space uniquely. The combination of topological and order-theoretic arguments is a characteristic feature of the theory of spectral spaces. In fact, this combination can be used as the basis for an alternative approach to spectral spaces, which is due to H. Priestley. She considers sets with a Boolean topology and a partial order satisfying a compatibility condition. These partially ordered topological spaces are now called Priestley spaces and are an important tool in (at least) lattice theory, logic, and the theory of computation. The theory started with the papers [Pri70] and [Pri72] and led to extensive research activities. We show in 1.5.11 and 1.5.15 that Priestley spaces and spectral spaces are exactly the same structures, but viewed from different perspectives, cf. [Pri70, Introduction] and [Fle00]. In our book Priestley’s approach is used mostly, and proves to be particularly valuable for the construction of examples. In 1.5.17 and 1.5.18 we give a complete description of all spectral spaces whose patch space is a given fixed Boolean space, a result first proved by Hochster, [Hoc69, Proposition 7]. We start the analysis of the specialization order with a Separation Lemma, 1.5.3, which is a key technical tool. It has a crucial role in: (a) proving separation properties for disjoint subsets of spectral spaces (see, e.g., 4.1.7); (b) the characterization of inversely quasi-compact subsets and quasicompact subsets, 4.1.3; (c) the characterization of quasi-compact open sets and of closed constructible sets in terms of proconstructible sets and the specialization relation, 1.5.4;

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(d) a characterization of spectral maps in terms of the constructible topology and the specialization order, 1.5.13. The separation results hinge on a general topological uniformization argument, which we explain at the outset. 1.5.1 Uniformization Lemma Let X be a set and L a lattice of subsets of X. Let T1 and T2 be topologies on X and suppose that L ⊆ A(T1 ) ∩ O(T2 ). Let Y1 ⊆ X be quasi-compact for T1 and let Y2 ⊆ X be quasi-compact for T2 . The following conditions are equivalent: (i) For all y1 ∈ Y1 and y2 ∈ Y2 there is some A ∈ L with y2 ∈ A and y1  A. (ii) There is some A ∈ L with Y2 ⊆ A and A ∩ Y1 = ∅. Proof (ii) ⇒ (i) The set A in (ii) serves as a separating set in (i), uniformly for all choices of y1 and y2 . This explains the name “Uniformization Lemma.” (i) ⇒ (ii) As a first step we pick any element z ∈ Y2 and claim that there is a set Az ∈ L with z ∈ Az and Az ∩ Y1 = ∅. To prove the claim, note that (by (i)) for every y ∈ Y1 there is a set By ∈ L with z ∈ By and y  By . Hence,  Y1 ⊆ y ∈Y1 (X \ By ). Since Y1 is T1 -quasi-compact and the X \ By are T1 -open, there are finitely many y1, . . . , yk ∈ Y1 with Y1 ⊆ (X \ By1 ) ∪ · · · ∪ (X \ Byk ). The set Az := By1 ∩ · · · ∩ Byk ∈ L is disjoint from Y1 and contains z. This completes the first step. Now we construct the set A. Using the sets Az (with z ∈ Y2 ) from the first  step, we have Y2 ⊆ z ∈Y2 Az . Since Y2 is quasi-compact in T2 and the sets Az are open in T2 , there are z1, . . . , zm ∈ Y2 with Y2 ⊆ A := Az1 ∪ · · · ∪ Azm ∈ L. By construction, A is disjoint from Y1 .  1.5.2 Corollary Let X be a spectral space and L  a sublattice of K(X). Let S ⊆ X be a subset and define L = {A ∈ L  | S ⊆ A}, which is a sublattice of L . The following conditions about a quasi-compact subset Y1 ⊆ X are equivalent: (i) For all y ∈ Y1 there is some A ∈ L such that y  A. (ii) There is some A ∈ L such that A ∩ Y1 = ∅ . Proof (ii) ⇒ (i) is trivial. (i) ⇒ (ii) We apply 1.5.1 with T1 the spectral topology of X, T2 the constructible con topology of X, and Y2 = S . The hypotheses of 1.5.1 are satisfied. The lattice L is contained in K(X) ⊆ A(X) ∩ O(Xcon ); the set Y1 is quasi-compact for T1 by hypothesis; the set Y2 is quasi-compact in Xcon since it is closed. The present condition (i) implies condition (i) in 1.5.1. Then condition (ii) in 1.5.1 holds, and this implies the present condition (ii). 

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29

1.5.3 Separation Lemma Let X be a spectral space and suppose that Y1 and Y2 are subsets such that Y1 is quasi-compact for the spectral topology and Y2 is quasi-compact for the inverse topology. The following conditions are equivalent:  y1 . (i) If y1 ∈ Y1 and y2 ∈ Y2 then y2  (ii) There is a set A ∈ K(X) with Y2 ⊆ A and A ∩ Y1 = ∅. ◦

(iii) There is a set U ∈ K (X) with Y2 ∩ U = ∅ and Y1 ⊆ U. Proof (i) ⇔ (ii) In 1.5.1 we set L = K(X), T1 the spectral topology, and T2 the inverse topology. Then the present conditions (i) and (ii) are exactly the conditions (i) and (ii) of 1.5.1, hence they are equivalent to each other. Obviously, (ii) and (iii) imply each other by forming complements.



A first application of 1.5.3, which will be generalized in 4.1.3, is a characterization of closedness for proconstructible sets and of openness for constructible sets in a spectral space. 1.5.4 Theorem Let S be a subset of a spectral space X. (i) If S is proconstructible, then the closure of S is Spez(S), the set of specializations of S. In particular, S is closed if and only if S is proconstructible and closed under specialization. (ii) S is closed and constructible if and only if S is constructible and closed under specialization. (iii) S is quasi-compact open if and only if S is constructible and closed under generalization. Proof (i) The inclusion Spez(S) ⊆ S holds trivially, 1.1.3. For the reverse inclusion, let S be proconstructible, which implies that S is inversely quasicompact (since the constructible topology refines the inverse topology). If z  Spez(S) then the sets Y1 = {z} and Y2 = S satisfy the hypotheses and condition (i) of 1.5.3. Hence, there is some A ∈ K(X) with S ⊆ A and z  A. We conclude that  Spez(S) = { A ∈ K(X) | S ⊆ A} = S. For the second assertion, first note that every closed set is proconstructible since the patch topology is finer than the spectral topology. So we may assume that S is proconstructible and have to show that S is closed if and only if S = Spez(S). But this is clear since it has been shown that S = Spez(S). (ii) follows directly from (i), cf. 1.3.18.

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(iii) follows from (ii) since a set is specialization closed if and only if its complement is generically closed and since complements of constructible sets are constructible.  1.5.5 Corollary Let X be a spectral space and S ⊆ X a subset. Then S=S S

inv

=S

con

= Spez(S

con inv

con

= Gen(S

),

con

).

In particular, if S is nonempty and irreducible, then the generic point of S is con contained in S . Proof The patch topology is finer than the spectral topology. Therefore the con

con

inclusions S ⊆ S ⊆ S imply S = S = Spez(S The remaining assertions follow immediately.

con

) (by 1.5.4(i)). 

In 4.1.3 we characterize those subsets S of X for which S = Spez(S). 1.5.6 Warning Considering the characterization of closed sets in 1.5.4 one may ask: is it true that a proconstructible set is closed under generalization if and only if it is open? Of course, an open set is always closed under generalization, but usually there are many generically closed proconstructible sets that are not open. For an extreme example, consider an infinite Boolean space X and let S ⊆ X be closed (i.e., proconstructible). Then S is trivially closed under generalization (since the specialization order is trivial), but not every closed subset of an infinite Boolean space is open. It is apparent from 1.5.4 that a spectral space is completely determined if one knows the patch topology and the specialization order. For then, the quasicompact open sets are the constructibly clopen sets that are generically closed (i.e., are down-sets for specialization). This observation leads us to the definition of a Priestley space. 1.5.7 Definition Let X be a Boolean space and ≤ a partial order on X. We call ≤ a spectral order if it satisfies the following Priestley Separation Axiom: For all y, z ∈ X with y  z there is a clopen (for the Boolean topology) subset Y ⊆ X that is an up-set for ≤ and contains y, but not z. A Boolean space with a spectral order is a Priestley space. Priestley spaces are also called totally order-disconnected spaces, cf. [Pri70, Section 2] or [Pri94b, Introduction]. In 1.5.11 we show that the Priestley spaces are the spectral spaces under a different guise (and vice versa).

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31

1.5.8 Example The specialization order of a spectral space is a spectral order on the patch space. To see this, let X be a spectral space and pick y, z ∈ X with y   z (i.e., z  {y}). Since K(X) is a basis of closed subsets of X, there is some Y ∈ K(X) with y ∈ Y and z  Y . The closed constructible set Y is an up-set for  and is clopen in Xcon . 1.5.9 Proposition Suppose that (X, ≤) is a Priestley space. The graph R = {(x, y) ∈ X × X | x ≤ y} of the spectral order relation is a closed subset of X × X. Proof Let y, z ∈ X be such that y  z. There is a clopen up-set Y with y ∈ Y and z  Y . Thus, X \ Y is a clopen down-set containing z, but not y. The subset Y × (X \ Y ) ⊆ X × X is clopen and disjoint from R, since (u, v) ∈ R ∩ (Y × (X \ Y )) implies u ≤ v and u ∈ Y , hence v ∈ Y (note that Y is an up-set), a contradiction.  1.5.10 Spectral Orders vs. Closed Partial Orders Let X be a topological space and let ≤ be a partial order on X. We call ≤ a closed partial order if its graph R is closed in X × X. By 1.5.9, the spectral order of a Priestley space (X, ≤) is a closed partial order on the Boolean space X. If ≤ is only a closed (not necessarily spectral) partial order then the statement of the Priestley Separation Axiom (1.5.7) holds with the weaker condition “Y is closed” (instead of “Y is clopen”). We show in 8.2.5 that this weaker condition is not enough to ensure that ≤ is a spectral order On the other hand, there are many Boolean spaces for which the closedness condition is sufficient for an order to be spectral. This is the case, for example, if ≤ is total (see 1.6.8), or if X is countable (see 8.2.3 and 8.2.4). 1.5.11 Theorem Let X be a Boolean space, and let T be the set of all spectral topologies τ on X such that (X, τ)con = X. Then the map T → { ≤ | ≤ is a spectral order on X }, τ →  τ is bijective. The inverse map sends a spectral order ≤ on X to the topology τ≤ with the following data: (i) τ≤ = O(X, τ≤ ) = {Y ∈ O(X) | Y is a down-set for ≤}; (ii) A(X, τ≤ ) = {Z ∈ A(X) | Z is an up-set for ≤}; ◦

(iii) τ≤ ∩ K(X) = K (X, τ≤ ) = {Y ∈ Clop(X) | Y is a down-set for ≤}; (iv) K(X, τ≤ ) = {Z ∈ Clop(X) | Z is an up-set for ≤}. The correspondence between spectral topologies and spectral orders is inclusion reversing – finer topologies belong to weaker specialization orders.

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Proof The map is well-defined by 1.5.8 and reverses inclusion (by the definition of specialization). It remains to show that every spectral order ≤ is the specialization order of exactly one spectral topology. = Injectivity is a consequence of 1.5.4. Suppose that σ, τ ∈ T and that  σ  . Then a subset U ⊆ X is quasi-compact open for σ if and only if it is clopen τ =  , if and only if it is in (X, σ)con = (X, τ)con and is a down-set for  σ τ quasi-compact open for τ. The topologies coincide since they have a common basis. For surjectivity, we pick a spectral order ≤ and define τ = τ≤ as in item (i). Both O(X) and the set of down-sets of (X, ≤) are bounded sublattices of P(X) and are closed under finite intersections and arbitrary unions. Thus, τ is a topology. The closed sets as in (ii) are obtained by complementation. The topology τ is coarser than the Boolean topology of X, hence (X, τ) is quasicompact. The definition of a spectral order implies that τ is T0 . Thus, axiom S1 is satisfied. The specialization relation  coincides with ≤. For, trivially, x ≤ y implies τ y. And, whenever x  y, the Priestley Separation Axiom tells us that x x τ does not specialize to y in τ. We define τ0 = τ ∩ K(X), which is an intersection of two bounded sublattices of P(X), hence is also a bounded sublattice. Every element of τ0 is compact in X, hence is quasicompact for τ. Claim τ0 is a basis for τ. Proof of Claim We show that for every set Y ∈ τ and every point y ∈ Y there is a set U ∈ τ0 with y ∈ U ⊆ Y . Pick an element z ∈ X \ Y . Then z  y, hence there is a set Uz ∈ K(X) that is a down-set for ≤ and contains y, but not z.  It follows that ( z ∈X\Y Uz ) ∩ (X \ Y ) = ∅. The set X \ Y and the sets Uz are all closed in X. Compactness implies that there is a finite subset F ⊆ X \ Y   such that ( z ∈F Uz ) ∩ (X \ Y ) = ∅. But then the set U = z ∈F Uz meets the requirements, and the claim is proved.  ◦

In view of 1.1.8, the claim implies that τ0 = K (X, τ≤ ), axioms S2 and S3 are satisfied, and item (iii) holds true. Item (iv) follows by taking complements. Finally we show that (X, τ) is sober. Suppose that V  ∅ is nonempty closed and irreducible. We have to show that there is a generic point. Assume that this is false. Then for every point x ∈ V there is a set Ux ∈ τ0 such that x  Ux and  Ux ∩ V  ∅. The intersection x ∈V Ux ∩ V is empty by construction. On the other hand, irreducibility of V implies that the family (Ux ∩ V)x ∈V has the FIP.

1.5 Specialization and Priestley Spaces

33

The set V and the sets Ux are all closed in X. By compactness it follows that   x ∈V Ux ∩ V  ∅, and we have reached a contradiction. 1.5.12 Spectral Orders and their Inverses Let X be a spectral space with specialization relation  . Then Xcon and Xinv are also spectral spaces. All three spaces have the same patch space. The specialization order of Xcon is the trivial order – every element is comparable only with itself. The specialization order of Xinv is the inverse partial order of  , x  inv y ⇐⇒ y  x, see 1.4.3 and A.1(vii). 1.5.13 Theorem The following conditions about a map f : X → Y between spectral spaces are equivalent: (i) f is spectral. (ii) f is continuous for the constructible topologies and is monotone for specialization (i.e., x  x  implies f (x)  f (x )). Proof (i) ⇒ (ii) Since f is continuous, it is monotone for specialization, 1.1.3. Being spectral, f is continuous for the constructible topologies, 1.3.21. (ii) ⇒ (i) It suffices to show that f −1 (C) is closed and constructible whenever C ⊆ Y is closed and constructible. As f is continuous for the constructible topologies, it is clear that f −1 (C) is constructible. By 1.5.4(ii) it remains to show that f −1 (C) is closed under specialization. This is clear since f preserves specialization.  1.5.14 Definition Suppose that (X, ≤) and (Y, ≤) are Priestley spaces. A map f : X → Y is a Priestley map if it is continuous as a map of the Boolean spaces X, Y and is monotone for the spectral orders. 1.5.15 Categories: Spectral Spaces vs. Priestley Spaces It is clear that a composition of Priestley maps is a Priestley map. Hence, the Priestley spaces and the Priestley maps form a category, which is denoted by Priestley. The results 1.5.11 and 1.5.13 yield the following comparison of the categories Spec and Priestley. There is a covariant functor Spec → Priestley. It sends a spectral space ) and a spectral map f to the Priestley map X to the Priestley space (Xcon,  X fcon : Xcon → Ycon . The functor is an isomorphism of categories. Thus, spectral spaces and Priestley spaces are alternative presentations of one and the same notion; the study of spectral spaces is then equivalent to the study of Priestley spaces. Even though our focus is on the topology of spectral

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Spectral Spaces and Spectral Maps

spaces, we shall frequently use the viewpoint provided by Priestley spaces. It is particularly valuable for the construction of examples. Our first application is a result by Hochster, which provides a criterion to recognize whether a topological space is spectral. The following lemma is needed for the proof. 1.5.16 Lemma Let X be a spectral space. (i) Pick a subset S ⊆ K(X) and let L be the sublattice of K(X) generated by S. Suppose that, given x, y ∈ X with x   y, there is some A ∈ S with x ∈ A and y  A. Then K(X) is contained in L. (ii) Suppose that τ is a topology on X generated by a sublattice B ⊆ K(X). If the specialization relation  is weaker than  , then τ is finer than τ the spectral topology. Proof (i) Pick an element B ∈ K(X). We apply 1.5.1 with the following data: • T1 is the topology determined by the subbasis {X \ S | S ∈ S}. Note that T1 is coarser than the patch topology. • T2 is the patch topology. • Y1 = X \ B, which is compact for the patch topology, hence quasi-compact for T1 . • Y2 = B, which is quasi-compact for T2 . The present hypotheses imply condition (i) of 1.5.1. Thus, 1.5.1 yields an element A ∈ L with B ⊆ A and A ∩ (X \ B) = ∅; this means B = A ∈ L. ◦

(ii) We wish to show that K (X) ⊆ B. We define L = {X \ B | B ∈ B}, which is a sublattice of K(X). The hypothesis says that x   y implies x   y. Hence τ

there is some A ∈ L with x ∈ A and y  A. Now (i) implies that K(X) ⊆ L, ◦

equivalently K (X) ⊆ B.



1.5.17 Theorem ([Hoc69, Proposition 7]) Let X be a compact space. Suppose that L is a bounded sublattice of the Boolean algebra Clop(X) and that L separates points in X. Then X is Boolean, L is a basis for a spectral topology ◦

τ on X with (X, τ)con = X, and L = K (X, τ). Proof The space X is totally disconnected since its points are separated by clopen sets. It is also compact, hence it is Boolean (cf. 1.3.2). The lattice L is a basis for a topology τ on X. Every element of L is closed in X, hence compact. Thus, τ has a basis that consists of quasi-compact sets and is closed ◦

under finite unions. By 1.1.8 there is only one such basis, namely K (X, τ), ◦ hence L = K (X, τ). The separation condition says that (X, τ) is a T0 -space. We

1.5 Specialization and Priestley Spaces

35

is spectral. Pick points x   y. There is a show that the specialization order  τ τ basic open set U ∈ L that contains y, but not x. Being open, U is a down-set for specialization. By 1.5.11 there is a unique spectral topology σ with X = (X, σ)con and  =  . The open sets of τ are open for σ by 1.5.11. On the other hand, σ τ 1.5.16(ii) shows that τ is finer than σ. We conclude that τ = σ is spectral.  1.5.18 Remark We may rephrase 1.5.17 in the following way. Let X be a compact space (hence Hausdorff) and T a coarsening of the topology of X. Consider the lattice L = Clop(X) ∩ O(X, T ). Suppose that L separates points in X and is a basis of T . Then T is a spectral topology on X with patch space ◦

X and K (X, T ) = L. 1.5.19 Example We show that in 1.5.18 the hypothesis “L is a basis of T ” cannot be omitted. To provide an example, let S be an infinite set and consider the spectral space S∞ exhibited in 1.6.13 below. We set X = (S∞ )con and construct a topology τ on X that lies between the spectral topology and the constructible topology and is not spectral. Note that Clop(X) = K (X) = {F | F ⊆ S finite} ∪ {{∞} ∪ C | C ⊆ S cofinite} and ◦

O(S∞ ) = {S∞ \ F | F ⊆ S finite} ∪ { ∅ } = K (S∞ ), see 1.6.14. Let τ be the cofinite topology on S∞ (i.e., the closed sets are the finite subsets and the entire space). It is clear that O(S∞ )  O(τ). Moreover, O(τ) ⊆ O(X) since every finite subset of S∞ is closed in X, but O(τ)  O(X) since each subset {∞} ∪ T, with T  S infinite, is proconstructible, but is not closed for τ. With L = K(X) ∩ O(τ), as in 1.5.18, we see that O(S∞ ) ⊆ L, hence L separates the points of X. However, τ is not spectral. As X is infinite, it is irreducible (and closed) for τ. But τ is a T1 -topology, so that all points are closed and there is no generic point for (X, τ). 1.5.20 Conclusion Let X be a Boolean space. We see from 1.5.17 and 1.5.11 that there are canonical bijective correspondences between: • spectral spaces with patch space X; • spectral orders on X; • bounded sublattices of K(X) that separate points.

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1.6 Examples Summary After defining spectral spaces we described and analyzed various fundamental structural features: the spectral topology, the patch topology, the inverse topology, the specialization order, the Boolean algebra of constructible sets, and the sublattices of quasi-compact open sets and of closed and constructible sets. At this point we interrupt the further development of the theory and present several elementary examples and constructions requiring only modest technical tools. The examples exhibit concrete instances of the theoretical concepts. Later they will be useful to produce counterexamples and to illustrate various phenomena. Finite spectral spaces and Boolean spaces were presented in Sections 1.1 and 1.3. Therefore, we consider infinite spaces with a nontrivial specialization order now. We start with spaces whose specialization order is total (Part A) and then merge two such spaces by identifying their closed points (Part B). In Part C we show that a minimal nontrivial partial order on a Boolean space is always spectral. (Note that a partial order ≤ is minimal nontrivial if there is exactly one relation x < y.) Finally, in Part D, we take the one-point compactification of a discrete space (which is Boolean, cf. 1.3.3) and equip it with a spectral order so that the point at infinity specializes to every point of the space.

A.

Totally Ordered Spectral Spaces

We examine T0 -spaces for which the specialization order is total. Then the topological notions discussed in Section 1.1 can be interpreted in terms of the order relation alone. In 1.6.4 we give an order-theoretic characterization of spectral spaces with a total specialization order. We refer to the Poset Zoo (Appendix, p. 579 ff) for terminology and notation on posets. 1.6.1 Proposition Let X be a T0 -space with specialization order  . (i) The following statements are equivalent: (a) (b) (c) (d)

 is a total order. There is a subbasis B of closed sets totally ordered by inclusion. The family A(X) of closed sets is totally ordered by inclusion. Every subset of X is irreducible.

Now suppose X is totally ordered by specialization.

1.6 Examples

37

(ii) A nonempty subset of X is quasi-compact if and only if it has a largest element for specialization. In particular, X satisfies axiom S1 if and only if X has a largest element. (iii) Let U ⊆ X be nonempty, then: ◦

(a) U ∈ K (X) ⇒ ∃x ∈ X : U = x ↓ . ◦

(b) U ∈ K (X) and X \ U has a generic point if and only if U = x ↓ for some x ∈ X that is the lower point of a jump. ◦

(iv) K (X) is closed under finite intersections (i.e., X satisfies axiom S3). Proof (i) (a) ⇒ (b) The set of  -up-sets is totally ordered by inclusion and contains B. (b) ⇒ (c) The subbasis B is closed under finite unions and finite intersections, hence is even a basis of closed sets. Assume that there are closed sets A, A and elements x ∈ A \ A and x  ∈ A \ A. Then, there are B, B  ∈ B with A ⊆ B, A ⊆ B  and x   B, x  B . But then B is not totally ordered, a contradiction. (c) ⇒ (d) is trivial. (d) ⇒ (a) Given x, y ∈ X the set {x}∪{y} is irreducible, hence {x} = {x}∪{y} or {y} = {x} ∪ {y}. This implies x  y or y  x. (ii) ( ⇐ ) Suppose that Y ⊆ X has a largest element y w.r.t.  . Then Y ⊆ U for every open set U containing y. Thus Y is quasi-compact. ( ⇒ ) If Y does not have a largest element then the collection {Y ∩ {y} | y ∈ Y } has the FIP, but the intersection is empty. Thus Y is not quasi-compact. (iii) (a) By (ii) and quasi-compactness, U has a largest element, say x. Thus U ⊆ x ↓ , and, being open, U is a down-set, hence x ↓ ⊆ U. (b) ( ⇒ ) By (a) there is some x ∈ X with U = x ↓ . The generic point of X \ U is the minimum of X \ U, hence is an immediate successor of x. ( ⇐ ) By (ii) we know that U = x ↓ is quasi-compact. If y is the immediate successor of x, then X \ U = y ↑ = {y} is closed, y is its generic point, and U is open. ◦

(iv) Given U,V ∈ K (X) there are x, y ∈ X with U = x ↓ and V = y ↓ (by (iii)(a)).  Hence U ∩ V = z ↓ with z = min{x, y}. By (ii), U ∩ V is quasi-compact. 1.6.2 The Coarse Lower Topology of a Totally Ordered Set Let P = (P, ≤) be a poset. The coarse lower topology τ  (≤) of P is described in A.8(ii). The set of finite unions of principal up-sets x ↑ forms a basis of closed sets for τ  (≤). The specialization relation of τ  (≤) is the partial order ≤. In fact, τ  (≤) is the coarsest topology on P with specialization order ≤.

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If (P, ≤) is totally ordered, then the set {x ↑ | x ∈ P} ∪ { ∅ } is totally ordered by inclusion, and is a sublattice of P(P). Thus, the principal upsets are a basis of closed sets for the coarse lower topology, {x  | x ∈ P} ∪ {P} is a sublattice of P(P), and is a basis of open sets. Now we ask how axioms S2 and S4 for τ  (≤) are expressed by properties of the total order ≤. The answer is used in 1.6.4 to show that, if X is a spectral space with total specialization order, then its topology coincides with τ  (  ). 1.6.3 Proposition Let P = (P, ≤) be a totally ordered set and let τ = τ  (≤) be the coarse lower topology. (i) The nonempty quasi-compact open sets of τ are precisely the sets x ↓ , where x is the lower point of a jump or the largest element of P. (ii) τ satisfies S2 if and only if ≤ is jump-dense. (iii) P has a largest element and τ satisfies S4 if and only if ≤ is complete. In this case, each nonempty closed set is the up-set of a singleton. Proof (i) The sets x ↓ , with x the lower point of a jump or x the largest element of P, are quasi-compact open by 1.6.1, items (ii) and (iii)(b). Since x ∈ x ↓ , these sets are nonempty. Conversely, let U ⊆ P be nonempty and quasi-compact open for τ. By 1.6.1(iii)(a) we know that U = x ↓ for some x ∈ P. If U = P, then x is the largest element of P. Otherwise, as U is open, the definition of τ gives a y ∈ P so that x ∈ y  ⊆ U. Since U = x ↓ , this is possible only if y is an immediate successor of x. (ii) ( ⇒ ) Pick elements x < y in P. We must find a jump between x and y. ◦

Since x ∈ y  , there is U ∈ K (P, τ) such that x ∈ U ⊆ y  . By (i), there is a jump u < v such that U = u ↓ . Then x ≤ u < v ≤ y, as required. ◦

( ⇐ ) Pick O ∈ τ and x ∈ O. We must find U ∈ K (X) such that x ∈ U ⊆ O. If x is the largest element of P, then U = x ↓ = O = P is quasi-compact open, 1.6.1(ii). If x is not the largest element of P then, by the definition of the topology, there is some y ∈ P such that x ∈ y  ⊆ O. There is a jump in between x and y, say x ≤ u < v ≤ y. Then u ↓ is quasi-compact open, 1.6.1(iii)(a), and x ∈ u ↓ ⊆ U. (iii) ( ⇒ ) We prove that every subset S ⊆ P has an infimum. If S = ∅ then the largest element of P is the infimum of S. Suppose that S  ∅ . Then S is nonempty closed and irreducible by 1.6.1(i). Soberness yields a generic point x ∈ S. Assume that x is not the infimum of S (i.e., there is y with x < y ≤ S). Then x  {y} = y ↑ and S ⊆ y ↑ . Thus x  S, a contradiction.

1.6 Examples

39

( ⇐ ) The empty set has an infimum, and this is the largest element of P. Now suppose S ⊆ P is nonempty closed (and irreducible, see 1.6.1(i)). By completeness, S has an infimum x, which implies that S ⊆ x ↑ = {x}. As S is closed it remains to show that x ∈ S. Assume that this is false. Since P \ S is open there is some y ∈ P with x ∈ y  ⊆ P \ S. But this means x < y ≤ S, contradicting the choice of x.  1.6.4 Theorem Let (X, τ) be a nonempty T0 -space whose specialization order  is total. (i) X is spectral if and only if the following three conditions hold: (a) X is complete for  ; (b) X is jump-dense for  ; (c) τ = τ  (  ). (ii) If X is spectral, then: (a) The inverse space of X is (X, τ  (  inv )). (b) The patch topology of X is the interval topology for  . 9 Proof

The topology τ is finer than the coarse lower topology of  , cf. 1.6.2.

(i) First assume that (a), (b), and (c) hold, in particular τ = τ  (  ); we claim that τ  (  ) is spectral. Axiom S1 follows from 1.6.1(ii) using (a), and S3 holds by 1.6.1(iv). Moreover, τ  (  ) satisfies S2 and S4 by 1.6.3(ii), (iii), using (a) and (b). Conversely, suppose X is spectral. It suffices to show τ = τ  (  ), since then 1.6.3(ii), (iii) imply (a) and (b). Only the inclusion τ ⊆ τ  (  ) must be proved. So, pick O ∈ τ and a point x ∈ O. We show that there is a τ  (  )-open set U ◦

with x ∈ U ⊆ O. As τ satisfies S2 there is some U ∈ K (X) with x ∈ U ⊆ O. If U = X then U = O = X ∈ τ  (  ). Otherwise, X \ U is nonempty closed and irreducible for τ. By S4 there is a generic point x (i.e., x ↑ = X \ U), but then U = x  ∈ τ  (  ). (ii) (a) By 1.4.3 we know that Xinv is spectral with total specialization order  inv . Thus (i)(c), applied to Xinv , yields the assertion. (ii) (b) First we prove that every constructible subset of X is open in the interval ◦

topology. It suffices to show that each element of the subbasis K (X) ∪ K(X) of the patch topology is open in the interval topology, see 1.3.11. But this is clear ◦

◦

from the description of K (τ  (  )) = K (τ) in 1.6.3(i). 9

For the interval topology, see A.8(iv). Since X is totally ordered it follows that {x  | x ∈ X } ∪ {y  | y ∈ X } is a subbasis of open sets for the interval topology.

40

Spectral Spaces and Spectral Maps

Conversely, for all x, y ∈ X, jump-density yields  {u ↓ | u < x and u is the lower element of a jump}, x =  {v  | y ≤ v and v is the lower element of a jump}. y = ◦

If w is the lower point of a jump, then w ↓ ∈ K (X) and w  = X \ w ↓ ∈ K(X). Thus, w ↓ and w  are clopen for the constructible topology, hence x  and y  are open.  1.6.5 Corollary Let (P, ≤) be a totally ordered set and let X ⊆ P(P) be the set of all down-sets of P. Then there is a unique spectral topology on X whose specialization order is inclusion. Proof Inclusion is a complete total order on X. It is also jump-dense, because the jumps are precisely the pairs a < a ↓ , where a ∈ P. Now apply 1.6.4(i).  1.6.6 Corollary Let (λ, ≤) be a well-ordered set. Then there is a unique topology on λ whose specialization order is ≤. If λ has a largest element, then this topology is spectral. Proof Let τ be a topology on λ with specialization order ≤. By definition of the coarse lower topology we know τ  (  ) ⊆ τ. Suppose ∅  C ⊆ λ is closed for τ. Since λ is well-ordered, the set C contains a smallest element x, hence C = x ↑ is closed for τ  (  ), proving τ = τ  (  ). If λ has a largest element, then λ is complete. Since λ is also jump-dense we may apply 1.6.4(i) again.  1.6.7 Proposition Let (X, ≤) be a totally ordered set and let τ L = τ L (≤) be the fine lower topology on X (see A.8(ii). Then: (i) (ii) (iii) (iv)

◦

K (X, τ L ) = {x ↓ | x ∈ X } ∪ { ∅ }. τ L satisfies axiom S1 if and only if ≤ has a largest element. τ L satisfies axioms S2 and S3 of a spectral space. If ≤ has a largest element, then the following are equivalent: (a) τ L satisfies axiom S4 (i.e., (X, τ L ) is sober). (b) τ L is spectral. (c) (X, ≤) is isomorphic to a successor ordinal. 10

Proof (i) By 1.6.1(iii)(a) we know that the inclusion “⊆” holds. Every set of the form x ↓ is open for τ L , hence equality follows from 1.6.1(ii). (ii) holds by 1.6.1(ii). 10

See A.4(v) or [Cie97, Section 4.2].

1.6 Examples

41

(iii) Axiom S3 holds by 1.6.1(iv). Axiom S2 follows from (i) because every down-set is a union of sets of the form x ↓ . (iv) (a) ⇔ (b) follows from (ii) and (iii), because we assume that ≤ has a largest element. Hence it remains to show that ≤ is well-ordered if and only if τ L is sober. Soberness says that all nonempty closed and irreducible subsets have a generic point. For τ L the nonempty closed and irreducible subsets are just the nonempty up-sets (cf. 1.6.1(i)). A nonempty up-set has a generic point if and only if it has a smallest point. Thus, soberness is equivalent to the statement that every nonempty up-set has a smallest element (i.e., X is well-ordered, A.4(ii)).  1.6.8 Proposition Let X be a Boolean space and ≤ be a total order on X such that x ↑ and x ↓ are closed for every x ∈ X. Then (X, ≤) is complete and jump-dense. Furthermore, ≤ is a spectral order on X and the topology on X is the interval topology given by ≤. Proof We may assume that X  ∅ . To prove completeness, consider a subset C ⊆ X. Then (c ↑ )c ∈C is a family of closed subsets of X and has the FIP. Let  D = c ∈C c ↑ be the set of upper bounds of C, which is closed and  ∅ . Now (d ↓ ∩ D)d ∈D is a family of closed sets and has the FIP. It follows that  ↓ d ∈D d ∩ D  ∅ . In fact, this is a singleton, say {z}, and z = sup(C). For jump-density, take u < v. Since u ↓ and v ↑ are closed and disjoint there is a clopen subset C with u ↓ ⊆ C and C < v. The family (c ↑ ∩ C)c ∈C consists of closed sets and has the FIP. Therefore the intersection is nonempty and is a singleton, say {x}. Clearly, x is the largest element of C. The set D = x ↑ ∩(X \C) of proper upper bounds of C is closed and u < D. It has been shown above that D has a smallest element y. Now it follows that x < y is a direct specialization lying between u and v. By 1.6.4, completeness and jump-density of ≤ imply that τ  (≤) is spectral and its patch space is the set X with the interval topology of ≤. Clearly, the interval topology coarsens the given Boolean topology on X. This is possible only if the topologies coincide. In particular, ≤ is a spectral order of X.  In Section 3.6 we return to spectral spaces with totally ordered specialization when we consider spectra of totally ordered sets. Moreover, the present considerations will be used for the study of specialization chains in Section 4.2 (i.e., totally ordered (under specialization) subsets of spectral spaces). In particular, the conditions in 1.6.4(i) play a role for the Kaplansky Problem, 4.2.13.

42

Spectral Spaces and Spectral Maps

B.

The Wedge Construction

We start with two disjoint spectral spaces X and Y whose specialization orders are both total. Both X and Y have a top element by 1.6.1(ii). A new spectral space is created by identifying the two top elements and leaving X and Y unchanged otherwise. The pictorial representation of the specialization order  of the new space has shape and is therefore called the wedge of X and Y . 1.6.9 Construction Let X and Y be disjoint spectral spaces with total specialization order and top elements x1 and y1 . We form the new set  X Y = (X \ {x1 }) ∪ (Y \ {y1 }) ∪ {z} with z  X ∪ Y . Identifying x1 = z = y1  we consider X and Y as subsets of X Y with inclusion maps eX and eY . Then  τ = {W ⊆ X Y | W ∩ X is open in X and W ∩ Y is open in Y } is clearly (the set of open sets of) a topology on X ∧ Y . The topological space  (X Y, τ) is called the wedge of X and Y . 1.6.10 Properties of the Wedge Continuing with the above notation we describe the basic topological properties of the wedge, showing, in particular, that it is a spectral space. As always (A.3(ii)), for x ∈ X the up-set and the down-set generated by x are denoted by x ↑ and x ↓ , respectively; similarly for y ∈ Y.  (i) The lattice of open sets is given by the definition, O(X Y ) = τ. Taking complements we get   A(X Y ) = {C ⊆ X Y | C ∩ X ∈ A(X) and C ∩ Y ∈ A(Y )}. In particular, X and Y are closed subspaces. Every nonempty closed subset of X, or of Y , is the set of specializations of a point, 1.6.3(iii). Thus, the nonempty closed sets are of the form x ↑ ∪ y ↑ with x ∈ X and y ∈ Y.  (ii) It follows that X Y is a T0 -space, and p  q holds in the wedge if and only if p, q ∈ X and p  q holds in X or p, q ∈ Y and p  q holds in Y . This confirms the wedge shape of the poset.  (iii) A set C ⊆ X Y is closed and irreducible if and only if it is contained in one of X and Y and is closed and irreducible in this subspace. If it is nonempty, then it is the closure of a single point in X or in Y . Since these are both spectral spaces it follows that C is the closure of a single  point. We conclude that X Y is sober. (iv) Let τ  be the coarse lower topology for the specialization order. The sets  p↑ with p ∈ X Y are a subbasis of closed sets for τ  and belong to

1.6 Examples 43   A(X Y ). Conversely, if C ∈ A(X Y ), say C = x ↑ ∪ y ↑ , see (i), then  C is closed for τ  . We see that τ  = O(X Y ).   (v) As X and Y are closed in X Y , a set C ⊆ X Y is quasi-compact if and only if C ∩ X and C ∩ Y are both quasi-compact. ◦  (vi) A set U ⊆ X Y is quasi-compact open if and only if U ∩ X ∈ K (X) and ◦

U ∩ Y ∈ K (Y ) (by (v) and the definition of the topology). Thus, a proper nonempty subset U is quasi-compact open if and only if U = x ↓ ∪ y ↓ or U = x ↓ or U = y ↓ , where x ∈ X and y ∈ Y are lower points of jumps, 1.6.3(i). ◦   (vii) It is clear from (vi) that K (X Y ) is a bounded sublattice of O(X Y ) and is a basis of the topology. In connection with (ii) and (iii) one  concludes that X Y is a spectral space.   (viii) The closed constructible sets of X Y are ∅, X Y and the sets x ↑ ∪ y ↑ , where x is an upper point of a jump in X or x is the generic point of X, and the same for y ∈ Y .

C.

Spectral Spaces with a Single Specialization

To apply the Priestley method for the construction of spectral spaces one must specify a spectral order on a Boolean space X, cf. 1.5.11. Experience shows that it may be hard to recognize whether a given partial order is spectral. But sometimes this is a tractable problem. We show that a minimal nontrivial partial order on a Boolean space is always spectral. 1.6.11 Construction Suppose that X is a Boolean space and let a, b ∈ X be distinct elements. A partial order on X is defined by: x ≤ y ⇐⇒ x = y or (x = a and y = b). We claim that ≤ is a spectral order. Pick x, y ∈ X such that x  y. We must find a ≤-up-set C ∈ K(X) with x ∈ C and y  C: • If x  a there is some C ∈ K(X) such that x ∈ C and a, y  C. This is an up-set since a  C. • If x = a, then x  y implies y  b. There is some C ∈ K(X) with a, b ∈ C and y  C. Again, C meets the requirements. Let τ be the topology of the spectral space associated with the Priestley space (X, ≤). Then ◦

K (X, τ) = {U ⊆ X | U is clopen in X and b ∈ U ⇒ a ∈ U}.

44

Spectral Spaces and Spectral Maps

The Spectral Space S∞ , where S is an Infinite Set

D.

Let S be an infinite set considered as a discrete topological space. The one-point compactification of S is denoted by S ∗ , which is a Boolean space, cf. 1.3.3. The point at infinity is denoted by ∞. A partial order on S ∗ is defined by setting ∞ < a for all a ∈ S. We show that this is a spectral order on S ∗ , hence yields a Priestley space and the associated spectral space, 1.5.11, which is denoted by S∞ . We give a detailed description of S∞ . 1.6.12 Reminder First recall the following facts about the Boolean space S ∗ : (i) O(S ∗ ) = P(S) ∪ {S ∗ \ F | F ⊆ S finite}; (ii) A(S ∗ ) = {T ∪ {∞} | T ∈ P(S)} ∪ {F ⊆ S | F finite }; (iii) Clop(S ∗ ) = {S ∗ \ F | F ⊆ S finite } ∪ {F ⊆ S | F finite}. 1.6.13 A Spectral Partial Order on S ∗ partial order on the set S ∗ = S ∪ {∞}:

The following conditions define a

s ≤ t ⇐⇒ s = ∞ or s = t ∈ S. The poset (S ∗, ≤) is depicted by the following Hasse diagram: ··· • i

•f

•]

•O

A•

8•

5 • · · · (s ∈ S)

• ∞ The specialization order of S∞ Claim

The partial order ≤ is spectral on the Boolean space S ∗ .

Proof of Claim Pick s, t ∈ S ∗ with s  t. Then s  ∞ and so s ∈ S. The singleton {s} is clopen in S ∗ , see 1.6.12, it is an up-set for ≤, and contains s, but not t.  The Priestley space (S ∗, ≤) defines a spectral space (see 1.5.11), which is denoted by S∞ . Specialization in S∞ is the partial order ≤. The figure in 1.6.13 shows that the points s ∈ S are closed, and the point ∞ is generic (i.e., dense) in S∞ . It is clear from the construction via Priestley spaces that (S∞ )con = S ∗ . ◦

Next, we determine the lattices K (S∞ ) and K(S∞ ). 1.6.14 Proposition Let S be an infinite set. If A ⊆ S ∗ , then: ◦

(i) A ∈ K (S∞ ) ⇐⇒ A = ∅, or S ∗ \ A is a finite subset of S. (ii) A ∈ K(S∞ ) ⇐⇒ A ⊆ S is finite, or A = S ∗ .

1.6 Examples

45

(iii) A ∈ K(S∞ ) ⇐⇒ A ⊆ S is finite, or S ∗ \ A is a finite subset of S. It follows that: (iv) Every open subset of S∞ is quasi-compact and every closed subset is constructible. Proof (i) and (ii) are equivalent (taking complements). We prove (ii). The closed and constructible sets are precisely those that are clopen in S ∗ = (S∞ )con and specialization-closed, see 1.5.4(ii). Therefore the assertion follows by inspection of the clopen subsets of S ∗ , see 1.6.12(iii). (iii) is just a repetition of the description of Clop(S ∗ ) in 1.6.12, using spectral space terminology. (iv) Note that K(S∞ ) is a basis of closed sets. It follows from (ii) that K(S∞ ) is closed under arbitrary intersections. Thus, every closed set belongs to K(S∞ ). The assertion about open sets follows by taking complements.  1.6.15 The Inverse Topology of S∞ The Hasse diagram of the specialization order of (S∞ )inv is the one from 1.6.13 turned upside down. Proposition 1.6.14 and Theorem 1.4.3(i) yield the following facts about subsets A ⊆ S ∗ : (i) A ∈ K((S∞ )inv ) ⇐⇒ A = ∅, or S ∗ \ A is a finite subset of S. ◦

(ii) A ∈ K ((S∞ )inv ) ⇐⇒ A ⊆ S is finite, or A = S ∗ . (iii) A ∈ K((S∞ )inv ) ⇐⇒ A ⊆ S is finite, or S ∗ \ A is a finite subset of S. (iv) O((S∞ )inv ) = P(S) ∪ {S ∗ ). (v) A((S∞ )inv ) = {T ⊆ S ∗ | ∞ ∈ T } ∪ {∅). Observe that not every open subset of (S∞ )inv is quasi-compact. For example, the subset S is open, discrete, and infinite, whence not quasi-compact. Incidentally, note that (S∞ )inv satisfies the separation axiom T5 . 11 To see inv inv this, let A, B ⊆ (S∞ )inv be nonempty sets with A ∩ B = A ∩ B = ∅ . We must find disjoint open sets O A , O B such that A ⊆ O A , B ⊆ O B . Since A  ∅ inv and every element specializes to ∞ in (S∞ )inv , we conclude ∞ ∈ A , and then inv inv A ∩ B = ∅ forces B ⊆ S. Likewise, A ∩ B = ∅ and B  ∅ imply A ⊆ S. Since the singletons {s} (s ∈ S) are open in (S∞ )inv (cf. (iv)), it follows that A, B are open, and we may take O A = A , O B = B. 11

The separation axiom T5 , see [StSe70, p. 12], says that all sets A, B with A ∩ B = A ∩ B = ∅ have disjoint neighborhoods. This property is sometimes called complete normality.

46

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1.7 Further Reading 1.7.1 Well-Filtered Spaces A T0 -space X = (X, τ) is well-filtered if, for every down-directed (for inclusion) family (Qi )i ∈I of quasi-compact saturated  subsets and every closed set A ⊆ X, the intersection A ∩ i ∈I Qi is nonempty if A ∩ Qi  ∅ for all i. This condition may be viewed as a weak form of compactness for the dual topology, 1.4.7. (i) Sober spaces are well-filtered, cf. [GL13, Thm. 8.3.5, p. 366]. (ii) If X is locally quasi-compact (cf. [GL13, Definition 4.4.1, p. 67], e.g. when X satisfies axiom S2), then X is sober if and only if X is wellfiltered, cf. [GL13, Thm. 8.3.8, p. 367]. 1.7.2 The Patch Topology of a T0 -Space Let (X, τ) be a T0 -space. In 1.4.7 the dual topology τ ∂ is compared with the inverse topology. The inverse topology is always coarser than τ ∂ , and the topologies coincide if axiom S2 holds. Therefore, it is natural to define the patch topology of X to be the topology generated by τ and τ ∂ . (If X is spectral then it is clear that this patch topology is the same as the constructible topology, 1.3.11 and 1.4.3.) One can show that the patch topology is quasi-compact if and only if X is quasi-compact, well-filtered, and coherent, cf. [GL13, Prop. 9.1.27, p. 404]. 1.7.3 Ordered Topological Spaces Let X = (X, τ) be a Hausdorff space equipped with a partial order ≤. If the partial order is closed (cf. 1.5.10), then (X, ≤) = (X, τ, ≤) is called an ordered (topological) space. Ordered spaces are studied in [Nac65]. For each ordered space (X, ≤) the set of closed up-sets of X is the set of closed sets of a topology, which we denote by τ(X, ≤). In particular, if X is a spectral space then (Xcon,  X ) is an ordered topological space (by 1.5.11) and τ(Xcon,  X ) = τ. 1.7.4 Stably Compact Spaces A T0 -space X is stably compact if it is quasicompact, locally quasi-compact, coherent, and sober (see [GL13, Definition 4.4.1, p. 67], 1.1.20, and 1.4.7 for these notions). Let (X, ≤) be a compact ordered topological space (in particular, Hausdorff, 1.7.3). Then τ(X, ≤) is a stably compact topology. Examples of stably compact spaces include: spectral spaces, compact spaces, and complete totally ordered sets equipped with the coarse lower topology, for example (R, τ  (≤)). Now assume that Y = (Y, τ) is stably compact. Then its dual space (Y, τ ∂ ) is stably compact as well and τ ∂∂ = τ. This correspondence between τ and

1.7 Further Reading

47

its dual is called de Groot duality. Let X be the patch space of Y . Then X is compact and (X,  Y ) is an ordered space with τ(X,  Y ) = τ. Stably compact spaces can also be characterized as retractions of spectral spaces. A space X is stably compact if and only if there is a spectral space Y containing X as a subspace together with a continuous map r : Y −→ X such that r ◦ i = idX , where i : X → Y is the inclusion. This has been proved by Harold Simmons in [Sim82]. 12 1.7.5 Nachbin Duality Assume that X is a compact space, not necessarily Boolean. Then the stably compact topologies with patch space X (see 1.7.2) correspond bijectively to the closed partial orders. This correspondence is sometimes referred to as Nachbin duality. If X is even Boolean then Nachbin duality restricts to the bijection (exhibited in 1.5.11) between spectral topologies with patch space X and spectral orders on X. However, we shall see in 8.2.5 that there exist Boolean spaces with a nonspectral closed partial order. Thus, the Nachbin dual of such a closed partial order is not a spectral topology. 1.7.6 Perfect Maps As a last topic in our further reading list we mention morphisms in the wider context of stably compact spaces. A map f : X → Y between arbitrary topological spaces is called perfect, cf. [GL13, 9.4.1, p. 419], if it is continuous and for all quasi-compact saturated sets Q ⊆ Y , the set f −1 (Q) is quasi-compact (it is saturated by continuity). In other words, f is perfect just if it is both continuous and continuous for the dual topologies, see 1.4.7. Hence by 1.4.6, when X,Y are spectral then f is perfect if and only if it is spectral. A perfect map between stably compact spaces is the same as a continuous map that is also continuous for the patch topologies, see [GL13, 9.4.5, p. 419]. The list above reveals parallels between the category of stably compact spaces with perfect maps as morphisms, and the category of spectral spaces. In fact, a sizable amount of topological results for spectral spaces in this book can be established in the stably compact context. For details we refer to [Law11], [GL13, Chapter 9], [AMJK04], and [Nac65].

12

For a proof also see [GL13, Theorem 9.5.14, p. 427], which attributes the result to Johnstone. However, Johnstone in [Joh83, top of p. 48] refers to Simmons.

2 Basic Constructions

There are many ways in which a spectral space, or a collection of spectral spaces, can be used to produce a new spectral space. In every category it is a matter of routine to ask for an appropriate notion of substructures or subobjects, for the existence of products, fiber products, projective limits, coproducts, quotient objects, and so on. Some of these questions can be answered with rather elementary means, others need more refined methods. We start with a collection of constructions that are easily accessible and are essential for the further analysis of spectral spaces. These are the formation of • spectral subspaces, Section 2.1, • products, Section 2.2, • finite coproducts, Section 2.4. The results imply that the category Spec is complete, 2.3.8. Combining spectral subspaces with products, we show that every spectral space is homeomorphic to a spectral subspace of a power of the Sierpiński space, 2.3.1. Thus, all spectral spaces can be obtained from the innocent-looking Sierpiński space by applying two of the most elementary constructions. Moreover, every spectral space is homeomorphic to a projective limit of finite spectral spaces, 2.3.10. In Section 2.5 we exhibit a general method for associating, in a functorial way, spectral spaces with the objects of different categories of algebraic structures. An immensely important functor is the Zariski spectrum, or prime spectrum, of a commutative unital ring. The constructions presented in Sections 2.1 to 2.3 lead to a simple method for proving that the Zariski spectrum of a ring is a spectral space. The same method yields various other functors into the category Spec. A more thorough analysis of the Zariski spectrum and the real spectrum of a ring is presented in Chapter 12 and Chapter 13, respectively. Further constructions with spectral spaces are presented in Chapter 6 (quotients) and in Chapter 10 (colimits), inter alia. 48

2.1 Spectral Subspaces

49

Hochster’s seminal paper [Hoc69] contains results about various constructions with spectral spaces (see, e.g., his Theorem 7). He shows in particular that every spectral space is a subspace of a power of the Sierpiński space (Proposition 9), and that every spectral space is a projective limit of finite T0 -spaces (Proposition 10).

2.1 Spectral Subspaces Summary Suppose that Y is a spectral space and X ⊆ Y is a subset. Then X carries the subspace topology, O(X) = {U ∩ X | U ∈ O(Y )}. We are concerned with the question of whether the subspace X is also spectral. Obviously, the general answer is “no”. For, if the subset X is not quasi-compact, then X cannot possibly become a spectral space in this way. However, there is also another more subtle problem. Assume that X is indeed a spectral space. It is necessary to relate X and Y to each other in the category Spec, that is, the inclusion map ought to be spectral. But Spec is not a full subcategory of Top, cf. 1.2.3, hence the continuous inclusion map is not automatically spectral. We exhibit an example of this phenomenon, cf. 2.1.2. On the basis of these considerations, a definition of spectral subspaces is given in 2.1.1. It is not always easy to decide directly from the definition whether a subset of a spectral space is a spectral subspace. But we give a characterization in 2.1.3 which solves this problem completely: the spectral subspaces are exactly the proconstructible subsets. It is a consequence of this characterization that images and inverse images of spectral subspaces under spectral maps are always spectral subspaces, see 2.1.4. 2.1.1 Definition Suppose that Y is a spectral space and X is a subset. We call X a spectral subspace if X with the subspace topology is a spectral space and the inclusion map X → Y is spectral. 2.1.2 Example The condition of 2.1.1 about the inclusion map is essential. We construct a spectral space Y and a subset X ⊆ Y such that X is Boolean for the subspace topology (hence also spectral, 1.3.4), but the inclusion map X → Y is not spectral. The set S = N ∪ {ω} is endowed with the discrete topology. We form the one-point compactification S ∗ = S ∪ {∞}, which is a Boolean space, cf. 1.3.3 and 1.6.12. By 1.6.11 the following partial order on S ∗ is spectral: x ≤ y ⇐⇒ (x = y) ∨ (x = ∞ ∧ y = ω).

50

Basic Constructions

The following Hasse diagram depicts the poset: ω •O •

•

•

•

•

•

···

• ∞ Hasse diagram of S ∗ = N ∪ {ω, ∞} The Priestley space (S ∗, ≤) yields a spectral space which we denote by Y . The specialization order is ≤, and S ∗ is the patch space, cf. 1.5.11. We define X to be the subspace of Y with underlying set S = N ∪ {ω}. Note that S, as a subset of S ∗ , is open (since S = S ∗ \ {∞}), and not closed. The inclusion map f : X → Y is continuous. We show that X is Boolean. More precisely, X is the one-point compactification of the discrete space N with limit point ω: every singleton {n} with n ∈ N is clopen in S ∗ and is both an up-set and a down-set with respect to ≤. Thus, {n} is clopen in Y , hence is also clopen in X. Suppose that O ∈ O(X) contains ω. There is some U ∈ O(Y ) with O = X ∩ U. Since ∞ ≤ ω and U is a down-set for ≤, we get ∞ ∈ U. Then U is cofinite, and O is cofinite as well. This proves that X is the one-point compactification of N, hence is Boolean. Now assume that the inclusion map f is spectral. Then X = f (X) ⊆ Y is proconstructible by 1.3.23 (i.e., S ⊆ Ycon = S ∗ is a closed subset), a contradiction. The next result provides a very satisfactory characterization of spectral subspaces. Quite often the criterion is easy to check. 2.1.3 Theorem Let Y be a spectral space. The following conditions about a subset X ⊆ Y are equivalent: (i) X is a spectral subspace. (ii) X is a proconstructible subset (i.e., X is closed in Ycon ). If X is a spectral subspace, then ◦

◦

K (X) = {U ∩ X | U ∈ K (Y )}, K(X) = {V ∩ X | V ∈ K(Y )}, K (X) = {C ∩ X | C ∈ K (Y )}. The patch space Xcon is the set X with the subspace topology inherited from Ycon . Proof The implication (i) ⇒ (ii) holds by 1.3.23, which says that images of spectral maps are proconstructible.

2.1 Spectral Subspaces

51

(ii) ⇒ (i) We consider X as a subspace of Y and first show that (∗)

◦

◦

K (X) = {U ∩ X | U ∈ K (Y )}. ◦

The sets U ∩ X with U ∈ K (Y ) are a basis of open sets for X (by definition of the subspace topology) and are a bounded sublattice of P(X). By 1.1.8 we only have to show that each U ∩ X is quasi-compact. Let τ be the topology on X induced by Ycon . The subspace (X, τ) ⊆ Ycon is closed, hence is a Boolean space ◦

(since Ycon is Boolean). If U ∈ K (Y ), then U is closed for the patch topology on Y , hence U ∩ X is closed in (X, τ). Thus U ∩ X is compact for τ. As τ is finer than the restriction of the spectral topology of Y , it follows that U ∩ X is quasi-compact in X. The equality (∗) immediately shows that the topological space X inherits axioms S1–S3 from the spectral space Y (cf. 1.1.5). To show that X is a spectral space it remains to prove soberness. So, let V ⊆ X be a nonempty closed and irreducible subset. Since V is closed in X and X is proconstructible in Y , V is also proconstructible in Y . Moreover, V is irreducible in Y as well. Therefore the set ◦

S = {U ∩ V | U ∈ K (Y ) and U ∩ V  ∅} has the FIP. Every set in S is proconstructible in Y . Hence, by compactness of X  Ycon , there exists a point z ∈ S ⊆ V. It is clear that {z} ⊆ V. On the other X hand, the definition of S shows that V ⊆ {z} , which establishes S4. We conclude that X is a spectral space and, by (∗), the inclusion f : X → Y is a spectral map, thus proving (ii) ⇒ (i). To show the remaining statements, notice that condition (∗) implies K(X) = {V ∩ X | V ∈ K(Y )} by taking complements. The inclusion map f : X → Y is spectral, as we have shown, hence fcon : Xcon → Ycon is a continuous injective map between Boolean spaces. Thus, fcon is a homeomorphism onto the image (i.e., Xcon is the set X with the topology inherited from Ycon ). The homomor◦

phism K( f ) : K(Y ) → K(X) : C → C ∩ X maps the generating set K (Y ) of ◦ the Boolean algebra K(Y ) onto the generating set K (X) of K(X). Thus, K( f ) is surjective.  2.1.4 Corollary Let f : X → Y be a spectral map. Images of spectral subspaces of X and preimages of spectral subspaces of Y under f are again spectral subspaces. In particular, the image of f is a spectral subspace of Y and all fibers of f are spectral subspaces of X. Proof

By 1.3.23 and 2.1.3.



52

Basic Constructions

2.1.5 Corollary Suppose that f : X → Y is a spectral map and Y  ⊆ Y is a spectral subspace containing f (X). Then the map f  : X → Y , obtained by restriction of the codomain, is a spectral map. ◦

◦

Given U  ∈ K (Y ) there is some U ∈ K (Y ) with U  = Y  ∩ U. Then ◦ ( f )−1 (U ) = f −1 (U) ∈ K (X). 

Proof

2.2 Products of Spectral Spaces Summary It is well known that the category Top has products and that the explicit description of products is elementary and simple. We prove that the category Spec has products as well and that these coincide with the products formed in the ambient category Top, 2.2.1. This fact was first proved by Hochster, [Hoc69, Theorem 7]. We give a detailed description of powers of the Sierpiński space, 2.2.3. Familiarity with products of topological spaces is assumed (see, e.g., [Kel75, p. 88 ff], [Bou71b, Chapitre I §4]). In particular we shall use Tychonoff’s Theorem, which asserts that a product of nonempty spaces is (quasi-)compact if and only if each factor is (quasi-)compact, [Bou71b, Chapitre I, p. 63]. Notation Let (Xi )i ∈I be a family of sets (or topological spaces). Given a subset   J ⊆ I, the projection i ∈I Xi → i ∈J Xi , (xi )i ∈I → (xi )i ∈J is denoted by pJ , or by pi if J = {i}. 2.2.1 Theorem Let I be an index set, (Xi )i ∈I a family of spectral spaces, and  X = i ∈I Xi the product in the category Top. Then: (i) X is spectral. ◦

(ii) The lattice K (X) is generated by the sets pi −1 (Ui ), where i ∈ I and ◦ Ui ∈ K (Xi ). Thus, a set U ⊆ X is quasi-compact open if and only if U is a finite union of sets ◦  Ui , where J ⊆ I is finite and Ui ∈ K (Xi ). p−1 J i ∈J

(iii) The lattice K(X) is generated by the sets p−1 i (Vi ), where i ∈ I and Vi ∈ K(Xi ). Thus, a set V ⊆ X is closed and constructible if and only if V is a finite union of sets  Vi , where J ⊆ I is finite and Vi ∈ K(Xi ). p−1 J i ∈J

2.2 Products of Spectral Spaces

53

(iv) Every projection map pJ , in particular the projections pi onto the factors, is spectral. (v) Specialization in X is determined componentwise: (xi )i ∈I X (xi)i ∈I ⇐⇒ ∀i ∈ I : xi Xi xi . (vi) The space X is the product of the Xi in the category Spec (i.e., for every spectral space Y and every family ( fi : Y → Xi )i ∈I of spectral maps there is a unique spectral map h : Y → X with fi = pi ◦ h for all i ∈ I).  (vii) Xcon = i ∈I (Xi )con . The Boolean algebra K(X) is the lattice generated by the sets p−1 i (Ci ), i ∈ I, and Ci ∈ K(Xi ). More explicitly, a set C ⊆ X is constructible if and only if C is a finite union of sets of the form  p−1 Ci , where J ⊆ I is finite and Ci ∈ K(Xi ). J i ∈J

(viii) Xinv =



i ∈I (Xi )inv .

Proof We assume that I  ∅ and that the spaces Xi are all nonempty. For, if I = ∅, then X = 1. And if I  ∅ and Xi = ∅ for some i ∈ I, then X = ∅. These cases are trivial. Products of T0 -spaces are T0 -spaces. Thus, X is a T0 -space. The definition of the product topology shows that specialization is determined componentwise, which proves (v).  Let T = i ∈I (Xi )con be the product of the patch spaces. This is a compact space by Tychonoff’s Theorem. The definition of the product topology shows that T has a basis of clopen sets (i.e., T is Boolean). Let B the set of all sets  p−1 J ( i ∈J Ci ), where J ⊆ I is finite and Ci ∈ K(Xi ). Clearly, B is closed under finite intersections, and B ⊆ K(T). We define B ◦ = B ∩ O(X) and ◦  ◦ B = B ∩ A(X). Thus, p−1 J ( i ∈J Ci ) ∈ B if and only if Ci ∈ K (Xi ) for each  i ∈ J, and p−1 J ( i ∈J Ci ) ∈ B if and only if Ci ∈ K(Xi ) for each i ∈ J. The sets B ◦ and B are closed under finite intersections. The elements of B are a basis of the topology of T, B ◦ is a basis of the topology of X, and B is  a basis of the topology of i ∈I (Xi )inv . The elements of B are compact in T,  hence are quasi-compact in X and in i ∈I (Xi )inv . Moreover, B and B ◦ and B  all separate points in the product set. Now 1.5.17 says that X and i ∈I (Xi )inv ◦

are spectral spaces with patch space T, that K (X) is the lattice generated by ◦  B ◦ , and that K ( i ∈I (Xi )inv ) is the lattice generated by B. These arguments prove (i), (ii), (iii) (from (ii) by taking complements), and  (vii). The spaces Xinv and i ∈I (Xi )inv have the same underlying set and the

54

Basic Constructions

same quasi-compact open sets, hence (viii) follows. It remains to prove (iv) and (vi). ◦

For (iv), pick a subset J ⊆ I, a finite subset K ⊆ J, and a set Ui ∈ K (Xi ) for each i ∈ K. Then







p−1 = U × X U × Xi i i i J i ∈K

i ∈J\K

i ∈K

i ∈I\K

shows that inverse images of quasi-compact open sets are quasi-compact open. Thus, pJ is spectral. For (vi), the mapping property of the product in Top yields the existence of a unique continuous map h : Y → X such that fi = pi ◦ h for all i ∈ I. We have to ◦

show that h is a spectral map (i.e., that h−1 (U) ∈ K (Y ) for all U ∈ B ◦ ). Since ◦   −1 −1 −1 p−1 i ∈J pi (Ui ) it suffices to show that h (pi (Ui )) ∈ K (Y ). J ( i ∈J Ui ) = But this follows from fi = pi ◦ h together with the hypothesis that each fi is spectral.  2.2.2 Corollary Suppose that (Xi )i ∈I and (Yi )i ∈I are families of spectral spaces and that ( fi : Xi → Yi )i ∈I is a family of spectral maps. Then



fi : Xi → Yi , (xi )i ∈I → ( fi (xi ))i ∈I i ∈I

i ∈I

i ∈I



is a spectral map.

2.2.3 Powers of the Sierpiński Space As an application of 2.2.1 we form powers of the Sierpiński space 2. A priori this is a narrow special case of the general construction. But we shall soon show (in Section 2.3) that every spectral space is a subspace of a suitable power of 2. So this rather innocentlooking construction is, indeed, very significant. Let S be any set, nonempty to avoid trivial considerations. The elements of the set 2S can be viewed either as families of 0s and 1s or as maps S → 2. For s ∈ S, let s : 2S → 2 be the evaluation map at s (i.e., s( f ) = f (s)). This is the same as the projection S ps : 2 → 2 onto the sth component. The following facts are immediate from 2.2.1. (a) The product space 2S is spectral with (2S )con = (2con )S , (2S )inv = (2inv )S . (b) The evaluation maps s : 2S → 2 are spectral. For k ∈ {0, 1}, an element s ∈ S, and a subset T ⊆ S we adopt the notation s −1 (k), [[ s = k ]] := { f ∈ 2S | f (s) = k} =

2.2 Products of Spectral Spaces = k ]] := { f ∈ 2S | ∀s ∈ T : f (s) = k} = [[ T



55 [[ t = k ]].

t ∈T

(c) As {0} is the only nontrivial open set in 2, using the notation in (b), the   basic quasi-compact open subsets of 2S are the sets p−1 s ∈R {0} = R = 0 ]], where R varies among the finite subsets of S. Thus, the open [[ R constructible sets and the closed constructible sets can be written as: n  ◦ i = 0 ]] | 0 ≤ n, R1, . . . , Rn ⊆ S finite}, [[ R K (2S ) = { i=1

K(2S ) = {

n 

[[ T i = 1 ]] | 0 ≤ n, T1, . . . ,Tn ⊆ S finite}.

i=1

Each constructible set has a presentation n  i = 0 ]] ∩ [[ T i = 1 ]] [[ R i=1

where n ∈ N0 and the Ri ,Ti ⊆ S are finite. (d) Recall that the specialization order of 2 is 0  1. In view of the componentwise nature of the specialization order in 2S , we see that f  g in 2S is equivalent to f ≤ g. (e) The set 2S with the componentwise partial order is a Boolean algebra. The meet and the join operations are maps ∧, ∨ : 2S × 2S → 2S . Both are spectral maps; we check this for the meet operation. Thus, it is claimed that ◦

s = 0 ]]) ∈ K (2S × 2S ). ∧−1 ([[ The following equation shows this: ∧−1 ([[ s = 0 ]]) = {( f , g) ∈ 2S × 2S | f (s) = 0 or g(s) = 0}  

s = 0 ]] . = [[ s = 0 ]] × 2S ∪ 2S × [[ (f) The construction of powers of 2 is a functor from Sets, the category of sets, to the category Spec. Namely, let F : S → T be a map of sets. Then the mapping property of products, 2.2.1(vi), yields the spectral map 2F : 2T → 2S ; 2F ( f ) = f ◦ F. 2.2.4 The Spectral Topology on a Power Set As in 2.2.3, let S be any set. The characteristic functions map χ : P(S) → 2S , sending R ⊆ S to the characteristic function χR , is bijective. Let τ be the topology of P(S) making χ a homeomorphism.

56

Basic Constructions

(i) The following facts about the topology of P(S) are translations of the corresponding statements in 2.2.3. (a) P(S) is a spectral space. ◦

(b) K (P(S)) is the bounded sublattice of the power set of P(S) generated by the sets {R ⊆ S | s  R}, where s ∈ S. (c) K(P(S)) is the bounded sublattice of the power set of P(S) generated by the sets {R ⊆ S | s ∈ R}, where s ∈ S. (d) For R,T ⊆ S we have R  T ⇐⇒ R ⊆ T. (ii) The underlying specialization poset of the spectral space P(S), cf. (i), is the Boolean algebra P(S). We claim that the spectral topology of P(S) coincides with the coarse lower topology of the poset (P(S), ⊆). Proof It is clear from (i)(d) that the specialization relation of the spectral topology coincides with the specialization relation of the coarse lower topology. Thus, the spectral topology is finer than the coarse lower topology, A.8(ii). It remains to show that every closed set of the spectral topology is closed for the coarse lower topology. It is sufficient to prove this for each subbasic closed set {R ∈ P(S) | s ∈ R}, see (i)(c). Thus, to wrap things up, just note that {R ∈ P(S) | s ∈ R} = {s} ↑ ⊆ P(S), hence is subbasic closed for the coarse lower topology of P(S).  It is a well-known fact that every complete and atomic Boolean algebra is isomorphic to the power set of its set of atoms, [Kop89, Corollary 2.7]. Thus the claim above can also be expressed in more abstract language by speaking about complete atomic Boolean algebras. (iii) The inverse topology of the coarse lower topology on P(S) is the coarse upper topology, A.8(ii). Proof The inverse topology is trivially finer than the coarse upper topology. Conversely, the sets {R ⊆ S | s ∈ R} (where s varies in S) are a subbasis of open sets for the inverse topology, cf. (i)(c). Therefore it suffices to note that each such set is open for the coarse upper topology. Namely, {R ⊆ S | s ∈ R} = P(S) \ (S \ {s})↓, and (S \ {s})↓ is subbasic closed for the coarse upper topology.



2.3 Spectral Subspaces of Products

57

(iv) The coarse lower topology of P(S) is spectral, (ii). This is the only spectral topology on P(S) with specialization order ⊆. T if and only Proof Let τ be a spectral topology on P(S) such that R  τ if R ⊆ T. Then τ  (P(S)) ⊆ τ. By 1.4.7 and (iii) we know that τ u (P(S)) = τ  (P(S))inv = τ  (P(S))∂ ⊇ τ ∂ = τinv . The specialization orders of τ u (P(S)) and τinv are equal, hence τ  (P(S))inv = τ u (P(S) = τinv . Forming inverse topologies again, we conclude that τ  (P(S)) = τ, 1.4.5. 

2.3 Spectral Subspaces of Products Summary The combination of the constructions in Section 2.1 and Section 2.2 leads to the consideration of spectral subspaces of products. A significant special case are spectral subspaces of powers of the Sierpiński space. We show that every spectral space is homeomorphic to such a subspace, 2.3.1. This result provides an important tool for the production of spectral spaces. In particular, it will be used in Section 2.5 to construct spectrum functors. A large number of important and useful constructs arise as subspaces of products, among them fiber products, 2.3.6, projective limits, 2.3.9, graphs of various relations, 2.3.3, and of spectral maps, 2.3.5. Fiber products are particularly significant since a category with products and fiber products is complete, 2.3.8. In fact, limits in the category Spec coincide with limits in the ambient category Top, which was first noted in [Hoc69, Theorem 7]. In Section 11.1 we show that Spec is a reflective subcategory of Top (i.e., the inclusion functor Spec → Top has a left adjoint functor, [HeSt79, Definition 36.1]). General category theory shows that right adjoint functors preserve limits and left adjoint functors preserve colimits, [HeSt79, Definition 24.1, Theorem 27.7]. Therefore, these category-theoretic facts imply Hochster’s result that limits in Spec coincide with limits in Top. Our first result follows rather naturally from the existence of products, Section 2.2, and the description of spectral maps into the Sierpińksi space, cf. 1.2.6. Part (i) of the theorem is due to Hochster, [Hoc69, Proposition 9]. 2.3.1 Theorem Suppose that X is a spectral space. (i) The map χ : X → 2 K(X) with χ(x)(V) = χV (x) is a homeomorphism onto a spectral subspace, where χV is the characteristic function of V.

58

Basic Constructions

(ii) The image of χ is the set of bounded lattice homomorphisms from K(X) to the Boolean algebra 2. Proof (i) By 1.2.6 the characteristic functions χV with V ∈ K(X) are exactly the spectral maps from X into 2. The maps can be combined to yield the spectral map χ : X → 2 K(X) , 2.2.1(vi). Therefore χ(X) ⊆ 2 K(X) is proconstructible, hence a spectral subspace, see 2.1.4. The map χ is injective since the closed constructible sets separate points in X (by the T0 -property). We show that χ is a closed map onto the image. It suffices to prove that χ(V) is closed in χ(X) for each V ∈ K(X). So, pick a closed constructible set V and an element x ∈ X. Then x ∈ V if and only if χ(x)(V) = χV (x) = 1, if and only if = 1 ]]. Thus, χ(V) = χ(X) ∩ [[ V = 1 ]], and this is a closed χ(x) ∈ χ(X) ∩ [[ V subset of χ(X). (ii) First we show that each map χ(x) : K(X) → 2 is a homomorphism of bounded lattices. Note that χ(x)(∅) = 0 and χ(x)(X) = 1. If V1,V2 ∈ K(X), then χ(x)(V1 ∪ V2 ) = 1 if and only if x ∈ V1 or x ∈ V2 , if and only if χ(x)(V1 ) ∨ χ(x)(V2 ) = 1. This proves that χ(x) is a homomorphism for the join operation. The proof for the meet operation is similar. Now suppose that ϕ : K(X) → 2 is a homomorphism of bounded lattices. We must find some x ∈ X with ϕ = χ(x). Note that ϕ−1 (1) is a prime filter in ◦

K(X), hence the set {X \ V | ϕ(V) = 1} is a prime ideal in K (X). By 1.1.12 the  set C = ϕ(V )=1 V ⊆ X is nonempty closed and irreducible. Soberness yields a generic point x for C. For each W ∈ K(X) the following equivalences hold: W ∈ χ(x)−1 (1) if and only if x ∈ W, if and only if C ⊆ W. We claim that C ⊆ W is equivalent to W ∈ ϕ−1 (1). Proof of Claim The inclusion C ⊆ W is clear if ϕ(W) = 1. For the reverse implication, assume C ⊆ W and note that X \ W is quasi-compact. As (X \  W) ∩ ϕ(V )=1 V = (X \ W) ∩ C = ∅ there are finitely many V1, . . . ,Vr ∈ ϕ−1 (1) with (X \ W) ∩ (V1 ∩ · · · ∩ Vr ) = ∅. Set V = V1 ∩ · · · ∩ Vr and note that ϕ−1 (1) is a filter. Thus, ϕ(V) = 1, and V ⊆ W implies ϕ(W) = 1.  Altogether we have shown that ϕ(W) = χ(x)(W) for all W ∈ K(X), which proves the assertion.  2.3.2 Remark As every spectral space is a subspace of a power of 2 (by 2.3.1(i)) it follows that every spectral space is a spectral subspace of a spectral space carrying the coarse lower topology. In 11.4.1(i) we show that every T0 space is homeomorphic to a subspace of a spectral space. Thus, every T0 -space is homeomorphic to a subspace of a spectral space carrying the coarse lower topology. Related to this theme, see also 7.1.5.

2.3 Spectral Subspaces of Products

59

2.3.3 Graphs of Relations We examine relations on spectral spaces (e.g., the graph of a spectral map or the graph of the specialization order). There are many important relations that are spectral subspaces of product spaces. First we fix some notation. Let X and Y be sets. • If X = Y , then the relation Δ(X) = {(x, x) | x ∈ X } is the diagonal of X. The diagonal is the image of the diagonal map ΔX : X → X × X. The map ΔX is continuous if X is a topological space and is spectral if X is a spectral space, by 2.2.1. It is well known that a topological space is Hausdorff if and only if the diagonal is closed in the product space, [Bou71b, p. I.52]. • Suppose that X = Y and ≤ is a partial order on X with graph X ≤ = {(x, y) | x ≤ y}. The case that X is a T0 -space with specialization order  is particularly significant for us. The graph is then denoted by X  . • Suppose that f : X → Y is a map and Γ( f ) = {(x, y) | f (x) = y} is its graph. We recall the following connection between Γ( f ) and the diagonal of Y . Consider the map f × idY : X × Y → Y × Y , (x, y) → ( f (x), y), then Γ( f ) = ( f × idY )−1 (ΔY ). 2.3.4 Proposition Let X be a spectral space. Then: (i) The sets ΔX and X  are proconstructible and ΔX ⊆ X  ⊆ ΔX = {(y, z) ∈ X × X | Gen(y) ∩ Gen(z)  ∅}. (ii) The following conditions are equivalent: (a) X is Boolean. (b) ΔX is closed. (c) X  is closed. (iii) The following conditions are equivalent: (a) X is irreducible. (b) ΔX is dense in X × X. (c) X  is dense in X × X. Proof (i) The inclusion ΔX ⊆ X  holds trivially. The diagonal is the image of a spectral map, cf. 2.3.3, hence is a proconstructible set. The set X  is proconstructible by 1.5.9. By 1.5.4(i) we know that ΔX = Spez(Δ(X)). Specialization in X × X is determined componentwise, see 2.2.1(v); hence (y, z) ∈ X × X belongs to Spez(Δ(X)) if and only if there is some x ∈ X such that x  y and x  z (i.e., if and only if Gen(y) ∩ Gen(z)  ∅). The inclusion X  ⊆ ΔX is a trivial consequence.

60

Basic Constructions

(ii) (a) ⇔ (b) We noted that ΔX is closed if and only if X is Hausdorff, which, for a spectral space, is the same as being Boolean (cf. 1.3.4). (a) ⇒ (c) The diagonal and the specialization relation coincide. Since the diagonal is closed, so is the specialization relation. (c) ⇒ (a) If X is not Boolean, then X has at least one proper specialization y, 1.3.20. Then (x, x)  (x, y), (y, x) in X × X. But (y, x)  X  , hence x  X  is not closed under specialization, hence it cannot be closed. (iii) (b) and (c) are equivalent since, by (i), ΔX = X  . (a) ⇒ (b) This follows from (i) since any two points have a common generalization, namely the generic point of X. (b) ⇒ (a) It is enough to show that any two nonempty sets U,V ∈ O(X) have a nonempty intersection. The diagonal is dense in X × X, and U × V ⊆ X × X is nonempty and open. Thus, there is a point (x, x) ∈ ΔX ∩ (U × V). But then x ∈ U ∩ V, as required.  2.3.5 Corollary Let f : X → Y be a spectral map. Then the graph Γ( f ) is a proconstructible subset of X × Y . Proof The graph is the inverse image of the proconstructible subset ΔY ⊆ Y ×Y under the spectral map f × idY , 2.3.3. The claim now follows from 1.3.23.  2.3.6 Fiber Products Let C be any category and consider a diagram (∗)

f

X

/So

g

Y

in C. An object Z, together with morphisms f , g  making the diagram Z (∗∗)

f

/Y

g

g

 X

f

 /S

commutative, is the fiber product (or the pull-back) of X and Y over S (or of the diagram (∗)) if it has the following universal property: For all objects T and all pairs of morphisms h : T → X, j : T → Y with f ◦ h = g ◦ j there is a unique morphism k : T → Z with h = g  ◦ k and j = f  ◦ k. Fiber products do not exist in every category; in many categories some fiber products exist, others don’t. If the fiber product of (∗) exists then it is frequently

2.3 Spectral Subspaces of Products

61

denoted by X ×S Y . The diagram (∗∗) is then called a pull-back diagram (or a cartesian square, [Mac71, p. 71]). Arbitrary fiber products exist in the category Top, and their construction is well known. If f : X → S and g : Y → S are continuous maps, then X ×S Y is the subspace {(x, y) ∈ X × Y | f (x) = g(y)} ⊆ X × Y , and f  and g  are the restrictions of the projections pX and pY from the product onto the factors. We show that fiber products also exist in the category Spec and coincide with the fiber products in Top. 2.3.7 Theorem Let f : X → S and g : Y → S be spectral maps. Then: (i) The set X ×S Y = {(x, y) ∈ X × Y | f (x) = g(y)} is proconstructible in X × Y. (ii) The restrictions pX : X ×S Y → X and pY : X ×S Y → Y of the projection maps are spectral. (iii) The following universal mapping property holds. Let f1 : Z → X and g1 : Z → Y be spectral maps such that f ◦ f1 = g ◦ g1 . Then there is a unique spectral map h : Z → X ×S Y making the diagram .X ;

f1 pX

Z

h

f



/ X ×S Y pY g1

# 0Y

?S

g

commutative. (iv) The following equalities hold for the fiber product of Xcon and Ycon over S (with respect to conS ◦ fcon and conS ◦ gcon ) and over Scon (with respect to fcon and gcon ): Xcon ×S Ycon = Xcon ×Scon Ycon = (X ×S Y )con . (v) Xinv ×Sinv Yinv = (X ×S Y )inv . Proof (i) The map f × g : X × Y → S × S is spectral, 2.2.2. By 2.3.4, the set Δ(S) ⊆ S × S is proconstructible. Consequently, X ×S Y = ( f × g)−1 (ΔS ) is a proconstructible subset of X ×S Y , 1.3.23. (ii) The maps pX and pY are restrictions of spectral maps to a spectral subspace, hence they are spectral. (iii) By 2.2.1(vi), there is a unique spectral map h : Z → X × Y such that

62

Basic Constructions

f1 = pX ◦ h and g1 = pY ◦ h, and we only need to check that h(Z) ⊆ X ×S Y . This follows readily from f ◦ f1 = g ◦ g1 . (iv) It is clear from the definition that the underlying sets of Xcon ×S Ycon , Xcon ×Scon Ycon , and (X ×S Y )con coincide. They coincide as topological spaces since they are all subspaces of the Boolean space Xcon × Ycon . (v) The underlying sets of Xinv ×Sinv Yinv and (X ×S Y )inv coincide. Their topologies coincide since both are spectral subspaces of Xinv × Yinv = (X × Y )inv , cf. 2.2.1(viii).  2.3.8 Corollary ([Hoc69, Theorem 7]) The category Spec is complete. Suppose that D : I → Spec is a diagram with limit (L, ( fi : L → D(i))i ∈I ). Then (L, ( fi : L → D(i))i ∈I ) is also a limit for the diagram D : I → Spec ⊆ Top (i.e., all limits of diagrams of spectral spaces coincide with their limits in the category of topological spaces). Proof The first statement follows from [HeSt79, Theorem 23.8], the second from [HeSt79, Theorem 24.3].  In every complete category there is a large number of constructions that are special instances of limits of diagrams (e.g., equalizers and projective limits). We give a detailed explanation of projective limits in Spec. 2.3.9 Projective Limits Let I = (I, ≤) be an up-directed poset. A projective system in Spec over I is • a family (Xi )i ∈I of spectral spaces together with • a family fi j : X j → Xi , i ≤ j, of spectral maps such that fii = idXi and fik = fi j ◦ f jk if i ≤ j ≤ k. If we consider I as a category with objects i ∈ I and one morphism j → i whenever i ≤ j, then a projective system is a functor I → Spec. By completeness, the limit (i.e., the projective limit) exists and can be computed in the category Top. Explicitly, the subset

Xi | ∀i, j ∈ I : i ≤ j ⇒ fi j (x j ) = xi } X = {(xi )i ∈I ∈ i ∈I

of the product space is proconstructible, hence is a spectral subspace. The space X, together with the restrictions fi = pi |X : X → Xi of the projection maps from the product to the factors, is the projective limit. Now everything follows immediately from properties of products and spectral subspaces. The patch space Xcon is the projective limit of the projective system

2.3 Spectral Subspaces of Products

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(((Xi )con )i ∈I , (( fi j )con )i ≤ j ). The inverse space Xinv is the projective limit of the projective system (((Xi )inv )i ∈I , (( fi j )inv )i ≤ j ). The projection maps are the spectral maps ( fi )con for the patch spaces and the spectral maps ( fi )inv for the inverse spaces. ◦

The elements of K (X) (resp., K(X), K(X)) are the finite unions of finite ◦ intersections of sets fi−1 (Ui ) with i ∈ I and Ui ∈ K (Xi ) (resp., Ui ∈ K(Xi ), Ui ∈ K(Xi )). Specialization in X is computed componentwise. Usually the projective limit is denoted by lim Xi and the projection maps by ←−− fi : lim Xi → Xi . ←−− 2.3.10 Proposition ([Hoc69, Proposition 10]) The class of spectral spaces is the class of projective limits of finite T0 -spaces. Proof Finite spectral spaces are exactly the finite T0 -spaces, 1.1.15. Thus, 2.3.9 shows that every projective limit of finite T0 -spaces is a spectral space. Conversely, let X be a spectral space. From 2.3.1 we get a spectral map f : X → 2I (with I = K(X)) that is a homeomorphism onto the image. We identify X = f (X). The set Pfin (I) of finite subsets of I is up-directed by inclusion. Further, for each J ∈ Pfin (I) the projection pJ : 2I → 2J maps X onto a spectral subspace XJ ⊆ 2J . Let hJ : X → XJ be the restriction of pJ . For J, K ∈ Pfin (I) with J ⊆ K the projection map pJ K : 2K → 2J restricts to a spectral map gJ K : XK → XJ . Now ((XJ )J , (gJ K )J ,K ) is a projective system of finite spaces and has a projective limit Z with projection maps (gJ : Z → XJ )J . Note that the special projection maps gi : Z → 2 with i ∈ I define an injective spectral map g : Z → 2I . Moreover, the universal mapping property of the projective limit, together with the spectral maps hJ : X → XJ , yields a spectral map h : X → Z. It is clear that g ◦ h : X → 2I is the inclusion map, hence it follows that h is a homeomorphism.  In 6.4.12(iv) we show that every spectral space is a projective limit of finite T0 spaces, where the transition maps and the projection maps are even identifying. 2.3.11 The Specialization Fiber Product Let f : X → S, g : Y → S be continuous maps between topological spaces. We define a topological space, called the specialization fiber product of X and Y along f and g. The underlying set is X ×S Y = {(x, y) ∈ X × Y | f (x)  g(y)} = ( f × g)−1 (S  ) together with the projection maps pX : X ×S Y → X and pY : X ×S Y → Y . We endow X ×S Y with the relative topology inherited from X × Y .

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2.3.12 Proposition With notation as above, we have: (i) X ×S Y has the following universal mapping property. If f1 : Z → X and g1 : Z → Y are continuous maps such that f ◦ f1 (z)  g ◦ g1 (z) for all z ∈ Z, then there is a unique continuous map h : Z → X ×S Y such that f1 = pX ◦ h and g1 = pY ◦ h. (ii) If X, Y , and S are spectral spaces and f and g are spectral maps, then X ×S Y is a proconstructible subset of X ×Y , hence a spectral subspace. If f1 : Z → X and g1 : Z → Y are spectral maps such that f ◦ f1 (z)  f ◦ g1 (z) for all z ∈ Z, then the map h : Z → X ×S Y of item (i) is spectral. Further, (X ×S Y )con is a subspace of Xcon × Ycon . Proof (i) We define h(z) := ( f1 (z), g1 (z)) to be the unique map Z → X × Y such that f1 = pX ◦ h and g1 = pY ◦ h. The assumptions on f1 and g1 imply h(Z) ⊆ X ×S Y , which proves (i). (ii) The map h : Z → X × Y is spectral since f1 and g1 are spectral. By 2.1.5 it suffices to show that X ×S Y ⊆ X × Y is a proconstructible subset, but this is immediate from the fact that f × g : X ×Y → S × S is spectral and S  ⊆ S × S is proconstructible, 1.3.23. For the constructible topology, observe that Xcon × Ycon = (X × Y )con by  2.2.1(vii) and that X ×S Y is a proconstructible subset.

2.4 Finite Coproducts Summary The formation of coproducts is another important construction in topology. Coproducts in Top, also called topological sums or direct sums, have a simple and explicit description, see 2.4.1. Spectral spaces also have coproducts, since we shall see that Spec is a co-complete category, 5.2.9. But the construction of general coproducts is much more involved in Spec than in Top, see Section 10.1. In the present section we consider the special case of finite coproducts of spectral spaces. By [Hoc69, Theorem 7] finite coproducts can be computed in the category Top, hence are topological sums and are simple. We determine the main data of the direct sum of two spectral spaces (i.e., the lattice of quasi-compact open subsets, the lattice of closed and constructible subsets, the Boolean algebra of constructible sets, as well as the specialization order, 2.4.2 and 2.4.3). The results are easily extended to any finite number of summands.

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2.4.1 Reminder on Topological Sums Consider a family (Xi )i ∈I of disjoint nonempty topological spaces, cf. [Bou71b, p. I.15]. We build a new space  i ∈I Xi (or X1 ⊕ X2 if I = {1, 2}), called the topological sum or direct sum.  • The underlying set is the (disjoint) union i ∈I Xi .  • A subset U ⊆ i ∈I Xi is open if and only if U ∩ Xi ∈ O(Xi ) for each i ∈ I.  Each space X j is a subspace of i ; the inclusion maps are denoted by i ∈I X   X . The topology of dj : Xj → i ∈I i i ∈I Xi is the finest topology on the  set i ∈I Xi such that the inclusion maps are all continuous. We give a list of elementary properties of the topological sum.  (i) A set A ⊆ i ∈I Xi is closed if and only if A ∩ Xi ∈ A(Xi ) for each i ∈ I. if C ∩ Xi is clopen in Xi for (ii) A subset C ⊆ i ∈I Xi is clopen if and only  each i ∈ I. In particular, each subset X j ⊆ i ∈I Xi is clopen.  (iii) i ∈I Xi is T0 (resp., T1 , Hausdorff) if and only if all summands are T0 (resp., T1 , Hausdorff).  (iv) A set ∅  C ⊆ i ∈I Xi is irreducible if and only if there is some i ∈ I (necessarily unique) such that C ⊆ Xi and C is irreducible in Xi .  is quasi-compact if and only if there is a finite (v) A set ∅  C ⊆ i ∈I X i subset J ⊆ I with C ⊆ i ∈J Xi and C ∩ Xi is quasi-compact for all i ∈ J.  (vi) i ∈I Xi is Boolean if and only if I is finite and each Xi is Boolean. (vii) The topological sum has the following universal mapping property, ex pressing that i ∈I Xi is the coproduct of the family (Xi )i ∈I in the category Top. Suppose that ( f j : X j → Y ) j ∈I is a family of continuous  maps. Then there is a unique continuous map f : i ∈I Xi → Y with f j = f ◦ d j for all j ∈ I. 2.4.2 Lemma Let X1 and X2 be disjoint topological spaces. Then: (i) (ii) (iii) (iv)

◦

◦

K (X1 ⊕ X2 ) = { A1 ∪ A2 | Ai ∈ K (Xi ), i = 1, 2}; K(X1 ⊕ X2 ) = { A1 ∪ A2 | Ai ∈ K(Xi ), i = 1, 2}; X1 ⊕X2 Xi = {x} ; if x ∈ Xi then {x} (X1 ⊕ X2 )  = X1 ∪ X2 .

Proof (i) is an immediate consequence of 2.4.1(v), and (ii) follows by taking complements in (i). (iii) holds because Xi is closed in X1 ⊕ X2 , and (iv) is a consequence of (iii).  2.4.3 Theorem The topological sum of two spaces X1 and X2 is spectral if and only if both spaces are spectral. If this is the case, then: ◦

◦

◦

(i) K (X1 ⊕ X2 )  K (X1 ) × K (X2 );

66 (ii) (iii) (iv) (v) (vi)

Basic Constructions K(X1 ⊕ X2 )  K(X1 ) × K(X2 ); K(X1 ⊕ X2 )  K(X1 ) × K(X2 ); (X1 ⊕ X2 )con = (X1 )con ⊕ (X2 )con ; (X1 ⊕ X2 )inv = (X1 )inv ⊕ (X2 )inv . The inclusion maps d j : X j → X1 ⊕ X2 ( j = 1, 2) are spectral.

Proof First assume that both X1 and X2 are spectral. Then X1 ⊕ X2 is spectral: axiom S1 holds by 2.4.1(iii) and (v). Axioms S2 and S3 are obvious from 2.4.2. Axiom S4 follows from 2.4.1(iv) and 2.4.2(iii). Now assume X1 ⊕ X2 is spectral. Then the subsets X1 and X2 are closed, hence proconstructible. Thus, they are spectral subspaces, which also yields item (vi). ◦

◦

◦

The maps K (d j ) : K (X1 ⊕ X2 ) → K (X j ) are homomorphisms of bounded ◦

◦

◦

lattices. Hence they yield a homomorphism K (X1 ⊕ X2 ) → K (X1 ) × K (X2 ), which is clearly bijective, hence an isomorphism, proving (i). Items (ii) and (iii) are done in exactly the same way. The universal mapping property of (X1 )con ⊕ (X2 )con , applied to the maps (d1 )con : (X1 )con → (X1 ⊕ X2 )con and (d2 )con : (X2 )con → (X1 ⊕ X2 )con , yields a continuous bijective map (X1 )con ⊕ (X2 )con → (X1 ⊕ X2 )con . The map is a homeomorphism since the spaces are Boolean by 2.4.1(vi), which proves (iv). For item (v), note that (X1 ⊕ X2 )inv and (X1 )inv ⊕ (X2 )inv are spectral and have the same underlying set. The sets of quasi-compact open subsets are the same by 2.4.2, hence the spaces coincide.  2.4.4 Corollary The topological sum of two spectral spaces is their coproduct in the category Spec.  2.4.5 Corollary A spectral space X is a direct sum of two subspaces if and only if there exists a nontrivial clopen set C ⊆ X. Proof Assuming X = X1 ⊕ X2 , it is clear from 2.4.1(ii) that X1 and X2 are both nontrivial clopen subsets. Conversely, if there is a nontrivial clopen subset C then the construction of the topological sum shows that X = C ⊕ (X \ C). 

2.5 Zariski, Real, and Other Spectra Summary Many important applications of spectral spaces are constructions that assign spectral spaces to the elements of some class of algebraic structures. The spectral space associated with a structure C is then called the spectrum of C. Frequently the algebraic structures are the objects of a category C and the

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construction of the spectrum can be extended to morphisms so that one obtains a functor C → Spec, which is typically contravariant. We present several such functorial constructions of spectra. By far the most widely used and most important construction is the Zariski spectrum, which is presented in Part A. It assigns the spectral space of prime ideals to a ring 1 and plays an absolutely fundamental role in modern algebraic geometry. Another construction, which is of great importance in real algebra and real algebraic geometry, is the real spectrum of a ring, see Part B. Both spectra are introduced here, but a more thorough exposition is deferred to Chapter 12 and Chapter 13. It is important to note that for a given class of structures usually there are many different possibilities to construct spectra for its members. For example, both the Zariski spectrum and the real spectrum assign a spectral space to any ring. It always depends on the applications one has in mind, which of the different possible constructions is most useful in a particular context. Our constructions of the Zariski spectrum and the real spectrum both follow the same pattern. In Part C we explain how the same method can be used to produce other spectra for rings or other classes of structures. We conclude with further reading about spectra attached to non-commutative rings in 2.5.14.

A.

The Zariski Spectrum

The set of prime ideals of a ring R is denoted by Spec(R). The set is empty if and only if R is the zero ring; it has just one element if R is a field. We introduce the Zariski topology on Spec(R), which makes it a spectral space, called the prime spectrum or the Zariski spectrum. The Zariski topology is defined in many texts about commutative algebra or algebraic geometry. We refer to [AtMa69, p. 12] and [Bou61, Chapitre II, §4.3], where basic topological properties of the Zariski spectrum are discussed. Hochster showed in [Hoc69] that every spectral space is the prime spectrum of some ring. This will be addressed in detail in Section 12.6. 2.5.1 Theorem Let R be a unital commutative ring. The set of all D(a) = {p ∈ Spec(R) | a  p} with a ∈ R, is a basis for a topology, called the Zariski topology. The set Spec(R) with the Zariski topology is a spectral space and is called the Zariski spectrum, or the prime spectrum, of the ring. The sets D(a) are quasi-compact. 1

By default, rings are commutative and unital throughout the book. Homomorphisms map 1 to 1. The category of rings is denoted by Rings. For general ring-theoretic notation and terminology we refer to Chapter 12, in particular to 12.1.1.

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Remark Our proof differs from the traditional proof in classical algebra textbooks and does not use any algebra except the definition of a prime ideal (and a ring). The method we use can be vastly generalized and serves as a model for the construction of many other spectra, see Part B and Part C below. Further, the strategy of our proof gives functoriality of the construction R → Spec(R) practically for free. 2 Proof We form the power 2R of the Sierpiński space, which is a spectral space, 2.2.3. The map χ : P(R) → 2R , M → χM (the characteristic function of M) is bijective, and we use χ to transfer the spectral topology of 2R to the power set, 2.2.4. Then the sets {M ∈ P(R) | a  M }, a ∈ R, are a subbasis of open sets for P(R). We restrict this topology to the subset Spec(R) ⊆ P(R) and obtain a topology that has the sets D(a), a ∈ R, as, a priori, a subbasis. The map χ restricts to a homeomorphism Spec(R) → χ(Spec(R)) ⊆ 2R . First we show that χ(Spec(R)) is a proconstructible subset of 2R . Suppose that R \ χ(Spec(R)). Then M = f −1 (1) violates one of the defining f = χM ∈ 2con conditions of a prime ideal (i.e., at least one of the following conditions is satisfied): (i) (ii) (iii) (iv) (v)

0  M (i.e., f (0) = 0). 1 ∈ M (i.e., f (1) = 1). M + M  M (i.e., ∃a, b ∈ R : f (a) = 1 = f (b) & f (a + b) = 0). R · M  M (i.e., ∃a, b ∈ R : f (a) = 1 & f (a · b) = 0). (R\M)·(R\M)  R\M, (i.e., ∃a, b ∈ R : f (a) = 0 = f (b) & f (a·b) = 1).

R ) that contains f and is We wish to show that there is some set U ∈ O(2con disjoint from χ(Spec(R)). Suitable choices for such open sets are:

(i) [[ 0 = 0 ]]: If p ⊆ R is an ideal then χp (0) = 1. (ii) [[ 1 = 1 ]]: If p is a proper ideal then χp (1) = 0. (iii) [[ a = 1 ]]∩[[ b = 1 ]]∩[[ a + b = 0 ]]: If p is an ideal then χp (a) = 1 = χp (b) implies χp (a + b) = 1. (iv) [[ a = 1 ]] ∩ [[ a · b = 0 ]]: If p is an ideal and χp (a) = 1 then χp (a · b) = 1 for all b ∈ R. (v) [[ a = 0 ]] ∩ [[ b = 0 ]] ∩ [[ a · b = 1 ]]: If p is a prime ideal then χp (a · b) = 1 implies χp (a) = 1 or χp (b) = 1. It has been shown so far that Spec(R) with the topology inherited from P(R)  is a spectral subspace and the sets D(a) are a subbasis of open sets. Next

2R 2

We emphasize that the method we use is not of our invention. However, it is difficult to give a reference where it has first been applied.

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we show that they are even a basis by proving that D(a) ∩ D(b) = D(a · b). Let p be a prime ideal and a, b ∈ R. Then the following statements are equivalent: • • • •

p ∈ D(a · b), a · b  p, a  p and b  p, p ∈ D(a) ∩ D(b).

Finally we note that each set D(a) is quasi-compact. This follows from χ(D(a)) = χ(Spec(R)) ∩ [[ a = 0 ]], for χ(Spec(R)) ⊆ 2R is proconstructible, R and [[ a = 0 ]] ⊆ 2 is constructible.  2.5.2 Main Topological Features of the Zariski Spectrum Let R be a ring. In our discussion of Zariski spectra we shall frequently identify the power set P(R) with 2R via χ and the Zariski spectrum with its image χ(Spec(R)) ⊆ 2R . The sets D(a) are a basis of the Zariski topology and are quasi-compact. It ◦

follows that K (Spec(R)) is the set of finite unions D(a1 ) ∪ · · · ∪ D(ar ), r ∈ N, ◦

ai ∈ R, cf. 1.1.8. The bottom element of K (Spec(R)) is D(0) = ∅, the top element is D(1) = Spec(R). The collection of sets V(a) = {p ∈ Spec(R) | a ∈ p} = Spec(R)\D(a), a ∈ R, is a basis of closed sets for the Zariski topology. They are constructible (since their complements are quasi-compact open), hence the elements of K(Spec(R)) are finite intersections V(a1 ) ∩ · · · ∩ V(ar ), r ∈ N, ai ∈ R. Every closed set  C ⊆ Spec(R) is an intersection C = a ∈S V(a) = {p ∈ Spec(R) | S ⊆ p} with a suitable set S ⊆ R. Alternatively we can write  C = {p ∈ Spec(R) | (S) ⊆ p} = {p ∈ Spec(R) | (S) ⊆ p},  where (S) ⊆ R is the ideal generated by S, and (S) is its radical. The Boolean algebra K(Spec(R)) is generated by the sets D(a) and V(b), a, b ∈ R. Hence, every constructible set is a finite union of sets of the form D(a) ∩ V(b1 ) ∩ · · · ∩ V(bn ). For specialization, consider two prime ideals p,q. Then the following conditions are equivalent: • q ∈ {p}, • ∀a ∈ R : q ∈ D(a) ⇒ p ∈ D(a), • R \ q ⊆ R \ p, i.e., p ⊆ q. Thus, {p} = {q ∈ Spec(R) | p ⊆ q}.

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So far we have described the basic data of the prime spectrum of one given ring. Now we include homomorphisms in our considerations and show that the construction of the Zariski spectrum is a functor. 2.5.3 Theorem Suppose that ϕ : R → S is a homomorphism of rings. Then the spectral map 2ϕ : 2S → 2R , 2.2.3(f), restricts to a spectral map Spec(ϕ) : Spec(S) → Spec(R) between the Zariski spectra. Thus, the construction of the Zariski spectrum is a functor Rings → Spec. Proof It suffices to show that the spectral map 2ϕ restricts to a map Spec(S) → Spec(R). But 2ϕ (q) = ϕ−1 (q), and it is well known (or easy to prove) that inverse images of prime ideals under homomorphisms are prime ideals.  2.5.4 The Zariski Spectrum of the Integers Let P be the set of prime numbers in N. We claim that the spectral space Spec(Z) is homeomorphic to the space P∞ (i.e., to S∞ where S = P, cf. Section 1.6.D). It is clear that, as a set, Spec(Z) = {(p) | p ∈ P} ∪ {(0)}. First note that Spec(Z)con is (homeomorphic to) the one-point compactification of the discrete space P, where the prime ideal (0) is the point at infinity. Each singleton {(p)} = V(p) with p ∈ P is constructible, hence clopen in Spec(Z)con . The only remaining point is (0), which, therefore, must be the point at infinity. These observations show that the map Spec(Z)con → P∗ , with (p) → p for p ∈ P and (0) → ∞ is a homeomorphism of Boolean spaces. Every nontrivial specialization in Spec(Z) is of the form (0)  (p), 2.5.2. This corresponds exactly to the specializations in P∞ . The claim now follows from 1.5.11. 2.5.5 A Ring whose Zariski Spectrum is the Sierpiński Space Consider the localization Z(p) = { nk | p  k} of the ring Z at a prime ideal (p) with p ∈ P. The spectrum of Z(p) has only two points, namely (0) and p · Z(p) . The specialization is given by (0)  p · Z(p) , which shows that Spec(Z(p) )  2. 2.5.6 Rings with Boolean Zariski Spectrum A ring A is von Neumann regular if, for each a ∈ A, there is some b ∈ A such that a2 · b = a. (In the literature these rings are also called absolutely flat, [Gil72, p. 111], [Bou61, p. 64].) We claim that the Zariski spectrum of a von Neumann regular ring is Boolean (i.e., there are no proper specializations in Spec(A), that is, every prime ideal is maximal). For the proof, pick a prime ideal p ⊆ A and any element a  p. There is an element b ∈ A with a · (a · b − 1) = 0. Then a · b − 1 ∈ p, hence a is a unit modulo p, which means that p is maximal. A ring is Boolean if every element is idempotent, [Kop89, p. 19]. Recall that Boolean algebras and Boolean rings are the same structures. Let A be

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a Boolean algebra. Then A is a Boolean ring, where symmetric difference is addition, a  a  = (a∧¬a )∨(a  ∧¬a), and the meet operation is multiplication, a · a  = a ∧ a . Conversely, if A is a Boolean ring then A with the join operation a ∨ a  = a + a  + a · a  and the meet operation a ∧ a  = a · a  is a Boolean algebra. Boolean rings are clearly von Neumann regular, hence have a Boolean Zariski spectrum. The converse is also true. If X is a Boolean space then there is a Boolean ring whose Zariski spectrum is (homeomorphic to) X. To see this let A be the ring of continuous functions X → F2 , where F2 is the discretely topologized field with two elements. The ring is clearly Boolean. For each x ∈ X the set px = {a ∈ A | a(x) = 0} is a prime ideal. One checks that the map X → Spec(A) : x → px is a homeomorphism. We will discuss these and other examples in a more algebraic context in 12.1.13.

B.

The Real Spectrum

The real spectrum is another construction that associates a spectral space with a ring. It was introduced by M. Coste and M.-F. Roy; see [CoRo82] for an early account. Here we describe only the construction and postpone a more extensive treatment to Chapter 13. 2.5.7 The Real Spectrum of a Ring We continue with a ring R. A subset α ⊆ R is a prime cone if it satisfies the following conditions: (i) ∀a, b ∈ α : a + b ∈ α. (ii) ∀a, b ∈ α : a · b ∈ α. (iii) ∀a ∈ R : a ∈ α or − a ∈ α. (iv) −1  α. (v) The set supp(α) = α ∩ −α = {a ∈ R | a ∈ α and − a ∈ α} is a prime ideal, which is called the support of P. Trivial examples of prime cones are the positive cones of total orders on integral domains. In fact, if α is any prime cone, the canonical homomorphism πα : R → R/supp(α) maps α onto the positive cone of a total order of the factor domain, and α = πα−1 (πα (α)). Thus, the prime cones of a ring R are the inverse images of positive cones of total orders under homomorphisms from R to integral domains. It is a consequence of conditions (i), (ii), and (iii) that the support is an ideal of R. By condition (iv) it is a proper ideal. Thus, if (i)–(iv) hold, then condition

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(v) is equivalent to the statement (v)

∀a, b ∈ R : (a · b ∈ α and − a · b ∈ α) ⇒ ((a ∈ α and − a ∈ α) or (b ∈ α and − b ∈ α)).

The set of prime cones of R is denoted by Sper(R) and is called the real spectrum of R. For the moment this is only a set. But we shall equip the real spectrum with a topology that makes it into a spectral space. The construction of the real spectrum is functorial, similar to the Zariski spectrum. 2.5.8 Theorem Let R be a ring. The sets HR>0 (a1, . . . , ar ) = {α ∈ Sper(R) | −a1, . . . , −ar  α}, r ∈ N, a1, . . . , ar ∈ R, are the basis for a topology on Sper(R), called the ◦

Harrison topology. The real spectrum is a spectral space and K (Sper(R)) is the set of finite unions of sets of the form HR>0 (a1, . . . , ar ). Proof We proceed as in the proof of 2.5.1. Again, let χ : P(R) → 2R be the map that sends subsets of R to their characteristic functions. We restrict χ to a bijective map Sper(R) → χ(Sper(R)). First we show that χ(Sper(R)) ⊆ 2R is a proconstructible subset. For the proof, pick an element f ∈ 2R \ χ(Sper(R)). The fact that f −1 (1) violates one of the defining conditions of a prime cone implies that at least one of the following conditions is satisfied: (i) (ii) (iii) (iv) (v)

∃a, b ∈ R : f (a) = 1 = f (b) and f (a + b) = 0. ∃a, b ∈ R : f (a) = 1 = f (b) and f (a · b) = 0. ∃a ∈ R : f (a) = 0 = f (−a). f (−1) = 1. ∃a, b ∈ R : f (a · b) = 1 = f (−a · b) and f (a)  f (−a) and f (b)  f (−b).

R that contains f and is disjoint We show that there is an open subset U ⊆ 2con R are suitable: from χ(Sper(R)). The following open subsets of 2con

(i) (ii) (iii) (iv) (v)

[[ a = 1 ]] ∩ [[ b = 1 ]] ∩ [[ a + b = 0 ]].  [[ a = 1 ]] ∩ [[ b = 1 ]] ∩ [[ a · b = 0 ]]. [[ a = 0 ]] ∩ [[ − a = 0 ]].  = 1 ]]. [[ −1 The intersection of the three open sets  [[ a · b = 1 ]] ∩ [[ −a · b = 1 ]], ([[ a = 1 ]] ∩ [[ − a = 0 ]]) ∪ ([[ a = 0 ]] ∩ [[ − a = 1 ]]),  = 0 ]]) ∪ ([[  = 1 ]]). ([[ b = 1 ]] ∩ [[ −b b = 0 ]] ∩ [[ −b

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The real spectrum is now a spectral space and the sets  ar = 0 ]], χ(Sper(R)) ∩ [[ −  a1 = 0 ]] ∩ · · · ∩ [[ − r ∈ N, a1, . . . , ar ∈ R are a basis of quasi-compact open sets. The inverse image of such a set under χ is HR>0 (a1, . . . , ar ) = {α ∈ Sper(R) | −a1, . . . , −ar  α}, and the proof is finished.



2.5.9 Basic Quasi-Compact Open Sets – Real Spectrum vs. Zariski Spectrum The Harrison basis of quasi-compact open sets in the real spectrum cannot be simplified as for the Zariski spectrum (i.e., the sets HR>0 (a) with a ∈ R are usually not a basis of open sets). This will be explained in Chapter 13. The sets Sper(R) \ HR>0 (a1, . . . , ar ) = {α ∈ Sper(R) | −a1 ∈ α or . . . or − ar ∈ α} are a basis of closed and constructible sets. 2.5.10 Specialization in the Real Spectrum For specialization, consider prime cones α and β. Suppose that β ∈ {α}. Then a  β implies β ∈ HR>0 (−a), hence α ∈ HR>0 (−a), hence a  α. Thus, we see that α ⊆ β. Conversely, suppose that α ⊆ β. Then β ∈ HR>0 (a1, . . . , ar ) implies −a1, . . . , −ar  β, hence −a1, . . . , −ar  α, hence α ∈ HR>0 (a1, . . . , ar ). We conclude that β ∈ {α}. Finally we record the basic functorial properties of the real spectrum. 2.5.11 Theorem Let R and S be rings and ϕ : R → S a homomorphism. (i) The spectral map 2ϕ : 2S → 2R restricts to a spectral map Sper(ϕ) : Sper(S) → Sper(R). The construction of the real spectrum is a functor Rings → Spec. (ii) The map supp : Sper(R) → Spec(R) : P → supp(P) is spectral. The support is a natural transformation Sper → Spec of functors. Proof (i) It suffices to show that 2ϕ restricts to a map between the real spectra. Then it is automatically a spectral map, and the real spectrum is a functor. So, one needs to show that ϕ−1 (β) is a prime cone of R if β is a prime cone of S. But this is evident from the definition of prime cones. (ii) To show that the map supp is spectral, note that supp−1 (D(a)) = H >0 (a) ∪ H >0 (−a) is quasi-compact open. The equality supp ◦ Sper(ϕ) = Spec(ϕ)◦supp is obvious and expresses that supp is a natural transformation. 

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C.

Further Examples

The Zariski spectrum and the real spectrum are functorial constructions that associate spectral spaces and spectral maps with rings and ring homomorphisms. Both constructions follow the same strategy. Exactly the same method can also be used in other situations to produce a functor from some category to the category of spectral spaces. In 2.5.12 we exhibit the general method and then illustrate it in 2.5.13 with several constructions. 2.5.12 A Construction Method for Spectrum Functors Let C be a category whose objects are sets equipped with some structure and whose morphisms are, in particular, maps between the underlying sets. Suppose that two morphisms coincide if they agree on the underlying sets. Many categories of algebraic structures are of this type. If A is an object of C, we write set(A) for the underlying set. Every C-morphism ϕ : A → B induces a spectral map 2ϕ : 2set(B) → 2set(A) . Suppose there is a construction that associates a proconstructible subset s(A) ⊆ 2set(A) with each object of C, so that 2ϕ (s(B)) ⊆ s(A). Then the restriction s(ϕ) : s(B) → s(A) is a spectral map, and s : C → Spec is a contravariant functor. A different method for the construction of spectrum functors is presented in 7.2.13. In the case of complicated algebraic structures the method of 7.2.13, if it is applicable, may be much simpler than the one exhibited here. 2.5.13 Examples The Zariski spectrum and the real spectrum are both functors constructed exactly as explained in 2.5.12. Now we show a few other applications of the general principles. They suggest that 2.5.12 has a wide range of applications. Right now we do not analyze any of the functors we define. The sole purpose is to illustrate the construction method. (a) The spectrum of subsets of a set The most trivial construction is the functor P: Sets → Sets, X → P(X). We identify P(X) with 2X via characteristic functions, cf. 2.2.4. The construction is a functor by 2.2.3(f). We view 2X as the spectrum of subsets of X. (b) The spectrum of ideals in a ring Given a ring R, the set of ideals is denoted by I(R). Using the characteristic functions map χ : P(R) → 2R , one checks that χ(I(R)) ⊆ 2R is a proconstructible subset. A homomorphism ϕ : R → S induces the map I(S) → I(R), J → ϕ−1 (J). Thus, 2ϕ ( χJ ) = χϕ −1 (J) ∈ χ(I(R)) (i.e., the map 2ϕ restricts to a map χ(I(S)) → χ(I(R))), and we have defined the ideal spectrum functor.

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(c) The spectrum of (normal) subgroups of a group The set of subgroups and the set of normal subgroups of a group G are denoted by U(G) and by N (G). As above, the characteristic functions map χ : P(G) → 2G sends U(G) and N (G) both to proconstructible subsets of 2G . The inverse image of a subgroup is a subgroup, and the inverse image of a normal subgroup is a normal subgroup. Thus we obtain two functors, the subgroup spectrum and the normal subgroup spectrum. (d) The up-sets of a poset We consider the category PoSets of posets, see A.1(vi). Recall that its morphisms are the monotonic maps. The set of up-sets of X is denoted by Up(X). Again we use the characteristic functions map χ : P(X) → 2X . If f : X → Y is a monotonic map then 2 f : 2Y → 2X restricts to a well-defined map χ(Up(Y )) → χ(Up(X)). By 2.5.12, we get a spectrum functor if χ(Up(X)) is a proconstructible subset of 2X . To prove this, pick some g ∈ 2X that does not belong to χ(Up(X)) (i.e., g −1 (1) ⊆ X is not an up-set). There are elements x ≤ x  in X such that g(x) = 1 and g(x ) = 0. X is open and contains g, but is disjoint The subset [[ x = 1 ]] ∩ [[ x  = 0 ]] ⊆ 2con from χ(Up(X)). This proves the claim. (e) The equivalence relations on a set Let S be a set and consider the set E(S) of equivalence relations. The subset E(S) ⊆ P(S × S) is a complete lattice and is not distributive (except for trivial cases). The top element is S × S, the bottom element is the diagonal ΔS , infima are intersections, and suprema are the infima of the sets of upper bounds. Alternatively, the supremum of a family (Ei )i ∈I of  equivalence relations is the transitive hull of i ∈I Ei . The lattice E(S) is also called the partition lattice of S, since, via equivalence classes, the equivalence relations correspond bijectively to the partitions into nonempty subsets. The partition lattice plays an important role in the representation theory of lattices, cf. [Grä98, IV, 4]. We shall return to the partition lattice in 14.3.6. Via the characteristic functions map χ : P(S×S) → 2S×S we consider subsets of S × S as elements of the spectral space 2S×S . It is claimed that E(S) ⊆ 2S×S is a proconstructible subset, hence is a spectral space. For the proof, let R ∈ 2S×S be a relation, but not an equivalence relation. Then, either • there is some a ∈ S with R(a, a) = 0, or • there are a, b ∈ S with R(a, b) = 1 and R(a, b) = 0, or • there are a, b, c ∈ S with R(a, b) = 1, R(b, c) = 1, and R(a, c) = 0. As in the proofs of 2.5.1 and 2.5.8, in each case there is a constructible set C ⊆ 2S×S with R ∈ C and C ∩ E(S) = ∅ . This proves the claim, and E(S) is a spectral space.

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Specialization in the space E(S) is the same as containment of the equivalence relations or coarsening of the corresponding partitions. The construction of E(S) is functorial. For each map ϕ : S → T of sets there is the map ϕ × ϕ : S × S → T × T, which induces the map 2ϕ×ϕ : 2T ×T → 2S×S , N → (ϕ × ϕ)−1 (N). As 2ϕ×ϕ maps equivalence relations to equivalence relations, it follows that 2ϕ×ϕ restricts to a spectral map E(ϕ) : E(T) → E(S). (f) The congruences on a bounded distributive lattice Let L be a bounded distributive lattice. An equivalence relation E on L is a congruence 3 if E ⊆ L × L is a bounded sublattice. The set of congruences is denoted by Cong(L) and is a subset of E(L). It is a complete lattice, but not a sublattice of E(L). The bottom element is the diagonal again, and the top element is L × L. Infima are intersections, and suprema are derived from these (as for equivalence relations). The congruences of a lattice L correspond bijectively to the isomorphism classes of surjective lattice homomorphisms onto bounded distributive lattices M. 4 Namely, for a homomorphism ϕ : L → M of bounded distributive lattices, the relation Rϕ = (ϕ × ϕ)−1 (Δ M ) = {(a, b) ∈ L | ϕ(a) = ϕ(b)} is a congruence. The same congruence is defined by the co-restriction ϕ  : L → ϕ(L). Conversely, let E be a congruence on L and L/E the set of equivalence classes. The class of a ∈ L is denoted by [a]E , and πE : L → L/E is the canonical map a → [a]E . One defines • • • •

⊥ L/E = [⊥]E ,  L/E = []E , [a]E ∨ [b]E = [a ∨ b]E , [a]E ∧ [b]E = [a ∧ b]E .

The operations are well-defined and turn L/E into a bounded distributive lattice such that the canonical map πE is a surjective homomorphism. Note that RπE = E. (For these constructions and their properties we refer to [Grä98, p. 25 ff] or, more generally, [Coh81, p. 57 ff].) With the same methods as above one shows that Cong(L) is a spectral subspace of E(L) and that the construction is functorial. Let E and F be two congruences. Then E  F if and only if E ⊆ F, if and only if there is a homomorphism h : L/E → L/F with πF = h ◦ πE . 3

4

The notion of congruences exists in far greater generality, namely for arbitrary algebras, [Coh81, p. 57]. We restrict ourselves to congruences on bounded distributive lattices since more general situations do not occur in this book. In any category C, given a fixed object A and morphisms f : A → B and g : A → C, a morphism over A from f to g is a morphism h : B → C with g = h ◦ f . If h is an isomorphism then f and g are said to be isomorphic over A.

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2.5.14 Further Reading: Spectra of Non-commutative Rings In the literature one can find various spaces attached to a non-commutative ring R, which resemble in one way or another the Zariski spectrum of a commutative ring. In each case these spaces share some features with the Zariski spectrum, but many of them lack significant characteristics. Some are too small to carry useful information about R, or they are too complicated to be exploitable, or they have few quasi-compact subsets (in particular they are not spectral); yet other spectra are not functorial in terms of ring homomorphisms. Here are some examples. (i) The construction of the Zariski spectrum given in paragraph 2.5A. also produces a spectral space when applied to a non-commutative unital ring R. The points are proper two-sided ideals I of R satisfying a · b ∈ I ⇒ a ∈ I or b ∈ I. In the non-commutative context, these ideals are called completely prime. The proof of 2.5.1 gives a spectral space again. (ii) Prime ideals in the non-commutative context are two-sided ideals satisfying a · R · b ⊆ I ⇒ a ∈ I or b ∈ I; in the commutative case this implies (completely) prime. One can define a topology on the set of prime ideals by declaring the sets D(a) = {p | a  p} as open; however, this space is not spectral in general. There are certain enlargements of the space, which are called extended prime spectra and are indeed spectral spaces; see [Bel97] and [KlTr07]. The spaces in (i) and (ii) frequently do not encode much information about the ring R. For example, in matrix rings over fields, (0) is the unique proper two-sided ideal. (iii) The construction that is conceptually closest to the Zariski spectrum construction is Cohn’s matrix spectrum, introduced in [Coh72]; also see [Coh95, Chapter 4]. Points are homomorphisms into skew fields modulo an appropriate equivalence relation, similarly to Zariski spectra, see the summary of Chapter 12. Cohn’s spectrum is notoriously complicated, but it is spectral by [Coh79]. A framework explaining the strategy of the construction for any first-order structure may be found in Section 14.3. (iv) In [Bra74], the space of primitive ideals of an approximately finitedimensional C ∗ -algebra is studied and its topological type is characterized; also see [BrEl78]. (v) We conclude the list by pointing to the Ziegler spectrum of a noncommutative ring, which is briefly explained at the end of 3.7.2. This construction is not functorial in ring homomorphisms and indeed is merely seen as an important invariant of the module category of R, rather than a tool to understand the ring R.

3 Stone Duality

In Section 2.5 we presented a general method for the construction of spectrum functors. The same method is used now to associate a spectral space with every bounded distributive lattice, Section 3.1. The construction is a contravariant functor again. This functor has an exceptional property distinguishing it sharply from the spectrum functors presented so far – the construction of the spectrum of a bounded distributive lattice can be reversed. In fact, the reverse construction ◦ is the functor K , which appeared already in 1.2.3. It is clear from Chapter 1 ◦

that both K (X) and K(X) are essential parts of the spectral space X. Now we show that the lattices even determine the space. This correspondence between bounded distributive lattices and spectral spaces is the content of Stone Duality Theory. Historically, duality theory started with M. H. Stone’s results about the correspondence between Boolean algebras and Boolean spaces, [Sto36], see also [Kop89, p. 101], which he extended to distributive lattices in [Sto37b]. (Stone mentions, [Sto37b, p. 1,2], that Birkhoff (in [Bir33]) and MacNeille (in [Mac36]) also found this result in some form.) Later the duality was extended to relate various other classes of partially ordered structures to suitable classes of topological spaces, see for example [HoKe72]. Nowadays the Stone duality of bounded distributive lattices and spectral spaces is frequently cast in terms of Priestley spaces (see, e.g., [Grä98, Chapter B] (authored by B. Davey and H. Priestley), [DaPr02, p. 256 ff], and note [Fle00]). We give a topological presentation of the duality, that is, in terms of spectral spaces. The spectrum of a bounded distributive lattice L, defined in 3.1.4, is a spectral space and is denoted by Spec(L). We show that: ◦

• any lattice L is canonically isomorphic to K (Spec(L)) (we say that L is represented via quasi-compact open sets); 78

3.1 The Spectrum of a Bounded Distributive Lattice

79

◦

• any spectral space X is canonically homeomorphic to Spec(K (X)) (we say that X is represented via quasi-compact open sets). ◦

The lattices K (X) and K(X) determine each other. Therefore the representations can also be expressed in terms of closed and constructible sets. Altogether, there are four representation theorems, which are presented in Section 3.2. Stone duality provides us with new tools for the study of spectral spaces. Every statement about a spectral space can be translated into a statement about its corresponding lattice, and vice versa. Difficult properties of spectral spaces may become tractable if expressed in terms of bounded distributive lattices. The spectrum of a lattice L is defined via homomorphisms into the Boolean algebra 2, see 3.1.4. Every homomorphism ϕ : L → 2 is completely determined by the prime ideal ϕ−1 (0) and also by the prime filter ϕ−1 (1). Thus, the spectrum can also be considered as a space of prime ideals or as a space of prime filters. These different descriptions of the spectrum appear in applications. Therefore, we give a detailed explanation of the spectrum both in terms of prime ideals and in terms of prime filters, Section 3.3. The spectral space Spec(L) has a patch space and an inverse space. The corresponding bounded distributive lattices are analyzed in Section 3.4 and Section 3.5. The patch space corresponds to a Boolean algebra, which contains the lattice L and is called the Boolean envelope of L, 3.4.2. Independently from spectral spaces, this is an important lattice-theoretic construction, which has been known for a long time, cf. [Mac39] and [Grä98, p. 116]. Finally, we use the duality theory to study the spectrum of a bounded totally ordered set in Section 3.6.

3.1 The Spectrum of a Bounded Distributive Lattice Summary We introduce the spectrum of a bounded distributive lattice. Starting with a lattice L we use the method of Section 2.5 to construct a spectral subspace of 2 L , cf. 3.1.1. The construction is a contravariant functor BDLat → Spec, see 3.1.2. But then we equip the subspace of 2 L with the inverse topology, call it the spectrum of L and denote it by Spec(L), 3.1.3, 3.1.4. The basic data of the spectral space Spec(L) are described in 3.1.6. 3.1.1 Theorem Let L be a bounded distributive lattice. Then HomBDLat (L, 2) ⊆ 2 L is a spectral subspace.

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Proof We show that HomBDLat (L, 2) is patch-closed in 2 L . So, suppose that ϕ ∈ 2 L \ HomBDLat (L, 2). Then ϕ violates one of the defining conditions of a bounded lattice homomorphism, hence it satisfies one of the following conditions: (i) (ii) (iii) (iv)

ϕ(⊥) = 1, or ϕ() = 0, or ∃a, b ∈ L : ϕ(a ∧ b)  ϕ(a) ∧ ϕ(b), or ∃a, b ∈ L : ϕ(a ∨ b)  ϕ(a) ∨ ϕ(b).

L of ϕ that is We show that in each case there is an open neighborhood U ⊆ 2con disjoint from HomBDLat (L, 2). The following sets are suitable choices for U in the different cases:

= 1 ]]. (i) [[ ⊥ = 0 ]]. (ii) [[  (iii) The union of [[ a ∧ b = 0 ]] ∩ [[ a = 1 ]] ∩ [[ b = 1 ]],  [[ a ∧ b = 1 ]] ∩ ([[ a = 0 ]] ∪ [[ b = 0 ]]). (iv) The union of [[ a ∧ b = 1 ]] ∩ [[ a = 0 ]] ∩ [[ b = 0 ]],  [[ a ∧ b = 0 ]] ∩ ([[ a = 1 ]] ∪ [[ b = 1 ]]). The proof is finished by 2.1.3.



3.1.2 Corollary Let ϕ : L → M be a homomorphism between bounded distributive lattices. (i) The map ϕ∗ : HomBDLat (M, 2) → HomBDLat (L, 2), ψ → ψ ◦ ϕ is spectral. (ii) The assignment L → HomBDLat (L, 2) extends to a contravariant functor BDLat → Spec.  = k ]]. a = k ]]) = [[ ϕ(a) (iii) (ϕ∗ )−1 ([[ Proof (i), (ii) The map ϕ∗ : HomBDLat (M, 2) → HomBDLat (L, 2) is the restriction of the spectral map 2ϕ : 2 M → 2 L , hence the assertion is clear, cf. 2.5.12. (iii) follows immediately from the definitions.  3.1.3 Spectral Topologies on HomBDLat (L, 2) Let L be a bounded distributive lattice. As HomBDLat (L, 2) is a spectral space, 3.1.1, its underlying set carries the spectral topology, the patch topology, and the inverse topology. Thus, there are three spectral spaces that are associated with L. For each of them the

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81

construction is functorial. The underlying sets are the same, but the topologies are different. We want to single out one of the three spaces as the spectrum of L. The spectral topology and the inverse topology carry the same amount of information, whereas the patch topology contains less information. Therefore it is natural to choose the set HomBDLat (L, 2) with the spectral topology or with the inverse topology. Our choice is fixed in the next definition. The reason for the choice is explained in 3.1.7. 3.1.4 Definition Let L be a bounded distributive lattice. The spectrum of L L . The is the set HomBDLat (L, 2) with the subspace topology inherited from 2inv spectrum is denoted by Spec(L). If ϕ : L → M is a homomorphism of bounded distributive lattices, the spectral map ϕ∗ , cf. 3.1.2(i), is denoted by Spec(ϕ). The construction is a functor Spec : BDLat → Spec. 3.1.5 Description of the Spectrum of a Bounded Distributive Lattice The next step is to collect the basic data of the spectrum of a bounded distributive lattice. To start with, this is just a reformulation of our general results about the product topology and spectral subspaces of products. But the description is simplified in 3.1.6 below. ◦

(i) K (Spec(L)) is the lattice of finite unions of sets [[ S = 1 ]] ∩ Spec(L), where S ⊆ L is finite. = 0 ]] ∩ Spec(L), (ii) K(Spec(L)) is the lattice of finite unions of sets [[ T where T ⊆ L is finite. = 0 ]] ∩ (iii) K(Spec(L)) is the lattice generated by the sets [[ S = 1 ]] ∩ [[ T Spec(L), where S,T ⊆ L are finite. (iv) If ϕ, ψ ∈ Spec(L) then ϕ  ψ if and only if ϕ ≥ ψ for the componentwise partial order derived from the natural total order 0 < 1 in 2. To simplify the notation, from now on we just write [[ S = 1 ]] instead of [[ S = 1 ]] ∩ Spec(L), and so on (if there is no risk of confusion). We show that every quasi-compact open set, as well as every closed and constructible set, is even a principal set (i.e., can be written as [[ a = 1 ]] or [[ b = 0 ]], respectively). 3.1.6 Theorem Suppose that L is a bounded distributive lattice. For a, b ∈ L we have: (i) [[ a = 1 ]] ∩ [[ b = 1 ]] = [[ a ∧ b = 1 ]]. (ii) [[ a = 1 ]] ∪ [[ b = 1 ]] = [[ a ∨ b = 1 ]]. ◦

(iii) K (Spec(L)) = {[[ a = 1 ]] | a ∈ L}.

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(iv) (v) (vi) (vii)

[[ a = 0 ]] ∩ [[ b = 0 ]] = [[ a ∨ b = 0 ]].  [[ a = 0 ]] ∪ [[ b = 0 ]] = [[ a ∧ b = 0 ]]. K(Spec(L)) = {[[ a = 0 ]] | a ∈ L}. K(Spec(L)) consists of finite unions of sets [[ a = 1 ]] ∩ [[ b = 0 ]].

Proof (iii) is immediate from (i) and (ii), and (vi) follows from (iv) and (v). Moreover, (vii) is a simple consequence of (iii) and (vi). For the proof of (i), observe that, for ϕ ∈ Spec(L), the following statements are equivalent: • • • •

ϕ ∈ [[ a = 1 ]] ∩ [[ b = 1 ]], ϕ(a) = 1 = ϕ(b), ϕ(a ∧ b) = 1, ϕ ∈ [[ a ∧ b = 1 ]]. 

The other assertions are proved in exactly the same way.

3.1.7 The spectral topology of Spec(L) By 3.1.6 the lattices of quasicompact open sets and of closed and constructible sets in Spec(L) can now be presented in the form ◦

a = 1 ]] | a ∈ L}, K (Spec(L)) = {[[ K(Spec(L)) = {[[ a = 0 ]] | a ∈ L}. ◦

Also, 3.1.6 shows that the set-theoretic operations in K (Spec(L)) correspond exactly to the lattice operations in L. This is the reason for our choice of the spectral topology on Spec(L) in 3.1.4. The other choice would have given us ◦

the lattice K (Spec(L)inv ) = K(Spec(L)). The set-theoretic operations would then correspond to the inverse lattice operations.

3.2 Stone Duality ◦

Summary In Sections 1.1 and 3.1 we presented the functor K : Spec → BDLat, which is essentially a part of the definition of spectral spaces, and the spectrum functor Spec : BDLat → Spec, respectively. Both functors are contravariant. They are not quite inverse to each other, but we show that both their compositions are naturally isomorphic (see, e.g., [HeSt79, 13.1] or [AHS90, 6.5]) to the identity functors of the categories. Thus, the categories are antiequivalent to each other. The Stone Representation Theorems give an explicit description of the natural isomorphisms, 3.2.5, 3.2.6, 3.2.8, and 3.2.9.

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We start with an important and useful fact about lattices. 3.2.1 Theorem ([Grä98, p. 84, Theorem 15]) Suppose that L is a bounded distributive lattice, let i be an ideal and f a filter such that i ∩ f = ∅. Then there is a prime ideal j such that i ⊆ j and j ∩ f = ∅. Proof Consider the set I of all ideals j that contain i and are disjoint from f. By Zorn’s Lemma there is a maximal element j ∈ I, which is a prime ideal, as we show. Assume that this is not the case. There are a, b ∈ L \ j with a ∧ b ∈ j. The sets ja = {x ∨ (u ∧ a) | x ∈ j, u ∈ L} and jb = {x ∨ (u ∧ b) | x ∈ j, u ∈ L} are ideals containing j properly, hence do not belong to I. There are elements z = x ∨ (u ∧ a) ∈ ja ∩ f and t = y ∨ (v ∧ b) ∈ jb ∩ f. Then z ∧ t ∈ f and z ∧ t = (x ∧ y) ∨ (x ∧ v ∧ b) ∨ (u ∧ a ∧ y) ∨ (a ∧ b ∧ u ∧ v) ∈ j, which is a contradiction.  3.2.2 Corollary Let L be a bounded distributive lattice. Then every ideal is an intersection of prime ideals. Proof Let i ⊆ L be an ideal, and consider P, the set of prime ideals containing i. For each a ∈ L \ i the principal filter f(a) generated by a is disjoint from i. Hence there is some pa ∈ P with a  pa , 3.2.1. The claim follows from   i = a ∈L\i pa . 3.2.3 Corollary Let L be a bounded distributive lattice, and pick elements a, b ∈ L such that a  b. Then there is an element λ ∈ HomBDLat (L, 2) such that λ(a) = 1 and λ(b) = 0. Proof The hypothesis implies that a  i(b) (the principal ideal generated by b). By 3.2.2 there is a prime ideal p with a  p and i(b) ⊆ p. The map λ : L → 2  with λ−1 (0) = p is a homomorphism and has the desired properties. 3.2.4 Corollary Let M be a bounded distributive lattice and L ⊆ M a bounded sublattice. Then every homomorphism λ ∈ HomBDLat (L, 2) can be extended to a homomorphism μ ∈ HomBDLat (M, 2) (i.e., the map HomBDLat (M, 2) → HomBDLat (L, 2), μ → μ| L is surjective). Proof The ideal i = i(λ−1 (0)) ⊆ M (generated by λ−1 (0) in M) and the filter f = f(λ−1 (1)) ⊆ M (generated by λ−1 (1) in M) are disjoint. By 3.2.1 there is a prime ideal j ⊆ M that contains i and is disjoint from f. Then the  homomorphism μ : M → 2 given by the condition μ−1 (0) = j extends λ. We are now ready to state and prove the Stone Duality Theorems. There are four different versions:

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• Lattices are represented via spectral spaces, and this can be done in two different ways – using either quasi-compact open sets or closed and constructible sets. • Spectral spaces are represented via lattices, and again this can be done using either quasi-compact open sets or closed and constructible sets. There is a certain amount of duplication. But all the different versions do occur in applications. With our detailed exposition we wish to do justice to the great importance of the Stone Representation Theorems. They contribute in a fundamental way to our intuitive understanding of spectral spaces. And they provide extremely useful technical tools. 3.2.5 Stone Representation – Lattices via Quasi-Compact Open Sets Let L and M be bounded distributive lattices and let ϕ ∈ HomBDLat (L, M). (i) The map ◦

a = 1 ]] ε L : L → K (Spec(L)), a → [[ is an isomorphism of lattices. (ii) The diagram L

εL

◦

/ K (Spec(L)) ◦

ϕ

 M

εM

◦



K (Spec(ϕ))

/ K (Spec(M))

of lattice homomorphisms is commutative. Proof (i) From 3.1.6 we know that ε L preserves finite infima and suprema. = 1 ]] = ∅ is the It also preserves the bottom and top elements, since [[ ⊥ ◦ = 1 ]] = Spec L is the top element of bottom element of K (Spec(L)) and [[  ◦

K (Spec(L)). Thus, ε L is a homomorphism of bounded lattices. By 3.1.6(iii), ε L is surjective. For injectivity, pick a, b ∈ L, a  b. By 3.2.3, there is some λ ∈ Spec(L) with λ(a)  λ(b), hence [[ a = 1 ]]  [[ b = 1 ]]. (ii) For each a ∈ L, ◦

K (Spec(ϕ)) ◦ ε L (a) ◦

a = 1 ]]) a = 1 ]]) = Spec(ϕ)−1 ([[ K (Spec(ϕ))([[ 3.1.2(iii)  = 1 ]] = ε M (ϕ(a)), = [[ ϕ(a) =

which proves the assertion.



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We can rewrite 3.2.5 in terms of the functor inv ◦ Spec. Note that Spec(L)inv has the underlying set HomBDLat (L, 2) and carries the subspace topology inherited from 2 L , cf. 3.1.4. In view of the equalities ◦

K (Spec(L)) = K(Spec(L)inv ), ◦

K (Spec(L)inv ) = K(Spec(L)), our first Stone Representation Theorem 3.2.5 takes the following form. Note that no proof is required since the lattices and maps are the same as in 3.2.5. 3.2.6 Stone Representation – Lattices via Closed Constructible Sets Let L and M be bounded distributive lattices and let ϕ ∈ HomBDLat (L, M). (i) The map ε L : L → K(Spec(L)inv ), a → [[ a = 1 ]] is an isomorphism of lattices. (ii) The diagram εL

L

/ K(Spec(L)inv )

ϕ

K(Spec(ϕ)inv )

 M

 / K(Spec(M)inv )

εM



of lattice homomorphisms is commutative.

3.2.7 Setting up the Stone Representation of Spectral Spaces Now we start with a spectral space X and want to show that X is homeomorphic to ◦

Spec(K (X)) and Spec(K(X))inv . As before, we use the characteristic functions isomorphism χ : P(X) → 2X : S → χS of Boolean algebras and the evaluation maps (or projection maps) x : 2X → 2 : f → f (x), cf. 2.2.3. The composition x ◦ χ is also denoted by x. The maps x are homomorphisms of Boolean algebras. Therefore, if L ⊆ P(X) ◦

x | L : L → 2 is a is a bounded sublattice (e.g., L = K (X) or L = K(X)), then homomorphism of bounded lattices. Thus, we have defined a map Λ X : X → ◦

◦

HomBDLat (L, 2). We write Λ X if L = K (X) and Λ X if L = K(X). 3.2.8 Stone Representation – Spectral Spaces via Quasi-Compact Open Sets Let X and Y be spectral spaces and f : X → Y a spectral map. (i) The map ◦

◦

x|◦ Λ X : X → Spec(K (X)), x → (X) K

86

Stone Duality is a homeomorphism. ◦

◦

◦

(ii) If U ∈ K (X) then Λ X (U) = {λ ∈ Spec(K (X)) | λ(U) = 1}. ◦

◦

(iii) If λ ∈ Spec(K (X)) then (Λ X )−1 (λ) is the generic point of the closed and  irreducible set U ∈λ−1 (0) X \ U. (iv) The diagram ◦

ΛX

X

◦

/ Spec(K (X)) ◦

Spec( K ( f ))

f

 ◦ / Spec(K (Y ))

◦

 Y

ΛY

of spectral maps is commutative. ◦

Proof (i) First we show that Λ X is a spectral map. By 3.1.6(iii), the sets ◦

◦

= 1 ]] = {λ ∈ Spec(K (X)) | λ(U) = 1} , U ∈ K (X), [[ U ◦

are the quasi-compact open sets of Spec(K (X)). Suppose that x ∈ X. Then the following statements are equivalent: • • • •

◦

= 1 ]]), x ∈ (Λ X )−1 ([[ U = 1 ]], x ∈ [[ U x (U) = 1, x ∈ U. ◦

◦

= 1 ]]); hence Λ X is spectral. This shows that U = (Λ X )−1 ([[ U For injectivity, suppose that x = y with x, y ∈ X. The preceding argument shows that x and y are contained in the same quasi-compact open sets of X. The T0 -property of X implies x = y. ◦

◦

For surjectivity, let λ ∈ Spec(K (X)). Then λ−1 (0) is a prime ideal of K (X).  The set C = X \ ( U ∈λ−1 (0) U) is nonempty closed and irreducible by 1.1.11. Let x be the generic point of C. Then the following conditions about an element ◦

V ∈ K (X) are equivalent: • V ∈ λ−1 (0), • x  V, • x (V) = 0. ◦

We have shown that λ = Λ X (x), and the proof of (i) is finished. (ii) and (iii) follow from the proof of (i).

3.2 Stone Duality

87

◦

(iv) Pick a point x ∈ X and an element B ∈ K (Y ). We wish to show that ◦

◦

◦

Spec(K ( f )) ◦ Λ X (x)(B) = ΛY ( f (x))(B). ◦

Note that ΛY ( f (x))(B) = f (x)(B) = 1 if and only if f (x) ∈ B. On the other hand, ◦

◦

◦

x ◦ K ( f )(B) = Spec(K ( f )) ◦ Λ X (x)(B) = x ( f −1 (B)) = 1 if and only if x ∈ f −1 (B), which proves the claim.



Finally we address the Stone representation of spectral spaces in terms of closed and constructible sets. One would expect that 3.2.8 translates into the following statement: 3.2.9 Stone Representation – Spectral Spaces via Closed Constructible Sets Let X and Y be spectral spaces and f : X → Y a spectral map. (i) The map Λ X : X → Spec(K(X))inv, x → x | K(X) is a homeomorphism. (ii) If A ∈ K(X), then Λ X (A) = {λ ∈ Spec(K(X))inv | λ(A) = 1}. (iii) If λ ∈ Spec(K(X))inv , then (Λ X )−1 (λ) is the unique generic point of the  closed and irreducible set A∈λ−1 (1) A. (iv) The diagram X

ΛX

Spec(K( f ))inv

f

 Y

/ Spec(K(X))inv

ΛY

 / Spec(K(Y ))inv

of spectral maps is commutative. Remark The relationship between 3.2.8 and 3.2.9 is not the same as that between 3.2.5 and 3.2.6. We observed that 3.2.5 and 3.2.6 speak about the same result, which is only phrased differently. Given a spectral space X, the ◦

lattices K (X) and K(X) do not coincide. Therefore, the spaces and maps in 3.2.9 are not identical to those in 3.2.8; hence, 3.2.9 is not merely a reformulation of 3.2.8 and needs a proof. Proof

Most of the work has already been done in 2.3.1.

(i) Note that Spec(K(X)) is the set HomBDLat (K(X), 2) viewed as a spectral

88

Stone Duality

K(X) , 3.1.4. Thus, Spec(K(X))inv is the set HomBDLat (K(X), 2) subspace of 2inv

viewed as a spectral subspace of 2 K(X) , and ΛX is the map χ of 2.3.1 with restricted codomain. This proves (i). (ii) and (iii) are clear from the proof of 2.3.1. (iv) Pick a point x ∈ X and an element B ∈ K(Y ). We wish to show that Spec(K( f )) ◦ Λ X (x)(B) = ΛY ( f (x))(B). Now ΛY ( f (x))(B) = f (x)(B) = 1 if and only if f (x) ∈ B. On the other hand, Spec(K( f )) ◦ Λ X (x)(B) = x ◦ K( f )(B) = x ( f −1 (B)) = 1 if and only if x ∈ f −1 (B), which proves the claim.



3.2.10 Conclusion The category-theoretic content of the Stone Representation Theorems 3.2.5 and 3.2.8 can be expressed in the following way. The categories Spec and BDLat are anti-equivalent via the mutually quasi-inverse functors ◦

◦

K and Spec. The maps Λ X are the components of a natural isomorphism ◦

IdSpec → Spec ◦ K ; the maps ε L are the components of a natural isomorphism ◦

IdBDLat → K ◦ Spec. Similarly, the Stone Representation Theorems 3.2.6 and 3.2.9 show that the categories Spec and BDLat are anti-equivalent via the mutually quasi-inverse functors K and inv ◦ Spec. The maps Λ X are the components of a natural isomorphism IdSpec → inv ◦ Spec ◦ K; the maps ε L are the components of a natural isomorphism IdBDLat → K ◦inv ◦ Spec. It follows from 1.3.4 (also see 1.3.17) that Stone duality restricts to an antiequivalence between the categories BoolSp of Boolean spaces and BoolAlg of Boolean algebras.

3.3 Spectral Spaces via Prime Ideals and Prime Filters Summary According to 3.1.4, the spectrum of a bounded distributive lattice L . A bounded lattice homomorL is the spectral subspace HomBDLat (L, 2) ⊆ 2inv phism ϕ : L → 2 determines the prime ideal ϕ−1 (0) and the prime filter ϕ−1 (1). The correspondence between homomorphisms and prime ideals, as well as between homomorphisms and prime filters, is bijective. Thus, instead of using homomorphisms as the elements of the underlying set of the spectrum, we can as well use prime ideals or prime filters. The mathematical content is no different if we view spectra in this way.

3.3 Spectral Spaces via Prime Ideals and Prime Filters

89

However, this perspective sometimes contributes to the intuitive understanding of spectra. In some applications it is natural to consider the points of a spectrum as prime ideals, in others it is more natural to consider them as prime filters. Moreover, each spectral space has three topologies (patch, spectral, and inverse). One needs to understand the different topologies for spectra of prime ideals and for spectra of prime filters. To anticipate all conceivable applications, we give a presentation of the spectrum of a lattice both in terms of prime ideals and prime filters. 3.3.1 The Spectrum of a Lattice via Prime Ideals and Prime Filters We define PrimI(L) to be the set of prime ideals and PrimF(L) to be the set of prime filters of L. The following canonical maps are bijective: Ker I : HomBDLat (L, 2) → PrimI(L), ϕ → ϕ−1 (0), KerF : HomBDLat (L, 2) → PrimF(L), ϕ → ϕ−1 (1). They are used to transfer all three spectral topologies from HomBDLat (L, 2) to the prime ideals and to the prime filters. We define PrimI(L) to be the spectral space that is homeomorphic to Spec(L) via Ker I and call it the prime ideal spectrum of L. Similarly, we define PrimF(L) to be the spectral space that is homeomorphic to Spec(L)inv via KerF and call it the prime filter spectrum of L. The characteristic functions map χ : P(L) → 2 L sends PrimI(L) homeoL and PrimF(L) onto a subspace of 2 L , 3.1.4, morphically onto a subspace of 2inv 3.1.7. We give a detailed exposition of the main features of each of the two spectral spaces and explain the connections between them. 3.3.2 The Prime Filter Spectrum Prime filters of L are denoted by f, g, and so on. The homomorphism L → 2 that corresponds to the prime filter f is denoted by λf . For a ∈ L we define D(a) = D LF (a) = {f ∈ PrimF(L) | a  f}, V(a) = VLF (a) = {f ∈ PrimF(L) | a ∈ f}. The equality (KerF )−1 (D(a)) = {ϕ ∈ Spec(L)inv | a  ϕ−1 (1)} = [[ a = 0 ]] ◦

shows that D(a) ∈ K (PrimF(L)), 3.1.6(vi). It follows that V(a) = PrimF(L) \ D(a) is closed and constructible and corresponds to the set [[ a = 1 ]] ⊆ Spec(L)inv . The map ◦

D : L → K (PrimF(L)), a → D(a)

90

Stone Duality

is an anti-isomorphism, 3.1.6(iv), (v). The formation of complements in the Boolean algebra P(PrimF(L)) is also an anti-automorphism. Therefore the map V : L → K(PrimF(L)), a → V(a) is an isomorphism. The Boolean algebra of constructible sets is the set of finite unions of sets D(a) ∩ V(b), a, b ∈ L. For specialization, note that the following conditions about f, g ∈ PrimF(L) are equivalent: • • • •

f  g, g ∈ D(a) ⇒ f ∈ D(a) for all a ∈ L, a  g ⇒ a  f for all a ∈ L, f ⊆ g.

3.3.3 The Prime Ideal Spectrum Prime ideals of L are denoted by i, j, and so on. The homomorphism L → 2 that corresponds to the prime ideal i is denoted by λi . For a ∈ L we define D(a) = D LI (a) = {i ∈ PrimI(L) | a  i}, V(a) = VLI (a) = {i ∈ PrimI(L) | a ∈ i}. The equality (Ker I )−1 (D(a)) = {ϕ ∈ Spec(L) | a  ϕ−1 (0)} = [[ a = 1 ]] ◦

gives D(a) ∈ K (PrimI(L)), 3.1.6(iii). It follows that V(a) = PrimI(L) \ D(a) is closed and constructible and corresponds to [[ a = 0 ]] ⊆ Spec(L). The map ◦

D : L → K (PrimI(L)), a → D(a) is an isomorphism, 3.1.6(i), (ii). The formation of complements in the Boolean algebra P(PrimI(L)) is an anti-automorphism. Therefore the map V : L → K(PrimI(L)), a → V(a) is an anti-isomorphism. The Boolean algebra of constructible sets is the set of finite unions of sets D(a) ∩ V(b), a, b ∈ L. Finally specialization. The following conditions about i, j ∈ PrimI(L) are equivalent: • • • •

i  j, j ∈ D(a) ⇒ i ∈ D(a) for all a ∈ L, a  j ⇒ a  i for all a ∈ L, i ⊆ j.

3.3 Spectral Spaces via Prime Ideals and Prime Filters

91

3.3.4 Maps between Prime Filter Spectra and between Prime Ideal Spectra Let ϕ : L → M be a homomorphism of bounded distributive lattices. Inverse images of prime filters in M are prime filters in L. The same holds for prime ideals. Thus, there are maps PrimF(ϕ) : PrimF(M) → PrimF(L), f → ϕ−1 (f), PrimI(ϕ) : PrimI(M) → PrimI(L), i → ϕ−1 (i) of the underlying sets of the prime filter spectra and the prime ideal spectra. If a ∈ L, then PrimF(ϕ)−1 (D LF (a)) = D FM (ϕ(a)), I (ϕ(a)), PrimI(ϕ)−1 (D LI (a)) = D M

which shows that both PrimF(ϕ) and PrimI(ϕ) are spectral maps. The construction of the prime filter spectrum and the construction of the prime ideal spectrum are both functors BDLat → Spec. 3.3.5 Example We illustrate the material in this section by considering the spaces S∞ and (S∞ )inv of 1.6.14 and 1.6.15. Let S be an infinite set. (i) The set L = {F ⊆ S | F is finite or F = S} is a bounded sublattice of P(S) with ⊥= ∅ and  = S. (a) The prime ideals of L are L\{} and all ideals i(s) = {T ∈ L | s  T }, for s ∈ S. (b) The prime filters of L are {} and all principal filters f({s}) generated in L by singleton subsets of S. (c) The maps (S∞ )inv → PrimI(L), s → i(s), ∞ → L \ {}, S∞ → PrimF(L), s → f({s}), ∞ → {} are homeomorphisms.

◦

(d) The following lattices are identical: L = K(S∞ ) = K ((S∞ )inv ). (ii) The set M = {C ⊆ S | C is cofinite or C = ∅ } is a bounded sublattice of P(S) with ⊥= ∅ and  = S. (a) The prime ideals of M are {⊥} and the principal ideals i(S \ {s}) generated by S \ {s} with s ∈ S. (b) The prime filters of M are L\{⊥} and the filters f(s) = {C ∈ M | s ∈ C} for s ∈ S.

92

Stone Duality (c) The maps S∞ → PrimI(M), s → i(S \ {s}), ∞ → {⊥}, (S∞ )inv → PrimF(M), s → f(s), ∞ → L \ {⊥} are homeomorphisms.

◦

(d) The following lattices are identical: M = K (S∞ ) = K((S∞ )inv ).

3.4 The Boolean Envelope of a Bounded Distributive Lattice Summary The Boolean algebras and their homomorphisms form the category BoolAlg, which is a full subcategory of BDLat, cf. 1.3.17. For basic facts about Boolean algebras we refer to [Kop89]. Given a bounded distributive lattice L, there is a unique smallest Boolean algebra containing L, which is called the Boolean hull or the Boolean envelope of L, cf. [Mac39], [Grä98, p. 116]. We construct the Boolean envelope using Stone duality. Recall from 3.2.10 that Stone duality restricts to a correspondence between Boolean algebras and Boolean spaces. Suppose that L is a bounded distributive lattice with spectrum Spec(L). The ◦

patch space Spec(L)con is Boolean, and K (Spec(L)con ) = K(Spec(L)) is the corresponding Boolean algebra. The relationship between this Boolean algebra and the lattice L reflects the relationship between Spec(L) and its patch space. ◦

The isomorphism ε L : L → K (Spec(L)), a → [[ a = 1 ]] of 3.2.5(i) yields an embedding of L into the Boolean algebra K(Spec(L)). We show in 3.4.1 that this is the Boolean envelope of L. The category-theoretic content of the construction is explained in 3.4.4. 3.4.1 Theorem Let L be a bounded distributive lattice. There exist a Boolean algebra ba(L) and a homomorphism ba L : L → ba(L) of bounded lattices such that the following universal mapping property holds. If B is a Boolean algebra and if ϕ : L → B is a homomorphism of bounded lattices, then there exists a unique homomorphism ψ : ba(L) → B such that the following diagram is commutative: L

ba L

ϕ

/ ba(L) ψ

!  B.

3.4 The Boolean Envelope of a Bounded Distributive Lattice

93

Proof Let B be a Boolean algebra and ϕ : L → B a homomorphism of bounded lattices. The functor Spec yields the spectral map f = Spec(ϕ) : Y = Spec(B) → X = Spec(L). ◦

Note that Y is a Boolean space since B  K (Y ), by 3.2.5 and 1.3.4. The universal mapping property of 1.3.24(iii) gives a unique spectral map g : Y → Xcon such ◦

that f = conX ◦ g. The functor K and the Stone Representation Theorem 3.2.5 yield the following commutative diagram of bounded distributive lattices and Boolean algebras: L

◦

◦

εL

K (con X )

/ K (X)

◦

/ K (Xcon )

◦

ϕ

K( f )

 B

◦

εB



/ K (Y )

◦

y

K (g)

◦

The homomorphisms ε L and εB are isomorphisms, and K (Xcon ) = K(X) is ◦ a Boolean algebra. We define ba(L) = K(X) and ba L = K (conX ) ◦ ε L . Then ◦

−1 ◦ (g) provides the desired factorization of ϕ. ψ = εB K Uniqueness follows from the fact that ε L is an isomorphism and that K(X)  is the Boolean algebra generated by ε L (L).

3.4.2 Definition Let L be a bounded distributive lattice. The Boolean algebra ba(L) = K(Spec(L)), together with the canonical homomorphism ◦

ba L = K (conSpec(L) ) ◦ ε L : L → ba(L), is the Boolean envelope of L. 3.4.3 Uniqueness of the Boolean Envelope We continue with the notation of 3.4.1 and show that the Boolean algebra ba(L) is unique up to unique isomorphism. This means: suppose that there is another Boolean algebra A together with a homomorphism β : L → A of bounded lattices such that the same universal mapping property holds for β. Then there is a unique isomorphism ϕ : ba(L) → A such that β = ϕ ◦ ba L . To see this, note that the universal mapping property applied with ba L : L → ba(L) and β : L → A yields a unique homomorphism ϕ : ba(L) → A such that β = ϕ ◦ ba L . We can also apply the universal mapping property with β : L → A and ba L : L → ba(L). There is a unique homomorphism ψ : A → ba(L) such that ba L = ψ ◦ β. Then id A ◦β = ϕ ◦ (ψ ◦ β) = (ϕ ◦ ψ) ◦ β. Uniqueness of the

94

Stone Duality

factorization implies ϕ◦ψ = id A. The same argument shows that ψ◦ϕ = idba(L) . Thus, ϕ and ψ are mutually inverse isomorphisms. 3.4.4 Category-Theoretic Interpretation of the Boolean Envelope The universal mapping property of the Boolean envelope can be expressed in category-theoretic language. (This is similar to, in fact is the Stone dual of, 1.3.24(iii).) First we show that the construction of the Boolean envelope is functorial, which means that the Boolean envelope also acts on bounded lattice homomorphisms: a homomorphism ϕ : L → M of bounded lattices yields a unique homomorphism ba(ϕ) : ba(L) → ba(M) of Boolean algebras such that ba(ϕ) ◦ ba L = ba M ◦ϕ. This follows from the universal mapping property. The construction ba is a functor now. The canonical homomorphisms ba L : L → ba(L) are the components of a natural transformation from the identity functor of BDLat to the composition of ba and the inclusion functor BoolAlg → BDLat. The functor ba is idempotent (i.e., baba(L) : ba(L) → ba(ba(L)) is an isomorphism). This follows from general category-theoretic principles, but it can be seen most easily by direct inspection of the construction. The subcategory BoolAlg ⊆ BDLat is a reflective subcategory (i.e., the inclusion functor has a left adjoint functor, which is called the reflector). The functor ba is the reflector, and the maps ba L : Lcon → ba(L) are the reflection maps.

3.5 Inverse Spaces and Inverse Lattices Summary Stone duality establishes a perfect correspondence between spectral spaces and bounded distributive lattices. Therefore every construction with spectral spaces must correspond to a construction with lattices. We explain what the formation of the inverse of a spectral space, cf. Section 1.4, means in terms of the corresponding lattice. 3.5.1 Inverse Posets and Lattices We start with some preliminary comments about inverse posets and lattices. Given a poset P = (P, ≤), the inverse poset Pinv is defined in A.1(vii). If L = (L, ≤) is a lattice, or a bounded lattice, then the inverse poset is a lattice, or a bounded lattice, as well, called the inverse lattice, and is denoted by Linv , A.6(iv) The lattice operations are: • x ∧inv y = x ∨ y, • x ∨inv y = x ∧ y, • ⊥inv =  and inv = ⊥.

3.5 Inverse Spaces and Inverse Lattices

95

If the lattice L is distributive, then so is Linv . Now suppose that A is a Boolean algebra. Then Ainv is a Boolean algebra as well. The complementation map ¬ = ¬ A : A → A is the same for A and Ainv . As a map from A to itself, ¬ is an anti-automorphism (by the de Morgan laws). Viewed as a map from A to Ainv , it is an isomorphism. Let L ⊆ A be a bounded sublattice. Then ¬(L) is a sublattice of A and also of Ainv . The restriction ¬ : L → ¬(L) is an anti-isomorphism if we consider ¬(L) as a sublattice of A, and is an isomorphism if we consider ¬(L) as a sublattice of Ainv . Accordingly, ¬ as a map Linv → ¬(L) is an isomorphism or an anti-isomorphism if ¬(L) is considered as a sublattice of A or as a sublattice of Ainv , respectively. The constructions of inverse posets, or of inverse lattices, are functors PoSets → PoSets and BDLat → BDLat. The functors are involutions. 3.5.2 Inverse Lattices and Inverse Spaces under Stone Duality Let X be a ◦

spectral space. The lattices K (X) and K(X) are sublattices of the Boolean al◦

gebra K(X). The complementation ¬ : K(X) → K(X) maps K (X) onto K(X), ◦

◦

which implies K (X)inv  K(X). We form the spectral spaces Spec(K (X)) and ◦

Spec(K(X)), as well as their inverse spaces Spec(K (X))inv and Spec(K(X))inv . Stone duality says that ◦

◦

X  Spec(K (X))  Spec(K(X))inv  Spec(K (X)inv )inv, see 3.2.8, 3.2.9. It follows that ◦

◦

Xinv  Spec(K (X))inv  Spec(K(X))  Spec(K (X)inv ). Conversely, we start with a bounded distributive lattice L. Then Stone duality, see 3.2.5 and 3.2.6, yields the isomorphisms ◦

◦

L  K (Spec(L)) = K(Spec(L)inv )  K (Spec(L)inv )inv, ◦

◦

Linv  K (Spec(L))inv  K(Spec(L)) = K (Spec(L)inv ). 3.5.3 Prime Ideal Spectrum, Prime Filter Spectrum, and Inverse Spaces In 3.3.1 the spectrum and the inverse spectrum of a bounded distributive lattice L are presented in terms of prime ideals and prime filters. Explicitly: PrimI(L)  Spec(L) and PrimF(L)  Spec(L)inv . The prime ideals of L are the prime filters of Linv , and vice versa. Moreover, by restriction, the complementation map ¬ : ba(L) → ba(L) yields an isomorphism ¬ : Linv → ¬(L), cf. 3.5.1. Thus, prime filters of Linv correspond to prime filters of ¬(L), and prime ideals correspond to prime ideals. We obtain

96

Stone Duality

the following isomorphisms between spectral spaces: PrimI(L) = PrimF(Linv )  PrimF(¬(L)) = PrimI(¬(L)inv ), PrimF(L) = PrimI(Linv )  PrimI(¬(L)) = PrimF(¬(L)inv ).

3.6 The Spectrum of a Totally Ordered Set Summary Every bounded totally ordered set (L, ≤) is a bounded distributive lattice with lattice operations s ∧ t = min{s, t} and s ∨ t = max{s, t}. Under Stone duality, Section 3.2, it corresponds to the spectral space Spec(L), which is homeomorphic to the prime ideal spectrum PrimI(L), 3.3.1. The prime ideals of L have a very simple description, cf. 3.6.2. Therefore we choose to give a presentation in terms of PrimI(L). The main data of the prime ideal spectrum of L are exhibited in 3.6.4, It follows that the Boolean algebra of constructible sets is the interval algebra of L, 3.6.5. It is important to note the difference between the present considerations and those in Section 1.6.A. In both cases we work with a totally ordered set. But this set plays a quite different role. In Section 1.6.A a totally ordered set (P, ≤) was equipped with a T0 -topology and we analyzed what it means that this topology is spectral. Now the bounded totally ordered set (L, ≤) is viewed as a lattice and, by Section 3.1, has a spectrum, which is the object of our curiosity. However, we show that the spectrum of L is totally ordered by specialization, 3.6.1. Therefore the considerations of Section 1.6.A can be applied to Spec(L). The totally ordered sets L and Spec(L) are related to each other in 3.6.6. 3.6.1 Proposition Suppose that L is a bounded distributive lattice. The following conditions are equivalent: (i) The lattice is totally ordered. (ii) The specialization order of Spec(L) is total. ◦

Proof (i) ⇒ (ii) By Stone duality, 3.2.5, the lattice K (Spec(L)) is isomorphic to L, hence is totally ordered. It follows that O(Spec(L)), hence also A(Spec(L)), is totally ordered by inclusion. Thus, 1.6.1(i) shows that the specialization order is total. (ii) ⇒ (i) Assume a, b ∈ L are incomparable. Then the sets D(a), D(b) ∈ ◦

K (Spec(L)) are incomparable, hence O(Spec(L)) and A(Spec(L)) are not to-

3.6 The Spectrum of a Totally Ordered Set

97

tally ordered by inclusion. It follows from 1.6.1(i) that the specialization order is not total.  3.6.2 Lemma Suppose that L is a bounded totally ordered set. Then the following conditions about a subset I ⊆ L are equivalent: (i) I is a proper ideal. (ii) I is a proper nonempty down-set. (iii) I is a prime ideal. Proof (iii) ⇒ (i) is trivial and (i) ⇒ (ii) holds, since ideals are nonempty downsets by definition. (ii) ⇒ (iii) The only condition we need to check is that a ∧ b ∈ I implies that a ∈ I or b ∈ I. But this is clear since a ∧ b ∈ {a, b}.  3.6.3 Remark We have, in fact, encountered the spectral space PrimI(L) in Section 1.6.A. To make the connection, let T be the totally ordered set L\{⊥, }. By 1.6.5, the set X of all down-sets of T carries a unique spectral topology whose specialization order is inclusion. The map PrimI(L) → X, i → i ∩ T is a homeomorphism. 3.6.4 Conclusion We know that specialization in PrimI(L) is the same as inclusion (i.e., i  j is equivalent to i ⊆ j, 3.3.2). The main data of PrimI(L) are obtained from the general results about the prime ideal spectrum of a bounded distributive lattice, Section 3.3. We record them briefly; no proofs are needed. (a) (b) (c) (d) (e) (f) (g) (h) (i)

If a ∈ L, then D(a) = {j ∈ PrimI(L) | j ∩ a ↑ = ∅}. D() = PrimI(L), D(⊥) = ∅. If a, b ∈ L, a ≤ b, then D(a) ∩ D(b) = D(a) and D(a) ∪ D(b) = D(b). The set U ⊆ PrimI(L) is open if and only if there is a subset S ⊆ L such  that U = {j ∈ PrimI(L) | ∃a ∈ S : j ∩ a ↑ = ∅} = a ∈S D(a). If a ∈ L, then V(a) = {j ∈ PrimI(L) | a ↓ ⊆ j}. V() = ∅, V(⊥) = PrimI(L). If a, b ∈ L, a ≤ b, then V(a) ∩ V(b) = V(b) and V(a) ∪ V(b) = V(a). The set V ⊆ PrimI(L) is closed if and only if there is a subset S ⊆ L such  that V = {j ∈ PrimI(L) | ∀a ∈ S : a ↓ ⊆ j} = a ∈S V(a). The Boolean algebra K(PrimI(L)) is the lattice generated by the intersections V(a) ∩ D(b) = {j ∈ PrimI(L) | a ↓ ⊆ j and j ∩ b↑ = ∅}

(a, b ∈ L).

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3.6.5 Interval Algebras The interval algebra of L provides another presentation of the Boolean algebra K(PrimI(L)). Consider the half-open intervals [a, b) ⊆ L. Their finite unions form a Boolean algebra IntAlg(L), called the interval algebra of L (cf. [Kop89, Chapter 6, Section 15]). Its bottom element is ∅ = [⊥, ⊥), and its top element is L \ {} = [⊥, ). The unique homomorphism IntAlg(L) → K(PrimI(L)) that sends [a, b) to V(a) ∩ D(b) is an isomorphism. 3.6.6 The Dedekind–MacNeille Completion bounded totally ordered set L. The map

We continue to consider a

ι L : L → I(L) = PrimI(L) ∪ {L} : a → a ↓ (where I(L) is the set of ideals of L) is an embedding into the totally ordered set of all ideals, hence into a complete totally ordered set. We ask whether there is a minimal complete subset of I(L) containing ι L (L). The following considerations show how such a set can be found. (i) First let (S, ≤) be any complete totally ordered set, and let J ⊆ S \ {} be a set of lower elements of jumps, A.2(iii). Then (S \ J, ≤) is complete as well. (ii) Now we use ι L to consider L as a subset of the complete totally ordered set I(L). Let J be the set of all lower points in I(L) that are not contained in ι L (L) (i.e., that are not principal ideals generated by the lower point of a jump in L). Then I(L) \ J is a minimal complete set containing ι L (L). It coincides with the Dedekind–MacNeille completion of L, cf. 9.1.3 and [Bir79, p. 126], [BaDw74, p. 236 ff], [DaPr02, p. 166 ff], [Rom08, p. 86].

3.7 Further Reading 3.7.1 Duality Theories We mentioned that, besides Stone duality, there are many other dualities between categories of (algebraic) structures and categories of topological spaces. In category theory dualities can be expressed as adjunctions of functors, cf. [Mac71, p. 91], [HeSt79, p. 92]. A general theory of dualities, including constructions and a large number of examples, is contained in [ClDa98]. Posets, semilattices, and similar structures are in the focus of [HoKe72], where various adjunctions and dualities are studied, with Stone duality as a special case. Recently the class of stably compact spaces and

3.7 Further Reading

99

suitable dualities have received considerable attention; we mention [JuSü96], [Law11], [vG12], and [vG14] as sources. 3.7.2 Distributive Semilattices We single out one generalization of Stone duality, and describe the appearance of the participating spaces and posets in algebra. The duality can essentially be obtained by following the development in this chapter; the missing details may be found in [Grä98, II.5]. ◦

Given any topological space X, the set K (X) of quasi-compact opens is ◦ closed under finite unions, hence (K (X), ∪) is a join-subsemilattice of P(X), ◦

cf. A.6(iii). We call a space X compactly based if K (X) is a basis. Spectral spaces are compactly based by axiom S2. In general, axiom S3 fails in compactly based spaces. For example, let P be a poset and let τ L be the fine lower topology of P, cf. A.8(ii). Then X = (P, τ L ) is compactly based since every down-set is ◦

◦

a union of sets from K (X) = {F ↓ | F ⊆ P finite}. But, clearly, K (X) need not be closed under finite intersection. ◦ When X is compactly based, then K (X) is a distributive join-semilattice, where a join-semilattice S = (S, ∨) is called distributive if for all a, b, c ∈ S with c ≤ a ∨ b there are a0 ≤ a, b0 ≤ b such that c = a0 ∨ b0 , cf. [Grä98, p. 99]. One verifies without difficulty that a lattice is distributive as a join-semilattice just if it is a distributive lattice (i.e., satisfies the distributivity law for lattices). Stone duality extends to bounded distributive join-semilattices, but care has to be taken when it comes to morphisms. We state this extension with reference to [Grä98, II.5, Theorems 5 and 8 on p. 101 f]. If X is compactly based, quasi-compact, T0 , and sober, then we call it semispectral. Hence, semi-spectral spaces are characterized by the axioms S1, S2, and S4 of spectral spaces, but possibly lack the coherence axiom S3. As for an example, if P is a poset with top element, then the sobrification of (P, τ L ), cf. 11.2.1 and 11.2.7, is semi-spectral but it is not spectral in general. Let semiSpec be the category of semi-spectral spaces where the morphisms are continuous maps such that preimages of open quasi-compact sets are again quasi-compact. Hence, Spec is a full subcategory of semiSpec. Let BDSLat be the category whose objects are bounded distributive joinsemilattices and whose morphisms ϕ : S → T are join and bound-preserving maps with the following property: for all t ∈ T the set ϕ−1 (t ↑ ) is down-directed. One checks that a BDSLat-morphism between bounded distributive lattices is a morphism in BDLat. The functor F : semiSpec → BDSLat that sends ◦

X to K (X) and a semiSpec-morphism f : X → Y to the map defined by

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F( f )(V) = f −1 (V) is an anti-equivalence of categories. A quasi-inverse of F and representation theorems can be obtained similar to 3.2.5 and 3.2.8. Given a bounded distributive join-semilattice S, we write PrimI(S) for the set of prime ideals of S (cf. A.7(iii)) equipped with the sets D(a) = {p | a  p} as subbasis. It turns out that PrimI(S) is semi-spectral and the map S → O(PrimI(S)), a → ◦

D(a) is a poset isomorphism onto K (PrimI(S)). We mention two occurrences of compactly based spaces in algebra. Firstly, the property of a space X being compactly based is equivalent to saying that the poset O(X) of open sets is an algebraic frame, cf. 9.5.9; furthermore, every algebraic frame occurs in this way, see 9.5.10. Algebraic frames play a fundamental role in universal algebra related to the Congruence Lattice Problem. This problem was posed by R. P. Dilworth some 70 years ago, who asked whether every algebraic frame is the congruence lattice of a lattice (the converse is an easy consequence of [FuNa42, Theorem 1]). It was answered in the negative by F. Wehrung in [Weh07]. For an overview of the development see [Plo08]. Secondly, compactly based spaces occur as an important invariant and organizing tool attached to specific Abelian categories, cf. [Pre09]. We focus on the category of right modules Mod-R over a (not necessarily commutative) ring R. One defines the so-called Ziegler spectrum ZgR introduced in [Zie84]. The points are isomorphism classes of indecomposable pureinjective R-modules. A subbasis of open sets is the set of all sets of the form {M ∈ ZgR | ϕ(M)  ψ(M)}, where ϕ and ψ are positive primitive formulas in one free variable, in the first-order language of R-modules; see [Pre09] for terminology. The space ZgR is quasi-compact and compactly based because the subbasis above is indeed a basis and consists of quasi-compact opens. However, in general ZgR is neither T0 nor coherent. It is an open question whether ZgR (or better its T0 -quotient) is sober, cf. [Gre13] and [Her93]. ◦

If we send K (ZgR ) through the duality above we obtain the semi-spectral ◦

space PrimI(K (ZgR )), which is the sobrification of the T0 -quotient of ZgR . 3.7.3 Transfer of Information between Bounded Distributive Lattices and Spectral Spaces As pointed out, Stone duality provides a powerful tool for the study of spectral spaces. Every property of a spectral space must correspond to a property of the corresponding bounded distributive lattice. The class of bounded distributive lattices is a variety in the sense of universal algebra, [Coh81, p. 162] or [Hod93, p. 426]. Categories whose object classes are varieties have many favorable properties, cf. [AHS90, Section 23], [HeSt79, Section 32]. Stone duality can be used to transfer properties back and forth between BDLat and

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101

Spec. The benefit can go in both directions – the algebra can help understand the topology, and vice versa. For example, co-completeness of BDLat is the same as completeness of Spec. Thus, a coproduct, or a fiber sum, in BDLat corresponds to a product, or a fiber product, in Spec, respectively (cf. [Sch80]).

4 Subsets of Spectral Spaces

Spectral subspaces were introduced in Section 2.1 and have been used heavily ever since. They are the proconstructible subsets of a spectral space, 2.1.3. This chapter is devoted to a detailed study of several important classes of subsets of spectral spaces, not necessarily proconstructible. The specialization order plays a decisive role, both for the definition and for the analysis of the subsets. Quasi-compact subsets are studied in Section 4.1. The notion itself does not refer to specialization, but there is a very useful characterization of quasicompactness in arbitrary T0 -spaces that uses specialization, 4.1.2. For spectral spaces it is one of the main results that a subset is quasi-compact if and only if its set of generalizations is proconstructible, 4.1.5. In Section 4.2 we use directed and totally ordered subsets to study properties of the specialization order. We obtain several completeness results (e.g., 4.2.6). In Section 4.3 we use the specialization order, in particular chains, to introduce and discuss the notions of dimension and rank in a spectral space. Every spectral space has minimal points and maximal points. In Section 4.4 we study topological properties of the set of minimal points, as well as of the set of maximal points as subspaces of a spectral space. The results include a characterization (originally proved in [Hoc71]) of those spaces that are homeomorphic to the space of minimal points of some spectral space, 4.4.11. It is an important issue to compare the restrictions of the different topologies (spectral, inverse, and constructible) of the ambient spectral space to the sets of maximal or minimal points. In Section 4.5 we consider convex subsets of spectral spaces. Every locally closed subset is convex. The converse is not true, but the two classes of sets are closely related, 4.5.6. As an important special case we study the locally closed points (i.e., the points whose singleton subsets are locally closed). These are particularly important for applications in commutative algebra and algebraic geometry. 102

4.1 Quasi-Compact Subsets, Closure, and Generalization

103

4.1 Quasi-Compact Subsets, Closure, and Generalization Summary A first key result in this section is a characterization of quasicompact T0 -spaces in terms of the subspace of maximal (= closed) points and the specialization relation, see 4.1.2. For a spectral space Y we show that a subset X ⊆ Y is quasi-compact for the inverse topology if and only if the closure of X for the spectral topology is the set of its specializations, 4.1.3. Using these results we obtain various characterizations of the subsets that are quasi-compact for the spectral topology, 4.1.5. As a consequence, we strengthen and supplement earlier results about the characterization of closed subsets (see 4.1.6 and compare with 1.5.4) and about the separation of disjoint sets by quasicompact open sets or by closed and constructible sets (see 4.1.7 and compare with 1.5.3). 4.1.1 Maximal Specialization Chains Exist Subsets of a T0 -space that are totally ordered for specialization are called specialization chains. Let S ⊆ X be a subset and let CS be the set of specialization chains containing S. The set CS is partially ordered by inclusion. The union of a totally ordered subset of CS also belongs to CS . Thus, Zorn’s Lemma shows that CS contains a maximal element if it is nonempty. 4.1.2 Proposition Let X be a T0 -space. Then X is quasi-compact if and only if X max is quasi-compact and for every x ∈ X there is some y ∈ X max with x  y, which means that X = Gen(X max ). Proof First suppose X is quasi-compact. We pick a point x ∈ X and claim that {x} ∩ X max  ∅. Let Cx be the set of specialization chains of X passing through x. It is clear that Cx  ∅. Hence there is a maximal element C ∈ Cx . The family ({c})c ∈C has the FIP, which implies  Y= {c}  ∅ c ∈C

(since X is quasi-compact). We pick an element y ∈ Y and claim that y ∈ {x} ∩ X max . From x ∈ C it is clear that y ∈ Y ⊆ {x}. To show that y ∈ X max , let z ∈ X with y  z. Since every point in C specializes to y, every point of C specializes to z as well. Thus, C ∪ {z} is a specialization chain, and maximality of C implies z ∈ C, hence z  y. Since specialization is a partial order we conclude that y = z. It has been shown that every point of X specializes to a point of X max (i.e., X =  Gen(X max )). Now we show that X max is quasi-compact. Let X max ⊆ i ∈I Ui  be an open cover. Then X = Gen(X max ) ⊆ i ∈I Ui . As X is quasi-compact

104

Subsets of Spectral Spaces  there is a finite subset J ⊆ I with X max ⊆ X ⊆ i ∈J Ui , showing that X max is quasi-compact. Conversely, suppose that X max is quasi-compact and X = Gen(X max ). To  show that X is quasi-compact, let X ⊆ i ∈I Ui be an open cover. Then X max ⊆  max ⊆  i ∈I Ui and there is a finite subset J of I with X i ∈J Ui . Now X =  max  Gen(X ) implies X ⊆ i ∈J Ui , as desired. The following result should be compared with the description of the closure of a subset of a spectral space in 1.5.4 and 1.5.5. 4.1.3 Theorem Let X be a spectral space and let Y be a subset. Then Y is  quasi-compact for the inverse topology if and only if Y = y ∈Y {y} = Spez(Y ). Proof First let Y be quasi-compact for the inverse topology. As Spez(Y ) ⊆ Y holds trivially, only the other inclusion has to be proved. Given z ∈ Y , the Separation Lemma 1.5.3 (applied with Y1 = {z} and Y2 = Y ) yields an element y ∈ Y with y  z. Hence Y = Spez(Y ). ◦  Conversely, suppose Y = Spez(Y ) and let Y ⊆ i ∈I Ai with Ai ∈ K (Xinv ) = K(X) (cf. 1.4.1). Every point y ∈ Y is contained in some set Ai and, as Ai is closed in X, it follows that {y} ⊆ Ai . Thus Y = Spez(Y ) implies that  Y ⊆ i ∈I Ai . The set Y is proconstructible and each set Ai is constructible. By  compactness of Xcon there is a finite subset J ⊆ I with Y ⊆ Y ⊆ i ∈J Ai (i.e., Y is quasi-compact for the inverse topology).  The previous results hold for every spectral space. Hence they can be translated into statements about the inverse of a spectral space. Recall that, given points x, y in a spectral space X, x  y in X if and only if y  x in Xinv, cf. 1.4.3. Therefore, the following statements about a set Y ⊆ X are equivalent: • • • •

Y is specialization-closed in X. X \ Y is generically closed in X. X \ Y is specialization-closed in Xinv . Y is generically closed in Xinv .

We obtain: 4.1.4 Corollary Let X be a spectral space, S ⊆ X a subset, and x an upper bound of S for specialization. Then there is a minimal upper bound

4.1 Quasi-Compact Subsets, Closure, and Generalization

105

y ∈ Gen(x). 1 Setting S = ∅ , this says that for every x ∈ X there is a point y ∈ X min with y  x.  Proof The set of upper bounds of S is Z = s ∈S {s}, a proconstructible set, hence a spectral subspace of X. Replacing X with Z it is therefore enough to show that there is a point y ∈ X min with y  x. By 4.1.2 there is a point y ∈ (Xinv )max such that x  inv y. The above reminder shows that y ∈ (Xinv )max =  X min and y  x. 4.1.5 Theorem Let X be a spectral space. The following statements about a subset Y ⊆ X are equivalent: (i) Y is quasi-compact. inv (ii) Y = Gen(Y ). ◦  (iii) Gen(Y ) = {U ∈ K (X) | Y ⊆ U} (i.e., Gen(Y ) is inversely closed). (iv) Gen(Y ) is proconstructible. (v) There is a quasi-compact set Z ⊆ X with Y ⊆ Z and Z max ⊆ Y . con (vi) (Y )max = Y max . Proof Recall that S ⊆ Gen(S) ⊆ S S ⊆ X.

inv

and S max = Gen(S)max for any subset

(i) ⇔ (ii) is 4.1.3 applied to Xinv . ◦

(ii) ⇒ (iii) Note that K (X) is a basis of closed sets for Xinv . (iii) ⇒ (iv) Inversely closed sets are proconstructible. (iv) ⇒ (v) Set Z = Gen(Y ) and use the reminder at the beginning of the proof. con

(v) ⇒ (vi) It suffices to show Gen(Y max ) = Gen((Y )max ). The hypotheses and 4.1.2 imply that Y max = Z max and Y is quasi-compact. From 4.1.3 we conclude Gen(Y max ) = Y max Gen(Y As Y

inv

= Gen(Y

con

con

inv

) = Gen((Y

=Y

inv

,

con max

)

).

) by 1.5.5, we get the claim. con

(vi) ⇒ (i) Using 4.1.2 it follows from quasi-compactness of Y that con max con max =Y is quasi-compact and every point of Y , hence every point (Y ) of Y , specializes to some point in Y max . Therefore, Y is quasi-compact by 4.1.2.  1

Thus, the poset (X,  ) has the mub-property (= minimal upper bound property), which asserts that a finite subset with an upper bound has a minimal upper bound, cf. [GHK+ 03, Definition III-5.3, pp. 253–254].

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4.1.6 Corollary Let X be a spectral space and let Y ⊆ X be a subset. (i) The following conditions are equivalent: (a) Y is closed. (This can also be expressed by saying that Y is proconstructible and specialization-closed, see 1.5.4.) (b) Y is quasi-compact in the inverse topology and specialization-closed. (ii) The following conditions are equivalent: (a) Y is closed in Xinv (i.e., Y is proconstructible and generically closed). (b) Y is quasi-compact and generically closed. Proof It suffices to prove (i) since (ii) is the inverse version of (i). (a) ⇒ (b) holds trivially, and (b) ⇒ (a) follows from 4.1.3.  We conclude this section with several separation properties for disjoint subsets of a spectral space. The results supplement the consequences of the Separation Lemma 1.5.3 and will be useful in particular for the study of normal spectral spaces, Section 8.4. 4.1.7 Proposition Let X be a spectral space and let Y, Z ⊆ X be disjoint. (i) If Y is closed and Z is quasi-compact, then there is a closed constructible subset A of X with Y ⊆ A and A ∩ Z = ∅. (ii) If Y and Z are quasi-compact in the inverse topology and if {y} ∩ {z} = ∅ for all y ∈ Y and z ∈ Z, then there are disjoint sets A, B ∈ K(X) with Y ⊆ A and Z ⊆ B. (iii) If Y and Z are quasi-compact and if Gen(y) ∩ Gen(z) = ∅ for all y ∈ Y ◦

and z ∈ Z, then there are disjoint sets U,V ∈ K (X) with Y ⊆ U and Z ⊆ V. Proof (i) follows immediately from 1.5.3 with Y1 = Z and Y2 = Y , noting that closed sets are inversely quasi-compact. (ii) From 4.1.3 we have Y = Spez(Y ) and Z = Spez(Z). Thus, the hypothesis implies Y ∩ Z = ∅, and by (i) there is a closed constructible set A ⊆ X with Y ⊆ A and A ∩ Z = ∅. Applying (i) with Z and A we obtain the desired set B. (iii) is the inverse version of (ii).



4.2 Directed Subsets and Specialization Chains

107

4.1.8 Proposition Let Y ⊆ X be a quasi-compact subset of a spectral space. The following conditions are equivalent: (i) Y has a basis of closed neighborhoods (i.e., for every O ∈ O(X) containing Y there are A ∈ A(X) and O  ∈ O(X) such that Y ⊆ O  ⊆ A ⊆ O). inv (ii) The inverse closure Y is closed also for the spectral topology. (iii) If x ∈ Gen(Y ), then {x} ⊆ Gen(Y ). Proof

(i) ⇒ (ii) Let x ∈ Y

inv

. We need to show that x ∈ Y ◦

inv

◦

(i.e., if U ∈ K (X)

and Y ⊆ U, then x ∈ U). So, pick U ∈ K (X) with Y ⊆ U. Condition (i) gives A ∈ A(X) and O  ∈ O(X) with Y ⊆ O  ⊆ A ⊆ U. Since Y is quasi-compact, we may assume that O  is quasi-compact, hence closed for the inverse topology. Thus Y

inv

⊆ O  ⊆ A, and x ∈ Y

inv

⊆ A ⊆ U.

(ii) ⇒ (iii) Suppose that x ∈ Gen(Y ) = Y that {x} ⊆ Gen(Y ).

inv

, cf. 4.1.5. Condition (ii) implies

(iii) ⇒ (i) Suppose that O ∈ O(X) and Y ⊆ O. Since Y is quasi-compact, inv Y = Gen(Y ) ⊆ O is proconstructible (cf. 4.1.5). The sets Gen(Y ) and X \ O are disjoint. Condition (iii) says that there is no point x with specializations ◦

in both sets. So, by 4.1.7(iii), there are disjoint sets U, U  ∈ K (X) such that inv Y ⊆ U  and X \ O ⊆ U. Setting A = X \ U we obtain the inclusions  Y ⊆ U  ⊆ A ⊆ O. 4.1.9 Corollary A point y of a spectral space X has a basis of closed neighborhoods if and only if y ∈ X max and, for each x ∈ Gen(y), y is the unique specialization of x in X max . Proof

The assertion follows from 4.1.8(i) ⇔ (iii), setting Y = {y}.



4.2 Directed Subsets and Specialization Chains Summary We study up-directed and down-directed sets in the specialization poset of a spectral space. A special case are the specialization chains, cf. 4.1.1. The order-theoretic property of down-directedness is closely connected to the topological property of irreducibility, 4.2.1. Therefore, we start with some elementary facts about irreducible sets. The main result is Theorem 4.2.6, which says that the patch closure of an up-directed set (or a down-directed set, or a totally ordered set) is also updirected (or down-directed, or totally ordered). Up-directed sets have suprema and down-directed sets have infima. It follows that every spectral space is a

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dcpo, 4.2.7. Nonempty specialization chains have both suprema and infima. Every maximal specialization chain is proconstructible, hence has a minimal point and a maximal point, 4.2.11. In connection with 1.6.4 we obtain two order-theoretic conditions that are satisfied by the specialization poset of every spectral space, 4.2.12, and are called Kaplansky conditions, 4.2.13. In Section 4.3 we shall use specialization chains to define the rank and the Krull dimension of a spectral space. 4.2.1 Proposition Let X be a T0 -space. (i) If Y ⊆ X is down-directed for specialization (in particular, if Y is a chain), then Y is irreducible. (ii) If Y ⊆ X is irreducible and x ∈ X is a generic point for Y , then x is the infimum of Y for specialization. (iii) Let (Zi )i ∈I be a family of irreducible subsets of X that is up-directed under  inclusion. Then Z = i ∈I Zi is irreducible, as well. Proof (i) Assume that Y ⊆ A∪ B where A, B ∈ A(X) with Y  A and Y  B. Pick a ∈ Y \ A and b ∈ Y \ B. There is a common generalization c ∈ Y of a and b. We may assume that c ∈ A (since Y ⊆ A ∪ B). Closedness of A implies that a ∈ {c} ⊆ A, a contradiction. (ii) It is clear that x is a lower bound for Y . If z ∈ X is any lower bound then Y ⊆ {z}. Therefore {x} = Y ⊆ {z}, which implies z  x. (iii) Assume that Z ⊆ A ∪ B, where A, B ∈ A(X), Z  A, and Z  B. There are Zi  A and Z j  B. The family is up-directed under inclusion, hence there is some k such that Zi ∪ Z j ⊆ Zk . Irreducibility of Zk and Zk ⊆ A ∪ B imply Zk ⊆ A or Zk ⊆ B, hence Zi ∪ Z j ⊆ A or Zi ∪ Z j ⊆ B, a contradiction.  4.2.2 In Sober T0 -Spaces, Irreducible Sets Have Infima Let X be a sober T0 -space. A nonempty subset Y is irreducible if and only if Y is irreducible, see 1.1.4(b), and this is equivalent to the existence of a (unique) generic point in Y . We know from 4.2.1(ii) that the generic point of Y is the infimum of Y in X. It follows from 4.2.1(i) that every down-directed nonempty set has an infimum. Now assume that X is a spectral space. Then the generic point (= infimum) con of Y even belongs to Y , 1.5.5. 4.2.3 Example We have seen that down-directed sets in T0 -spaces are irreducible, 4.2.1. Sometimes the converse is also true, see 4.2.4, although it is not true in general, not even for spectral spaces. For an example we use the space N∞ of Section 1.6.D. The entire space is irreducible with generic point ∞. It follows from N = N∞ that N is an irreducible subset. But N is trivially ordered by specialization, hence is not down-directed.

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109

The same example also shows that a set may have an infimum without being irreducible. The subset {1, 2} ⊆ N∞ is not irreducible, but has the infimum ∞ with respect to specialization. 4.2.4 Proposition Let (X, ≤) be a poset, τ L the fine lower topology, see A.8(ii), and A ⊆ X a nonempty subset. Then A is irreducible for τ L if and only if A is down-directed. Proof If A is down-directed then the claim has been shown in 4.2.1(i). Conversely, let A be irreducible for τ L and a, b ∈ A. Then a ↓ ∩ A  ∅  b↓ ∩ A. Since a ↓ and b↓ are open for τ L , irreducibility of A entails a ↓ ∩ b↓ ∩ A  ∅ . Hence a, b have a common lower bound in A.  Now we restrict our attention to spectral spaces. Note that every general statement about down-directed subsets in spectral spaces translates into a statement about up-directed subsets, using the inverse topology. Sometimes we state the corresponding facts explicitly because of their importance, even though usually no extra proof is needed. 4.2.5 Lemma Suppose that X is a spectral space. Let (Zi )i ∈I be a family of nonempty proconstructible subsets that is down-directed under inclusion and  set Z = i ∈I Zi . Then: (i) Z is nonempty. (ii) If each Zi is irreducible, then Z is irreducible. Proof (i) The family (Zi )i ∈I has the FIP. Compactness of the patch topology implies that Z  ∅. (ii) Assume that Z is not irreducible (i.e., there are A, B ∈ K(X) with Z ⊆ A∪B, Z  A and Z  B). The family (Zi ∩ (X \ (A ∪ B)))i ∈I is down-directed under inclusion and its members are proconstructible. If they are all nonempty then the family has the FIP, hence  Zi ∩ (X \ (A ∪ B))  ∅, Z ∩ (X \ (A ∪ B)) = i ∈I

a contradiction. So there is some i ∈ I with Zi ⊆ A ∪ B, which implies  Z ⊆ Zi ⊆ A or Z ⊆ Zi ⊆ B (by irreducibility of Zi ). 4.2.6 Theorem Let X be a spectral space and let Y be a nonempty subset. con

(i) If Y is down-directed for specialization, then Y is down-directed as well. The set Y has an infimum in X, which is the generic point of Y and con belongs to Y .

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Subsets of Spectral Spaces con

(ii) If Y is up-directed for specialization, then Y is up-directed as well. The inv set Y has a supremum, which is the generic point of Y and belongs to con Y . con (iii) If Y is a specialization chain, then Y is a chain. The set Y has an infimum con ⊆ X, and a supremum in X, which in X, which is the generic point of Y con is the closed point of Y ⊆ X. Proof (i) The nonempty down-directed set Y is irreducible, 4.2.1(i). Then con Y is irreducible as well, and the generic point x belongs to Y , 1.5.5. Thus, con Y has a smallest element, hence is down-directed. The generic point is the infimum of Y , see 4.2.1(ii). (ii) is the inverse version of (i). con

is proconstructible, hence a spectral subspace of X. To see (iii) The set Y con con is a chain it suffices to show that any two elements of K(Y ) are that Y con comparable under inclusion, cf. 1.6.1(i). Assume that A, B ∈ K(Y ) are incomparable (i.e., A \ B and B \ A are constructible and nonempty). Then the sets (A \ B) ∩ Y and (B \ A) ∩ Y are both nonempty, which implies that A ∩ Y, B ∩ Y ∈ A(Y ) are incomparable, and Y is not a chain by 1.6.1(i).  4.2.7 Spectral Spaces are Dcpos It is an immediate consequence of 4.2.6 that every spectral space X is a dcpo (for the notion see A.6(xi)). Of course, Xinv is a dcpo as well (i.e., X is an fcpo, A.6(xiii)). The supremum of an up-directed nonempty subset of X, or of Xinv , is described in 4.2.6(ii), respectively 4.2.6(i). 4.2.8 Example Spectral spaces are usually not Dedekind-complete (cf. A.6(x)). A subset Y ⊆ X of a spectral space need not have a supremum, even if it has an upper bound. By 4.1.4 there are minimal upper bounds, but the point is that they need not be unique. Consider the spectral space with the following specialization relation (see 1.1.16): xO _

? yO

z

t.

The set {z, t} is bounded from above by x and y, but does not have a supremum. 4.2.9 Corollary Let f : X → Y be a spectral map and let D ⊆ X be nonempty and up-directed for specialization. Then f (D) is nonempty and up-directed in Y , and f (sup(D)) = sup( f (D)). Proof Clearly, f (D) is up-directed since f , being continuous, is monotonic for specialization. By 4.2.6(ii) there are a supremum x for D and a supremum

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111

y for f (D). Monotonicity of f implies y  f (x). It follows from 4.2.6(ii) that con con x ∈ D , hence f (x) ∈ f (D ). Continuity of f for the patch topology con con con implies f (D ) = f (D) . Since y is the unique maximal point of f (D) we  conclude that f (x)  y, 4.2.6(ii), hence f (x) = y. 4.2.10 Corollary Let Y be a subset of a spectral space X. (i) Y is the closure of a minimal point of X if and only if Y is down-directed under specialization and maximal with this property. (ii) Y is the set of generalizations of a maximal point of X if and only if Y is up-directed under specialization and maximal with this property. Proof (i) Obviously, the closure of a minimal point is maximally downdirected under specialization. Conversely, suppose that Y is down-directed under specialization and is maximal with this property. Then 4.2.6(i) shows con con that Y = Y and Y has a generic point x ∈ Y . Maximality and 4.1.4 imply x ∈ X min . 

(ii) is (i) for the inverse topology (or apply 4.2.6(ii)).

4.2.11 Corollary Let X be a spectral space and Y a maximal (under inclusion) specialization chain. Then Y is proconstructible, the generic point of Y is also generic in X, and the closed point of Y is also closed in X. con

Proof By maximality of Y and 4.2.6(iii) we know that Y = Y . Let x, y ∈ Y be the generic point and the closed point. There are a generic point x  ∈ X and a closed point y  ∈ X such that x   x and y  y  (see 4.1.2 and 4.1.4). If x  x  or y  y , then the chain Y can be enlarged, which contradicts maximality.  4.2.12 Corollary (Jump-density of the specialization order) Consider a spectral space X and its specialization poset (X,  ). Then: (i) Every nonempty specialization chain has a supremum and an infimum. (ii) (X,  ) is jump-dense (cf. A.5). y. Proof (i) has been shown in 4.2.6(iii). For (ii), pick x, y ∈ X with x   There is a maximal specialization chain Y passing through x and y. The chain has a jump between x and y, cf. 1.6.4(i), say x  x   y   y with x , y  ∈ Y .    z y , then z  Y , and Y ∪ {z} is a If there is an element z ∈ X such that x    longer chain, contradicting the maximality of Y .  4.2.13 The Kaplansky Problem The two properties of the poset (X,  ) shown in 4.2.12 were noted by Kaplansky, [Kap70, p. 6]. He raised the question whether every poset having these properties is the specialization poset of a spectral space. (In fact, Kaplansky posed the question in terms of prime spectra

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of rings. But in view of [Hoc69], also see Section 12.6, this is only a different form of the same question.) Lewis and Ohm exhibited a third necessary property of specialization orders of spectral spaces and gave an example of a topological space having all three properties without being spectral, [LeOh76, Example 2.1]. The additional property is: (H) Let F ⊆ P(X) be a subset of {Spez(x) | x ∈ X } or a subset of  {Gen(x) | x ∈ X }. If F has the FIP, then F  ∅. In fact, the example of Lewis and Ohm also satisfies the stronger condition (H + ) If F ⊆ {Spez(x) | x ∈ X } ∪ {Gen(x) | x ∈ X } has the FIP, then  F  ∅. For any poset (P, ≤) the set {x ↑ | x ∈ P} ∪ {y ↓ | y ∈ P} is a subbasis of the interval topology, A.8(iv). Therefore the Alexander Subbasis Theorem, [Kel75, p. 139], shows that (H + ) is equivalent to quasi-compactness of the interval topology. If X is a spectral space then the constructible topology is finer than the interval topology of (X,  ), 1.4.4, hence the interval topology is quasi-compact, and (H + ) holds. We present the example given by Lewis and Ohm, [LeOh76, Example 2.1], and show that it satisfies also condition (H + ). Let Y be a compact space that is · not Boolean, set X = Y ∪A(Y ) and equip X with the partial order whose only nontrivial relations are C < y for y ∈ C ∈ A(Y ). Then X satisfies conditions (i) and (ii) of 4.2.12 and condition (H), but is not the specialization poset of a spectral space, [LeOh76, Example 2.1]. We prove that (H + ) also holds. Pick a set F as in (H + ) satisfying the FIP. If F ⊆ {Spez(x) | x ∈ X }, or  F ⊆ {Gen(x) | x ∈ X }, then F  ∅ by (H). If F contains a singleton,  then clearly F  ∅ . Hence we may assume that F contains sets Gen(y) and Spez(C) for some y ∈ Y and some C ∈ A(Y ). But then Gen(y) ∩ Spez(C) ⊆  {y, C} is finite, and the FIP implies F  ∅ . We also note that every poset P = (P, ≤) with quasi-compact interval topology has the generalized mub-property of 4.1.4. To see this, let u ∈ P be an  upper bound for a set S ⊆ P. The set T = u ↓ ∩ s ∈S s ↑ is nonempty and closed for the interval topology of P, hence is quasi-compact for the interval topology and also for the coarse upper topology τ u (P). By 4.1.2 the set T contains closed points for τ u (P), which are maximal in T for the specialization order ≤inv of the coarse upper topology, hence are minimal in T for ≤. We see that Kaplansky’s original question has been answered in the negative, but the following challenge remains and is quoted as the Kaplansky Problem. Give a characterization of the partial orders that are specialization orders of spectral spaces. The problem is still open (despite various partial results). We

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113

return to the Kaplansky Problem in connection with Noetherian spaces (8.1.22) and with real spectra (13.6.2). 4.2.14 The Kaplansky Problem – When Does there Exist a Unique Spectral Topology with Given Specialization Relation? Continuing with the Kaplansky Problem, we consider a poset P = (P, ≤) and assume that there is a spectral topology τ with specialization order ≤. When is it true that there is only one such spectral topology? The question was studied in [Geh94]. In general there are several spectral topologies with the same specialization order, see 8.1.23 for an example. But there is a simple sufficient condition ensuring uniqueness, cf. [Geh94, p. 33 ff]. The interval topology, which we denote by τP , is coarser than the patch topology τcon , hence is quasi-compact, 1.4.4. Now assume that the interval topology is Hausdorff. Then the identity map (P, τ)con → (P, τP ) is a continuous bijective map between compact Hausdorff spaces, hence is a homeomorphism. Thus, the interval topology is Boolean and (P, τP ) is the patch space of (P, τ). It follows that (P, τ) is the unique spectral space with patch space (P, τP ) and specialization order ≤, 1.5.11, hence the unique spectral space with specialization order ≤. For example, assume that the partial order ≤ is total. Then one checks easily that the interval topology is Hausdorff. It follows that there is at most one spectral topology compatible with ≤. Thus, we recover the uniqueness statement in 1.6.4. Or, let P be an atomic Boolean algebra and pick x  y in P. We may assume that there is an atom a with a ≤ x and a∧ y = ⊥. Then P = a ↑ ∪(¬a)↓ is a partition into subsets that are closed for the interval topology. Hence they are both open, hence clopen. Moreover, x ∈ a ↑ and y ∈ (¬a)↓ . Thus, the interval topology is Hausdorff. In the case that P = P(S) for a set S, we recover the uniqueness result of 2.2.4(iv).

4.3 Rank and Dimension Summary In algebraic geometry the most fundamental invariant of a variety is its dimension. A non-singular variety V over C has a dimension as a complex manifold. This dimension is equal to the maximal length of specialization chains in the prime spectrum of the coordinate ring C[V] (see [Mum76, Section I.1A] and [Mat80, 14.G]). This algebraic description of the dimension of a variety inspired the definition of a dimension for arbitrary rings and spectral spaces. The Krull dimension, Kdim(X), of a nonempty spectral space X is the supremum in N0 ∪ {+∞}

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x1  ···  xn with n ∈ N. The of the lengths of specialization chains x0     Krull dimension of a ring A (as always, commutative and unital) is the Krull dimension of its Zariski spectrum and is denoted by Kdim(A). We start by introducing the Cantor–Bendixson rank of a topological space and its points, 4.3.1, and then use it to define the rank of a poset and its elements, 4.3.3. Thus, applying this definition to the specialization poset, every spectral space and all its points have a rank. The rank takes values in the class of ordinals or is equal to the symbol ∞, which is defined to be larger than all ordinals. Similar to the Krull dimension, the rank depends on the length and structure of specialization chains. If the Krull dimension is finite, then it coincides with the rank. But in the case of infinite Krull dimension the rank is more discriminating, as we explain in 4.3.8. The (transfinite) numerical invariants provided by the Cantor–Bendixson rank and the rank can be used to study the internal structure of a spectral space, which is particularly useful in the case of Noetherian spaces and Zariski spectra of Noetherian rings, see 8.1.25 f and 12.4.10 ff for details. In the present section we present basic properties of the Cantor–Bendixson rank on a topological space, as well as of the rank on a poset, and illustrate the notions with examples, 4.3.2 and 4.3.5. 4.3.1 Definition Let ∞ be an element larger than every ordinal. 2 For an arbitrary topological space X we write δX for the set of all non-isolated (i.e., accumulation) points of X. The set δX is called the Cantor–Bendixson derivative of X, cf. [Kop89, p. 274]. If δX = X  ∅ , then X is called perfect. For an ordinal α we define subsets δα X of X by transfinite recursion 3 on α: • δ0 X = X. • δ α+1 X = δδ α X.  • If α is a limit ordinal, then δ α X = β rk(x1 ) > β. We repeat the construction with x1 in place of x and obtain some x2 ∈ x1↑ with α > rk(x1 ) > rk(x2 ) > β. Iteration yields a properly decreasing sequence rk(x1 ) > rk(x2 ) >  rk(x3 ) > · · · of ordinals, a contradiction. (viii) Suppose X  ∅ is anti-well-ordered of order type λinv for an ordinal λ. Then a simple (transfinite) induction on α ∈ X shows that rk(α) = α. Hence Rk(X) = λ if λ is a limit ordinal and Rk(X) = β if λ = β + 1. This should also be compared with 4.3.2(vi). 4.3.6 Proposition For a poset X the following conditions are equivalent: (i) Rk(X) < ∞. (ii) X has the ACC (i.e., every nonempty chain has a maximum). Proof (i) ⇒ (ii) Assume there is a chain in X without maximum. Then the chain contains a strictly increasing sequence x1 < x2 < x3 < · · · . It follows from 4.3.5(vi) that rk(x1 ) = ∞, which implies Rk(X) = ∞ (by 4.3.2(ii) and 4.3.5(iv)). (ii) ⇒ (i) Suppose rk(x) = ∞ for some x ∈ X. Pick any ordinal α with card(α) > card(X). There is some x1 ∈ x ↑ \ {x} with rk(x1 ) ≥ α, 4.3.3. Now 4.3.2(xi) implies rk(x1 ) = ∞ (since card(rk(x1 )) = card(Rk(x1↑ )) > card(x1↑ )). Iterating this procedure we obtain a strictly increasing sequence x < x1 < x2 < · · · , which is a chain without maximum.  4.3.7 Examples (i) Let X be a spectral space. Then Rk(X) = 0 if and only if X is Boolean. (ii) The Sierpiński space 2 has rank 1, as does the space S∞ of 1.6.13. (iii) In Section 8.1 we study Noetherian spectral spaces. Their rank is never ∞, but every ordinal is the rank of a Noetherian spectral space, see 8.1.4(iii) and 4.3.5(viii). (iv) There are spectral spaces of rank ω that have no infinite chain. The construction refines the one used for the space S∞ , see 1.6.13. Consider the family (Sn )1≤n∈N of sets with Sn = {xn0, . . . , xnn }. We  define S = 1≤n∈N Sn and endow S with the discrete topology. Let S ∗ be the one-point compactification, which is a Boolean space. The limit point is denoted by ∞ again. A partial order is defined by xn0 < xn1 < · · · < xnn

118

Subsets of Spectral Spaces and ∞ < x for every x ∈ S. One checks easily that this is a spectral order, hence we obtain a spectral space X with patch space S ∗ and specialization order ≤, 1.5.11. Clearly, there is no infinite chain in X, but there are specialization chains of arbitrary finite length. Thus, Rk(X) = ω.

4.3.8 Krull Dimension vs. Rank We compare the Krull dimension of a spectral space X with its rank. First suppose that the Krull dimension is finite, say equal to n ∈ N. Every strictly ascending chain in (X,  ) has length at most n, and there is a chain with length n. The same is true about the descending chains, and the definition of the rank shows that Rk(X) = n. Now suppose that the Krull dimension is ∞ (i.e., there are finite ascending specialization chains of arbitrary length). Then there are also finite descending chains of arbitrary finite length, which implies that Rk(X) ≥ ω. It may happen that (X,  ) has the ACC and the DCC, cf. 4.3.7(iv), or that (X,  ) has the ACC, but not the DCC, cf. 4.3.5(viii), or that X has the DCC, but not the ACC, cf. 1.6.6, or that X has both infinite ascending and descending chains (e.g., the spectrum of the lattice R (extended real line), cf. Section 3.6). The Krull dimension is always ∞, but the rank differs widely, hence is more discriminating.

4.4 Minimal Points and Maximal Points Summary We study the set of maximal points in a spectral space X, denoted by X max , and the set of minimal points, denoted by X min , cf. 1.1.6. There are plenty of maximal and minimal points since it has been shown in 4.1.2 and 4.1.4 that Spez(x) ∩ X max and Gen(x) ∩ X min are both nonempty for every point x ∈ X. From previous results we know much more about X max than about X min . Therefore we focus our attention primarily on X min . But as a side benefit we also gain additional information about X max (using inverse spaces). For X max , a summary of the previously established properties and an outlook are given in 4.4.2. Following a characterization of the points of X belonging to X min , 4.4.4, we analyze properties of the topology of X min and the relationship between X min and the ambient space X. The restrictions of the spectral topology and the patch topology coincide on X min , 4.4.6, and are completely regular and totally disconnected, 4.4.8. The space X min is compact if and only if it is a spectral subspace of X, 4.4.16. The patch closure of X min is described in 4.4.19. In 4.4.11 we present a characterization (originally proved by [Hoc71]) of the

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119

topological spaces that are minspectral (i.e., are homeomorphic to X min for some spectral space X). 4.4.1 Minimal and Maximal Points in a Spectral Space and its Inverse It is clear from 1.4.3(iii) that X min = (Xinv )max and X max = (Xinv )min . But at this point a warning is in order: these equalities are only identities of sets, not of topological spaces! The relationship between the different topologies will be analyzed below. Despite the warning, these identities entail that every general statement about the spaces X max and X min corresponds to an “ inverse statement” about the spaces X min and X max . For example, a statement about the spectral topology restricted to X max that holds for every spectral space is also true for the inverse topology restricted to X min . Recall that the maximal points of a proconstructible subset Y ⊆ X determine the inverse closure of Y , and the minimal points determine the closure, 4.1.5 and 4.1.3. If Y is any subset of X, then con min

min

=Y

inv max

= (Y

Y

= Spez(Y

con min

)

= (Y

con min

)

and (Y

)

con inv max

)

= Gen(Y

con max

)

= (Y

con max

)

.

4.4.2 The Space of Maximal Points in a Spectral Space Let X be a spectral space. Many of the main properties of the space X max have been proved already, which we summarize here together with a preview of things to come (in 4.4.3, 4.4.9, and 4.4.17). (i) The space X max is quasi-compact and every point of X specializes to a maximal point, see 4.1.2. (ii) The subspace X max ⊆ X is T1 . For, recall that the T1 -property is equivalent to the statement that every point is closed, [Eng89, p. 37]. By 1.1.3, a point x ∈ X is maximal for specialization if and only if {x} ∈ A(X), if and only if {x} ∈ A(X max ). (iii) We shall see later that every quasi-compact T1 -space is the space of maximal points of a suitable spectral space, cf. 11.4.6; also see [Hoc69, Proposition 11]. (iv) The restriction of the constructible topology to X max is trivially finer than the restriction of the spectral topology. We shall see below that the restriction of the inverse topology to X max coincides with the restriction of the constructible topology, 4.4.9(i). The meaning of quasi-compactness of X max for the inverse topology is described in 4.4.17.

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(v) By Stone duality, the map ◦

◦

x|◦ Λ X : X → Spec(K (X)), x → (X) K

◦

is a homeomorphism, 3.2.8. The prime ideal spectrum PrimI(K (X)) is ◦

homeomorphic to Spec(K (X)) by definition, 3.3.1. Explicitly, the home◦

◦

omorphism Ker I ◦ Λ X : X → PrimI(K (X)) sends a point x to the prime ideal ◦

i(X \ {x}) = {U ∈ K (X) | x  U}, see 1.1.11. The maximal points correspond to the maximal ideals. Lattice◦

theoretic characterizations of maximal ideals in K (X) are given in 4.4.3. ◦

4.4.3 Proposition Let p ⊆ K (X) be a prime ideal and let x ∈ X be the corresponding point. The following are equivalent: ◦

(i) x is a maximal point (i.e., p is a maximal ideal of K (X)). ◦

(ii) The ideal generated by p in the Boolean envelope ba(K (X)) = K(X) is prime. (iii) The set p generates K(X) as a Boolean algebra. (iv) The set X  ∪ (X × {x}) is a spectral order on Xcon , and ◦

◦

K (Y ) = p ∪ {X } ⊆ K (X), where Y is the corresponding spectral space. Proof (i) ⇔ (ii) Let i be the ideal generated by p in the Boolean envelope. In the Boolean algebra K(X) every prime ideal is maximal, and every ideal is the intersection of the maximal ideals that contain it, 3.2.2. The maximal ideals of K(X) containing i correspond to the points of con−1 X ({x}), where conX : Xcon → X is the canonical map. Thus, i is prime if and only if {x} contains only one point, if and only if x is a maximal point. (i) ⇒ (iv) Since x is a maximal point it is a routine exercise to show that R := X  ∪ (X × {x}) is a partial order of X. The clopen subsets of Xcon that are down-sets for this relation are precisely the elements of p ∪ {X }. Therefore, the assertion follows from 1.5.11 if R is a spectral order. To prove this, pick (y, z)  R. Then y   z and x   z (since z  x and x is maximal). We use spectrality of the specialization order  (cf. 1.5.8) to find some U ∈ p with ◦

z ∈ U and some V ∈ K (X) with z ∈ V and y  V. It follows that U ∩ V ∈ p and y  U ∩ V.

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121

◦

(iv) ⇒ (iii) Since Y has patch space Xcon , we know that K (Y ) generates K(X) ◦

as a Boolean algebra, 1.3.15. As K (Y ) = p ∪ {X } we obtain (iii). (iii) ⇒ (i) Assume there is some y with x  y. Then x, y  U for all U ∈ p,  hence no Boolean combination of sets from p can separate x from y. But there is an element in K(X) separating x from y, and we conclude that p does not generate K(X) as a Boolean algebra.  Now we turn to the space of minimal points in a spectral space X. The minimality of a point is an order-theoretic property. The first task is to give a topological characterization. This can be done using the interior of subsets of X. The following notation will be used. Given a topological space X and a subset Y ⊆ X, the set intX (Y ) is the interior of Y in X (i.e., the largest open subset of X contained in Y ). 4.4.4 Proposition Let X be a spectral space. The following conditions about a point x ∈ X are equivalent: (i) x ∈ X min . (ii) If x ∈ V ∈ K(X), then x ∈ intX (V) (i.e., the point x is interior to any closed constructible set containing it). (iii) If x ∈ Z ∈ O(Xcon ), then x ∈ intX (Z) (i.e., the point x is interior (in X) to any constructibly open set containing it). ◦

Proof (i) ⇒ (iii) If x  intX (Z), then U ∩ (X \ Z)  ∅ for all U ∈ K (X) containing x. Since Xcon is compact and X \ Z is proconstructible, we get  ◦ (X \ Z) ∩ {U ∈ K (X) | x ∈ U}  ∅ . Any element in this set is a proper generalization of x, contrary to (i). (iii) ⇒ (ii) is clear since K(X) ⊆ O(Xcon ). (ii) ⇒ (i) We pick y ∈ Gen(x) and claim that y = x. It suffices to show that y ∈ V for all V ∈ K(X) containing x. But if x ∈ V ∈ K(X) then x ∈ intX (V),  by hypothesis, hence y ∈ Gen(x) ⊆ intX (V) ⊆ V. 4.4.5 Corollary Let X be a spectral space. The following conditions about a point x ∈ X are equivalent: (i) x is isolated in X (i.e., the singleton {x} is open). (ii) x is isolated in Xcon and belongs to X min .



We begin with the study of X min as a subspace of the spectral space X. In particular, we study topological properties of the minimal spectrum and explain

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how X min sits inside X. Following Hochster, cf. [Hoc71, Introduction], we call a topological space minspectral if it is homeomorphic to X min for some spectral space X. First there are a few facts that follow immediately from the previous results. 4.4.6 Corollary subset. Then:

Let X be a spectral space and let Y ⊆ X be an arbitrary

(i) intXcon (Y ) ∩ X min = intX (Y ) ∩ X min . Consequently, the spectral topology and the constructible topology of X restrict to the same topology on X min . (ii) If X min ⊆ Y , then Y is dense in X. (iii) If Y is proconstructible and dense in X, then X min ⊆ Y . (iv) If Y is open in X, then the closure of Y ∩ X min in X min is Y ∩ X min . (v) Suppose that Y ∈ O(Xcon ). Then X min ∩ Y = ∅ if and only if intX (Y ) = ∅. con (vi) The set X \ X min is the largest set Z ∈ O(Xcon ) with intX (Z) = ∅. Proof (i) The inclusion intXcon (Y ) ∩ X min ⊇ intX (Y ) ∩ X min holds trivially; the other one follows from 4.4.4(i) ⇒ (iii). (ii) The hypothesis and 4.1.4 imply that X = Spez(X min ) ⊆ Spez(Y ) ⊆ Y . (iii) Clearly, if X min ⊆ Spez(Y ) then X min ⊆ Y . So, the assertion follows from the inclusion X min ⊆ X = Y = Spez(Y ), which holds by hypothesis and 4.1.3. (iv) Let Z be the closure of Y ∩X min in X min . The inclusion Y ∩X min ⊇ Z is clear. For the other inclusion let x ∈ Y ∩ X min . We have to show U ∩ (Y ∩ X min )  ∅ for each open neighborhood U of x. But x ∈ Y implies U ∩ Y  ∅ , and, as Y is open, we get U ∩ Y ∩ X min  ∅ . (v) follows from (ii), (iii), and 4.4.4 applied with X \ Y . (vi) is immediate from (v).



4.4.7 Completely Regular Spaces and the Stone–Čech Compactification Completely regular spaces appear in various places in the book. Here they hit the stage for the first time. Therefore we give a short explanation and mention the Stone–Čech compactification, which will also be used repeatedly. A topological space X is said to be completely regular, or a Tychonoff space, if it is T1 and satisfies the following separation condition. Given a closed set A ⊆ X and a point x ∈ X \ A there is a continuous function f : X → R with x ∈ f −1 (1) and A ⊆ f −1 (0), [Eng89, p. 39], [GiJe60, Chapter 3]. Note that a completely regular space is clearly Hausdorff. Every 0dimensional space (which means that there is a basis of clopen sets, [Eng89, p. 360]) is completely regular (and also totally disconnected). Trivially, subspaces of completely regular spaces are completely regular. Products of completely regular spaces are completely regular, [GiJe60, p. 42]. Every normal T1 -space,

4.4 Minimal Points and Maximal Points

123

hence every compact space, is obviously completely regular. Thus, all subspaces of compact spaces are completely regular. It is a remarkable fact that the converse of this statement is also true (i.e., completely regular spaces are exactly the subspaces of compact spaces). More precisely, for each completely regular space X there is an embedding βX : X → βX into a compact space such that the following mapping property holds: if f : X → Y is any continuous map to a compact space then there is a unique continuous map g : βX → Y with f = g ◦ βX , see [GiJe60, Chapter 6] and 8.4.16. The space βX together with the embedding βX is the Stone–Čech compactification of X. We describe the construction in 8.4.15. 4.4.8 Corollary Let X be a spectral space. Then X min is 0-dimensional, hence completely regular and totally disconnected. Proof By 4.4.6(i), X and Xcon induce the same topology on X min . Thus, all  properties are inherited from the Boolean space Xcon . We interrupt the study of the minimal spectrum briefly to record a few facts about X max that can be derived from 4.4.6(i) and 4.4.8 using the inverse topology. 4.4.9 Corollary Let X be a spectral space. Then: (i) The inverse topology and the constructible topology of X restrict to the same topology on X max . (ii) X max is completely regular for the inverse topology. (iii) X max with the inverse topology has a basis of clopen sets and is totally disconnected.  4.4.10 Properties of Minspectral Spaces The next main result is a characterization of minspectral spaces discovered by Hochster, [Hoc71]. The following comments motivate the conditions appearing in the characterization. Suppose that Y is a spectral space. The previous considerations yield the following properties of Y min : (a) Y min is Hausdorff.

◦

(b) The sets Y min ∩U with U ∈ K (Y ) are a basis of open sets for Y min and also a basis of closed sets for Y min with the restriction of the inverse topology. The restriction of the inverse topology is quasi-compact by 4.4.1 and 4.4.2(i).

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Note the following formulation of the Alexander Subbasis Theorem, [Kel75, p. 139, Theorem 6], in terms of closed sets: a topological space Z is quasicompact if and only if there is a subbasis S of closed sets such that every subset C ⊆ S with the FIP has nonempty intersection. 4.4.11 Theorem (cf. [Hoc71, Proposition 1 and Theorem 1]) The following statements about a T1 -space X are equivalent: (i) X is minspectral. (ii) There is a bounded sublattice L ⊆ O(X) which is a basis of the topology  and has the following property: If C ⊆ L has the FIP then C  ∅. (iii) There is a subbasis S of open sets with the following property: if C ⊆ S  has the FIP then C  ∅. If the equivalent conditions hold, let L be any sublattice as in (ii) and set Y = PrimI(L). Then f : X → Y, f (x) = {U ∈ L | x  U} is a homeomorphism onto Y min and induces an isomorphism ◦

L → {V ∩ Y min | V ∈ K (Y )}. The subspace Y min ⊆ Y is dense for the patch topology. Proof The implication (ii) ⇒ (iii) is trivial and (i) ⇒ (ii) is clear from 4.4.10 ◦

(choose L = {U ∩ X | U ∈ K (Z)}, where Z is spectral and X = Z min ). (iii) ⇒ (ii) We define L to be the bounded sublattice of P(X) generated by S; thus L is a basis of open sets. Now define τ to be the topology that has S as a subbasis of closed sets. The Alexander Subbasis Theorem, cf. 4.4.10, says that (X, τ) is quasi-compact. Since L is a basis of closed sets for τ it follows that condition (ii) is satisfied. The assertion following the equivalence clearly implies (i). Therefore, to finish the proof it suffices to show that (ii) implies this assertion. To start with, note that f (x) = {U ∈ L | x  U} is obviously a prime ideal of L. Claim For every minimal prime ideal i of L there is a unique x ∈ X with f (x) = i. Proof of Claim As L is a basis of X, uniqueness follows from the hypothesis that X is T1 . To find x ∈ X as claimed, set C = L \ i, which is a prime filter in  L, hence has the FIP. By (ii) there is some x ∈ C. Clearly f (x) ⊆ i and by minimality of i we get f (x) = i as required for the claim.  The claim also implies that each f (x) is a minimal prime ideal of L. Pick a

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minimal prime ideal i ⊆ L contained in f (x) and choose some y ∈ X with f (y) = i as in the claim. Thus f (y) ⊆ f (x). Again, as L is a basis of X and X is T1 , we get x = y, hence f (x) = f (y) = i is minimal. So far it has been shown that f is a bijection onto the set of minimal prime ideals of L. For U ∈ L and x ∈ X the equivalence f (x) ∈ D(U) ⇐⇒ U  f (x) ⇐⇒ x ∈ U ⇐⇒ f (x) ∈ f (U) implies that f (U) = D(U) ∩ f (X) = D(U) ∩ Y min (i.e., f is a homeomorphism onto its image and f induces a lattice isomorphism as claimed). For the final assertion we must show that every nonempty basic constructible set D(U) \ D(V), with U,V ∈ L, contains an element f (x). By 3.3.3 the map ◦

D : L → K (Y ), U → D(U) is an isomorphism, hence extends to an isomor◦ phism ba(D) : ba(L) → ba(K (Y )) = K(Y ), which takes a basic constructible set U \ V to D(U) \ D(V), 3.4.1. But then D(U) \ D(V)  ∅ implies U \ V  ∅ , and f (x) ∈ D(U) \ D(V) for every point x ∈ U \ V.  4.4.12 The Homeomorphism Type of Spectral Spaces with Given Minimal Spectrum We consider the question whether the spectral space Y constructed in 4.4.11 is uniquely determined by the minspectral space X (up to homeomorphism). Thus, assume that there are two lattices L1 and L2 as in 4.4.11(ii) and let Y1 and Y2 be their prime ideal spectra. Then, is it true that L1 and L2 are isomorphic? The following example shows that the answer is negative in general. Let X = N with the discrete topology, let L1 = {N} ∪ {F ⊆ N | F is finite}, and let L2 be the lattice of closed sets of any compact Hausdorff topology on X (e.g., consider N as the one-point compactification of N \ {1} with 1 as the point at infinity). Then L1 and L2 both satisfy the conditions of 4.4.11(ii), but are clearly not isomorphic as bounded distributive lattices. 4.4.13 Examples Using 4.4.11 one can check for various spaces whether they are minspectral. We cite a few examples from [Hoc71]. (i) Let X be a complete metric space that satisfies the ultrametric triangle inequality. 4 Then X is minspectral by [Hoc71, Proposition 4]. A subbasis with condition (iii) of 4.4.11 is given by the open balls. (ii) Arbitrary products and arbitrary topological sums of minspectral spaces are minspectral by [Hoc71, Proposition 5]. In particular, discrete spaces are minspectral. (iii) The set of irrational numbers with the topology inherited from R is homeomorphic to a countable infinite product of countable discrete spaces, 4

For the definition and basic properties of ultrametric spaces, see [Rib96].

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hence is minspectral (cf. [Hoc71, bottom of p. 753]). Notice also that this space is complete metrizable as it is a G δ -set (i.e., a countable intersection of open subsets of R), and any G δ -set in a complete metric space has a complete metric. (iv) Every metrizable minspectral space has a metric in which it is complete, cf. [Hoc71, bottom of p. 753]. As the Baire category theorem holds for complete metric spaces (see, e.g., [GiJe60, 16.25]), but fails for Q, the 0-dimensional space Q is not minspectral. (v) According to [Hoc71, bottom of p. 753] the space Q can be embedded as a closed subspace into 2ℵ0 copies of the irrationals. Hence, closed subspaces of minspectral spaces are in general not minspectral. Example 4.4.13(iv) implies that the Baire Category Theorem holds in every metrizable minspectral space. In fact, the Baire theorem holds in all minspectral spaces. Recall that a space has the Baire property if every countable intersection of open and dense subsets is dense. By [GL13, Proposition 8.3.24, p. 373], every locally quasi-compact and sober space has the Baire property. In particular, spectral spaces have the Baire property. More generally: 4.4.14 Baire Category Theorem Every inversely quasi-compact subset of a spectral space X has the Baire property. In particular, every subspace of X containing X min satisfies the Baire property. Proof Let Y ⊆ X be inversely quasi-compact. Let U1, U2, ... ⊆ Y be open and dense in Y . Choose Oi ⊆ X open in X with Ui = Oi ∩ Y . Note that, in every topological space, finite intersections of open and dense sets are again open and dense. Hence we may replace Oi by O1 ∩ · · · ∩ Oi and Ui by U1 ∩ · · · ∩ Ui and assume that O1 ⊇ O2 ⊇ · · · . Let O be an open subset of X with O ∩ Y  ∅ . It suffices to show that  O ∩ Y ∩ n On  ∅ . Claim There is a sequence of quasi-compact open subsets Vn ⊆ X with (a) (b) (c) (d)

V1 ⊆ O, Vn ⊆ On , Vn ∩ Y  ∅ , and Vn ⊇ Vn+1 .

Proof of Claim The construction is recursive. To start with, choose V1 ⊆ O1 ∩ O quasi-compact open such that V1 ∩ Y  ∅ . This is possible, since O1 ∩ O ∩ Y = O ∩ U1  ∅ (as O ∩ Y  ∅ and U1 is dense in Y ). Now suppose

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V1, . . . ,Vn with (a)–(d) have already been defined. Since On+1 ∩ Y = Un+1 is dense in Y and Vn ∩Y  ∅ we have Vn ∩On+1 ∩Y  ∅ . Choose a quasi-compact open subset Vn+1 of Vn ∩On+1 with Vn+1 ∩Y  ∅ . This finishes the construction  of the Vn . Since Y is inversely quasi-compact, the Vn are inversely closed, and Vn ∩Y  ∅ ,  we see that n∈N Vn ∩ Y is nonempty. Any point in this intersection belongs to  O ∩ Y ∩ n On  ∅ .  4.4.15 Remark Recall that by condition (vi) of 4.4.6, every spectral space X has a largest constructibly open subset with empty interior, namely X \ con X min . It follows that an arbitrary union of closed and constructible sets with empty interior also has empty interior. The requirement of constructibility here is essential: every Boolean space without isolated points is the union of its singleton subsets, which are all closed with empty interior. 4.4.16 Proposition equivalent:

Let X be a spectral space. The following conditions are

(i) X min is quasi-compact. (ii) X min is Boolean. (iii) X min is proconstructible, hence is a spectral subspace. ◦  (iv) X min = {U ∈ K (X) | U is dense in X }. (v) The spectral topology and the inverse topology of X restrict to the same topology on X min . Proof (i) ⇒ (ii) is clear since X min is always Hausdorff and totally disconnected. (ii) ⇒ (iii) Condition (ii) says that X min is a spectral space. Moreover, the inclusion map X min → X is a spectral map by 4.4.6(i) and 1.3.21. We know that the image of a spectral map is proconstructible, 1.3.23. For (iii) ⇒ (i), just not that proconstructible subsets are quasi-compact. (i) ⇒ (iv) It is always true that X min = Gen(X min ). Therefore 4.1.5(iii) implies ◦  that X min = {U ∈ K (X) | X min ⊆ U}. The claim follows since a set ◦

U ∈ K (X) is dense if and only if it contains X min , see 4.4.6(ii) and (iii). (iv) ⇒ (v) By 4.4.6(i) we know that O(X)|X min = O(Xcon )|X min ⊇ O(Xinv )|X min . ◦

Pick U ∈ K (X). We must show that X min ∩ U ∈ O(Xinv )|X min . As X \ U ∈ K(X) it follows that X min ∩ (X \ U) = X min ∩ intX (X \ U), see 4.4.6(i). Since X min ⊆ U ∪ intX (X \ U) and X \ (U ∪ intX (X \ U)) is closed, the hypothesis yields

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some U  ∈ K (X) with X min ⊆ U  ⊆ U ∪ intX (X \ U). Then U  ∩ intX (X \ U) = U  ∩ (X \ U) is open and constructible. It follows that X min ∩ U = X min ∩ (X \ (U  ∩ (X \ U))) ∈ O(Xinv )|X min . (v) ⇒ (i) By hypothesis, the spectral topology and the inverse topology restrict to the same topology on X min . Now just note that X min = (Xinv )max is quasicompact for the inverse topology, see 4.4.2(i).  4.4.17 Corollary equivalent: (i) (ii) (iii) (iv) (v)

Let X be a spectral space. The following conditions are

X max is quasi-compact for the inverse topology. X max is Boolean for the inverse topology. X max is proconstructible.  X max = {V ∈ K(X) | X max ⊆ V }. The spectral topology and the inverse topology of X restrict to the same topology on X max .

Proof



This is the statement of 4.4.16 for Xinv .

4.4.18 Example Let X be a spectral space. By 4.1.5 the set X min is inversely closed if it is proconstructible. Similarly, X max is a closed subset if it is proconstructible, cf. 4.1.3. Now one may ask whether the constructible closure of X min is always inversely closed and the constructible closure of X max is always closed. These questions are inverses of each other. We show with an example that the general answer is “no” for X max , hence also for X min . Let A be the ring C([0, 1], R) of real-valued continuous functions on the unit interval and let X = Spec(A). Then X max is dense, but not constructibly dense in X (see [ScTr10, Example 8.8]) – the following nonempty constructible set is disjoint from X max : Y = V( f ) \ V(e

−

1 f2

),

where f : [0, 1] → R is the inclusion and, for any function g, V(g) = {p ∈ X | con g ∈ p}. Thus, X max ⊆ X \ Y  X = X max . More generally, let Z be any completely regular topological space. The space Z may be identified with a subspace of Spec(C(Z, R))max , which is clearly con is the set of prime z-ideals of C(Z, R), dense in Spec(C(Z, R)). Then Z [Sch97, Theorem 3.2]. If there are any proper specializations in Spec(C(Z, R)), then the set of prime z-ideals is a proper subset of Spec(C(Z, R)), [GiJe60, 14.13].

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In 8.4.13 and 8.4.14 we study the question of what it means that X max is Boolean in a normal spectral space. 4.4.19 Proposition Let X be a spectral space and let x ∈ X. (ii) (iii)

con

if and only if intX (C)  ∅ for every C ∈ K(X) with x ∈ C. inv min x∈X if and only if intX (V)  ∅ for each V ∈ K(X) with x ∈ V. con inv con min X = X min (i.e., X min is inversely closed) if and only if for every

(i) x ∈ X min

C ∈ K(X) with intX (C) = ∅ there is a set V ∈ K(X) such that C ⊆ V and intX (V) = ∅. Proof Suppose that C ∈ K(X). Then 4.4.6(v) says that X min ∩ C = ∅ if and only if intX (C) = ∅ . con

if and only if X min ∩ C  ∅ for each (i) The point x belongs to X min C ∈ K(X) with x ∈ C, if and only if intX (C)  ∅ for each such C. (ii) is proved similarly, replacing C ∈ K(X) by V ∈ K(X). con

(iii) The set X \ X min is inversely open if and only if for every C ∈ K(X), disjoint from X min (i.e., with intX (C) = ∅ ), there is some V ∈ K(X) with C ⊆ V and X min ∩ V = ∅ (i.e., with intX (V) = ∅ ).  4.4.20 Corollary If X is a spectral space, the following are equivalent: (i) X min is constructibly dense. (ii) Every open and quasi-compact set U is regular open. 5 ◦

Proof (i) ⇒ (ii) Let U ∈ K (X) and suppose U  int(U). Then int(U) \ U is nonempty and constructibly open. By (i) and 4.4.19(i), int(U)\U has nonempty interior, which is impossible. ◦

(ii) ⇒ (i) Let U ∩ V ⊆ X be nonempty and basic constructible with U ∈ K (X) and V ∈ K(X). By 4.4.19(i) it suffices to show that U ∩V has nonempty interior. By (ii), X \ V is regular open, hence V is regular closed and U ∩ int(V) = U ∩ V  ∅ . It follows that ∅  U ∩ int(V) ⊆ int(U ∩ V).



4.4.21 Open and Closed Regularization in Spectral Spaces In spectral spaces, as in most other topological spaces, open sets need not be regular open and closed sets need not be regular closed. Let X be a topological space. For each O ∈ O(X) the set N(O) = int(O) is the open regularization of O, 5

Recall that a subset C of a topological space is regular open (or regular closed, cf. [Kop89, p. 25, p. 28]) if C is the interior of its closure (or if C is the closure of its interior). The complement of a regular open set is regular closed, and vice versa.

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the smallest regular open subset containing O, [Kop89, p. 25]. Similarly, for A ∈ A(X), the set N(A) = int(A) is the closed regularization of A. This is the largest regular closed set contained in A. For each O ∈ O(X) we have X \ N(O) = N(X \ O). The subset RO(X) ⊆ O(X) of regular open sets is a complete Boolean algebra, [Kop89, Theorem 1.37]. Similarly, RC(X) ⊆ A(X), the set of regular closed sets, is a complete Boolean algebra. Via complementation the Boolean algebras RO(X) and RC(X) are anti-isomorphic to each other. However, note that they are not sublattices of O(X) and A(X), respectively. The Boolean operations in RO(X) are described in [Kop89, Theorem 1.37]. The Boolean operations of RC(X) can be derived from those of RO(X) using complementation. The maps N : O(X) → RO(X) and N : A(X) → RC(X) are homomorphisms of bounded lattices. But if they are considered as maps into O(X) and A(X) then they are not lattice homomorphisms. (We mention that N : O(X) → O(X) is a nucleus of the frame O(X), see 9.1.1 and 9.1.2. More on this map in 9.4.16(ii).) Now we assume that X is a spectral space and we consider the restrictions ◦

N : K (X) → RO(X) and N : K(X) → RC(X) of N and N, which are homomorphisms of bounded lattices. Their images are denoted by L and L. These are bounded sublattices of RO(X) and RC(X), but not of O(X) and A(X). The antiisomorphism RO(X) → RC(X), O → X \ O restricts to an anti-isomorphism L → L. It is important to note that N(U) and N(A) need not be constructible if U and A are constructible. (But in Section 8.3 we shall see an important class of spectral spaces, the semi-Heyting spaces, where these sets are constructible, 8.3.3.) The spectral topology and the constructible topology of X restrict to the same topology on X min , 4.4.6(i). Thus, for A ∈ K(X) the set A ∩ X min is open in X min and there is a largest open set O ⊆ X with O ∩ X min = A ∩ X min . It is clear that O = int(A) (i.e., N(A) = int(A) = Spez(A ∩ X min )). It follows that A ∈ K(X) is regular closed if and only if A = Spez(A ∩ X min ). Let Z ⊆ X be the constructible closure of X min and i : Z → X the inclusion. If A ∈ K(X) then A∩Z ∈ K(Z) , and N(A) = A ∩ Z = Spez(A∩Z). It follows that N(A)∩ Z = A∩ Z. We claim that the restriction map ρ : L → K(Z), B → B ∩ Z is an isomorphism. It is clear that ρ is a homomorphism of bounded lattices. For injectivity, consider B, B  ∈ L with B ∩ Z = ρ(B) = ρ(B ) = B  ∩ Z. But then B = Spez(B ∩ Z) = Spez(B  ∩ Z) = B . For surjective, pick C ∈ K(Z) and A ∈ K(X) with C = A ∩ Z. Then C = N(A) ∩ Z = ρ(N(A)). The complementation map can be used to transfer these considerations to the

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◦

open regularization map. In particular, the map ρ : L → K (Z) : O → O ∩ Z is an isomorphism. The following diagrams exhibit the various lattices and maps of this discussion. The solid arrows are lattice homomorphisms, whereas the dashed arrows are poset homomorphisms: A(X) O

N

⊆

⊆

K(X) K(i)

/ RC(X) O

N

⊆ ⊆

/ A(X) ;

⊆ N

K (X) ◦

K (i) ◦



/ RO(X) O ⊆

◦

/L

ρ

 | K(Z)

N

O(X) O

⊆ ⊆

/ O(X) ;

/L

ρ

 {



K (Z). ◦

Note that L is a Boolean algebra if and only if K (Z) is a Boolean algebra, if and only if Z is Boolean (i.e., Z = X min and X min is proconstructible), if and only if K(Z) is a Boolean algebra, if and only if L is a Boolean algebra. Then ◦

the Boolean algebras K (Z) and K(Z) coincide. Assume that X min is proconstructible. Let X min = U ∪ V be a partition into ◦

clopen subspaces and pick U ,V  ∈ K (X) with U  ∩ X min = U, V  ∩ X min = V. X Then U = U  = X \ N(V ) is regular closed.

4.5 Convexity and Locally Closed Sets and Points Summary Convexity is an important condition for subsets of a poset. (For terminology and notation on convexity in posets, see A.3(viii) and (ix).) Every subset of a poset is contained in a smallest convex subset, its convex hull. In T0 -spaces, hence in every spectral space, the specialization order provides a notion of convex subsets. We interpret convexity in terms of the classical topological notion of local closedness. A constructible set in a spectral space is convex if and only if it is locally closed, if and only if it is a basic constructible set, 4.5.6(ii). In general, proconstructible convex sets are not locally closed, but they are always intersections of locally closed sets, 4.5.6(i). The convex hull of a proconstructible subset is the intersection of all locally closed constructible sets that contain it. A point x ∈ X is locally closed if the corresponding singleton subset is locally closed. In a T1 -space every point is closed, hence locally closed. But

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most spectral spaces contain points that are not locally closed. In 4.5.17 we give a characterization of locally closed points in a spectral space X. The set of locally closed points is the smallest very dense subset, 4.5.21 (i.e., the smallest subset having nonempty intersection with every nonempty locally closed set). A subset Y ⊆ X is very dense if and only if the inclusion Y → X induces an isomorphism O(X) → O(Y ), 4.5.23. (The results about locally closed points and very dense subsets are from [GrDi71, §0.2.6 and §0.2.7] and [BEPS03].) 4.5.1 Convexity in T0 -Spaces A subset S ⊆ Z of a T0 -space is convex if it is convex for the specialization order, cf. A.3(viii). If S ⊆ X is any subset then the convex hull is conv(S) = Spez(S) ∩ Gen(S). For each continuous map f : X → Y between T0 -spaces, inverse images of convex subsets of Y are convex in X (since f is monotonic for the specialization order). 4.5.2 Locally Closed Sets in a Topological Space Local closedness is a classical condition about subsets of topological spaces, cf. [Bou71b, §3 No. 3]. We briefly mention the main facts that we need. Let X be a topological space. A subset C ⊆ X is said to be locally closed if every element x ∈ C has a neighborhood Ux ∈ O(X) such that C ∩ Ux is closed in Ux . Open subsets and closed subsets are locally closed. Finite intersections of locally closed sets are locally closed. The following conditions about a subset C ⊆ X are equivalent: • C is locally closed. • C is an open subset of its closure C. • There are sets U ∈ O(X) and A ∈ A(X) such that C = U ∩ A. • There are U, U  ∈ O(X) such that C = U \ U . The following fact is an immediate consequence of these characterizations. Let C ⊆ X be locally closed and consider a subset A ⊆ C. Then A is locally closed in C if and only if it is locally closed in X. 4.5.3 Convex Sets vs. Locally Closed Sets Let X be a T0 -space. Open subsets are down-sets for specialization, closed subsets are up-sets. Thus, both open subsets and closed subsets are convex. Arbitrary intersections of open and closed subsets are convex as well. In particular, locally closed sets are convex. Singletons are trivially convex, but in general they are not locally closed, as the following example shows. Thus, convex sets need not be locally closed. Let S be a discrete infinite space, and consider the spectral space S∞ from Section 1.6.D. Then the singleton subset {∞} is convex (and also proconstructible), but it is not locally closed.

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4.5.4 Proposition Let X be a spectral space and suppose that S ⊆ X is quasi-compact both for the spectral topology and for the inverse topology. Then inv

conv(S) = S ∩ S  ◦ = {V ∩ U | V ∈ K(X) , U ∈ K (X) and S ⊆ V ∩ U}. Proof Since S is quasi-compact in the inverse topology, 4.1.3 yields Spez(S) =  S = {A ∈ K(X) | S ⊆ A}. Quasi-compactness for the spectral topology ◦  inv implies Gen(S) = S = {U ∈ K (X) | S ⊆ U}, by 4.1.5. Now the assertion follows from the equality conv(S) = Spez(S) ∩ Gen(S), see 4.5.1.  4.5.5 Corollary Let X be a spectral space and suppose that S ⊆ X is quasicompact both for the spectral topology and for the inverse topology. Then conv(S) is proconstructible.  4.5.6 Corollary Let X be a spectral space and suppose that S ⊆ X is quasicompact both for the spectral topology and for the inverse topology. (i) The following conditions are equivalent: (a) S is convex. (b) S is an intersection of constructible convex sets, that is,  ◦ S= {V ∩ U | V ∈ K(X) , U ∈ K (X) and S ⊆ V ∩ U}. (ii) Assume that S is constructible. Then the following conditions are equivalent: (a) S is convex.

◦

(b) There are V ∈ K(X) and U ∈ K (X) such that S = V ∩ U. (c) S is locally closed. Proof (i) The set S is convex if and only if S = conv(S). The claim follows from the description of conv(S) in 4.5.4. (ii) The implications (b) ⇒ (c) ⇒ (a) hold trivially, cf. 4.5.3. To prove (a) ⇒ (b) we write  ◦ S= {V ∩ U | V ∈ K(X) , U ∈ K (X) and S ⊆ V ∩ U} (as in (i)(b)). The claim follows from the fact that X \ S is quasi-compact and ◦

both {V ∈ K(X) | S ⊆ V } and {U ∈ K (X) | S ⊆ U} are down-directed for inclusion. 

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4.5.7 Convex Sets and Proconstructible Equivalence Relations We present a construction that will be needed for the study of the extension of spectral maps in Section 5.6. Let X ⊆ Y be a spectral subspace and let E ⊆ X × X be a proconstructible equivalence relation (i.e., an equivalence relation that is a proconstructible subset of X × X). Let X/E be the set of equivalence classes. The equivalence class of x ∈ X is denoted by [x]E . We note that this is a proconstructible set. For, if p1, p2 : X × X → X are the projections onto the components, then [x]E = p2 (E ∩ p−1 1 (x)). It follows from 4.5.5 that for each A ∈ X/E, the convex hull convY (A) of A in the ambient space Y is  proconstructible. We show that A∈X/E convY (A) is proconstructible as well. 4.5.8 Proposition Let Y be a spectral space, X ⊆ Y a spectral subspace, and E ⊆ X × X a proconstructible equivalence relation on X. Then the set  C= convY (A) = {y ∈ Y | ∃(x, x ) ∈ E : x  y  x  } A∈X/E

is a proconstructible subset of Y . Proof The equivalence relation is a proconstructible subset of Ycon × Y . According to 4.5.5 the convex hull convYcon ×Y (E) in Ycon × Y is proconstructible as well. Therefore, p2 (convYcon ×Y (E)) ⊆ Y is proconstructible. It remains to show that C = p2 (convYcon ×Y (E)). First suppose that y ∈ C and x  y  x  with (x, x ) ∈ E. Then (x, x)  (x, y)  (x, x ) in Ycon ×Y (cf. 2.2.1), which shows (x, y) ∈ convYcon ×Y (E). Conversely, pick y ∈ p2 (convYcon ×Y (E)), say (x, x1 )  (z, y)  (x , x1 ) in Ycon × Y with (x, x1 ), (x , x1 ) ∈ E. The first component is Boolean, hence the specializations x  z  x  imply x = z = x . It follows that (x1, x), (x, x1 ) ∈ E and (x1, x1 ) ∈ E (by transitivity). Now the specializations x1  y  x1 in the second component (i.e., in Y ) yield y ∈ C.  4.5.9 Corollary Let X be a subspace of a spectral space Y and consider a continuous map f : Xcon → T to a Hausdorff space. Then the set C = {y ∈ Y | ∃t ∈ T ∃x, x  ∈ f −1 (t) : x  y  x  } is proconstructible in Y . Proof The equivalence relation E = ( f × f )−1 (ΔT ) = {(x, x ) ∈ Xcon × Xcon | f (x) = f (x )} is closed in Xcon × Xcon since T is Hausdorff and f is continuous. The hypotheses of 4.5.8 are satisfied; hence C is proconstructible. 

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135

4.5.10 Locally Closed Points and TD -Spaces A point x in a topological space X is locally closed if the singleton subset {x} is locally closed. In the literature such points are also called Goldman points, cf. [BEPS03, p. 3220]. The set of locally closed points is denoted by LocCl(X). One calls X a TD -space if every point is locally closed, see [AuTh62], [Thr62], [Bru62], [PiPu12, Chap. I, 2.1, p. 5]. Note that every TD -space is T0 . Let C ⊆ X be a locally closed subset. It follows from 4.5.2 that LocCl(X) ∩ C = LocCl(C). A locally closed point can always be separated from every other point by an open set. Therefore, without serious loss of generality, one may frequently restrict the study of locally closed points to T0 -spaces. 4.5.11 Proposition (cf. [BEPS03, Section 3]) For a T0 -space X and a point x ∈ X, the following conditions are equivalent: (i) (ii) (iii) (iv)

x is locally closed. x is isolated in {x}. There is O ∈ O(X) such that x ∈ O max .  If {x} = i ∈I Ai with Ai ∈ A(X) then there is i ∈ I such that {x} = Ai . 6

Proof

(i) ⇔ (ii) is immediate from 4.5.2.

(ii) ⇒ (iii) If {x} is relatively open in {x}, there is a set O ∈ O(X) with {x} = {x} ∩ O. This shows that x is a closed point of O. (iii) ⇒ (iv) It suffices to show that there is some i ∈ I with x ∈ Ai . For then, the inclusion Ai ⊆ {x} implies {x} = Ai . Use (iii) to pick a set O ∈ O(X) such  that x is closed in O. As O is a neighborhood of x and x ∈ i ∈I Ai there is some i ∈ I with O ∩ Ai  ∅. Then O ∩ Ai ⊆ O ∩ {x} = {x} implies x ∈ Ai , as asserted. (iv) ⇒ (i) For each y ∈ I := {x} \ {x} we set Ay = {y}. Then x  Ay for all y,  and (iv) shows y ∈I Ay  {x}, that is, {x} = {x} ∩ (X \



Ay )

y ∈I

is locally closed.



4.5.12 Corollary Let X be a set with T0 topologies σ ⊆ τ. If x ∈ X is locally closed for σ then it is locally closed for τ. 6

If property (iv) is satisfied then the set {x } is called strong irreducibility in [BEPS03]. In ring theory there is a notion of strongly irreducible ideals, [Sch16]. Thus, considering the set of ideals of a ring as a spectral space, cf. 2.5.13(b), there are two different notions of strong irreducibility, which must not be confused.

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, 4.5.11. By hypothProof There is some O ∈ σ with x maximal in O for  σ then there is some y ∈ O with esis, x ∈ O ∈ τ. If x is not maximal in O for  τ y. But  strengthens  , hence x  y ∈ O, a contradiction.  y  x and x  τ σ τ σ 4.5.13 Corollary Let X be a T0 -space, C a subset closed under specialization, and x ∈ C. Then x is locally closed in X if and only if x is locally closed in C. Proof We apply 4.5.11 several times. First assume that x is locally closed in X, say x ∈ O max with O ∈ O(X). Then it is clear that x ∈ (C ∩ O)max with C ∩ O ∈ O(C), hence x is locally closed in C. Conversely, pick U ∈ O(C) such that x ∈ U max . There is some O ∈ O(X) with U = C ∩ O. Since C is closed under specialization it follows that x ∈ O max .   4.5.14 Corollary Let X be a T0 -space and X = i ∈I Ci a partition into subsets that are closed for specialization and generalization. Then LocCl(X) =   i ∈I LocCl(Ci ). 4.5.15 Examples

Let X be a T0 -space.

(i) Every maximal (= closed) point is locally closed. In particular, every quasi-compact space has locally closed points, 4.1.2. (ii) If X is a T1 -space, then every point is closed; hence X is a TD -space. (iii) Let X be the totally ordered set Q with the coarse lower topology, 1.6.2. This is a T0 -space, and the sets r ↑ with r ∈ Q are a basis of closed sets. We claim that LocCl(X) = ∅. Assume that there is a locally closed point q (i.e., {q} = {q} ∩ O for some O ∈ O(X)). It follows that X \ O = q is closed, hence is the intersection of all basic closed sets containing it. There is a smallest such set, namely q ↑ . But then q ↑ = X \ O = q , a contradiction. (iv) Assume that X carries the fine lower topology for the specialization order. Then every point is locally closed, since x ∈ X is maximal in the open set x ↓ , 4.5.11. 4.5.16 Examples We display the space of locally closed points in some spectral spaces that are important in algebraic geometry. (i) If X = Spec(C[T1, ...,Tn ]), then LocCl(X) = Cn as spaces, where Cn is equipped with the Zariski topology (use 12.3.12(i) and 12.3.11, also see 11.2.14). (ii) If X is a spectral space such that the closures of points are specialization

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137

chains, 7 then one readily checks that LocCl(X) = X max ∪ {x ∈ X | x has an immediate specialization}. (iii) If X is the real spectrum of R[T1, ...,Tn ], then LocCl(X) = X. This follows from (ii) because the closure of every point is a finite specialization chain, see 13.5.14(ii). 4.5.17 Corollary (cf. [BEPS03, Section 3]) Let X be a spectral space. The following conditions about a point x ∈ X are equivalent: (i) x is a locally closed point. (ii) x is isolated in {x} for the patch topology. ◦

(iii) There is a set U ∈ K (X) such that x ∈ U max . (iv) There is a set C ∈ K(X) such that x ∈ C max . In particular, LocCl(X) is patch dense in X. Proof (i) ⇒ (ii) is clear since the patch topology is finer than the spectral topology. ◦

(ii) ⇒ (iii) By hypothesis there are U ∈ K (X) and V ∈ K(X) with {x} = U ∩V, which implies that x ∈ U max . (iii) ⇒ (iv) is trivial. (iv) ⇒ (i) There is a basic constructible set U ∩ V ⊆ C containing x. Then (iv) implies x ∈ U max , and the equivalence (i) ⇔ (iii) in 4.5.11 yields (i). For the last statement, pick a constructible set C. Every point of C max is locally  closed (by the equivalence), and C max  ∅ by 4.1.2. 4.5.18 Corollary Let X be a spectral space. Then X max ⊆ LocCl(X). If x ∈ LocCl(X) \ X max and x  y, then there is an immediate specialization z of  x with z  y. Proof The first assertion is 4.5.15(i). For the second assertion pick a nonmaximal locally closed point x and some y ∈ {x} \ {x}. The set {x} \ {x} is closed, hence a spectral subspace of X, hence has a minimal point z (i.e., an immediate specialization of x) with z  y, 4.1.4.  4.5.19 Corollary Let X be a spectral space. The following conditions about a point x ∈ X are equivalent: (i) x is a locally closed point for the inverse topology. 7

This means that X is a spectral root system, cf. Section 8.5 In particular, this is the case if X is totally ordered for specialization.

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(ii) x is isolated in Gen(x) for the patch topology. (iii) There is some V ∈ K(X) with x ∈ V min . (iv) There is some C ∈ K(X) with x ∈ C min . Proof



This is the inverse version of 4.5.17.

4.5.20 The b-Topology and Very Dense Sets (i) Given any topological space X, the locally closed subsets are a basis for a new, of course finer, topology, which is called the b-topology in [Sku69, 2.2]. Every locally closed set is a difference O \ O  of open sets, hence belongs to the enveloping Boolean algebra ba(O(X)). In fact, the enveloping Boolean algebra is generated by the locally closed sets, hence every element of ba(O(X)) is a finite union of locally closed sets. A subset S ⊆ X of a topological space is very dense 8 if it is dense for the b-topology. This is equivalent to each of the following properties: (a) S ∩ C  ∅ for each nonempty locally closed subset C ⊆ X, [GrDi71, §0.2.6]. (b) The restriction map O(X) → O(S) : O → O ∩ S is injective (equivalently: a poset isomorphism, or a lattice isomorphism). Assume that X is a T0 -space. Then, by [Bar68], these conditions are also equivalent to the statement that the inclusion map S → X is an epimorphism in the category T0 Top of T0 -spaces (that has continuous maps as morphisms). Also see the discussion of epimorphisms in 5.2.4. Assume that S is very dense in X. Then the poset-isomorphism O(X) → ◦

◦

O(S) restricts to a poset-isomorphism K (X) → K (S) (whether or not X ◦ is spectral). For, if Z is a topological space, then the subset K (Z) ⊆ O(Z) is determined by the poset structure of O(Z). Namely, O ∈ O(X) is quasi-compact if and only if for all C ⊆ O(X), if O is the supremum of C in O(X), then there is a finite subset F ⊆ C with supremum O. It is immediate from the definition that a very dense subset contains all locally closed points. Moreover, if S ⊆ X is very dense and S ⊆ T ⊆ X, then T is also very dense. Every topological space contains very dense subsets (e.g., the entire space). So, if X is a TD -space, then X is the only very dense subset, 4.5.10. (ii) Now let X be a spectral space. Then a very dense subset is, in particular, ◦

constructibly dense, since the sets U∩V with U ∈ K (X) and V ∈ K(X) are 8

Very dense subsets are also called strongly dense in [BEPS03, p. 3217].

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139

a basis of the patch topology and are locally closed. In general, being very dense is a much stronger condition than denseness for the constructible topology. But constructibly dense subsets are very dense if X has the following property: (∗) If C is a nonempty locally closed set then the constructible interior is nonempty. 4.5.21 Proposition (cf. [BEPS03, Proposition 3.6]) Let X be a T0 -space such ◦

that K (X) is a basis of the topology. Then LocCl(X) is the smallest very dense subset of X. Proof By 4.5.20(i), every very dense subset contains LocCl(X). It remains to show that LocCl(X) is very dense. Let C ⊆ X be a nonempty locally closed ◦

subset (i.e., C = O ∩ C for some O ∈ O(X)). Since K (X) is a basis, we may ◦

assume that O ∈ K (X). If x ∈ O ∩ C, then, by 4.1.2, x specializes to a maximal point y of O, which implies y ∈ LocCl(X), see 4.5.11. Obviously y ∈ C, so y ∈ C, hence y ∈ C ∩ LocCl(X).  4.5.22 Definition A continuous map f : X → Y between topological spaces is a quasi-homeomorphism if the lattice homomorphism O( f ) : O(Y ) → O(X) is an isomorphism, [GrDi71, §0.2.7]. 4.5.23 Proposition ([GrDi71, 0.2.7.1]) Let f : X → Y be a continuous map between topological spaces. (i) If f is a quasi-homeomorphism and X  ⊆ X is a subspace, then the restriction f  : X  → f (X ) is a quasi-homeomorphism. (ii) f is a quasi-homeomorphism if and only if O( f ) is surjective and f (X) is very dense in Y . (iii) If f is a surjective quasi-homeomorphism and X is a T0 -space then f is a homeomorphism. Proof (i) Let g : f (X ) → Y be the inclusion. Then O( f ) = O( f ) ◦ O(g). Surjectivity of O( f ) follows from surjectivity of O( f ), and injectivity follows from surjectivity of f . (ii) Let h : X → f (X) be f with restricted codomain, and let j : f (X) → Y be the inclusion. Then O( f ) = O(h) ◦ O( j), and: • O(h) is injective because h is surjective; • O( j) is surjective because j is the inclusion of a subspace. Suppose f is a quasi-homeomorphism (i.e., O( f ) is an isomorphism). Then O( j) is also injective, hence an isomorphism. We show that f (X) ⊆ Y is

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very dense. Let C be a nonempty locally closed subset of Y , say C = O  \ O = (O ∪ O ) \ O for some O, O  ∈ O(X), 4.5.2. Injectivity of O( j) implies f (X) ∩O  f (X) ∩ (O ∪O ) (i.e., there is some x ∈ X with f (x) ∈ O  \O = C). Conversely, assume that O( f ) is surjective and f (X) ⊆ Y is very dense. To show that O( f ) is also injective, pick O  O  in O(Y ), say O   O. Then f (X) ∩ (O  \ O)  ∅, which yields f −1 (O )  f −1 (O). (iii) It suffices to show that f is injective. Pick x, x  ∈ X with f (x) = f (x ). Then x and x  cannot be separated by open sets (since O( f ) is an isomorphism),  hence x = x  (since X is T0 ). We shall meet quasi-homeomorphisms again in 11.1.3 and in Section 11.2.

5 Properties of Spectral Maps

Spectral maps are the morphisms of the category Spec. They relate different spectral spaces to each other. So far we have mostly analyzed properties of spectral spaces. Now it is time to study spectral maps more closely. Various questions need to be addressed. In our category-theoretic context the characterization of monomorphisms and epimorphisms in Spec is a fundamental task. We present a complete answer in Section 5.2, showing that the monomorphisms are the injective spectral maps and the epimorphisms are the surjective spectral maps. The analysis of monomorphisms is continued in Section 5.4, where we consider spectral maps that are homeomorphisms onto the image. Similar more detailed questions about epimorphisms are studied in Chapter 6, particularly in Section 6.4. Other questions about spectral maps are concerned with topological properties. Closed maps and open maps are studied in Section 5.3, dominant and irreducible maps in Section 5.5. It is an important problem in general topology to study extensions of continuous maps, cf. [GiJe60], [Kel75, p. 115, p. 242], [Eng89, p. 69]. We consider the extension problem for spectral maps and ask: given a spectral map defined on a spectral subspace of a spectral space, does there exist an extension to a spectral map defined on a larger subspace, or even on the entire ambient space? A first answer is presented in Section 5.6. In Section 5.1 we describe images of spectral subspaces under spectral maps, which provides useful tools for the later sections. By Stone duality every spectral map corresponds to a unique homomorphism of bounded distributive lattices, see Section 3.2. Therefore every property of a spectral map may be considered as the topological expression of an algebraic property of the corresponding lattice homomorphism, and conversely. Throughout we keep an eye on these connections and find several useful translations between algebraic and topological properties. 141

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Properties of Spectral Maps

The notion of a spectral map was introduced in Section 1.2. Later we proved several alternative characterizations of spectral maps. As a reminder and a starting point we collect the equivalent conditions here. Reminder Let f : X → Y be a map between spectral spaces. The following conditions are equivalent: (i) f is a spectral map (i.e., the formation of preimages yields a homomor◦

(ii) (iii) (iv) (v)

◦

◦

phism K ( f ) : K (Y ) → K (X) of bounded distributive lattices). finv is a spectral map (i.e., the formation of preimages yields a homomorphism K( f ) : K(Y ) → K(X) of bounded distributive lattices). f is continuous for the spectral topology, the inverse topology, and the constructible topology. f is continuous for any two of the spectral topology, the inverse topology, and the constructible topology. f is continuous for the constructible topology and is monotonic for the specialization order.

If f is spectral, then fcon denotes the map f viewed as a spectral map Xcon → Ycon , and finv denotes the map f viewed as a spectral map Xinv → Yinv . Condition (i) is the definition of a spectral map, see 1.2.2. Conditions (i) and (ii) are equivalent by taking complements. They are equivalent to (iii) and (iv) by 1.4.6. Finally, 1.5.13 says that (i) and (v) are equivalent.

5.1 Images of Proconstructible Sets under Spectral Maps Summary A spectral map sends a proconstructible subset of the domain to a proconstructible subset of the codomain, 1.3.23. Every proconstructible set is the intersection of all constructible sets U ∪ V containing it, where U is quasicompact open and V is closed and constructible. This leads to a description of the image of a proconstructible set in terms of constructible subsets of the domain, as well as of the codomain, 5.1.2. Prime ideal spectra, Section 3.3, are used to translate the result into lattice-theoretic language, 5.1.7. 5.1.1 Lemma Let f : X → Y be a spectral map between spectral spaces and suppose that C is a down-directed (for inclusion) set of proconstructible sets. Then    f ( C) = f (C) = { f (C) | C ∈ C}.   Proof The inclusion f ( C) ⊆ f (C) holds trivially. For the other inclusion

5.1 Images of Proconstructible Sets under Spectral Maps 143  pick y ∈ f (C). Then {C ∩ f −1 (y) | C ∈ C} is a down-directed set of nonempty proconstructible sets, hence is nonempty, 4.2.5(i).  5.1.2 Proposition Let f : X → Y be a spectral map between spectral spaces and let S ⊆ X be proconstructible. Then f (S) = A = B, where  ◦ A= { f (U ∪ V) | U ∈ K (X) , V ∈ K(X) and S ⊆ U ∪ V },  ◦ B= { U  ∪ V  | U  ∈ K (Y ) , V  ∈ K(Y ) and f (S) ⊆ U  ∪ V  }. Proof

The equality f (S) = A is a special case of 5.1.1 with ◦

C = {U ∪ V | U ∈ K (X) and V ∈ K(X) and S ⊆ U ∪ V }. The equality f (S) = B is immediate from the fact that f (S) ⊆ Ycon is closed ◦

and the sets U  ∪ V , with U  ∈ K (Y ), V  ∈ K(Y ), are a basis of closed sets for  Ycon . 5.1.3 Corollary Let f : X → Y be a spectral map between spectral spaces and let S ⊆ X be a subset. Then: ◦  ) = { f (U ∪ V) | U ∈ K (X) , V ∈ K(X), and S ⊆ U ∪ V }.  (ii) f (S) = { f (V) | V ∈ K(X) and S ⊆ V }.

(i) f (S

con

Proof Both claims follow from the equality f (S) = A in 5.1.2 together with the equalities  ◦ con S = {U ∪ V | U ∈ K (X) , V ∈ K(X) and S ⊆ U ∪ V } and S =



{V ∈ K(X) | S ⊆ V }.



5.1.4 Observations We continue with the notation of 5.1.2 and 5.1.3.  (i) Suppose that S ⊆ X is closed. Then f (S) = { f (V) | S ⊆ V ∈ K(X)}. (ii) Similarly, if S ⊆ X is proconstructible and generically closed then f (S) = ◦  { f (U) | S ⊆ U ∈ K (X)}. ◦

(iii) The equality f (S) = B of 5.1.2 can be rephrased in terms of K alone or in terms of K alone. The set f (S) coincides with either of the following sets:  ◦ {U  ∪ (Y \ U ) | U , U  ∈ K (Y ) and f −1 (U ) ∩ S ⊆ f −1 (U )},  {(Y \ V ) ∪ V  | V ,V  ∈ K(Y ) and f −1 (V ) ∩ S ⊆ f −1 (V )}.

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Now we consider a homomorphism ϕ : L → M of bounded distributive lattices and rewrite the description of the image of a proconstructible subset of the spectrum of M in terms of lattice elements. The results are expressed using the prime ideal spectrum, see Section 3.3. Recall from 3.3.4 that for a ∈ L we have PrimI(ϕ)−1 (D L (a)) = D M (ϕ(a)), PrimI(ϕ)−1 (VL (a)) = VM (ϕ(a)). 5.1.5 Corollary Let ϕ : L → M be a homomorphism of bounded distributive lattices and let S ⊆ PrimI(M) be proconstructible. Then (PrimI(ϕ))(S)  = {D L (a) ∪ VL (b) | a, b ∈ L , S ⊆ D M (ϕ(a)) ∪ VM (ϕ(b))}. Proof

This is a special case of the equality f (S) = B in 5.1.2.



5.1.6 Lemma Let L be a bounded distributive lattice and consider elements x, y, z ∈ L. Then VL (x) ⊆ D L (y) ∪ VL (z) if and only if z ≤ x ∨ y, and D L (x) ⊆ D L (y) ∪ VL (z) if and only if x ∧ z ≤ y. Proof The equivalence of the following conditions implies the first assertion: • • • •

VL (x) ⊆ D L (y) ∪ VL (z). VL (y) ∩ D L (z) ⊆ D L (x) (by forming complements). D L (z) ⊆ D L (x) ∪ D L (y). z ≤ x ∨ y (by 3.3.3).

The second assertion is proved similarly.



5.1.7 Corollary Let ϕ : L → M be a homomorphism of bounded distributive lattices, suppose that c ∈ M and that S ⊆ PrimI(M) is proconstructible. (i) PrimI(ϕ)(VM (c)) is the intersection of all D L (a) ∪VL (b) with a, b ∈ L and ϕ(b) ≤ c ∨ ϕ(a). (ii) PrimI(ϕ)(D M (c)) is the intersection of all D L (a) ∪ VL (b) with a, b ∈ L and c ∧ ϕ(b) ≤ ϕ(a). (iii) The closure of PrimI(ϕ)(S) (in PrimI(L)) is the intersection of all VL (a) with a ∈ L and S ⊆ VM (ϕ(a)). (iv) PrimI(ϕ)(S) is closed in PrimI(L) if and only if, for all a, b ∈ L, the inclusion S ⊆ D M (ϕ(a)) ∪ VM (ϕ(b)) implies that there is some x ∈ L with S ⊆ VM (ϕ(x)) and b ≤ x ∨ a. (v) PrimI(ϕ)(VM (c)) is closed in PrimI(L) if and only if for all a, b ∈ L with ϕ(b) ≤ c ∨ ϕ(a) there is some x ∈ L such that ϕ(x) ≤ c and b ≤ x ∨ a.

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145

(vi) The image of PrimI(ϕ) is  {D L (a) ∪ VL (b) | a, b ∈ L and ϕ(b) ≤ ϕ(a)}. Proof (i) is 5.1.5 with S = VM (c). We use 5.1.6 to translate the condition VM (c) ⊆ D M (ϕ(a)) ∪ VM (ϕ(b)) into the statement ϕ(b) ≤ c ∨ ϕ(a). (ii) is 5.1.5 with S = D M (c). Now, see 5.1.6 again, the condition D M (c) ⊆ D M (ϕ(a)) ∪ VM (ϕ(b)) translates into c ∧ ϕ(b) ≤ ϕ(a). (iii) holds because {VL (a) | a ∈ L} is a basis of closed sets for PrimI(L). (iv) The closure of PrimI(ϕ)(S) has been computed in (iii). Thus, it follows from 5.1.5 that PrimI(ϕ)(S) is closed if and only if  {VL (x) | S ⊆ VM (ϕ(x))} ⊆ D L (a) ∪ VL (b) for all a, b ∈ L with S ⊆ D M (ϕ(a)) ∪ VM (ϕ(b)). The set D L (a) ∪ VL (b) is constructible and the collection of sets VL (x) with S ⊆ VM (ϕ(x)) is down-directed. Thus, S ⊆ D M (ϕ(a)) ∪ VM (ϕ(b)) if and only if there is some x ∈ L such that S ⊆ VM (ϕ(x)) and VL (x) ⊆ D L (a) ∪ VL (b). This last condition translates into b ≤ x ∨ a, 5.1.6. (v) is (iv) applied with S = VM (c) (using 5.1.6 again). (vi) is (i) with c = ⊥ (or (ii) with c = ).



5.1.8 Corollary Let L be a bounded sublattice of M and ϕ : L → M the inclusion. Then for c ∈ M we have  (i) PrimI(ϕ)(VM (c)) = {VL (a) ∪ D L (b) | a, b ∈ L and b ≤ c ∨ a},  (ii) PrimI(ϕ)(D M (c)) = {VL (a) ∪ D L (b) | a, b ∈ L and c ∧ b ≤ a}. 

5.2 Monomorphisms and Epimorphisms Summary Monomorphisms and epimorphisms are of fundamental importance in every category. It is always an important task to give practicable criteria to decide whether a given morphism is a monomorphism or an epimorphism. We study this question for the category Spec. To recall the notions, let C be any category and consider a morphism f : C → D. • f is a monomorphism if we have g = h for all morphisms g, h : B → C with f ◦ g = f ◦ h. • f is an epimorphism if we have j = k for all morphisms j, k : D → E with j ◦ f = k ◦ f.

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The main results are characterizations of the monomorphisms and of the epimorphisms in Spec, see 5.2.2 and 5.2.5. The monomorphisms are exactly the injective spectral maps and the epimorphisms are exactly the surjective spectral maps. Moreover, using Stone duality, the properties are translated into conditions about lattice homomorphisms. The characterization of monomorphisms and epimorphisms leads to additional information about the category Spec, 5.2.8 and 5.2.9. This is the last piece of information needed to see that the complete category Spec, see 2.3.8, is also co-complete, 5.2.9. 5.2.1 Injective and Surjective Spectral Maps vs. Monomorphisms and Epimorphisms We start with a general category-theoretic remark. Suppose that C is a category such that • every object is a set carrying some additional structure, and • every morphism f : C → D is a map of the underlying sets that is compatible with the additional structure. Then it is immediately clear that any injective morphism is a monomorphism and any surjective morphism is an epimorphism. In many categories arising in algebra and topology the monomorphisms are exactly the injective morphisms. Quite often epimorphisms are more delicate – frequently there are epimorphisms that are not surjective. For example, the monomorphisms of BDLat are the injective homomorphisms. But epimorphisms need not be surjective: it is clear from the universal mapping property of the Boolean envelope of a bounded distributive lattice L that the canonical homomorphism ba L : L → ba(L) is an epimorphism, see 3.4.1. But ba L is surjective only if L is a Boolean algebra. The spectral counterpart of the lattice homomorphism ba L is the canonical spectral map conX : Xcon → X, which is bijective, hence is both a monomorphism and an epimorphism. But it is an isomorphism if and only if X is a Boolean space. The general considerations apply to spectral maps. Thus, injective spectral maps are monomorphisms, and surjective spectral maps are epimorphisms. We shall see that the converse is also true in both cases. The only surprise is that epimorphisms are always surjective, which was first noted by Picavet, cf. [Pic75, Proposition 5]. 5.2.2 Theorem are equivalent:

Let f : X → Y be a spectral map. The following conditions

(i) f is injective. (ii) f is a monomorphism in Spec.

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147

(iii) fcon is a monomorphism in BoolSp. ◦

◦

◦

(iv) K ( f ) : K (Y ) → K (X) is an epimorphism in BDLat. (v) K( f ) : K(Y ) → K(X) is surjective. Proof

(i) ⇒ (ii) follows from the discussion in 5.2.1.

(ii) ⇒ (iii) Since f and conX are monomorphisms it follows that f ◦ conX = conY ◦ fcon is a monomorphism as well. It is a general fact that, if a composition of maps is a monomorphism, then the first map of the composition is a monomorphism. Thus, fcon is a monomorphism in Spec, hence also in the subcategory BoolSp. (iii) ⇒ (i) Suppose that f (x) = f (x ) with x, x  ∈ X. We consider the maps fx , fx : 1 → Xcon between Boolean spaces defined by fx (0) = x and fx (0) = x , cf. 1.2.5. We conclude that fx = fx , hence x = x , since fcon ◦ fx = fcon ◦ fx and fcon is a monomorphism. The equivalence (ii) ⇔ (iv) holds by Stone duality, see 3.2.10. (i) ⇒ (v) As f is injective, it follows that fcon is an injective continuous map between Boolean spaces, hence is a homeomorphism onto the closed subspace f (X)con = f (Xcon ) ⊆ Ycon . Thus, the clopen subsets of f (X)con are the restrictions of clopen subsets of Ycon , that is, K( f ) : K(Y ) → K( f (X))  K(X) is surjective. (v) ⇒ (i) Assume that f is not injective, say f (x) = f (x ) with x  x . There exists a constructible set C ∈ K(X) such that x ∈ C and x   C. Then C is not in the image of K( f ), whence K( f ) is not surjective.  5.2.3 Other Characterizations of Monomorphisms The characterization of monomorphisms in 5.2.2 can be supplemented by numerous other equivalent statements, for example: • finv is a monomorphism. • K( f ) : K(Y ) → K(X) is an epimorphism in BDLat. • K( f ) : K(Y ) → K(X) is an epimorphism in BoolAlg. The proofs are simple applications of 5.2.2 and Stone duality. 5.2.4 Epimorphisms in Topological Categories In every category of topological spaces the surjective continuous maps are epimorphisms, 5.2.1. There are scores of subcategories in Top, and the characterization of epimorphisms may be quite different in different subcategories. It is a well-known fact that

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a continuous map f : X → Y is an epimorphism in the category of Hausdorff spaces if and only if f (X) ⊆ Y is dense, [Bou71b, p. I.53, Corollaire 1]. On the other hand, f is an epimorphism in Top if and only if f is surjective. For, if f is not surjective then let Z = {a, b} be an indiscrete space with a  b. The maps g, h : Y → Z, defined by g(y) = a for all y and h(y) = a for y ∈ f (X) and h(y) = b otherwise, are both continuous; however g  h, whereas g ◦ f = h ◦ f . Also, if f is a map between Boolean spaces and f is an epimorphism in BoolSp then f is surjective. For, if f is not surjective then there is a nonempty clopen subset C ⊆ Y with f (X) ∩ C = ∅ . Define g, h : Y → 2con by g(y) = 0 for all y ∈ Y and h(y) = 0 if and only if y  C. It is clear that g  h, but g ◦ f = h ◦ f (since both compositions are the constant map with value 0). The epimorphisms in T0 Top are characterized in [Bar68]. They are the continuous maps f : X → Y such that f (X) is very dense in Y (i.e., is dense in Y for the b-topology, cf. 4.5.20 for the notion of the b-topology). Thus, f is an epimorphism in T0 Top if and only if O( f ) : O(Y ) → O(X) is injective. In fact, a slightly more general statement holds: Claim Let f : X → Y be a continuous map into a T0 -space such that im( f ) is very dense in Y . If g, h : Y → Z are continuous maps into a T0 space such that g ◦ f = h ◦ f then g = h. Proof of Claim Assume there is some y ∈ Y with g(y)  h(y). Pick an open set O ∈ O(Z) with g(y) ∈ O and h(y)  O (or vice versa). Then y ∈ g −1 (O) \ h−1 (O) (i.e., this is a nonempty locally closed set). By hypothesis there is some x ∈ X with f (x) ∈ g −1 (O) \ h−1 (O). But then g( f (x)) ∈ O and h( f (x))  O, a contradiction.  5.2.5 Theorem Suppose that f : X → Y is a spectral map. The following conditions are equivalent: (i) f is surjective. (ii) f is an epimorphism in Spec. (iii) fcon is an epimorphism in BoolSp. ◦

◦

◦

(iv) K ( f ) : K (Y ) → K (X) is injective. (v) K( f ) : K(Y ) → K(X) is injective. Proof The implication (i) ⇒ (ii) and the equivalence (i) ⇔ (iii) follow from the discussion in 5.2.1. (ii) ⇒ (i) Assume by way of contradiction that f is not surjective. There is ◦

a nonempty basic constructible set V \ U ∈ K(Y ) with U,V ∈ K (Y ) and f (X) ∩ (V \ U) = ∅ . Replacing V by U ∪ V one may assume that U ⊆ V. We define spectral maps h, h  : Y → 2 (cf. 1.2.6) by the conditions that h−1 (0) = U

5.2 Monomorphisms and Epimorphisms

149

and (h )−1 (0) = V. Then h  h , but h ◦ f = h  ◦ f (since f −1 (U) = f −1 (V)). Thus, f is not an epimorphism. ◦

(ii) ⇔ (iv) The spectral map f is an epimorphism if and only if K ( f ) is a ◦ monomorphism if and only if K ( f ) is injective, using Stone duality, see 3.2.10. (i) ⇒ (v) Surjectivity of f is equivalent to injectivity of P( f ) : P(Y ) → P(X). As K( f ) is a restriction of P( f ), we see that K( f ) is injective. (v) ⇒ (i) Assume that f is not surjective. The proconstructible subset f (X) ⊆ Y is an intersection of constructible sets, hence there is some nonempty C ∈ K(Y ) such that f −1 (C) = ∅. But then K( f )(C) = K( f )(∅), and K( f ) is not injective.  5.2.6 Other Characterizations of Epimorphisms As in 5.2.3, the characterization of epimorphisms in 5.2.5 can be supplemented by several other equivalent statements, for example: • finv is an epimorphism. • K( f ) : K(Y ) → K(X) is injective. • K( f ) : K(Y ) → K(X) is a monomorphism in BoolAlg. Again the proofs are trivial applications of 5.2.5 and Stone duality. 5.2.7 Corollary The spectral map f : X → Y is a monomorphism and an epimorphism if and only if K( f ) : K(Y ) → K(X) is an isomorphism.  The statement of 5.2.7 can be rephrased in the following form. The spectral map f : X → Y is a monomorphism and an epimorphism if and only if fcon is a homeomorphism. Assuming that this is the case, one may identify Xcon and Ycon via fcon , and then the specialization order of Y refines the specialization order of X. We derive an important category-theoretic fact from 5.2.2 and 5.2.5. Recall the following terminology: a category C is said to be well-powered if, for any object C, the class of isomorphism classes of monomorphisms with codomain C is a set. Dually, the category is co-well-powered if the class of isomorphism classes of epimorphisms with domain C is a set. 5.2.8 Corollary The category Spec is well-powered and co-well-powered. Proof Suppose that f : X → Y is a monomorphism of spectral spaces. By 5.2.2, X is homeomorphic onto the set f (X) with a spectral topology that refines the subspace topology inherited from Y . There is only a set of subsets of Y and each subset carries only a set of different topologies. Thus, the class of isomorphism classes of monomorphisms with codomain Y is a set.

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Now consider an epimorphism f : Y → Z. Let E f = {(y, y ) ∈ Y ×Y | f (y) = f (y )} be the equivalence relation defined by f . By 5.2.5, Z is homeomorphic to the set Y /E f with a suitable spectral topology. The set Y has only a set of equivalence relations and, for each equivalence relation E ⊆ Y ×Y , the set Y /E carries only a set of different (spectral) topologies. Thus, there is only a set of isomorphism classes of epimorphisms.  5.2.9 Corollary The category Spec is co-complete. Proof It is known from category theory that Spec is co-complete if it is complete, well-powered, and has a co-separator, [HeSt79, Theorem 23.14]. All these conditions are satisfied, see 2.3.8, 5.2.8, and 1.2.8.  Co-completeness of Spec can also be deduced using Stone duality (cf. the remarks in 3.7.3). The categories Spec and BDLat are anti-equivalent to each other, see 3.2.10. The object class of BDLat is a variety in the sense of universal algebra, cf. [Coh81, p. 162]. It is a well-known fact that BDLat is then both complete and co-complete. Thus, by duality, Spec is also co-complete and complete. This argument gives only a general existence result and does not lead to any description of colimits. A more explicit study of colimits is contained in Chapter 10. 5.2.10 Monomorphisms and Epimorphisms – Basic Facts We record several elementary facts about monomorphisms and epimorphisms that are true both in Spec and BDLat. In both categories the monomorphisms are the injective morphisms, 5.2.2. In both categories surjective morphisms are epimorphisms, 5.2.5. In Spec, but not in BDLat, every epimorphism is surjective, cf. 5.2.1. Note that Spec and BDLat are complete and co-complete. The underlying set of a fiber product is the fiber product of the underlying sets. For Spec this has been shown in 2.3.7. For BDLat this is easy to prove or is well known. Let C be either category. The easy proofs of the following facts are omitted: (i) A composition of two injective (or surjective) morphisms is injective (resp., surjective); a composition of two epimorphisms is an epimorphism. If the composition of two morphisms a : A → B and b: B → C is injective, then a is injective. If the composition is surjective or an epimorphism, then b is surjective or an epimorphism. (ii) Let a : A → C and b: B → C be morphisms in C and consider the following pull-back diagram, cf. 2.3.6:

5.3 Closed and Open Spectral Maps a

A ×C B b

151

/B b

 A

a

 /C.

Then: (a) If a is injective or surjective, then so is a . (b) If b is surjective and a  is injective, then a is injective. (iii) Suppose that A

b

c

c

 C

/B

b

 /D

is a push-out diagram in C. 1 If b is an epimorphism, then so is b. 5.2.11 Projective Limits and Epimorphisms Let ((Yi )i ∈I , (gi j : Yj → Yi )i ≤ j ) be a projective system of spectral spaces with projective limit (Y, (gi : Y → Yi )i ∈I ), 2.3.9. Let ( fi : X → Yi )i ∈I be a cone of epimorphisms with limit map f : X → Y . We claim that f is also an epimorphism. By 5.2.5 the epimorphisms are exactly the surjective spectral maps. For y ∈ Y set yi = gi (y) ∈ Yi . The sets fi−1 (yi ) form a down-directed family of nonempty (since fi is surjective) proconstructible subsets of X. Compactness implies that    −1 −1 −1 i ∈I fi (yi )  ∅. If x ∈ i ∈I fi (yi ), then f (x) ∈ i ∈I gi (yi ) = {y}. The analysis of monomorphisms and epimorphisms is continued in Section 5.4 and Chapter 6.

5.3 Closed and Open Spectral Maps Summary Closed maps and open maps play an important role in general topology and are equally important in the category Spec. In 5.3.3 and 5.3.5 we prove various characterizations of closed and open spectral maps. Particularly important among the equivalent conditions are the going-up property and the going-down property, which use the specialization order to express closedness 1

The notion of a push-out diagram (also known as a co-cartesian square, [Mac71, p. 65]) is the dual notion of a pull-back diagram, see 2.3.6. The object D, together with the morphisms b and c , is the fiber sum (or the push-out) and is the colimit of the diagram C ← A → B.

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Properties of Spectral Maps

or openness. The names are transferred from commutative algebra, where the corresponding conditions are known as the Going-up Theorem and the Goingdown Theorem, [Mat80, Section 5]. 5.3.1 Definition A spectral map f : X → Y has the (i) going-up property if, given y  y  in Y and x ∈ X with f (x) = y, there exists an element x  ∈ X such that x  x  and f (x ) = y : /o _x

X  Y

/o

/o

/o / x  _

f

 / y .

 y

(ii) going-down property if, given y   y in Y and x ∈ X with f (x) = y, there exists an element x  ∈ X such that x   x and f (x ) = y : x_ /o

X  Y

f

/o

/o

 y

/o / x _ / y .

We observe that a spectral map f : X → Y has going-up if and only if finv has going-down. Thus, every general result about the going-up property implies a corresponding result about the going-down property. First we note some elementary facts about going-up and going-down. 5.3.2 Proposition (i) Suppose that f : X → Y and g : Y → Z both have going-up (or goingdown). Then so does g ◦ f . (ii) Let f : X → Y and g : Z → Y be spectral maps and consider the following pull-back diagram: f

X ×Y Z g

/Z g

 X

f

 /Y.

If f has going-up or going-down, then so does f . Proof (i) is straightforward. (ii) For basic properties of the fiber product, see 2.3.7. Suppose that f has going-up. Pick elements (x, z) ∈ X ×Y Z and z  ∈ Z such that z = f (x, z)  z . It follows that f (x) = g(z)  g(z ). Since f has going-up there is some x  ∈ X

5.3 Closed and Open Spectral Maps

153

with x  x  and f (x ) = g(z ). But then (x , z ) ∈ X ×Y Z, (x, z)  (x , z ), and z  = f (x , z ), which proves the claim. The going-down property is proved similarly, or alternatively, one applies the going-up property to the inverse spaces and maps.  5.3.3 Theorem are equivalent: (i) (ii) (iii) (iv) (v)

The following conditions about a spectral map f : X → Y

f is closed (i.e., if A ∈ A(X), then f (A) ∈ A(Y )). If V ∈ K(X), then f (V) ∈ A(Y ). f has going-up, in other words, f ({x}) is closed for all x ∈ X. If P ⊆ Y is proconstructible, then f −1 (Gen(P)) = Gen( f −1 (P)). If C ⊆ Y is basic constructible, then f −1 (Gen(C)) = Gen( f −1 (C)).

Proof We prove the following chain of implications: (i) ⇒ (iii) ⇒ (iv) ⇒ (v) ⇒ (ii) ⇒ (i). (i) ⇒ (iii) is trivial. (iii) ⇒ (iv) Since f is spectral and Gen(P) is inversely closed, f −1 (Gen(P)) is inversely closed as well. Consequently, f −1 (Gen(P)) ⊇ Gen( f −1 (P)). To prove the other inclusion, pick x ∈ X \Gen( f −1 (P)). Then f −1 (P) is disjoint from {x}, thus f ({x}) ∩ P = ∅ . By (iii), f ({x}) is closed, hence f ({x}) ∩ Gen(P) = ∅ , in particular x  f −1 (Gen(P)). (iv) ⇒ (v) is trivial. (v) ⇒ (ii) Take A ∈ K(X), then f (A) is proconstructible. To prove closedness it suffices to show that f (A) is closed under specialization. So, pick x ∈ A, write y = f (x), and let y  ∈ {y}. We must show that A ∩ f −1 (y )  ∅ . Let ◦

B = {U ∩ V | U ∈ K (Y ), V ∈ K(Y ) and y  ∈ U ∩ V }. Since B is a basis of neighborhoods of y  in Ycon we see that f −1 (y ) =  −1 −1 C ∈B f (C). Note that f (C) is constructible and A is proconstructible. Hence it suffices to show that A ∩ f −1 (C)  ∅ for each C ∈ B. By y  y  we have y ∈ Gen(C), which implies x ∈ f −1 (Gen(C)). It follows from (v) that x ∈ Gen( f −1 (C)) (i.e., there is some x  ∈ f −1 (C) with x  x ). Since A is closed and x ∈ A, this means x  ∈ A ∩ f −1 (C). (ii) ⇒ (i) follows from 5.1.1, since K(X) is a basis of closed sets.



We spell out 5.3.3 for finv . 5.3.4 Corollary are equivalent:

The following conditions about a spectral map f : X → Y

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Properties of Spectral Maps

(i) finv is closed. (ii) (iii) (iv) (v)

◦

If U ∈ K (X), then f (U) ∈ A(Yinv ). f has going-down. If P ⊆ Y is proconstructible, then f −1 (P) = f −1 (P). If C ⊆ Y is basic constructible, then f −1 (C) = f −1 (C).

5.3.5 Corollary are equivalent:



The following conditions about a spectral map f : X → Y

(i) f is open (i.e., O ∈ O(X) implies f (O) ∈ O(Y )). ◦

◦

(ii) If U ∈ K (X), then f (U) ∈ K (Y ).

◦

(iii) f has going-down and f (U) ∈ K(Y ) for all U ∈ K (X). In particular, if f is open then finv is closed. ◦

Proof (i) ⇒ (ii) If U ∈ K (X) then f (U) is quasi-compact by continuity and is open by hypothesis. ◦

◦

(ii) ⇒ (i) follows from the fact that K (X) and K (Y ) are bases of the spectral topologies. ◦

◦

(ii) ⇒ (iii) If U ∈ K (X) then f (U) ∈ K (Y ) = K(Y ) ∩ A(Yinv ). So the second condition of (iii) is satisfied. Moreover, 5.3.4 shows that f has going down. ◦

(iii) ⇒ (ii) If U ∈ K (X) then f (U) ∈ K(Y ) by assumption and is closed under generalization by the going-down property. It follows from 1.5.4 that ◦

f (U) ∈ K (Y ). If f is open then 5.3.4 and the equivalence (i) ⇔ (iii) show that finv is closed.  5.3.6 Going-Down vs. Openness Suppose that f : X → Y is a spectral map with the going-down property (i.e., finv is closed, 5.3.4), equivalently, has going-up, 5.3.3. But this does not imply that f is open. (Of course, one should not expect f to be open. Note that closed maps need not map closed constructible sets to closed constructible sets either.) To produce examples of spectral maps that are not open, but have going-down, let Y be a spectral space and let X ⊆ Y be a spectral subspace that is closed under generalization, but is not open. Then the inclusion map X → Y has going-down, but is not open. For example, the singleton subset {∞} ⊂ S∞ , see Section 1.6.D, is closed under generalization, but is not open. 5.3.7 Properness of Spectral Maps vs. Closedness Properness is a condition about continuous maps related to closedness. We briefly review the meaning of properness in the context of spectral spaces. First, recall that a continuous

5.3 Closed and Open Spectral Maps

155

map f : X → Y between topological spaces is proper if it is closed and the fibers f −1 (y), y ∈ Y , are quasi-compact, cf. [Bou71b, p. TG I.75, Théorème 1]. Inverse images of quasi-compact sets under proper maps are quasi-compact, [Bou71b, p. TG I.77, Proposition 6]. (i) Let f : X → Y be a continuous map between spectral spaces. We claim that f is proper if and only if f is spectral and closed. Proof

First suppose that f is proper. Then f is closed by definition ◦

and if U ∈ K (Y ) then f −1 (U) is open by continuity and quasi-compact by properness. Conversely, fibers of spectral maps are quasi-compact by 1.3.23. Thus, a closed spectral map is proper.  (ii) In general topology there exist continuous maps that are not proper, but have the property that the inverse image of a quasi-compact set is quasicompact. The following example shows that the same is true for spectral maps. Suppose that X is finite and f : X → Y is a spectral map. Then every subset of X is quasi-compact, but f need not be closed. The simplest example is the spectral map 1 → 2, 0 → 0. (iii) Finally we note that usually inverse images of quasi-compact subsets of spectral spaces under spectral maps are not quasi-compact. Let Y be a spectral space and K a quasi-compact subset. Then K = conY−1 (K) is (quasi-)compact in Ycon if and only if K is closed in Ycon , if and only if K is proconstructible in Y . Thus, examples are provided by quasi-compact subsets that are not proconstructible. Explicitly: • Consider the spectral space S∞ , see Section 1.6.D, together with the canonical map conS∞ : (S∞ )con = S ∗ → S∞ . The set S is quasi-compact in S∞ (it is the set of closed points), but is not quasi-compact in S ∗ . • Let Y be a spectral specialization chain (see Section 1.6.A). Then a subset K ⊆ Y is quasi-compact if and only if it has a largest element, see 1.6.1(ii). The preimage of K under the canonical map conY : Ycon → Y is not quasi-compact in general – for example, if K does not have a smallest element. 5.3.8 Characterization of Closedness of a Spectral Map in Terms of Lattices Let ϕ : L → M be a homomorphism between bounded distributive lattices. In view of 5.1.7(v), 5.3.3, and 5.3.7, the following conditions are equivalent: (i) Spec(ϕ) is a closed map. (ii) Spec(ϕ) maps closed and constructible sets onto closed sets. (iii) Spec(ϕ) has the going-up property.

156

Properties of Spectral Maps

(iv) Spec(ϕ) is a proper map. (v) For all a, b ∈ L and c ∈ M with ϕ(b) ≤ c ∨ ϕ(a) there is x ∈ L such that ϕ(x) ≤ c and b ≤ x ∨ a.

5.4 Embeddings Summary The characterization of monomorphisms in 5.2.2 leads to a followup problem. Given a monomorphism f : X → Y in Spec, by 5.2.2 the corre◦

◦

◦

sponding lattice homomorphism K ( f ) : K (Y ) → K (X) is an epimorphism in BDLat, but need not be surjective, cf. 5.2.1. Now we address the question: for ◦

which monomorphisms f is it true that the lattice homomorphism K ( f ) is surjective? In 5.4.3 we give both a topological answer ( f is an embedding, i.e., a homeomorphism onto the image) and a category-theoretic answer ( f is an extremal monomorphism). In 5.4.10, open embeddings and closed embeddings are characterized in terms of their lattice homomorphisms. We finish with a list of useful elementary properties of embeddings: their behavior under composition, in pull-back diagrams, and in push-out diagrams. 5.4.1 Definition Let f : X → Y be a spectral map. (i) f is an embedding if f is a homeomorphism onto a spectral subspace. (ii) f is a closed embedding (resp., an open embedding) if it is an embedding and is a closed map (resp., an open map). (iii) f is an extremal monomorphism if it satisfies the following condition: whenever f = h◦g and g is an epimorphism, then g is a homeomorphism. 5.4.2 Remarks Let f : X → Y be a spectral map. (i) f is an embedding if and only if f is injective and f (x)  f (x ) implies x  x . It is clear that this is equivalent to finv being an embedding. (ii) Let f be an embedding. Then f is a closed embedding (resp., an open embedding) if and only if f (X) ⊆ Y is closed (resp., open, even quasicompact open). Moreover, f is an embedding onto a generically closed subspace if and only if finv is a closed embedding. (iii) In general category theory, an extremal monomorphism is also assumed to be a monomorphism. In our context this assumption is not needed. We claim that 5.4.1(iii) implies injectivity. To prove this, write f = h ◦ g, where g : X → f (X) is f with restricted codomain and h : f (X) → Y is the inclusion. Now 5.4.1(iii) says that g is a homeomorphism, hence is

5.4 Embeddings

157

injective. Thus, f , being a composition of two injective maps, is injective as well. Let f : X → Y be a spectral map. The following conditions

5.4.3 Theorem are equivalent:

(i) f is an embedding. ◦

(ii) K ( f ) is surjective. (iii) f is an extremal monomorphism. Proof

(i) ⇒ (ii) As f (X) ⊆ Y is a spectral subspace it follows from 2.1.3 ◦

◦

that the restriction map K (Y ) → K ( f (X)), U → U ∩ f (X) is surjective. Since ◦

X  f (X) it follows that K ( f ) is surjective. (ii) ⇒ (iii) Suppose that f = h ◦ g is a factorization with an epimorphism ◦

◦

◦

g : X → Z and a spectral map h : Z → Y . Then K ( f ) = K (g) ◦ K (h), and ◦ ◦ surjectivity of K ( f ) implies that K (g) is surjective. As g is an epimorphism, ◦

5.2.5 shows that K (g) is injective. In BDLat, bijective homomorphisms are ◦ isomorphisms, hence K (g) is an isomorphism of lattices. Stone duality shows that g is a homeomorphism in Spec, 3.2.10. (iii) ⇒ (i) We factor f = h ◦ g, where g : X → f (X) is f with restricted codomain and h : f (X) → Y is the inclusion. As g is surjective we conclude that g is a homeomorphism (i.e., f is an embedding).  5.4.4 Corollary Let f : X → Y be a spectral map. (i) The map f factors into the surjective spectral map g : X → f (X), which is f with restricted codomain, and the extremal monomorphism h : f (X) → Y , which is the inclusion of the spectral subspace. ◦

◦

◦

(ii) The image of K ( f ) : K (Y ) → K (X) is canonically isomorphic to ◦  K ( f (X)). 5.4.5 Spectral Subspaces vs. Congruences of Lattices Let X be a spectral map. The isomorphism classes of embeddings Y → X correspond bijectively to the spectral subspaces (i.e., to A(Xcon )). By 5.4.3 the spectral subspaces correspond to the isomorphism classes of surjective lattice homomorphisms ◦

◦

K (X) → M, hence also to the congruences of K (X), cf. 2.5.13(f). Let Y,Y  ⊂ X be two spectral subspaces with inclusion maps e : Y → X and e  : Y  → X. The ◦

◦

◦

◦

◦

◦

congruences of K (X) defined by K (e) : K (X) → K (Y ) and K (e ) : K (X) → ◦ K (Y ) are denoted by C and C , respectively. Then the following conditions are equivalent:

158

Properties of Spectral Maps

• Y ⊆ Y . ◦ ◦ ◦ ◦ • There is a homomorphism h : K (Y ) → K (Y ) with K (e) = h ◦ K (e ). • C  ⊆ C. ◦

Thus, the posets A(Xcon ) and Cong(K (X)) are anti-isomorphic to each other. ◦

Both are lattices, where A(Xcon ) is even distributive. It follows that Cong(K (X)) is distributive as well. ◦ Moreover, the bijective map between A(Xcon ) and Cong(K (X)) can be used ◦

to transfer the spectral space structure from Cong(K (X)) to A(Xcon ). Note that the specialization relation of A(Xcon ) is then the inverse of the inclusion relation (i.e., Y  Y  if and only if Y  ⊆ Y ). 5.4.6 Definition Let X ⊆ Y be a spectral subspace. A spectral map r : Y → X is a spectral retraction if r |X = idX . Let f : Y → Z be a spectral map such that there is a spectral map s : Z → Y with f ◦ s = id Z . Then s is a spectral section of f . If r : Y → X is a retraction, then the inclusion X → Y is a spectral section for r. 5.4.7 Corollary Let f : X → Y be a spectral map and suppose that g : Y → X is a spectral section for f . Then g is an embedding. ◦

◦

◦

Proof The hypothesis implies that K (g) ◦ K ( f ) = id ◦ . But then K (g) is K (Y) surjective, and the claim follows from 5.4.3.  5.4.8 Example Spectral sections of surjective spectral maps are embeddings, 5.4.7. The converse is not true: there are embeddings that are not sections for any spectral map. For an example we exhibit a spectral subspace such that there is no retraction from the ambient space to the subspace. The following finite graph represents a poset, hence a finite spectral space: d^ Y:

?e @c_

a

b.

A sub-poset, hence a subspace of the spectral space, is given by the following diagram: dO ]

A eO

a

b.

X:

5.4 Embeddings

159

We show that there is no poset retraction from the ambient poset to the subposet. Assuming that there is a retraction r, it must map c to one of a, b, d, or e. Suppose, for example, that r(c) = a. Then b ≤ c implies that b = r(b) ≤ r(c) = a, which is false. All other choices for r(c) lead to the same type of contradiction. Thus, there is no spectral retraction from the ambient spectral space to its subspace. 5.4.9 Lattices Modulo Ideals and Filters We have shown that embeddings of spectral spaces correspond to surjective lattice homomorphisms under Stone duality. The next step is to characterize, in lattice-theoretic terms, closed embeddings and embeddings onto generically closed subspaces. To prepare for the characterization in 5.4.10 we give a brief reminder of factor lattices modulo ideals and filters. For factor lattices modulo congruences, see 2.5.13(f) and [Grä98, p. 20 ff]. Let L be a bounded distributive lattice with an ideal i and a filter f. Then • ≡i := {(a, b) ∈ L × L | ∃x ∈ i : a ∨ x = b ∨ x}, • ≡f := {(a, b) ∈ L × L | ∃y ∈ f : a ∧ y = b ∧ y} are congruences on L. Using the Boolean envelope (see Section 3.4), the defining conditions can also be expressed in the following form: • a ≡i b is equivalent to a  b ∈ i , • a ≡f b is equivalent to a  b ∈ f , where a  b is symmetric difference in ba(L) and i  (resp., f ) is the ideal (resp., the filter) generated by i (resp., f) in ba(L) (cf. [Kop89, p. 74 ff]). The factor lattices modulo the congruences are denoted by L/i and L/f, respectively; the congruence classes of a ∈ L are denoted by [a]i and [a]f . Let πi : L → L/i and πf : L → L/f be the canonical homomorphisms. Note that i = πi−1 (⊥) and f = πf−1 (). A lattice homomorphism ϕ : L → M factors uniquely through L/i (resp., through L/f) if and only if i ⊆ ϕ−1 (⊥) (resp., f ⊆ ϕ−1 ()). The induced homomorphism L/i → M (resp., L/f → M) is denoted by ϕi (resp., ϕf ). ◦

◦

5.4.10 Theorem Let f : X → Y be a spectral map, set ϕ = K ( f ) : K (Y ) → ◦ ◦ K (X) and consider the ideal i = ϕ−1 (⊥) ⊆ K (Y ) and the filter f = ϕ−1 () ⊆ ◦

◦

◦

◦

◦

K (Y ). Let ϕi : K (Y )/i → K (X) and ϕf : K (Y )/f → K (X) be the unique homomorphisms with ϕ = ϕi ◦ πi and ϕ = ϕf ◦ πf , cf. 5.4.9. Then: (i) f is an embedding if and only if ϕ is surjective, if and only if ϕi is surjective, if and only if ϕf is surjective. ◦

(ii) Spec(πi ) is a homeomorphism from Spec(K (Y )/i) onto f (X).

160

Properties of Spectral Maps ◦

(iii) f is a closed embedding if and only if the homomorphism ϕi : K (Y )/i → ◦ K (X) is an isomorphism. ◦

(iv) Spec(πf ) is a homeomorphism from Spec(K (Y )/f) onto Gen( f (X)). (v) f is an embedding onto an inversely closed subspace if and only if the ◦

◦

homomorphism ϕf : K (Y )/f → K (X) is an isomorphism. Proof (i) follows from 5.4.3 and the fact that πi and πf are surjective. ◦

(ii) By 1.1.9 the ideals of K (Y ) correspond bijectively to O(Y ). In particular, i corresponds to the open set Y \ f (X) = intY (Y \ f (X)). Thus, if ψ is the ◦

◦

surjective lattice homomorphism K (Y ) → K ( f (X)) induced by the inclusion map f (X) → Y , then it follows that ψ −1 (⊥) = i. Hence there is a unique ◦

◦

homomorphism ψi : K (Y )/i → K ( f (X)) with ψ = ψi ◦ πi . As ψi is surjective it ◦ suffices to prove injectivity. So, pick U, U  ∈ K (Y ) with U ∩ f (X) = U  ∩ f (X) (i.e., (U  U ) ∩ f (X) = ∅). The set U  U  is constructible and is contained in ◦

Y \ f (X). Hence there is some W ∈ K (Y ) such that U  U  ⊆ W ⊆ Y \ f (X). This proves U ≡i U  (since W ∈ i and U ∪ W = U  ∪ W), and ψi is injective. ◦

(iii) Let g : X → f (X) be f with restricted codomain and set ρ = K (g). The following commutative diagram shows the lattice homomorphisms under consideration: ◦

0 K ( f (X)) ;

ψ ψi ◦

πi

K (Y )

◦

/ K (Y )/i

ρ

ϕi

$

ϕ

◦



. K (X) .

As ψi is an isomorphism, see (ii), it follows that ϕi is an isomorphism if and only if ρ is an isomorphism. By Stone duality this is equivalent to g being a homeomorphism. (iv) and (v) can be proved similarly to (ii) and (iii), mutatis mutandis. Or, alternatively, they are the inverse statements of (ii) and (iii).  5.4.11 We continue with the notation of 5.4.9 and 5.4.10. Then ϕ splits into a sequence of homomorphisms: ◦

K (Y )

πi

◦

/ K (Y )/i

ϕi

◦

/ ϕ(K (Y ))

⊆

◦

/ K (X) .

5.4 Embeddings

161

◦

The image ϕ(K (Y )) is the lattice that corresponds to the spectral subspace ◦ f (X) ⊆ Y , see 5.4.4, and K (Y )/i is the lattice that corresponds to f (X) ⊆ Y , see 5.4.10. Thus, the sequence of lattice homomorphisms corresponds to the following sequence of spectral maps: g

X

⊆

/ f (X)

/ f (X)

⊆

/ Y,

where g is f with restricted codomain and the other maps are inclusions of spectral subspaces. Hence, f (X) is a closed subspace of Y if and only if the ◦

◦

homomorphism ϕi : K (Y )/i → ϕ(K (Y )) is an isomorphism, if and only if ◦ ◦ ϕi : K (Y )/i → K (X) is injective. Similarly, we can factor ϕ into ◦

πf

K (Y )

◦

◦

ϕf

/ K (Y )/f

⊆

/ ϕ(K (Y ))

◦

/ K (X) .

◦

Now K (Y )/f is the lattice that corresponds to Gen( f (X)) ⊆ Y (see 5.4.10), and the sequence of lattice homomorphisms corresponds to the following sequence of spectral maps: g

X

/ f (X)

⊆

⊆

/ Gen( f (X))

/Y.

These considerations lead to another characterization of closed spectral maps and inversely closed spectral maps. 5.4.12 Corollary Let f : X → Y be a spectral map. For a spectral subspace C ⊆ X, let fC : C → Y be the composition of f with the inclusion map C → X. ◦

◦

◦

We define ϕC = K ( fC ) : K (Y ) → K (C). (⊥). The map f is closed if and only if the (i) For A ∈ A(X), let i A = ϕ−1 A ◦

◦

◦

◦

induced homomorphism (ϕ A )i A : K (Y )/i A → K (A) is injective for every A ∈ A(X). (). The map finv is closed if and only if the (ii) For B ∈ A(Xinv ), let fB = ϕ−1 B induced homomorphism (ϕ B )fB : K (Y )/fB → K (B) is injective for every B ∈ A(Xinv ).  Finally, we record some useful elementary information about embeddings, which supplements 5.2.10. 5.4.13 Proposition (i) Let f : X → Y and g : Y → Z be embeddings. Then g◦ f is an embedding.

162

Properties of Spectral Maps

(ii) Let f : X → Y and g : Z → Y be spectral maps and consider the following pull-back diagram: f

X ×Y Z g

/Z g

 X

 / Y.

f

If f is an embedding (resp., a closed embedding, or an embedding onto a generically closed subspace), then so is f . (iii) Let f

X g

 Z

f

/Y  /T

g

be a push-out diagram in the category Spec. If f is an embedding (resp., a closed embedding, or an inversely closed embedding), then so is f . Proof (i) is obvious. (ii) It follows from 5.2.10(ii) that f  is a monomorphism. To see that f  is an embedding we show that z = f (x, z)  f (x , z ) = z  implies (x, z)  (x , z ). Since specialization in the fiber product is equivalent to specialization in both components, it suffices to show that x  x . Since f is an embedding, this follows from f (x) = g(z)  g(z ) = f (x ). If f is a closed embedding then f has going-up, 5.3.3, and f  has going-up as well, 5.3.2, hence is closed, 5.3.3. Now let f be a homeomorphism onto a  generically closed subspace. Then finv is a closed embedding. It follows that finv  is also a closed embedding, hence f is a homeomorphism onto a generically closed subspace. (iii) Consider the Stone dual of the push-out diagram in the statement of (iii): ◦

M × L N = K (T)

◦

ϕ  := K ( f  )

◦

/ K (Z) =: N

◦

ψ := K (g )

◦



◦

M := K (Y )

◦

ϕ:= K ( f )

◦



ψ:= K (g)

/ K (X) =: L ,

which is a pull-back in BDLat. We use the abbreviations shown in the daigram. Since f is an embedding we know that ϕ is surjective, 5.4.3. Then ϕ  is surjective as well, 5.2.10(ii), hence f  is an embedding, 5.4.3.

5.5 Irreducible Maps and Dominant Maps

163

Now let f be a closed embedding. It follows from 5.4.10(iii) that the induced map ϕi : M/i → L is an isomorphism, where i = ϕ−1 (⊥). Setting j = (ϕ )−1 (⊥) = i × {⊥} we must show that ϕ (b, c) = ϕ (b, c ) if and only if there is some (x, ⊥) ∈ j with (b, c) ∨ (x, ⊥) = (b, c ) ∨ (x, ⊥). First assume that c = ϕ (b, c) = ϕ (b, c ) = c . Then ϕ(b) = ψ(c) = ψ(c ) = ϕ(b). The hypothesis yields an element x ∈ i with b ∨ x = b ∨ x. It follows that (x, ⊥) ∈ j and (b, c) ∨ (x, ⊥) = (b, c ) ∨ (x, ⊥). Conversely, if (b, c) ∨ (x, ⊥) = (b, c ) ∨ (x, ⊥) with (x, ⊥) ∈ j then it is clear that ϕ (b, c) = c = c  = ϕ (b, c ). The assertion about inversely closed embeddings follows from the one about closed embeddings. 

5.5 Irreducible Maps and Dominant Maps Summary

A map f : X → Y between topological spaces is said to be

• irreducible if it is surjective and f (C)  Y for every proper closed subset C  X, • dominant if f (X) is dense in Y . Both properties concern the image of a map. Therefore, the results of the previous sections provide tools to analyze their meaning for spectral maps. In 5.5.1 and in 5.5.4 we characterize irreducible maps and dominant maps using the corresponding lattice homomorphisms. 5.5.1 Proposition The following conditions about a spectral map f : X → Y are equivalent: (i) f is irreducible. (ii) f is surjective, and f (V)  Y for every V ∈ K(X) \ {X }. (iii) The homomorphism K( f ) is injective, and for each V ∈ K(X) \ {X } there is some C ∈ K(Y ) \ {Y } such that f (V) ⊆ C. (iv) The homomorphism K( f ) is injective, and for each V ∈ K(X) \ {X } there are W1, W2 ∈ K(Y ) such that W2  W1 and f (V) ∩ W2 ⊆ W1 . Proof (ii) is a particular case of (i). The converse follows from the fact that K(X) is a basis of closed sets for X. (ii) ⇒ (iii) By 5.2.6 we know that K( f ) is injective. Given a set V ∈ K(X)\{X }, the hypothesis implies that f (V) is a proper proconstructible subset of Y . Every

164

Properties of Spectral Maps

proconstructible subset is an intersection of constructible sets. Hence there is some C ∈ K(Y ) \ {Y } such that f (V) ⊆ C. (iii) ⇒ (iv) Given V ∈ K(X) \ {X }, pick C as in (iii). There are W1, W2 ∈ K(Y ) such that ∅  W2 \ W1 ⊆ Y \ C. Then W2  W1 and f (V) ⊆ C ⊆ W1 ∪ (Y \ W2 ), which is equivalent to f (V) ∩ W2 ⊆ W1 . (iv) ⇒ (ii) Since K( f ) is injective, 5.2.5 and 5.2.6 imply that f is surjective. If V ∈ K(X) \ {X }, then there are W1 and W2 as in (iv), hence f (V) ⊆  W1 ∪ (Y \ W2 )  Y . 5.5.2 Lattice Homomorphisms and Irreducible Maps The equivalences of 5.5.1 can be restated in terms of lattices, using Stone duality. The following conditions about a homomorphism ϕ : L → M of bounded distributive lattices are equivalent: (i) Spec(ϕ) is irreducible. (ii) ϕ is injective and (Spec(ϕ))(V)  Spec(L) for all V ∈ K(Spec(M)) with V  Spec(M). (iii) ϕ is injective and for every b ∈ M \ {⊥} there is some x ∈ ba(L) such that ⊥ < ba(ϕ)(x) ≤ b (in ba(M)). (iv) ϕ is injective and for every b ∈ M \ {⊥} there are a1, a2 ∈ L such that a1  a2 and ϕ(a1 ) ≤ b ∨ ϕ(a2 ) (equivalently, ⊥ < ϕ(a1 ) ∧ ¬ϕ(a2 ) ≤ b in ba(M)). 5.5.3 Proposition Let f : X → Y be a surjective spectral map. Then there is a closed subset C ⊆ X such that the restriction f |C : C → Y is irreducible. Proof Let A be the set of closed subsets A ⊆ X such that f (A) = Y , and let   D ⊆ A be a chain. It follows from 5.1.1 that f ( D) = D ∈ D f (D) = Y . Therefore Zorn’s Lemma yields a minimal element C ∈ A, and it is clear that  f |C : C → Y is irreducible. 5.5.4 Proposition Let f : X → Y be a spectral map. The following conditions are equivalent: (i) f is dominant. (ii) If C ∈ A(Y ) \ {Y }, then f −1 (C)  X. (iii) If V ∈ K(Y ) \ {Y }, then f −1 (V)  X. ◦

◦

(iv) K ( f )−1 (⊥) is the trivial ideal {⊥} ⊆ K (Y ). (v) Y min ⊆ f (X). (vi) For every ideal i  K(Y ) there is an ideal j  K(X) such that K( f )(i) ⊆ j.

5.6 Extending Spectral Maps

165

Proof The equivalence (i) ⇔ (ii) is the definition of dominance. The equivalence (ii) ⇔ (iii) follows from the fact that K(Y ) is a basis of closed sets for Y . Conditions (iii) and (iv) are equivalent by forming complements. (Alternatively, (i) ⇔ (iv) follows from 5.4.11.) (i) ⇔ (v) follows from the fact that the proconstructible set f (X) ⊆ Y is dense if and only if it contains Y min , see 4.4.6. (iii) ⇒ (vi) Assume that i ⊆ K(Y ) is a proper ideal, but K( f )(i) is not contained in a proper ideal. Then there are V1, . . . ,Vr ∈ i such that X = f −1 (V1 ) ∪ · · · ∪ f −1 (Vr ). Setting V = V1 ∪ · · · ∪ Vr we have V ∈ i, hence V  Y , but K( f )(V) = f −1 (V) = X, which contradicts (iii). (vi) ⇒ (iii) Assume there is some V ∈ K(Y ), V  Y , such that K( f )(V) = X. Then the image of the principal ideal i(V) ⊆ K(Y ) (which is proper) contains X, hence is not contained in a proper ideal, a contradiction. 

5.6 Extending Spectral Maps Summary Let X be a spectral subspace of Y and consider a spectral map f : X → Z. We ask whether f can be extended to a spectral map f  : X  → Z defined on a larger spectral subspace of Y , or even on all of Y . This is the spectral version of a problem that is ubiquitous in general topology, [Kel75], [GiJe60], [Eng89]. In 5.6.2 we describe a spectral subspace X  ⊆ Y containing X such that f can be extended, even uniquely, to X . However, it may happen that X = X , and that there is no proper extension of f at all. An important special case of the general extension problem asks whether there exists a retraction r : Y → X from a spectral space onto a spectral subspace. It has been shown in 5.4.8 that retractions do not always exist. We exhibit a necessary condition for the existence of a retraction and show that it is also sufficient if Y \ X has only one element, 5.6.4. If the subspace X is empty then the extension problem is trivial. Therefore, we assume throughout that X  ∅ . 5.6.1 Uniqueness of Extensions First suppose that Y is a T0 -space, X ⊆ Y is a subspace, and f : X → Z is a continuous map. Pick a point y ∈ Y \ X and set X  = X ∪ {y}. In general it is not possible to extend f to a continuous map f  : X  → Z, cf. Example 5.4.8. Even if there is an extension f , it need not be unique. For an example, let Y be a finite (= discrete) Boolean space, X  Y a proper subspace, and Z a space

166

Properties of Spectral Maps

with at least two points. Then every map X  → Z of the underlying sets is continuous. Hence there are as many different extensions of f as Z has points. Now assume that Y has proper specializations. Then, if y is suitably positioned with respect to X, there is only one possible extension. Suppose that there are points x, x1 ∈ X such that x  y  x1 and f (x) = f (x1 ). If there is a continuous extension f , then f (x)  f (y)  f (x1 ) = f (x), and it follows that f (y) = f (x). The condition about the point y can be expressed in terms of convex hulls, cf. Section 4.5, namely y ∈ convY f −1 ( f (x)). Therefore, convexity enters into our considerations here. Turning to spectral spaces, consider a spectral subspace X ⊆ Y , a spectral space Z, and a spectral map f : X → Z. For the next result, recall that the set  C = z ∈Z convY ( f −1 (z)) is a proconstructible subset of Y , cf. 4.5.9. 5.6.2 Theorem Let Y be a spectral space, let X ⊆ Y be a spectral subspace,  and let f : X → Z be a spectral map. Setting C = z ∈Z convY ( f −1 (z)), there is a unique spectral map g : C → Z extending f . Proof By 5.6.1 we know that there is at most one spectral map g : C → Z that extends f , namely g(y) = z if y ∈ convY ( f −1 (z)). First we must show that g, as described, is a well-defined map. This is the case if and only if convY ( f −1 (z)) and convY ( f −1 (z )) are disjoint for z  z  in Z. Assume this is false and pick y ∈ convY ( f −1 (z)) ∩ convY ( f −1 (z )), say x  y  x1 and x   y  x1 with x, x1 ∈ f −1 (z) and x , x1 ∈ f −1 (z ). Then x  x1 and x   x1 implies z = f (x)  f (x1 ) = z  and z  = f (x )  f (x1 ) = z, hence z = z , a contradiction. We claim that the map g is continuous for the patch topology. So, pick a  set K ∈ K(Z). The sets g −1 (K) = z ∈K convY ( f −1 (z)) and g −1 (Z \ K) =  −1 z ∈Z\K convY ( f (z)) are proconstructible by 4.5.9. Since they are disjoint and C = g −1 (K) ∪ g −1 (Z \ K), we conclude that g −1 (K) ∈ K(C). It remains to show that g preserves specialization, 1.5.13. So, pick elements y  y1 in C, say x  y  x  with x, x  ∈ f −1 (z) and x1  y1  x1 with x1, x1 ∈ f −1 (z ). Then x  y  y1  x1 , which implies g(y) = f (x)  f (x1 ) = g(y1 ).  5.6.3 Existence of Spectral Retractions It will be possible to say more about the extension of spectral maps when fiber sums become available, cf. 10.2.19. Then the extension problem can be reduced to the question of whether, for a spectral subspace X ⊆ Y , there exists a spectral retraction r : Y → X (i.e., whether the identity map idX : X → X can be extended to a spectral map defined on all of Y ). Here we consider this special case of the extension problem.

5.6 Extending Spectral Maps

167

We determine a necessary condition for the existence of a retraction. Note that a retraction r : Y → X yields the retraction rcon : Ycon → Xcon of the patch spaces. The existence of a retraction from a Boolean space onto a closed subspace can be a difficult problem. It is related to projective Boolean algebras, cf. [MB89b, Chapter 20]. It is shown in [MB89b, Chapter 20, Thm. 1.6] that every countable Boolean algebra is projective. In our situation this means that there is a retraction at the level of patch spaces if Xcon has a countable basis. The question of whether a spectral retraction exists can be split into two problems: • First one looks for a retraction of the patch spaces. • Then one asks whether a retraction of the patch spaces is monotonic for the specialization order. If Y \ X consists of a single element then the issue is trivial for the patch spaces, and we obtain a complete description of the spectral retractions. 5.6.4 Proposition Let X be a spectral subspace of the spectral space Y and let y ∈ Y . If r : X ∪ {y} → X is a retraction, then   Spez(x) ∩ Gen(x ). r(y) ∈ S(y) = X ∩ x ∈Gen(y)∩X

x  ∈Spez(y)∩X

Further, for each z ∈ S(y) there is a retraction r : X ∪ {y} → X such that r(y) = z. Proof Assume that there is a spectral retraction r : X ∪ {y} → X. Then for each x ∈ Gen(y) ∩ X and all x  ∈ Spez(y) ∩ X the image r(y) must belong to Spez(x) ∩ Gen(x ). Thus, r(y) ∈ S(y). Conversely, pick an element z ∈ S(y) and define r : X ∪ {y} → X to be the retraction map with r(y) = z. The condition z ∈ S(y) expresses the fact that r respects specialization. Moreover, r is spectral since the patch space of X ∪ {y}  is the topological sum of Xcon and {y} (if y  X).  Continuing with the notation of 5.6.4, note that x ∈Gen(y)∩X Spez(x) = Y if Gen(y) ∩ X = ∅, and similarly for the other intersection. Therefore S(y)  ∅ holds trivially whenever y does not have any generalizations and specializations in X. As an illustration, consider Example 5.4.8, where we showed that retractions do not exist in general. In the example we have X ∩ Spez(c) = {d, e}, X ∩ Gen(c) = {a, b} and X ∩ Gen(d) ∩ Gen(e) = {a, b}, X ∩ Spez(a) ∩ Spez(b) = {d, e}. The intersection in 5.6.4 is empty, confirming that there is no retraction. The relationship with the general extension problem will be dealt with in 10.2.19.

6 Quotient Constructions

The formation of quotient spaces is an important and well-known construction in topology. Let X be a topological space and let E ⊆ X × X be an equivalence relation, let pE : X → X/E be the canonical map onto the set of equivalence classes. There is a finest topology on X/E such that pE is continuous. It is called the quotient topology and X/E is the quotient space of X modulo E, [Bou71b, p. TG I.20 f], [Eng89, p. 90 ff], [Kel75, p. 94 ff]. We ask whether the formation of quotient topologies is also possible in the category Spec. Examples show that the topological quotient space of a spectral space modulo an equivalence relation need not be spectral. However, we show that it is possible to construct spectral quotients having the same universal mapping property in Spec as topological quotient spaces in Top. The approach to quotients of spectral spaces is far more general than the familiar construction in general topology. We construct a quotient of a spectral space X with respect to any relation R ⊆ X × X, Section 6.1, and call it the spectral quotient modulo R. The construction is so general that for every surjective spectral map f : X → Y there is always a relation R ⊆ X × X such that Y is homeomorphic to the quotient of X modulo R. Different relations on a spectral space may lead to the same spectral quotient. For every relation R there is a largest one yielding the same quotient; it is called the saturation of R. Saturations are studied in Section 6.2. The definition of spectral quotients includes, as a special case, quotients modulo equivalence relations, which are studied in Section 6.4. These have distinguished topological and category-theoretic properties. An important instance of this construction is the formation of the space of connected components of a spectral space, which is discussed in Section 6.6. As always, Stone duality allows us to translate the topological constructions into lattice-theoretic language, which is explained in Section 6.5. In Section 168

6.1 Spectral Quotients Modulo Relations

169

6.3 we use the method of spectral quotients to present a new construction of a spectral space from a given finite collection of spectral spaces. To set the stage, we recall a few facts from topology. Let f : X → Y be a surjective map from a topological space onto a set. Usually there are many different topologies τ on Y such that f : X → (Y, τ) is continuous. But there is always a finest one, which is called the final topology with respect to f , or the quotient topology. Its open sets are the subsets O ⊆ Y such that f −1 (O) ∈ O(X), cf. [Bou71b, p. TG I.14]. Suppose that Y is a topological space and that f : X → Y is surjective and continuous. The map is called identifying if the topology of Y is the quotient topology for f . The following conditions are equivalent, cf. [Bou71b, p. TG I.21], [Eng89, p. 90 ff], [Kel75, p. 94 ff]: (i) f is identifying. (ii) If Z is a topological space and g : X → Z and h : Y → Z are maps between the underlying sets such that g = h ◦ f , then g is continuous if and only if h is continuous. (iii) For every topological space Z and every continuous map g : X → Z with ( f × f )−1 (ΔY ) ⊆ (g ×g)−1 (Δ Z ) there is a unique continuous map h : Y → Z such that g = h ◦ f .

6.1 Spectral Quotients Modulo Relations Summary Given a spectral space X and an arbitrary binary relation R ⊆ X × X we introduce the spectral quotient of X modulo R, 6.1.1. The definition stipulates that a universal mapping property is satisfied, similar to the mapping property of quotient spaces in topology. The construction of quotient spaces in topology is quite simple. Quotients also exist for spectral spaces, but the proof is less obvious, 6.1.6. Spectral quotients are so general that the codomain of any surjective spectral map is the spectral quotient of the domain for a suitable relation, 6.1.7. For example, a spectral space is the spectral quotient of its patch space modulo the specialization order, 6.1.8. 6.1.1 Definition Suppose that X is a topological space and that R ⊆ X × X is a relation. (i) A continuous map f : X → Y to a T0 -space is R-compatible if (x, x ) ∈ R implies f (x)  f (x ).

170

Quotient Constructions

(ii) A T0 -space Y together with an R-compatible continuous map q : X → Y is a T0 -quotient modulo R if the following universal property holds: for every T0 -space T and every R-compatible continuous map f : X → T there is a unique continuous map g : Y → T such that f = g ◦ q. (iii) Suppose that X is a spectral space. Then an R-compatible spectral map q : X → Y is a spectral quotient modulo R if the following universal mapping property holds: for every spectral space T and every R-compatible spectral map f : X → T there is a unique spectral map g : Y → T such that f = g ◦ q. Hochster mentions that spectral quotients modulo equivalence relations always exist, [Hoc69, Theorem 7]. Our construction is a vast generalization with many applications. The following notation is used frequently. If S is a set and E ⊆ S × S is an equivalence relation then [s] = [s]E is the equivalence class of s ∈ S. The set of equivalence classes is denoted by X/Set E. First we record a few simple observations about properties of T0 -quotients and spectral quotients modulo a relation. It will be shown in 6.1.5 and 6.1.6 that the quotients exist. 6.1.2 Basic Facts about R-Compatible Maps and Quotients topological space and let R ⊆ X × X be a binary relation.

Let X be a

(i) Suppose that X has a T0 -quotient modulo R. Then any two T0 -quotients are homeomorphic via a unique homeomorphism (i.e., if q : X → Y and q  : X → Y  are both T0 -quotients of X, there is a unique homeomorphism g : Y → Y  such that q  = g ◦ g). This follows at once from the universal mapping property. Therefore we speak about the T0 -quotient (once we have shown it exists, 6.1.5) and denote it by qR,T0 : X → X/T0 R. (ii) Similarly, if X is a spectral space and a spectral quotient modulo R exists, then any two spectral quotients are uniquely homeomorphic. We speak about the spectral quotient and denote it by qR : X → X/R (once we have shown it exists, 6.1.6). (iii) Let X be a spectral map with a binary relation R such that both quotients exist. Then there is a unique continuous map g : X/T0 R → X/R with qR = g ◦ qR,T0 . This follows from the mapping property of the T0 -quotient (since the spectral quotient qR is R-compatible). (iv) Suppose that f : X → Y is an R-compatible map. Then the map g : X → f (X) obtained by restriction of the codomain is R-compatible as well. Also, if h : Y → Z is continuous then h ◦ f is R-compatible.

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(v) Let ( fi : X → Yi )i ∈I be a family of R-compatible maps. Consider the  product i ∈I Yi together with the projection maps pi onto the components.  The map f : X → i ∈I Yi defined by fi = pi ◦ f is R-compatible as well (since specialization in the product is determined componentwise). (vi) Let X and Y be spectral spaces and f : X → Y a map that is patch -compatible, continuous. Then f is a spectral map if and only if f is  X 1.5.13. 6.1.3 Proposition Let X be a topological space (resp., a spectral space) with a relation R ⊆ X × X. The T0 -quotient map (resp., the spectral quotient map) is surjective, assuming that the quotients exist. Proof The arguments are identical for both quotients. We present the proof for the spectral quotient. Let q : X → qR (X) be the map q with restricted codomain and let i : qR (X) → X/R be the inclusion. Note that q is R-compatible, 6.1.2(iv). We claim that q has the same universal mapping property as qR . So, let f : X → T be an R-compatible spectral map. There exists a unique spectral map g : X/R → T such that f = g ◦ qR . Then f also factors through qR (X), namely f = (g ◦ i) ◦ q. The factorization is unique since q is surjective. Thus, q has the mapping property of the spectral quotient. Now it follows from 6.1.2(i) and (ii) that i is a homeomorphism (i.e., qR is surjective).  6.1.4 The T0 -Reflection of a Topological Space We recall a construction from general topology. Let X be a topological space. For any point x ∈ X, let Ox (X) be the set of open neighborhoods of x. Then E = {(x, x ) | Ox (X) = Ox (X)} is an equivalence relation. Let q : X → X/E be the canonical map onto the set of equivalence classes. If we equip X/E with the quotient topology then q is a quasi-homeomorphism, 4.5.22 (i.e., the lattice homomorphism O(X/E) → O(X) : O → q−1 (O) is an isomorphism). The inverse isomorphism is given by U → q(U). One checks easily that X/E is a T0 -space. The quotient map q has the following universal mapping property. Let f : X → Z be any continuous map to a T0 -space. Then there is a unique continuous map g : X/E → Z such that f = g ◦ q. The map q : X → X/E is called the T0 -reflection of X. Let τ ⊆ O(X) be a coarser topology. Then, in exactly the same way, τ defines an equivalence relation Eτ . The T0 -reflection qτ : (X, τ) → X/Eτ exists as well. There is a unique continuous map g : X/E → X/Eτ such that qτ = g ◦ q.

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6.1.5 Proposition Suppose that X is a topological space with a relation R ⊆ X × X. Then the T0 -quotient modulo R exists. Proof

We define τ = {U ∈ O(X) | ∀(x, x ) ∈ R : x  ∈ U ⇒ x ∈ U}.

One checks easily that τ is a topology on X. Trivially, τ is coarser than O(X). Let qτ : (X, τ) → X/Eτ be the T0 -reflection of (X, τ) (with notation as in 6.1.4). We claim that qτ is R-compatible. So, let (x, x ) ∈ R. The definition of τ implies {U ∈ τ | x  ∈ U} ⊆ {U ∈ τ | x ∈ U}, which yields the inclusion {qτ (U) | U ∈ τ and [x ] ∈ qτ (U)} ⊆ {qτ (U) | U ∈ τ and [x] ∈ qτ (U)} and the specialization [x]  [x ]. It remains to check that qτ : X → X/Eτ has the required universal mapping property. Let f : X → T be any continuous and R-compatible map into a T0 space. The image of the lattice homomorphism O( f ) : O(T) → O(X), O → f −1 (O) is contained in τ, hence f : (X, τ) → T is continuous. The universal property of the T0 -reflection yields a unique continuous map g : X/Eτ → T  such that f = g ◦ qτ . 6.1.6 Theorem The spectral quotient of X modulo R exists for any spectral space X and any relation R ⊆ X × X. Proof In Section 6.5 we shall use Stone duality to give a lattice-theoretic description of spectral quotients. Proving 6.5.2 we then obtain an alternative proof of the existence of spectral quotients. Here we use topological and categorytheoretic methods. Let E(X, R) be the set of isomorphism classes of R-compatible epimorphisms with domain X, cf. 5.2.8. Note that the map to the final object 1 is trivially R-compatible. Hence the set is nonempty. We pick a set of representatives for the isomorphism classes and write them as a family (qi : X → Zi )i ∈I . The index set I is made into a poset by defining: i ≤ j if there is a spectral map qi j : Z j → Zi such that qi = qi j ◦ q j . The map qi j is unique, since q j is an epimorphism. It follows from 6.1.2(v) that the poset is up-directed. Given X → Zi and X → Z j a common upper bound is obtained from the map X → Zi × Z j , x → (qi (x), q j (x)) by restricting the codomain to the image. Thus, the family (Zi )i ∈I of codomains together with the

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maps qi j is a projective system. Let Z be the projective limit with projection maps pi : Z → Zi (see 2.3.9). The universal property of the projective limit provides a map q : X → Z such that qi = pi ◦ q for each i ∈ I. We show that q : X → Z is the spectral quotient. As a first step we show: Claim

The map q : X → Z is surjective.

Proof of Claim It follows from 6.1.2(v) that q is R-compatible (since the  projective limit is a spectral subspace of the product i ∈I Zi and since each map qi is R-compatible). The map q  : X → q(X) obtained by restriction of the codomain is also R-compatible by 6.1.2(iv), and is an epimorphism. By definition of the family (Zi )i ∈I , there are a unique element i0 ∈ I and a unique homeomorphism r : q(X) → Zi0 such that qi0 = r ◦ q . Thus, i0 is the largest element of I and pi0 : Z  Zi0 is a homeomorphism. It follows that q(X) = Z.  Finally we check that q has the mapping property of the spectral quotient. Suppose that the spectral map f : X → Y is R-compatible. There is at most one map g : Z → Y such that f = g ◦ q (since q is an epimorphism). For the existence, note that the map f  : X → f (X) (restricted codomain) is an Rcompatible epimorphism. Therefore there exist some i ∈ I and an isomorphism h : Zi → f (X) such that f  = h ◦ qi . Let e : f (X) → Y be the inclusion map.  Then f = e ◦ f  = (e ◦ h ◦ pi ) ◦ q is the desired factorization. 6.1.7 Corollary Let f : X → Y be a surjective spectral map. Then X/R  Y , where R is the preimage of Y  under f × f . Proof Obviously, f is R-compatible. Let g : X/R → Y be the unique spectral map with f = g ◦ qR . Note that g is surjective since f is surjective. We claim that g is also injective. For, if f (x) = g ◦ qR (x) = g ◦ qR (x ) = f (x ) then (x, x ), (x , x) ∈ R, hence qR (x)  qR (x ) and qR (x )  qR (x), which yields qR (x) = qR (x ). We have shown that g is a bijective spectral map. To show that g is a homeomorphism we must prove that g reflects specialization (i.e., g(qR (x))  g(qR (x )) implies qR (x)  qR (x )). The hypothesis says that f (x) = g(qR (x))  g(qR (x )) = f (x ). But then (x, x ) ∈ R and qR (x)  qR (x ) (by Rcompatibility).  We emphasize the following important special case of 6.1.7, which is obtained by setting f = conX : Xcon → X. 6.1.8 Corollary The spectral space X is the spectral quotient of the Boolean  space Xcon modulo the specialization relation X  .

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6.1.9 Examples (i) Suppose that X is a spectral space with a binary relation R. We can form the T0 -quotient qR,T0 : X → X/T0 R and the spectral quotient qR : X → X/R. Let g : X/T0 R → X/R be the unique continuous map such that qR = g ◦ qR,T0 , 6.1.2(iii). We show with an example that X/T0 R may be a spectral space without being the spectral quotient. Let X be the spectral space ω + 1, 1.6.6, and R = {(n + 1, n) | n ∈ N0 }. Let f : X → Y be a continuous R-compatible map into a T0 -space. For each n ∈ N0 we have f (n)  f (n + 1) (by continuity) and f (n + 1)  f (n) (by R-compatibility). Thus, f is constant on N0 . Now let Y = 2 and define f (n) = 0 for all n ∈ N0 and f (ω) = 1. This map is the T0 -quotient, as one checks easily. The domain and the codomain are both spectral spaces. But f is not a spectral map since f −1 (0) = N0 is open, but not quasi-compact in X, 1.6.3(i). We conclude that the spectral quotient is the final object 1. (ii) Let X be a spectral space with an equivalence relation R. Then the T0 quotient need not coincide with the topological quotient. For an example, let X = 3, which is essentially the totally ordered set 0  1  2. With R = ΔX ∪ {(0, 2), (2, 0)} the topological quotient space is the indiscrete space on the set {[0]R , [1]R }. The T0 -quotient, which is the same as the spectral quotient, is the space 1.

6.2 Saturated Relations Summary As shown in 6.1.6, every binary relation R ⊆ X × X on a spectral space determines the spectral quotient qR : X → X/R. Usually there are many different relations that define the same spectral quotient. Among these there is a largest relation, namely (qR × qR )−1 ((X/R)  ), which we denote by Rsat and call the saturation of R, cf. 6.1.7. A relation is saturated if it agrees with its saturation. The formation of the saturation is a closure operator, [Coh81, p. 42], on the power set P(X × X) – that is, it is a map P(X × X) → P(X × X) with the following properties: (i) R ⊆ S implies Rsat ⊆ S sat , (ii) R ⊆ Rsat , and (iii) Rsat = (Rsat )sat .

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By 6.1.7 the set of isomorphism classes of epimorphisms with domain X, cf. 5.2.8, corresponds bijectively to the set of saturated relations. We would like to find a characterization of saturated relations or to construct the saturation of a given relation. This goal is extremely ambitious. We know from 6.1.8 that any spectral space is the quotient of its patch space modulo the specialization relation. In fact, given a Boolean space X, the saturated partial orders are exactly the spectral orders on X, 6.2.1. Thus, any general characterization of saturated relations includes, as a special case, a characterization of specialization orders. This, however, is an open problem known to be hard, cf. the discussion of the Kaplansky Problem in 4.2.13. So, we aim at more modest objectives. We take steps in the direction outlined above, but do not expect to reach the ultimate goals. Let R ⊆ X × X be any relation. There is a smallest proconstructible transitive relation S containing R ∪ X  . Clearly, S is contained in the saturation of R, and one may ask whether it actually coincides with Rsat . Example 6.2.2 shows that this question has a negative answer in general. On the other hand, if S is constructible, then we show that the answer is positive, 6.2.4. So, constructible saturated relations are accessible. The best we achieve for general saturated relations is to show that they are intersections of constructible saturated relations, 6.2.7. 6.2.1 Proposition Let X be a Boolean space and R a partial order on X. Then R is a spectral order if and only if R is saturated. Proof Assume that R is spectral and let Y be the spectral space with patch space X and specialization order R, cf. 1.5.11. Then Y = X/R, see 6.1.7, shows that R is saturated. Conversely, assume that R is saturated (i.e., R = (qR × qR )−1 ((X/R)  )). Antisymmetry of R implies that qR is injective, hence  (X/R)con = X and R coincides with the spectral order (X/R)  . 6.2.2 Example We exhibit a spectral space X together with a closed equivalence relation E ⊆ X × X such that the topological quotient space X/E is not spectral. This implies that E is not saturated. Let Y be a compact space that is not Boolean and let X = PrimF(P(Y )) be the space of ultrafilters of Y (which is a Boolean space by 1.3.4 and 3.3.2). If  u ∈ X, then, by compactness of Y , there is a unique point g(u) ∈ Z ∈u Z.  Claim For any closed subset A of Y we have g −1 (A) = {V(B) | B ⊆ Y closed and A ⊆ int(B)}. In particular, g : X → Y is continuous. Proof of Claim For the inclusion ⊆, take u ∈ g −1 (A) and assume A ⊆ int(B). It follows that B ∈ u. For, otherwise, Y \ B ∈ u, hence g(u) ∈ Y \ B, contradicting

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g(u) ∈ int(B). Conversely, if u is in the intersection, then g(u) ∈ B for all closed neighborhoods of A. Since Y is normal, g(u) ∈ A.  As X and Y are compact, the continuous surjective map g is identifying. Moreover, the equivalence relation E = (g × g)−1 (ΔY ) is closed in X × X. The spectral quotient qE : X → X/E yields the continuous and E-compatible map (qE )con : X = Xcon → (X/E)con of the patch spaces. The mapping property defining X/E gives a spectral map X/E → (X/E)con , which is the inverse of conX/E . Hence X/E is Boolean. Since Y carries the quotient topology with respect to g, there is a unique continuous and surjective map h : Y → X/E such that qE = h ◦ g. The space Y is not Boolean, hence h is not a homeomorphism, hence E is not saturated. Taking Y = [0, 1] ⊆ R, the Boolean space X/E is connected, hence it follows that X/E  1. 6.2.3 The Quotient Order Defined by a Relation Suppose that the relation R ⊆ X × X is proconstructible, transitive, and contains X  . With R−1 = {(y, x) ∈ X × X | (x, y) ∈ R} the relation E = R ∩ R−1 is a proconstructible equivalence relation. We define a binary relation on X/Set E (the set of equivalence classes) by setting [x] ≤R [y] if there exist x  ∈ [x] and y  ∈ [y] such that (x , y ) ∈ R (which is equivalent to (x , y ) ∈ R for all x  ∈ [x] and all y  ∈ [y]). The relation ≤R is a partial order, which we call the quotient order on X/Set E. If the relation R is saturated then it is clearly true that X/Set E is the underlying set of the spectral space X/R and ≤R is the specialization order of X/R. 6.2.4 Theorem Let X be a spectral space with a proconstructible and transitive relation R ⊆ X × X that contains X  . Let E be the equivalence relation R ∩ R−1 . Then the following conditions are equivalent: (i) R is constructible in X × X. (ii) E is constructible in X × X. (iii) The set X/Set E is finite. If the equivalent conditions are satisfied, then the relation R is saturated. The spectral quotient X/R is finite with underlying set X/Set E. The specialization relation of X/R is the quotient order of R on X/Set E. Proof The implication (i) ⇒ (ii) holds because intersections of constructible sets are constructible. (ii) ⇒ (iii) By compactness it suffices to show that every equivalence class is clopen for the patch topology. Closedness is clear since the equivalence relation is proconstructible. For any equivalence class A modulo E and any

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x ∈ A there is some constructible neighborhood C of x such that C × C ⊆ E (since (x, x) ∈ E). It follows that C ⊆ A, hence A is open. (iii) ⇒ (i) The set X/Set E with the quotient order is a finite poset, hence is the specialization poset of a unique spectral topology on X/Set E. We denote this spectral space by Y . The set X/Set E with the quotient topology of Xcon is a finite Hausdorff space (since E ⊆ Xcon × Xcon is proconstructible, i.e., closed, [Bou71b, p. TG I.55, Proposition 8]), hence is discrete. Thus, the canonical map q : X → Y : x → [x]E is continuous for the patch topology. The definition of the quotient order shows that q(x) ≤R q(x ) if and only if (x, x ) ∈ R (i.e., R = (q × q)−1 (Y  )). Since Y  ⊆ Ycon × Ycon is clopen it follows that R ⊆ Xcon × Xcon is clopen, proving (i). Assume that the equivalent conditions (i)–(iii) hold. It remains to show that R is saturated and that Y = X/R. The map q is spectral since it is R-compatible and X  ⊆ R. Saturatedness follows from R = (q × q)−1 (Y  ). It is clear that q has the universal mapping property of the spectral quotient.  6.2.5 Corollary Let X be a spectral space and R a constructible transitive relation on X containing the specialization relation. Then the spectral quotient modulo R coincides with the T0 -quotient. Proof The T0 -quotient and the spectral quotient are both finite and coincide as posets, see 6.2.4.  We know from 6.2.2 that proconstructible and transitive relations are not saturated in general. So the assertions of 6.2.4 are not true for relations that are only proconstructible (instead of being constructible). But we show that every saturated proconstructible relation is an intersection of saturated constructible relations. The next lemma is needed as a tool. 6.2.6 Lemma Let X be a spectral space. Suppose that (Ri )i ∈I is a family of  saturated relations on X. Then R = i ∈I Ri is saturated as well. Proof Let q : X → X/R be the quotient map. For x, y ∈ X with q(x)  q(y) we must show (x, y) ∈ R. We form the quotient spectra qi : X → X/Ri . The maps qi are R-compatible, hence the universal property of q yields unique spectral maps gi : X/R → X/Ri such that qi = gi ◦ q. Then q(x)  q(y) implies qi (x)  qi (y) for all i. As Ri is saturated, it follows that (x, y) ∈ Ri for all i ∈ I (i.e., (x, y) ∈ R).  6.2.7 Proposition Let X be a spectral space with a proconstructible, reflexive, and transitive relation R ⊆ X × X. Then the following conditions are equivalent:

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(i) R is saturated. (ii) There is a family (Ri )i ∈I of constructible transitive relations containing  X  such that R = i ∈I Ri . (iii) There is a family (Ri )i ∈I of constructible transitive relations containing  X  such that R = i ∈I Ri and each equivalence relation Ri ∩ Ri−1 has only two classes. Proof (i) ⇒ (iii) By 2.3.1 there are a set I and a spectral map f : X/R → 2I that is a homeomorphism onto the image. Let pi : 2I → 2 be the projections onto the components. Then (2.3.1 and 2.2.3)  ((pi ◦ f ) × (pi ◦ f ))−1 (2  ). (X/R)  = i ∈I

Since R is saturated, we get R = (qR × qR )−1 ((X/R)  ) =



Ri ,

i ∈I

where

 −1 Ri = (pi ◦ f ◦ qR ) × (pi ◦ f ◦ qR ) (2  ).

Clearly, each Ri is constructible, transitive, and contains X  . Furthermore, Ri ∩ Ri−1 has only two classes. (iii) ⇒ (ii) is trivial.

 (ii) ⇒ (i) We write R = i ∈I Ri with a family (Ri )i ∈I of constructible and transitive relations that contain X  . By 6.2.4, the Ri are saturated. By 6.2.6, their intersection R is saturated as well.  To finish the section we review what has been achieved. Let R ⊆ X × X be a relation on a spectral space. The saturation Rsat contains R and the specialization relation of X, is transitive and proconstructible. So, to determine the saturation one considers the relation R ∪ X  , forms its transitive hull, and then the patch closure. But the patch closure need not be transitive anymore. So, one forms the transitive hull again, which, however, is not proconstructible in general. The process must be repeated over and over again. Conceivably one needs a transfinite iteration to reach a transitive and proconstructible relation. But even then the saturation has not been reached, in general, as Example 6.2.2 shows. So, building up the saturation from the inside is a hopelessly difficult and complicated process. The only exception is the case where R ∪ X  is a constructible relation, see 6.2.4. Another strategy is to approach the saturation from outside. We have shown in 6.2.7 that the saturation of a relation is an intersection of a family (Ri )i ∈I of

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constructible relations such that the equivalence relations Ri ∩ Ri−1 have two classes. We can make this statement a little more precise by specifying the index set I. Theorem 2.3.1 tells us that I may be chosen as K(X/R). However, not knowing X/R we also do not know K(X/R). Therefore another suitable set I is needed. We may choose I to be the set of all R-compatible spectral maps qi : X → X/Ri  2 onto the Sierpiński space.

6.3 Spectral Orders and Spectral Relations Summary In general, it is a difficult task to determine the saturation of a relation on a spectral space, or to find out whether a given relation is saturated, Section 6.2. In this section the problem is studied for the case that the relation is a partial order. We show, see 6.3.4, that a partial order on a Boolean space Z is spectral (i.e., is saturated, see 6.2.1) if and only if its graph is closed in Z × Z and there is a finite partition of Z into clopen subspaces such that the restriction of the partial order to each clopen piece is spectral. We apply this result to exhibit a new construction of spectral spaces. Let X and Y be disjoint spectral spaces and R ⊆ X × Y a subset. Then R naturally defines a partial order on the topological sum Z = X ⊕ Y , cf. 6.3.5. If the partial order is spectral, then we call R a spectral relation on (X,Y ), cf. 6.3.7 and 6.3.8. We show that the relation is spectral if R is proconstructible in X × Y , 6.3.9. The construction of spectral orders via spectral relations can be iterated, which is described in 6.3.12. 6.3.1 Relations on a Sum of Two Boolean Spaces We start by introducing notation to be used throughout this section. Let X and Y be disjoint Boolean spaces and define Z = X ⊕ Y to be the topological sum, which is also a Boolean space (cf. 2.4.3). We consider a relation R ⊆ Z × Z. The projections Z × Z → Z onto the components are denoted by p1 and p2 . Let R−1 = {(y, x) ∈ Y × X | (x, y) ∈ R} be the relation obtained by swapping the components. For A, B ⊆ Z we set R(A) = {z  ∈ Z | ∃z ∈ A: (z, z ) ∈ R} = p2 (p−1 1 (A) ∩ R), R−1 (B) = {z ∈ Z | ∃z  ∈ B : (z, z ) ∈ R} = p1 (p−1 2 (B) ∩ R). We partition R into the following subsets, which will be used repeatedly: • RX,X = R ∩ (X × X), RY ,Y = R ∩ (Y × Y ), • RX,Y = R ∩ (X × Y ), RY ,X = R ∩ (Y × X). Assume that A ⊆ X and B ⊆ Y . Then RX,Y (A) ⊆ Y and RY ,X (B) ⊆ X.

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6.3.2 Partial Orders on a Sum of Two Boolean Spaces We continue with the setup of 6.3.1 and assume in addition that R is the graph of a partial order ≤ on Z. Then RX,X and RY ,Y are the graphs of the restrictions ≤|X and ≤|Y . For A, B ⊆ Z we have R(A) = A↑ and R−1 (B) = B ↓ . Consider a subset A ⊆ X. Then RX,Y (A) = R(A) ∩ Y = A↑ ∩ Y . If A is an up-set in X then it follows that RY ,X (RX,Y (A)) = R(A↑ ∩ Y ) ∩ X = (A↑ ∩ Y )↑ ∩ X ⊆ A, hence A ∪ RX,Y (A) is an up-set in Z. Similarly, if B ⊆ Y is an up-set then RX,Y (RY ,X (B)) = R(B ↑ ∩ X) ∩ Y = (B ↑ ∩ X)↑ ∩ Y ⊆ B, and B ∪ RY ,X (B) is an up-set in Z. Let A ⊆ X and B ⊆ Y be up-sets with RX,Y (A) ⊆ B and RY ,X (B) ⊆ A. Then A ∪ B is an up-set in Z. Assume that the partial order is even closed (i.e., R ⊆ Z × Z is a closed subset, 1.5.10). The projection maps p1 and p2 are both continuous and closed. Thus, if A, B ⊆ Z are closed subsets then R(A) and R−1 (B) are closed as well, 6.3.1. But note that R(A) or R−1 (B) need not be clopen if A or B is clopen. 6.3.3 Proposition With the notation of 6.3.1 and 6.3.2, let R be the graph of a closed partial order ≤ on Z. Assume that ≤|X and ≤|Y are spectral orders on X and Y , let X  and Y  be the corresponding spectral spaces, see 1.5.11. Then, for each A ∈ K(X ) there is some B ∈ K(Y ) with RX,Y (A) ⊆ B and RY ,X (B) ⊆ A. Thus, A ∪ B ∈ K(Z) is a constructible up-set. Proof Note that RX,Y (A) ⊆ Y is a closed up-set (i.e., RX,Y (A) is closed in Y ).  Thus, RX,Y (A) = B ∈B B where B = {B ∈ K(Y ) | RX,Y (A) ⊆ B}. It suffices to show that there is some B ∈ B with RY ,X (B) ⊆ A. Assume the claim is false. The family (RY ,X (B)∩(X \A))B ∈B has the FIP since B is down-directed by inclusion. The members of the family are all closed in the  compact set X \ A. Compactness yields an element x ∈ B ∈B RY ,X (B)∩(X \ A). Note that RY−1,X (x) = x ↓ ∩Y . Thus, for each B ∈ B the set RY−1,X (x) ∩ B is closed and nonempty in the compact set RY−1,X (x). Hence the family (RY−1,X (x) ∩ B)B ∈B has the FIP as well, and there is some  B = RY−1,X (x) ∩ RX,Y (A), y ∈ RY−1,X (x) ∩ B ∈B

which implies x ∈ RY ,X (RX,Y (A)) ⊆ A (see 6.3.2), a contradiction. The final statement follows from 6.3.2.



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6.3.4 Theorem ([BMM02, Lemma 3.1]) Let Z be a Boolean space with a partial order ≤, whose graph is denoted by R. The following conditions are equivalent: (i) ≤ is a spectral order. (ii) R ⊆ Z × Z is closed, and there is a partition Z = Z1 ∪ · · · ∪ Zn into clopen subsets such that each restriction ≤| Zi is a spectral order. Proof (i) ⇒ (ii) is trivial. Recall that the graph R of the spectral order is closed in Z, cf. 1.5.9. (ii) ⇒ (i) By a trivial induction it suffices to show the claim for a partition into two clopen subsets. So, let R ⊆ Z × Z be closed and let Z = X ∪ Y , where X,Y ∈ K(Z) and X ∩ Y = ∅. Suppose that ≤|X and ≤|Y are both spectral orders on X and Y . The corresponding spectral spaces are denoted by X  and Y . To show that ≤ is spectral, pick z, z  ∈ Z with z  z . We have to produce a constructible ≤-up-set C ⊆ Z with z ∈ C and z   C. We prove the claim for z ∈ X. The case that z ∈ Y then follows by symmetry. First assume that also z  ∈ X and pick A ∈ K(X ) with z ∈ A and z   A. Then 6.3.3 yields a set B ∈ K(Y ) with RX,Y (A) ⊆ B and RY ,X (B) ⊆ A. Hence A ∪ B is a constructible ≤-up-set of Z meeting the requirements. −1 (z  ) = (z  ) ↓ ∩ X, Now suppose that z  ∈ Y . Note that z does not belong to RX,Y  which is a generically closed proconstructible subset of X . Hence there is some −1 (z  ) = ∅ (i.e., z   R A ∈ K(X ) with z ∈ A and A ∩ RX,Y X,Y (A)). Now 6.3.3

yields a set B ∈ K(Y ) with RX,Y (A) ⊆ B and RY ,X (B) ⊆ A. The proof of 6.3.3 shows that there is even such a set B not containing z . Thus, A ∪ B is a constructible up-set as needed.  6.3.5 Construction of Spectral Spaces by Spectral Relations – Preparations In the rest of the section we present a new construction of spectral spaces. It produces a new space from finitely many disjoint spectral spaces together with a partial order on their union so that the new space contains each of the original spaces as a spectral subspace. The main step is to construct a new space from two given spaces. The general case of finitely many spaces is then done recursively, see 6.3.12. So, let X and Y be disjoint spectral spaces and let R ⊆ X × Y be a relation. The topological sum Z = X ⊕ Y is a spectral space. We define the relation S ⊆ Z × Z to be the transitive hull of X  ∪ Y  ∪ R. It is a routine matter to show that S is the graph of a partial order ≤ = ≤X,Y ,R on Z.

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Quotient Constructions

Explicitly, z ≤ z  if and only if • (z, z ) ∈ X × X and z  X z , or • (z, z ) ∈ Y × Y and z  Y z , or • (z, z ) ∈ X × Y and there is (x, y) ∈ X × Y such that z  X x R y  Y z . We adorn ≤ by suitable indices whenever it is necessary to emphasize dependence on R, X, and Y . Note that SX,X = X  , SY ,Y = Y  , R ⊆ SX,Y , and SY ,X = ∅, using the notation of 6.3.1. Thus, S is a spectral order if and only if SX,Y is a proconstructible subset of X × Y , 6.3.4. We want to find conditions, in terms of the relation R, for S to be a spectral order. 6.3.6 Lemma Let X, Y , Z and R, S, ≤ be as in 6.3.5. We consider R as a relation on Z. Then: (i) For each y ∈ Y , y ↓ = GenX (R−1 (GenY (y))) ∪ GenY (y). (ii) For each x ∈ X, x ↑ = SpezX (x) ∪ SpezY (R(SpezX (x))). (iii) A subset C ⊆ Z is a constructible up-set for ≤ if and only if C ∩ X ∈ K(X), C ∩ Y ∈ K(Y ), and R(C ∩ X) ⊆ C ∩ Y . Proof (i), (ii) For x ∈ X and y ∈ Y it is clearly true that x ↑ ∩ X = SpezX (x) and y ↓ ∩ Y = GenY (y). The rest of the assertions follow from the following equivalent conditions about x ∈ X and y ∈ Y : • • • •

y ∈ x ↑. There are x  ∈ X and y  ∈ Y with x  x , (x , y ) ∈ R and y   y. There is y  ∈ R(SpezX (x)) with y ∈ SpezY (y ). There is x  ∈ R−1 (GenY (y)) with x ∈ GenX (x ).

(iii) Let C ⊆ Z be a constructible ≤-up-set. Then C ∩ X and C ∩ Y are constructible up-sets in X and Y (i.e., C ∩ X ∈ K(X), C ∩ Y ∈ K(Y )). If y ∈ R(C ∩ X) then there is some x ∈ C ∩ X with (x, y) ∈ R, hence x ≤ y. As C is an up-set it follows that y ∈ C ∩ Y . Conversely, suppose that C∩X ∈ K(X), C∩Y ∈ K(Y ), and R(C∩X) ⊆ C∩Y . Then C = (C ∩ X) ∪ (C ∩ Y ) is constructible in Z. Suppose that z ≤ z  and z ∈ C. If z, z  ∈ X or z, z  ∈ Y then it is clear that z  ∈ C. If z ∈ X and z  ∈ Y then pick x ∈ X and y ∈ Y with z  x, (x, y) ∈ R and y  z . Then x ∈ C ∩ X,  hence y ∈ R(C ∩ X) ⊆ C ∩ Y , hence z  ∈ C ∩ Y .

6.3 Spectral Orders and Spectral Relations 6.3.7 Proposition equivalent:

183

With the notation of 6.3.6 the following conditions are

(i) ≤ is a spectral order on Zcon . (ii) (a) R−1 (GenY (y)) is quasi-compact in X for all y ∈ Y , and (b) R(A) is inversely quasi-compact in Y for all A ∈ K(X). (iii) (a) (b) (iv) (a) (b)

◦

R−1 (U) is quasi-compact in X for all U ∈ K (Y ), and R(SpezX (x)) is inversely quasi-compact in Y for all x ∈ X. R−1 (B) is quasi-compact in X for B ⊆ Y inversely closed, and R(A) is inversely quasi-compact in Y for all closed A ⊆ X.

Proof (i) ⇒ (iv) Let A ⊆ X be closed and B ⊆ Y inversely closed. Both sets are proconstructible. It follows that S(A) ∩ Y = p2 (p−1 1 (A) ∩ S)) ∩ Y = SpezY (R(A)), −1 S −1 (B) ∩ X = p1 (p−1 2 (B) ∩ S) ∩ X = Gen X (R (B))

are proconstructible in Y and X. By 4.1.3 and 4.1.5 it follows that R(A) is inversely quasi-compact in Y and R−1 (B) is quasi-compact in X. Obviously, (iv) implies both (ii) and (iii). (ii) ⇒ (i) We have to show that SX,Y is proconstructible in X ×Y , see 6.3.4. So, suppose that (x, y) ∈ X ×Y \ SX,Y . It suffices to show that there is a constructible ≤-up-set C ⊆ Z with x ∈ C and y  C, for then (x, y) ∈ C × (Z \ C) and (C × (Z \ C)) ∩ S = ∅. As R−1 (GenY (y)) is quasi-compact it follows that y ↓ ∩ X = S −1 (GenY (y)) ∩ X = GenX (R−1 (GenY (y))) is inversely closed in X (by (ii)(a) and 4.1.5) and does not contain x. Thus, there is some A ∈ K(X) with x ∈ A and y ↓ ∩ A = ∅ . As R(A) ⊆ Y is inversely quasi-compact (by (ii)(b)) it follows that S(A) ∩ Y = SpezY (R(A)) is closed in Y . Note that y  SpezY (R(A)) since y ↓ ∩ A = ∅ . Hence there is some B ∈ K(Y ) with SpezY (R(A)) ⊆ B and y  B. By 6.3.6(iii) the set A ∪ B is a constructible up-set for ≤ with x ∈ C, but y  C. (iii) ⇒ (i) is shown in exactly the same way, or one appeals to the inverse topology and the implication (ii) ⇒ (i).  6.3.8 Definition Let X and Y be disjoint spectral spaces. A subset R ⊆ X ×Y is a spectral relation on (X,Y ) if the equivalent conditions of 6.3.7 are satisfied. 6.3.9 Corollary Let X and Y be disjoint spectral spaces and R ⊆ X × Y a proconstructible subset. Then ≤X,Y ,R is a spectral order. For example, this is the case if X  ⊆ X is proconstructible and R is the graph of a continuous  →Y . function f : Xcon con

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Quotient Constructions

Proof For a closed subset A ⊆ X and an inversely closed subset B ⊆ Y the sets R(A) and R−1 (B) are proconstructible, hence the conditions of 6.3.7(iv) are satisfied.  6.3.10 Corollary Let X and Y be disjoint Boolean spaces and let R ⊆ X × Y be a relation. Set Z = X ⊕ Y and let ≤ be the partial order ≤X,Y ,R . Then the following conditions are equivalent: (i) ≤ is a spectral order. (ii) R ⊆ X × Y is closed. (iii) For all (x, y) ∈ (X × Y ) \ R there is some A ∈ K(X) such that x ∈ A, y  R(A), and R(A) is closed in Y . Proof Let S be the graph of the partial order ≤ again. Since X and Y are Boolean it follows that S = ΔX ∪ ΔY ∪ R and SX,Y = R. Thus, ≤ is spectral if and only if S ⊆ Z × Z is closed (by 6.3.4), if and only if R ⊆ X × Y is closed. This proves (i) ⇔ (ii). (ii) ⇒ (iii) For (x, y) ∈ (X × Y ) \ R there are A ∈ K(X) and B ∈ K(Y ) with (x, y) ∈ A × B and (A × B) ∩ R = ∅. It follows that R(A) ∩ B = ∅, hence y  R(A). Moreover, R(A) is closed since R is a closed relation. (iii) ⇒ (ii) Suppose that (x, y) ∈ (X × Y ) \ R and pick A ∈ K(X) according to condition (iii). There is some B ∈ K(Y ) with R(A) ⊆ B and y  B. It follows that A × (Y \ B) is clopen in X × Y , contains (x, y), and is disjoint from R.  6.3.11 Examples (i) Let X and Y be any pair of disjoint spectral spaces, let R = X × Y . Then 6.3.9 shows that ≤ is a spectral relation. Intuitively, “Y is stacked on top of X” in such a way that every point of X specializes to every point of Y . (ii) Let λ be a successor ordinal. The spectral space of 1.6.6 is also denoted by λ. Let λ  be another disjoint copy of the same space. The elements of λ are denoted by α, β, . . ., those of λ  are denoted by α , β , . . . The map f : λ → λ  : α → α  is a homeomorphism. By 6.3.9 the graph of f yields a spectral order on λ ⊕ λ , which is the ladder depicted in the following diagram: 0O 

/ 1 O

/ ···

/ α O

/ (α + 1) O

/ ···

0

/1

/ ···

/α

/ α+1

/ ···

6.3 Spectral Orders and Spectral Relations

185

(iii) We consider two disjoint copies X and X  of the one-point compactification N∗0 of the discrete space N0 . The spaces are both Boolean. Hence the spectral topology and the inverse topology coincide, and a subset is quasi-compact for either topology if and only if it is closed. The elements of X are denoted by n and ∞, where n ∈ N0 , and those of X  are denoted by n  and ∞. The map g : X → X  with g(0) = ∞, g(n) = n  for n ∈ N and g(∞) = 0 is bijective. Note that the map g is not continuous. Let R be the graph of g. Then ΔX ∪ ΔY ∪ R is a partial order on X ⊕ X . Properties (ii)(a) and (iii)(b) of 6.3.7 are trivially satisfied, whereas (ii)(b) and (iii)(a) are not satisfied (since, e.g., X\{0} ∈ K(X), but R(X \ {0}) = g(X \ {0}) = X  \ {∞ } is not closed, and X  \ {0 } ∈ K(X ), but R−1 (X  \ {0 }) = g −1 (X  \ {0 }) = X \ {∞} is not closed). 6.3.12 Construction of Spectral Spaces by Spectral Relations – The General Case We extend the construction of spectral spaces via spectral relations by iteration. Let (I, #) be a finite poset. For each i ∈ I let Xi be a spectral space, and for each jump i  j in I let Ri, j ⊆ Xi × X j be a spectral relation. We extend the notation by setting Ri,i = Xi and Ri, j = ∅ if i  j and the pair (i, j) is not a jump in I.  The topological sum Z = i ∈I Xi is a spectral space. We show that the transitive hull of the relation R = i, j ∈I Ri, j is a spectral order on Zcon . Set N = card(I) and let (in )n ≤ N be an enumeration of I such that ik  il  implies k < l. We set Zk = X for k = 0, . . . , N. Let Sk ⊆ Zk × Zk be  j ≤k i j the transitive hull of Rk = p,q ≤k Ri p ,iq . We argue by induction. So, assume that Sk is a spectral order on Zk . We show that Sk+1 is a spectral order on Zk+1 . First note that Zk+1 = Zk ⊕ Xik+1 and that Sk+1 is the transitive hull of  Ri j ,ik+1 ∪ Xi . Sk ∪ k+1 j ≤k

 The subset j ≤k Ri j ,ik+1 ⊆ Sk × Xik+1 is proconstructible, hence is a spectral relation, 6.3.9. It follows that Sk+1 is a spectral order, and the claim follows by induction. For an explicit description of the spectral order on Zcon , consider z, z  ∈ Z. Then z ≤ z  if and only if: • there is some i with z, z  ∈ Xi and z  z  in Xi , or, • z ∈ Xi , z  ∈ X j with i  j, there are a maximal chain i = r0  r1  · · ·  rt = j and elements x0 = z  x0 in Xr0 , x1  x1 in Xr1 , . . . , xt  xt = z  in Xrt with (xs , xs+1 ) ∈ Rrs ,rs+1 for all s = 0, . . . , t − 1.

186

Quotient Constructions

The following diagram depicts the situation: / x = z t

xO t

rt $

Rr t −1 ,r t

rt−1

xt−1 O

$

/ x t−1

Rr t −2 ,r t −1

O $

Rr1 ,r2

r1

xO 1

$ r0

/ x1

Rr0 ,r1

z = x0

/ x 0

For an application where this construction is used, see 8.1.14.

6.4 Quotients Modulo Equivalence Relations and Identifying Maps Summary Spectral quotients exist for all spectral spaces and all binary relations, see 6.1.6. This section is devoted to the important special case of spectral quotients modulo equivalence relations. We start with a description of the construction of such quotients, 6.4.3. The notion of an identifying spectral map, 6.4.5, is an obvious adaptation of identifying maps in the category Top. Theorem 6.4.9 characterizes spectral quotients modulo equivalence relations in topological terms (the quotient map is an identifying spectral map), in latticetheoretic terms and in category-theoretic terms. We show that a spectral map that is identifying in Top is also identifying as a spectral map, but not vice versa, 6.4.10 and 6.4.18. Similar to the going-up property and the going-down property, cf. 5.3.1, we discuss a lifting property for spectral maps. It implies that the map is identifying, 6.4.13. 6.4.1 The Mapping Property of Quotients Modulo Equivalence Relations Let X be a spectral space and E ⊆ X × X an equivalence relation. A spectral

6.4 Equivalence Relations and Identifying Maps

187

map f : X → Y is E-compatible if and only if f (x)  f (x ) for all (x, x ) ∈ E, if and only if f (x) = f (x ) for all (x, x ) ∈ E (since E is symmetric). Therefore, the universal mapping property of the spectral quotient qE : X → X/E can also be expressed in the following form: Let f : X → Y be a spectral map such that (x, x ) ∈ E implies f (x) = f (x ). Then there is a unique spectral map g : X/E → Y such that f = g ◦ qE . We start with a two-step construction of the spectral quotient of X modulo the equivalence relation E. The first step is to form the quotient of Xcon modulo E. The second step shows the impact of the specialization order of X on the construction. For the first step we need the following fact about spectral quotients of Boolean spaces. 6.4.2 Proposition Let X be a Boolean space and E an equivalence relation. Then the spectral quotient X/E is Boolean. Proof The map (qE )con : X = Xcon → (X/E)con is E-compatible. Therefore, there is a unique spectral map g : X/E → (X/E)con such that (qE )con = g ◦ qE . It is clear that g is the identity on the underlying sets. Hence the spectral topology X/E coincides with the patch topology. (This argument has been used already in Example 6.2.2.)  6.4.3 The Spectral Quotient Modulo an Equivalence Relation, Constructed in Two Steps Let X be a spectral space with an equivalence relation E. First we form the quotient spectrum qE : Xcon → Xcon /E. We abbreviate Y = Xcon /E and note that this is a Boolean space by 6.4.2. The image of X  under qE is a relation on Y , which we denote by R. It is clear that R is reflexive, but R need not be antisymmetric or transitive. The composition of qE with the quotient spectrum qR : Y → Y /R yields the spectral map qR ◦ qE : Xcon → Y /R. The definition of R implies that qR ◦ qE is X  -compatible. Since X = Xcon /X  (by 6.1.8), there is a unique spectral map g : X → Y /R such that qR ◦ qE = g ◦ conX . Clearly, qR ◦ qE = g on the underlying sets. Note that g is surjective since both qE and qR are surjective. We claim that g : X → Y /R is the quotient spectrum of X modulo E (i.e., g has the universal mapping property). So, let f : X → Z be an E-compatible spectral map. Then f ◦ conX : Xcon → Z is also E-compatible, and there exists a unique spectral map h : Y → Z such that f ◦ conX = h ◦ qE . Note that h is R-compatible since f is spectral. Therefore, there is a unique spectral map k : Y /R → Z with h = k ◦ qR . The equality f = k ◦ g follows from f ◦ conX = k ◦ qR ◦ qE = k ◦ g ◦ conX . Finally, k is unique by surjectivity of g, thus proving the claim.

188

Quotient Constructions

6.4.4 The Spectral Quotient Modulo a Constructible Equivalence Relation To illustrate the two-step construction we consider the case of a constructible equivalence relation E ⊆ X × X. The equivalence classes are clopen in Xcon , cf. the proof of 6.2.4(ii) ⇒ (iii). Thus, Y = Xcon /E is a finite discrete space. Its elements are the equivalence classes modulo E, and the quotient map qE : Xcon → Y sends every element of X to its equivalence class. The second step uses only the finite discrete space Y and the relation R = (qE × qE )(X  ) on Y , which is reflexive, but need not be transitive or antisymmetric. The quotient spectrum Y /R is a finite spectral space (i.e., a finite poset, 1.1.16). It has the following mapping property. Let h : Y → Z be a map to a poset such that h(y) ≤ h(y ) for (y, y ) ∈ R. Then there is a unique poset homomorphism k : Y /R → Z with h = k ◦ qR . To construct qR , let T be the transitive hull of R. This is a quasi-order (i.e., A = T ∩ T −1 is an equivalence relation on Y , and the image of T under the canonical map q A : Y → Y /A: y → [y] A is a partial order, which is denoted by T/A). Let qR be the map q A interpreted as a poset homomorphism onto (Y /A,T/A). One checks that qR has the required mapping property. For a concrete example with finite spectral spaces (= finite posets), consider the poset X shown in the following diagram: cO

eO

bO

d

a Let E be the equivalence relation with classes [a] = {a, c}, [b] = {b, d}, and [e] = {e}. Then the quotient Y = Xcon /E is the set of equivalence classes, and R = ΔY ∪ {([a], [b]), ([b], [a]), ([b], [e])}, which is neither transitive nor antisymmetric. The transitive hull is T = R ∪ {([a], [e])}, which yields the equivalence relation A = T ∩ T −1 = ΔY ∪ {([a], [b])}. It has two classes, namely [a] A = {[a], [b]} and [e] A = {[e]}. The partial order T/A on Y /A is given by [a] A ≤ [e] A. Thus, X/E  (Y /A,T/A)  2. 6.4.5 Definition A surjective spectral map f : X → Y is said to be identifying ◦

if the following condition holds: a subset D ⊆ Y belongs to K (Y ) if and only if ◦

f −1 (D) ∈ K (X). 6.4.6 Example Suppose that X is a spectral space and that f : X → Y is a surjective map onto a set. The map f becomes identifying in the category

6.4 Equivalence Relations and Identifying Maps

189

Top if we endow Y with the quotient topology. We know that Y need not be a spectral space, see 6.2.2. But, even if Y is spectral, the map f need not be spectral, cf. 6.1.9. We present yet another example of this phenomenon. Recall that continuity implies that f preserves specialization. Thus, if f is not a spectral map, then it has to be discontinuous for the patch topology, see 1.5.13. We use a construction that has been used in 1.3.8 already and define f : (S∞ )inv → 2 by f −1 (1) = {∞}, see 1.6.15. The map is clearly identifying in Top, but it is not spectral, since {∞} is not constructible. 6.4.7 Definition A spectral map f : X → Y is an extremal epimorphism if it satisfies the following condition: whenever f = h◦g and h is a monomorphism, then h is a homeomorphism. 6.4.8 Remarks Let f : X → Y be a spectral map. (i) The notion of extremal epimorphisms is dual (in the category-theoretic sense) to extremal monomorphisms, cf. 5.4.1. (ii) Our definition of an extremal epimorphism does not explicitly require that f be surjective. But we show that this is an easy consequence of the definition. Suppose that f : X → Y is an extremal epimorphism. Then we write f = h ◦ g, where g : X → f (X) is the map f with restricted codomain and h : f (X) → Y is the inclusion of the image. Extremality of f implies that h is a homeomorphism. Thus, h is surjective, and so is f . (iii) Suppose that f is an epimorphism, and consider a factorization f = h ◦ g, where h : Z → Y is a monomorphism. Then h is bijective, hence is a homeomorphism for the patch topology. If f is an extremal epimorphism, then h is a homeomorphism also for the spectral topology, which is equivalent to the additional condition that h(z)  h(z ) implies z  z . 6.4.9 Theorem Let f : X → Y be a surjective spectral map. The following conditions are equivalent: (i) f is an identifying spectral map. (ii) For every subset C of Y , if f −1 (C) ∈ K(X), then C ∈ K(Y ). ◦

◦

(iii) For every C ∈ K(Y ), if f −1 (C) ∈ K (X), then C ∈ K (Y ). (iv) For every C ∈ K(Y ), if f −1 (C) ∈ K(X), then C ∈ K(Y ). (v) finv : Xinv → Yinv is an identifying spectral map. (vi) Let Z be any spectral space, let g : X → Z and h : Y → Z be maps of

190

Quotient Constructions the underlying sets such that f

X g



Z

/Y 

h

commutes. Then h is spectral if g is spectral. (vii) ( f × f )−1 (Y  ) is the saturation of ( f × f )−1 (ΔY ). (viii) There is an equivalence relation E on X such that the spectral quotient qE : X → X/E is isomorphic to f : X → Y (i.e., there is a homeomorphism g : X/E → Y such that f = g ◦ qE ). (ix) f is an extremal epimorphism. Proof Conditions (i) and (ii) are equivalent by taking complements. In the same way, (iii) and (iv) are equivalent. Condition (v) is merely a reformulation of condition (ii). The implication (i) ⇒ (iii) holds trivially. ◦

(iii) ⇒ (i) Take any set C ⊆ Y with f −1 (C) ∈ K (X). The map fcon : Xcon → Ycon is identifying in the category Top since it is a surjective continuous map between ◦

compact spaces. It follows that C ∈ K(Y ), hence (iii) says that C ∈ K (Y ). Thus, (i)–(v) are equivalent. (i) ⇒ (ix) Suppose that f = h ◦ g, where g : X → Z is some spectral map and h : Z → Y is a monomorphism. Since f is surjective, h is bijective. It remains to show that h maps quasi-compact open sets to quasi-compact open sets. So, ◦

◦

pick U ∈ K (Z). Then f −1 (h(U)) = g −1 (U) ∈ K (X). Since f is identifying, we ◦ conclude that h(U) ∈ K (Y ). (ix) ⇒ (viii) Let E be the equivalence relation ( f × f )−1 (ΔY ). The universal property of the spectral quotient qE : X → X/E yields a unique spectral map g : X/E → Y such that f = g ◦ qE . The definition of E shows that g is injective (i.e., a monomorphism). Since f is extremal, we conclude that g is a homeomorphism. (viii) ⇒ (vii) By (viii) we identify f with the spectral quotient map qE : X → X/E. The definition of the saturation says that E sat = ( f × f )−1 (Y  ). If R is any relation between E and E sat , then Rsat = E sat . Thus the claim follows from the inclusions E ⊆ ( f × f )−1 (ΔY ) ⊆ E sat . (vii) ⇒ (vi) Let g : X → Z, h : Y → Z be set maps to a spectral space Z such that g = h ◦ f . We need to show that h is spectral if g is spectral. Spectrality of g implies that (g×g)−1 (Z  ) is saturated. The equality g = h◦ f

6.4 Equivalence Relations and Identifying Maps

191

implies ( f × f )−1 (ΔY ) ⊆ (g × g)−1 (Z  ). Hence ( f × f )−1 (Y  ), being the saturation of ( f × f )−1 (ΔY ) (by (vii)), is contained in (g × g)−1 (Z  ). We conclude that g is ( f × f )−1 (Y  )-compatible. By 6.1.7 there is a spectral map k : Y → Z with g = k ◦ f . Since f is surjective, we see that k = h on the underlying sets. ◦

(vi) ⇒ (i) Pick any set C ⊆ Y with f −1 (C) ∈ K (X). We define set maps g : X → 2 and h : Y → 2 by the conditions g −1 (1) = f −1 (C) and h−1 (1) = C. It is clear that g = h ◦ f and that g is a spectral map. Thus, h is spectral as well, ◦

and we conclude C ∈ K (Y ).



In 6.5.6 we supplement 6.4.9 with a characterization of identifying spectral maps in terms of the corresponding lattice homomorphisms. 6.4.10 Corollary Let f : X → Y be a surjective spectral map that is identifying in Top. Then f is also identifying in Spec. ◦

Proof Take C ∈ K(Y ) with f −1 (C) ∈ K (X). Then C = f ( f −1 (C)) (since f is surjective) and C ∈ O(Y ) (since Y carries the quotient topology for f ).  6.4.11 Corollary Spectral maps have extremal epi-mono factorizations, which means that, given a spectral map f : X → Y , there are a spectral space Z, an extremal epimorphism g : X → Z, and a monomorphism h : Z → Y such that f = h ◦ g. Proof We define E = ( f × f )−1 (ΔY ). Then f = h ◦ qE , where qE : X → X/E is the spectral quotient modulo E, hence is an extremal epimorphism by the equivalence (viii) ⇔ (ix) in 6.4.9. Clearly, h : X/E → Y is injective.  We exhibit various elementary properties of identifying spectral maps, similar to lists of basic properties of monomorphisms, epimorphisms, and embeddings, see 5.2.10 and 5.4.13. 6.4.12 Proposition All spaces and all maps are spectral. (i) If f : X → Y and g : Y → Z are identifying, then g ◦ f : X → Z is identifying. (ii) If g ◦ f : X → Y → Z is identifying, then g is identifying. (iii) Suppose that X

f

/Y g

g

 Z

f

 /T

192

Quotient Constructions

is a push-out diagram. If f is identifying, then so is f . (iv) Given a spectral space X, let (qi : X → Xi )i ∈I be a set of representatives for the isomorphism classes of identifying maps g : X → F with F finite. Define i ≤ j for i, j ∈ I by the condition that there is a map gi j : X j → Xi with qi = gi j ◦ q j . Then ((Xi )i ∈I , (gi j )i ≤ j ) is a projective system, and X is its projective limit. (v) Let ((Yi )i ∈I , (gi j : Yj → Yi )i ≤ j ) be a projective system with limit (Y, (gi : Y → Yi )i ∈I ) and suppose that ( fi : X → Yi )i ∈I is a family of identifying maps with fi = gi j ◦ f j for i ≤ j. Then the limit map f : X → Y is identifying. (vi) An epimorphism f : X → Y is identifying if and only if g ◦ f is identifying for every identifying map g : Y → Z onto a finite spectral space. Proof (i) and (ii) follow immediately from the definition of identifying spectral maps, or from 6.4.9. (iii) Let W be a spectral space and k : T → W a map of the underlying sets such that h = k ◦ f  : Z → W is a spectral map. We must show that k is spectral as well. Since h is spectral it follows that h ◦ g = (k ◦ f ) ◦ g = (k ◦ g ) ◦ f is spectral. The hypothesis that f is identifying implies that k ◦ g  is a spectral map. The mapping property of the push-out diagram yields a spectral map l : T → W with l ◦ g  = k ◦ g  and l ◦ f  = h. Thus, l ◦ f  = k ◦ f  on the underlying sets. Since f  is surjective (by 5.2.10(iii)) we conclude that k = l (i.e., k is spectral). (iv) It has been shown in 2.3.10 that X is the projective limit of a projective system of finite spectral spaces. Explicitly, let (qk : X → Xk )k ∈K be a set of representatives of the isomorphism classes of surjective spectral maps. The index set K is partially ordered in the same way as I, namely, k ≤ l if there is a spectral map gkl : Xl → Xk with qk = gkl ◦ ql . Then X is the limit of the projective system ((Xk )k ∈K , (gkl : Xl → Xk )k ≤l ). We may assume that I is the subset of those k ∈ K for which qk : X → Xk is identifying. It suffices to show that I is cofinal in K. So, pick any element k ∈ K and define Ek = (qk × qk )−1 (ΔXk ), which is an equivalence relation on X. Since qk is Ek -compatible there is a unique spectral map hk : X/Ek → Xk with qk = hk ◦ qEk . The definition of Ek implies that hk is bijective on the underlying sets. Hence there is some i ∈ I such that qEk is isomorphic to qi . It follows that k ≤ i, which proves the claim. (v) With E = ( f × f )−1 (ΔY ) we obtain the spectral quotient qE : X → X/E and the extremal epi-mono factorization f = h ◦ qE . If h is a homeomorphism then f is an extremal epimorphism, hence is identifying, 6.4.9. The identifying maps fi : X → Yi are all E-compatible, hence factor uniquely as fi = hi ◦ qE

6.4 Equivalence Relations and Identifying Maps

193

with hi : X/E → Yi . Note that each hi is identifying by (ii) and hi = gi j ◦ h j for i ≤ j. It is clear that h : X/E → Y is the limit map of the family (hi )i ∈I . We may now assume that E = ΔX , hence X = X/E, and f = h. Thus, f is bijective (injective by assumption and surjective by 5.2.11), and we identify the ◦

◦

underlying sets of X and Y . It follows from 5.2.2 and 5.2.5 that K (Y ) ⊆ K (X) ◦ ◦ and K(X) = K(Y ). It remains to prove K (Y ) = K (X). So, pick an element ◦

U ∈ K (X) ⊆ K(X) = K(Y ). By 2.3.9 there are a finite subset J ⊆ I and a   constructible set CJ ⊆ j ∈J Yj such that U = p−1 j ∈J Yj J (C), where pJ : Y → is the projection. As I is directed there is an upper bound k ∈ I for the finite −1 (C), where g : Y →  set J. Let Ck = gJk Jk k j ∈J Yj is defined by the maps g jk : Yk → Yj with j ∈ J. We conclude that U = fk−1 (Ck ), which is open in ◦

X, hence Ck ∈ K (Yk ) (since fk is identifying). Finally, U = f −1 (gk−1 (Ck )) = ◦

gk−1 (Ck ) implies U ∈ K (Y ). (vi) If f is identifying and g : Y → Z is identifying then g ◦ f is identifying by (i). For the converse we write Y as the projective limit of a projective system ((Yi )i ∈I , (gi j )i ≤ j ) with finite spaces Yi and identifying projections qi : Y → Yi , which can be done by (iv). The hypothesis says that the maps qi ◦ f : X → Yi are identifying. Hence we can apply (v) and see that f is identifying.  6.4.13 Proposition Let f : X → Y be a surjective spectral map. Suppose that f has the following lifting property: for all y, y  ∈ Y with y  y  there are x, x  ∈ X with x  x  and f (x) = y, f (x ) = y . Let S ⊆ Y be a subset. (i) S is closed under specialization in Y if f −1 (S) is closed under specialization in X. (ii) S is generically closed in Y if f −1 (S) is generically closed in X. (iii) f is identifying in the category Top, hence is also identifying as a spectral map, cf. 6.4.10. Proof (i) Suppose that y, y  ∈ Y , y ∈ S, and y  y . The lifting property yields x, x  ∈ X such that x  x  and f (x) = y and f (x ) = y . Thus, x ∈ f −1 (S) and x  x , which implies x  ∈ f −1 (S) (i.e., y  = f (x ) ∈ S). (ii) is the inverse version of (i). (iii) Suppose that C ⊆ Y and f −1 (C) ∈ A(X). Then f −1 (C) is closed under specialization, hence C is closed under specialization. Moreover, C = f ( f −1 (C)) is proconstructible since f is spectral and surjective. Thus C ∈ A(Y ), 1.5.4.  6.4.14 Corollary Let f : X → Y be a surjective spectral map closed for the spectral topology or for the inverse topology. Then f is identifying in Top and therefore also in Spec.

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Quotient Constructions

Proof Since f is surjective the going-up property (or the going-down property) implies the lifting property, cf. 5.3.3 and 5.3.4. Thus, 6.4.13 yields the assertion.  6.4.15 Example We describe the identifying spectral maps onto 2. From 6.4.9 we know that a spectral map f : X → Y is identifying if and only if it is isomorphic to the spectral quotient modulo the equivalence relation E = ( f × f )−1 (ΔY ). Now suppose that Y is finite. Then E is a constructible equivalence relation, and the construction of the spectral quotient has been described in 6.4.4. The underlying sets of X/E and Y are identified. The relation R = ( f × f )(X  ) is contained in Y  , and Y  is the smallest transitive relation containing R. In particular, if Y = 2 then E = ( f −1 (0) × f −1 (0)) ∪ ( f −1 (1) × f −1 (1)). Hence f is identifying if and only if there are x ∈ f −1 (0) and x  ∈ f −1 (1) with x  x . The next few examples illustrate the relationship between the formation of quotient spaces in Top and in Spec. 6.4.16 Example Let X be a spectral space and let f : X → Y be a surjective map onto a set Y . We know from 6.2.2 that the quotient topology need not be spectral. We present an example with small finite spectral spaces. Let X = {a, b, c, d} be the spectral space with four points and the only nontrivial specializations a  b and c  d. The following diagram depicts a map onto the set 2: /b a X f

 2

do

c

 0



1

The quotient topology on 2 is the indiscrete topology, hence is not a T0 -topology. Thus, the quotient space is not spectral. 6.4.17 Example Let f : X → Y be a surjective spectral map. If the lifting property holds, then f is identifying in Top, hence also in Spec, 6.4.13. We present an identifying map without the lifting property. Let X = {a, b, c, d} be the same space as in 6.4.16 (i.e., the nontrivial specializations are a  b and c  d). Recall that 3 is the spectral space with elements 0, 1, 2 and specializations 0  1  2. The diagram depicts a surjective spectral

6.4 Equivalence Relations and Identifying Maps

195

map f : X → 3 that is identifying in Top, hence in Spec, 6.4.10, but does not have the lifting property: a

X

/b

/d

c

f

 0

 3

 /1

 / 2.

Note that the map f also satisfies conditions (i) and (ii) of 6.4.13, but is neither closed nor open. 6.4.18 Example We present a surjective spectral map f : X → Y that is identifying in Spec, but is not identifying in Top. Thus, if f is identifying in Spec then, in general, the quotient topology on Y is finer than the spectral topology. Let Y be the totally ordered spectral space (ω + 1)inv , cf. 1.6.6. Thus, Y is a specialization chain with minimal point ω and maximal point 0. The patch space Ycon is the one-point compactification of the discrete space N0 with limit point ω. To define the space X we start with the Boolean space Ycon and define a partial order by setting k ≤ l if and only if k = l, or there is some n ∈ N0 with k = 2 · n + 1 and l = 2 · n. It is straightforward to check that the partial order is spectral, hence defines a spectral space X with Xcon = Ycon , 1.5.11. Note that ω is a closed point of X. We define a map f : X → Y by setting f (ω) = ω, f (0) = 0, and f (2 · n) = f (2 · n − 1) = n for all n ∈ N: ω

 ω /

6

···

5



/3

/4

3



/2

/2

1



/1

/0

 / 0.

The map f preserves specialization and is continuous for the patch topology (since the inverse image of a finite subset of N0 is a finite subset of N0 ), hence is spectral. Claim 1 f is identifying in the category Spec. Proof of Claim 1 Pick a subset C ⊆ Y such that f −1 (C) ∈ K(X). We show that C ∈ K(Y ), see 6.4.9(ii). First assume that ω ∈ C. We claim that C = Y : there is a smallest element n ∈ N0 such that [2 · n, ω] ⊆ f −1 (C). Then [n, ω] = f ([2 · n, ω]) ⊆ C. If n = 0

196

Quotient Constructions

then C = Y . So, assume n > 0. Then 2 · n − 1 ∈ f −1 (n) ⊆ f −1 (C). This implies 2 · n − 2 ∈ f −1 (C) (since f −1 (C) is closed under specialization), contradicting minimality of n, and we are finished with this case. Next, assume that ω  C, hence ω  f −1 (C). But then f −1 (C) is a finite closed subset of X. There is a largest element n ∈ N such that 2 · n − 1 ∈ f −1 (C), hence such that n ∈ C. We have to show that C = [0, n]. Suppose that 0 < k ∈ C. Then 2 · k − 1 ∈ f −1 (k) ⊆ f −1 (C), hence 2 · k − 2 ∈ f −1 (C) (since f −1 (C) is closed under specialization), hence k − 1 ∈ f ( f −1 (C)) = C, finishing the second case.  Claim 2 f is not identifying in Top. Proof of Claim 2 Just note that {ω} ⊆ Y is not closed, whereas f −1 ({ω}) = {ω} ⊆ X is closed.  Claim 3 The inverse map finv is identifying in Top. −1 (D) is closed in X . Proof of Claim 3 Pick a subset D ⊆ Yinv such that finv inv −1 We claim that D is closed. If finv (D) = {ω}, then D = {ω}, hence D is closed in Yinv . So, assume that there is some n ∈ D ∩ N0 . Pick the smallest such −1 (D) (by definition of f ) and 2 · n + 1 ∈ f −1 (D) (since element. Then 2 · n ∈ finv inv 2 · n  inv 2 · n + 1 in Xinv ). Thus, n + 1 ∈ D, and we conclude that [n, ω) ⊆ D, −1 (D). It follows that ω is a limit point, by closedness even hence [2 · n, ω) ⊆ finv −1 (D). But then D = [n, ω] is closed.  an element, of finv Finally we note that a minor modification of this construction yields a spectral map g : Z → T such that g and ginv are both identifying in Spec and are both not identifying in Top: just take Z = X ⊕ Xinv , T = Y ⊕ Yinv , and g = f ⊕ finv . Proposition 6.4.12 shows that identifying maps are well-behaved under some constructions with spectral spaces. The next two examples show that there are other constructions that are not compatible with identifying maps. 6.4.19 Example Suppose that f : X → Y is an identifying spectral map, and let X  ⊆ X be a spectral subspace. We show that the map f  : X  → f (X ) onto the subspace f (X ) ⊆ Y need not be identifying. Let X = {x, y, z} with the nontrivial specializations x, y  z and define f : X → 2 by f (x) = 0 and f (y) = f (z) = 1. Clearly, f is identifying (even closed). The subspace X  = {x, y} ⊂ X is discrete, and the restriction f  : X  → f (X ) = 2 is bijective, but is clearly not identifying. In fact, the spectral quotient of X  modulo ( f  × f )−1 (Δ2 ) is X   2con . 6.4.20 Example Suppose that f : X → Y and g : X → Z are identifying maps. In general, the induced map ( f , g) : X → Y × Z, x → ( f (x), g(x)) is not identifying onto its image. Let X = {x, y, z, t} with the nontrivial specialization y  z, and define

6.5 Spectral Quotients and Lattices

197

• f : X → 2 by f (x) = f (y) = 0 and f (z) = f (t) = 1, • g : X → 2con by g(x) = g(t) = 0 and g(y) = g(z) = 1. Then ( f , g) : X → 2 × 2con is given by x → (0, 0), y → (0, 1), z → (1, 1), and t → (1, 0), hence is bijective. Both f and g have the lifting property, hence are identifying, 6.4.13. If a bijective map is identifying then it is a homeomorphism, which, clearly, is not the case with ( f , g).

6.5 Spectral Quotients and Lattices Summary In the spirit of Stone duality, we relate spectral quotients to lattices. If X is a spectral space with a binary relation R then the quotient map qR : X → ◦

◦

X/R is an epimorphism. By 5.2.5, the homomorphism K (qR ) : K (X/R) → ◦ ◦ ◦ K (X) is injective. We describe the sublattice K (X/R) ⊆ K (X) in terms of R, 6.5.2, and give a characterization of those sublattices that correspond to equivalence relations and identifying maps, 6.5.6. 6.5.1 Lemma Suppose that L is a bounded distributive lattice and that i and j are prime ideals. Then the set (L \ i) ∪ j is a bounded sublattice of L. Proof As ⊥ ∈ j and   i we have ⊥,  ∈ (L \ i) ∪ j. Pick a, b ∈ (L \ i) ∪ j. If both elements belong to j or to L \ i then so do a ∨ b and a ∧ b. If a ∈ L \ i and b ∈ j, then a ∨ b ∈ L \ i and a ∧ b ∈ j.  6.5.2 Proposition Let L be a lattice and let X = PrimI(L). Suppose that R ⊆ X × X is a relation. Then  M= (L \ i) ∪ j (i,j)∈R

is a bounded sublattice of L and the inclusion homomorphism corresponds to the spectral quotient qR : X → X/R. Proof By 6.5.1, M is a bounded sublattice of L. Let ϕ : M → L be the inclusion. We show that the spectral map PrimI(ϕ) : X → PrimI(M) has the mapping property of the spectral quotient. Let f : X → Y be a spectral map such that f (i)  f (j) whenever (i, j) ∈ R. ◦

◦

◦

If the image of K ( f ) : K (Y ) → K (X)  L is contained in M then the claim ◦ ◦ follows by Stone duality, 3.2.8. Suppose that U ∈ K (Y ) and write K ( f )(U) = f −1 (U) = D(a) with a ∈ L, 3.3.3. We must prove that a ∈ M (i.e., a ∈ (L \ i) ∪ j for all (i, j) ∈ R). Assuming this is false, there is some (i, j) ∈ R such that

198

Quotient Constructions

a ∈ (L \ j) ∩ i. Then i  D(a) and j ∈ D(a), which implies f (i)  U and  f (j) ∈ U, contradicting the condition f (i)  f (j). 6.5.3 Example Let L be a bounded distributive lattice and let i, j ∈ PrimI(L) be two distinct prime ideals. By 6.5.2 we know that (L \ i) ∪ j ⊆ L is the lattice corresponding to the spectral quotient modulo the relation R = {(i, j)}. In Section 1.6.C we considered a special case of this construction. We started with a Boolean space X and then introduced a single nontrivial specialization, thus producing a spectral space Y . Now we can describe the construction in terms of lattices and prime ideal spectra. The Boolean space X ◦

is identified with PrimI(B), where B = K (X) = K(X). Pick two distinct prime (i.e., maximal) ideals i, j ∈ X and add the specialization i  j to obtain the space ◦

Y . Then K (Y ) = (B \ i) ∪ j is the lattice corresponding to Y under Stone duality. Every spectral space is the spectral quotient of its patch space modulo the specialization relation, cf. 6.1.8. Thus, we obtain 6.5.4 Corollary Let L be a lattice with enveloping Boolean algebra ba(L) and X = PrimI(L). Then

  L = ba(L) \ {i \ j | i, j ∈ PrimI(ba(L)), i ∩ L  j ∩ L} .  We turn to spectral quotients modulo equivalence relations. First we record the following special case of 6.5.2. 6.5.5 Corollary Let L be a lattice and E ⊆ X × X an equivalence relation,  where X = PrimI(L). Then the sublattice (i,j)∈E L \ (i  j) ⊆ L corresponds  to the spectral quotient qE : X → X/E. 6.5.6 Theorem Let ϕ : L → M be a lattice homomorphism with corresponding spectral map f = PrimI(ϕ) : Y = PrimI(M) → PrimI(L) = X. Then the following conditions are equivalent: (i) f is identifying as a spectral map. (ii) ϕ is injective, and the following diagram is a fiber product in BDLat: ϕ

L ba L

⊆

 ba(L)

/M ⊆ ba M

ba(ϕ)

 / ba(M)

(i.e., L = ba(ϕ)−1 (M), or L = M ∩ ba(L), if ϕ and ba(ϕ) are considered as inclusions).

6.6 The Space of Connected Components

199

Proof First note that, by 5.2.5, f is an epimorphism if and only if ϕ is injective, if and only if ba(ϕ) is injective. (i) ⇒ (ii) The diagram is clearly commutative. We must show that, given c ∈ ba(L) and b ∈ M such that ba(ϕ)(c) = ba M (b), there exists a ∈ L with ba L (a) = c and ϕ(a) = b. Using Stone duality we restate this assertion in ◦

terms of the prime ideal spectra. Suppose that C ∈ K(X) and U ∈ K (Y ) with ◦ −1 (C) = con−1 (U). Then there exists O ∈ fcon K (X) such that con−1 Y X (O) = C and f −1 (O) = U. To prove this, note that, on the underlying sets, f = fcon , conX = idX , and ◦

conY = idY . Therefore we need to show that, given C ∈ K(X) and U ∈ K (Y ) ◦ with f −1 (C) = U, the set C belongs to K (X). This is exactly condition 6.4.9(iii), and we are done since f is identifying. ◦

(ii) ⇒ (i) By 6.4.9(iii) we have to show that C ∈ K (X) if C ∈ K(X) with ◦  f −1 (C) ∈ K (Y ). We write C = ri=1 D(ai ) ∩V(ci ) with ai , ci ∈ L and f −1 (C) =  D(b) with b ∈ M. Stone duality translates the equality f −1 ( ri=1 D(ai ) ∩ r V(ci )) = D(b) into the equality ba(ϕ)( i=1 ba L (ai ) ∧ ¬ ba L (ci )) = ba M (b). By  condition (ii) there is an element a ∈ L with ba L (a) = ri=1 ba L (ai ) ∧ ¬ ba L (ci ) and ϕ(a) = b. It follows (using surjectivity of f ) that C = f ( f −1 (C)) = ◦

f (D(ϕ(a))) = f ( f −1 (D(a))) = D(a) ∈ K (X).



6.6 The Space of Connected Components Summary Every topological space X can be partitioned into its maximal connected subsets, the connected components, [Bou71b, TG I.83 ff], [Kel75, p. 53 ff], [Eng89, p. 352 ff]. Every connected set, in particular every point, is contained in a unique component. The connected components are closed, and two distinct components are disjoint. The set of connected components is denoted by Γ(X). Let ΓX : X → Γ(X) be the canonical map that sends a point to the connected component containing it. The set Γ(X), equipped with the quotient topology, is a T1 -space (since the connected components are closed in X) and totally disconnected, [Bou71b, p. TG I.84, Proposition 8]. It follows that every connected component is closed both under specialization and under generalization. We start our study of the connected components of spectral spaces by showing that connectedness can be established using only quasi-compact open sets (or closed and constructible sets), 6.6.1. It follows that the connected components

200

Quotient Constructions

are proconstructible and closed under specialization and generalization, 6.6.2. The description of the connected component of a point in 6.6.5 implies that Γ(X) is a Boolean space and the canonical map ΓX is identifying, 6.6.6. Supplementing the earlier result that the subcategory BoolSp ⊆ Spec is coreflective, 1.3.24, we show that BoolSp ⊆ Spec is also a reflective subcategory, where the canonical map ΓX : X → Γ(X) is the reflection map, 6.6.8. The Boolean algebra corresponding to the Boolean space Γ(X) is canonically isomorphic to ◦

Clop(X), the largest Boolean algebra contained in K (X), 6.6.9. 6.6.1 Proposition Let X be a spectral space and let S ⊆ X be a proconstructible subset. The following conditions are equivalent: (i) S is connected.

◦

(ii) For all U, U  ∈ K (X), it follows from S ∩ U ∩ U  = ∅ and S ⊆ U ∪ U  that S ⊆ U or S ⊆ U . (iii) For all V,V  ∈ K(X), it follows from S ∩ V ∩ V  = ∅ and S ⊆ V ∪ V  that S ⊆ V or S ⊆ V . (iv) S is connected for the inverse topology. ◦

◦

Proof We may assume that S = X, since K (S) = {U ∩ S | U ∈ K (X)} and K(S) = {U ∩ S | U ∈ K(X)}, 2.1.3. (i) ⇔ (ii) If X is the disjoint union of two open subsets, then these sets are quasi-compact. ◦

(ii) ⇔ (iii) is clear since K(X) is the set of complements of K (X). (iii) ⇔ (iv) is the equivalence of (i) and (ii) applied to Xinv .



6.6.2 Corollary Let X be a spectral space and let C ⊆ X be a connected component. Then C is closed both for the spectral topology and for the inverse topology. In particular, C is proconstructible, and C = Spez(C) = Gen(C).  6.6.3 Connectedness vs. Maximal Points and Minimal Points The spectral space X is connected if the subspace X max is connected, since any proper partition of X into two closed sets restricts to a proper partition of X max into closed sets. The converse fails in general (i.e., there are connected spectral spaces X with X max not connected). For example, let X be the space with three points 0, 1, 2 and the nontrivial specializations 0  1 and 0  2. However, let X be a normal spectral space, which means that Gen(x) is closed for each x ∈ X max , see 8.4.5. Then there is a continuous retraction rX : X → X max , 8.4.7. Thus, connectedness of X implies that X max is connected.

6.6 The Space of Connected Components

201

Recall from 4.4.8 that the space of minimal points of any spectral space is totally disconnected. We proceed as in [GiJe60, 16.13 and 16.15] to give a description of the connected components of a spectral space. Note that Clop(X) = K(X) ∩ A(X) ∩ A(Xinv ). The following lemma is needed as a tool. 6.6.4 Lemma Let X be a spectral space. Given families (Ki )i ∈I , (Ap ) p ∈P , and (Bq )q ∈Q of constructible subsets of X, we set: (i) K :=



i ∈I



Ki , A :=



p ∈P



Ap , and B :=



q ∈Q

(ii) KI  = i ∈I  Ki , AP = p ∈P Ap , and BQ = P  ⊆ P, and Q  ⊆ Q are finite subsets.

Bq . 

q ∈Q

Bq , where I  ⊆ I,

Suppose that A ∩ B = ∅ and K = A ∪ B. Then there are finite subsets J ⊆ I, R ⊆ P, and S ⊆ Q such that AR ∩ BS = ∅, KJ ⊆ AR ∪ BS , AR ∩ K = A, and BS ∩ K = B. Proof It follows from A ∩ B = ∅ and compactness of Xcon that there are finite subsets R ⊆ P and S ⊆ Q with AR ∩ BS = ∅. Then K is disjoint from the constructible set X \ (AR ∪ BS ), hence (by compactness) there is a finite subset J ⊆ I such that X \ (AR ∪ BS ) is disjoint from KJ . Note that  AR ∩ K = AR ∩ (A ∪ B) = AR ∩ A = A and, similarly, BS ∩ K = B. 6.6.5 Theorem Let X be a spectral space and consider a point x ∈ X. Then  ΓX (x) = {C ∈ Clop(X) | x ∈ C}.  Proof We set I = {C ∈ Clop(X) | x ∈ C} and K = C ∈I C. The inclusion  ΓX (x) ⊆ K is obvious. For the other inclusion we show that K = C ∈I C is connected. Then x ∈ K implies K ⊆ ΓX (x). So, let K = A ∪ A be a partition of K into two disjoint closed subsets. Assuming x ∈ A we claim that A = ∅ . In order to apply 6.6.4 we set P = {V ∈ K(X) | A ⊆ V } and Q = {V  ∈ K(X) | A ⊆ V  } and note that   A = V ∈P V and A = V  ∈Q V . Then there are finite subsets J ⊆ I, R ⊆ P,    and S ⊆ Q such that KJ = C ∈J C, AR = V ∈R V, and AS = V  ∈S V  satisfy the conditions in 6.6.4. Thus, KJ ∈ Clop(X) and KJ = (KJ ∩ AR ) ∪ (KJ ∩ AS ) is a partition into clopen sets. Since x ∈ A it follows that KJ ∩ AR is a clopen set containing x, hence an element of I, which implies K ⊆ KJ ∩ AR . But then  A ⊆ AR ∩ AS = ∅. 6.6.6 Corollary Let X be a spectral space. The space Γ(X) of connected components is Boolean, and the continuous map ΓX : X → Γ(X) is an identifying spectral map.

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Quotient Constructions

Proof Let ΓX (x) and ΓX (x ) be distinct connected components. By 6.6.5 there is a set U ∈ Clop(X) that contains x, but not x . Thus, ΓX (U) ⊆ Γ(X) is a clopen subset that separates ΓX (x) and ΓX (x ), showing that Γ(X) is Hausdorff. Moreover, Γ(X) is totally disconnected by [Bou71b, p. TG I.84, Proposition 9]. Compactness follows from the fact that Γ(X) is a continuous image of the quasi-compact space X. Altogether, Γ(X) is Boolean. The map ΓX is spectral by 1.3.6, and is identifying as a spectral map since it is identifying as a continuous map (by 6.4.10).  6.6.7 The Connected Components of a Spectral Space and of its Inverse Space Let X be a spectral space and E = (ΓX × ΓX )−1 (ΔΓ(X) ). Since ΓX is identifying it follows that the map g : X/E → Γ(X) with ΓX = g ◦ qE is a homeomorphism, 6.4.9. The equivalence classes of E are the connected components of X. By 6.6.5 the inverse space Xinv has the same connected components as X, since both spaces have the same clopen subsets. Both ΓX ◦ conX : Xcon → Γ(X) and ΓXinv ◦conXinv : (Xinv )con → Γ(Xinv ) are continuous surjective maps between Boolean spaces. Therefore, the codomains carry the quotient topology. Note that Xcon = (Xinv )con , the underlying sets of Γ(X) and Γ(Xinv ) coincide, and ΓX ◦ conX = ΓXinv ◦ conXinv . Therefore Γ(Xinv ) = Γ(X). 6.6.8 BoolSp is Reflective in Spec The spectral quotient map ΓX : X → Γ(X) has the following universal property. Let f : X → Y be a spectral map into a Boolean space. Then there is a unique spectral map g : Γ(X) → Y such that f = g ◦ ΓX . To see this it suffices to show that f is E-compatible (with E as in 6.6.7). But if (x, x ) ∈ E then x and x  belong to the same connected component C of X. The image of a connected set under f is connected. Hence f (C) is connected in Y . Since Y is Boolean the only connected subsets are singletons. Thus f (C) = {y} for some y ∈ Y , and it follows that f (x) = y = f (x ). The universal mapping property says that the subcategory BoolSp is reflective in the category Spec. The reflection map of a spectral space X is the canonical map ΓX . The reflector is the functor that assigns Γ(X) to the spectral space X and sends a spectral map f : X → Y to the unique spectral map Γ( f ) : Γ(X) → Γ(Y ) with ΓY ◦ f = Γ( f )◦ΓX . The reflector is clearly idempotent (i.e., ΓΓ(X) : Γ(X) → Γ(Γ(X)) is an isomorphism). This also follows from the fact that BoolSp ⊆ Spec is a full subcategory, 1.3.9 and [HeSt79, Proposition 36.5]. Recall from 1.3.24(iv) that BoolSp is also coreflective in the category Spec. 6.6.9 The Stone Dual of the Space of Connected Components By Stone duality the construction of the space Γ(X) of connected components of X

6.6 The Space of Connected Components

203

has a counterpart in bounded lattices. The map ΓX : X → Γ(X) is surjective. ◦

◦

◦

Therefore the corresponding homomorphism K (ΓX ) : K (Γ(X)) → K (X) is ◦ injective. As Γ(X) is a Boolean space the lattice K (Γ(X)) is a Boolean algebra. It has the following universal mapping property (the dual of 6.6.7). ◦

Suppose that B is a Boolean algebra and ϕ : B → K (X) is a homomorphism ◦

of bounded lattices. Then there is a unique homomorphism ψ : B → K (Γ(X)) ◦ ◦ ◦ such that ϕ = K (ΓX ) ◦ ψ. Identifying K (Γ(X)) with its image in K (X) we see ◦

◦

that K (Γ(X)) is the largest Boolean subalgebra in the lattice K (X). It coincides ◦

with Clop(X) = K (X) ∩ K(X). 6.6.10 Connected Components vs. Graph Components in Spectral Spaces Every spectral space X is a poset via the specialization order, hence defines a graph and is the union of its graph components, A.3(vii). We compare the graph components with the connected components. Every connected component ΓX (x) is closed for the spectral topology and the inverse topology, hence belongs to the complete lattice A(X) ∩ A(Xinv ), 6.6.2. For each point x ∈ X there is a smallest set γX (x) ∈ A(X) ∩ A(Xinv ) containing x. Clearly, γX (x) ⊆ ΓX (x), and the sets γX (x) form a partition of X. Trivially, there are spectral spaces with γX (x) = ΓX (x) for all x ∈ X. But there are also spectral spaces X and points x ∈ X such that γX (x)  ΓX (x). For an example of this phenomenon, let X be a normal spectral space, see Section 8.4. Then γX (x) = Gen(x) for each x ∈ X max . It may happen that X has many closed points and is connected, hence that Γ(X) has just one point. For example, let Z be a compact connected space. The Zariski spectrum Spec(C(Z, R)) of the ring C(Z, R) of real-valued continuous functions is a root system, Section 8.5 and [GiJe60, Chapter 14], hence is a normal spectral space, 8.5.1. The space Spec(C(Z, R))max is canonically homeomorphic to Z, [GiJe60, 4.9]. Thus, Spec(C(Z, R)) is connected, see 6.6.3. To compare the connected components with the graph components, we consider the specialization poset (X,  ) and write R = X  . We set S = R ∪ R−1 and let T be the transitive hull of S. It is clear that the relations R−1 , S, and T all have the same saturation, Section 6.2. The equivalence classes of T are the graph components, and %x& ⊆ γX (x) for each x ∈ X (since %x& is the smallest subset closed under specialization and generalization and containing x). The set γX (x) is proconstructible by construction, but %x& need not be proconstructible, [Sch11, Example 3.5]. Thus, in general the inclusion %x& ⊆ γX (x) is proper. However, if the recursive construction, cf. A.3(vii), of the graph component of x ∈ X stops after finitely many steps, then the graph component is proconstructible as well, hence %x& =

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γX (x). In particular, this is the case if the spectral space is a root system or a forest, cf. A.5. Assume that the graph components of X are proconstructible. Then the following conditions are clearly equivalent: (i) There are only finitely many graph components. (ii) Every graph component is constructible.

◦

(iii) Every graph component is clopen (i.e., belongs to K (X) ∩ K(X)). 6.6.11 Proposition Let X be a spectral space. We continue with the notation of 6.6.10. Then the spectral quotient of X modulo the graph components (i.e., X/T) is canonically homeomorphic to the space Γ(X) of connected components. Proof It was noted in 6.6.10 that the saturations of the relations R−1 , S, and T coincide (i.e., that the canonical maps X/(R−1 ) → X/S → X/T between the spectral quotients are homeomorphisms). The spectral map ΓX : X → Γ(X) is S-compatible, hence there is a unique spectral map f : X/S → Γ(X) with ΓX = f ◦ qS . It is claimed that f is a homeomorphism. To prove this it suffices to show that X/S is a Boolean space. For then, the mapping property of ΓX , see 6.6.8, yields a unique spectral map g : Γ(X) → X/S with qS = g ◦ ΓX . The universal mapping properties of ΓX and qS imply that f ◦ g = idΓ(X) and g ◦ f = idX/S , hence f and g are homeomorphisms. Thus, it remains to show that X/S is a Boolean space (i.e., that conX/S : (X/S)con → X/S is a homeomorphism). First we prove that (qS )con is S-compatible. The relation S is symmetric, hence (x, x ) ∈ S implies qS (x)  qS (x ) and qS (x )  qS (x). It follows that (qS )con (x) = qS (x) = qS (x ) = (qS )con (x ), which implies S-compatibility. Being S-compatible, (qS )con is R-compatible. Since X = Xcon /R (see 6.1.8) we obtain a spectral map h : X → (X/S)con with (qS )con = h ◦ conX . Note that h and (qS )con are identical as set maps. Thus g is also S-compatible. Hence there is a map k : X/S → (X/S)con such that h = k ◦ qS . The maps are shown in the following diagram: Xcon (qS )con

 { (X/S)con o

con X

h

con X /S k

/X qS

 / X/S .

One checks (by a routine diagram chase) that k and conX/S are mutually inverse homeomorphisms. 

7 Scott Topology and Coarse Lower Topology

We study upper and lower topologies on a given poset X = (X, ≤), A.8(i). In [Pri94a, p. 78] a topology τ on X is called intrinsic if its open sets are defined in terms of the partial order or its inverse. The coarse lower topology and the fine lower topology are intrinsic and have specialization order ≤. The coarse upper topology, the Scott topology (to be defined in 7.1.6), and the fine upper topology are intrinsic and have specialization order ≤inv . The Scott topology (which is defined only if X is a dcpo) arose in the 1970s with the development of the mathematical theory of computation and domain theory. Several important notions were introduced in this context. We mention, in particular, the way-below relation, 7.1.1, which plays a big role in our studies. Our main source for facts from domain theory is the monograph [GHK+ 03], where some motivation from the theory of computation is also provided. 1 We focus our attention on the questions of whether the coarse lower topology on a poset and the Scott topology on a dcpo are spectral. This is also one of the main problems studied in [Pri94a] (see p. 80, in particular). Such investigations may be viewed as a variant of the Kaplansky Problem, cf. 4.2.13. The Kaplansky Problem asks for a characterization of the posets that are the specialization posets of spectral spaces. Now we ask for a characterization of the posets for which special constructions (i.e., the Scott topology and the coarse lower topology) lead to spectral spaces. In Section 7.1 we study basic properties of the coarse lower topology and the Scott topology (if the poset is a dcpo). The Scott topology is finer than the dual of the coarse lower topology. If the Scott topology is spectral, then so is the coarse lower topology, and the topologies are inverse to each other. This, and much more about both topologies and the connections between them, is the content of 7.1.21. In particular, it contains a characterization of the dcpos 1

For the emergence of the Scott topology, see especially [GHK+ 03, pp. xxiii–xxxiv].

205

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whose Scott topology is spectral. The key to these results is provided by the way-below relation, which is decisive for the description of the quasi-compact open sets for the Scott topology, 7.1.12. The study of spectrality of the coarse lower topology and the Scott topology is continued in Sections 7.2 and 7.3, where we apply 7.1.21 to special classes of posets and dcpos. The notion of a fine coherent poset (cf. 7.2.1) proves to be very useful. Conditional join-semilattices (which includes lattices, forests, and many root systems) are fine coherent. For fine coherent dcpos the Scott topology is spectral if and only if the coarse lower topology is spectral, 7.2.5. This leads to versions of 7.1.21 for complete lattices, 7.2.8, forests, and root systems, 7.3.8. Finally, in Section 7.4 we apply the previous results to the tree of infinite words built on an alphabet Γ. Their Scott topology is spectral, and the quasi-compact open sets have a particularly explicit description. It is a simple consequence of the order structure that word trees on alphabets of different cardinality are not homeomorphic to each other, 7.4.5. But the patch spaces (for the Scott topology) are homeomorphic to each other for all finite alphabets with at least two elements, 7.4.11. We mention [Pri94a] as a principal reference for this chapter and we give references en route for major overlaps with this paper.

7.1 When Scott is Spectral Summary We start with a poset X = (X, ≤) and ask whether the coarse lower topology (see A.8 (ii)) and the Scott topology (which will be defined in 7.1.6 and is an upper topology in the sense of A.8(i)) are spectral. The Scott topology is defined only under the hypothesis that X is a dcpo, 7.1.6. In view of the questions we study, this is not a serious restriction since the specialization poset of any spectral space is a dcpo and an fcpo, 4.2.7. We start with a review of elementary properties of the coarse lower topology and the Scott topology, see 7.1.4 and 7.1.8. It is a particularly important task to determine the quasicompact open sets. For the Scott topology a characterization is achieved in 7.1.12. It implies that the coarse lower topology is spectral if the Scott topology is spectral, and then both topologies are inverse to each other, 7.1.13. After analyzing the quasi-compact open sets for the Scott topology, always with a view toward the spectral space axioms S1 to S4, see 7.1.18, 7.1.19, and 7.1.20, we reach the main result, Theorem 7.1.21. It gives a characterization of those dcpos whose Scott topology is spectral and shows that, with some restrictions,

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207

spectrality of the Scott topology is equivalent to spectrality of the coarse lower topology. Some posets with spectral coarse lower topology have been encountered already. In Theorem 1.6.4 we gave a characterization of those totally ordered sets whose coarse lower topology is spectral. In 2.2.4(ii) we showed that the coarse lower topology on the power set of any set (with inclusion as partial order) is spectral. Looking ahead, we mention that the topology of any Noetherian spectral space is a coarse lower topology, Theorem 8.1.11. The development of the theory starts with the definition of the way-below relation on subsets of a poset. It is needed for the definition of the Scott topology and for the study of its quasi-compact open sets. 7.1.1 Definition Let X = (X, ≤) be a poset and F, G ⊆ X. (i) We write F 'X G, or F ' G, and say that F is way below G if the following condition holds: given an up-directed subset D ⊆ X with Ub(D) ⊆ G ↑ (where Ub(D) is the set of upper bounds of D, A.3(iii)) then D ∩ F ↑  ∅ . We omit the braces in the notation whenever F = {x} or G = {y} is a singleton. (ii) An element x is compact if x ' x. The set of compact elements of X is denoted by K(X). 7.1.2 Remarks Let X = (X, ≤) be a poset. (i) The way-below relation between elements can be found in [GHK+ 03, p. 49 f], where some motivation is also supplied, and in [PiPu12, 5.1, p. 134] under the name “well below”. It has applications in domain theory and in frame theory (pointfree topology). (ii) The way-below notion between subsets of a poset, as defined in 7.1.1(i), is more general than the one used in domain theory, [GHK+ 03, p. 226], where X is assumed to be a dcpo. (Just note that, if X is a dcpo, F, G are subsets of X and D ⊆ X is up-directed then Ub(D) = sup(D)↑ , hence Ub(D) ⊆ G ↑ if and only if sup(D) ∈ G ↑ . Therefore, F ' G if and only if sup(D) ∈ G ↑ implies D ∩ F ↑  ∅ . This is exactly the condition used in [GHK+ 03, p. 226] to define the way-below relation.) (iii) Without proof we record the following simple properties of the way-below relation between subsets of a poset X (generalizing [GHK+ 03, p. 50]). Let F, G, H, K ⊆ X. (a) Obviously, X is always way below itself. (b) F ' G ⇐⇒ F ↑ ' G ↑ =⇒ F ↑ ⊇ G ↑ . (c) H ↑ ⊇ F ↑ ⊇ G ↑ ⊇ K ↑ and F ' G imply H ' K.

208

Scott Topology and Coarse Lower Topology (d) ' is transitive. (e) ' is antisymmetric on the set of anti-chains and on the set of upsets. That is, let F, G be both anti-chains or both up-sets and assume F ' G and G ' F. Then it follows that F = G. (f) Usually ' is not reflexive, neither for singletons, nor for general subsets. (g) Let X be a dcpo and (Fi )i ∈I a family of subsets with Fi ' Fi for   all i ∈ I. Then i Fi ' i Fi . For, given an up-directed D with  sup(D) ∈ ( i ∈I Fi )↑ , there is some i ∈ I with sup(D) ∈ Fi↑ . Then  ∅  Fi↑ ∩ D ⊆ ( i ∈I Fi )↑ ∩ D. In particular, if F ⊆ X and x ' x for all x ∈ F, then F ' F. 2 (h) Let X be a dcpo and F, G up-sets with F ' F and G ' G. Then F ∩ G ' F ∩ G. For, if D ⊆ X is up-directed with sup(D) ∈ F ∩ G, then there are x ∈ D ∩ F and y ∈ D ∩G. Let z ∈ D be an upper bound for x and y. Since F and G are up-sets it follows that z ∈ D ∩ (F ∩ G).

(iv) If X has a bottom element ⊥, then ⊥ is compact. Now assume that X is totally ordered and pick x, y ∈ X. Then clearly x ' y if and only if x < y, or x = y is minimal, or x = y is an upper point of a jump. (v) If ∅ ' G then G = ∅ . Moreover, ∅ ' ∅ if and only if every up-directed set has an upper bound. Proof Assume that ∅ ' G and G  ∅ , say x ∈ G. Then {x} is up-directed and Ub({x}) ⊆ G ↑ , but {x} ∩ ∅ = ∅ , a contradiction. Now assume that ∅ ' ∅ and let D ⊆ X be up-directed. If Ub(D) = ∅ then Ub(D) ⊆ ∅ ↑ , but D ∩ ∅ = ∅ , a contradiction. Finally, if every up-directed set has an upper bound then the inclusion Ub(D) ⊆ ∅ ↑ = ∅ does not hold for any up-directed set D, hence the implication “Ub(D) ⊆ ∅ ↑ → D ∩ ∅ ↑  ∅ ” is true.  By definition, every up-directed subset of a dcpo has an upper bound. Hence, in a dcpo it is always true that ∅ ' ∅ .  (vi) Let X = i ∈I Ci be a partition into subsets that are both up-sets and down-sets, cf. A.3(vii). For F, G ⊆ X the following are equivalent: (a) F ' G in X. (b) For each i ∈ I, Ci ∩ F ' Ci ∩ G in Ci . Proof (a) ⇒ (b) Pick i ∈ I and D ⊆ Ci with UbCi (D) ⊆ (Ci ∩ G)↑ . Then UbX (D) = UbCi (D) ⊆ G ↑ . Hence D ∩ (Ci ∩ F)↑ = D ∩ F ↑  ∅ , and it follows that Ci ∩ F ' Ci ∩ G in Ci . 2

The dcpo assumption is critical here. For example, in the chain ω + ωinv , all points n in ωinv satisfy n ' n, but ωinv  ωinv ; cf. 7.1.2(ii).

7.1 When Scott is Spectral

209

(b) ⇒ (a) Let D ⊆ X be up-directed with UbX (D) ⊆ G ↑ . By A.3(vii), there is some i ∈ I with D ⊆ Ci . Thus, UbCi (D) = UbX (D) ⊆ Ci ∩ G ↑ = (Ci ∩ G)↑ , which implies D ∩ F ↑ = D ∩ (Ci ∩ F)↑  ∅ . This proves (a).  (c) Assume that the equivalent conditions (a) and (b) hold and Ci ∩F = ∅ for some i. Then Ci ∩ G = ∅ and every up-directed set contained in Ci has an upper bound. (d) In particular, if x ∈ X, say x ∈ Ci , then x 'X x if and only if x 'Ci x and ∅ 'C j ∅ for j  i. (e) If X is a dcpo and x ∈ Ci then x 'X x if and only if x 'Ci x. The following discussion motivates the notion of compactness for elements in a poset. In a topological space X, quasi-compactness of a subset K is defined by the Heine–Borel covering property, using the complete lattice of open sets. Explicitly, K is quasi-compact if and only if for all families (Ui )i ∈I of open sets   with K ⊆ i ∈I Ui there is a finite subset J ⊆ I with K ⊆ i ∈J Ui . This definition can be copied verbatim to posets, assuming a certain amount of completeness (which is satisfied in complete lattices, for example). Using the description of the way-below relation in a dcpo, cf. 7.1.2(ii), we prove (cf. [GHK+ 03, p. 115 and p. 9], also see [Ban81, p. 2]): 7.1.3 Lemma Let P be a dcpo such that every nonempty subset has an infimum. Then the following conditions about an element a ∈ P are equivalent: (i) a is compact (in the sense of 7.1.1(ii)).  (ii) For every S ⊆ L bounded from above and with a ≤ S there is a finite  subset F ⊆ S with a ≤ F . Proof (i) ⇒ (ii) First note that every set bounded from above has a supremum, namely the infimum of its (nonempty) set of upper bounds. Let F be the set of   finite subsets of S. Then the family ( F)F ∈ F is up-directed, and a ≤ S =   sup{ F | F ∈ F }. By compactness of a there is some F ∈ F with a ≤ F. (ii) ⇒ (i) Let D ⊆ P be up-directed with a ≤ sup(D). By (ii) there is a finite  subset F ⊆ D with a ≤ F. Let d ∈ D be an upper bound for F. Then it follows that a ≤ d, proving compactness.  The coarse lower topology on a poset is defined in A.8(ii). One of our main objectives is to determine when it is spectral. We start by compiling a list of elementary properties of the coarse lower topology. 7.1.4 Generalities on the Coarse Lower Topology Let X = (X, ≤) be a poset with coarse lower topology τ  (X).

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(i) Assume X is trivially ordered (i.e., x ≤ y if and only if x = y). Then: (a) τ  (X) is the cofinite topology, cf. 1.5.19 (i.e., the closed sets are X and the finite subsets of X). (b) A subset Y ⊆ X is irreducible if and only if Y is infinite or has at most one element. Thus: (c) The cofinite topology is not spectral if X is infinite, cf. 1.5.19. (d) Every subset of X is clearly quasi-compact for τ  (X). Moreover, every subset is also saturated (i.e., a down-set). It follows that: (e) (τ  (X))∂ is discrete. (ii) Let Y be a subset of X. Then τ  (Y ) ⊆ τ  (X)|Y . Equality holds under each of the following hypotheses: (a) For each x ∈ X there is a finite F ⊆ Y with x ↑ ∩ Y = F ↑ ∩ Y . (b) For all y ∈ Y and x ∈ X, if x ↓ ∩ Y ≤ y, then x ≤ y. (This condition is satisfied, for example, if X is the poset of open subsets of a space Z and Y is a basis of Z. For then, each x ∈ X is even the supremum of x ↓ ∩ Y in X.) Proof The sets y ↑ ∩ Y (with y ∈ Y ) are a subbasis of closed sets for τ  (Y ). They are restrictions of subbasic closed sets of τ  (X). It follows that τ  (Y ) ⊆ τ  (X)|Y . Assume (a) and note that the sets x ↑ ∩ Y with x ∈ X are a subbasis of closed sets for τ  (X)|Y . The hypothesis says that these are also closed for τ  (Y ), which implies the claim. Assuming (b), we show that, for each x ∈ X, the set Y ∩ (X \ x ↑ ) is open for τ  (Y ). Take y ∈ Y ∩ (X \ x ↑ ). Then x  y and, by (b), there is some y∗ ∈ x ↓ ∩Y with y∗  y. It follows that y ∈ Y \(y∗ )↑ ⊆ Y \x ↑ = Y ∩(X \x ↑ ), as required.    (iii) If Y ⊆ X is a down-set, then τ (Y ) = τ (X)|Y . For, given x ∈ X, the condition of (ii)(a) is satisfied by ∅ ⊆ Y if x  Y and by {x} ⊆ Y if x ∈ Y . In particular, τ  (X) restricts to the cofinite topology on the down-set X min (by (i)). For arbitrary subsets Y ⊆ X the topologies τ  (X)|Y and τ  (Y ) need not coincide. For example, let Y be the set of maximal points in the following poset: •

•

•

•

•

•

• •

•

7.1 When Scott is Spectral

(iv)

(v)

(vi)

(vii)

(viii)

211

Note that the point on the right defines a singleton subset of Y , which is open and nonempty, but not cofinite. Also recall from 2.2.4(ii) and 2.3.1 that every T0 -space is a subspace of a space with coarse lower topology. But not every T0 -topology is a coarse lower topology. A subset Y ⊆ X is quasi-compact in τ  (X) if and only if the following condition holds for every subset Z ⊆ X: if Y ∩ Ub(F)  ∅ for each finite F ⊆ Z, then Y ∩ Ub(Z)  ∅ . (See [Kop95, Theorem 4.7(b), p. 24] and [Pri94a, proof of Proposition 2.1(i), p. 83].) Proof The claim is exactly Alexander’s Subbasis Theorem ([Kel75, Ch. 5, Thm. 6]) applied to the subbasis {x ↑ | x ∈ X } of closed sets of τ  (X). Namely, Y is quasi-compact if and only if for all Z ⊆ X the following  implication holds: if Y ∩ Ub(F) = Y ∩ z ∈F z ↑  ∅ for all finite F ⊆ Z,   then Y ∩ Ub(Z) = Y ∩ z ∈Z z ↑  ∅ . If Q is a down-set and is quasi-compact for τ  (X) (equivalently, Q is quasi-compact saturated) then X \ Q ' X \ Q. In particular, if F ⊆ X and X \ F ↑ is quasi-compact then F ' F, cf. 7.1.2(iii)(b). (With additional hypotheses about X the implication we prove here becomes an equivalence, see 7.3.5 and 7.3.2(v).) Proof Let D ⊆ Q be up-directed. Then the family (Q ∩ d ↑ )d ∈D has the  FIP, hence has nonempty intersection (i.e., Q ∩ Ub(D) = Q ∩ d ∈D d ↑  ∅ ). This implies X \ Q ' X \ Q.   A set Y ⊆ X is irreducible for τ (X) if and only if for all finite F ⊆ X with Y ⊆ F ↑ there is some x ∈ F with Y ⊆ x ↑ . Proof The sets F ↑ with finite F are a basis of closed sets for τ  (X). The assertion follows since irreducibility can always be tested with a basis of closed sets.  Let x ∈ X and let Y ⊆ X be nonempty and τ  (X)-irreducible. Then x is the generic point of the τ  (X)-closure of Y if and only if x is the infimum of Y . Proof It has been shown in 4.2.1(ii) that x = inf(Y ) if x is the generic point of the closure of Y . Conversely, if x is the infimum of Y , then x ↑ contains Y , hence also the closure of Y . It remains to show that x belongs to the closure of Y . To prove this, let F ⊆ X be finite with Y ⊆ F ↑ . By (vi) there is some y ∈ F such that Y ⊆ y ↑ . Then y ≤ Y , hence y ≤ x,  hence x ∈ F ↑ , as required. Assume that τ  (X) is sober. Then X is an fcpo, X = (X min )↑ and X min is finite. Proof It follows from 4.2.1 that X is an fcpo. In particular, every maximal chain has an infimum, which implies X = (X min )↑ . If X min

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is infinite then it is irreducible, see (i) and (iii). It follows that X is irreducible (being the closure of X min ), but does not have a generic point, a contradiction.  (ix) Let γ be a spectral topology on X with specialization relation ≤inv . Then γ ⊆ (τ  (X))∂ . Indeed, since τ  (X) is the coarsest T0 -topology with specialization order ≤, we have τ  (X) ⊆ γ ∂ = γinv , and so γ = γ ∂∂ ⊆ (τ  (X))∂ , cf. 1.4.7. Every T0 -space occurs as a subspace of a poset with its coarse lower topology in many ways (e.g. see 2.3.2). For later use we record the following instance. 7.1.5 Proposition Let X be a T0 -space. Then the map f : X → O(X), f (x) = X \ {x} is a homeomorphism onto its image, where O(X) is equipped with its coarse lower topology τ  (⊆). Dually, the map g : X → A(X), g(x) = {x} is a homeomorphism onto its image, where A(X) is equipped with its coarse upper topology τ u (⊆). 3 Proof Since X is T0 , the map g is injective. If A ∈ A(X), then clearly g(A) = {{x} | x ∈ A} = im(g) ∩ A↓ . The sets A↓ are a subbasis of closed sets for τ u (A(X), ⊆), hence g is a homeomorphism onto its image. The statement for f is proved similarly, or we may use f = c◦g, where c is the homeomorphism (A(X) τ u ) → (O(X), τ  ) given by taking complements.  Next we define the Scott topology on a dcpo. It plays a central role in domain theory. As a general source for domain-theoretic facts, we refer to [GHK+ 03]. We shall see that the Scott topology is closely related to the coarse lower topology. 7.1.6 Definition (cf. [GHK+ 03, Definition II-1.3, p. 134]) Let (X, ≤) be a dcpo. A subset U ⊆ X is open for the Scott topology on (X, ≤) if it satisfies the following conditions: (i) U is an up-set for ≤. (ii) U ' U, that is, for every up-directed set D ⊆ X, sup(D) ∈ U implies D ∩ U  ∅ .4 It is clear from the definitions that these are indeed the open sets of a topology. The Scott topology is denoted by σ(X, ≤) = σ(X) = σ. 3

4

The coarse upper topology of the poset A(X) (partially ordered by inclusion) is also known as the (Vietoris) hit-topology. For, given any O ∈ O(X), the set of A ∈ A(X) that hit O is subbasic open for the coarse upper topology. Recall that D  ∅ by our convention.

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7.1.7 Specialization and Domain Theory – a Precaution In topology there are two different traditions concerning the specialization relation. Let X be a T0 -space and consider x, y ∈ X. • In this book y is a specialization of x if y ∈ {x} and we write x  y, see 1.1.3. The specialization relation is a partial order. • Other authors (see e.g., [GHK+ 03, Definition O-5.2, p. 42], [GL13, Definition 4.2.1, p. 54]) define the specialization order ≤ to be the inverse of our specialization relation (i.e., x ≤ y if x ∈ {y}). This difference is important here since [GHK+ 03] is our main source for domain-theoretic facts. Quite obviously, there is a danger of confusion. For example, in [GHK+ 03] an open (or closed) subset is an up-set (resp., a downset) for the specialization order, which is opposite to our usage. Let Y ⊆ X and y ∈ Y . Then y is a closed point of Y for the Scott topology if and only if y is a minimal point for ≤ (i.e., (Y,  )max = Y min = (Y, ≤)min and, similarly, σ min max max ) =Y = (Y, ≤) ). (Y,  σ The notation x  y is not used in [GHK+ 03]. Of course, we continue to use it consistently throughout this book. Thus, the notation has a clear meaning and should not lead to any confusion. Also, phrases such as “x specializes to y”, or “y is a specialization of x”, do not occur in [GHK+ 03]. So, misunderstandings can be avoided if we use our words carefully. Note that the definition of the Scott topology in 7.1.6 refers only to the given partial order, hence is unmistakable. (There is no specialization relation before the topology is defined.) The Scott topology defined here is the same as in [GHK+ 03, Definition II-1.3, p. 134]. Moreover, the coarse lower topology (or the coarse upper topology) in this book is the lower topology (resp., upper topology) of [GHK+ 03, Definition O-5.4, p. 43]. The only difference is the direction of the specialization relation once a topology has been defined. 7.1.8 Generalities on the Scott Topology The Scott topology is defined only for dcpos, not for arbitrary posets. So, for the following considerations we assume that (X, ≤) is a dcpo. The proofs of most of the following statements are easy consequences of the definitions and are therefore omitted, see also [GHK+ 03, § II-1, pp. 134 ff]. (i) For Y ⊆ X we have Y ' Y if and only if Y ↑ ' Y ↑ (7.1.2(iii)(b)), if and only if Y ↑ is Scott-open. (ii) A set C ⊆ X is Scott-closed if and only if it is a down-set and is closed under up-directed suprema (i.e., is a sub-dcpo, A.6(xii)). In particular, every principal down-set x ↓ is the Scott closure of {x}.

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(iii) The Scott topology is an upper topology for ≤, A.8(i) For, τ u (X) ⊆ σ(X) ⊆ τU (X) (since open sets are up-sets and the principal down-sets, which are a subbasis of closed sets for τ u (X), are closed for the Scott = ≤inv . topology, (ii)). Thus, σ(X) is a T0 -topology and  σ (iv) If T is an upper topology then each up-set is the intersection of the open sets containing it. So, by (iii), the up-sets are the saturated sets for σ(X), 1.1.20. In particular, the principal up-set x ↑ is the smallest saturated set containing x. (v) Let U ∈ σ(X) and x ∈ X. Then x ∈ U if and only if U ' x. (vi) If C is Scott-closed, then C = (C max )↓ (because every maximal chain in C has a supremum in X, which even belongs to C). (vii) The Scott topology restricts to the discrete topology on X min . For, it is clear from (ii) that every subset of X min is Scott-closed. Or, one notes that X min is discrete already in the coarser topology (τ  (X))∂ , by 7.1.4(i) and (iii) in connection with (ix) below. (viii) X is quasi-compact in the Scott topology if and only if X is the up-set of a finite set (i.e., X = (X min )↑ and X min is finite), if and only if X is quasi-compact for every upper topology. Proof By (iii) and 4.1.2, X is quasi-compact in the Scott topology if and only if X = (X min )↑ and X min is quasi-compact. Thus it suffices to remember that the Scott topology restricts to the discrete topology on X min , (vii). Finally, 4.1.2 says that X is quasi-compact for the fine upper topology (hence for every upper topology) if and only if X is the up-set of a finite set.   (ix) Assume that Q ⊆ X is a quasi-compact saturated set for τ (X), 1.1.20. In particular, Q is a down-set, and 7.1.4(v) shows that X \ Q is Scott-open. The quasi-compact saturated sets for τ  (X) are a basis of closed sets for the dual topology (τ  (X))∂ , 1.4.7, hence (τ  (X))∂ ⊆ σ(X). Now let T be any topology with specialization order ≤. Then 1.4.7 shows that T ∂ ⊆ (τ  (X))∂ , and it follows that T ∂ ⊆ σ(X). (x) Let τ be some spectral upper topology on X. Then τ ⊆ σ(X), cf. the abstract of [Pri94a]. Note that τinv is a spectral lower topology, and the equality τinv = τ ∂ holds by 1.4.7. Now (ix) shows that τ = (τinv )inv = (τinv )∂ ⊆ σ(X). (xi) Assume that σ(X) is spectral. It is always true that (τ  (X))∂ ⊆ σ(X), (ix). Since τ  (X) ⊆ σ(X)inv it follows from 1.4.7 that σ(X) = (σ(X)inv )inv = (σ(X)inv )∂ ⊆ (τ  (X))∂ . Hence (τ  (X))∂ = σ(X).

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(xii) The equality σ(X) = τU (X) holds if and only if every element of X is compact (by definition of the topologies). (xiii) Suppose that τU (X) is spectral. Then it follows from (x) that σ(X) = τU (X) and from (xi) that (τ  (X))∂ = σ(X). Moreover, τ  (X) = τU (X)inv . To see this, note that the inclusion τ  (X) ⊆ τU (X)inv is clear, since both topologies have specialization order ≤. The other inclusion follows from ◦

K(τU (X)inv ) = K (τU (X)) = {F ↑ | F ⊆ X finite} ⊆ A(τ  (X)). 7.1.9 Proposition Let X = (X, ≤) be a dcpo and τ a spectral topology with specialization order ≤. Then τinv = σ(X) ∩ τcon . Proof It is clear that τinv ⊆ τcon . Moreover, 1.4.7 and 7.1.8(ix) show that τinv ⊆ (τ  )∂ ⊆ σ(X). For the inclusion σ(X) ∩ τcon ⊆ τinv , pick a proconstructible subset U ⊆ X (i.e., closed for τcon ) that is closed for σ(X), hence is closed for σ(X) ∩ τcon . Then U is a proconstructible down-set (7.1.8(ii)), hence is closed  for τinv , 4.1.6(ii). 7.1.10 Proposition Let X = (X, ≤) be a dcpo and let S ⊆ X be a subset. Let τ = σ(X)|S be the subspace topology on S inherited from σ(X). Then: (i) If S is a sub-dcpo of X, then τ is contained in σ(S), the Scott topology of S. (ii) Let S be locally closed for σ(X). Then S is a sub-dcpo and τ is the Scott topology of S. Proof (i) Let C ⊆ S be closed for τ, say C = A ∩ S with A ⊆ X Scott-closed. Then C is a down-set in S and is a sub-dcpo, hence is Scott-closed in S, see 7.1.8(ii). (ii) Let S = C ∩ U with C Scott-closed and U Scott-open in X. Note that both C and U are sub-dcpos, hence so is S, A.6(xii). Now the inclusion τ ⊆ σ(S) follows from (i), and it remains to show that σ(S) ⊆ τ. This is done in two steps, first for the case that S is Scott-open, then for the case that S is Scott-closed. Applying the cases consecutively we obtain the claim. Case 1 S is Scott-open. Pick V ∈ σ(S). Then V is an up-set in X, and it suffices to prove that V is way below itself in X. So, let D ⊆ X be up-directed with supX (D) ∈ V. Then supX (D) ∈ S, hence D ∩ S  ∅ . Since S is an up-set it follows that D ∩ S is up-directed and has supremum supX (D) ∈ S. As V is way below itself in S we see that D ∩ V = (D ∩ S) ∩ V  ∅ , proving the claim. Case 2 S is Scott-closed. Let A ⊆ S be closed for σ(S). Then A is a down-set in X and a sub-dcpo, hence is Scott-closed in X, 7.1.8(ii), hence is closed for τ. 

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7.1.11 Corollary Let (X, ≤) be a dcpo and let S ⊆ X be locally closed for the Scott topology. (i) S is quasi-compact for σ(X) if and only if S = (S min )↑ ∩ S and S min is finite, if and only if S is quasi-compact for every upper topology of X. (ii) If C ⊆ S is closed for the Scott topology of S, then C = (C max )↓ ∩ S. Proof By 7.1.10(ii) we may assume S = X. Then (i) and (ii) hold by 7.1.8(viii) and (vi), respectively.  In preparation for the main result (Theorem 7.1.21, which characterizes the dcpos with spectral Scott topology) we analyze the quasi-compact Scott-open subsets of a dcpo. 7.1.12 Corollary If (X, ≤) is a dcpo then ◦

K (σ(X)) = {F ↑ | F finite anti-chain and F ' F}. Proof If F ⊆ X is finite with F ' F then F ↑ is quasi-compact by 7.1.11(i) ◦

and is Scott-open by 7.1.8(i). Conversely, if U ∈ K (σ(X)) then 7.1.11(i) shows that U min is a finite anti-chain and U = (U min )↑ .  ◦

7.1.13 Corollary Let (X, ≤) be a dcpo. Every U ∈ K (σ(X)) is closed for the coarse lower topology. In particular, if σ(X) is spectral then τ  (X) = σ(X)inv , and τ  (X) is spectral. Proof

In view of 7.1.12 the first assertion is clear by the definition of the ◦

coarse lower topology. It follows that the basic open sets X \ U, U ∈ K (σ(X)), of σ(X)inv are open for τ  (X), hence that σ(X)inv ⊆ τ  (X). To finish the proof, recall that the specialization orders of τ  (X) and σ(X)inv coincide, 7.1.8(iii),  hence τ  (X) ⊆ σ(X)inv . 7.1.14 Corollary For a dcpo (X, ≤) the following conditions are equivalent: ◦

= ≤inv . (i) K (σ(X)) is a subbasis for a T0 -topology τ with  τ (ii) For all x, y ∈ X with x  y there is a finite anti-chain F ⊆ X such that F ' F, x ∈ F ↑ , and y  F ↑ .  x, hence x  Proof (i) ⇒ (ii) Pick x, y as in (ii). Then y inv x, hence y  τ

τ

{y} . There is some τ-open set U with x ∈ U, but y  U. Then there is also ◦

some subbasic open set F ↑ ∈ K (σ(X)) (where F is a finite anti-chain and F ' F) with x ∈ F ↑ and y  F ↑ .

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217

is a (ii) ⇒ (i) We show that ≤inv is the specialization relation of τ. Then  τ partial order, hence τ is T0 , 1.1.6. is equal to ≤inv and τ ⊆ σ(X). This implies ≤inv =  ⊆  . First note that  σ σ τ For the other inclusion pick x, y with y inv x (i.e., x  y). Then (ii) yields a ◦

finite anti-chain F such that F ' F, x ∈ F ↑ , and y  F ↑ . As F ↑ ∈ K (σ(X)) ⊆ τ τ we conclude that x  {y} (i.e., y   x).  τ

Next we state Rudin’s Lemma (without proof), which is a classical result about posets and will be used repeatedly in the sequel. For a topological generalization we refer to [HeKe13, Lemma 3.1]. 7.1.15 Rudin’s Lemma (cf. [GHK+ 03, Lemma III-3.3, p. 227]) Consider a poset X = (X, ≤) and let F be a set of finite nonempty subsets such that for all F, G ∈ F there is some H ∈ F with H ↑ ⊆ G ↑ ∩ F ↑ . Then there is an   up-directed set D ⊆ F ∈ F F such that D ∩ F  ∅ for all F ∈ F . 7.1.16 Lemma Let (X, ≤) be a dcpo, x ∈ X, and F a set of nonempty finitely generated up-sets such that  (a) x ↑ = F and (b) F is down-directed for inclusion. Then every up-set U with U ' x contains some element of F . Proof Assume, by way of contradiction, that V \ U  ∅ for every V ∈ F . Thus, for each V ∈ F the set FV = V min \ U is finite and nonempty. (For, V min is finite since V is finitely generated and V min is the smallest generating set. If FV = ∅ then V min ⊆ U, hence V ⊆ U, contradicting the assumption.) ↑ Claim If V ⊆ W with V, W ∈ F then FV↑ ⊆ FW . ↑ Proof It suffices to show that FV ⊆ FW . So, pick y ∈ FV . Since FV ⊆ W \ U there is some w ∈ W min with w ≤ y. As U is an up-set and y ∈ w ↑ \U it follows ↑ .  that w ∈ FW , hence y ∈ FW

The set {FV↑ | V ∈ F } is down-directed for inclusion by (b) and the claim. The FV s are finite and nonempty, hence Rudin’s Lemma (7.1.15) applies, and there  is an up-directed set D ⊆ V ∈ F FV ⊆ X \ U such that D ∩ FV  ∅ for all V ∈ F , say dV ∈ D ∩ FV . It follows from (a) that    d↑ ⊆ dV↑ ⊆ V = x ↑. sup(D) ∈ d ∈D

V ∈F

V ∈F

Now U ' x implies D ∩ U  ∅ , contradicting the choice of D.



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7.1.17 Terminology Let (X, ≤) be a dcpo. (i) If, for all x ∈ X, the set {F ↑ | F finite and F ' x} satisfies conditions (a) and (b) of 7.1.16 then X is called quasi-continuous; see [GLS83, Definition 1.2]. 5 (ii) If, for all x ∈ X, the set {F ↑ | F finite and F ' F ' x} satisfies conditions (a) and (b) of 7.1.16 then X is called quasi-algebraic, cf. [GHK+ 03, Definition III-3.23, p. 237]. 6 (iii) If X is quasi-algebraic then it is quasi-continuous. This follows easily from the observation that, for each x ∈ X, the subset {G ↑ | G finite G ' G ' x} ⊆ {F ↑ | F finite and F ' x} is co-initial for inclusion (by 7.1.16). 7.1.18 Lemma Let X = (X, ≤) be a dcpo and let τ be a lower topology on X. Suppose that there is a subset (cf. 7.1.12) ◦

F ⊆ K (σ(X)) = {F ↑ | F finite anti-chain and F ' F} that is a basis of closed sets for τ and is closed under finite intersections. Then: (i) F is a basis for the Scott topology σ(X). ◦

(ii) K (σ(X)) is a basis of σ(X) (i.e., axiom S2 holds). (iii) The inverse topology of σ(X) coincides with the dual topology. ◦

(iv) If F is also closed under finite unions, then F = K (σ(X)), hence σ(X) satisfies axiom S3. Proof (i) Let U be a Scott-open set and pick x ∈ U. By 7.1.8(v) and 7.1.16 it suffices to show that Fx = {V ∈ F | x ∈ V } has properties (a) and (b) of 7.1.16. Condition (a) holds as x ↑ is τ-closed and F is a basis of closed sets for τ. For (b), note that F is down-directed for inclusion, being closed under finite intersections. (ii) is immediate from (i), and (iii) follows from 1.4.7. ◦

(iv) By (i), F is a basis for σ(X). Since it is contained in K (σ(X)) and is ◦ closed under finite unions, it follows from 1.1.8 that F = K (σ(X)). Axiom S3 holds since, by hypothesis, F is closed under finite intersections.  5 6

In [GHK+ 03, Definition III-3.2, p. 226] such posets are called quasi-continuous domains. But note that they need not be domains, as remarked in [GHK+ 03, Remark on p. 227]. Again, in [GHK+ 03] a quasi-algebraic dcpo is called a quasi-algebraic domain, although, in general, it is not a domain. In [Pri94a, p. 85] the quasi-algebraic dcpos are called generalized-algebraic.

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◦

7.1.19 Proposition Let (X, ≤) be a dcpo. Then K (σ(X)) is a basis for σ(X) (i.e., σ(X) satisfies axiom S2) if and only if X is quasi-algebraic. Proof

For x ∈ X, let ◦

Fx ⊆ K (σ(X)) = {F ↑ | F finite anti-chain and F ' F} be the set of quasi-compact open sets containing x, cf. 7.1.12. ◦

First assume that K (σ(X)) is a basis of open sets. This implies immediately that each Fx is down-directed for inclusion and is an open neighborhood basis  for x. But then x ↑ = Fx , and we see that for all x ∈ X, Fx satisfies conditions (a) and (b) of 7.1.16. Conversely, assume X is quasi-algebraic, let U be Scott-open and x ∈ U. Note that U ' x (by 7.1.8(v)). Hence 7.1.16 yields some Vx ∈ Fx with x ∈ Vx ⊆ U. ◦

It follows that K (σ(X)) is a basis.



Proposition 7.1.19 was first proved in [Ern09, Prop. 7, Cor. 7, p. 2068] and [Yok09, Prop. 1.2, p. 331], answering a question initially raised by [Pri94a, p. 93]. Also see [GHK+ 03, Exercise III-3.24(iv), p. 237]. Compare the next result with [GHK+ 03, Prop. III-3.7(i), p. 229]. ◦

7.1.20 Proposition Let (X, ≤) be a dcpo and assume that K (σ(X)) is a basis of σ(X). Then: (i) σ(X) is sober.  (ii) LocCl(σ(X)) = {F ⊆ X | F finite anti-chain with F ' F}. Proof

(i) Let ∅  A ⊆ X be closed and irreducible for σ(X). Note that the ◦

set CA = {U ∈ K (σ(X)) | A ∩ U  ∅ } is closed under finite intersections ◦ (by irreducibility of A) and is down-directed under inclusion (since K (σ(X)) is a basis). Now 7.1.12 shows that each U min is finite and U = (U min )↑ . Since A is a down-set it follows that A ∩ U = A ∩ (A ∩ U min )↑ . The sets A ∩ U min with U ∈ CA satisfy the hypotheses of Rudin’s Lemma 7.1.15. Thus there is an  up-directed set D ⊆ U ∈ CA A ∩ U min with D ∩ (A ∩ U min )  ∅ for all U ∈ CA.  It follows that sup(D) ∈ A ∩ CA, which proves the claim. (ii) According to 4.5.11(iii), a point x is locally closed if and only if there is )max (i.e., x is a closed point of O), if some Scott-open set O with x ∈ (O,  σ ◦

◦

and only if x ∈ (U,  )max for some U ∈ K (σ(X)) (as K (σ(X)) is a basis). σ ◦

For U = F ↑ ∈ K (σ(X)) (where F is a finite anti-chain, cf. 7.1.12) we have )max = U min = F, and the assertion follows.  (U,  σ

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For quasi-continuous dcpos the next result is essentially [Pri94a, Thm 3.7]. 7.1.21 Theorem Let X = (X, ≤) be a poset. The following are equivalent: (i) X is a dcpo and the Scott topology σ(X) is spectral. (ii) The coarse lower topology τ  (X) is spectral and every A ∈ K(τ  (X)) is the up-set of a finite set. (iii) X has the following properties: (a) X is a dcpo and an fcpo. (b) For all x, y ∈ X with x  y there is a finite anti-chain F ⊆ X with F ' F, x ∈ F ↑ and y  F ↑ . 7 (c) Let F, G be finite anti-chains with F ' F and G ' G. Then there is a finite set H with F ↑ ∩ G ↑ = H ↑ . 8 (d) X min is finite. If these conditions hold, then: ◦

(iv) K (τ  (X)) = {X \ F ↑ | F finite anti-chain with F ' F}. (v) σ(X) = (τ  (X))inv . (vi) The constructible topology of τ  (X) is the join of the topologies σ(X) and τ  (X). 9 (vii) X is quasi-algebraic.  (viii) LocCl(σ(X)) = {F ⊆ X | F finite anti-chain with F ' F}. Proof Each of the conditions (i), (ii), and (iii) implies that X is a dcpo and an fcpo, cf. 4.2.7. Hence we may assume that X has these properties all along. (i) ⇒ (ii) We know from 7.1.13 that τ  (X) is spectral and σ(X) = τ  (X)inv . ◦

And 7.1.12 shows that every A ∈ K(τ  (X)) = K (σ(X)) is the up-set of a finite set. (ii) ⇒ (iii) Condition (a) is assumed (see the beginning of the proof). Condition (d) follows from 7.1.4(viii). For conditions (b) and (c) we first prove: Claim

◦

K(τ  (X)) ⊆ K (σ(X)). ◦

Proof of Claim If U ∈ K (τ  (X)) then X \ U ' X \ U (by 7.1.4(v)) and there is a finite anti-chain F with F ↑ = X \ U (by the assumption in (ii)). It follows ◦

from 7.1.12 that X \ U ∈ K (σ(X)). 7 8

See 7.1.14(i) for an equivalent topological condition. This condition implies H ' H, cf. 7.1.2(iii)(b) and (h). In view of 7.1.12, it says that

9

K (σ(X)) is closed under finite intersections. For a dcpo X the join of σ(X) and τ  (X) is called the Lawson topology, cf. [GHK+ 03, Definition III-1.5, p. 211], which is studied intensively in domain theory.

◦



7.2 Fine Coherent Posets and Complete Lattices

221

◦

Now we apply 7.1.18 with F = K(τ  (X)) ⊆ K (σ(X)), which is a sublattice of P(X) and a basis of closed sets for τ  (X), hence satisfies the hypotheses of 7.1.18 and 7.1.18(iv). Now 7.1.18(ii) shows that σ(X) satisfies axiom S2, which implies that X is quasi-algebraic, 7.1.19. The implication (i) ⇒ (ii) of 7.1.14 shows that condition (b) holds. Moreover, it follows from 7.1.18(iv) that ◦

K (σ(X)) is closed under finite intersections (axiom S3), which, in view of 7.1.12, yields condition (c). (iii) ⇒ (i) It is clear that σ(X) is T0 . Quasi-compactness follows from 4.1.2 )max is finite. It is an immediate consince X is an fcpo and X min = (X,  σ sequence of 7.1.12 and condition (c) that σ(X) satisfies axiom S3. Condition ◦

(b) and 7.1.14(i) show that K (σ(X)) is a subbasis of a T0 -topology. Thus, the ◦

inverse topology σ(X)inv (see 1.4.1) is a T0 -topology having K (σ(X)) as a basis ◦

of closed sets. It follows from 7.1.18 that K (σ(X)) is a basis of σ(X). Finally, soberness follows from 7.1.20. So far the equivalence of (i), (ii), and (iii) has been established. Assuming the equivalent conditions we turn to the remaining assertions. Since σ(X) is spectral it follows from 7.1.13 that (v) holds. Both (iv) and (vi) follow immediately (cf. 7.1.12). Quasi-algebraicity (i.e., (vii)) holds by 7.1.19 and has also been shown in the proof of the equivalence. Finally, (viii) holds by 7.1.20(ii). 

7.2 Fine Coherent Posets and Complete Lattices Summary We continue the study of the coarse lower topology and the Scott topology on posets and dcpos. The main result of the previous section, Theorem 7.1.21, is the point of departure. We show that the equivalences of 7.1.21 can be strengthened under suitable additional assumptions about the posets we consider. The notion of fine coherence, cf. 7.2.1, is a key condition. Every conditional join-semilattice is fine coherent. Fine coherence has an impact on the existence and the description of quasi-compact open sets for the coarse lower topology and the Scott topology, 7.2.4. This leads to the characterization of fine coherent dcpos with spectral coarse lower topology (equivalently: with spectral Scott topology), 7.2.5, and to related results for complete lattices and algebraic lattices, 7.2.8. Finally, these results are applied to algebraic lattices (of subsets of a set), 7.2.12. We obtain a method for the construction of spectral spaces whose points are substructures of algebraic structures (e.g., ideals in a ring, sub-vector

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spaces of a vector space), Example 7.2.13. The construction supplements the methods exhibited in Section 2.5. 7.2.1 Definition A poset X = (X, ≤) is fine coherent if for all x, y ∈ X there is a finite set F ⊆ X with x ↑ ∩ y ↑ = F ↑ . In a fine coherent poset, the set of upper bounds of a nonempty finite set F is the up-set of a finite set (by a trivial induction on the size of F). If, in addition, the poset itself is the up-set of a finite set, then it is said to have Property M in [Pri94a]. 7.2.2 Fine Coherence – Elementary Facts Let X = (X, ≤) be a poset. (i) First we note that the following conditions are equivalent to fine coherence: ◦

(a) Axiom S3 holds for the fine upper topology, that is, K (τU (X)) is closed under finite intersections. To see this, observe that Y ⊆ X is quasi-compact open for τU (X) if and only if Y = F ↑ for some finite subset F ⊆ X. (b) The fine upper topology is coherent. (This property accounts for the name “fine coherent”. Note that the quasi-compact saturated sets for τU (X) are exactly the quasi-compact open sets.) (c) The set of finite anti-chains, partially ordered by F ≤ G if and only ◦

if F ↑ ⊇ G ↑ , is a lattice and is isomorphic to K (τU (X)). (d) The intersection of two (or finitely many) basic closed sets for τ  (X) is basic closed. (ii) If X is fine coherent then condition (c) of 7.1.21(iii) holds. (iii) We shall see in 8.1.11(i) ⇒ (x) that the specialization poset of a Noetherian spectral space is fine coherent. More on Noetherian spectral spaces and their specialization posets may be found in Section 8.1. (iv) If X is a conditional join-semilattice (for the notion, see A.6(ii)) then it is fine coherent. In fact, either x ↑ ∩ y ↑ = ∅ = ∅ ↑ or x ↑ ∩ y ↑ is the principal up-set sup(x, y)↑ . We display a few other classes of conditional join-semilattices. For the notions of root systems and forests we refer to A.5. They are studied more closely in Section 7.3. 7.2.3 Proposition Let X = (X, ≤) be a poset. (i) If X is a lattice or a forest then it is a conditional join-semilattice. (ii) If X is a root system then the following are equivalent: (a) X is a conditional join-semilattice.

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(b) X is fine coherent. (c) The coarse upper topology τ u (X) is coherent. This holds, for example, if X is an fcpo root system. Proof (i) Trivially, lattices are conditional join-semilattices. Let X be a forest and x, y ∈ X with an upper bound z ≥ x, y. As z ↓ is totally ordered it follows that x ∨ y exists and is equal to max{x, y}. (ii) (a) ⇒ (b) holds unconditionally, as observed in 7.2.2(iv). (b) ⇒ (a) For x, y ∈ X there is a finite set F with x ↑ ∩ y ↑ = F ↑ . In the root system X the sets x ↑ and y ↑ are totally ordered, hence there is a smallest element z in the finite set F, and it is clear that z = sup{x, y}. (a) ⇒ (c) As X is a root system the inverse poset Xinv is a forest and τ u (X) = τ  (Xinv ). We claim that, if both Y1 and Y2 are quasi-compact and saturated for τ  (Xinv ), then so is Y1 ∩ Y2 . This is clear for saturatedness, only quasicompactness needs to be discussed. We prove this using 7.1.4(iv). So, let Z ⊆ Xinv be a subset such that Y1 ∩ Y2 ∩ UbXinv (F)  ∅ for all finite subsets F ⊆ Z. As Xinv is a forest it follows that each of these finite subsets is totally ordered (being bounded from above). Thus, Z is totally ordered and there are y1 ∈ Y1 ∩UbXinv (Z) and y2 ∈ Y2 ∩UbXinv (Z). It follows that y1 ∨ y2 = y1 ∧inv y2 ∈ Y1 ∩ Y2 ∩ UbXinv (Z), proving the claim. (c) ⇒ (a) Assume x, y ∈ X have an upper bound z (i.e., z ∈ x ↑ ∩ y ↑ ). The set x ↑ ∩ y ↑ is totally ordered (since X is a root system). Thus, if there is a minimal element t ∈ x ↑ ∩ y ↑ then it is unique (i.e., t = x ∨ y). For the existence of a minimal element, note that x ↑ and y ↑ are quasi-compact saturated for τ u (X), hence x ↑ ∩ y ↑ is quasi-compact saturated for τ u (X) and )max = (x ↑ ∩ y ↑ )min with contains z. By 4.1.2 there is some t ∈ (x ↑ ∩ y ↑,  τu t, hence t ≤ z. Thus, t is the desired minimal element. z u τ

For the last assertion, suppose the set Ub(x, y) of common upper bounds of x, y ∈ X is nonempty. Then it is a totally ordered set, hence has an infimum (by the fcpo property), which is x ∨ y.  We describe topological data of fine coherent posets that are related to spectral spaces. 7.2.4 Proposition Let X = (X, ≤) be a fine coherent poset. ◦

(i) K (τ  (X)) ⊆ { ∅ } ∪ {X \ F ↑ | F finite anti-chain with F ' F}. If X is a dcpo then: (ii) Q ⊆ X is quasi-compact for τ  (X) if and only if Q ↓ is Scott-closed.

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(iii) σ(X) = (τ  (X))∂ and τ  (X) is coherent, hence satisfies axiom S3. (iv) τ  (X) is quasi-compact, hence satisfies axiom S1. ◦

◦

(v) The map K (τ  (X)) → K (σ(X)) ∪ {X }, U → X \ U is bijective. Hence ◦ ◦ K (τ  (X)) and K (σ(X)) ∪ {X } are both sublattices of P(X). (vi) The following conditions are equivalent: ◦

(a) K (τ  (X)) is a basis of τ  (X). ◦

(b) K (σ(X)) is a basis of σ(X) (i.e., X is quasi-algebraic, cf. 7.1.19). (c) For all x  y there is a finite anti-chain F with F ' F with x ∈ F ↑ and y  F ↑ (cf. 7.1.14). ◦

Proof (i) Pick ∅  U ∈ K (τ  (X)) and note that X \ U ' X \ U, 7.1.4(v). It remains to show that the up-set X \ U is finitely generated. By the definition of  τ  (X) there is a family (Fi )i ∈I of finite subsets of X with U = i ∈I X \ Fi↑ . As  U is quasi-compact we may assume that I is finite. Then X \ U = i ∈I Fi↑ , and fine coherence yields a finite set (even an anti-chain) F with X \ U = F ↑ . (ii) Note that Q is quasi-compact for τ  (X) if and only if Q ↓ is quasi-compact, 4.1.2. Therefore we may assume that Q is a down-set. First let Q be quasi-compact for τ  (X) and consider an up-directed subset D ⊆  Q. The set {Q∩d ↑ | d ∈ D} has the FIP, hence Q∩Ub(D) = Q∩ d ∈D d ↑  ∅ . Therefore sup(D) ∈ Q. For the converse we use 7.1.4(iv). So, pick Z ⊆ X such that Q ∩ Ub(F)  ∅ for all finite subsets F ⊆ Z. We have to show that Q ∩ Ub(Z)  ∅ . For each finite set F ⊆ Z there is a finite anti-chain CF with Ub(F) = CF↑ (by fine ↑ , hence (Q ∩ CF )↑ ⊇ (Q ∩ CG )↑ . coherence). For F ⊆ G we have CF↑ ⊇ CG The sets Q ∩ CF are all finite and nonempty. Thus {Q ∩ CF | F ⊆ Z finite} satisfies the hypotheses of Rudin’s Lemma, 7.1.15. There is an up-directed set  D ⊆ F Q ∩ CF with D ∩ (Q ∩ CF )  ∅ for all F. Since Q is Scott-closed it follows that sup(D) ∈ Q. The construction shows that sup(D) is an upper bound for every finite subset of Z, hence of Z itself. (iii) The set of quasi-compact saturated sets for τ  (X) is a basis of closed sets for the dual topology, 1.4.7. By (ii) it coincides with the set of Scott-closed sets, which yields the assertion. (iv) It is clear that τ  (X) is T0 . Quasi-compactness follows from (ii). (v) The map is well-defined (by (i) and 7.1.12) and is clearly injective. For ◦

surjectivity, pick F ↑ ∈ K (σ(X)) \ {X } (where F is a finite anti-chain). Then X\F ↑ ∈ τ  (X) (by definition of the coarse lower topology) and is quasi-compact by (ii).

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The last assertion follows from the fact that both sets are closed under finite unions. ◦

(vi) (a) ⇒ (b) Note that K (σ(X)) is closed under finite intersections and is a basis of closed sets for τ  (X) (by (v)). Hence 7.1.18 yields (b). (b) ⇒ (c) follows from 7.1.14. ◦

(c) ⇒ (a) By 7.1.14 we know that K (σ(X)) is a subbasis of an upper topol◦ ogy. The complements are a basis of a lower topology, say τ (as K (σ(X)) is closed under finite unions). The definition of the coarse lower topology implies τ  (X) ⊆ τ, and (v) yields equality.  Assume in 7.2.4 that X is totally ordered. Then we recover 1.6.3(ii) from (vi). Note that x ' x holds if and only if x is minimal or the upper point of a jump, see 7.1.2(iv). The next theorem characterizes those fine coherent dcpos whose Scott topology is spectral (i.e., we spell out Theorem 7.1.21 for fine coherent posets). 7.2.5 Theorem (cf. [Pri94a, Thm. 3.11]) For a fine coherent dcpo X = (X, ≤) the following are equivalent: (i) σ(X) = (τ  (X))∂ (cf. 7.2.4(iii)) is spectral. (ii) τ  (X) is spectral. (iii) (a) X is an fcpo. (b) X is the up-set of a finite set. (c) For all x, y ∈ X with x  y there is a finite anti-chain F ⊆ X such that F ' F, x ∈ F ↑ , and y  F ↑ . If these conditions hold, then (iv) (v) (vi) (vii) (viii)

◦

K (τ  (X)) = {X \ F ↑ | F finite anti-chain with F ' F}. σ(X) = (τ  (X))inv . The patch topology of τ  (X) is the join of σ(X) and τ  (X). X is quasi-algebraic.  LocCl(σ) = {F ⊆ X | F finite anti-chain with F ' F}.

Proof (i) ⇒ (ii) follows from 7.1.21. Conversely, if τ  (X) is spectral then (τ  (X))∂ = τ  (X)inv is spectral, cf. 1.4.7 and 1.4.3 (i.e., (ii) ⇒ (i) holds). The implication (i) ⇒ (iii) follows from 7.1.21. For the converse, note that condition (c) of 7.1.21(iii) is a consequence of fine coherence, whereas conditions (a), (b), and (d) are satisfied by (iii). Now 7.1.21 shows that σ(X) is spectral. The remaining statements are taken from 7.1.21. 

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Let X be a complete lattice. Then X is a fine coherent dcpo, 7.2.3, and the preceding results are applicable to X. The set of compact elements of X is denoted by K(X), 7.1.1(ii). By 7.1.21(iii)(c), footnote, K(X) is a sub-joinsemilattice of X. Hence, for each x ∈ X the set x ↓ ∩ K(X) is up-directed. 7.2.6 Definition A complete lattice X is said to be algebraic if each x ∈ X is the supremum of x ↓ ∩ K(X), cf. [GHK+ 03, Definition I-4.2, p. 115 f]. 7.2.7 Lemma Every algebraic lattice is quasi-algebraic. Proof Let X be an algebraic lattice. Pick x ∈ X and consider the set F := {F ↑ | F finite and F ' F ' x}, cf. 7.1.17(ii). By fine coherence the set is  F. down-directed for inclusion. It is clear from the definition that x ↑ ⊆ ↑ ↓ Algebraicity implies that the subset {y | y ∈ x ∩ K(X)} ⊆ F has intersection x ↑ , finishing the proof.  7.2.8 Theorem (cf. [Pri94a, Thm. 4.1, 4.2, and 4.3]) Let X = (X, ≤) be a complete lattice. (i) The following conditions are equivalent: (a) σ(X) = (τ  (X))∂ (cf. 7.2.4(iii)) is spectral. (b) τ  (X) is spectral. (c) X is quasi-algebraic as a dcpo. ◦

(d) K (σ(X)) generates a T0 -topology with specialization order ≤inv . (ii) The lattice X is algebraic if and only if (a) τ  (X) is spectral, and (b) An element x ∈ X is compact if the following implication holds: 10  (*) For each subset S ⊆ X, if x = S then there is a finite subset  F ⊆ S with x = F. If X is an algebraic lattice, then (c) LocCl(τ  (X)inv ) = K(X), and ◦

(d) K (τ  (X)) = {X \ F ↑ | F ⊆ K(X) finite}. Remark A description of the locally closed points in a spectral root system, similar to 7.2.8(ii)(c), is given in 8.5.8. Proof (i) The equivalence (a) ⇔ (b) and the implication (a) ⇒ (c) have been shown in 7.2.5. The implication (c) ⇒ (d) follows from 7.1.19. For the implication (d) ⇒ (a), note that 7.2.5(iii)(a), (b) hold in every complete lattice, and 10

By 7.1.3 it is also true that condition (*) follows from compactness of x. Moreover, if S  ∅ then one may replace S by the up-directed set of suprema of its finite subsets, A.6(xiv), hence one may assume that S is up-directed.

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7.2.5(iii)(c) is the present condition (d) (7.1.19). Thus, (d) ⇒ (a) also follows from 7.2.5. (ii) First assume that X is algebraic. For the proof of (a) it suffices to check condition (i)(d). The definition of an algebraic lattice and 7.1.12 show that, for ◦

each x ∈ X, the set {y ↑ | y ∈ x ↓ ∩ K(X)} is contained in K (σ(X)) and is a neighborhood basis of x for σ(X). This proves (i)(d). To show (b), pick x ∈ X and assume that (*) holds. We use 7.1.3 to see that x is compact. This is clear if x is the bottom element, 7.1.2(iv). So, assume that  x  ⊥ and pick S ⊆ X with x ≤ S. Note that S  ∅ since x  ⊥, hence we may assume that S is up-directed, A.6(xiv).   Claim x = (x ∧ S) = {x ∧ s | s ∈ S}. (Thus, in algebraic lattices the frame law, A.6(xv), holds for up-directed sets.)  Proof of Claim It is clear that (x ∧ S) ≤ x. For each y ∈ x ↓ ∩ K(X) we have  y ≤ x ≤ S. Since y ' y there is some s ∈ S with y ≤ s, and it follows that    y ≤ x ∧ s, hence y ≤ (x ∧ S). Thus, x = (x ↓ ∩ K(X)) ≤ (x ∧ S) ≤ x.    Using (*), there is some finite F ⊆ S with x = (x ∧ F) ≤ F, showing (b). Conversely, we assume conditions (a) and (b) and show that X is algebraic.  For x ∈ X we define y = (x ↓ ∩ K(X)) and then prove x = y. It is clear that y ≤ x. Assume, by way of contradiction, that y < x. It follows from (a) and (i) that σ(X) is spectral, hence, by 7.1.21, there is a finite anti-chain F with F ' F, x ∈ F ↑ , and y  F ↑ . Pick t ∈ F such that t ≤ x. Then t  y, hence t  K(X) (by the definition of y). Condition (b) yields an up-directed set S with   t = S, but s < t for all s ∈ S. As S = t ∈ F ' F there is some s ∈ S ∩ F ↑ . Since F is an anti-chain we see that t = s ∈ S, a contradiction. (c), (d) Assume X is algebraic. Let F be an anti-chain with F ' F. The assertions follow from 7.2.5(iv), (v), and (viii) if we show that F ⊆ K(X). Pick  x ∈ F and let D be an up-directed set with x ≤ D. The claim above shows   that x = (x ∧ D). Hence we may assume that x = D (since x ∧ D is also up-directed). It follows from F ' F that there are y ∈ F and d ∈ D with y ≤ d. But then y ≤ d ≤ x implies x = y = d ∈ D (since F is an anti-chain).  7.2.9 Corollary A frame (cf. A.6(xv)) is algebraic if and only if its coarse lower topology is spectral. Proof The coarse lower topology of an algebraic frame is spectral by 7.2.8(ii). Conversely, if L is any frame, then the frame law, A.6(xv), implies the equality in the claim in the proof of 7.2.8(ii). Hence, condition (*) in 7.2.8(ii)(b) yields compactness. 

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7.2.10 Example The well-ordered set ω + 1 = (ω + 1, ≤) has a unique topology with specialization order ≤, namely τ  (ω + 1), and this topology is  spectral, 1.6.6. The coarse lower topology of the wedge X = (ω + 1) (ω + 1) is spectral as well, 1.6.10. It is clear that X is a root system and is an fcpo, hence satisfies the equivalent conditions of 7.2.5 (cf. 7.2.3). However, there are no elements x, y ∈ X with x ' y, in particular X has no compact elements. 7.2.11 Example There are complete lattices with spectral coarse lower topology, hence with many finite anti-chains F satisfying F ' F, cf. 7.1.12, but with very few compact elements. To see an example, let X be the spectral root system of 7.2.10. Enlarging the poset X by adding a bottom element we obtain a complete lattice, say Y . In a lattice the bottom element is always compact, 7.1.2(iv). It follows from 7.2.10 that x ' y if and only if x is the bottom element. This is the smallest possible way-below relation for a bounded lattice. To prove that τ  (Y ) is spectral, we show that the conditions in 7.2.5(iii) are satisfied. It is clear that 7.2.5(iii)(a) and (b) hold. To check condition (c), we exhibit the finite anti-chains F with F ' F. Anti-chains are either singletons or sets {a, b} ⊆ Y , where a, b are in different branches of X. We know already that {⊥} is the only singleton subset that is way below itself. Now consider an anti-chain {a, b} and let D be up-directed with sup(D) ∈ {a, b} ↑ . If ω ∈ D then a, b ≤ ω ∈ D. Assume that ω  D. Then D is contained in one of the two legs, say in the leg containing a. Since a is the upper element of a jump in ω + 1 it follows that there is some d ∈ D with a ≤ d. Thus we see that {a, b} ' {a, b}. This characterization of the finite anti-chains F with F ' F and the description ◦

◦

of K (σ(Y )) in 7.1.12 show that K (σ(Y )) generates a T0 -topology (of course with specialization relation ≤inv ). Thus τ  (Y ) is spectral. 7.2.12 Algebraic Lattices of Sets For any set S the power set P(S) is a complete lattice and has spectral coarse lower topology by 2.2.4(ii). Moreover, an element T ∈ P(S) is compact if and only if it is finite, as one checks easily. Now let L ⊆ P(S) be a subset that is closed under arbitrary intersections and under up-directed unions. Then L = (L, ⊆) is a complete poset, hence a complete lattice, A.6(ix). Clearly, it is a meet-subsemilattice of P(S), but in general not a join-subsemilattice.  Every subset T ⊆ S generates an element of L, namely (T) L = {B ∈ L | T ⊆ B}. We say that A ∈ L is finitely generated if there is a finite subset F ⊆ A such that A = (F) L . Then: (i) The compact elements of L are the finitely generated elements. (ii) The lattice L is algebraic.

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(iii) L in its coarse lower topology is a spectral subspace of P(S) in its coarse lower topology. In particular, the coarse lower topology of P(S) restricts to the coarse lower topology of L. Proof (i) and (ii) Let A ∈ L. For each finite set F ⊆ A it is clear that (F) L ⊆ A. Then  (∗) A= (F) L = sup L {(F) L | F ⊆ A finite} F ⊆ A finite

and the union is up-directed. It follows that A is a compact element of L if and only if this union is finite if and only if A is finitely generated, showing (i). Now equality (∗) says that each element of L is the supremum of compact elements, hence L is algebraic. (iii) The coarse lower topology of P(S) restricts to the coarse lower topology of L, as follows from 7.1.4(ii)(a). Given T ∈ P(S) we have T ↑ ∩ L = ((T) L )↑ ∩ L. By 7.2.8(ii), the coarse lower topology of the algebraic lattice L is spectral. Hence it remains to show that the inclusion map L → P(S) is spectral. Consider a subbasic closed constructible subset {s} ↑ = {T ⊆ S | s ∈ T } of P(S) for some s ∈ S, cf. 2.2.4(i)(c). Then {s} ↑ ∩ L is the principal up-set in L generated by (s) L , which is closed and constructible for the coarse lower topology of L, see 7.2.8(ii)(d).  7.2.13 Examples The situation described in 7.2.12 occurs very frequently in algebra and other contexts. We present a short list of typical examples. Many of the spectral spaces we obtain in this way can also be constructed with the methods of Section 2.5 (see, in particular, 2.5.12 and 2.5.13). (a) Let K be a field and V a K-vector space. Then the set L(V) of linear subspaces is closed in P(V) under intersections and up-directed unions. (b) Let R be a ring and let I(R) be the set of ideals (including the improper ideal R). Then I(R) is closed under intersections and under up-directed unions. Hence I(R) ⊆ P(R) is a spectral subspace, reproving 2.5.13(b). If R contains a subfield K, then R is also a K-vector space and the inclusion I(R) ⊆ L(R) is a spectral map, because both are spectral subspaces of P(R). On the other hand, the set of prime ideals of a ring, or of a bounded distributive lattice, is, in general, not closed under arbitrary intersections. In fact, the topology of a spectral space X need not be the coarse lower topology of a poset (i.e., X is in general not obtained through the mechanism in 7.2.12); for example, think of Boolean spaces. (c) Generalizing (a) and (b), substructures of a given first-order structure

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satisfy the conditions of 7.2.12 and thus constitute examples of spectral subspaces of the power set of the ambient structure, equipped with the coarse lower topology. This applies to subrings of a given ring, submodules of a module, ordered subgroups of an ordered group, and so on. (d) Other examples satisfying 7.2.12 and thus having a spectral coarse lower topology are the sets of -

algebraically closed subfields of a field, ideals of a join-semilattice S, denoted by I(S). 11 down-sets of a poset (but not ideals of a poset), convex subsets of a poset, convex subsets of Rn .

It is clear that this list can be extended indefinitely. As in 2.5.12, often such constructions can be extended to functors, which requires a specification of how homomorphisms induce spectral maps. For example, consider (a) and let ϕ : V → W be a linear map of K-vector spaces. Then the spectral map P(ϕ) : P(W) → P(V) restricts to a map L(ϕ) : L(W) → L(V), which is necessarily spectral and completes the definition of the functor. 7.2.14 Remark One can generalize 7.2.12, for example by working within an algebraic lattice M instead of (P(S), ⊆). However, 7.2.12 already implies that case. Let M be any algebraic lattice. Then the sub-poset K(M) of compact elements of M is a join-subsemilattice. Further, the map ϕ : M −→ I(K(M)), x → x ↓ ∩ K(M) is a poset isomorphism; using algebraicity of M one readily  shows that the compositional inverse sends an ideal I to M I. Since the set of ideals of any join-subsemilattice satisfies the conditions in 7.2.12, we see that any algebraic lattice is isomorphic to an algebraic lattice produced by the mechanism in 7.2.12. Finally, in 14.2.14 we will see a model-theoretic perspective on spectral spaces obtained from the coarse lower topology of algebraic lattices.

7.3 The Coarse Lower Topology on Root Systems and Forests Summary We continue to explore the consequences of 7.1.21 and 7.2.5, considering the cases of root systems and forests. In 7.2.3 we saw that forests are always fine coherent and root systems are fine coherent under mild additional 11

Notice also that the map S → I(S), s → s↓ is a poset isomorphism onto the compact elements of I(S).

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assumptions. In 7.3.1, 7.3.2, and 7.3.3 we collect various basic properties of forests and of root systems needed for the study of their coarse lower topologies. The way-below relation plays an important role again. Its main properties for forests are exhibited in 7.3.5. Then the spectral space axioms S1 to S4 (for the coarse lower topology) are related to the partial order in a root system or a forest, 7.3.6 and 7.3.7. Finally, we arrive at a characterization of the root systems and forests whose coarse lower topology is spectral, 7.3.8. As a consequence we prove that a root system with spectral coarse lower topology does not admit any other spectral topologies, 7.3.10. 7.3.1 Generalities on Forests Let X = (X, ≤) be a forest. See A.5(iii) ff. for the notion of a forest and a few elementary properties. In particular, every subset of a forest is a forest, and the up-directed sets are the nonempty chains. (i) We know, for any poset X and any down-set Y ⊆ X, that τ  (X) restricts to the coarse lower topology on Y , 7.1.4(iii). If X is a forest then this is also true for every basic τ  -closed set. Proof Let Y = F ↑ with F ⊆ X a finite anti-chain. If F = ∅ then Y = ∅ and the assertion is clear. So assume that F  ∅ and pick x ∈ X. We apply the criterion of 7.1.4(ii)(a). If x ∈ F ↑ then there is nothing to show. Assume that x  F ↑ . Then x ↑ ∩ F ↑ = (x ↑ ∩ F)↑ . The assertion follows  since x ↑ ∩ F is finite. (ii) By 7.1.4(iv), a subset Y ⊆ X is τ  (X)-quasi-compact if and only if every nonempty chain contained in Y ↓ has an upper bound in Y . (iii) The forest X is a dcpo if and only if every nonempty chain has a supremum. (This equivalence is true for any poset, see the Proposition in A.6(xi), but it is obvious for forests.) Also, X is an fcpo if and only if every nonempty chain has an infimum. Proof for fcpo A nonempty chain is down-directed, hence it has an infimum if X is an fcpo. For the converse, let D be a down-directed set and pick d ∈ D. Then d ↓ ∩ D is a nonempty chain, hence has an infimum. Since d ↓ ∩ D is co-initial in D it follows that Lb(d ↓ ∩ D) = Lb(D) (cf.  A.3(vi) and A.3(iii), hence inf(d ↓ ∩ D) = inf(D). (iv) The subset X min is always τ  (X)-quasi-compact, cf. 7.1.4(i) and (iii). Now assume that X is a dcpo. Then (ii) and (iii) show that Y ⊆ X is τ  (X)-quasi-compact if and only if every chain in Y ↓ has its supremum in Y ↓ . In particular, X is τ  (X)-quasi-compact, and so is X max , by 4.1.2. However, general anti-chains in X need not be quasi-compact. For an example, let X be the dcpo forest

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•

•

•

•

•

∞

and let Y be the set of points in the top row. Note that Y is an anti-chain, hence Y ↓ = (Y max )↓ . Let C be the bottom row without the point ∞. This is a chain contained in Y ↓ , but without an upper bound in Y . 7.3.2 Generalities on Root Systems Let X = (X, ≤) be a root system, cf. A.5(i) and (ii). The notions of root systems and forests are inverse to each other. So, many facts about forests can be translated into statements about root systems, and vice versa. We mention that every subset of a root system is a root system and the down-directed sets are the nonempty chains. (i) For each nonempty Y ⊆ X the set Ub(Y ) of upper bounds is totally ordered. In particular, if Z ⊆ X is bounded from below then Z is totally ordered. (ii) If A ⊆ X is an anti-chain then τ  (X)| A = τ  (A). Proof First note that τ  (X) restricts to τ  (A↓ ) on A↓ , 7.1.4(iii). Therefore we may assume that A = X max and X = A↓ . Then, for each x ∈ X, there is a unique a ∈ A with x ≤ a, hence x ↑ ∩ A = a ↑ ∩ A. Now the claim follows from 7.1.4(ii)(a).  In particular, A is always quasi-compact for τ  (X) and is irreducible if it is infinite, 7.1.4(i). (iii) For Y ⊆ X, the following are equivalent: (a) Y is quasi-compact for τ  (X). (b) Each nonempty chain C ⊆ Y ↓ has an upper bound in Y . (c) Y ↓ = (Y max )↓ . Proof (a) ⇒ (b) follows directly from 7.1.4(iv). (b) ⇒ (c) We show that Y ⊆ (Y max )↓ . This is clear if Y = ∅ . If Y  ∅ we pick y ∈ Y , consider a maximal chain C ⊆ Y ↓ containing y, and let z ∈ Y be an upper bound. Maximality of C implies z ∈ Y max . (c) ⇒ (a) Since Y max is an anti-chain, it is quasi-compact by (ii). Now 4.1.2 applies.  In particular, X is quasi-compact if and only if it has a ceiling, A.2(v). (iv) The intersection of two τ  (X)-quasi-compact down-sets is quasi-compact (and a down-set, of course), that is, τ  (X) is coherent. In view of (i) this follows from the implication (b) ⇒ (a) in (iii).

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(v) Let A ⊆ X be an anti-chain. Then X \ A↑ is τ  (X)-quasi-compact if and only if A ' A. Proof The implication “ ⇒ ” holds by 7.1.4(v). Now assume that A ' A and let C ⊆ X \ A↑ be a nonempty chain. If all its upper bounds are in A↑ then C ∩ A↑  ∅ , a contradiction. Thus, Ub(C) ∩ (X \ A↑ )  ∅ , and the claim follows from the implication (b) ⇒ (a) in (iii).  (vi) If X has a ceiling then the coarse lower topology satisfies the spectral space axioms S1 (by (iii)) and S3 (by (iv)). (vii) A root system X is an fcpo if and only if every nonempty chain has an infimum and is a dcpo if and only if every nonempty chain has a supremum. This is the inverse statement of 7.3.1(iii). 7.3.3 Graph Components in Forests and Root Systems Being posets, forests and root systems have graph components, A.3(vii). The graph components in spectral spaces (for the specialization order) are discussed in 6.6.10 and 6.6.11 and are compared with the connected components. Here we recall some basic facts about the graph components in forests and root systems, see A.5(ii), (iv). Let X = (X, ≤) be a root system and Xinv = (X, ≤inv ) the inverse poset, which is a forest. Therefore, most facts about graph components can be treated simultaneously for forests and root systems. (i) For each x ∈ X its graph component in X coincides with its graph component in Xinv . In X it is the set of y ∈ X having a common upper bound with x. In terms of Xinv , it is the set of y ∈ X having a common lower bound with x. (ii) Every graph component is both an up-set and a down-set (in X and in Xinv ). So, elements in different graph components are incomparable. The graph components are the minimal subsets (for inclusion) that are up-sets and down-sets. (iii) Let D ⊆ X be up-directed or down-directed. Then D is contained in a single graph component. (iv) Let A ⊆ X be an anti-chain. Then (iii) shows that A ' A in X (resp., Xinv ) if and only if, for each graph component C, C ∩ A ' C ∩ A holds in (C, ≤) (resp., (C, ≤inv )), cf. 7.1.2(vi). (v) If X has a ceiling (equivalently, Xinv has a floor) then the graph components are the principal down-sets generated by the maximal elements of X (equivalently, the principal up-sets generated by the minimal elements of Xinv ). (vi) If X is a spectral space and a root system (equivalently, Xinv is a spectral space and a forest) then the connected components of X coincide with the connected components of Xinv . They are unions of graph components,

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but in general the graph component of an element is smaller than its connected component, 6.6.10. (vii) If X is a dcpo then the graph components are clopen for σ(X). Proof Let Y be a graph component. Every up-directed set D is contained in the connected component of sup(D) (since its elements are comparable with sup(D)). So, if sup(D) ∈ Y then D ∩ Y  ∅ , which shows that Y is Scott-open. On the other hand, if sup(D)  Y then D is contained in some other connected component, hence X \ Y is also Scott-open.  (viii) If Xinv is a dcpo (i.e., X is an fcpo) then every graph component of Xinv is clopen for the Scott topology σ(Xinv ). The proof is verbatim the same as that of (vii). We have seen in Section 7.1 and in Section 7.2 (see, in particular, 7.1.12, 7.1.18, 7.1.21, and 7.2.4) that, under suitable hypotheses, the quasi-compact open sets for the coarse lower topology and the Scott topology can be described using the way-below relation. It has been shown in 7.3.2(v) that, for a root system X, the τ  (X)-quasi-compact saturated sets are the anti-chains that are way below themselves. We continue in this direction and take a look at the way-below relation for forests. Also compare the following facts with 7.1.2(iv). 7.3.4 Lemma Let X be a forest, equipped with the coarse lower topology. Assume that X is quasi-compact, and pick x, y ∈ X. It is always true that x ' y implies x ≤ y, [GHK+ 03, p. 50]. In addition, we now have: (i) If x < y then x ' y. (ii) Consider the following conditions: (a) X \ y ↑ is quasi-compact. (b) y ' y. (c) y is minimal in X or is the upper point of a jump. The implications (a) ⇔ (b) ⇐ (c) are always true. If X is a dcpo then (b) ⇒ (c) holds as well. Proof (i) Let D be up-directed (hence a chain) with y ≤ sup(D). Then, x < y ≤ sup(D) and, since D is totally ordered, there is some d ∈ D so that x < d ≤ sup(D) (i.e., x ↑ ∩ D  ∅ ). (ii) (a) ⇒ (b) follows from 7.1.4(v). (b) ⇒ (a) Let C ⊆ X \ y ↑ be a nonempty chain. Since y ' y it follows that Ub(C) ∩ (X \ y ↑ )  ∅ . The claim follows from 7.3.1(ii). (c) ⇒ (b) Assume there is an up-directed set D with D ∩ y ↑ = ∅ and Ub(D) ⊆

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235

y ↑ . Note that D is a chain and has an upper bound by quasi-compactness, 7.3.1. If y is minimal then D ⊆ y ↑ , a contradiction. So, let z < y be a jump. Then D ⊆ z ↓ (i.e., z ∈ Ub(D)), a contradiction. (b) ⇒ (c) Now let X be a dcpo forest. (Note that this implies quasi-compactness, 7.3.1(iv)). Assume that y is not a minimal point and not the upper point of a jump. Then y ↓ \ {y} is a nonempty chain with supremum y, which is impossible since y ' y.  7.3.5 Theorem Let X = (X, ≤) be a dcpo forest equipped with the coarse lower topology. Let A ⊆ X be an anti-chain. Then the following conditions are equivalent: (i) (ii) (iii) (iv)

A ' A. X \ A↑ is quasi-compact. Every a ∈ A is compact. For every a ∈ A either a ∈ X min or a is the upper point of a jump.

In particular: (v) B ' B for every subset B ⊆ X min . Proof (i) ⇒ (ii) Let C ⊆ X \ A↑ be a nonempty chain. Then A ' A implies sup(C) ∈ X \ A↑ , and the claim follows from 7.3.1(ii). (ii) ⇒ (iii) Pick a ∈ A and let D be up-directed (hence a chain) with sup(D) ∈ a ↑ . Then Ub(D) = sup(D)↑ ⊆ a ↑ ⊆ A↑ . It follows from 7.3.1(ii) that D  X\A↑ . There is some b ∈ A with b↑ ∩ D  ∅ . Both a and b belong to the chain sup(D)↓ , hence are comparable. Since A is an anti-chain we see that a = b (i.e., a ↑ ∩ D  ∅ ). (iii) ⇒ (iv) has been shown in 7.3.4. (iv) ⇒ (i) Consider an up-directed set D with sup(D) ∈ A↑ , hence a ≤ sup(D) for some a ∈ A. Since a ' a (see 7.3.4) it follows that ∅  a ↑ ∩ D ⊆ A↑ ∩ D. (v) follows from the equivalence (i) ⇔ (iv).



Remark Continuing with a dcpo forest X, the quasi-compact sets X \ A↑ in 7.3.5 are even super quasi-compact 12 in the sense that every cover X \ A↑ ⊆  ↑ i ∈I X \ xi by subbasic open sets has a subcover by at most two sets. For, if there are two incomparable elements xi and x j then xi↑ ∩ x j↑ = ∅ , hence

X \ A↑ ⊆ X = (X \ xi↑ ) ∪ (X \ x j↑ ). On the other hand, if the xi form a chain then  the supremum, say x, exists and i ∈I xi↑ = x ↑ . Since x ↑ ⊆ A↑ there is some 12

The notion of super-compactness has been introduced by de Groot, see [FPT69]. For more on super-compact spaces we refer to [vM77].

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a ∈ A with a ≤ x. As a is compact (see 7.3.5) there is some i ∈ I with a ≤ xi , and it follows that X \ A↑ ⊆ X \ xi↑ . Next we explore the connections between the different spectral space axioms and properties of root systems and forests. 7.3.6 Proposition Let X = (X, ≤) be a root system, equipped with the coarse lower topology. (i) X is quasi-compact if and only if X = (X max )↓ , if and only if X satisfies axiom S1. (ii) X satisfies axiom S3. (iii) X is sober (i.e., satisfies axiom S4) if and only if X is an fcpo and X min is finite. Proof (i) and (ii) have been shown in 7.3.2(iii) and (iv). (iii) The implication ( ⇒ ) has been shown in 7.1.4(viii). For the converse, let A ⊆ X be nonempty, irreducible, and closed. Since X is an fcpo we know that  X = (X min )↑ = x ∈X min x ↑ is a finite cover by closed sets. Hence A ⊆ x ↑ for some x ∈ X min , which implies that A is a chain. Since X is an fcpo it follows that inf(A) exists, and the assertion follows from 7.1.4(vii).  7.3.7 Proposition Let X = (X, ≤) be a forest, equipped with the coarse lower topology. (i) X is quasi-compact (i.e., satisfies axiom S1), if and only if every chain has an upper bound. (ii) X is coherent (hence satisfies axiom S3) if and only if it is a conditional meet-semilattice. (iii) Assume X is quasi-compact and coherent. 13 Then X satisfies axiom S2 if and only if it is jump-dense. (iv) If X is a dcpo then X is sober (hence satisfies axiom S4) if and only if X is an fcpo and X min is finite. Proof

(i) has been proved in 7.3.1(ii).

(ii) holds by 7.2.3(ii) applied to Xinv . (iii) Suppose that X satisfies axiom S2, and pick x < y. The subset y ↓ is a chain and a quasi-compact down-set. Hence τ  (X) restricts to τ  (y ↓ ), 7.1.4(iii), and, by coherence, τ  (y ↓ ) satisfies axiom S2. Now 1.6.3(ii) shows that there is a jump between x and y. 13

The assumption holds if X is a dcpo forest. But the assumption does not imply the dcpo property: consider the chain ω + ωinv , which is not a dcpo. On the other hand, its coarse lower topology is quasi-compact, cf. 1.6.3(i), and coherent (by (ii)).

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237

Conversely, let X be jump-dense and pick x ∈ X and an open neighborhood U. We may assume that U is a basic open set for τ  (X), say U = X \ F ↑ with F finite. Pick y ∈ F and note that y  x. Case 1 Assume that z = x ∧ y exists. Then z < y, hence there is a jump between z and y, say z ≤ u < v ≤ y. It follows that x ∈ X \ v ↑ ⊆ X \ y ↑ , and X \ v ↑ is quasi-compact by 7.3.4. Case 2 Assume that the infimum of x and y does not exist. Hence x ↓ ∩ y ↓ = ∅ since X is a conditional meet-semilattice, cf. (ii). Then either y is minimal (hence y ' y and X \ y ↑ is a quasi-compact neighborhood of x, cf. 7.3.4) or there are an element z < y and (as in Case 1) a jump z ≤ u < v ≤ y (hence v ' v and X \ v ↑ ⊆ X \ y ↑ is a quasi-compact neighborhood of x, using 7.3.4 again). Thus, for each y ∈ F there exists y  ' y  ≤ y with x ∈ X \ (y )↑ . Coher ence implies that y ∈F X \ (y )↑ is a quasi-compact open neighborhood of x contained in X \ F ↑ . (iv) If X is sober then the assertion holds by 7.1.4(viii). For the converse, let A ⊆ X be nonempty, irreducible, and closed. Since X is an fcpo it follows that  X = (X min )↑ , hence X = x ∈X min x ↑ is a partition into closed subsets. Since A  is irreducible there is a unique x ∈ X min with A ⊆ x ↑ . The set {a ↓ | a ∈ A} is a nonempty chain, hence has a supremum, say z (as X is a dcpo). This is the infimum of A, hence its generic point, cf. 7.1.4(vii).  7.3.8 Theorem Let X = (X, ≤) be a root system or a forest. The following are equivalent: (i) The Scott topology σ(X) is spectral. (ii) The coarse lower topology τ  (X) is spectral. (iii) (a) (X, ≤) is a dcpo and an fcpo. (b) (X, ≤) is jump-dense. (c) X min is finite. If these conditions hold then σ(X) = τ  (X)inv . Proof If there is any spectral topology τ with specialization order ≤ or with specialization order ≤inv then we know that X is a dcpo, an fcpo, and is jumpdense, see 4.2.7 and 4.2.12. So we may add these properties to the hypotheses about X. Therefore X is fine coherent (by 7.2.3), hence (i) and (ii) are equivalent (see 7.2.5). Moreover, (ii) ⇒ (iii) follows from 7.1.4(viii). Finally it remains to show that τ  (X) is spectral if X min is finite. First let X be a forest. We apply 7.3.7: a dcpo forest is quasi-compact (by

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7.3.1(iv)) and a conditional meet-semilattice (by the inverse of 7.2.3(ii)). Jumpdensity holds by hypothesis and X min finite is assumed. Altogether, by 7.3.7, all spectral space axioms hold for τ  (X). Now assume that X is a root system. We check condition (c) of 7.2.5(iii). So, pick x, y ∈ X with x  y. We have to produce a finite anti-chain F ⊆ X such that F ' F, x ∈ F ↑ , and y  F ↑ . Let C be a graph component of X. The assumption about X min yields: • C ∩ X min is finite, hence: • C = (C ∩ X min )↑ is basic closed for τ  (X), and • There are only finitely many graph components, which are clopen and quasi-compact (since X is quasi-compact, see 7.3.2(iii)). Let Cx and Cy be the graph components of x and y. Case 1 Cx  Cy . Then F = Cx ∩ X min is a finite anti-chain with F ' F (by 7.3.2(v), since X \ F ↑ = X \ Cx is a union of graph components, hence is quasi-compact). Clearly x ∈ F ↑ and y  F ↑ . Case 2 Cx = Cy . There is some t ∈ X max with x, y ≤ t (by quasicompactness), hence z = x ∨ y exists (by 7.2.3(ii)). Pick a jump u < v between y and z, and set Y = X \ u ↓ . Then Y is a root system and a dcpo and is jumpdense. It is also an fcpo. For, let D ⊆ Y be down-directed, hence a chain. Then inf X (D) exists. If inf X (D)  Y then inf X (D) ≤ u and D ⊆ inf X (D)↑ ∩ Y = v ↑ . It follows that v ≤ inf X (D), a contradiction. Finally note that Y min is finite. Thus, Y satisfies all conditions in (iii). It is clear that C = Cx ∩ Y is a connected component of Y . We set F = C min and note that this is a finite anti-chain (both in X and in Y ). In X we have x ∈ F ↑ and y  F ↑ . Clearly, F ' F as a subset of Y . To show that F ' F also holds in X, let D ⊆ X be up-directed with sup(D) ∈ F ↑ . Then D ⊆ Cx . Assume that D ∩ F ↑ = ∅ (i.e., D ⊆ u ↓ ). This implies sup(D) ≤ u, a contradiction, and the proof is finished.  We record the following inverse version of 7.3.8, which deals with the coarse upper topology. 7.3.9 Corollary are equivalent:

Let X = (X, ≤) be a root system or a forest. The following

(i) τ u (X) is spectral. (ii) (a) (X, ≤) is a dcpo and an fcpo. (b) (X, ≤) is jump-dense. (c) (X, ≤) has a finite ceiling (cf. A.2(v)).



7.4 Finite and Infinite Words

239

7.3.10 Corollary Let X = (X, ≤) be a root system with spectral coarse lower topology. Then (i) τ  (Xinv ) is spectral, and τ  (Xinv ) = τ u (X) = τ  (X)inv = σ(X). (ii) τ  (X) is the only spectral topology with specialization order ≤. (iii) τ u (X) is the only spectral topology with specialization order ≤inv . Proof (i) Since card(X max ) ≤ card(X min ), the forest Xinv satisfies the equivalent conditions of 7.3.8; hence its coarse lower topology is spectral. We know that τ  (Xinv ) = τ u (X) ⊆ τ  (X)inv = σ(X), 7.2.5. For the other inclusion, let ◦

U ∈ K(τ  (X)inv ) = K (τ  (X)). Then U = (U max )↓ (by 7.3.2(iii)). Note that U max is finite since U min = U ∩ X min is finite. Hence U is closed for τ u (X), which implies τ u (X) = τ  (X)inv . = ≤. Then τ u (X) ⊆ τinv ⊆ σ(X) = (ii) Let τ be a spectral topology with  τ τ u (X) (by (i) and 7.1.8(x)). Thus, τ = σ(X)inv = τ  (X), 7.3.8. (iii) follows from (ii).



7.3.11 Remark The uniqueness statement in 7.3.10(ii) can also be proved using 4.2.14. For the proof we show that the interval topology of X is Hausdorff under the assumptions of 7.3.10. To show this, pick x  y in X. Case 1 Let x and y be incomparable for ≤. Then M = X min \ x ↓ and N = X min \ y ↓ cover X min (since X is a root system). The sets M and N are finite, hence X \ M ↑ and X \ N ↑ are open for τ  (X). It is clear from the construction that the sets are disjoint and x ∈ X \ M ↑ , y ∈ X \ N ↑ . Case 2 Assume that x and y are comparable for ≤, say x < y. There is a jump (u, v) with x ≤ u < v ≤ y. We set M = X min \ u ↓ . Then A = X \ v ↑ ∩ X \ M ↑ is open for τ  (X) and contains x. On the other hand, B = X \ u ↓ is open for τ u (X) and contains y. Thus, A and B are open neighborhoods of x and y for the interval topology. One checks easily that A ∩ B = ∅ .

7.4 Finite and Infinite Words Summary The formation of words from an alphabet Γ is a classical construction in algebra (see, e.g., [Bou71a, §7, No. 2]). The set of finite words, which we denote by Γ∗ , is the free monoid generated by Γ. It is a tree with respect to a natural partial order. We embed it in the larger tree Γ ≤ω which contains also the infinite words on Γ. Notation and basic properties of the construction are explained in 7.4.1, 7.4.2, and 7.4.3. The tree Γ ≤ω has the properties

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(studied in Section 7.3) that are needed to make the coarse lower topology and the Scott topology spectral, 7.4.4 (i). We study connections between the spectral spaces (Γ ≤ω , σ(Γ ≤ω )) and (Δ ≤ω , σ(Δ ≤ω )) where Γ and Δ are different alphabets. The spaces are never homeomorphic if the alphabets have different cardinality, 7.4.5. On the other hand, if both alphabets are finite and contain at least two elements, then their patch spaces are homeomorphic, 7.4.11. More recently, word constructions have played a role in topology and combinatorics, see [EcNa08] and the references therein, as well as in the mathematical theory of computation and denotational semantics, cf. [Vic89, p. 63] (where word trees are called “Kahn domains”). In these applications the topological features of the trees play a prominent role. They are briefly mentioned in [Eda97, § 3.1, pp. 407–408], under the name domain of streams. 7.4.1 Words of an Alphabet Let Γ be an arbitrary nonempty set, finite or infinite. The most important case is the case 2 ≤ |Γ| < ∞. Its elements are considered as the symbols of an alphabet. We use the alphabet to build words (i.e., sequences of symbols). We fix the notation and recall some of the main definitions in the process. • Recall that n, for n ∈ N0 , is the set {0, 1, . . . , n − 1}, A.4(iv). The smallest limit ordinal is ω, which we identify with N0 , A.4(v). For k ∈ N0 we define k + ω = ω. • For n ≤ ω the elements of Γn are called words of length n. If s ∈ Γn then (s) = n is its length. For n = 0 there is only one word, the empty word, which is denoted by Λ. A word s is finite or infinite according as (s) ∈ N0 or (s) = ω.   • For n ≤ ω we write Γ 0 (a)) = {(αi )i ∈I | −a j  α j } = f j−1 (H >0 (a j )) is quasi-compact open, proving the assertion. 

13.5 The Real Spectrum in Real Algebraic Geometry Summary The real spectrum is an essential tool in real algebra and real geometry. In this section we show how the real spectrum relates to geometry. This requires a certain amount of model theory, which is presented in 13.5.A. Then the constructible sets in the real spectrum of a polynomial ring over a real closed field K are connected with the semi-algebraic subsets of the affine space K n , 13.5.B. The Real Chevalley Theorem is presented in 13.5.C. It says that the map between real spectra induced by a finitely presentable algebra sends constructible sets to constructible sets. Real closed fields are used frequently in this chapter. We start by recalling a few basic algebraic properties of this important class of ordered fields. 13.5.1 A real closed field (abbreviated RCF) is an ordered field (K, ≤) satisfying the intermediate value property for polynomials in one variable: if P ∈ K[X] changes sign between two points a, b ∈ K (i.e., P(a)P(b) < 0), then P has a root in K between a and b. Obviously, the ordered field of real numbers is real closed. Here are some basic algebraic properties of real closed fields (see [BCR98, Chapter 1]): (a) Every real closed field has a unique, total ring order, where every positive element has a square root. √ (b) A field K is real closed if and only if −1  K 2 and K( −1) is algebraically closed. (c) Every ordered field (F, ≤) has an algebraic real closed extension ρ(F, ≤) whose order (given by ρ(F, ≤)2 ) extends ≤; ρ(F, ≤) is unique up to a unique F-isomorphism; it is called the real closure of (F, ≤). (d) Every embedding of ordered fields extends uniquely to an embedding of their respective real closures.

13.5 The Real Spectrum in Real Algebraic Geometry

A.

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Constructible Sets as Definable Sets

We study the structure of the constructible subsets of real spectra of rings. As a consequence of Tarski’s celebrated Quantifier-Elimination Theorem for the first-order theory of real closed fields, we show that the constructible subsets of Sper(A) are exactly the sets parametrically defined by a first-order formula of the language of ordered rings, in a sense made precise in the statement of Proposition 13.5.5. This characterization is used below, notably in the proof of the Real Chevalley Theorem, 13.5.17. 13.5.2 Definition and Notation We will be using first-order model theory, applied to the case of ordered fields. For a brief recap on terminology, see Section 14.1. We denote by Lor the first-order language for ordered rings obtained by augmenting the language Lr for rings with a binary predicate ≤ to name the order (i.e., Lor = {+, ·, −, 0, 1, ≤}). As is customary in first-order logic, formulas are built from the operations, relations, and constants specified in the language by means of connectives (“conjunction,” “disjunction,” “implication,” “negation”) and quantifiers (“there exists,” “for all”). For illustration, here are a couple of examples (expressed in standard mathematical notation): • (In the language Lor ) “Positive elements have square roots: ” ∀x (x ≥ 0 −→ ∃y(x = y 2 )). • Using infinitely many formulas in the language Lr , one can express that a field is algebraically closed. Take all the formulas ∀x0 . . . ∀xn ∃y(xn y n + · · · + x1 y + x0 = 0), where n ∈ N. • Similarly, using infinitely many formulas in the language Lor , one can express that an ordered field is real closed. Take all the formulas $ ∀x0 . . . ∀xn ∀y, z y < z ∧ xn y n + · · · + x1 y + x0 < 0 ∧ x n z n + · · · + x 1 z + x0 > 0

%

→ ∃u(y < u < z ∧ xn u + · · · + x1 u + x0 = 0) n

where n ∈ N. This shows that the class RCF of real closed fields is firstorder axiomatizable. On these matters the reader may consult any standard text on first-order logic or model theory, for example, [ChKe90] or [Hod93].

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The Real Spectrum

13.5.3 Semi-Algebraic Sets Let (K, ≤) be a totally ordered field and n ∈ N. A semi-algebraic subset of K n is a finite Boolean combination of sets of the form {x ∈ K n | P(x) > 0}, where P ∈ K[X1, . . . , Xn ]. Note that the sets obtained by replacing > by ≥ or = are also semi-algebraic. In 13.5.1 we introduced the class of real closed fields (RCF). These fields have the following remarkable property. 13.5.4 Elimination Theory for Real Closed Fields (i) Tarski’s Theorem says that the first-order theory of real closed fields has quantifier elimination, see [Tar31], [Tar51]. This can be spelled out in two equivalent ways. (Logical version) Any first-order formula ϕ(v1, . . . , vn ) (n ≥ 0) of the language Lor of ordered rings is equivalent, under the axioms for RCF, to an Lor -formula without quantifiers in the same free variables v1, . . . , vn . (Geometric version) If K is a real closed field then the projection of any semi-algebraic subset of K n+1 onto K n is semi-algebraic. Tarksi’s original proof is constructive. An alternative constructive proof is given in [BCR98, §1.4, pp. 17–21]. In [Dic85, Thm. 3.4, pp. 83–85] an easier non-constructive proof using model theory can be found. An important and frequently used consequence of quantifier elimination is the following. (ii) Transfer principle. 6 A ring homomorphism f : K −→ L between RCFs K, L is an elementary embedding (i.e., preserves and reflects the validity of any first-order formula ϕ(v1, . . . , vn ), with n ≥ 0, of the language Lor ). For all a1, . . . , an ∈ K: ϕ(a1, . . . , an ) holds in K if and only if ϕ( f (a1 ), . . . , f (an )) holds in L. For example, if K ⊆ L, a semi-algebraic subset S of K n is nonempty if and only if the corresponding semi-algebraic subset of L n , defined by the same formula, is nonempty. This reasoning is used several times below. Remarks and Examples The transfer principle has many uses and applications. Here are two classical examples of transfer of results from R to any real closed field, where the proof over R uses tools (e.g., from analysis) that are not available for other real closed fields. The statements expressing these tools can in general not be transferred, however the proved statements themselves can be expressed by a (set of) first-order formula(s) in the language Lor and they can be transferred. 6

Also known as model-completeness of the theory of RCFs.

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• In [GiJe60, 13.3, pp. 174–175] the continuity (for the interval topology) of the real (parts of the complex) roots of polynomials over R as functions of its coefficients is proved using complex analysis (Rouché’s theorem). • A second example is the Bott–Milnor–Kervaire theorem [BoMi58] which asserts that Rn admits a structure of a (not necessarily associative or commutative) R-algebra without zero-divisors only for n = 1, 2, 4, 8; examples in these dimensions are classical, but not unique for n = 4, 8. The proof for R uses a result from differential topology (the “non-parallelizability of spheres” in certain dimensions). In this case, the transfer only uses the (weaker) completeness of the first-order theory of RCFs. If associativity is required (n = 1, 2, 4), there is an algebraic proof due to Frobenius that applies to any RCF; no such proof is known for n = 8. If commutativity (but not associativity) is required, then n = 1, 2 (Hopf). • Another application of the transfer principle is the Artin–Lang homomorphism theorem, see [ABR96, Thm. 1.7, p. 8]. (iii) Open quantifier elimination (cf. [vdD82], [Pre07]) This result, obtained by Recio in [Rec77] some 45 years after Tarski’s, adds a subtle complement to the latter. Suppose we are given a semi-algebraic set S ⊆ K n for some real closed field K. Tarski’s Theorem says that S is described by a formula of the form ri si   (+) ( Pi j (v1, . . . , vn ) > 0 ∧ Qik (v1, . . . , vn ) = 0) i

j=1

k=1

with Pi j , Qik ∈ K[X1, . . . , Xn ]. Suppose in addition that S is open in the Euclidean topology. A priori, nontrivial equalities may occur in such a representation. The Open Quantifier-Elimination Theorem 7 shows that S can, alternatively, be defined by a quantifier-free formula   pr (†) r s=1 Rr s (v1, . . . , vn ) > 0, for some Rr s ∈ K[X1, . . . , Xn ]. (Notice that dropping the equalities in (+) will, in general, not give the desired result.) The representation (†) “certifies” unequivocally the openness of the set S. A similar result holds for closed semi-algebraic sets, upon replacing in (†) the strict inequalities > by weak ones ≥. Now for the promised description of the constructible subsets of Sper(A). 7

Sometimes also called Finiteness Theorem.

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The Real Spectrum

13.5.5 Proposition Let A be a ring. A set C ⊆ Sper(A) is constructible if and only if there is an Lor -formula ϕC (v1, . . . , vn ) and elements a1, . . . , an ∈ A such that C = {α ∈ Sper(A) | ϕC (πα (a1 ), . . . , πα (an )) is true in k(α)}. Proof “ ⇒ ” If C = H >0 (a1, . . . , an ), let ϕC (v1, . . . , vn ) be the formula v1 > 0 ∧ · · · ∧ vn > 0. The statement holds because, for α ∈ Sper(A) and a ∈ A, α ∈ H >0 (a) ⇐⇒ πα (a) > 0 in k(α) (see 13.1.5 and 13.1.6). For an arbitrary constructible set, proceed by induction on its structure as a Boolean combination of basic sets of this form. “ ⇐ ” We must show that the set {α ∈ Sper(A) | ϕC (πα (a1 ), . . . , πα (an )) is true in k(α)} is constructible, for any Lor -formula ϕ(v1, . . . , vn ) and all a1, . . . , an ∈ A. The Quantifier-Elimination Theorem for real closed fields (13.5.4(i)) reduces the problem to the case where ϕ is quantifier-free; by induction on the structure of (propositional) formulas it suffices to deal with the case where ϕ is atomic (i.e., of either form P(v1, . . . , vn ) > 0 or P(v1, . . . , vn ) = 0 with P ∈ Z[X1, . . . , Xn ]). Since πα is a ring homomorphism, we have P(πα (a1 ), . . . , πα (an )) > 0 in k(α) ⇐⇒ α ∈ H A>0 (P(a1, . . . , an )) and P(πα (a1 ), . . . , πα (an )) = 0 in k(α) ⇐⇒ P(πα (a1 ), . . . , πα (an )) ≯ 0 in k(α) and P(πα (a1 ), . . . , πα (an )) ≮ 0 in k(α) ⇐⇒ α ∈ H A≥0 (−P(a1, . . . , an )) ∩ H A≥0 (P(a1, . . . , an )). In either case we obtain a constructible subset of Sper(A).



As a particular case we obtain a recast of the characterizations of quasi-compact open and closed constructible sets given in 13.1.9(ii) and 13.1.11(ii). 13.5.6 Corollary Let A be a ring. (i) The quasi-compact open subsets of Sper(A) are those defined, in the sense of 13.5.5, by Lor -formulas of the form ri m  

Pi j (v1, . . . , vn ) > 0

i=1 j=1

for some n-tuple a1, . . . , an ∈ A and polynomials Pi j ∈ Z[X1, . . . , Xn ].

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(ii) The closed constructible subsets of Sper(A) are those defined, in the sense of 13.5.5, by Lor -formulas of the form r s  

Qk (v1, . . . , vn ) ≥ 0

=1 k=1

for some n-tuple a1, . . . , an ∈ A and polynomials Qk ∈ Z[X1, . . . , Xn ].  The representation of the constructible subsets of Sper(A) given by the preceding results is a powerful tool in the investigation of their topological properties, employing a combination of logical and topological techniques. As an illustration, we prove: 13.5.7 Proposition Let A be a ring. Let ϕ(v1, . . . , vn ) be an Lor -formula, and ¯ holds in R} is open for the Euclidean topolassume that the set {r¯ ∈ Rn | ϕ(r) ¯ holds in k(α)} ogy of Rn . Then, for a ∈ An , the set {α ∈ Sper(A) | ϕ(πα (a)) is quasi-compact open in the spectral topology of Sper(A). A similar statement holds for “closed and constructible” instead of “quasi-compact open.” By the transfer principle 13.5.4(ii), the field R can be replaced by any other real closed field. Proof The openness assumption, together with the Open QuantifierElimination Theorem for RCFs, 13.5.4(v), implies that ϕ(v1, . . . , vn ) is equiva i Pi j (v1, . . . , vn ) > 0, with lent, under the axioms for RCFs, to a formula i rj=1 Pi j ∈ Z[X1, . . . , Xn ]. By 13.5.6(i) the subset of Sper(A) defined (in the sense of 13.5.5) by this formula and the given n-tuple a ∈ An is quasi-compact and open for the spectral topology. 

B.

Constructible Sets and Semi-algebraic Sets

¯ We shall now apply the general structure theorems of 13.5.A to the rings K[ X], ¯ X = (X1, . . . , Xn ), of n-variable polynomials (n ≥ 1) over a real closed field K. 8 For such rings the presentation of constructible sets given in 13.5.5 yields ¯ and semi-algebraic a bijection between constructible subsets of Sper(K[ X]) n subsets of K ; this will be spelled out in 13.5.9. In analogy to the situation in algebraic geometry over algebraically closed ¯ max . For any fields, the points of K n are mapped injectively to (Sper(K[ X])) n ¯ −→ K at a¯ gives a point point a¯ ∈ K , the evaluation map eva¯ : K[ X] ≥0 ¯ Φ(a) ¯ = eva−1 ¯ (K ) ∈ Sper(K[ X]), 8

To be on more familiar ground, the reader can think, without essential loss of generality, of the case K = R.

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which is maximal, as its support is the maximal ideal m(a) ¯ = Ker(eva¯ ). The outstanding result in the present case is that, under the injection Φ, ¯ is the the topology induced on K n by the spectral topology of Sper(K[ X]) Euclidean topology (not the Zariski topology); a similar phenomenon will be ¯ exposed in 13.5.15 below. To see this, take a subbasic open set of Sper(K[ X]), which (according to 13.1.6) is of the form ¯ | P + supp(γ) >γ 0} H >0 (P) = {γ ∈ Sper(K[ X])

¯ with P ∈ K[ X].

Then (∗)

¯ >Φ(a) Φ−1 (H >0 (P)) = { a¯ ∈ K n | P + m(a) ¯ 0} = { a¯ ∈ K n | P(a) ¯ > 0} = P−1 ((0, +∞)).

¯ is a subbasis for the Euclidean Since the family {P−1 ((0, +∞)) | P ∈ K[ X]} n topology of K , the equality (∗) proves our assertion. Further, (∗) implies that the correspondence (∗∗)

H >0 (P) −→ P−1 ((0, +∞))

¯ (P ∈ K[ X])

is bijective. Incidentally, note that equality (∗) also shows that K n is dense in ¯ Summarizing: Sper(K[ X]). 13.5.8 Theorem With notation as above, let K be a real closed field. The ¯ in such a way that (identifying K n with its map Φ embeds K n into Sper(K[ X]) image), we have: ¯ induces the Euclidean topology on (i) The spectral topology of Sper(K[ X]) n K . ¯ (with P ∈ (ii) The subbasic quasi-compact open sets H >0 (P) of Sper(K[ X]) ¯ correspond bijectively to the semi-algebraic subsets of K n of the K[ X]) ¯ > 0}. form { a¯ ∈ K n | P(a)

¯ (iii) K n is constructibly dense in Sper(K[ X]).



Using quantifier elimination for RCFs (see 13.5.4(i)), one gets much more out of this bijection. Indeed, since inverse images commute with set-theoretic operations, taking finite Boolean combinations in (∗) gives, as in (∗∗), a one-toone correspondence between arbitrary semi-algebraic subsets of K n and finite Boolean combinations of (quasi-compact) subbasic open (i.e., constructible ¯ subsets) of Sper(K[ X]).

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13.5.9 Theorem Let K be a RCF, and let S be a semi-algebraic subset of K n . ¯ such that S = S∩ K n . (i) There is a unique constructible set S ⊆ Sper(K[ X])  (ii) The map S −→ S (called the tilde map) is an isomorphism between the Boolean algebra of semi-algebraic subsets of K n and the Boolean algebra ¯ of constructible subsets of Sper(K[ X]). (iii) The tilde map induces bijections between open (resp., closed) semialgebraic subsets of K n and quasi-compact open (resp., closed construct¯ ible) subsets of Sper(K[ X]). (iv) If S is an open (resp., closed) semi-algebraic subset of K n , then S is the ¯ whose largest (resp., smallest) open (resp., closed) subset of Sper(K[ X]) n intersection with K is S. (v) The tilde map commutes with the interior and closure operations for the ¯ Euclidean topology in K n and the spectral topology in Sper(K[ X]). Sketch of Proof To illustrate the main ideas of the argument, we comment only on items (i) and (iii). The other items then follow easily. (i) The existence of S comes from 13.5.5. If S is defined by    ( Pi j (a) ¯ =0∧ Qik (a) ¯ > 0) } , S = { a¯ ∈ K n | i

it suffices to set ¯ | S = {α ∈ Sper(K[ X])

j

 

(

i

k

πα (Pi j ) = 0 ∧

j



πα (Qik ) > 0) in k(α)}.

k

¯ As for uniqueness, if C1 ∩K n = C2 ∩K n with C1, C2 constructible in Sper(K[ X]), we must show C1 = C2 . By considering the symmetric difference C1  C2 , we reduce this to show C ∩ K n = ∅ ⇒ C = ∅ . Assume, by way of contradiction, that C ∩ K n = ∅ and C  ∅ . If C is defined (in the sense of 13.5.5) by a formula ϕ(v1, . . . , vm ) and elements ¯ (i = 1, . . . , m), condition C ∩ K n = ∅ implies that the formula Fi ∈ K[ X] ∀¯v ¬ϕ(F1 (¯v ), . . . , Fm (¯v )) holds in K (because for all a¯ ∈ K n , the prime cone Φ(a) ¯ is not in C). On the other hand, picking α ∈ C, the field k(α) is real closed and contains K. By the transfer principle 13.5.4(ii), this inclusion is elementary, and hence the formula holds in k(α) as well. However, using 13.5.5 again, the formula ϕ(πα (X1 ), . . . , πα (Xn )) holds in k(α), as α ∈ C.   (iii) If S is open, write it as S = i j H >0 (Pi j ) and apply equality (∗) preceding 13.5.8 to see that S ∩ K n is open in K n .

To prove that every open semi-algebraic subset U of K n is of the form C ∩ K n for some open constructible set C, we invoke the Open Quantifier-Elimination

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  ¯ > 0}, for Theorem 13.5.4(iii), which tells us that U = { a¯ ∈ K n | i j Pi j (a)   >0  some Pi j ∈ K[X1, . . . , Xn ]. It suffices to take C = i j H (Pi j ). Further References For more details on Theorems 13.5.8, 13.5.9, and the tilde operation, see [BCR98, §7.2]; especially Prop. 7.2.2, p. 143; Thm. 7.2.3 and Cor. 7.2.4, p. 144. The effect of the tilde operation in Sper(R[X]) is described in [BCR98, Ex. 7.2.6, p. 145]. The extension of the tilde operation to semialgebraic functions is explained in [BCR98, Prop. 7.2.8 and §7.3, pp. 145 ff]. 13.5.10 Corollary (Bröcker’s Ultrafilter Theorem, cf. [Brö82, §4]) Let K be an RCF. The Stone space of the Boolean algebra of semi-algebraic subsets of ¯ con . K n is homeomorphic to Sper(K[ X]) Proof By Stone duality (for Boolean algebras), the isomorphism of Boolean algebras in 13.5.9(ii) implies that their corresponding Stone spaces (of ultrafilters) are homeomorphic. ¯ are the clopens of Sper(K[ X]) ¯ con The constructible subsets of Sper(K[ X]) (see 1.3.15). By Stone duality of Boolean algebras again, the Stone space of the ¯ con .  ¯ con ) is homeomorphic to Sper(K[ X]) Boolean algebra Clop(Sper(K[ X]) ¯ is a Heyting spectral 13.5.11 Corollary Let K be an RCF. Then Sper(K[ X]) 9 space (cf. 8.3.1). ¯ By 13.5.9(ii), C = S for Let C be a constructible subset of Sper(K[ X]). n  and by 13.5.9(v), S = S& ∩K n . some semi-algebraic set S ⊆ K . Then C = S,  So, C is constructible by 13.5.9(i).

Proof

Remark The situation where the n-dimensional affine space K n (over some RCF K) is replaced by an arbitrary (nonempty) algebraic subvariety of K n is ¯ is replaced by the a particular case of the above. The polynomial ring K[ X] ¯ coordinate ring K[V] := K[ X]/I(V) of a given algebraic variety V, where: ¯ • V is defined, say, by polynomials P1, . . . , Pr ∈ K[ X]. n • The set V(K) = { a¯ ∈ K | P1 (a) ¯ = 0, . . . , Pr (a) ¯ = 0} of K-rational points of V is nonempty (this requirement guarantees that K[V] is a semi-real ring, i.e., Sper(K[V])  ∅ ). ¯ = 0 for a¯ ∈ V(K)} is the vanishing • I(V) = {F ∈ K[X1, . . . , Xn ] | F(a) ideal of V.  13.5.12 Dimension of Semi-Algebraic Sets The dimension theory of semialgebraic sets S ⊆ K n (K a real closed field) is exposed in [BCR98, §2.8, pp. 50 ff]. Briefly: 9

For an account of similar results for rings of power series and related rings, see 13.6.4.

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• dim(S) is the Krull dimension (cf. Section 4.3, Summary) of the ring K[X1, . . . , Xn ]/I(S) (as before, I(S) is the vanishing ideal of S). • For a constructible set C ⊆ Sper(K[V]), dim(C) is the Krull dimension of the spectral subspace C (see Section 4.3, Summary), that is, the maximum length of inclusion chains of prime precones of Sper(K[V]) belonging to C. 13.5.13 Proposition For any semi-algebraic set S ⊆ K n and any algebraic variety V over K, we have:  (i) ([BCR98, Prop. 7.5.6, p. 156]) dim(S) = dim(S). (ii) ([BCR98, Prop. 2.8.2, p. 50]) dim(S) = dim(VS ), where VS is the Zariski closure of S. Consequently: (iii) dim(Sper(K[V])) = dim(V).



13.5.14 Locally closed points of Sper(K[V]) Let V be an algebraic variety over a real closed field K. Then: (i) The specialization chains in Sper(K[V]) are finite, of length ≤ dim(V). (ii) All points of Sper(K[V]) are locally closed. Proof Item (i) is 13.5.13(iii). Item (ii) follows from (i) and the fact that Sper(K[V]) is a spectral root system, using 4.5.16(ii).  In 13.5.B we saw that Rn embeds naturally into Sper(R[T1, . . . ,Tn ]). This is also the case when we look at any completely regular topological space X, if R[T1, . . . ,Tn ] is replaced by the ring C(X) of continuous real-valued functions on X. 13.5.15 Proposition Every completely regular topological space X embeds naturally into the real spectrum of the ring C(X). 10 Proof As in 13.5.B, the evaluation function evx : C(X) −→ R, evx ( f ) = f (x),

for f ∈ C(X), x ∈ X

maps X into Sper(C(X))max . Thus, for x ∈ X we have ≥0 max Φ(x) := ev−1 . x (R ) ∈ Sper(C(X))

Since X is completely regular, we have: • The map Φ is injective ([GiJe60, §3.1, p. 36]). 10

Recall from 13.3.3 that Sper(C(X)) is homeomorphic to Spec(C(X)) via the support map.

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The Real Spectrum

• The family { f −1 ((0, +∞)) | f ∈ C(X)} is a (sub)basis for the topology of X ([GiJe60, Theorems 3.2 and 3.6]). As in 13.5.B(∗), for f ∈ C(X) we have >0 ( f )) = f −1 ((0, +∞)), Φ−1 (HC(X)

(†)

>0 ( f ) −→ f −1 ((0, +∞)) is bijective. Since, which implies that the map HC(X) >0 ( f ) | f ∈ C(X)} is a subbasis for the specby definition, the family {HC(X) tral topology of Sper(C(X)), we conclude that Sper(C(X)) induces the given topology on X. 

13.5.16 The Tilde Map in Rings of Continuous Functions There is no straightforward analog of the tilde map in the case of completely regular topological spaces. However, there is a map K(Sper(C(X))) −→ ba(Coz(X)) 11 sending C to Φ−1 (C), with Φ as in the proof of 13.5.15 (Φ−1 (C) can be thought of as C ∩ X). By 13.5.5, C is a finite Boolean combination of sets of the >0 ( f ), f ∈ C(X); by the equality (†) in 13.5.15, Φ−1 (C) is a finite form HC(X) Boolean combination of sets f −1 ((0, +∞)), and hence belongs to ba(Coz(X)), cf. [GiJe60, 1.11, p. 15]. One checks that the map F : PrimI(Coz(X)) −→ Spec(C(X)) p −→ { f ∈ C(X) | Coz( f ) ∈ p} is a homeomorphism onto the spectral subspace of so-called prime z-ideals, cf. [GiJe60, 2.7 ff, pp. 27–30]. Hence, after all, there is a tilde map, but only to the spectral subspace of prime z-ideals of C(X) (X is constructibly dense in the latter space). See also [Sch97, Theorem 3.2]. Remarks For more on tilde operators, see [ABR96, Ch. I, 3.4, pp. 14–15, and Ch. V, 5.2, pp. 139 ff]; also see the paragraph “The analytic tilde operators,” in 13.6.4 below.

C.

The Real Chevalley Theorem

This subsection is devoted to proving a real analog of the celebrated Chevalley Theorem of algebraic geometry. The classical Chevalley Theorem appears in [Gro64, Chap. IV, §1, Th. 1.8.4, p. 239]. 13.5.17 Real Chevalley Theorem ([CoRo82, Proposition 2.3, p. 34]) Let ϕ : A −→ B be a homomorphism of rings and suppose B is finitely presented as an A-algebra. Then Sper(ϕ) maps constructible sets to constructible sets. In other words, the map Sper(ϕ) is open for the constructible topologies. 11

That is, the Boolean envelope of the lattice of cozero sets of X; cf. 3.4.2, 8.4.15.

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Proof Let B = A[X1, . . . , Xn ]/I, with I generated by polynomials P1, . . . , Pr ∈ A[X1, . . . , Xn ]. Let C be a constructible subset of Sper(B) defined, in the sense of 13.5.5, by an Lor -formula θ(v1, . . . , vm ) and elements Qi /I (i = 1, . . . , m), where Q1, . . . , Q m ∈ A[X1, . . . , Xn ]: (∗)

C = {β ∈ Sper(B) | k(β) |= θ[πβ (Q1 /I), . . . , πβ (Q m /I)]}.

We prove that (Sper(ϕ))(C) is the set of all α ∈ Sper(A) with $ r (†) k(α) |= ∃y1 . . . yn (πα Pi )(y1, . . . , yn ) = 0 and i=1

%

θ[(πα Q1 )(y1, . . . , yn ), . . . , (πα Q m )(y1, . . . , yn )] , where πα F denotes the polynomial whose coefficients are the images under πα of the coefficients of F ∈ A[X1, . . . , Xn ]. We first take β ∈ C and show that α = (Sper(ϕ))(β) satisfies (†). By 13.3.2(ii), there is a homomorphism ψ : κ(α) −→ κ(β) of ordered fields with πβ ◦ ϕ = ψ ◦ πα . By 13.5.1(d), ψ extends to an embedding ψ : k(α) −→ k(β) of the real closures. Setting zi = πβ (Xi /I) we have ((πβ ◦ ϕ)F)(z1, . . . , zn ) = πβ (F/I) for F ∈ A[X1, . . . , Xn ], hence k(β) |= ((πβ ◦ ϕ)P j )(z1, . . . , zn ) = 0

( j = 1, . . . , r).

By (∗) we obtain k(β) |= θ[((πβ ◦ ϕ)Q1 )(z1, . . . , zn ), . . . , ((πβ ◦ ϕ)Q m )(z1, . . . , zn )], showing that the formula in (†) holds in k(β) with witnesses z1, . . . , zn . Substituting πβ ◦ ϕ by ψ ◦ πα and taking into account that the map ψ is elementary (13.5.4(iv)), we conclude that the formula in (†) holds in k(α). Conversely, take α ∈ Sper(A), assume that (†) holds, and pick witnesses y1, . . . , yn ∈ k(α). Then the equations (πα P j )(y1, . . . , yn ) = 0 ( j = 1, . . . , r) show that there is a ring homomorphism τ : B −→ k(α) with τ(a) = πα (a) (a ∈ A) and τ(Xi /I) = yi (i = 1, . . . , n). Setting β := τ −1 (k(α)+ ) we see that α = (Sper(ϕ))(β) and that the canonical map ψ : k(α) −→ k(β) is an isomorphism. Using τ ◦ ϕ = πα in (†) and composing with ψ yields k(β) |=  θ[πβ (Q1 /I), . . . , πβ (Q m /I)]; hence β ∈ C, by (∗). 13.5.18 Corollary Let F be a field, B a finitely generated F-algebra (i.e., B is the coordinate ring of some variety over F), and ϕ : F −→ B the inclusion map. Then the map Sper(ϕ) is open. Proof

From 13.5.17, since Sper(F) is Boolean.



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The Real Spectrum

13.5.19 Proposition Let K be a formally real field, let X be a single variable, and let ι : K[X] −→ K(X) be the inclusion map. Then Sper(ι) : Sper(K(X)) −→ Sper(K[X]) is a homeomorphism onto Sper(K[X])min . Hence its image is proconstructible, but it is neither constructible nor open. Proof Clearly Sper(ι) is a homeomorphism onto the prime cones of K[X] with support (0) and all these cones are minimal in Sper(K[X]). Conversely, if α ∈ Sper(K[X])min , then α has support (0), otherwise there is a nonzero polynomial P(X) ∈ supp(α) and some r in the real closure R of (K, α ∩ K) with P(r) = 0. But then, for any of the two orderings β of R(X) that specializes to αr , see 13.1.8, we have β ∩ K[X]  α, a contradiction. Hence Sper(ι) is a homeomorphism onto Sper(K[X])min and so Sper(K[X])min is proconstructible. To see that the image of Sper(ι) is not constructible/open we note that K(X) is a ring of fractions of K[X], but it is not a fraction ring of a single element. Now 13.3.8 proves the result.  The following result is of interest in the theory of ordered fields. 13.5.20 Proposition (Elman–Lam–Wadsworth, [ELW79]) Let F ⊆ L be fields such that L is finitely generated as a field over F and let ι : F −→ L be the inclusion map. Then Sper(ι) is an open map. Proof In general, L is not finitely generated as an F-algebra, so Chevalley’s Theorem cannot be applied directly to the map ι. However, L = F(X1, . . . , Xn )[y] for some n ∈ N and some y algebraic over F(X1, . . . , Xn ). By induction it suffices to deal with the following cases: (1) L = F[y], y algebraic over F. (2) L = F(X), where X is purely transcendental over F. Case (1) is a particular instance of 13.5.17. Case (2) requires a finer touch. Let ε0 : F −→ F[X] and ε : F[X] −→ F(X) be the inclusion maps. By 13.5.19, Sper(ε) is a homeomorphism onto Sper(F[X])min . Now take a constructible subset C of Sper(F(X)) and let D be its image under Sper(ε). Since Sper(F[X])min is proconstructible, there is an open quasi-compact subset U of Sper(F[X]) with U ∩ Sper(F[X])min = D. We then get (Sper(ε0 ))(D) = (Sper(ε0 ))(U), because if β ∈ U, then take δ ∈ Sper(F[X])min with δ ⊆ β; as U is open we get δ ∈ U, so δ ∈ D and, as Sper(F) is Boolean, we obtain ε0−1 (δ) = ε0−1 (β). By the Real Chevalley Theorem 13.5.17, (Sper(ε0 ))(D) is constructible. But this set is (Sper(ι))(C), as required. 

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Remark This result can be sharpened considerably; it can be shown that, under the given hypotheses, the spectral map Sper(ι) has a continuous section s : im(Sper(ι)) −→ Sper(L) (i.e., (Sper(ι)) ◦ s is the identity on im(Sper(ι))). This was first proved by Haran and Jarden in [HaJa85, Proposition 8.2].

13.6 Further Results and Reading 13.6.1 The Spectrum of an -Group The following books are general references for -groups: [BKW77], [AnFe88], [Dar95], and [Ste10]. In this book, a lattice-ordered group (or -group) is an Abelian group G = (G, +) together with a partial order ≤ such that (G, ≤) is a lattice and ≤ is compatible with the group operation (i.e., x ≤ y ⇒ x + z ≤ y + z holds). 12 It follows that (G, +) is torsion free and that (G, ≤) is a distributive lattice, [Dar95, Corollary 3.17, p. 15]. An -group homomorphism is a group homomorphism between -groups that preserves the lattice structures. Every totally ordered Abelian group Γ is an -group. Since every torsion-free Abelian group can be totally ordered, every such Abelian group is the group underlying an -group. If X is any set, then Γ X equipped with the product structure is obviously again an -group. One of the fundamental theorems of the area says that each -group is isomorphic to an -subgroup G of some Γ X , hence G is closed under addition and taking finite meets and joins; cf. [Dar95, Corollary 51.9, p. 341]. We have met -groups in this chapter already: if X is a topological space, then the set of continuous functions X −→ R carries an -subgroup of RX . The group is the additive group of the ring of continuous functions X −→ R. Another important source of examples comes from ring theory: if R is a Bézout domain (i.e., every finitely generated ideal of R is principal) with quotient field K, then the group K × /R× of divisibility of R equipped with the partial order a · R× ≤ b · R× ⇐⇒ a ∈ b · R is an -group. In fact, by the Jaffard–Ohm Theorem [AnFe88, Thm. 11.2], all -groups are of this form. (Recall: we are only talking about Abelian -groups.) The spectrum Spec(G) of an -group G consists of all kernels P ⊆ G of group homomorphisms G −→ Γ to totally ordered Abelian groups. These sets are called prime -ideals and an intrinsic algebraic description is the following: a subgroup P of G is a prime -ideal if and only if P is convex in G, and for 12

In general, -groups are not necessarily Abelian, see [Dar95, Ch. 1, §§2, 3, pp. 7, 9 ff], but we will not talk about this case.

532

The Real Spectrum

all a, b ∈ G with 0 ≤ a ∧ b ∈ P we have a ∈ P or b ∈ P. If P is a prime -ideal of G, then the order of G/P is given by a + P > 0 ⇐⇒ a > p for all p ∈ P. Using the technique from Section 2.5 (alternatively, one can apply 14.2.11) and the elementary description of prime -ideals above, the following can be shown without difficulty. Theorem Let G be an Abelian -group. Then Spec(G), equipped with the topology that has the sets H(a) = {P ∈ Spec(G) | a + P > 0} as a subbasis, is a spectral space. Specialization in Spec(G) is inclusion and ◦

K (Spec(G)) is the bounded sublattice of P(Spec(G)) generated by the sets H(a). In fact, writing D(a) = {P ∈ Spec(G) | a + P  0} and using H(a) = D(a ∨ 0), ◦

it is not difficult to see that K (Spec(G)) = {D(a) | a ∈ G}. Furthermore: (i) For any -group G, Spec(G) is a spectral root system. (ii) The set {G} ⊆ Spec(G) is constructible if and only if G has a strong order unit (i.e., an element u > 0 such that {nu | n ∈ N} is cofinal in G). (iii) If X is a topological space, then the Zariski spectrum of the ring C(X) of real-valued continuous functions is a proconstructible subset of the spectrum of the -group (C(X), +, ≤). (iv) Spec can be extended to a contravariant functor from the category of -groups and -group homomorphisms to the category Spec of spectral spaces: an -group homomorphism ϕ : G −→ H is mapped to  Spec(ϕ) : Spec(H) −→ Spec(G), P → ϕ−1 (P). As for references, we name [BKW77], [RuYa08], and [Sch13b]. The first two define the spectrum of an -group as the subspace of our Spec(G) consisting of proper prime -ideals; hence these spaces are in general not spectral. The paper [Sch13b] works with our definition and focuses on the study of the inverse space Spec(G)inv . The theorem above is contained in [Sch13b, Section 2]. 13.6.2 The Real Kaplansky Problem and the Real Homeomorphism Problem In 4.2.13 we stated and discussed the Kaplansky Problem, namely to characterize those partial orders that are the specialization orders of spectral spaces (equivalently, the specialization orders of spectra of rings). In 8.1.22 the specialization posets of Noetherian spectral spaces were characterized, but not the specialization posets of the Zariski spectra of Noetherian rings, for which the problem remains open. One can pose Kaplansky’s question in the realm of real spectra. That is,

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ask whether a root system (X, ≤) satisfying Kaplansky’s necessary conditions 4.2.12, namely (i) every nonempty chain of (X, ≤) has a supremum and an infimum, and (ii) it is jump-dense, is isomorphic to the specialization poset of the real spectrum of some commutative, unital ring. An affirmative answer to this question was given in [DGL00]. A second, related problem is the (real) homeomorphism problem, namely the question of whether any spectral root system is homeomorphic to the real spectrum of a ring. Note that the analogous problem with the words “real” and “root system” omitted has a positive answer, and this is precisely Hochster’s Main Theorem 12.6.6. The answer to the real homeomorphism problem is negative; in [DeMa94] Delzell and Madden gave a counterexample. In [MeTr12] Mellor and Tressl further refined the result by showing that the spectral space constructed in [DeMa94] is not homeomorphic to any subspace of the real spectrum of a ring. They also show that there is no description of the real spectra of rings in any language belonging to a very large family of infinitary first-order languages. After the Delzell–Madden counterexample was produced, the relevant problem in this area turned into the search for a topological characterization of the spectral spaces of the form Sper(A) for some ring A. There is still no candidate condition known today. 13 We note that the Delzell–Madden counterexample and the spaces constructed in [MeTr12] do not have a countable basis. Recently, Wehrung in [Weh18] has shown that a spectral root system with a countable basis is homeomorphic to the spectrum of an Abelian latticed-ordered group. We remark that the homeomorphism problem has a positive solution in the particular case of spectral spaces with a total specialization order. In fact, 1.6.4(i) shows that in this case the topology of the space X is the coarse lower . To get a ring whose real spectrum (and, topology determined by the order  X in fact, also its Zariski spectrum) is homeomorphic to X, we argue as follows: (1) Choose a divisible, totally ordered Abelian group G, whose set of convex subgroups, ordered by inclusion, is anti-isomorphic to (X,  ). For exX ample, one can choose the Hahn group with real coefficients for the chain ), see [PC83, Chapter I, §5]. (X,  X (2) An argument involving the order and the canonical valuation of the real closed field R((G)) of generalized power series with real coefficients and exponents in G, cf. [DaWo96, Definition 2.8, Theorem 2.15], shows that 13

However, see 13.6.5.

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the Zariski spectrum of the ring R[[ G ]] 14 is anti-isomorphic to the set of ); see [DGL00, convex subgroups of G, hence order-isomorphic to (X,  X §1, pp. 178–179] and [DaWo96, §2]. (3) Kaplansky’s property (ii) (see 1.6.4(i)(a)) entails that {V(P) | P ∈ Spec(R[[ G ]])} is a basis of closed sets for the Zariski topology; since V(P) is the up-set generated by P for inclusion, this coincides with the coarse lower topology determined by inclusion. (4) Finally, observe that the support map, 13.3.A, is a homeomorphism of Sper(R[[ G ]]) onto Spec(R[[ G ]]). This is because every prime ideal of R[[ G ]] is convex in the canonical order of this ring, and hence induces a unique order on R[[ G ]]/P, see 13.3.3. 13.6.3 Real Closed Rings By Hochster’s Main Theorem 12.6.6, the real spectrum of a ring A is homeomorphic to the Zariski spectrum of some other ring B. This can indeed be done functorially in a much stronger sense. Theorem There is a functor ρ : Rings −→ Rings together with a ring homomorphism ρ A : A −→ ρ(A) for each ring A such that (i) Sper(ρ A) : Sper(ρ(A)) −→ Sper(A) is a homeomorphism. (ii) The support map suppρ(A) : Sper(ρ(A)) −→ Spec(ρ(A)) is a homeomorphism.  Hence we obtain a commutative diagram of homeomorphisms Sper(ρ(A)) Sper(ρ A )

supp ρ( A)

Spec(ρ(A))

f

Sper(A) , where f := Sper(ρ A) ◦ supp−1 . We see that the real spectrum of A is naturally ρ(A) homeomorphic to the Zariski spectrum of ρ(A). Rings of the form ρ(A) are called real closed and ρ(A) is called the real closure of the ring A. We will not spell out the rigorous definition of the functor ρ and the morphism ρ A, as this is of technical nature and not instructive without the development of a broader context, which lies outside the scope of this book. However, we want to mention that ρ is indeed a reflector (cf. 3.4.4) with reflection maps ρ A, when co-restricted to the full subcategory of Rings whose objects are real closed 14

The ring R[[ G ]] is both the valuation ring of the canonical valuation on R((G)), and a real closed ring, cf. 13.6.3.

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rings. As such, the construction is a powerful tool for the analysis of the real spectrum of a ring. Prominent examples of real closed rings are real closed fields, convex subrings of real closed fields, rings of continuous functions X −→ R for some topological space X, and the ring R of continuous semi-algebraic functions Rn −→ R (hence continuous functions that have a semi-algebraic graph). In fact, R is the real closure of A = R[X1, . . . , Xn ] and ρ A is the inclusion map; one may consider A → R as a prototype of the construction of ρ A and ρ(A). 15 As for references: real closed rings were introduced by Niels Schwartz; see [Sch89], where the results above are proved as well. There are many different ways to define or characterize real closed rings and how they can be studied. For definitions and characterizations in various contexts, see [Sch89, Def. 4.1, p. 26], [Sch86, Theorems 19–23, p. 189 ff], [ScMa99, Section 12], and [Tre07, Def. (2.1)/Cor. (2.13)]. For rings of continuous functions as real closed rings, see [Sch97], [Tre07], [Tre10], and [Tre98]. 13.6.4 The Real Spectrum of Power Series Rings and Related Rings The first results on the topology of the real spectrum of formal power series rings with real coefficients, and quotients of them, were proved in [AGR85] (for two variables) and then in [AlAn87]. Here is an example. Theorem (Alonso–Andradas) Let A be a complete Noetherian local ring with formally real residue field F. Then: (i) Sper(A) is a Heyting spectral space (cf. 8.3.1). (ii) The connected components (in the spectral topology) of constructible subsets of Sper(A) are constructible (and finite in number) if and only if F has finitely many orders.  The proof reduces to the case when A = K[[ X1, . . . , Xn ]], the ring of formal power series in n variables with coefficients in a real closed field K, and then proceeds by induction on n using the Weierstrass Preparation Theorem; see [ABR96, Ch. VII, Thm. 8.3, pp. 205–207]. The same proof gives a similar result for the rings of convergent power series over R and of power series algebraic over polynomials over real closed fields; see [ABR96, p. 207]. Note that when A is a ring of polynomials over a real closed field, item (i) was proved in 13.5.11. 15

However, in general, ρ A is not injective (e.g. ρ(C) is the null ring). Also notice that real closures here are attached to rings and not to ordered rings. For example, the real closure of the √ ordered subfield Q( 2) of R is the field Ralg of real algebraic numbers, whereas the real √ √ closure of the field Q( 2) is the ring Ralg × Ralg (corresponding to the two orders of Q( 2)).

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The result was generalized in [ABR88] to the real spectra of excellent rings, and even to a wide class of their subspaces, see [ABR96, Ch. VII, Prop. 6.1, pp. 198–199; Thm. 8.4, p. 207, and Cor. 8.5, p. 208]. In fact, this extension covers the case of rings of germs of analytic functions on compact global semi-analytic subsets of real analytic manifolds, for which the reader will find pleasure in [ABR96, Ch. VIII, §8, pp. 244 ff]. On the other hand, the Heyting property fails for the real spectrum of the ring C(R) of continuous real-valued functions on R. An example is due to Gamboa, see [Gam88] and [ABR96, Ch. VII, Ex. 6.2, pp. 199–200]. The Analytic Tilde Operator The definability results from 13.5.A can also be applied to rings of formal power series and of (local and global) analytic functions. As an illustration, consider the ring of germs of analytic functions at a point a of an analytic variety X ⊆ Rn . 16 The set of such germs is denoted by OX,a . Via Taylor expansions, OX,a is isomorphic to the ring R{X1, . . . , Xn } of convergent power series. With the terminology of set germs (cf. [ABR96, Ch. VIII, 2.1, pp. 213–214]), a semi-analytic set germ (at a) is a set germ of the form n  {x ∈ Xa | fi1 (x) > 0, . . . , fik (x) > 0, gi (x) = 0} i=1

i

that is, a set germ of Xa quantifier-free, defined by equalities and inequalities on analytic functions on a neighborhood of a. As in the semi-algebraic case, 13.5.9(i), there is a tilde operator that sends Sa ⊆ Sper(OX,a ) each semi-analytic set germ Sa ⊆ Xa to the constructible set ' defined by the same Lor -formula, 13.5.2. An analytic version of the Artin–Lang theorem holds for the ring of convergent power series, proving that this tilde map is a bijection that preserves topological operations as in 13.5.9. For details we refer the reader to [ABR96, Ch. VIII] and [AnRu95, Ch. 3, pp. 24 ff]. 13.6.5 Real Semigroups and Abstract Real Spectra In 13.6.2 we saw that not every spectral root system is homeomorphic to the real spectrum of some ring. This situation naturally raises the question of whether there is a category of “algebraic” structures anti-equivalent to the category of spectral root systems with appropriate spectral morphisms. A positive answer to this query was given in [DiPe12]. The required “algebraic-like” objects had previously been introduced in [DiPe04] under the name real semigroups. A detailed description of these objects exceeds the scope of this book; to see an important example we briefly describe the real semigroup GA associated with a semi-real ring A, cf. 13.1.20(ii): 16

That is, the set of zeros of a finite set of real analytic functions in n variables.

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• The monoid GA consists of all functions a : Sper(A) −→ 3, for a ∈ A, defined by ⎧ 1 if a ∈ α \ (−α) ⎪ ⎪ ⎨ ⎪ a(α) = 0 if a ∈ α ∩ −α ⎪ ⎪ ⎪ −1 if a ∈ (−α) \ α ⎩ (α ∈ Sper(A)), with (well-defined) operation a · b = a · b. This is a commutative semigroup with unit 1 and additional constants 0 and −1, satisfying certain laws, called a ternary semigroup, cf. [DiPe04, §1, pp. 100–105]. • The trace of the addition of A in GA is not a well-defined operation, but determines a ternary relation Dt 17 defined as follows: for a, b, c ∈ A, GA

a∈

Dt (b, c) GA

⇐⇒ there are a , b, c  ∈ A with a = a , b = b, c = c  and a  = b + c  .

• The class of real semigroups is defined by a finite set of first-order axioms in the language {·, 1, 0, −1, Dt }, cf. [DiPe04, §2, p. 106 ff]. 18 The real semigroup corresponding to a given spectral root system, X, denoted Sp(X), consists of all spectral maps f : X −→ 3sp , where 3sp is the set 3 = {1, 0, −1} endowed with the spectral topology where the singletons {1}, {−1} are open and {0} is closed, hence with specialization as in the diagram 0

•A ] • 1

•

−1

We consider 3 also with multiplication of integers. The product operation and the ternary relation Dt in Sp(X) are pointwise induced by the corresponding operation and (unique) ternary relation D3t in 3, see [DiPe04, Cor. 2.4, p. 109], defined as follows: for i, j ∈ {0, 1, −1}, D3t (i,

j) =

D3t ( j, i)

⎧ {i} ⎪ ⎪ ⎨ ⎪ = {i + j} ⎪ ⎪ ⎪ {0, 1, −1} ⎩

if i = j if i or j is 0 if i = − j and j ∈ {1, −1}.

Real semigroups of the form Sp(X) are called spectral real semigroups; they 17 18

Called “transversal representation.” A second ternary relation D, interdefinable with D t , is also used in the formulation of the  axioms. In the example above, a ∈ D (b, c) ⇔ there are s, t ∈ A2 such that a = sb + tc. GA

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have been introduced and investigated in [DiPe12]. Other classes of real semigroups that have been studied in detail are the Post algebras and the fans, see [DiPe17a],[DiPe17b]. There is a general duality, proved in [DiPe04, Thm. 4.1, p. 115 ff], between real semigroups and certain topological structures generalizing the real spectra of rings, called abstract real spectra, introduced by Marshall in [Mar96] and investigated in Chapters 6–8 of that monograph. To any real semigroup G, it associates the set of real semigroup homomorphisms (characters) G −→ 3 endowed with the topology having as subbasis of opens the set of all homomorphisms g : G −→ 3 taking on value 1 at a fixed element a ∈ G. This topology is spectral and, in fact, a spectral root system (cf. [Mar96, Prop. 6.3.3(i), p. 113 and Prop. 6.4.1, p. 114]). Conversely, to any abstract real spectrum (X, G) is associated the real semigroup consisting of the monoid G = (G, 1, 0, −1) t defined by: for a, b, c ∈ G, together with a ternary relation DG t (b, c) ⇔ ∀h ∈ X(h(a) ∈ D3t (h(b), h(c)). a ∈ DG

This correspondence extends to morphisms, yielding the announced duality. In this context, the counterexample of [DeMa94] becomes an example of an abstract real spectrum non-isomorphic to the real spectrum of any ring (cf. [Mar96, §8.8, pp. 176–177]). Returning to the question raised at the beginning of 13.6.5, it is shown in [DiPe12, Prop. 3.7, p. 379] that the abstract real spectrum associated with the real semigroup Sp(X) is (homeomorphic to) the given spectral root system X. Further results from [DiPe12] relate, in a natural way, the present set of ideas with those from 13.6.3. Thus: • [DiPe12, Thm. 9.3, pp. 409–410] shows that the real semigroup associated with a real closed ring, and in fact with any lattice-ordered ring, is spectral. • [DiPe12, Prop. 9.4, pp. 410–411] proves that, for any semi-real ring A, the real semigroup Sp(Sper(A)) is isomorphic to G ρ(A) , the real semigroup associated with the real closure of the ring A (see Theorem in 13.6.3). Historical Notes The idea of setting up an axiomatic generalization of the real spectra of rings was initially put forward by Bröcker in [Brö94] as a tool to study constructible sets in real geometry – both algebraic and analytic – in sufficient generality. Bröcker’s axioms are essentially those appearing in [ABR96, Ch. III], where abstract real spectra occur under the name spaces of signs. In [Mar96, Chapters 6–8], Marshall gave a much simpler, equivalent, set of axioms for abstract real spectra and significantly developed their theory. In both Bröcker’s and Marshall’s approaches, the axioms essentially involve, in abstract disguise, concepts and results from the algebraic theory of quadratic forms.

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Marshall’s approach was recast in dual (i.e., categorically anti-equivalent) form by Dickmann–Petrovich [DiPe04] as the theory of real semigroups. This framework is better adapted to algebraic and model-theoretic treatment. Ternary semigroups naturally arose in the investigation of the semigroup structures underlying real semigroups (i.e., ignoring the ternary “representation” relations of the latter). Ternary semigroups have both a simple definition and a surprisingly rich structure.

14 Spectral Spaces via Model Theory

In this chapter we show how spectral spaces can be analyzed and constructed from a model-theoretic perspective. The reader is assumed to have seen firstorder predicate logic, including routine manipulations with formulas and structures; the precise setup is explained in Section 14.1. The central notion of the chapter is type space. In the classical meaning of the word, this is the spectrum (or “Stone-space”) S(T) of the Boolean algebra (called the Tarski–Lindenbaum algebra) of formulas, modulo equivalence with respect to a given theory T. Hence, in model theory, Boolean spaces are in the forefront. We will extend the setup by taking into account a given set Δ of formulas of interest, and adjust the type space to reflect this additional information. For example, in the language of rings one may want to focus on the set Δ of polynomial identities. Or, one considers instances of a given formula ϕ(x, y) (i.e., we choose Δ to be the set of all formulas ϕ(x, c), where c varies among the constants of a given language). One then essentially runs the classical construction of type spaces within Δ and obtains a set S Δ (T); see 14.2.1 and 14.2.4. In modern model theory, this construction is known under the name partial types and it is considered as a Boolean quotient space of S(T). In 14.2.5 we show that S Δ (T) carries a natural, in general not Boolean, spectral topology; the open quasi-compact sets reflect lattice combinations of formulas from Δ. The space S Δ (T) is examined in Section 14.2. To see how the spectrum of a bounded distributive lattice, and thus any spectral space, fits into the model-theoretic setup, see 14.2.10; for the Zariski spectrum, see 14.2.9. The second half of the chapter introduces a general method of attaching spectra to a given first-order structure A. One method is through expansions of A, see 14.2.12. The second, more general, method is exposed in Section 14.3 and defines spectra of A via morphisms into models of some axiomatizable class of structures (in a possibly larger language). For both methods the spectra are of the form S Δ (T) for suitably chosen data Δ and T. In 14.3.13 we show that 540

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541

the points of spectra attached to a structure A using the new methods admit an algebraic description in terms of expansions of A. Section 14.3 also contains a gallery of examples, showing how various concrete spectral spaces from this book can be seen with the new methods, most prominently how to obtain the spectrum of a bounded distributive lattice, the Zariski spectrum of a ring, and the real spectrum of a ring. Partial types are explicitly used in model theory (mainly without reference to the spectral topology) in various contexts: positive model theory (cf. [BYaPo07]), equational theories (cf. [PiSr84], [JuLa01]), and Zariski structures (cf. [Zil10]). Spectral spaces of the form S Δ (T) are studied in special cases in [Rob90] and in a more general form in [Tre96] and in [Ber99]. For readers familiar with topos theory, this chapter can also be compared with [Joh77].

14.1 The Model-Theoretic Setup Summary In this chapter we use a considerable amount of model theory. It is clearly impossible to recount all definitions and results that are needed. For a general reference we cite Hodges’ book [Hod93]. To avoid ambiguities, we start with a short rundown of basic terminology, notation, and facts. Occasional further explanations are given as we go along. 14.1.1 Languages, Structures, Theories, and Models (i) We use first-order languages of predicate logic with equality, function symbols, relation symbols, constant symbols, and countably many variables, [Hod93, p. 25]. The language is always single sorted if we do not explicitly make any other assumption. Languages are denoted by the letter L , possibly adorned by superscripts, subscripts, and so on. We assume that the notions of terms and formulas for a language L are familiar, [Hod93, p. 11 ff, p. 25 ff]. (ii) An L -structure is a nonempty set together with interpretations for the function symbols, relation symbols, and constant symbols of the language. The underlying set of a structure M is called its universe or domain. (iii) Let L be a language. Another language L  extends L if all symbols of L are contained in L . An L -structure N expands an L -structure M if the universes are equal and the interpretation of every L -symbol in N is the same as in M. We call M the underlying L -structure of N. (iv) The set of all L -formulas is denoted by Fml(L ). For ϕ ∈ Fml(L ) we write ϕ(x1, . . . , xn ) to indicate that the set of free variables of ϕ is

542

(v)

(vi)

(vii)

(viii)

(ix)

Spectral Spaces via Model Theory contained in {x1, . . . , xn }. An L -formula is a sentence if it does not have any free variables. The set of sentences is denoted by Sen(L ). Let M be an L -structure and T a set of L -sentences. Then M |= T stands for “every sentence ϕ ∈ T is true in M.” A model of a set T of L -sentences is an L -structure such that M |= T. In this book an L -theory is any set of L -sentences having a model (such sets are also said to be consistent). The class of all models of T is denoted by Mod(T). The fundamental theorem of model theory is the Compactness Theorem, which says that a set of L -sentences has a model if (and only if) every finite subset has a model. One proof uses Łoś’s Theorem on Ultraproducts. For ultraproducts and Łoś’s Theorem, see [Hod93, section 9.5, p. 449]. Let T be an L -theory and ϕ(x1, . . . , xk ) an L -formula. We write T |= ϕ if M |= ∀x1 . . . xk ϕ holds for every model M of T. If T = ∅ then we just write |= ϕ. The theory of a structure M is defined as Th(M) = {ϕ ∈ Sen(L ) | M |= ϕ}. These are precisely the complete theories. A theory T is complete if and only if it is maximally consistent, equivalently, if ϕ ∈ T or ψ ∈ T for all ϕ, ψ ∈ Sen(L ) with T |= ϕ ∨ ψ. 1 For a subset Δ ⊆ Sen(L ) we set ¬Δ := {¬δ | δ ∈ Δ} and define Δ to be the smallest subset of Sen(L ) that contains Δ and is closed under finite conjunctions and disjunctions. Recall from the normal form theorem of propositional logic that for each γ ∈ Δ there are n, k1, . . . , k n ∈ N and   n k i . δi j ∈ Δ (1 ≤ i ≤ n, 1 ≤ j ≤ k i ) with |= γ ↔ i=1 j=1 δi j Intuitively one may think of Δ as the lattice generated by Δ. Formally this is not correct though since, for example, δ ∧ δ is not equal to δ, but is merely logically equivalent to δ. See 14.2.8 for a description of the precise relationship between Δ and lattices.

14.1.2 Adding Constants to a Language Given a language L and a set C, we define L (C) to be the extended language containing a new (i.e., not contained in L ) constant symbol for each c ∈ C. If C is disjoint from L then we identify C with the set of new constant symbols. If C = {c1, c2, . . .} is countable, then all formulas of L can be understood as L (C)-sentences in the following sense: for L -formulas ϕ( x) ¯ and ψ( x) ¯ (where ¯ ↔ ψ( x) ¯ in L if and only if |= ϕ(c) ¯ ↔ ψ(c) ¯ x¯ = (x1, . . . , xn )) we have |= ϕ( x) in L (C) (where c¯ = (c1, . . . , cn )). The reason is that, for any L -structure M 1

This last condition is reminiscent of the definition of a prime filter in a lattice, A.7(iii) In fact, in 14.2.8 we explain how T can be seen as an ultrafilter (= prime filter) of a certain Boolean algebra.

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and any tuple a = (a1, . . . , an ) ∈ M n , there is an expansion N of M to an L (C)-structure such that ci is interpreted by ai . (This is possible since C is disjoint from L .) Let M be an L -structure and A a subset of M. For each a ∈ A we define a to be a symbol not in L (distinct symbols for distinct elements of A). Then we can form the extended language L (A) given by the new set of constants {a | a ∈ A}. The L -structure M is expanded to an L (A)-structure (M, A), where the new constant a is interpreted by a. 14.1.3 Homomorhisms and Embeddings A map f : M → N between (the domains of) two L -structures preserves an L -formula ϕ(x1, . . . , xk ) if, for all a1, . . . , ak ∈ M, M |= ϕ(a1, . . . , ak ) implies N |= ϕ( f (a1 ), . . . , f (ak )). The map f is called • a homomorphism if it preserves all atomic formulas, • an embedding if it preserves all quantifier-free formulas, and • an elementary embedding if it preserves all formulas. Let M be an L -structure. Another L -structure N is an elementary extension of M if its universe contains the universe of M and the inclusion map M → N is elementary. Then we also say that M is an elementary substructure of N.

14.2 Spectral Spaces of Types Summary The word “type” in the title of this section has two different (but related, as we shall see) interpretations. On the one hand, it refers to the modeltheoretic notion of types, [Hod93]; on the other hand, to the category-theoretic concept of isomorphism types of objects. Type spaces are defined in 14.2.1 by model-theoretic methods. In 14.2.3(i) (which is continued in 14.2.9) and in 14.2.10 we explain the concrete meaning of the model-theoretic point of view in an algebraic context. This brings isomorphism types into the picture. For example, the Zariski spectrum of a ring A is homeomorphic to a space of isomorphism types of A-algebras A → K, where K is a field generated (as a field) by the image of the structure homomorphism. Similarly, the spectrum of a bounded distributive lattice can be described as a type space. Type spaces appear, with many applications, in model theory. Some basic examples are given in 14.2.3(ii)–(iv). The main topological features of type spaces are exhibited in 14.2.5, 14.2.6, and 14.2.7. In particular, they are spectral spaces. Conversely, every spectral space is a type space (since Zariski spectra of rings and spectra of bounded

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distributive lattices are presentable as type spaces). We show how, under suitable assumptions, the expansions of a structure correspond bijectively to the elements of a type space, thus making the set of expansions a spectral space, 14.2.11 and 14.2.13. These model-theoretic methods are then used to show how the set of congruences of a universal algebra and the set of local subrings of a field can be made into spectral spaces, 14.2.14 and 14.2.15. We conclude with model-theoretic applications of type spaces in elimination theory and interpolation theory, 14.2.16 and 14.2.17. 14.2.1 Definition Let L be a language and consider subsets T, Δ ⊆ Sen(L ). (i) A complete L -theory p is a type of T if T ⊆ p. We write S(T) for the set of types of T (i.e., all complete theories containing T); its elements are denoted by p, q, r, . . . (ii) The elements of the set S Δ (T) = {p ∩ Δ | p ∈ S(T)} ⊆ P(Δ) are the Δ-types of T. The elements of S Δ (T) are denoted by p,q, r, . . ., similar to prime ideals and prime filters of bounded distributive lattices and prime ideals of rings. 2 (iii) The sets D(δ) := {p ∈ S Δ (T) | δ  p}, δ ∈ Δ are a subbasis of open sets for a topology on S Δ (T). Their complements are denoted by V(δ). More generally, for Σ ⊆ Δ we set V(Σ) = {p ∈  S Δ (T) | Σ ⊆ p} = σ ∈Σ V(σ). If we need to specify Δ or T, then we write DΔ,T and VΔ,T . (iv) The set S Δ (T) with the topology defined in (iii) is the space of Δ-types of T. If Δ = Sen(L ) then S(T) = S Δ (T) is the type space of T. 14.2.2 Remarks We continue with the notation of 14.2.1: (i) The set S Δ (T) is nonempty if and only if T is a theory, independently of whether Δ is empty or not. (ii) The notation S Δ (T) does not display the language L . This does not lead to any ambiguities. For, if L  is a language extending L , then T is an L -theory and Δ ⊆ Sen(L ). Hence the set of Δ-types of T is the same if we form it for L or for L . (The reason is that a complete theory in the language L is the same as a complete theory in the language L  intersected with the set of L -sentences.) But note that S(T) with respect to L is not the same as S(T) with respect to L  (since Sen(L )  Sen(L )). 2

The notation we introduce is inspired by that used for prime ideal spectra of bounded distributive lattices and for Zariski spectra of rings. Similarities are clearly visible. Formal connections appear in 14.2.10 and 14.2.3(i).

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(iii) In 2.2.4 we showed that the power set of any set S is a spectral space, where the sets DP(S) (s) = {R ⊆ S | s  R} (with s ∈ S) are a subbasis of open sets. Thus, P(Δ) is a spectral space with subbasis DP(Δ) (δ) = {Γ ⊆ Δ | δ  Γ}, δ ∈ Δ. By definition of the topology, S Δ (T) ⊆ P(Δ) is a subspace. We shall see in 14.2.5 that it is even a spectral subspace. (iv) The construction of the power set is a functor Sets → Spec, 2.5.13(a). Therefore the inclusion Δ → Sen(L ) yields the surjective spectral map P(Sen(L )) → P(Δ), Γ → Γ ∩ Δ. By definition of the type spaces it restricts to a continuous surjective map S(T) → S Δ (T). It will be clear that this is even a spectral map, once it has been shown that S(T) ⊆ P(Sen(L )) and S Δ (T) ⊆ P(Δ) are spectral subspaces. The construction of the type spaces may appear to be rather abstract in the eyes of somebody whose everyday fare does not include model theory. More complicated abstract constructions are to come. Therefore, at this point we interrupt the development and display a few examples, showing the meaning of the definitions in concrete situations, thus motivating our setup. The examples will accompany us to illustrate the further developments. 14.2.3 Examples (i) The Zariski spectrum. The Zariski (or prime) spectrum of a ring was defined in 2.5.1 and studied more closely in Chapter 12. Now we use type spaces to describe the underlying set of the prime spectrum. Topological considerations are deferred until 14.2.9 and 14.3.9(i). Let R be a ring (commutative and unital, as always). Let K be a field and consider a homomorphism ε : R → K. Then Ker(ε) = {r ∈ R | ε(r) = 0} is a prime ideal. In fact, every prime ideal is of this form, 12.2.5. To view this situation from a model-theoretic perspective, we consider the pair (K, ε) as a model of a particular theory T in a language attached to the ring R. Then Spec(R) can be identified with the set of Δ-types of T, where Δ is chosen suitably. We start by defining the theory and the language. Let L = {+, −, ·, 0, 1} be the language of rings and Tfields the L theory of fields. Then we enlarge the language L by adding constants r with r ∈ R, cf. 14.1.2, and obtain the extended language L (R). The homomorphism ε : R → K determines the following L (R)-structure M = MK ,ε : • The field K is the underlying L -structure of M, 14.1.1(iii).

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Spectral Spaces via Model Theory • For each r ∈ R the new constant symbol r is interpreted in M by r M = ε(r). Certainly, M is a model of Tfields . The definition of the interpretations r M shows that ε is a homomorphism if and only if 1 M = 1 (∈ K), r M + s M = (r + s) M , r M · s M = (r · s) M , for all r, s ∈ R. These conditions can be expressed by sentences in the language L (R), namely: 1 = 1 (the constant symbol of L ), r + s = r + s, r · s = r · s. Let TR be the set of these L (R)-sentences. Then it follows that M = MK ,ε |= Tfields ∪ TR = : T . One recovers the pair (K, ε) from M since K is the underlying L -structure and ε is the map R → K, r → r M . Conversely, let N be a model of T. Then N has an underlying L structure, which is a field, say L. Moreover, the map ε : R → L defined by ε(r) = r N is a ring homomorphism. It is clear from the definitions that N = ML,ε . Altogether we see that the map {(K, ε) | K a field and ε ∈ Hom(R, K)} −→ Mod(T) (K, ε) −→ MK ,ε is bijective, and we may consider any ring homomorphism R → K to a field as a model of T. In particular, let a  R be a prime ideal. Then we consider the natural map R → qf(R/a) to the quotient field of R/a as a model of T. Thus, we obtain a map f : Spec(R) −→ S(T) a −→ Th(R → qf(R/a)). It is easy to see that f is injective. But f is rarely inclusion-preserving (since the elements of S(T) are maximal consistent subsets of Sen(L (R)), 14.1.1(viii), which shows that the poset S(T) is trivially ordered) and need not be surjective. (For an example, let R = Z and consider the complete

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theory p of the embedding ε : Z → C. Then p contains the axioms for fields of characteristic 0 and the sentence ϕ : ∃x(x 2 + 1 = 0). Assume that p ∈ im( f ), i.e., p = f ((0)). Then p contains the theory of Q, hence the sentence ¬ϕ, a contradiction.) Now we choose a subset Δ ⊆ Sen(L ) such that the composition g = r ◦ f of f with the restriction map r : S(T) → S Δ (T) is bijective. Setting Δ = {r = 0 | r ∈ R} ⊆ Sen(L (R)) we prove the following. Claim The map g : Spec(R) → S Δ (T), a → {r = 0 | r ∈ a} is bijective. Proof of Claim It is clear that g is injective. For surjectivity, take a type p ∈ S Δ (T), say p = p ∩ Δ with p ∈ S(T). Pick a model M = MK ,ε of p, that is, a suitable field K together with a ring homomorphism ε : R → K. For each r ∈ R we see that r ∈ Ker(ε) if and only if MK ,ε |= r = 0. Therefore p = {r = 0 | r ∈ R and MK ,ε |= r = 0} = g(Ker(ε)), which proves p ∈ im(g).  We revisit this example in 14.2.11 and 14.3.9(i), where the seemingly random choice of L , T, and Δ will become understandable in a larger context. (ii) Type spaces in model theory. We consider classical type spaces as they occur in model theory. Fix a language L , an integer n ∈ N, and a tuple x = (x1, . . . , xn ) of n distinct variables. Let M be an L -structure and pick ¯ of a¯ in M is the collection an element a¯ ∈ M n . Then the type tp M (a) ¯ “knows” all of all L -formulas ϕ( x) ¯ with M |= ϕ(a). ¯ Intuitively, tp M (a) first-order properties of the tuple a¯ in M that can be expressed in the language L , which includes the complete theory of M. Now we also fix an L -theory T. The space of n-types of T, see [Hod93, p. 280], is a classical construction in model theory. It is a topological space whose underlying set is the collection of types of all tuples a ∈ M n , where M runs through the models of T. We show how our setup can be used to construct the space of n-types. First we choose a tuple c¯ = (c1, . . . , cn ) of distinct new constant symbols (i.e., symbols not in L ) and build the extended language L (c), 14.1.2. Then T is also an L (c)-theory ¯ and, by 14.2.1, there is the space ¯ (T) (i.e., S(T) in the language L (c)). S Sen(L (c)) ¯ We abbreviate ¯ Sn (T) = S Sen(L (c)) (T)

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Spectral Spaces via Model Theory and call Sn (T) the space of n-types of T. (Sometimes this space is also referred to as the space of complete n-types.) A structure in the language L (c) ¯ is an L -structure M together with an n-tuple a¯ ∈ M n , the components serving as interpretations of the ci . If p ∈ Sn (T) then a model of p consists of a model M of p ∩ Sen(L ) (in particular, a model of T) expanded (cf. 14.1.1(iii)) by a tuple a ∈ M n of interpretations of the ci , hence satisfying the conditions about c expressed in p. The model is denoted by (M, a), ¯ and a is a realization of p in M n , see [Hod93, p. 42]. Recall that L -formulas in the free variables x1, . . . , xn can be seen as sentences in the language L (c1, . . . , cn ) and vice versa, see 14.1.2. Therefore, p is uniquely determined by the set {ϕ( x) ¯ ∈ Fml(L ) | M |= ϕ(a)}. ¯ Thus we see that Sn (T) is indeed the classical set of n-types of T as used in model theory, see above. (Here we speak only about the set Sn (T), since these considerations do not yet include a comparison of topologies.) Alternatively, the elements of Sn (T) can be described as equivalence ¯ ¯ and (N, b) classes of models (M, a) ¯ of the L (c)-theory T, where (M, a) are defined to be equivalent if ¯ M |= ϕ(a) ¯ ⇐⇒ N |= ϕ(b)

for all ϕ( x) ¯ ∈ Fml(L ). (iii) The space of ϕ-types. We continue to work with an L -theory T and the language L (c) ¯ as in (ii). Fix an L -formula ϕ( x, ¯ y¯ ), x¯ = (x1, . . . , xn ), y¯ = (y1, . . . , yk ), n, k ∈ N, and define Δ = {ϕ(c, ¯ d) | d ∈ c(L )k }, 3 where c(L ) is the set of constant symbols of L . Thus, Δ can be seen as a family of L (c)-formulas parametrized by c(L )k . The space S Δ (T) = Sn (T) ∩ P(Δ) is called the space of ϕ-types. Consider pairs (M, a), where M |= T and a ∈ M n . As in (ii), every such pair gives rise to a type in Sn (T), hence to a ϕ-type in S Δ (T), which is realized ¯ yield the same point of S Δ (T) if by a. ¯ Two such pairs (M, a) ¯ and (N, b) and only if ¯ ⇐⇒ N |= ϕ(b, ¯ d) ¯ M |= ϕ(a, ¯ d) for all d ∈ c(L )k . There are many natural instances of type spaces where the set Δ is closed 3

If L has no constant symbols, this set is empty since c(L ) k = ∅ (or by convention). The components of a tuple d need not be distinct.

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under finite conjunctions or disjunctions, or both. However, in the case of ϕ-types usually the set Δ is neither closed under finite conjunctions, nor under finite disjunctions. In order to capture this important example, we did not impose any further conditions on Δ in the general setup. (iv) Spaces of ϕ-types and definable families. We discuss a special instance of (iii). Let L0 be a language and M0 an L0 -structure. The language L0 is extended by adding new constants for the elements of M0 , cf. 14.1.2, yielding the language L = L0 (M0 ). Let M = (M0, M0 ) be the expanded L -structure of M0 , where m (with m ∈ M0 ) is interpreted by m. Let T be the L -theory of M. Given an L -formula ϕ( x, ¯ y¯ ) there is the set Δ of L (c)-sentences as in (iii). We describe a sort of “physical” (or geometric) interpretation of the set of ϕ-types. The formula ϕ defines the set ϕ(M, M) := {(a, b) | M |= ϕ(a, b)} ⊆ M n × M k (the graph of ϕ over M). Moreover, for each tuple d = (d1, . . . , dk ) of constant symbols of L , the set ¯ := { a¯ ∈ M n | M |= ϕ(a, ¯ d1M , . . . , dkM )} ϕ(M, d) = ϕ(M, M) ∩ (M n × {d}) is the fiber of ϕ (or of ϕ(M, M)) over d (with respect to the projection ¯ d) is uniquely M n × M k → M k ). Note that, modulo T, the formula ϕ( x, determined by the fiber ϕ(M, d). Let L be the bounded sublattice of P(M n ) generated by the set of fibers. It is easy to verify that the map ¯ | ϕ(M, d) ¯ ∈ F} ¯ d) PrimF(L) −→ S Δ (T), F −→ {ϕ( x, is a homeomorphism. Thus, S Δ (T) is a spectral space and we can consider S Δ (T) as a spectrum associated with the definable family (ϕ(M, d)d of all fibers of ϕ. (In 14.2.5 below we prove that the space S Δ (T) is spectral for all T and all Δ.) Finally we show that, with a suitable choice of data L0 , M, n, and ϕ, every spectral space can be presented as the space of ϕ-types associated with a definable family. So, in a certain sense, the construction of spectral spaces in the present example is exhaustive. To see this, let Z be a spectral space and consider the two-sorted (for better readability) language L0 = {ε}, where ε = ε(x, y) is a binary relation symbol, x ranges in the first sort, and y in the second sort. Let M0 be the L0 -structure (Z, K(Z), ∈), that is, the first sort is the set underlying

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Spectral Spaces via Model Theory Z, the second sort is K(Z), and the symbol ε is interpreted as the element relation. We choose n = k = 1 and let ϕ(x, y) be the L -formula ε(x, y). Then we can show that the map Z −→ S Δ (T), z −→ {ϕ(x, b) | b ∈ K(Z) and z ∈ b} is a homeomorphism. The preimage of p ∈ S Δ (T) is the generic point of  {b ∈ K(Z) | ϕ(x, b) ∈ p}. See also 2.3.1.

Before studying the topology of S Δ (T), we characterize the points of S Δ (T) without reference to complete theories. 14.2.4 Proposition Let Δ ⊆ Sen(L ), let T be an L -theory, and let p be a subset of Δ. Consider the following statements: (i) p ∈ S Δ (T). (ii) T ∪ p ∪ {¬δ | δ ∈ Δ \ p} is consistent. (iii) For each n ∈ N and all δ1, . . . , δn ∈ Δ with T ∪ p |= δ1 ∨ · · · ∨ δn there is some i ∈ {1, . . . , n} with δi ∈ p. The implications (i) ⇔ (ii) ⇒ (iii) are always true. If p  Δ, then all statements are equivalent. Proof (i) ⇒ (ii) Take p ∈ S(T) with p = p ∩ Δ. As p is complete it contains the set in (ii). Thus, the set is consistent. (ii) ⇒ (i) There is a complete theory p containing the set in (ii). It follows that p = p ∩ Δ ∈ S Δ (T). (i) ⇒ (iii) Suppose T ∪ p |= δ1 ∨ · · · ∨ δn . Since p ∈ S Δ (T), there is a complete theory p ⊇ T with p = Δ ∩ p. Now δ1 ∨ · · · ∨ δn ∈ p and completeness of p imply δi ∈ p for some i. Hence δi ∈ p ∩ Δ = p, as required. Now, assuming (iii) and p  Δ, we prove (ii). By the Compactness Theorem (see 14.1.1(vi)) and the hypothesis p  Δ it suffices to show that for all δ1, . . . , δn ∈ Δ \ p the set T ∪ p ∪ {¬δ1 ∧ · · · ∧ ¬δn } is consistent. But if this set is inconsistent, then T ∪ p |= δ1 ∨ · · · ∨ δn , which  contradicts (iii) and the choice of δi ∈ Δ \ p. 14.2.5 Theorem Let L be a language and T, Δ ⊆ Sen(L ). Then: (i) S Δ (T) is a spectral space.

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◦

(ii) K (S Δ (T)) is the bounded sublattice of the power set of S Δ (T) generated by {D(δ) | δ ∈ Δ}. (iii) K(S Δ (T)) is the bounded sublattice of the power set of S Δ (T) generated by {V(δ) | δ ∈ Δ}. (iv) For p,q ∈ S Δ (T) we have p ⊆ q ⇐⇒ p  q (i.e., q ∈ {p}). Proof Recall that S Δ (T) is a subspace of the spectral space P(Δ), cf. 14.2.2(iii). The sets DP(Δ) (δ) = {p ⊆ Δ | δ  p}, δ ∈ Δ form a subbasis of quasi-compact open sets for P(Δ), and D(δ) = DP(Δ) (δ) ∩ S Δ (T). We write VP(Δ) (δ) = P(Δ) \ DP(Δ) (δ). (i) We claim that S Δ (T) ⊆ P(Δ) is a spectral subspace. By 2.1.3 it suffices to prove that S Δ (T) ⊆ P(Δ) is a proconstructible subset. Pick q ∈ P(Δ) \ S Δ (T). The set T ∪ q ∪ {¬δ | δ ∈ Δ \ q} is inconsistent (see 14.2.4(ii)). By the Compactness Theorem there are δ1, . . . , δn ∈ q and δ1 , . . . , δk ∈ Δ \q such that T ∪ {δ1, . . . , δn, ¬δ1 , . . . , ¬δk } is inconsistent, which means that V(δ1 ) ∩ · · · ∩ V(δn ) ∩ D(δ1 ) ∩ · · · ∩ D(δk ) = ∅ . But then VP(Δ) (δ1 ) ∩ · · · ∩ VP(Δ) (δn ) ∩ DP(Δ) (δ1 ) ∩ · · · ∩ VP(Δ) (δk ) is constructible in P(Δ), contains q, and is disjoint from S Δ (T). (ii), (iii), and (iv) are inherited from the corresponding facts about P(Δ), see 2.2.4(i).  14.2.6 Corollary Let L be a language and T,T , Γ, Δ ⊆ Sen(L ). (i) If Δ ⊆ Γ then the surjective map r : S Γ (T)  S Δ (T), r(q) = q ∩ Δ is spectral. For δ ∈ Δ we have r −1 (DΔ (δ)) = DΓ (δ) and r −1 (VΔ (δ)) = VΓ (δ). (ii) If T  |= T, then S Δ (T ) is a spectral subspace of S Δ (T). Proof (i) Surjectivity and the description of the preimages are clear from the definitions. Thus, r is spectral by 14.2.5. (ii) It follows immediately from T  |= T that S Δ (T ) ⊆ S Δ (T). Since both sets are proconstructible in P(Δ) it follows that S Δ (T ) ⊆ S Δ (T) is proconstructible, hence is a spectral subspace. 

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We use 14.2.6 to describe how the inverse space, the patch space, and spectral subspaces of a type space S Δ (T) can be presented as type spaces, modifying T and Δ. 14.2.7 Proposition With L , T, and Δ as before, let Z ⊆ S Δ (T) be a subset. Let F be any inconsistent sentence and T any tautology. (i) Z is closed if and only if Z = S Δ (T ∪ Σ) for some Σ ⊆ Δ ∪ {F}. (ii) Z is closed and constructible if and only if Z = S Δ (T ∪ {σ}) for some σ ∈ Δ ∪ {F, T}. (iii) Z is quasi-compact open if and only if Z = S Δ (T ∪ {σ}) for some σ ∈ ¬Δ ∪ {F, T}. (iv) Z is constructible if and only if Z = S Δ (T ∪ {σ}) for some σ ∈ (Δ ∪ ¬Δ) ∪ {F, T}. (v) Z is proconstructible if and only if Z = S Δ (T ∪ Σ)) for some Σ ⊆ (Δ ∪ ¬Δ) ∪ {F}. (vi) The map f : S Δ (T)inv → S ¬Δ (T), f (p) = ¬(Δ \ p) is a homeomorphism. (vii) The space S Δ∪¬Δ (T) is Boolean and the map r : S Δ∪¬Δ (T) → S Δ (T), r(q) = q ∩ Δ is a homeomorphism for the constructible topologies. Its inverse sends p to p ∪ ¬(Δ \ p). Proof (ii) It is clear that S Δ (T ∪ {F}) = ∅ and S Δ (T ∪ {T}) = S Δ (T), which covers the cases Z = ∅ and Z = S Δ (T). Now assume that ∅  Z  S Δ (T).   First let Z be closed and constructible, say Z = i j V(δi j ). Setting σ =   Δ i j δi j ∈ Δ we have Z = S (T ∪ {σ}), for a complete type p ⊇ T contains σ if and only if there is some i with {δi j | j} ⊆ p. Conversely, let Z = S Δ (T ∪ {σ}) with σ ∈ Δ. Then there are δi j ∈ Δ such     that σ = i j δi j ∈ Δ. It follows that Z = i j V(δi j ), hence Z is closed and constructible. (i) is a consequence of (ii) since every closed set is an intersection of closed and constructible sets. Note that the tautology T is not needed here; one can use Σ = ∅ if Z = S Δ (T). (vi) Pick p ∈ S Δ (T) and p ∈ S(T) with p = p ∩ Δ. Then γ ∈ f (p) if and only if ¬γ ∈ Δ \ p, if and only if ¬γ ∈ Δ \ p, if and only if γ ∈ p ∩ ¬Δ. This proves f (p) = p ∩ ¬Δ ∈ S ¬Δ (T), and f is well-defined. Obviously, f is a bijection. For δ ∈ Δ we have f (VΔ (δ)) = D¬Δ (¬δ), which implies that f is a homeomorphism. (vii) The space S Δ∪¬Δ (T) is Boolean since the subbasis {D(γ) | γ ∈ Δ ∪ ¬Δ} consists of clopen sets. We know that r is a surjective spectral map, 14.2.6(i). Therefore it suffices to prove injectivity. Suppose that q ∈ S Δ∪¬Δ (T) and let q

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be a complete type with q = q ∩ (Δ ∪ ¬Δ). We set p = r(q) = q ∩ Δ = q ∩ Δ. Then (using the proof of (vi)) we see that q = (q ∩ Δ) ∪ (q ∩ ¬Δ) = p ∪ ¬(Δ \p). (iii), (iv), and (v) can be proved with the same arguments as (i) and (ii), mutatis mutandis. Alternatively, they follow, respectively, from (ii) and (vi), from (ii) and (vii), and from (i) and (vii).  14.2.8 Identifying Sentences Modulo Logical Equivalence We fix an L theory T and consider the equivalence relation T |= ϕ ↔ ψ on L -sentences. The equivalence classes [ϕ] form a Boolean algebra, say A, where [ϕ] ∧ [ψ] = [ϕ ∧ ψ], [ϕ] ∨ [ψ] = [ϕ ∨ ψ], and [¬ϕ] is the complement of [ϕ] in A. The Boolean algebra A is called the Tarski–Lindenbaum algebra of T. Let π : Sen(L ) → A, π(ϕ) = [ϕ] be the residue map and take Δ ⊆ Sen(L ). Frequently Δ is identified with the set {[δ] | δ ∈ Δ} ⊆ A. However, some care has to be taken, for not every manipulation on the syntactical side is reflected by the passage to A via π (e.g., think of replacing constant symbols by other constant symbols). The Tarski–Lindenbaum algebra contains the algebra of constructible subsets of the type space. The map ε : Δ → K(S Δ (T)), ε(δ) = V(δ) extends to a surjective map ε¯ : Δ ∪ {F, T}  K(S Δ (T)), 14.2.7(ii), and there is a unique embedding ρ : K(S Δ (T)) → A of bounded distributive lattices making the diagram π

Sen(L )

A ρ

ε¯

Δ ∪ {F, T} Δ

ε

K(S Δ (T)) {V(δ) | δ ∈ Δ}

commutative. This can be verified readily, or one uses 14.2.6(i) with Γ = Sen(L ) and Stone duality. 14.2.9 The Zariski Spectrum, Continued Given a ring R, we saw in 14.2.3(i) how the underlying set of the Zariski spectrum Spec(R) can be described as the underlying set of a type space. We continue this discussion and show that the type space is homeomorphic to the Zariski spectrum. The notation is the same as in 14.2.3(i). We claim that the map g : Spec(R) → S Δ (T), a → {r = 0 | r ∈ a} is a homeomorphism (where Δ = {r = 0 | r ∈ R}). Pick a subbasic open set

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DΔ (δ) ⊆ S Δ (T) (with δ : (r = 0)). The equality g −1 (DΔ (δ) = {a ∈ Spec(R) | δ  Th(R → qf(R/a))} = DSpec(R) (r) proves the claim. 14.2.10 The Spectrum of a Bounded Distributive Lattice We show how the (prime ideal) spectrum of any bounded distributive lattice can be seen in the framework of types. By Stone duality, this shows that every spectral space occurs as a type space (which we know already from the discussion of the Zariski spectrum in 14.2.3(i) and 14.2.9, cf. Hochster’s Theorem, in Section 12.6, or from 14.2.3(iv)). The strategy is parallel to the presentation of the Zariski spectrum. Let L be the language {∧, ∨, ⊥, } of bounded lattices, where ∧ stands for meet and ∨ stands for join. The L -theory of the bounded distributive lattice 2 is denoted by Th(2). Let M be a bounded distributive lattice and let L (M) be the extension of the language L by adding a new constant symbol a for each a ∈ M, cf. 14.1.2. A model N of the theory Th(2) in the extended language L (M) is an expansion of 2, 14.1.1(iii) (i.e., the underlying lattice is 2 together with interpretations of the new constant symbols a). Thus, we can view N as the map M → 2, a → a N . Since 2 only has two elements, it is obvious that N is uniquely determined by Th(N) ∩ Δ where Δ = {a = ⊥ | a ∈ M }. It is clear that S Δ (Th(2)) is homeomorphic to 2 M and S(Th(2)) is homeomorphic M . To single out the homomorphisms among maps M → 2, we consider to 2con the set TM = {a ∧ b = a ∧ b | a, b ∈ M } ∪ {a ∨ b = a ∨ b | a, b ∈ M } ∪ {⊥ =⊥,  = } of L (M)-sentences and define T := Th(2) ∪ TM . Then a model of T is a homomorphism M → 2 of bounded distributive lattices, and the map PrimI(M) −→ S Δ (T), a → {a =⊥| a ∈ a} is a homeomorphism. 14.2.11 Spectral Spaces of Expansions of a Structure In the examples of Section 2.5, in particular the Zariski spectrum, real spectrum and ideal spectrum of a ring, subgroups of a group, partition lattices, and so on, we saw a common theme: consider a language L0 (e.g., the language of ring theory, or of lattice theory, or of posets) and an L0 -structure M. We use “certain properties” to single out subsets of M, or of powers of M, which serve as points of a spectral

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space (e.g., prime ideals, ideals, subgroups, equivalence relations, etc.). We exhibit a method explaining these examples in a very general setting. The main question is how to make the clause “certain properties” concrete and workable. We show how our model-theoretic tools can answer this question. In the examples, the “certain properties” can be formulated in a first-order language L , which extends L0 by a new unary relation symbol P. If N is an L -expansion of an L0 -structure M, then P can be used to describe a subset of M, namely {a ∈ M | N |= P(a)}. If P does not have any special properties, then every subset of M can be obtained from a suitable L -expansion. The class of subsets obtained in this way can be narrowed down by requiring that P satisfies “certain conditions.” For example, in a ring A we want to single out the ideals. Then we let L0 be the language of ring theory and consider an expansion B of A for the extended language L = L ∪ {P}. The set {a ∈ A | B |= P(a)} is an ideal if and only if the following three sentences hold in B: • P(0), • ∀x, y(P(x) → P(x · y)), • ∀x, y(P(x) ∧ P(y) → P(x + y)). Hence, the set I(A) of ideals of A corresponds bijectively to the set of L expansions of A satisfying the three sentences above. Thus, we see how L -expansions of L0 -structures and first-order model theory can be used to give concrete meaning to the vague expression “certain properties.” We introduce the notation and setup for the general discussion of this method. 14.2.12 Notation and Definition Let L0 be a language, let L be an extension of L0 by relation symbols, and T a set of universal L -sentences. Given an L0 structure A, we define a spectral space T-Specex (A) and call it the spectrum of T-expansions of A. The underlying set is T-Specex (A) = {M |= T | M expands A}. The topology on T-Specex (A) is described in 14.2.13 below. The strategy is to establish a bijection with a type space and then transfer the topology to T-Specex (A). For the type space we use the following data: (a) Δ A is the set of all positive quantifier-free L (A)-sentences. (b) diag(A), the diagram of A, is the set of all quantifier-free L0 (A)-sentences that are true in the L0 (A)-structure (A, A).

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14.2.13 Proposition The map T-Specex (A) −→ S Δ A (T ∪ diag(A)) M −→ Th(M, A) ∩ Δ A is bijective. Thus, T-Specex (A) is a spectral space if the topology is transferred from the type space. (i) The sets {M ∈ T-Specex (A) | M |= ¬R(a1, . . . , an )}, where R runs through the new relation symbols of L (n = arity of R) and a1, . . . , an ∈ A, form a subbasis of quasi-compact open sets. (ii) If M, N ∈ T-Specex (A) then M specializes to N if and only if, for every new relation symbol R, the interpretation of R in M is contained in the interpretation of R in N. Proof Let M be any L -expansion of A. It is clear that Th(M, A) |= diag(A), which implies that the map is well-defined. We have to show that, for each p ∈ S Δ A (T ∪ diag(A)), there is a unique expansion M of A with M |= T and p = Th(M, A) ∩ Δ A. For uniqueness, pick expansions M and N of A with Th(M, A) ∩ Δ A = Th(N, A) ∩ Δ A. It suffices to show that the interpretations of each new relation symbol R in M and N are equal. Let R have arity n and a1, . . . , an ∈ A. Then M |= R(a1, . . . , an ) ⇐⇒ R(a1, . . . , an ) ∈ Th(M, A) ∩ Δ A ⇐⇒ R(a1, . . . , an ) ∈ Th(N, A) ∩ Δ A ⇐⇒ N |= R(a1, . . . , an ). For surjectivity, let p ∈ S Δ A (T ∪ diag(A)) and pick a complete type p ⊇ T ∪ diag(A) with p = p ∩ Δ A. There is an L (A)-structure M with p = Th(M ). Let α : A → M be the map that sends a ∈ A to the interpretation of a in M . It follows from M |= diag(A) that α is an L0 -embedding. The only new symbols of L (compared with L0 ) are relation symbols. Therefore, the L -structure of M restricts to an L -substructure M on A. It is clear that (M, A) |= diag(A), hence that M is an L -expansion of A. Since T is a universal theory it follows that M |= T, hence M ∈ T-Specex (A) and (M, A) |= T ∪ diag(A). Finally, the definition of M shows that α is an L -embedding, and by definition of Δ A, we see that p = Th(M, A) ∩ Δ A (i.e., surjectivity holds). (i) and (ii) follow from the corresponding facts about S Δ A (T ∪ diag(A)) (since L has no new constant symbols and function symbols with respect to L0 ). 

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14.2.14 Algebraic Lattices and Congruences of Universal Algebras In 7.2.12 we saw that the examples in 2.5.13 can also be dealt with by special algebraic lattices of subsets of power sets. Recall from 7.2.8, that the coarse lower topology of any algebraic lattice is spectral. This can be connected to the expansion spectrum as follows. If L is an algebraic lattice, then, by [GrSc63, Theorem I], L is isomorphic to the lattice of congruences of a universal algebra (i.e., a first-order structure A in a language L0 without relation symbols other than the identity relation). Hence we may assume that L is the lattice of congruence relations of A. Now we add a binary predicate E to our language L0 and write L = L0 ∪ {E }. Consider L -expansions (A, ≡) of A with the property that ≡ is a congruence relation of A and interprets E. Let T be a set of L -sentences expressing that E is a congruence relation, which is a set of universal sentences. Then T-Specex (A) is defined and corresponds bijectively to L. Using 14.2.13(i) we conclude that L, with its coarse lower topology, is homeomorphic to the spectral space T-Specex (A): for a1, a2 ∈ A, let ≡a1 ,a2 be the smallest congruence of A containing (a1, a2 ). The sets (≡a1 ,a2 )↑ ⊆ L are a subbasis of closed sets for the coarse lower topology, hence their complements are a subbasis of open sets. These subbasic open sets of the coarse lower topology correspond exactly to the subbasic open sets of T-Specex (A) displayed in 14.2.13(i). 14.2.15 Example: Local Subrings of a Field Let K be a field. To speak about K we use the language of fields, denoted by L0 , which is the language of rings together with an additional unary function symbol ι, interpreted by the map ιK : K → K, ιK (a) = a−1 if a  0 and ι(0) = 0. We extend the language L0 by adding two new unary relation symbols P and Q and write L = L0 ∪ {P, Q}. Let K be an L0 -structure and M an L expansion. Then the realizations P M (of P) and Q M (of Q) in M are considered as subsets of K. Let T0 to be the L0 -theory of fields. Moreover, T is the following set of L -sentences. To explain their intended meaning, let K be a model of T0 (i.e., a field) and M an L -expansion of K. (a) P(1) ∧ ∀x, y(P(x) ∧ P(y) → P(x + y) ∧ P(x · y)). If M satisfies this sentence, then P M ⊆ K is a subring. (b) ∀x(Q(x) → P(x)). If M satisfies (a) and this sentence, then P M ⊆ K is a subring and contains Q M as a subset. (c) Q(0) ∧ ¬Q(1) ∧ ∀x, y, z(Q(x) ∧ Q(y) ∧ P(z) → Q(x + y) ∧ Q(x · z)). If M satisfies (a), (b), and this sentence, then Q M is a proper ideal in the subring P M ⊆ K. (d) ∀x(P(x) ∧ ¬Q(x) → P(ι(x))).

558

Spectral Spaces via Model Theory If M satisfies all conditions, then Q M ⊆ P M is a proper ideal, and P M \ Q M is the group of units of P M . Thus, P M is a local ring with maximal ideal Q M .

The set T consists of universal L -sentences. In fact, it is a theory since the following L -structure is a model of T0 ∪ T: pick a field K and define an L -expansion M by P M = K and Q M = {0}. The spectral space T-Specex (K) is defined since T is a universal theory. Its elements correspond bijectively to the local subrings of K. 4 Therefore, we write the elements of T-Specex (K) as pairs (A, m), where A ⊆ K is a local subring with maximal ideal m. Specialization in T-Specex (K) is described in 14.2.13(ii). Explicitly, (A, m)  (A, m ) if and only if A ⊆ A and m ⊆ m . If this is the case then it follows that m  ∩ A  A, hence that m  ∩ A = m (i.e., the local ring (A, m ) dominates the local subring (A, m), cf. [Bou98, p. 375, Definition 1]). A classical result in commutative algebra says that the maximal elements of T-Specex (K) are the pairs (A, m), where A is a valuation ring with quotient field K and m is the valuation ideal, [Bou98, p. 376, Theorem 1]. Finally we show how type spaces give us a very efficient way to present classical model-theoretic results related to elimination theory (e.g., [Hod93, Theorems 6.6.3, 8.4.9, and 9.4.7]). Most abstract tests for quantifier elimination, modelcompleteness, interpolation tests, but also model-theoretic characterizations of properties of theories like universality, inductivity, and rely on the following principle. Let T be an L -theory and assume that Δ ⊆ Sen(L ) is closed under finite conjunctions and disjunctions. Given two L -sentences ϕ and ψ, there is a local condition (in the type space S Δ (T)) describing precisely when we can find some δ ∈ Δ with T |= ϕ → δ → ψ. The local condition says, roughly, that at each point of S Δ (T) one can find some δ with this property. The precise implementation is given in 14.2.17. For a wide range of applications we formulate the following results in a slightly more general way. 14.2.16 Type Spaces in Elimination Theory Let T be an L -theory and assume that Δ ⊆ Sen(L ) is closed under finite conjunctions and disjunctions. 4

We point out that the set of local subrings of K cannot be axiomatized by a universal theory if we replace the language of fields by the language of rings and use just one unary relation to single out subsets of K. For this reason we need the function symbol ι in L0 .

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Let Φ, Ψ ⊆ Sen(L ) be subsets such that T ∪ Φ and T ∪ Ψ are consistent. Then the following conditions are equivalent: (i) There is some δ ∈ Δ with T ∪ Φ |= δ and T ∪ Ψ |= ¬δ. (ii) For all p, q ∈ S(T), if p ∩ Δ ⊆ q and Φ ⊆ p, then Ψ  q. Proof The implication (i) ⇒ (ii) is straightforward; the decisive point is to prove (ii) ⇒ (i). Let X be the spectral space X = S Δ (T) and consider the restriction map r : S(T) → S Δ (T), which is spectral, 14.2.6(i). Then Y1 := r(V(Ψ))  r(p1 ) if and Y2 := r(V(Φ)) are proconstructible. Condition (ii) says that r(p2 )   y1 for y1 ∈ Y1 and y2 ∈ Y2 . Applying p2 ∈ V(Φ) and p1 ∈ V(Ψ). Therefore y2  the Separation Lemma 1.5.3 we see that there is some A ∈ K(S Δ (T)) with Y2 ⊆ A and A ∩ Y1 = ∅. It follows from Y1  ∅  Y2 that ∅  A  S Δ (T). Using Δ = Δ and 14.2.5(iii) we find some δ ∈ Δ with A = VΔ (δ). Now r(V(Φ)) = Y2 ⊆ A = VΔ (δ) implies T ∪Φ |= δ, and VΔ (δ)∩r(V(Ψ)) = A∩Y1 = ∅ implies T ∪ Ψ |= ¬δ, as required.  As an application of 14.2.16 we obtain the following interpolation and elimination test, generalizing the classical Craig test, cf. [Hod93, Theorem 6.6.3]. 14.2.17 Interpolation Test Let T be an L -theory and let Δ ⊆ Sen(L ) be closed under finite conjunctions and disjunctions. Fix some inconsistent sentence F and let Φ, Ψ ⊆ Sen(L ). The following are equivalent: (i) There is a set Σ ⊆ Δ ∪ {F} such that T ∪ Φ |= Σ and T ∪ Σ |= Ψ. (ii) If p, q ∈ S(T) with p ∩ Δ ⊆ q, then Φ ⊆ p implies Ψ ⊆ q. Remark If Ψ = Φ this test becomes an elimination test, since (i) then says that Φ is logically equivalent to Σ modulo T. If Ψ is finite one can choose Σ in (i) to be a singleton. Proof The implication (i) ⇒ (ii) is obvious again. For the converse, first assume that T ∪ Φ is inconsistent. Then (i) holds with Σ = {F}. Therefore we assume now that T ∪ Φ is consistent. Pick some ψ ∈ Ψ such that T ∪ {¬ψ} is consistent. Then condition (ii) shows that ¬ψ  q for all p, q ∈ S(T) with p ∩ Δ ⊆ q. Therefore 14.2.16(ii) ⇒ (i) is applicable with Φ and {¬ψ}, that is, there is some δψ ∈ Δ with T ∪ Φ |= δψ and T ∪ {¬ψ} |= ¬δψ , hence T ∪ {δψ } |= ψ. Consequently, the set Σ := {δψ | ψ ∈ Ψ and T ∪ {¬ψ} consistent} has the interpolation property asserted in (i).



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Both 14.2.16 and 14.2.17 establish an equivalence between a global condition about S(T) and a local condition. The global condition, item (i) in both results, speaks about a property of the theory T, whereas the local condition, item (ii) in both cases, speaks about properties of elements of S(T). It is not unusual that local conditions are of a semantic nature (i.e., can be expressed in terms of structures). As an example, we mention the following theorem (which also shows that 14.2.17 can be used to prove axiomatization results): a theory T has an axiomatization by universal sentences (which is a global statement) if and only if every substructure of a model of T is also a model of T (which is a local statement), cf. [Hod93, Cor. 6.5.3, p. 295].

14.3 Spectra of Structures and their Elementary Description Summary In Section 14.2 we showed how model-theoretic tools, in particular types, can be used to build spectral spaces associated with first-order structures. The method is very general and produces various spectral spaces belonging to a given structure. To explain the ideas pursued in this section we consider a special situation, returning to the Zariski spectrum and the real spectrum of a ring. Both the Zariski spectrum and the real spectrum of a ring A were first introduced in Section 2.5. Both were presented as spectral subspaces of the power set P(A). Very little ring theory was needed for this construction. Accordingly, the constructions did not provide any information on whether, or why, these particular spectral spaces should be of any importance in ring theory. The reason comes to light if one looks at the spectra in a different way. In Chapters 12 and 13 we showed that both spectra are representation spaces for homomorphisms from A to a special class of structures: fields in the case of the Zariski spectrum, ordered real closed fields in the case of the real spectrum. Two homomorphisms ϕ : A → K and ψ : A → L yield the same point in the spectrum if they are equivalent over A in a suitable sense. The ring A is considered as a ring of functions defined on the spectrum, taking values in fields (or in real closed fields). The homomorphisms into the fields (or real closed fields) are considered as the evaluation maps at the points of the spectrum. The homomorphisms can be used to describe the points of the spectra as subsets of the ring A, the prime ideals (i.e., the kernels of homomorphisms to fields) in the case of the Zariski spectrum, the prime cones (i.e., the inverse images of the positive cones under homomorphisms to ordered real closed fields) in the case of the real spectrum. This presentation of the spectra is extremely helpful

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for their practical application. It brings us back to the original definitions in Section 2.5, but now with an increased understanding of the algebraic meaning. Now we take a new look at spectra of general first-order structures, following the strategy outlined for the Zariski spectrum and the real spectrum. Let A be a first-order structure and K an axiomatizable class of structures in a language extending the language of A. We consider all homomorphisms A → K (with K ∈ K) and introduce a notion of equivalence between them, 14.3.1. The equivalence classes form a set and correspond bijectively to the points of a suitable type space (as defined in Section 14.2), cf. 14.3.3. The set of equivalence classes is turned into a spectral space, called the T-spectrum of A, by transferring the topology from the type space. The T-spectrum can also be described using a suitable term algebra, 14.3.7. We present an extensive list of examples, cf. 14.3.6 and 14.3.9, illustrating the definitions and showing that these methods can help identify other useful classes of spectral spaces. Finally we show how the points of the T-spectrum of A can be described (via suitable relations) in terms of subsets of powers of An with n ∈ N, 14.3.13. 14.3.1 The Model-Theoretic Setup To start with, we describe the modeltheoretic tools and notation to be used in this section. We work with a first-order language L0 and an extension L , an L0 -theory T0 , and an L -theory T. It is not assumed that T0 ⊆ T. Fix a model A of T0 and consider L0 -homomorphisms α : A → M to models of T. The L -substructure of M generated by α(A) is denoted by %α(A)& M . Two L0 -homomorphisms α : A → M, β : A → N are equivalent, in writing: α ∼ β, if there is an L -isomorphism γ : %α(A)& M −→ %β(A)& N such that γ ◦ α = β, as depicted in the diagram: α(A)

%α(A)& M

M

α  γ

A β

β(A)

%β(A)& N

N.

Obviously ∼ is an equivalence relation on the class of all L0 -homomorphisms from A to models of T. We define T-Spec(A) = the equivalence classes of ∼, which is a class, to start with. But it follows from 14.3.3 below that T-Spec(A) is actually a set. The equivalence class of (M, α) (with α : A → M as above) is denoted by (M, α)∼ . The set T-Spec(A) is topologized using the methods of Section 14.2. We form

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the language L (A) which extends L by the set {a | a ∈ A} of new constant symbols, 14.1.2. Similar to 14.2.12, we define: (a) Δ A to be the set of all positive quantifier-free L (A)-sentences. (b) diag+ (A), the positive diagram of A, to be the set of all positive quantifier -free L0 (A)-sentences that are true in the L0 (A)-structure (A, A). For each L -structure M and each L0 -homomorphism α : A → M we write (M, α) for the L (A)-expansion of M interpreting the constant symbol a as α(a). The assumption that α is an L0 -homomorphism is equivalent to (M, α) being a model of diag+ (A), cf. [Hod93, Lemma 1.4.2, p. 17]. So, M |= T implies (M, α) |= T ∪ diag+ (A), and (M, α) gives rise to a point p(M, α) := Th(M, α) ∩ Δ A ∈ S Δ A (T ∪ diag+ (A)). 14.3.2 Lemma Let α : A → M |= T and β : A → N |= T be two L0 homomorphisms. Then p(M, α) ⊆ p(N, β) if and only if there is a, necessarily unique, L -homomorphism γ : %α(A)& M → %β(A)& N with γ ◦ α = β. Proof If a homomorphism γ exists as stated, then it is clear that p(M, α) ⊆ p(N, β). For the converse, we define the map γ as follows. For each r ∈ %α(A)& M there are an L -term t(x1, . . . , xk ) and elements a1, . . . , ak ∈ A with r = t M (α(a1 ), . . . , α(ak )). To make γ an L -homomorphism with γ ◦ α = β, we are forced to define γ(r) = t N (β(a1 ), . . . , β(ak )). First we have to check that γ is indeed well-defined. To see this, consider any other L -term s(y1, . . . , yl ) and elements b1, . . . , bl ∈ A with r = s M (α(b1 ), . . . , α(bl )). The positive and quantifier-free L (A)-sentence t(a1, . . . , ak ) = s(b1, . . . , bl ) is in p(M, α), but then p(M, α) ⊆ p(N, β) implies t N (β(a1 ), . . . , β(ak )) = s N (β(b1 ), . . . , β(bl )). Using p(M, α) ⊆ p(N, β), it is a routine exercise to show that γ is an L homomorphism, as required.  14.3.3 Theorem and Definition The map f : T-Spec(A) −→ S Δ A (T ∪ diag+ (A)), (M, α)∼ −→ p(M, α) is bijective. We use f to transfer the topology from the type space to T-Spec(A) and call the space T-Spec(A) the T-spectrum of A. 5 5

The type space S Δ A (T ∪ diag+ (A)) depends on L since, implicitly, L is present in Δ A . To improve readability the dependency of T-Spec(A) on L is not shown in the notation. But, of course, we spell it out explicitly if the language is not clear from the context.

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Proof First we show that f is well-defined. Let α : A → M |= T and β : A → N |= T be equivalent L0 -homomorphisms and let γ : %α(A)& M → %β(A)& N be an L -isomorphism with γ ◦ α = β. We apply 14.3.2 to γ and γ −1 and see that p(M, α) ⊆ p(N, β) and, respectively, p(N, β) ⊆ p(M, α), which proves well-definedness. For injectivity, assume that p(M, α) = p(N, β). By 14.3.2 there are unique γ : %α(A)& M → %β(A)& N with γ ◦ α = β and γ  : %α(A)& N → %β(A)& M with γ  ◦ β = α. The uniqueness statement in 14.3.2 implies that γ  = γ −1 (i.e., γ is an L -isomorphism and α ∼ β). Finally, to prove surjectivity, pick p ∈ S Δ A (T ∪ diag+ (A)) and a model M of T ∪ diag+ (A) with p = Th(M ) ∩ Δ A. We define M to be M , considered as an L -structure. Thus, M is a model of T. The map α : A → M, a → aM (the interpretation of a in M ) is an L0 -homomorphism. The construction shows that f (M, α) = p.  14.3.4 Topological Data of the T-Spectrum It follows from 14.3.3 and 14.2.5 that T-Spec(A) is a spectral space. We spell out the main topological data of T-Spec(A). The assertion of (i) is an immediate consequence of 14.3.2. Both (ii) and (iii) are transferred from the type space to the T-spectrum, 14.2.5. (i) If (M, α)∼, (N, β)∼ ∈ T-Spec(A), then (M, α)∼  (N, β)∼ if and only if there is an L -homomorphism γ : %α(A)& M −→ %β(A)& N with γ ◦ α = β. (ii) The sets D(ϕ) = {(M, α)∼ | (M, α) ϕ}, ϕ an atomic L (A)-sentence are a subbasis of quasi-compact open sets. According to the two types of atomic sentences, the subbasis contains two kinds of open sets, namely: (a) Let x = (x1, . . . , xk ) be a tuple of variables and a ∈ Ak . If t( x) ¯ and s( x) ¯ are L -terms then ¯  s(α(a)) ¯ in M } D(t(a) ¯ = s(a)) ¯ = {(M, α)∼ | t(α(a)) belongs to the subbasis (where α(a) ¯ = (α(a1 ), . . . , α(ak ))). (b) Given an n-ary relation symbol R ∈ L , a sequence of L -terms ¯ . . . , tn ( x) ¯ with x = (x1, . . . , xk ), and a tuple a¯ ∈ Ak , the set t1 ( x), ¯ . . . , tn (a))) ¯ D(R(t1 (a), = {(M, α)∼ | ¬R(t1 (α(a)), ¯ . . . , tn (α(a))) ¯ in M } belongs to the subbasis. (iii) The quasi-compact open subsets of T-Spec(A) are finite unions of finite intersections of sets from (ii).

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14.3.5 Comparison with the Expansion Spectrum of A We consider the general setup introduced above, but assume that L is an extension of L0 by relation symbols (no new constant symbols or function symbols) and that T consists of universal sentences. Then both the T-spectrum of A and the spectrum of T-expansions of A are defined, 14.2.12. We claim that the canonical map g : T-Specex (A) −→ T-Spec(A), M −→ (M, id)∼ is a homeomorphism onto a spectral subspace. Proof

This follows by considering the commutative diagram T-Specex (A)

g



S Δ A (T ∪ diag(A))

T-Spec(A) 

S Δ A (T ∪ diag+ (A)),

where the vertical maps are the homeomorphisms of 14.2.11 and 14.3.3 and the map at the bottom is the inclusion of the spectral subspace S Δ A (T ∪ diag(A)) ⊆  S Δ A (T ∪ diag+ (A)), see 14.2.6(ii). 14.3.6 The Partition Lattice As a first example we look at the case where L = L0 = ∅ and T = T0 = ∅ . Then the models of T0 and T are just sets, homomorphisms are maps, and substructures are subsets. In particular, if α : A → M is an L0 -homomorphism then %α(A)& M = α(A). Thus, if α : A → M and β : A → N are maps, then (M, α) ∼ (N, β) if and only if the set of nonempty fibers of α is the set of nonempty fibers of β. We conclude that T-Spec(A) corresponds bijectively to the set of partitions of A. We identify partitions with equivalence relations and recall that the set E(A) of equivalence relations of A is a spectral subspace of the power set P(A × A), 2.5.13(e). Indeed, we show that T-Spec(A) is homeomorphic to E(A). We define f : T-Spec(A) −→ P(A × A) (M, α)∼ ) −→ Rα = {(a, b) ∈ A × A | α(a) = α(b)}. Obviously, f is a well-defined bijection onto E(A). The sets D(a = b), where a, b ∈ A, 14.3.4, are a subbasis of quasi-compact open sets in T-Spec(A). It follows from f (D(a = b) = {R ∈ E(A) | ¬R(a, b)} that f is a homeomorphism onto the image. 14.3.7 T-Spectra and Term Algebras The spectral space T-Spec(A) can be constructed using term algebras. We do not use this fact later on (except in the discussion of examples) and, therefore, recall only the definition adapted to our context. Otherwise, we refer to [Hod93, p. 14] for details.

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Let A be an L0 -structure. The following L -structure TmAlgT (A) is the L -term algebra of A for T. (a) Let ctm(L (A)) be the set of constant L (A)-terms (i.e., terms without free variables) and define s, t ∈ ctm(L (A)) to be equivalent, in writing: s ≈ t, if T ∪diag+ (A) |= s = t. The universe of the term algebra TmAlgT (A), denoted by | TmAlgT (A)|, is the set of equivalence classes. The equivalence class of t ∈ ctm(L (A)) is denoted by [t]≈ . (b) An n-ary relation symbol R in L is interpreted in TmAlgT (A) as {([t1 ]≈, . . . , [tn ]≈ ) | ti ∈ ctm(L (A)) and T ∪ diag+ (A) |= R(t1, ..., tn )}. (c) The interpretation of an n-ary function symbol F in L is the map F TmAlgT (A) : | TmAlgT (A)| n −→ | TmAlgT (A)| ([t1 ]≈, . . . , [tn ]≈ ) −→ [F(t1, ..., tn )]≈ . (It is straightforward to see that this indeed is a well-defined map.) (d) Finally, for a constant symbol c of L we define cTmAlgT (A) := [c]≈ . In general, the term algebra TmAlgT (A) is not a model of T. If every function symbol and every constant symbol of L belong to L0 , then the universe of TmAlgT (A) is A. But TmAlgT (A) need not be an expansion of A, even in the case L = L0 . This is the case, for example, if L = L0 = {P} for a unary predicate P and A is any nonempty set, where P is interpreted as the empty set and T = {∀x P(x)}. The role of the term algebra in our context is explained by the following facts: (i) The map ι A : A −→ TmAlgT (A), ι(a) = [a]≈ is an L0 -homomorphism. (ii) If α : A −→ M is an L0 -homomorphism to a T-model M, then there is a unique L -homomorphism αˆ : TmAlgT (A) −→ M with α = αˆ ◦ ι A. The image of αˆ coincides with the L -substructure generated by α(A) in M. (iii) It follows from (ii) that the map ˆ ∼ T-Spec(A) −→ T-Spec(TmAlgT (A)), (M, α)∼ → (M, α) is a homeomorphism. In the construction of T-Spec(TmAlgT (A)) we work with the extended language L instead of L0 . Hence, if the construction of the term algebra is accessible, then (to some extent) the T-spectrum may become easier to understand. This is illustrated in example 14.3.9(ii).

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In our model-theoretic discussion of spectral spaces we have been concerned only with the construction of spaces so far. All previous chapters bear ample witness to the importance of a category-theoretic point of view. Therefore, we also show how the model-theoretic approach produces spectrum functors. 14.3.8 Proposition Every L0 -homomorphism λ : A → B between models of T0 induces a spectral map T-Spec(λ) : T-Spec(B) −→ T-Spec(A) (N, β)∼ −→ (N, β ◦ λ)∼ . Thus, T-Spec is a contravariant functor from the category of T0 -models with L0 -homomorphisms to the category of spectral spaces. Proof Clearly, the definition of the points of the spectra implies that T-Spec(λ) is well-defined. To show spectrality, let ϕ(x1, . . . , xn ) be an atomic L -formula and pick a1, . . . , an ∈ A. Then Spec(λ)−1 (DT-Spec(A) (ϕ(a1, . . . , an ))) =DT-Spec(B) (ϕ(λ(a1 ), . . . , λ(an ))) implies the claim, cf. 14.3.3(ii).



14.3.9 Examples We present a long list of examples illustrating the definition and construction of T-spectra. Some of the spectra we discuss were studied in previous chapters and sections, and are mentioned here to show how they can be captured by the method of T-spectra. (i) The Zariski spectrum of a ring A is homeomorphic to T-Spec(A), where: • L = L0 is the language {+, ·, −, 0, 1} of (unital) rings. • T0 is the L0 -theory of (commutative and unital) rings and T is the L -theory of fields. If A is a ring, then the map Θ : Spec(A) −→ T-Spec(A), a → (qf(A/a), πa )∼ is a homeomorphism, where πa : A → qf(A/a) is the residue map. The inverse of Θ sends (M, α)∼ ∈ T-Spec(A) to the kernel of α. This is a consequence of 14.2.3(i) and 14.2.9. (ii) The ideal spectrum of a univariate polynomial ring K[X], K a field, can be seen as a T-spectrum. We choose: • L0 to be the language {+, ·, −, 0, 1} of (unital) rings and let L = L0 ∪ {c} be the extension by one new constant symbol.

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• T0 is the L0 -theory of fields and T is the L -theory of (commutative and unital) rings. No additional assumption is made about the new constant symbol c. Let K be a field (i.e., K |= T0 ). The elements of T-Spec(K) correspond bijectively to the isomorphism classes of simply generated commutative K-algebras. 6 These correspond canonically to the ideals of the polynomial ring K[X]. In fact, routine checking shows that T-Spec(K) is isomorphic to the spectral space of ideals of K[X] (cf. 2.5.13(b)). (iii) The prime ideal spectrum PrimI(L) of a bounded distributive lattice L is homeomorphic to T-Spec(L), where: • L = L0 is the language {∧, ∨, ⊥, } of bounded lattices. • T0 is the L0 -theory of bounded distributive lattices and T is the L theory of the Boolean algebra 2 = {⊥, }. If L is a bounded distributive lattice, then the map Θ : PrimI(L) −→ T-Spec(L), a → (2, πa )∼ is a homeomorphism, where πa : L → 2 is the homomorphism with πa−1 (⊥) = a. The inverse of Θ sends (M, α)∼ to α−1 (⊥). Notice that all models M of T are isomorphic to 2. (In 14.2.10 we presented PrimI(L) as a type space.) (iv) Every poset P = (P, ≤) can be embedded (as a poset) into various distributive lattices L. (For example, P can be embedded into its power set by sending p ∈ P to p↓ . Or, P can be embedded into τ  (P) by p → P \ p↑ .) Given any such embedding α : P → L, the set α(P) generates a sublattice %α(P)& L ⊆ L. The isomorphism types (over P) of the sublattices %α(P)& L form a spectral space: • L0 = {≤, D} is the language of posets extended by a binary relation symbol D. Moreover, we set L = L0 (∧, ∨). • Let δ be the L0 -sentence ∀x, y(D(x, y) ↔ x  y). We define T0 to be the L0 -theory of posets enlarged by δ. Finally, T is the L -theory of distributive lattices together with δ. Given a poset P = (P, ≤), we expand it canonically to a model A of T0 . Then an L0 -homomorphism A → M |= T is a poset embedding into a lattice and, by definition, T-Spec(A) consists of the isomorphism classes (over P) of distributive lattices generated by P. 6

For, let (M , α) be a model of T together with a homomorphism α : K → M. Then the class (M , α)∼ ∈ T-Spec(K) is uniquely determined by the co-restriction K → %α(K)& M . Note that %α(K)& M = α(K)[c M ], the subring of M generated by α(K) and the interpretation c M of the new constant symbol. Instead, one can use the approach via term algebras, 14.3.7. Just note that the term algebra of K for T is L -isomorphic to the univariate polynomial ring K[X].

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(v) The real spectrum of a ring A is homeomorphic to T-Spec(A), where: • L0 is the language {+, ·, −, 0, 1} of (unital) rings and L = L0 ∪ {P} is the extension by a unary predicate P. • T0 is the L0 -theory of rings and T is the L -theory of real closed ordered fields. If R is a real closed field then P is interpreted as the positive cone (i.e., the set of non-negative elements, equivalently, the set of squares). Recall from 2.5.7 (also see 13.1.6 and 13.1.2) that the points of the real spectrum are the prime cones of A, that is, the subsets q satisfying the conditions • • • •

q + q ⊆ q, q · q ⊆ q, q ∪ −q = A, q ∩ −q ∈ Spec(A).

The description of the topology of Sper(A), cf. 2.5.8 or 13.1.6, shows that the map T-Spec(A) −→ Sper(A), (M, α)∼ → α−1 (M ≥0 ) is a homeomorphism. The inverse map sends q ∈ Sper(A) to (k(q), k q )∼ , where k(q) is the real closure of the totally ordered field κ(q) = (qf(A/(q∩ −q), q/(q∩−q))) and k q : A → k(q) is the canonical map A → A/(q∩−q) followed by the inclusion A/(q ∩ −q) → k(q). (vi) The space of real prime ideals of a ring A (cf. 13.3.A) is homeomorphic to T-Spec(A), where: • L = L0 is the language {+, ·, −, 0, 1} of (unital) rings. • T0 is the L0 -theory of rings and T is the L -theory of real closed fields. One checks that the map T-Spec(A) −→ Specre (A), (M, α)∼ → α−1 (0 M ) is a homeomorphism. The inverse map sends a ∈ Specre (A) to (k(q), k q )∼ , where q is a prime cone with support a, cf. 13.1.20(iii), k(q) is the real closure of κ(q), and k q : A → k(q) is as in (v). Remark on (v) and (vi). Given a ring A, both (v) and (vi) use the category of homomorphisms ρR : A → R to real closed fields for the construction of the T-spectra. Note that homomorphisms between real closed fields always preserve positivity (since squares are mapped to squares and the squares are the positive elements). However, the spectra we obtain are different. This is due

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to the different languages we use. In (v) the predicate P is used to name the set of non-negative elements in real closed fields, whereas in (vi) there is no name for the set of non-negative elements. Therefore, the difference of languages causes the class of substructures of real closed fields to be different. In (v) the substructures are totally ordered rings, whereas the substructures in (vi) are just subrings. Two totally ordered rings may be isomorphic as rings (disregarding the order relation) without being isomorphic as ordered rings. (vii) The valuation spectrum of a ring. The real spectrum of a ring A is one of the most important tools for the study of order-related properties of A, see Chapter 13. As we have seen in (v), the real spectrum can be constructed using homomorphisms to ordered real closed fields. Similarly, one can use homomorphisms from A to valued fields (see, e.g., [EnPr05]) to study certain arithmetic properties of A. This method leads to the notion of valuation spectra. There are different possibilities to build a valuation spectrum of a ring. We focus on a construction proposed by Huber and Knebusch in [HuKn94]. We recall the definition and state the link to the model-theoretic framework of this chapter. The reader should be familiar with the basic valuation theory of fields (see, e.g., [EnPr05]), but we recall some terminology and facts about the valuation theory of rings. A valuation of a ring A, cf. [Bou98, VI.3.1] or [KnZh02], is a map v : A → Γ ∪ {∞}, where Γ = (Γ, +, ≤) is a totally ordered Abelian group, ∞ is an element not in Γ, the order and the addition of Γ are extended to Γ ∪ {∞} by γ ≤ ∞, respectively, γ + ∞ = ∞ = ∞ + γ, for all γ ∈ Γ ∪ {∞}. If a ∈ A then v(a) is called the value of a. We require that v(x + y) ≥ min(v(x), v(y)), v(x · y) = v(x) + v(y) for all x, y ∈ A and v(0) = ∞, v(1) = 0. (Notice that the values of the units of A belong to Γ.) Let v be a valuation of A. Then v −1 (∞) is a prime ideal of A, called the support of v, also denoted by supp(v). There is a unique valuation v¯ : qf(A/v −1 (∞)) → Γ∪{∞} with v(a) = v¯ (a+v −1 (∞)) for all a ∈ A. The valued field (qf(A/v −1 (∞)), v¯ ) is denoted by κ(v). The valuation ring of v¯ is denoted by κ(v)v¯ , and the value group of v¯ is taken to be v¯ (κ(v) \ {0}) (i.e., the subgroup of Γ generated by v(A\ v −1 (∞))). Let αv : A → κ(v) be the canonical homomorphism, which we use to view κ(v) as an A-algebra. Note that v = v¯ ◦ αv . Two valuations v and w are said to be equivalent, denoted by v ≈ w, if κ(v) and κ(w) are A-isomorphic as valued fields. Following [HuKn94, p. 169 f], we define Spv(A), the valuation spectrum, to be the set of ≈-classes of valuations equipped with the topology

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Spectral Spaces via Model Theory T generated by the sets {[v]≈ ∈ Spv(A) | ∞  v(a) ≤ v(b)}, with a, b ∈ A (i.e., these sets are a subbasis of opens). The first task is to understand the points of Spv(A). The idea is to identify information inside A that can be used to determine the points of Spv(A) (similar to the Zariski spectrum, whose points are the prime ideals in A, and the real spectrum, whose points are the prime cones in A). Comparing with the real spectrum, it is tempting to expect that [v]≈ is determined by the subring αv−1 (κ(v)v¯ ) ⊆ A. However, this is not the case – the snag is that αv−1 (κ(v)v¯ ) does not determine the A-isomorphism type of the valued field κ(v). 7 So, some other information is needed to encode the valuation inside A. If A is a field, then the valuations correspond bijectively to the valuation rings and also to the divisibility relations determined by the valuation rings. Explicitly, let (K, w) be a valued field with valuation ring Kw . Then {(x, y) ∈ K × K | w(x) ≤ w(y)} = {(x, y) ∈ K × K | ∃z ∈ Kw : y = x · z} is the divisibility relation of (K, w), and we write x |w y if (x, y) belongs to the relation. Note that Kw = {y ∈ K | 1 |w y}. Now let v be a valuation of the ring A. The inverse image of the divisibility relation |v¯ of κ(v) under αv is called the divisibility relation of (A, v). As for fields, we write a |v b if (a, b) belongs to the relation. One can show that the divisibility relation of (A, v) determines the valued field κ(v) (as an A-algebra). To capture divisibility relations axiomatically, we consider binary relations | on A satisfying the following conditions: (a) (b) (c) (d) (e) (f)

0  1, a | b or b | a, if a | b and b | c then a | c, if a | b and a | c then a | b + c, if a | b then a · c | b · c, if a · c | b · c and 0  c then a | b.

By [HuKn94, Proposition (1.1.1)], the points of Spv(A) correspond bijectively to these relations, and Spv(A) is a spectral space. 7

For example, consider the ring Z and its quotient field Q. For each prime number p there is a valuation ring Q p ⊆ Q containing Z. The restrictions of the valuation rings all coincide, but the valuation rings are all non-isomorphic.

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From this description of the points, one can place Spv(A) in the modeltheoretic context. There is a bijection between the valuation spectrum Spv(A) and the T-spectrum of A determined by the following data: • L0 is the language {+, ·, −, 0, 1} of (unital) rings and L is L0 extended by a binary predicate symbol |. • T0 is the L0 -theory of rings and T is the L -theory of fields K having properties (a)–(f) above. Let K be a model of T. Then the interpretation | K of the relation symbol | is the divisibility relation of a unique valuation vK of K (i.e., x | K y is equivalent to vK (x) ≤ vK (y)). 8 One checks that the map h : T-Spec(A) −→ Spv(A), (M, α)∼ → [v M ◦ α]≈ is a homeomorphism for the constructible topologies. Concerning the spectral topology, there are other possibilities than the one defined above (but always with the same constructible topology), cf. [HuKn94] and [Sch90]. For example, in [HuKn94, beginning of section 1.3] the set Spv(A) is equipped with the topology T  generated by the sets {[v]≈ ∈ Spv(A) | v(a) < v(b)}, where a, b ∈ A. Under h, these sets correspond to the quasi-compact open sets D(a|b) ⊆ S Δ A (T ∪ diag+ (A))  T-Spec(A) (see 14.3.3). It follows that h is a homeomorphism if Spv(A) carries the topology T . Keeping the languages L0 and L as above, but replacing the theory T by another theory T , we obtain another spectral space T -Spec(A) and a map h  : T -Spec(A) −→ Spv(A), (M, α)∼ → [v M ◦ α]≈, which is a homeomorphism for the topology T on Spv(A), (cf. [Ber99, Example 3.6, p. 8]). To achieve this we must change the axioms (a)– (f) so that, in a valued field, the relation a | b expresses the condition ∞  v(a) ≤ v(b). Suitable axioms are: (a) (b) (c) (d) (e) (g) 8

1 | 0, If a  b, then a | b or b | a, if a | b and b | c then a | c, if a | b and a | c then a | b + c, if a | b and c  0, then a cot c | b · c, if x | y, then x  0, and if x  0, then x | x.

The valuation vK is trivial if and only if x | K y and y | K x both hold for all x, y ∈ K × .

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We refer to [GoMa10], [Zha02], and [Kne98] for other studies about spectra related to valuations. (viii) For a prime number p, the p-adic spectrum, of a ring A, Spec p (A) is defined to be T-Spec(A), where: • L0 is the language {+, ·, −, 0, 1} of (unital) rings and L = L0 ∪ {Pn | n ∈ N}, where each Pn is a unary predicate. One calls L the Macintyre language for Q p . • T0 is the theory of rings and T is the L -theory of the field Q p together with the set of axioms ∀x (Pn (x) ↔ ∃y : x = y n ), n ∈ N. The p-adic spectrum was introduced by E. Robinson in [Rob86]. For other sources and studies around the p-adic spectrum, see [BrSc86], [Bél87], [Bél90], and [PrRo84]. The points of Spec p (A) are given by ring homomorphisms α : A → M |= T (modulo equivalence as in 14.3.1). In the ring A they are represented by the sequences (p, (Pn )n∈N ), where p = ker(α) and Pn = α−1 ({x ∈ M | ∃y ∈ M : x = y n }. An axiomatic description of the sequences (p, (Pn )n∈N ), without reference to α or M, can be found in [Rob86, Section 4]; for local rings, see [Rob86, 4.3]. An alternative description is given in [BrSc86, §3, p. 18 f]. 14.3.10 Internal Description of T-Spectra of Structures Considering the examples of 14.3.9, we see that in several cases the points of the T-spectrum T-Spec(A) have an internal description inside A: • Consider the prime ideal spectrum of a bounded distributive lattice, the Zariski spectrum and the real spectrum of a ring. In each case the points of the spectrum are represented by suitable classes of subsets of the structure (prime ideals in the case of bounded distributive lattices, also prime ideals for the Zariski spectrum of a ring, and prime cones for the real spectrum). • For the valuation spectrum of a ring, points are given as binary relations satisfying properties (a)–(f) of 14.3.9(vii). • For the p-adic spectrum, the points are given by sequences of subsets of A satisfying certain algebraic conditions, 14.3.9(viii). In fact, such an algebraic description exists for all T-spectra, as we are going to explain now. Consider a first-order language L0 and an extension L . We work under the assumption that L0 has no relation symbols (as is the case in all our examples); the general case can be done similarly, but is notationally more complex. Let L † be the following language:

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L † extends L0 . L † contains a new binary relation symbol E. L † contains all relation symbols of L (with the same arity). If L has new function symbols or new constant symbols with respect to L0 , then, for each atomic L -formula ϕ with exactly the free distinct variables x1, . . . , xk , there is a new k-ary relation symbol Rϕ in L † .

Thus, L † is an extension of L0 by relation symbols. If L does not have new function symbols or constant symbols compared to L0 , then L † is just L ∪ {E }. As before, consider an L0 -theory T0 and an L -theory T. Let A and M be models of T0 and T, respectively, and let α : A → M be an L0 -homomorphism. We define an L † -structure A(M, α) as follows: (i) A(M, α) expands A. (ii) E is interpreted in A(M, α) by {(x, y) ∈ A × A | α(x) = α(y)}. (iii) Let R be a relation symbol of L with arity k. Then R is interpreted in A(M, α) as {(a1, . . . , ak ) ∈ Ak | (α(a1 ), . . . , α(ak )) ∈ R M }. (iv) Assume that L has new function symbols or constant symbols with respect to L0 and let ϕ = ϕ(x1, . . . , xk ) be an atomic L -formula with exactly the free variables x1, . . . , xk . Then the interpretation of Rϕ in A(M, α) is the relation {(a1, . . . , ak ) ∈ Ak | ϕ(α(a1 ), . . . , α(ak )) holds in M }. 9 Note that the relation symbol E is not needed if L has new function symbols with respect to L0 . For then, L † contains the new relation symbol Rx=y , which has the same interpretation as E. 14.3.11 Definition We define T-Spec(T0 ) to be the L † -theory consisting of the sentences ψ ∈ Sen(L † ) with: • ψ is universal. • If A |= T0 , M |= T, and α : A → M is an L0 -homomorphism, then A(M, α) |= ψ. 14.3.12 Remark The extension of the language L0 to L † is by relation symbols only, and T-Spec(T0 ) is a set of universal L † -sentences. Thus, by 14.2.12, the expansion spectrum (T-Spec(T0 ))-Specex (A) = {A ∈ Mod(T-Spec(T0 )) | A expands A} is defined for each L0 -structure A. 9

Hence, if y1 , . . . , yk is another set of distinct variables, then Rϕ(x1 , ..., x k ) and Rϕ(y1 , ..., y k ) have the same interpretation.

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14.3.13 Theorem For every model A of T0 , the map Λ A : T-Spec(A) −→ {A ∈ Mod(T-Spec(T0 )) | A expands A} (M, α)∼ −→ A(M, α) is a homeomorphism. Remark Since the set on the right-hand side is (T-Spec(T0 ))-Specex (A), this bijection informally says the following: the points in T-Spec(A) are described, independently of A |= T0 , in a first-order way by the sentences in T-Spec(T0 ). Since the sentences in T-Spec(T0 ) are universal, we may think of T-Spec(T0 ) as the promised “algebraic description of the points in T-Spec(A).” Proof Let T † be the intersection of the L † -theories of all expansions B(M, α) of L0 -structures B, where B |= T0 , M |= T, and α : B → M is an L0 homomorphism, 14.3.10. Thus, T-Spec(T0 ) is the universal part (T † )∀ of T † . Now we fix A |= T0 and prove the assertion in five steps: (a) Λ A is well-defined. (b) If C |= (T † )∀ , then there are D |= T0 , M |= T, and an L0 -homomorphism α : D −→ M such that C is isomorphic to an L † -substructure of D(M, α). (c) Λ A is surjective. (d) Λ A is injective. 10 (e) Λ is a homeomorphism. (a) Assume that (M, α) and (N, β) are equivalent, 14.3.1, and let γ : %α(A)& M → %β(A)& N be an L -isomorphism with γ ◦ α = β. It is a matter of routine to check that A(M, α) = A(N, β). Moreover, A(M, α) |= T-Spec(T0 ) since A(M, α) |= T † (by definition of T † ), and T-Spec(T0 ) ⊆ T † . (b) By [Hod93, Corollary 6.5.3, p. 295], C, being a model of (T † )∀ , is a substructure of a model of T † . Therefore, we may assume that C itself is a model of T † . We define K to be the class of all L † -structures that can be embedded elementarily into an ultraproduct of structures of the form B(M, α), where B |= T0 , M |= T, and α : B → M is an L0 -homomorphism. It follows from [Hod93, Corollary 9.5.10, p. 454] that C can be embedded into an ultraproduct of a family (Ai (Mi , αi ))i ∈I . So we may assume that

Ai (Mi , αi )/U|. |C| ⊆ | i ∈I 10

This is the only step where we need to pull back atomic L -formulas. In example 14.3.14(ii) one can observe that it is insufficient to pull back just graphs of new primitive L -functions.

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575

It remains to show that there are D, M, and α such that

Ai (Mi , αi )/U. D(M, α) = 

We define D = map given by

i ∈I

i ∈I

Ai /U and M =



i ∈I

Mi /U and let α : D → M be the

α((ai )i ∈I /U) = (αi (ai ))i ∈I /U. Łoś’s Theorem ([Hod93, Theorem 9.5.1, p. 450]) shows that D |= T0 and M |= T. Moreover, it is a routine matter to show that α is an L0 -homomorphism.   It is clear that i ∈I Ai (Mi , αi )/U is an L † -expansion of i ∈I Ai /U = D. By definition, D(M, α) is an L † -expansion of D as well. Therefore, it remains to  show that D(M, α) and i ∈I Ai (Mi , αi )/U coincide as L † -structures. As an example we check this for an n-ary relation symbol R ∈ L and a tuple  (a1, . . . , an ) in i ∈I Ai . The components of the a j are denoted by a ji ∈ Ai . The following equivalences prove the claim: D(M, α) |= R((a1i )i ∈I /U, . . . , (ani )i ∈I /U) ⇐⇒ M |= R(α((a1i )i ∈I /U), . . . , α((ani )i ∈I /U)) ⇐⇒ M |= R((αi (a1i ))i ∈I /U, . . . , (αi (ani ))i ∈I /U) ⇐⇒ {i ∈ I | Mi |= R(αi (a1i ), . . . , αi (ani ))} ∈ U ⇐⇒ {i ∈ I | Ai (Mi , αi ) |= R(a1i , . . . , ani )} ∈ U

Ai (Mi , αi )/U |= R((a1i )i ∈I /U, . . . , (ani )i ∈I /U). ⇐⇒ i ∈I

To finish the proof of (b), we just note that all other cases are done similarly. (c) Suppose A |= (T † )∀ expands A |= T0 . According to (b) there are D |= T0 , M |= T, an L0 -homomorphism β : D → M, and an isomorphism ϕ : A → D(M, β) onto an L † -substructure. Setting α = β ◦ ϕ : A → M, it is routine to check that A = Λ A((M, α)∼ ). (d) Take L0 -homomorphisms α : A −→ M |= T and β : A −→ N |= T with A(M, α) = A(N, β). We need to find an L -isomorphism γ : %α(A)& M → %β(A)& N with γ ◦ α = β. To define γ, pick an element r ∈ %α(A)& M . There are an L -term t(x1, . . . , xk ) and elements a1, . . . , ak ∈ A with r = t M (α(a1 ), . . . , α(ak )). We set γ(r) = t N (β(a1 ), . . . , β(ak )). Claim

γ is well-defined.

Proof of Claim

Assume there is another representation of r (i.e., there

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are an L -term s(y1, . . . , yl ) and elements b1, . . . , bl ∈ A with r = s M (α(b1 ), . . . , α(bl ))). We must consider two cases. Case 1 All function symbols and constants symbols of L are in L0 . (Here we need the relation symbol E of L † .) The terms t and s can be evaluated in A and we have (t A(a1, . . . , ak ), s A(b1, . . . , bl )) ∈ E A(M ,α) . It follows from E A(M ,α) = E A(N ,β) that t N (β(a1 ), . . . , β(ak )) = s N (β(b1 ), . . . , β(bl )), 

as required.

Case 2 L has a new function symbol or a new constant symbol with respect to L0 . (Here we need the relations Rϕ of L † .) Let ϕ be the L -formula s(x1, . . . , xk ) = t(y1, . . . , yl ), where all variables are distinct. The interpretation of Rϕ in A(M, α) and in A(N, β), 14.3.10, shows that A(N ,β)

(a1, . . . , ak , b1, . . . , bl ) ∈ RϕA(M ,α) = Rϕ

,

hence t N (β(a1 ), . . . , β(ak )) = s N (β(b1 ), . . . , β(bl )), finishing Case 2.



γ :

In exactly the same way we also find a well-defined map %β(A)& N → %α(A)& M with γ  ◦ β = α. It is clear that γ  is the compositional inverse of γ. Finally, it is a routine exercise (using the assumption that A(M, α) = A(N, β)) to show that γ and γ  are L -homomorphisms. (e) At this point we know that Λ A is bijective. One checks that Λ A is a homeomorphism by comparing the quasi-compact open sets of T-Spec(A), cf. 14.3.4(ii), with those of the expansion spectrum, cf. 14.2.13 (of course using the definition of L † ).  14.3.14 Examples Revisited We return to some of the examples in 14.3.9 and illustrate what 14.3.13 tells us about these. In particular, we record the language L † and give axioms for T-Spec(T0 ). Note that 14.3.13 does not give a recipe for finding concrete axioms of T-Spec(T0 ). Rather, we choose axioms in an ad hoc way, using the description of T-Spec(A) in the different examples of 14.3.9. Throughout, L0 is the language {+, ·, −, 0, 1} of (unital) rings. (i) For the Zariski spectrum, see 14.3.9(i), we have L † = L ∪ {E } = L0 ∪ {E }, and T-Spec(T0 ) is axiomatized by T0 together with the universal

14.3 Spectra of Structures and their Elementary Description

577

closures of E(x, y) ↔ E(x − y, 0), E(x, 0) and E(y, 0) → E(x + y, 0), E(x, 0) → E(x · y, 0), E(x · y, 0) → E(x, 0) or E(y, 0), ¬E(1, 0). Let A be an L † -structure satisfying T0 and these conditions. Then the interpretation of E in A is the equivalence relation {(a, b) ∈ A × A | a − b ∈ a}, where a is a prime ideal. Thus, the expansions of A |= T0 to models of T-Spec(T0 ) correspond bijectively to the prime ideals of A. (ii) In 14.3.9(ii) we chose: • L = L0 ∪ {c}, where c is a new constant symbol. • T0 is the L0 -theory of fields and T is the L -theory of commutative unital rings. (No additional assumptions about the new constant symbol c.) For a field K, we saw that T-Spec(K) is isomorphic to the spectral space of ideals of K[X]. The language L † here is L0 together with a k-ary predicate Rs,t for all L -terms s, t, where x1, . . . , xk are exactly the free variables of the formula s = t. (Recall that E is not needed here since Rx=y serves exactly the same purpose as E). The universal closures of the following formulas are an axiom system for T-Spec(T0 ): Rs=t (x1, . . . , xk ) ↔ Rs−t=0 (x1, . . . , xk ), Rs=0 (x1, . . . , xk ) and Rt=0 (x1, . . . , xk ) → Rs+t=0 (x1, . . . , xk ), Rs=0 (x1, . . . , xk ) → Rs ·t=0 (x1, . . . , xk ). The axioms describe expansions of the field K to L † , and these correspond bijectively to the ideals of K[X]. (iii) In the case of the real spectrum, see 14.3.9(v), we have L † = L ∪ {E } = L0 ∪ {P, E }. An axiom system for T-Spec(T0 ) is the extension of the theory of rings by the L † -axioms expressing that P is a prime cone (as stated in 14.3.9(v)) and E is the equivalence relation given by the support

578

Spectral Spaces via Model Theory of P. Explicitly these axioms are the universal closures of: P(x) and P(y) → P(x + y) and P(x · y), P(x) or P(−x), P(x · y) and P(−x · y) → (P(x) and P(−x)) or (P(y) and P(−y)), ¬P(−1), E(x, y) ↔ P(x − y) and P(y − x).

(iv) For the valuation spectrum, see 14.3.9(vii), we have L † = L ∪ {E } = L0 ∪ {|, E }. An axiom system for T-Spec(T0 ) is the extension of the theory of rings by the L † -axioms expressing that | is a binary relation with properties (a)–(f), cf. 14.3.9(vii), which are universal L † axioms, and the axiom ∀x y (E(x, y) ↔ 0 | x − y). 14.3.15 The language L † , 14.3.10, is a massive extension of L0 when L has new function symbols or constant symbols. For applications it is essential to find an explicit axiomatization of T-Spec(T0 ), or to find an explicit description of the models of the theory. There is no general method for this problem. In principle, one can try to use the term algebra for this purpose – recall from 14.3.7 that for each A |= T0 there is a natural homeomorphism T-Spec(A) → T-Spec(TmAlgT (A)). The term algebra may be difficult to construct and understand, but if we really have a grasp on it, then the description of the points of T-Spec(TmAlgT (A)) should be more accessible. See 14.3.9(ii) for a case, where this strategy actually works: the term algebra is a univariate polynomial ring over a field, and the points of the spectrum are its ideals.

Appendix The Poset Zoo

Partially ordered sets are among the most important tools for our topological studies and are used extensively. For easy reference we collect basic terminology and notation here in one place. A.1 Ordered Sets and Monotone Maps (i) Let P be a set with a binary relation ≤, whose graph is denoted by P ≤ . The relation is a quasi-order if the relation is reflexive and transitive (not necessarily antisymmetric), and then the pair (P, ≤) is a quasi-ordered set. Usually we write just P, the order relation being understood. The set P ≤ = {(p, q) ∈ P × P | p ≤ q} is the graph of ≤. (ii) For a topological space X the specialization relation  is introduced in 1.1.3. The pair (X,  ) is a quasi-ordered set, called the specialization set of X, cf. 1.1.3. This construction is the main reason for our need of, and interest in, ordered sets. General notation and terminology for ordered sets is sometimes different from that used for specialization sets. We point out such differences, including additional comments, and put the explanations applying to specialization sets in square brackets. (iii) A quasi-ordered set P is partially ordered, and the relation is a partial order, if antisymmetry holds. Partially ordered sets are also called posets. [In the case of a topological space X, the specialization set is a poset if and only if X is a T0 -space, 1.1.6.] (iv) A poset P is totally ordered, also called a chain, if p ≤ q or q ≤ p for all p, q ∈ P. The order relation is then a total order. An anti-chain is a poset in which two different elements are always incomparable. 579

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(v) A map f : P → Q of quasi-ordered sets is called monotone if p ≤ p implies f (p) ≤ f (p). [Let f : X → Y be a continuous map. Then f , considered as a map of the specialization sets, is monotone.] (vi) The quasi-ordered sets with the monotone maps form the category of quasi-ordered sets, which is denoted by QoSets. The posets and the totally ordered sets form full subcategories, which are denoted by PoSets and ToSets, respectively. [The construction sending a topological space to its specialization set is a functor Top → QoSets. It restricts to a functor T0 Top → PoSets.] (vii) For each poset P = (P, ≤) the inverse order ≤inv is defined by: p ≤inv p if and only if p ≤ p. The pair (P, ≤inv ) is a poset again, called the inverse poset of P, and is denoted by Pinv . Let f : P → Q be a monotone map between posets. Then f , considered as a map between the inverse posets Pinv and Qinv , is denoted by finv . [For a T0 -space X the inverse of the specialization order is denoted by  inv and is called the inverse specialization order. The inverse specialization set is denoted by (X,  inv ). In 1.4.1 we also introduce the inverse Xinv of a topological space. Usually the posets (X,  inv ) and (Xinv,  ) do not agree. But if X is a spectral space, then they do, 1.4.3.] (viii) If P is a poset and Q is a subset then the partial order restricts to a partial order of Q. We call (Q, ≤|Q ) a sub-poset. The same holds for subsets of quasi-ordered sets and totally ordered sets. A.2 Special Relations between Elements of Posets (i) If p ≤ q then q is a successor of p and p is a predecessor of q. [In a topological space X, if x  y then y is a specialization of x and x is y and a generalization of y, 1.1.3. If x  y and x  y then we write x   call y a proper specialization of x and x a proper generalization of y.] (ii) Let p < q and assume there is no r with p < r < q. Then q is an immediate successor of p and p is an immediate predecessor of q. [Let X be a topological space. If x  y and there is no z with x  z y    then y is an immediate specialization of x and x is an immediate generalization of y.] (iii) A jump is a pair (p, q), where p < q is immediate. Then p is called a lower element and q an upper element of the poset. (iv) In a poset (P, ≤) the subsets of minimal elements and maximal elements are denoted by Pmin and Pmax , respectively. Both sets may be empty. [Let (X, τ) be a topological space. The sets of minimal and maximal

Appendix: The Poset Zoo

581

)min and X max = elements of (X,  ) are denoted by X min = (X,  τ max ) , respectively. Both sets may be empty. The elements of X min (X,  τ are called minimal points, or generic points. The elements of X max are the maximal points, or closed points. The name “closed points” is justified since x ∈ X is maximal if and only if {x} ⊆ X is a closed subset, provided X is T0 .] (v) A poset P has a ceiling if every element has a successor in Pmax . It has a finite ceiling if it has a ceiling and Pmax is finite. Similarly, P has a floor if every element has a predecessor in Pmin . It has a finite floor if it has a floor and Pmin is finite. (vi) If P has a smallest element, or a largest element, then it is called, respectively, the bottom element denoted by ⊥ = ⊥ P or the top element denoted by  =  P . A.3 Up-Sets, Down-Sets, and Convexity (i) A subset Q ⊆ P is an up-set, or a down-set, if, respectively, q ∈ Q and q ≤ p imply p ∈ Q, or q ∈ Q and p ≤ q imply p ∈ Q. [In a topological space X, a subset Y ⊆ X is closed under specialization, or is closed under generalization (= generically closed), if it is an up-set or a down-set in (X,  ), 1.1.3.] (ii) For p ∈ P the sets p↑ = {q ∈ P | p ≤ q} and p↓ = {q ∈ P | q ≤ p} are the principal up-set and the principal down-set defined by p. Moreover, we set p = p↑ \ {p} = {q ∈ P | p < q} and p = p↓ \ {p} = {q ∈ P | q < p}. (iii) If Q ⊆ P we write Q ≤ p or p ≤ Q to indicate that p is an upper bound or a lower bound of Q (i.e., p↓ ⊇ Q or p↑ ⊇ Q). If Q has an upper bound, or a lower bound, then Q is said to be bounded from above, or bounded from below. It is bounded if it is bounded from above and below. The sets of upper bounds and lower bounds of Q are denoted by Ub(Q) = {p ∈ P | ∀q ∈ Q : q ≤ p} =



q ↑,

q ∈Q

Lb(Q) = {p ∈ P | ∀q ∈ Q : p ≤ q} =



q ∈Q

In particular, Ub( ∅ ) = P = Lb( ∅ ).

q↓.

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(iv) If Q ⊆ P, then Q ↑ = {p ∈ P | ∃q ∈ Q : q ≤ p} =



q ↑,

q ∈Q

Q ↓ = {p ∈ P | ∃q ∈ Q : p ≤ q} =



q↓

q ∈Q

are the up-set generated by Q and the down-set generated by Q. Note that ∅ ↑ = ∅ = ∅ ↓ . [LetY ⊆ X be a subset of a topological space. Then we writeY ↑ = Spez(Y ) and Y ↓ = Gen(Y ), 1.1.3.] (v) A nonempty subset Q ⊆ P is up-directed if p↑ ∩ q ↑ ∩ Q  ∅ for all p, q ∈ Q (i.e., if any two elements from Q have a common upper bound in Q). Similarly, Q is down-directed if p↓ ∩ q ↓ ∩ Q  ∅ for all p, q ∈ Q (i.e., if any two elements from Q have a common lower bound in Q). If Q is up-directed then Qmax contains at most one element. Similarly, if Q is down-directed then Qmin contains at most one element. (vi) Let R ⊆ Q ⊆ P be subsets. Then R is cofinal in Q if R ↓ = Q ↓ . If R is cofinal in Q then Ub(R) = Ub(Q). The set Q is up-directed if and only if q ↑ ∩ Q is cofinal in Q for all q ∈ Q. The subset R is co-initial if R ↑ = Q ↑ . This is the inverse notion of cofinality. (vii) Every poset P defines an undirected graph with the elements of P as vertices and an edge %p, q& = %q, p& between p and q if p ≤ q or q ≤ p. Each element p ∈ P defines a graph component, which is the set of elements of P that can be connected with p by a finite path. The graph component of p, denoted by %p&, is the smallest set containing p that is both an up-set and a down-set. The graph components form a partition  of the graph. If P = i ∈I Ci is a partition into sets that are both upsets and down-sets, then each Ci is a union of graph components. Every up-directed set or down-directed set is contained in a graph component. Using up-sets and down-sets there is a recursive construction of the graph component of an element p: one defines %p&0 = {p} and, for n ∈ N0 , %p&n+1 = (%p&n )↑ if n is even, and %p&n+1 = (%p&n )↓ if n is odd.  Then the graph component is %p& = n∈N0 %p&n . It is the smallest subset of P that contains p and is an up-set and a down-set. (For graph theory we refer to [GY04].) (viii) A subset Q of P is convex if q1 ≤ p ≤ q2 and q1, q2 ∈ Q implies p ∈ Q. Up-sets, down-sets and singletons are convex. Preimages of convex sets, in particular fibers, under monotone maps are convex. [A subset S in a T0 -space is convex if it is convex for the specialization

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order. Let f : X → Y be a continuous map of T0 -spaces. Since it is monotone for specialization, inverse images of convex sets, in particular fibers, are convex.] (ix) Arbitrary intersections of convex sets are convex. Therefore, for any subset Q ⊆ P there is a smallest convex set containing it, its convex hull, which is denoted by conv(Q) = conv P (Q). It can be constructed as follows: conv(Q) = {r ∈ P | ∃p, q ∈ Q : p ≤ r ≤ q} = Q ↑ ∩ Q ↓ . If p ≤ q ∈ P, then the convex hull of {p, q} is p ↑ ∩ q ↓ and we denote it by [p, q]. [Let Y ⊆ X be a subset in a T0 -space. Then conv(Y ) = Spez(Y ) ∩ Gen(Y ), see 4.5.1.] A.4 Chain Conditions and Ordinals (i) A poset has the ascending chain condition, abbreviated ACC, if there is no infinite strictly increasing sequence p1 < p2 < p3 < · · · . The descending chain condition, abbreviated DCC, holds if there is no infinite strictly decreasing sequence p1 > p2 > p3 > · · · . A poset with the DCC is called well-founded, see [MB89a, p. 640]. Inversely, a poset with the ACC is anti-well-founded. If a poset has the ACC or the DCC then every sub-poset inherits the property. (ii) A totally ordered set is well-ordered if it has the DCC. Subsets of wellordered sets are well-ordered. (iii) The naturally ordered sets N = {1, 2, . . .} (natural numbers) and N0 = {0} ∪ N (non-negative integers) are well-ordered. As ordered sets N and N0 are isomorphic. For each n ∈ N0 we set n = {0, 1, . . . , n − 1} ⊆ N0 . (iv) An ordinal number, or ordinal, is a set α with the property α ⊆ P(α) (consequently γ ∈ β ∈ α ⇒ γ ∈ α) and is well-ordered by the element relation, cf. [Cie97, Section 4.2]. Each well-ordered set is order-isomorphic to a unique ordinal by a unique poset isomorphism. We are mainly interested in the isomorphism type of ordinals. Therefore, one may think of an ordinal as (the isomorphism class of) a well-ordered set. In particular, frequently we identify isomorphic ordinals. For two ordinals α and β the following are equivalent: (a) (b) (c) (d)

α ∈ β ∪ {β}. α ⊆ β. α is a down-set of β. There is a poset embedding α → β.

584

Appendix: The Poset Zoo If the conditions are satisfied then we write α ≤ β. For any two ordinals α and β (at least) one of the relations α ≤ β and β ≤ α holds. The relations α ≤ β and β ≤ α hold simultaneously if and only if α = β.

(v) An ordinal α is a successor ordinal if there is some ordinal β with α = β ∪ {β}. Then β is the largest element of α, we call α the successor of β and write α = β + 1 = β+ . Every ordinal α has a unique successor ordinal, namely α + 1. A nonempty ordinal is a limit ordinal if it does not have a largest element (i.e., is not a successor). In particular, limit ordinals are infinite. There is a smallest infinite ordinal, which is denoted by ω. Clearly ω is a limit ordinal and can be identified with N0 . The elements of ω are the finite ordinals. A.5 Root Systems and Forests (i) A poset P = (P, ≤) is a root system if every principal up-set p↑ (i.e., the set of successors of p), is totally ordered. Let Q ⊆ P be a subset and assume that * is a partial order on Q such that q * q  implies q ≤ q . Then (Q, *) is a root system. In particular, every sub-poset of a root system is a root system. (ii) Let P be a root system. The graph component of p ∈ P, A.3(vii) is the set (p↑ )↓ (i.e., %p& = %p&2 , with notation as in A.3(vii)). Thus, p, q belong to the same graph component if and only if they have a common upper bound. Every graph component is a root of the root system. If there is only one root (i.e., if any two elements have an upper bound, say, if P is up-directed, see A.3(v)), then the root system is also called a root. Every root has at most one maximal element. (iii) A poset P = (P, ≤) is a forest if every principal down-set p↓ (i.e, the set of predecessors of p) is totally ordered. Similar to root systems, if Q ⊆ P is a subset and * is a partial order on Q such that q * q  implies q ≤ q  then (Q, *) is a forest. In particular, every sub-poset of a forest is a forest. Note that the notion of a forest is the order-theoretic inverse of the notion of a root system, A.1(vii) (iv) Let P be a forest. The graph component of p ∈ P, A.3(vii) is the set (p↓ )↑ (i.e., %p& = %p&3 , with notation as in A.3(vii)). Thus, p, q belong to the same graph component if and only if they have a common lower bound. Every graph component is a tree of the forest. If there is only one tree (i.e., if any two elements have a lower bound, say, if P is down-directed,

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see A.3(v)), then the forest is also called a tree. Every tree has at most one minimal element. (v) In a root system every down-directed set is totally ordered, and in a forest every up-directed set is totally ordered. A.6 Suprema and Infima, Lattices and Completeness (i) If a subset Q ⊆ P has a smallest upper bound then it is denoted by  sup(Q) = Q and is called the supremum, or the join of Q. If Q has a  largest lower bound then it is denoted by inf(Q) = Q and is called the infimum, or the meet of Q. Suprema and infima are unique if they exist. If R ⊆ Q is a cofinal subset (A.3(vi)) then sup(Q) exists if and only if sup(R) exists, and the suprema coincide if they exist. The empty set has a supremum (or an infimum) if and only if P has a bottom element (or a top element). If this is the case then sup( ∅ ) = ⊥ (or inf( ∅ ) = ). If Q = {p1, . . . , pn } is finite and the supremum exists, then it is also denoted by p1 ∨ · · · ∨ pn . If the infimum exists, then we also write p1 ∧ · · · ∧ p n . Suprema and infima need not exist in posets. Conditions about the existence of suprema and infima lead to a large number of different classes of posets, some of which we describe since they are used in the book. Conditions about the existence of suprema and infima in a poset P imply corresponding conditions about the existence of infima and suprema in the inverse poset Pinv . The transition from a poset to its inverse transforms suprema to infima, and conversely. Therefore, the inverse versions of the following definitions are obvious and are mostly not discussed. (ii) A poset P is a conditional ∨-semilattice, also called a conditional joinsemilattice, if p ∨ q exists whenever {p, q} has an upper bound. Similarly, P is a conditional ∧-semilattice, or conditional meet-semilattice, if p ∧ q exists whenever {p, q} has a lower bound. (iii) A poset P is a ∨-semilattice, or join-semilattice, if p ∨ q exists for all p, q ∈ P. Similarly, P is a ∧-semilattice, or meet-semilattice, if p ∧ q exists for all p, q ∈ P. Thus, the formation of joins (or meets) of two elements are binary operations on P, which are denoted by ∨ (or ∧). The inverse poset Pinv is a meet-semilattice (or a join-semilattice) with meet operation ∧inv defined by p ∧inv q = p ∨ q (or with join operation ∨inv defined by p ∨inv q = p ∧ q).

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(iv) A lattice is a poset P that is both a ∨-semilattice and a ∧-semilattice (i.e., if any two elements, or any nonempty finite subset, has both a supremum and an infimum). The inverse poset of a lattice is a lattice as well and is called the inverse lattice. (v) A lattice P is bounded if it has a top element and a bottom element, cf. A.2(vi) This is the same as to say P is bounded, A.3(iii), or that sup(P) and inf(P) exist, or that inf( ∅ ) and sup( ∅ ) exist. The inverse poset of a bounded lattice is a bounded lattice. (vi) A lattice P is distributive if the following equivalent conditions are satisfied: • ∀p, q, r : p ∨ (q ∧ r) = (p ∨ q) ∧ (p ∨ r), • ∀p, q, r : p ∧ (q ∨ r) = (p ∧ q) ∨ (p ∧ r). The inverse poset of a distributive lattice is a distributive lattice. A lattice that is both bounded and distributive is a bounded distributive lattice. (vii) Let P be a lattice. An element p ∈ P is join-irreducible (or ∨irreducible) if p = q ∨ r implies p = q or p = r. The inverse notion is meet-irreducibility (i.e., the element p is meet-irreducible, or ∧-irreducible), if p = q ∧ r implies p = q, or p = r). If the lattice P is distributive then join-irreducibility is equivalent to the following condition: if p ≤ q ∨ r then p ≤ q or p ≤ r. A corresponding equivalence holds for meet-irreducibility. (viii) Let P and Q be lattices. A map f : P → Q is a lattice homomorphism if f (p ∨ p) = f (p) ∨ f (p) and f (p ∧ p) = f (p) ∧ f (p) for all p, p ∈ P. If this is the case then f is monotone. But note that monotone maps need not satisfy these equations. If P and Q are even bounded lattices then f is a bounded lattice homomorphism if f ( P ) = Q and f (⊥ P ) = ⊥Q hold in addition. The bounded distributive lattices with the bounded lattice homomorphisms form the category BDLat. If f is a homomorphism of (bounded) lattices then finv is a homomorphism of the inverse (bounded) lattices. (ix) A poset is complete if every subset has a supremum. Equivalently, one may require that every subset has an infimum. Thus, a complete poset is a bounded lattice, though not necessarily distributive. (x) A poset is Dedekind-complete, or conditionally complete, if every nonempty subset that is bounded from above has a supremum; equivalently, every nonempty subset that is bounded from below has an infimum, cf.

Appendix: The Poset Zoo

587

[BaDw74, p. 46]. A Dedekind-complete poset need not be a lattice. The inverse poset of a Dedekind-complete poset is Dedekind-complete. (xi) A poset is a directed complete partial order, abbreviated dcpo, if every up-directed subset has a supremum (cf. [GHK+ 03, Def. 0-2.1, p. 9]); also recall that up-directed sets are nonempty by definition. In particular, in a dcpo every nonempty chain has a supremum. It follows from A.5(v) that a forest is a dcpo if and only if nonempty chains have a supremum. The same is true in a root system P. For, if Q ⊆ P is up-directed and q ∈ Q, then the chain q ↑ ∩ Q is cofinal in Q, A.3(vi), and A.6(i) shows that Q has a supremum if and only if q ↑ ∩Q has a supremum. Without proof we record the following surprising result, which holds for arbitrary posets. It is not used in the book, but may help to understand the notion of a dcpo. Proposition (cf. [AbJu94, Proposition 2.1.15, p. 15] and [Mar76]) A poset (X, ≤) is a dcpo if and only if every nonempty chain in X has a supremum. Remark This is a subtle and nontrivial result. The snag is that the supremum of an up-directed set D in a dcpo need not be the supremum of any chain C that is contained in D. (xii) Let P and Q be dcpos. A monotone map f : P → Q is a dcpo homomorphism if f (sup(D)) = sup( f (D)) for every up-directed set D ⊆ P. A subset M ⊆ P is a sub-dcpo if (M, ≤ | M ) is a dcpo and the inclusion map M → P is a dcpo homomorphism, equivalently: if D ⊆ M is up-directed then sup P (D) ∈ M (i.e., M is closed in P under up-directed suprema). We mention the following examples of sub-dcpos: • Every anti-chain is a sub-dcpo. In particular, Pmin and singleton subsets are sub-dcpos. • Chains of P that are maximal for inclusion (also called maximal chains) are sub-dcpos. • All up-sets are sub-dcpos. • The set of sub-dcpos of P is closed under finite unions and arbitrary intersections. (xiii) A poset is a filtered complete partial order, abbreviated fcpo, if every down-directed subset has an infimum. This is the inverse of the notion of a dcpo. (xiv) Let P be a lattice. Then a nonempty subset S has a supremum if and only  if the up-directed set T = { F | F ⊆ P finite} has a supremum. If this   is the case then S = T. Thus, P is a dcpo if and only if it has a top element and is conditionally complete.

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(xv) Let P be a complete and bounded distributive lattice. Then P is called a frame if, for each a ∈ P and every family (bi )i ∈I in P, the equality   a ∧ i ∈I bi = i ∈I (a ∧ bi ) holds. We refer to this rule as the frame law. Given another frame Q, a map ϕ : P → Q is a frame homomorphism  if ϕ is a homomorphism of bounded lattices and satisfies ϕ( i ∈I bi ) =  i ∈I ϕ(bi ) (where (bi )i ∈I is a family in P). Let X and Y be topological spaces. Then O(X) and O(Y ) are frames. The map O( f ) : O(Y ) → O(X), O → f −1 (O) induced by a continuous map f : X → Y is a frame homomorphism. The lattice P is an inverse frame if Pinv is a frame. A map ϕ : P → Q between inverse frames is an inverse frame homomorphism if ϕinv : Pinv → Qinv is a frame homomorphism. Moreover, P is a bi-frame if it is a frame and an inverse frame. A bi-frame homomorphism is a map of bi-frames that is both a frame homomorphism and an inverse frame homomorphism. If X is a topological space then A(X), the lattice of closed subsets, is an inverse frame. Every complete Boolean algebra is a bi-frame, cf. [Kop89, p. 22, Lemma]. Trivially, every finite distributive lattice is a bi-frame. The frames, the inverse frames, and the bi-frames form subcategories of BDLat that are denoted by Fr, InvFr, and BiFr. The construction of O(X) and O( f ) (where X is a topological space and f a continuous map) is a functor Top → Fr. Since Fr ⊆ BDLat is a subcategory, O is also considered as a functor Top → BDLat. Detailed treatments of frames and their homomorphisms may be found in [Joh86] and in [PiPu12]. A.7 Ideals and Filters in Posets and Lattices (i) An ideal in a poset P is an up-directed down-set. (Recall that up-directed sets are assumed to be nonempty, A.3 (v)) Hence, in a join-semilattice an ideal is a nonempty down-set closed under finite joins. If i is an ideal in P then we write i  P. The set of ideals of P is denoted by I(P). The notion of a filter is the inverse of the notion of an ideal. Thus, a filter in a poset is a down-directed up-set. Hence, in a meet-semilattice a filter is a nonempty up-set closed under finite meets. (ii) A subset Q ⊆ P is up-directed if and only if Q ↓ is an ideal. If this is the case then Q ↓ is the ideal generated by Q. If P is a join-semilattice then every nonempty subset R generates a smallest directed subset Q containing R (the set of finite joins of elements of R). Thus, Q ↓ is the smallest ideal containing R and is called the ideal generated by R.

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(iii) Let P be a poset and let Q be a proper subset. Then Q is a prime ideal if it is an ideal and P \ Q is a filter of P. The inverse notion of a prime ideal is a prime filter: the subset Q is a prime filter if it is a filter and P \ Q is an ideal. When P is a lattice, a proper ideal is prime just if p ∧ q ∈ Q implies p ∈ Q or q ∈ Q; a proper filter is prime just if p ∨ q ∈ Q implies p ∈ Q or q ∈ Q. A.8 Topologies on Posets (i) A topology τ on a poset (P, ≤) is a lower topology if ≤ is the specialization , cf. 1.1.3. We call τ an upper topology if the inverse partial order  τ . In a lower topology, the open sets order ≤inv is the specialization order  τ are down-sets; in an upper topology, the open sets are up-sets. For each subset S ⊆ P, every lower topology (or upper topology) on P restricts to a lower topology (resp., upper topology) on S. (ii) For each poset P there are a coarsest and a finest lower topology. The coarse lower topology, which is denoted by τ  = τ  (≤) = τ  (P) = τ  (P, ≤), has the principal up-sets p↑ , A.3(ii), as a subbasis (in general not closed under finite unions or intersections) of closed sets. Some authors (e.g., [GHK+ 03, Def. III-1.1, p. 210]) also call τ  simply the lower topology. The open sets for the fine lower topology are the down-sets in P. The fine lower topology is also called the Alexandroff topology and is denoted by τ L = τ L (≤) = τ L (P) = τ L (P, ≤). (iii) Similarly, there are a coarsest and a finest upper topology. The coarse upper topology, which is denoted by τ u = τ u (≤) = τ u (P) = τ u (P, ≤), has the principal down-sets p↓ , A.3(ii), as a subbasis (in general not closed under finite unions or intersections) of closed sets. The open sets for the fine upper topology, which is denoted by τU = τU (≤) = τU (P) = τU (P, ≤), are the up-sets in P. The coarse upper topology coincides with the coarse lower topology for the inverse order of P, and the fine upper topology is the fine lower topology for the inverse order. (iv) The interval topology on P is the join of the coarse lower topology and the coarse upper topology. The sets x ↑ and y ↓ with x, y ∈ X are a subbasis of closed sets for the interval topology, cf. [Ern80]. The interval topology has the T1 -property, since the singleton subsets {x} = x ↑ ∩ x ↓ are closed. The interval topology of P coincides with the interval topology of the inverse poset Pinv .

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[Yok09] [Zar47] [Zar52]

605

Marcus Tressl. The real spectrum of continuous definable functions in o-minimal structures. In Publ. Math. Univ. Paris VII, Séminaire sur les Structures Algébriques Ordonnées, Univerité Paris VII, vol. 68, p. 1–15. 1998. 535 Marcus Tressl. Computation of the z-radical in C(X). Adv. Geom., 6(1):139–175, 2006. 400 Marcus Tressl. Super real closed rings. Fund. Math., 194(2):121–177, 2007. 506, 535 Marcus Tressl. Bounded super real closed rings. In Logic Colloquium 2007, volume 35 of Lecture Notes in Logic, p. 220–237. Association for Symbolic Logic, La Jolla, CA, 2010. 506, 535 Lou van den Dries. Some applications of a model theoretic fact to (semi-) algebraic geometry. Nederl. Akad. Wetensch. Indag. Math., 44(4):397–401, 1982. 521 Sam J. van Gool. Duality and canonical extensions for stably compact spaces. Topology Appl., 159(1):341–359, 2012. 13, 99 Sam J. van Gool. On sheaves and duality. PhD thesis, Universiteit Nijmegen, 2014. 13, 99 Steven Vickers. Topology via logic, volume 5 of Cambridge Tracts in Theoretical Computer Science. Cambridge University Press, Cambridge, 1989. 240 Jan van Mill. Supercompactness and Wallman spaces. Mathematisch Centrum, Amsterdam, 1977. Mathematical Centre Tracts, No. 85. 235 Jan van Mill. An introduction to βω. In Handbook of set-theoretic topology, p. 503–567. North-Holland, Amsterdam, 1984. 289 Robert J. Walker. Algebraic curves. Springer-Verlag, New York, 1978. Reprint of the 1950 edition. 289 Friedrich Wehrung. A solution of Dilworth’s congruence lattice problem. Adv. Math., 216(2):610–625, 2007. 100 Friedrich Wehrung. Spectral spaces of countable abelian lattice-ordered groups. Trans. Amer. Math. Soc., to appear, 2018. 533 Roger Wiegand and Sylvia Wiegand. Prime ideals and decompositions of modules. In Non-Noetherian commutative ring theory, volume 520 of Mathematics and its Applications, p. 403–428. Kluwer Academic, Dordrecht, 2000. 258 Roger Wiegand and Sylvia Wiegand. Prime ideals in Noetherian rings: a survey. In Ring and module theory, Trends in Mathematics, p. 175–193. Birkhäuser/Springer, Basel, 2010. 258 Tomoo Yokoyama. A poset with spectral Scott topology is a quasialgebraic domain. Order, 26(4):331–335, 2009. 219 Oscar Zariski. A new proof of Hilbert’s Nullstellensatz. Bull. Amer. Math. Soc., 53:362–368, 1947. 446 Oscar Zariski. The fundamental ideas of abstract algebraic geometry. In Proceedings of the International Congress of Mathematicians, Cambridge, MA, 1950, vol. 2, p. 77–89. American Mathematical Society, Providence, RI, 1952. xiv

606 [ZaSa58]

[Zha02] [Zie84] [Zil10]

References Oscar Zariski and Pierre Samuel. Commutative algebra, Volume I. The University Series in Higher Mathematics. D. Van Nostrand, Princeton, NJ, 1958. With the cooperation of I. S. Cohen. 418, 450, 451, 452 Digen Zhang. The M-valuation spectrum of a commutative ring. Comm. Algebra, 30(6):2883–2896, 2002. 572 Martin Ziegler. Model theory of modules. Ann. Pure Appl. Logic, 26(2):149–213, 1984. 100 Boris Zilber. Zariski geometries. Geometry from the logician’s point of view. Volume 360 of London Mathematical Society Lecture Note Series. Cambridge University Press, Cambridge, 2010. 541

Index of Categories and Functors Categories are printed in typewriter font

Categories BDLat, bounded distributive lattices, 586 BDSLat, bounded distributive join-semilattices, 99 BiFr, bi-frames, 588 BiLocSp, bi-localic spaces, 313 BoolAlg, Boolean algebras, 19 BoolSp, Boolean spaces, 16 CReg, completely regular spaces, 289 Comp, compact spaces, 289 Frop , opposite category of Fr, 299 Fr, frames, 588 HeytAlg, Heyting algebras, 278 ISprings, category of indexed springs, 483 InvFr, inverse frames, 588 InvLocSp, inverse localic spaces, 313 LocSp, localic spaces, 313 PoSets, posets = partially ordered sets, 580 Priestley, Priestley spaces, 33 QoSets, quasi-ordered sets, 580 RedRings, reduced rings, 439 Rings, rings (commutative, unital), 67 Sets, sets, 55 Spec, spectral spaces, 11 Springs, category of springs, 483 T0 Sob, sober T0 -spaces, 384 T0 Top, T0 -spaces, 138 ToSets, totally ordered sets, 580 Top, topological spaces, 11 U, category of spaces with indeterminates, 473 semiSpec, semi-spectral spaces, 99

Functors λ : Rings → BDLat, reticulation functor, 431 O : Top → BDLat, 588 O : Top → Fr, 588

607

◦

K : Spec → BDLat, 11 con : Spec → BoolSp, 22 inv : PoSets → PoSets , BDLat → BDLat, 95 inv : Spec → Spec, 25 red : Rings → RedRings, 439 PrimF : BDLat → Spec, 91 PrimI : BDLat → Spec, 91 Spec : BDLat → Spec, 81 Spec : Rings → Spec, 70 Sper : Rings → Spec, 73 ba : BDLat → BoolAlg, 94 P : T0 Top → PoSets, 408 Q : Spec → PoSets, 408 L : Top → Loc, 370 R : Spec → Loc, 370 S : Top → Spec, 370

Index of Examples By order of appearance

1.1.7: Ideals in the lattice of open subsets of [0, 1] ⊆ R, 5 1.1.18: Topologies on {0, 1}, 8 1.1.19: Topologies on {0, . . . , n − 1}, where n ∈ N, 9 1.2.5: Spectral maps with domain n, 12 1.2.6: Spectral maps with codomain n, 12 1.3.3: The one-point compactification of a discrete space is Boolean, 14 1.3.5: 1 is a Boolean space, 15 1.3.8: Continuous maps between spectral spaces that are not spectral maps, 16 1.3.12: The patch space of a finite spectral space, 17 1.4.2: The inverse topology, first examples, 23 1.5.6: Generically closed proconstructible sets in spectral spaces need not be open, 30 1.5.8: Specialization in a spectral space is a spectral order, 31 1.5.19: The cofinite topology on an infinite set is not spectral, 35 1.6.11: Spectral space with a single nontrivial specialization, 43 1.7.4: Stably compact spaces, 46 2.1.2: A subspace of a spectral space that is spectral, but is not a spectral subspace, 49 2.5.4: The Zariski spectrum of the integers, 70 2.5.5: The Sierpiński space as a Zariski spectrum, 70 2.5.6: Boolean Zariski spectra, 70 2.5.13(a): Spectrum of subsets of a set, 74 2.5.13(b): Spectrum of ideals in a ring, 74 2.5.13(c): Spectrum of (normal) subgroups of a group, 75 2.5.13(d): Spectrum of up-sets in a poset, 75

2.5.13(e): Spectrum of equivalence relations on a set, 75 2.5.13(f): Spectrum of congruence relations on a bounded distributive lattice, 76 3.3.5: Prime ideal spectrum and prime filter spectrum of S∞ and its inverse space, 91 4.2.3: In a spectral space irreducible sets need not be down-directed by specialization, 108 4.2.8: Spectral spaces need not be Dedekind-complete, 110 4.2.13: A poset satisfying the Kaplansky conditions and the Lewis-Ohm condition without being the specialization poset of a spectral space, 112 4.3.2(i): Cantor–Bendixson rank of space ω + 1 with interval topology, 115 4.3.2(i): Densely totally ordered sets with interval topology have Cantor–Bendixson rank ∞, 115 4.3.2(vi): Cantor–Bendixson rank of spectral space λ + 1, λ an ordinal, 115 4.3.5(viii): Rank of poset λinv , where λ is an ordinal, 117 4.3.7: Rank of spectral spaces, 117 4.3.8: Comparison of Krull dimension and rank for a spectral space, 118 4.4.12: Non-isomorphic spectral spaces with homoeomorphic subspaces of minimal spectra, 125 4.4.13: Minspectral spaces, 125 4.4.18: The constructible closure of X max in a spectral space X need not be closed, 128 4.5.3: Convex sets in a spectral space need not be locally closed, 132

608

Index of Examples 4.5.15: Locally closed points in topological spaces, 136 4.5.16: Locally closed points in spectral spaces, 136 5.2.1: Epimorphisms in BLat need not be surjective, 146 5.3.6: Spectral maps with going-down need not be open, 154 5.3.7(ii): A non-proper spectral map such that inverse images of quasi-compact sets are quasi-compact, 155 5.3.7(iii): Inverse images of quasi-compact sets need not be quasi-compact, 155 5.4.8: Spectral subspaces need not be retractions of the ambient space, 158 5.6.1: Extensions of continuous maps need not be unique, 165 6.1.9: A spectral space with a T0 -quotient that is a spectral space, but not a spectral quotient, 174 6.1.9: A spectral space with an equivalence relation such that the T0 -quotient is not the topological quotient, 174 6.2.2: A spectral space with a closed equivalence relation that is not saturated, 175 6.3.11: Spectral relations, 184 6.4.4: Construction of a spectral quotient modulo an equivalence relation, 188 6.4.6: A continuous map between spectral spaces that is identifying in Top, but is not a spectral map, 188 6.4.15: Identifying spectral maps onto 2, 194 6.4.16: A topological quotient of a spectral space need not be a spectral space, 194 6.4.17: An identifying spectral map without the lifting property, 194 6.4.18: An identifying spectral map that is not topologically identifying, 195 6.4.19: Identifying spectral maps need not restrict to identifying spectral maps on spectral subspaces, 196 6.4.20: Products of identifying spectral maps need not be identifying, 196 6.5.3: The Stone dual of a spectral space with a single nontrivial specialization, 198 6.6.3: A connected spectral space whose space of closed points is not connected, 200 6.6.10: Minimal sets that are both closed and inversely closed need not be connected components, 203

609

7.1.2(iii)(f), footnote: A poset with a subset that is not way below itself, but is a union of subsets that are all way below themselves, 208 7.1.4(iii): Subset Y of a poset (X, ≤) such that τ  (X) does not restrict to τ  (Y), 211 7.2.8: Algebraic lattice has spectral coarse lower topology, 226 7.2.10: A spectral root system with empty way-below relation, 228 7.2.11: A complete lattice with spectral coarse lower topology and trivial way-below relation, 228 7.2.13: Algebraic lattices in algebra, 229 7.3.1(iv): A dcpo-forest with an anti-chain that is not τ  (X)-quasi-compact, 232 7.3.7(iii), footnote: A quasi-compact chain with coherent coarse lower topology which is not a dcpo, 236 8.1.4: Noetherian and non-Noetherian spaces, 248 8.1.8: Noetherian spaces that are not spectral, 249 8.1.14: A non-Noetherian spectral space and an open map onto a Noetherian spectral space with Noetherian fibers, 254 8.1.16: A non-Noetherian spectral space X, such that each C ∈ A(X) has only finitely many generic points and there are no infinite strictly decreasing chains of irreducible sets in K(X), 256 8.1.23: A non-Noetherian spectral space such that the coarse lower topology for the specialization order is spectral and Noetherian, 259 8.1.24: A Noetherian spectral space such that the interval topology for the specialization order is not the patch topology, 259 8.1.26: Noetherian and non-Noetherian spectral spaces, comparison of rank and Cantor–Bendixson rank of the patch space, 260 8.2.5: A Boolean space with non-spectral closed partial order, 266 8.3.2: Semi-Heyting spaces and Heyting spaces, 267 8.3.8: Frames as Heyting algebras, 270 8.3.8: Frame homomorphisms need not be pseudo-complemented homomorphisms, 270 8.3.11(i): A non-semi-Heyting spectral space

610

Index of Examples

in which all open sets are regular open, 272 8.3.11(ii): A non-semi-Heyting spectral space such that every subset has at most two generic points, 272 8.3.11(iii): A semi-Heyting space that is not Heyting, 272 8.3.14: Heyting algebras of open semilinear sets, 274 8.3.14: Heyting algebras of open semi-algebraic sets, 274 8.3.18: A spectral map between semi-Heyting maps that is not semi-Heyting, but behaves well on generic points, 276 8.3.21: A semi-Heyting map between two Heyting spaces that is not a Heyting map, 277 8.4.1: A normal spectral space that is not regular, 280 8.4.4: The smallest non-normal spectral space, 281 8.4.4: A normal spectral space containing a non-normal open subspace, 281 8.4.15: The Stone–Čech compactification of a discrete space, 288 8.5.2: The spectral space PrimI(Coz(X)) is a root system if X is completely regular, 291 8.5.3(i): The Tychonoff plank, a non-normal subspace of a spectral root system, 292 8.5.3(iii): Word trees, with the Scott topology, are hereditarily normal spectral root systems, 292 8.5.3(iv): Real spectra are spectral root systems, but need not be hereditarily normal, 293 9.1.1: A bounded lattice homomorphism between frames that is not a frame homomorphism, 300 9.1.1: A bounded lattice homomorphism between bi-frames that is not a frame homomorphism and not an inverse frame homomorphism, 301 9.1.2(a): A nucleus of a frame that is a lattice homomorphism, but not a bounded lattice homomorphism, 302 9.1.2(b): A nucleus of a frame that is not a lattice homomorphism, 302 9.1.2: Open regularization in a topological space is a nucleus, 302 9.1.3: A bounded distributive lattice with a

completion such that not all infinite joins are preserved by the embedding in the completion, 303 9.2.3(i): Noetherian spectral spaces are localic spaces, 308 9.2.3(i): A Noetherian spectral space that is not inverse localic, 308 9.2.3(ii): Extremally disconnected Boolean spaces are localic, 308 9.2.3(iii): A well-ordered set having a largest element is a localic space if equipped with the coarse lower topology, 308 9.2.4(i): A localic space that is not extremally disconnected, 308 9.2.4(ii): An extremally disconnected spectral space that is not localic, 308 9.2.9: A localic space whose patch space is not extremally disconnected, 311 9.3.6: A non-localic spectral map between localic spaces, 313 9.3.8: Two localic spaces whose spectral product is not the localic product, 314 9.4.5(i): Constructible subsets in a localic space are localic subspaces, 317 9.4.5(ii): Closed subsets in a localic space are localic subspaces if and only if they are closed constructible, 317 9.4.5(iii): Finite unions of localic subspaces are localic, 317 9.4.5(iv): Two localic subspaces of a localic space whose intersection is not localic, 318 9.4.8: A localic map and a localic subspace of the codomain whose inverse image is not a localic subspace, 319 9.5.4(ii): A localic space without localic points, 324 9.5.8: Noetherian spectral spaces have enough points, 325 9.5.8: Extremally disconnected Boolean spaces do not have enough points, 325 10.1.5: The coproduct of a family of singleton spaces, 332 10.1.9: Zariski spectra of rings yield bounded distributive lattices with designated subsets, 334 10.1.12: A product of lattices with elements of infinite complexity, 336 10.1.15: A subset that is separating for closed sets in the bounded distributive lattice of a local spectral space, 337

Index of Examples 10.1.15: The natural basis of the Zariski spectrum of a ring is separating for closed sets, 337 10.2.9: Fiber sums of finite spectral spaces, 345 10.2.18: The wedge of two totally ordered spectral spaces is a fiber sum, 353 10.4.4: A spectral root system whose specialization is not total and does not have a proper refinement by a total spectral order, 364 10.4.9: A weakening of the specialization order on a spectral subspace that does not yield a spectral order, 368 11.1.8: A topological space whose spectral reflection map is not a monomorphism, 377 11.1.8: A topological space whose spectral reflection map is not an epimorphism, 378 11.1.11: The spectral reflection of an ordinal with its coarse lower topology, 379 11.2.14: The affine space K n over an algebraically closed field is the set of locally closed points of its sobrification, 390 11.4.2: A T0 -space whose spectral reflection map is not an epimorphism of T0 -spaces, 394 11.4.15: The minimal spectrum of a spectral root system need not be the Gleason cover of the space of closed points, 400 11.6.1, footnote: Monotonic maps between posets that are not continuous for the coarse lower topology, 409 12.1.13(i): Null ring has empty Zariski spectrum, 427 12.1.13(ii): Rings whose Zariski spectrum is a singleton, 427 12.1.13(iii): Finite rings have finite Boolean Zariski spectrum, 428 12.1.13(iv): Non-isomorphic rings with infinite homeomorphic Zariski spectrum, 428 12.1.13(v): Every Boolean space is a Zariski spectrum in many different ways, 428 12.1.13(vi): Zariski spectra of valuation rings are totally ordered by specialization, 428 12.1.20(c): A quasi-compact open set in a Zariski spectrum that is not principal open, 433 12.2.1: A spectral map between Zariski spectra

611

that is not induced by a ring homomorphism, 435 12.2.12: An injective ring homomorphism such that the image of the map between the Zariski spectra is the set of minimal prime ideals, 440 12.3.3: G-domains, 443 12.3.13: Jacobson rings, 448 12.4.9: A ring with quasi-compact open sets of prescribed complexity, 455 12.4.14: Stratification by rank of the Zariski spectrum of a polynomial ring over an algebraically closed field, 457 12.4.15: A Noetherian spectral space with finite Krull dimension with constructibly dense set of maximal points, but rank different from the Cantor–Bendixson rank for the patch topology, 458 12.5.4, proof of (ii): Two non-isomorphic field amalgamations of two fields, 461 12.5.6(i): A family of rings such that the coproduct of the Zariski spectra does not map onto the Zariski spectrum of the product of the rings, 462 12.5.9: Fiber products of rings where the Zariski spectrum is not homeomorphic to the fiber sums of the Zariski spectra of the components, 467 12.6.2: Subcategories C ⊆ Spec such that the functor Spec does not have a section over C, 470 12.6.5: Spaces with indeterminates for a given spectral space, 473 12.6.8: A category of spectral spaces such that the Zariski spectrum functor has a section, but no space-preserving functor, 475 13.1.3: Examples of preorders, 490 13.1.7 Real spectra of fields, 491 13.1.8 Real spectra of polynomial rings, 491 13.1.12(i): Example of proper inclusion between a preorder and its saturation, 495 13.1.14: Example of closed subset of Sper(A) ≥0 not in the image of map H A , 495 13.1.23: Examples of q-modules, 499 13.3.4: The support map may not reflect specialization, 506 13.3.14: A non-convex spectral map between real spectra of rings, 512 13.3.20: A real spectral morphism without the “going down” property, 514

612

Index of Examples

13.6.5: The real semigroup associated with a semi-real ring, 536 13.6.5: The real semigroup structure on 3 = {0, 1, −1}, 537 14.2.3(i): Zariski spectrum of a ring via model theory, 545 14.2.3(i): Homomorphism from a ring to a field whose theory is not determined by a prime ideal, 546 14.2.3(ii): Type spaces in model theory, 547 14.2.3(iv): Definable families in a structure, 549 14.2.9: Zariski spectrum of a ring via model theory, continuation of 14.2.3(i), 553 14.2.10: Spectrum of a bounded distributive lattice via model theory, 554 14.2.14: The spectral space of congruences of a universal algebra, 557 14.2.15: The spectral space of local subrings of a field, 557

14.3.9(i) and 14.3.14(i): Zariski spectrum as a T -spectrum, 566 14.3.9(ii) and 14.3.14(ii): Ideal spectrum of a polynomial ring over a field as a T -spectrum, 566 14.3.9(iii): Prime ideal spectrum of a bounded distributive lattice as a T -spectrum, 567 14.3.9(iv): Spectral space of isomorphism classes of distributive lattices generated by a poset, 567 14.3.9(v) and 14.3.14(iii): Real spectrum of a ring as a T -spectrum, 568 14.3.9(vi): Space of real prime ideals of a ring as a T -spectrum, 568 14.3.9(vii) and 14.3.14(iv): Valuation spectrum of a ring, 569 14.3.9(viii): The p-adic spectrum of a ring, 572

Symbol Index The first entries contain symbols with no or ambiguous alphabetical value, sorted in order of appearance. After these, the symbols are ordered alphabetically.

Order, see Appendix: The Poset Zoo, 579 ,  Symbols without Alphabetic Value, in Order of Appearance , ,   , specialization, 3  , specialization (in topology τ), 3 τ Xcon , constructible (= patch) topology of X, 16 con S , constructible closure of S, 16 fcon , map induced by f on patch spaces, 21 Xinv , inverse topology on X, 23 inv C , inverse closure of C, 23 finv , map induced by f on inverse spaces, 25 = k ]], set of maps in 2 S with value k on [[ T T ⊆ S, 54 [[ s = k ]], set of maps in 2 S with value k at s ∈ S, 54 X  , graph of specialization of X, 59 X × S Y, fiber product (= pull-back) of X and Y over S, 61 X × S Y, specialization fiber product of X, Y over S, 63 X1 ⊕ X2 , topological sum of two spaces, 65  i∈I Xi , topological sum (= coproduct) of Xi , 65 , symmetric difference in Boolean algebra, 71 A-isomorphism between morphisms in a category, 76 A-morphism between morphisms in a category, 76 ¬ = ¬ A , complementation map in Boolean algebra A, 95 0-dimensional space, 122 [·] E , equivalence class for E, 134 ≡f , lattice filter congruence, 159 ≡i , lattice ideal congruence, 159

[·]i , [·]f , congruence classes in lattices, 159 X/R, spectral quotient of X modulo R, 170 X/T0 R, T0 -quotient of X modulo R, 170 X/Set E, 170 R sat , saturation of relation R, 174 ≤ R , quotient order on X/Set E, 176 ', way below (for subsets of poset), 207 · → ·, implication (Heyting algebra), 269 ∼a, pseudo-complement of a, 269 %S & L , sub-frame generated by S ⊆ L, 301 0i∈I Xi , spectral coproduct of Xi , 329 Y clcon , set of closed and constructible points in spectral space Y, 397 a  A, a ideal of ring A, 421 A S , ring of fractions with denominators in S, 435 a S , ideal in ring of fractions A S generated by canonical image of ideal a, 435 Ap , localization of A at prime ideal p, 436 ap , ideal generated in localization Ap by canonical image of ideal a  A, 436 b : c, quotient of ideals b, c in a ring, 439 ≤α , order determined by α ∈ Sper(A) on field qf(A/supp(α)), 490 ¬Δ = {¬δ | δ ∈ Δ}, 542 |=, 542 X ≤ , graph of order relation ≤ on X, 579 P max , set of maximal elements of poset (P, ≤), 580 P min , set of minimal elements of poset (P, ≤), 580 Pinv , inverse of poset P, 580 )max , set of closed points of X max = (X,  τ the topological space (X, τ), 581 X min = (X,  )min , set of generic points of τ the topological space (X, τ), 581

613

614

Symbol Index

≤inv , inverse of order relation ≤, 580  , proper specialization, 580  Q ≤ p, ∀q ∈ Q : q ≤ p, 581 ⊥ = ⊥ P , smallest element of poset P, 581  =  P , largest element in poset P, 581 p ≤ Q, ∀q ∈ Q : p ≤ q, 581 p↓ , principal down-set of p, 581 p  , 581 p↑ , principal up-set of p, 581 p  , 581 Q↓ , down-set generated by Q, 582 Q↑ , up-set generated by Q, 582 %p&, graph component of vertex p in a graph, 582 [·, ·], closed interval = convex hull of two points, 583  Q = sup(Q), supremum of Q, 585  Q = inf(Q), infimum of Q, 585 ∨, join operation on lattice, 585 ∨-semilattice = join-semilattice, 585 ∨inv , join operation in inverse of a ∧-semilattice, 585 ∧, meet operation on lattice, 585 ∧-semilattice = meet-semilattice, 585 ∧inv , meet operation in inverse of a ∨-semilattice, 585 ∨-irreducible = join-irreducible, 586 ∧-irreducible = meet-irreducible, 586 ,  Alphabetic Symbol List, ,  1, one-element space, 8 2, Boolean algebra with two elements, 15 2, Sierpiński space, 8 3, = n for n = 3 , 9 A(E; L), closed elements (of completion E of L), 303 Ared , reduced ring associated with ring A, 439 A× , group of units in ring A, 421 A(X), set of closed sets of X, 2 ba(L), Boolean envelope of L, 93 ba L , embedding L → ba(L), 93 ba(ϕ), extension of lattice morphism ϕ to Boolean envelope, 94 β X : X → βX, Stone–Čech compactification of the completely regular space X, 123 CB(X), Cantor–Bendixson rank of space X, 114 CB X (x) = CB(x), Cantor–Bendixson rank of x ∈ X, 114

χ, characteristic functions map P(X) → 2X , 55 Clop(X), set of clopen subsets of X, 14 Cong(L), set of congruences of lattice L, 76 conv(Q) = conv P (Q), convex hull of Q in P, 583 con X , identity Xcon → X, 21 Coz(X), lattice of cozero sets in topological space X, 287 C(X, R), ring of continuous functions on topological space X with values in R, 287 D(·), basic opens of Zariski topology, 67 F (·), basic opens of PrimF(L), 89 D(·) = D L I D(·), D L (·), basic opens of PrimI(L), 90 D(δ), subbasic open sets of the space of Δ-types, 544 δX, set of non-isolated points of X, 114 δ α X, δ ∞ X, iterations of δX, 114 Δ(X), diagonal of X, 59 Δ X , diagonal X → X × X, 59 Δ A , positive quantifier-free L (A)-sentences, 555 diag(A), diagram of structure A, 555 diag+ (A), positive diagram of structure A, 562 E(S), set of equivalence relations on S, 75 E(X), canonical extension of X, 304 F2 , field with two elements, 71 f(a), principal filter generated by a, 83 finv , map f between posets considered as a map between the inverse posets, 580 Fml(L ), formulas of language L , 542 Γ( f ), graph of map f , 59 Γ∗ , finite words, 240 Γ ≤n , words of length at most n, 240 Γ < n , words of length < n, 240 ΓX , canonical map X → Γ(X), 199 Γ(X), set of connected components of X, 199 Gen(A), set of generalizations of elements of A, 3 HL( A) , Harrison topology on L(A), 500 HomBDLat (L, 2), set of bounded distributive lattice homomorphisms of L to 2, 79 >0 (a , . . . , a ), basic opens of real HR r 1 spectrum, 72 ht(a), height of ideal a in a ring, 452 Ifin (A), finitely generated ideals in ring A, 422 I rad (A), set of radical ideals of A, 421 rad (A), radical ideals generated by finite Ifin subsets of ring A, 422 I(R), set of ideals of ring R, 74

Symbol Index I(P), set of ideals of poset P, 588 i  P, ideal in a poset, 588 i(a), principal ideal generated by a in a lattice, 83  inf(Q) = Q, infimum of Q, 585 IntAlg(L), interval algebra of L, 98 int X (Y), interior of Y in space X, 121 ιp , localizing homomorphism at p, 436 ι S , canonical homomorphism to a ring of fractions, 435 k(α), real closure of field κ(α), 515 κ(α), ordered field qf(A/supp(α), ≤α ), for α ∈ Sper(A), 490 κ(v), valued field determined by valuation v of a ring, 569 Kdim(A), Krull dimension of ring A, 114 Kdim(X), Krull dimension of space X, 113 Ker F , transfer map HomBDLat (L, 2) → PrimF(L), 89 Ker I , transfer map HomBDLat (L, 2) → PrimI(L), 89 K(X), set of compact elements in a poset X, 207 ◦ K (X), set of quasi-compact opens of X, 3 ◦

K (τ), set of quasi-compact opens (in topology τ), 3 ◦

◦

K ( f ), restriction of P( f ) to K (·), 11 K(X), set of constructible sets of X (clopens of Xcon ), 16 K( f ), map K(Y) → K(X) induced by f : X → Y, 21 K(X), set of closed constructible sets of X, 16 K( f ), map K(Y) → K(X) induced by f : X → Y, 21 K( f ), map K(Y) → K(X) induced by f : x → Y, 25 L(A), denotes either Qmod(A), Preord(A), or Satpre(A), 500 λ A , reticulation map of ring A, 429 λf , homomorphism L → 2 defined by prime filter f, 89 λi , homomorphism L → 2 defined by prime ideal i, 90 Λ, empty word, 240 ◦

Λ X , Stone representation, 85 Λ X , Stone representation, 85 Lb(Q), set of lower bounds of Q, 581 L (C), language L extended by new constants, 542 L/f, factor lattice modulo filter f ⊆ L, 159

615

L/i, factor lattice modulo ideal i ⊆ L, 159 Δ, closure of set Δ of sentences under conjunction and disjunction, 542 (S), length of word s, 240 lim Xi , projective limit of Xi , 63 ←−− LocCl(X), set of locally closed points of X, 135 Lor , language for ordered rings, 519 (M , α), expansion of structure A determined by homomorphism α, 562 M · A = (M), ideal generated by subset M in ring A, 421 (M) = (M) A = M · A, ideal generated by subset M in ring A, 421 μ(M), multiplicative set generated by M, 421 μs (M), saturated multiplicative set generated by M, 421 N∗ , inclusion of the image of nucleus N in the frame, 302 N ∗ , nucleus N with restricted codomain, 302 N(G), set of normal subgroups of group G, 75 nil(A), nilradical in ring A, 421 N = {1, 2, . . . }, natural numbers, 583 N0 = {0} ∪ N, non-negative integers, 583 N (O), open regularization of O ∈ O(X), 129 n, set {0, 1, . . . , n − 1}, naturally ordered, 583 n, spectral space with elements 0, 1, . . . , n − 1 in natural order, 9 NY , X , nucleus associated with localic subspace Y of X, 319 O(E; L), open elements (of completion E of L), 303 O( f ), restriction of P( f ) to O(·), 11 ω = N0 , smallest infinite ordinal, 584 N (A), closed regularization of A ∈ A(X), 130 O(X), set of open subsets of X, 2 O x (X), filter of open neighborhoods of x ∈ X, 171 Pfin (I), set of finite subsets of I, 63 P( f ), power set dual of f , 11 πα , canonical map A −→ κ(α) for α ∈ Sper(A), 490 πa , canonical homomorphism from a ring to the factor ring modulo a, 435 πf : L → L/f, canonical quotient homomorphism, 159 πi : L → L/i, canonical quotient homomorphism, 159 Preord(A), set of preorders of ring A, 499

616

Symbol Index

PrimF(L), set of prime filters of bounded distributive lattice L, 89 PrimF(ϕ), spectral dual of lattice morphism ϕ, 91 PrimI(L), set of prime ideals of bounded distributive lattice L, 89 PrimI(ϕ), spectral dual of lattice morphism ϕ, 91 P, set of prime numbers, 70 ptbi (Y), set of bi-localic points of Y, 315 ptinv (Y), set of inverse localic points of Y, 315 pt(Y), set of localic points of Y, 315 P(X), power set of X, 2 qf(A), quotient field of domains A, 436 Qmod(A), set of quadratic modules of ring A, 499 q R ,T0 , canonical map X → X/T0 R to T0 -quotient, 170 q R : X → X/R, canonical spectral quotient, 170 q : X → X/E, T0 -reflection of X, 171 R−1 , inversion of relation R, 176 RCF = real closed field, 518 RC(X), set of regular closed sets in topological space X, 130 Rk α (X), elements in a poset with rank ≥ α, 116 Rk(X), rank of a poset or of a spectral space, 116 rk(x) = rk(X ,≤) (x), rank of point x in poset (X, ≤), 116 RO(X), set of regular open sets in space X, 130 r = rX , retraction of normal spectral space X onto X max , 283 R X , localic coreflection map, 404 Satpre(A), set of saturated preorders of ring A, 499 Sd , S with the discrete topology, 415 S Δ (T ), Δ-types of T , 544 Sen(L ), sentences of language L , 542 S( f ), spectral reflection of f : X −→ X, 372 σ(X, ≤) = σ(X) = σ, Scott topology on poset (X, ≤), 212 S∞ , 44 Sloc (Y), set of localic subspaces of Y, 315 Sn (T ), space of n-types of theory T , 548 Sob X , sobrification map of topological space X, 384 Sob(X), sobrification of space X, 384

Sob( f ), sobrification of continuous map f , 385 Spec(L), spectrum of bounded distributive lattice L, 81 Spec p (A), p-adic spectrum of ring A, 572 Spec(ϕ), spectral dual of lattice homomorphism ϕ, 81 Spec(ϕ), map between Zariski spectra induced by a ring homomorphism ϕ, 70, 434 Specre (ϕ), map between real prime ideals induced by a ring homomorphism ϕ, 507 Specre (R), space of real prime ideals of R, 505 Spec(R), Zariski spectrum (or prime spectrum) of ring R, 67 Sper(ϕ), real spectral dual of ring morphism ϕ, 73 Sper(R), real spectrum of ring R, 72, 491 Spez(A), set of specializations of elements of A, 3 Spv(A), valuation spectrum of ring A, 569  (M), radical ideal generated by M, 421 S ∗ , one-point compactification of discrete space S, 44 S(T ), set of types of T , 544 supp, support of (prime) cone, 71, 489 supp(v), support of valuation v, 569  sup(Q) = Q, supremum of Q, 585 S X , reflection map of X, 372 S X , reflection X → L(X), 372 S(X), spectral reflection of X, 372 T0 , 4 T1 , 4 T5 , 45 τ ∂ , dual (= co-compact) topology, 26 τ L (P, ≤), fine lower topology on poset (P, ≤), 589 τ U (P, ≤), fine lower topology on poset (P, ≤), 589 τ  (P, ≤), coarse lower topology on poset (P, ≤), 589 τ u (P, ≤), coarse upper topology on poset (P, ≤), 589 τ≤ , spectral topology with specialization ≤, 31 TD -space, all points are locally closed, 135 Th(M), first-order theory of structure M, 542 T |= ϕ, 542 tp M ( a), ¯ 547 T-Specex (A), space of expansions of A, 555 T-Spec(A), T -spectrum of structure A, 562 T-Spec(T0 ), 573 Ub(Q), set of upper bounds of Q, 581 U(G), set of subgroups of group G, 75

Symbol Index Up(X), family of up-sets of (X, ≤), 75 V (·), basic closed sets of Zariski topology, 69 V (·) = VLF (·), basic closed sets of PrimF(L), 89 V (·), VLI (·), basic closed sets of PrimI(L), 90

617

V (δ), subbasic closed sets for the space of Δ-types, 544 Z(X), lattice of zero sets in topological space X, 287

Subject Index Boldface page numbers indicate definitions or explanations

A ,

B ,

, 

,  b-topology, 138, 257, 374 Baire category theorem, 126 Baire property, 126 basic constructible set, 17 Bézout domain, 531 bi-frame, 588 bi-frame homomorphism, 588 bi-localic map, 312 point, 315 space, 307, 358 subspace, 315 Boolean envelope of bounded distributive lattice, 93 ring, 70, 432 Zariski spectrum, 71 Boolean algebra, 432 = Boolean ring, 70 atom, 244 atomic, 244 atomless, 244 super atomic, 115 Boolean space, 14, 30 = patch space, 19 = spectral + T1 , 20 = spectral Hausdorff, 15 connected components of spectral space, 201 quasi-compact open = clopen, 14 with total spectral order, 41 Boolean spaces category, 16 bottom element, 581 bounded

abstract real spectrum, 538 ACC = ascending chain condition, 248, 583 adjunction between Spec and BoolSp, co-unit, 23 adjunction space of spectral spaces, 352, 464 adjunction space of topological spaces, 344 affine spring, 477 Alexandroff topology = fine lower topology, 589 algebra closure, 274 co-Heyting, 270 finitely generated, 422 finitely presentable, 422 Heyting, 269, 301 homomorphism, 422 interval, 98 Post, 538 ring as, 422 structure map, 422 algebraic frame, 325 has spectral coarse lower topology, 227 is spatial (i.e., is a topology), 326 algebraic lattice, 226 from spectral tree, 296 has spectral coarse lower topology, 226 algebraic set, 448 amalgamation field, 440 anti-chain, 579 anti-well-founded poset, 583 ascending chain condition = ACC, 583 atom in a Boolean algebra, 244 atomic Boolean algebra, 244 atomless Boolean algebra, 244

618

Subject Index from above, 581 from below, 581 lattice, 586 lattice homomorphism, 586 set in a poset, 581 sublattice, 2, 6 bounded distributive lattice, 586 Boolean envelope, 93 completion, 302 with designated subset, 334 Brouwerian lattice, 269 , 

C , ,  canonical extension of a bounded distributive lattice, 303, 360, 410, 412 canonical homomorphism to a ring of fractions, 435 Cantor space, generalized, 324 Cantor–Bendixson derivative of a topological space, 114 Cantor–Bendixson rank in a Noetherian spectral space, 260 of a point in a space, 114 of a space, 114 cartesian square, 61 categories, list of, 607 category bi-localic spaces, 313 co-well-powered, 149 inverse localic spaces, 313 localic spaces, 313 well-powered, 149 category of quasi-ordered sets, 580 category Spec, 11 co-complete, 150 co-well-powered, 149 complete, 62 has co-separator, 13 has coproducts, 329 has fiber products, 61 has final object, 12 has initial object, 12 has products, 52 has separator, 13 well-powered, 149 ceiling, 581 finite, 581 chain = totally ordered set, 579 patch closure in a spectral space, 109 clopen set, 14

619

closed generically, 3 under generalization, 3 under specialization, 3 closed constructible Stone representation, 85 closed constructible set, 16 in spectrum of bounded distributive lattice, 82 in Zariski spectrum of ring, 69 Stone representation, 87 closed element in a completion, 303 closed embedding, 156 closed partial order, 31, 265 closed point, 581 closed regularization, 130 in semi-Heyting space, 268 closed sets characterization in spectral space, 106 coarse upper topology, 212 Vietoris hit-topology, 212 closure algebra, 274 homomorphism, 275 closure operator, 174 co-cartesian square, 151 co-compact topology, 26 co-epicomplete object, 412 co-Heyting algebra, 270 co-initial subset, 582 co-separator in a category, 13 co-unit of adjunction between Spec and BoolSp, 23 co-well-powered category, 149 coarse lower topology, 42, 209, 589 for a totally ordered set, 37 is spectral on algebraic lattices, 226 of a Noetherian spectral space, 251 on a frame, 227 on a topology, 212 spectral and Noetherian, 259 coarse upper topology, 387, 589 on closed sets of a space, 212 cofinal subset, 582 cofinite topology, 35, 210 coherent frame, 325 coherent topological space, 10 Cohn’s matrix spectrum, 77 compact completion of a bounded distributive lattice, 303 compact element, 207 in a forest, 295

620

Subject Index

in a frame, 325 in root system or forest, 295 compact space, 3 spectral reflection, 398 compactly based space, 99 Compactness Theorem of model theory, 542 compatible family of subsets, 356 complete poset, 38, 586 complete theory, 542 completely normal topological space, 45 prime ideals, 77 regular topological space, 122, 287, 291 spectral reflection, 398 completion Dedekind–MacNeille, 302 completion of a bounded distributive lattice, 302 canonical extension, 303 compact, 303 dense, 303 complexity of an element in a bounded distributive lattice, 335, 455, 463 component irreducible, 251 concatenation of words, 240 conditional ∨-semilattice = conditional join-semilattice, 585 conditional ∧-semilattice = conditional meet-semilattice, 585 conditional join-semilattice is fine coherent, 222 conditional join-semilattice = conditional ∨-semilattice, 585 conditional meet-semilattice = conditional ∧-semilattice, 585 conditionally complete, 586 cone, 489 prime, 71, 489 support of, 489 congruence lattice, modulo filter, 159 lattice, modulo ideal, 159 on lattice, vs. spectral subspace, 157 congruence on bounded distributive lattice, 76, 159 congruences lattice of, 76 connected components in a spectral space, 285 constructible set, 16 basic, 17

characterization, 18 in spectrum of bounded distributive lattice, 82 constructible topology, 16 constructibly closed set, 16 constructibly open set, 16 constructions T0 -quotient modulo a relation, 170 adjunction space of spectral spaces, 352 adjunction spaces of topological spaces, 344 coarsening the topology on a spectral subspace, 362 contracting a spectral subspace to a point, 361 fiber product (= pull-back), 61 gluing two spectral spaces along a common closed subspace, 362 gluing two spectral spaces along a common quasi-compact open subspace, 361 product of spectral spaces, 52 refining the spectral topology on a subspace, 365 spectral quotient modulo a relation, 170 topological (direct) sum = coproduct, 65 wedge of spectral spaces, 42, 353, 364 continuous map identifying, 169 proper, 155 convex, 509  -subgroup of an -group, 299 hull of a subset in a poset, 583 ideal, 501 in a poset, 582 map, 509 subset in a T0 -space, 582 coproduct = direct sum of spectral spaces, 329, 354 of topological spaces, 65 coreflection map from patch space to its spectral space, 23 coreflective subcategory BoolSp in Spec, 23 coreflector Spec −→ BoolSp, 23 cozero set of a continuous function, 287 , 

D , ,  DCC = descending chain condition, 248, 583 dcpo (= directed complete partial order), 587 homomorphism, 587 quasi-algebraic, 218

Subject Index quasi-continuous, 218 sub-, 587 de Groot dual of a topological space, 26 de Groot duality, 47 Dedekind–MacNeille completion, 98, 302 Dedekind-complete, 586 Δ-type of T , 544 dense completion of a bounded distributive lattice, 303 descending chain condition = DCC, 583 designated subset in a bounded distributive lattice, 334 diagonal map, 59 diagonal of a set, 59 diagram category, 328 of a structure, 555 pull-back, 61 push-out, 151 dimension Krull, of a ring, 114 Krull, of a spectral space, 113 direct limit = direct limit cone, 358 direct sum (= copoduct) of spectral spaces, 65 of topological spaces, 65 direct system, 358 directed complete partial order (= dcpo), 587 directed set, 582 patch closure in a spectral space, 109 suprema and infima in a spectral space, 109 distributive lattice, 586 distributive semilattice, 99 divisibility relation of a valued field, 570 of a valued ring, 570 dominant map, 163 domination relation between local rings, 558 down-directed set is irreducible, 108 down-directed subset of a poset, 582 down-set generated by a subset Q in a poset, 582 in a poset, 581 principal, 581 dual space of stably compact space, 47 dual topology, 26, 46, 212 vs. inverse topology, 26 , 

621

E , ,  element in lattice join-irreducible, 586 meet-irreducible, 586 elementary embedding of L -structures, 543 embedding of real closed fields, 520 extension of an L -structure, 543 substructure of anL -structure, 543 elimination of quantifiers (for RCFs), 520 embedding, 156 closed, 156 open, 156 embedding of L -structures, 543 empty word, 240 enough points in a localic space, 324, 374, 382, 388 envelope Boolean of bounded distributive lattice, 93 epimorphism in a category, 145 extremal, 189 epireflective subcategory, 377 epireflector, 377 equivalence relation proconstructible, 134 equivalent L0 -homomorphisms, 561 Esakia duality, 278 Esakia space, 278 examples of spectra also see Index of Examples, 608 algebraic lattice with coarse lower topology, 226 congruence relations on a bounded distributive lattice, 76 equivalence relations on a set, 75 expansions of a structure, 555 ideals in a ring, 74 local subrings of a field, 557 preorders of a ring, 499 quadratic modules of a ring, 499 real spectrum of a ring, 71 saturated preorders of a ring, 499 space of Δ-types of a theory, 544 space of ϕ-types, 548 space of n-types, 548 space of types of a theory, 544 (normal) subgroups of a group, 75 subsets of a set, 74 T -spectrum of a structure, 562 up-sets in a poset, 75

622

Subject Index

Zariski spectrum of a ring, 67 expansion of a structure, 541 extended prime spectrum, 77 extension of a language, 541 extension of spectral maps, 166 extremal epi-mono factorization, 191 epimorphism, 189 monomorphism, 156 extremally disconnected Boolean space not enough points, 325 extremally disconnected space, 306, 308, 311, 414 , 

F , ,  fan, 538 fcpo = filtered complete partial order, 587 fiber of real spectral morphism, 515 fiber product, 60 specialization, 63 fiber sum, 151 field amalgamation, 440, 461 real closed, 518 open quantifier elimination, 521 quantifier elimination, 520 real closure of ordered, 518 filter prime, in a poset, 589 principal, 83 filter in a poset, 588 filtered complete partial order = fcpo, 587 final topology, 169 fine coherent poset, 222, 259 fine lower topology, 360, 589 on a totally ordered set, 40 fine upper topology, 589 finite ceiling, 581 floor, 581 ordinal, 584 word, 240 finite intersection property (FIP), 17 finitely generated algebra, 422 element, 228 ideal in a ring, 422 preorder, 494 finitely presentable algebra, 422

FIP, finite intersection property, 17 first-order axiomatizable, 519 floor, 581 finite, 581 forest, 584 fine coherent, 222 spectral, 290 frame, 270, 300, 588 algebraic, 227, 325 bi-, 588 coherent, 325 enough points, 324 homomorphism, 588 inverse, 588 law, 588 nucleus, 130, 301 point, 322 spatial, 324 with spectral coarse lower topology, 227 frame homomorphism, 270 free functor, 413 free object, 413 free spectral space, 415 functors, list of, 607 , 

G , ,  G-domain, 442 generalization, 3 immediate, 580 proper, 580 generalized Cantor space, 324 generalized-algebraic poset, 218 generic point, 7 generic point in a space, 581 generically closed, 3 Gleason cover, 399 gluing along a closed set, 362 along a quasi-compact open set, 361 going-down property, 152 going-up property, 152 Goldman point, 135 graph components, 582 in normal spectral space, 283 in spectral root system or forest, 295 vs. connected components, 203 graph of an order relation, 579 group lattice-ordered, 531 , 

Subject Index

H , ,  Harrison topology on real spectrum of a ring, 72 height of an ideal in a ring, 452 hereditarily disconnected space (= totally disconnected), 14 hereditarily normal space, 292 Heyting subspace, 278 Heyting algebra, 269, 301 Heyting algebras homomorphism, 269 Heyting map, 275 Heyting space, 267, 310 Hilbert ring, 447 homomorphism of L -structures, 543 of algebras, 422 of bi-frames, 588 of closure algebras, 275 of frames, 588 of Heyting algebras, 269 of inverse frames, 588 of pseudo-complemented lattices, 269 hull–kernel topology, 424 , 

I , ,  ideal N -closed ideal generated by a subset of a frame, 320 N -closed ideal in a frame, 320 prime, in a poset, 589 vanishing, 526 ideal generated by an up-directed set, 588 ideal in a join-semilattice of open sets, 5 ideal in a poset, 588 ideal of a ring, 421 convex, 501 real, 489 ideal quotient in a ring, 439 idempotent reflector, 377 identifying continuous map, 169 identifying spectral map, 188 in terms of lattices, 198 presentation of colimits, 355 image of proconstructible subset under spectral map, 143 immediate successor, 580

623

generalization, 580 predecessor, 580 specialization, 580 implication intuitionistic, 269 index extension of a spring with index, 481 index of a spring, 480 infimum (= meet) of Q, 585 infinite ordinal, 584 infinite word, 240 initial segment of a word, 240 injective object, 412 integral algebra, 442 integral element, 442 interior of a set, 121 intermediate value property, 518 interval algebra, 98 interval topology, 589 intuitionistic implication, 269 intuitionistic propositional logic, 274 inverse lattice, 586 frame, 588 frame homomorphism, 588 localic map, 312 localic point, 315 localic space, 307 inverse localic points are locally closed, 327 localic subspace, 315 order relation, 580 poset, 580 space, 23 of a Noetherian spectral space, 251 topology, 23 of product space, 52 inversely closed set, 23 characterization, 106 open set, 23 quasi-compact set vs. closure, 104 irreducible components, 251 irreducible map, 163 irreducible set characterization via ideals in O(X), 6 if specialization total order, 36 in a topological space, 4 in topological sum, 65 need not be down-directed, 108

624

Subject Index

isomorphism over A between morphisms in a category, 76 , 

J , ,  Jacobson ring, 447 join (= supremum), of Q, 585 join-irreducible element in lattice, 586 join-semilattice = ∨-semilattice, 585 jump in a poset, 580 lower element, 580 upper element, 580 jump-dense spectral space, 111 totally ordered set, 38 , 

K , ,  Kaplansky Problem, 112 Noetherian version, 258 Kolmogorov space, 4 Krull dimension of a ring, 114 of a spectral space, 113 Kuratowski identities, 274 , 

L , ,  lattice, 586 algebraic, 226 bounded, 586 Brouwerian, 269 co-Heyting algebra, 270 Dedekind–MacNeille completion, 302 distributive, 586 filter, principal, 83 ideal, principal, 83 inverse, 586 normal, 280 of congruences of bounded distributive lattice is distributive, 157 prime filter spectrum, 89 prime ideal spectrum, 89 prime ideals, existence, 83 principal ideal, 83 pseudo-complemented, 269 relatively pseudo-complemented, 269 lattice homomorphism, 586 bounded, 586

lattice of congruences on bounded distributive lattice, 76 lattice-ordered group, 299, 531 Lawson topology, 220 length of a word, 240 -group = lattice-ordered group, 531 -group=lattice-ordered group, 299 lifting property for spectral maps, 193 limit direct, 358 ordinal, 584 projective, 62 local ring, 436 local spectral space, 333 locale, 299 localic coreflection map, 404 coreflection of spectral map, 404 coreflection of spectral space, 404 coreflector, 404 map, 312 point, 315 in spectral reflection, 382 subspace, 315 associated nucleus, 319 localic space, 307 enough points, 324, 382 inverse, 307 spatial, 324 spectral reflection, 374 localic spaces colimits computed in Spec, 357 localization of A at p, 436 localizing map at p, 436 locally closed point, 135, 389 in a spectral root system, 295 in Noetherian spectral spaces, 257, 327 in non-Noetherian spectral space, 327 in spectral reflection, 374 in spectral root system or forest, 295 smallest very dense subset, 139 vs. localic point, 324 locally closed set in a topological space, 132 lower bound, 581 lower element of a jump, 580 lower topology, 589 , 

M , ,  Macintyre language for Q p , 572 map

Subject Index R-compatible, 169 bi-localic, 312 convex, 509 diagonal, 59 dominant, 163 Heyting, 275 inverse localic, 312 irreducible, 163 localic, 312 Priestley, 33 semi-Heyting, 275 spectral, see spectral map, 11 maximal ideal in a ring, 427 maximal point in a space, 581 maximal points exist in quasi-compact space, 103 meet (= infimum) of Q, 585 meet-irreducible element in lattice, 586 meet-semilattice = ∧-semilattice, 585 minimal point in a space, 581 minimal points exist in spectral spaces, 105 minimal prime ideal in a ring, 427 minimal upper bound property, 105 minspectral space, 122 characterization, 124 model of a set of sentences, 542 model-completeness, 520 monomorphism extremal, 156 in a category, 145 monoreflective subcategory, 377 monoreflector, 377 monotone map, 580 morphism of indexed springs, 483 morphism over A between morphisms in a category, 76 mub-property, 105 multiplicative set saturated, 421 multiplicative set in a ring, 421 , 

N , ,  Nachbin duality, 47 natural nucleus of a localic space, 309 nilpotent element in a ring, 421 nilradical, 421, 439 Noetherian inverse localic space, inverse localic point = locally closed point, 327 Noetherian ring, 450

625

Noetherian spectral space, 249, 379 characterization by localic points, 324 characterization of, 251, 255, 317 countable, 263 has enough points, 325 is localic, 308 localic points vs. locally closed points, 327 Noetherian topological space, 248, 381 characterization by sobrification, 389 normal bounded distributive lattice, 280 spectral space, 281 space of closed points Boolean, 286 topological space, 280 characterization via spectral reflection, 397 nucleus associated with localic subspace, 319 natural, of a localic space, 309 of a frame, 130, 301 Nullstellensatz, 425 , 

O , ,  one-point compactification, 14 open element in a completion, 303 open embedding, 156 open quantifier elimination for RCFs, 521 open regularization, 129, 302 in a localic space, 322 in semi-Heyting space, 268 vs. pseudo-complementation, 271 order quotient, 176 specialization, 3 spectral, 30 order relation graph, 579 ordered field real closure, 518 ordered ring language for, 519 ordered topological space, 46 ordinal, 583 finite, 584 infinite, 584 limit, 584 successor, 584 ordinal number = ordinal, 583 ordinals space of, 379

626

Subject Index

P , ,  p-adic spectrum of a ring, 572 partial order, 579 partial order (in ring), 489 partially ordered set = poset, 579 partition lattice of a set, 75, 564 patch = proconstructible set, 16 patch open set, 16 patch space (of a spectral space), 16 patch topology = constructible topology in a spectral space, 16 of a T0 -space, 46 of a spectral space, 16 of product space, 52 perfect map, 47 perfect topological space, 114 point bi-localic, 315 Cantor–Bendixson rank, 114 Goldman, 135 inverse localic, 315 localic, 315 locally closed in a topological space, 135 maximal = closed, 581 minimal = generic, 581 rational, 526 spectral, 315 point of a frame, 322 polynomial function on an algebraic set, 448 poset = partially ordered set, 579 also see Appendix: The Poset Zoo, 579 anti-chain, 579 anti-well-founded, 583 ceiling, 581 compact element of, 207 complete, 586 conditionally complete, 586 convex hull of a subset, 583 convex subset, 582 Dedekind-complete, 586 down-directed subset, 582 filter, 588 fine coherent, 222 finite ceiling, 581 finite floor, 581 floor, 581 forest, 584 generalized-algebraic, 218

ideal, 588 inverse, 580 rank, 116 root system, 584 totally ordered = chain, 579 up-directed subset, 582 well-founded, 583 positive diagram of A, 562 Post algebra, 538 predecessor, 580 immediate, 580 prefix of a word, 240 prefix order on words, 241 preorder, 489 finitely generated, 494 proper, 499 saturated, 495 total, 489 preorder generated, 490 preordered ring, 489 preservation of formulas, 543 Priestley map, 33 Separation Axiom, 30 space, 30 prime -ideal, 531 z-ideal, 128 cone, 489 prime cone, 71 prime filter in a poset, 589 prime filter spectrum of a bounded distributive lattice, 89, 95 prime ideal in a poset, 589 prime ideal in a ring, 421 support, 71 prime ideal spectrum of a bounded distributive lattice, 89, 95 prime ideals in lattices existence, 83 prime spectrum of a ring, 67 = Zariski spectrum, 67 principal down-set, 581 set, 81 up-set, 581 principal closed subset in Zariski spectrum, 423 principal filter, 83 principal ideal, 83 principal open subset in Zariski spectrum, 423

Subject Index proconstructible equivalence relation, 134 proconstructible set, 16 is spectral subspace, 50 projective limit, 62 identifying spectral maps, 192 object, 412 system, 62 proper generalization, 580 proper ideal in a ring, 421 proper map, 155 proper specialization, 580 propositional logic intuitionistic, 274 pseudo-complement, 269 pseudo-complement of an element in a Heyting algebra, 269 pseudo-complementation vs. open regularization, 271 pseudo-complemented lattice, 269 homomorphism, 269 pseudo-field, 503 pull-back, 60 pull-back diagram, 61 push-out, 151 push-out diagram, 151, 460 identifying spectral maps, 192 , 

Q , ,  q-module, 498 proper, 499 quadratic module, 498 proper, 499 quantifier elimination for RCFs, 520 quasi-algebraic dcpo, 218 quasi-compact, 3 quasi-compact open, 3 in a Boolean space, 14 in definition of spectral maps, 11 in definition of spectral spaces, 4 in product space, 52 in real spectrum of ring, 72 in spectrum of bounded distributive lattice, 82 in Zariski spectrum of ring, 69 Stone representation, 84 vs. closed and constructible sets, 19 quasi-compact open sets for inverse topology, 24 vs. basis of the topology, 5

627

quasi-compact set characterization in T0 -space, 103 vs. inverse closure, 105 quasi-compact space spectral reflection, 395 quasi-compact saturated set, 9, 26, 46 quasi-continuous dcpo, 218 quasi-homeomorphism, 139, 347, 373, 385, 389, 391 quasi-order, 579 quasi-ordered set, 579 quasi-ordered sets category, 580 quotient T0 -, modulo a relation, 170 spectral, modulo a relation, 170 quotient field of a domain, 436 quotient order, 176 quotient topology, 169 , 

R , ,  radical ideal in a ring, 421 rank Cantor–Bendixson in a Noetherian spectral space, 260 Cantor–Bendixson of a point, 114 Cantor–Bendixson of a space, 114 in a Noetherian spectral space, 260 of a poset, 116 of a spectral space, 116 of point x in a poset, 116 rational point, 526 real Chevalley Theorem, 528 ideal, 489 ring, 489 spectral morphism, 505 convexity, 510 fiber, 515 spectrum, 491 real closed field, 518 open quantifier elimination, 521 quantifier elimination, 520 transfer principle, 520 real closed ring, 534 real closure of a ring, 534 of ordered field, 518 real semigroup, 536 associated with semi-real ring, 536

628

Subject Index

spectral, 537 real spectral morphism, 505 convexity, 510 real spectrum, 72 abstract, 538 is spectral space, 72 specialization, 73, 501 realization of a type, 548 reduced ring, 439 reflection T0 -, 171, 341 reflection map bounded distributive lattice to Boolean envelope, 94 topological space to spectral reflection, 372 reflective subcategory, 57 BoolAlg in BDLat, 94 Spec in Top, 375 Boolean spaces in spectral spaces, 202 reflector BDLat to BoolAlg, 94 Top to Spec, 375 idempotent, 377 regular closed set, 129 open set, 129 topological space, 280 regularization closed, in a topological space, 130 open, in a topological space, 129 relation inverse order, 580 partial order, 579 quasi-order, 579 saturated, 174 total order, 579 relative pseudo-complementation, 269 relatively pseudo-complemented lattice, 269 reticulation functor, 431 reticulation of a ring, 429, 463 retraction spectral, 158 ring = commutative, unital ring, 67 Boolean, 70 Krull dimension, 114 partial order, 489 preordered, 489 real, 489 semi-real, 498 von Neumann regular, 70

von Neumann regular, Zariski spectrum, 70 ring homomorphism cofinal, 512 ring of fractions, 435 root, 584 root in a root system, 584 root system, 584 fine coherence, 222 spectral, 290 Rudin’s Lemma, 217, 249 , 

S , ,  saturated multiplicative set, 421 preorder, 495 relation, 174 set in a topological space, 9 saturation of a relation on a spectral space, 174 scattered Boolean space closed partial orders, 265 scattered compact countable space, 263 scattered space, 115, 263 Scott topology, 212 section spectral, 158 semi-algebraic, 520 semi-Heyting map, 275 semi-Heyting space, 267 semi-Heyting subspace, 276 semi-real, 498 semi-spectral, 99 semigroup real, 536 associated with semi-real ring, 536 ternary, 537 semilattice, 585 distributive, 99 sentence of a language, 542 separates points, 4 separating for closed sets, 336, 431 separation axiom T0 , 4 T1 , 4 T5 , 45 TD , 135 Priestley, 30 Separation Axioms normal, 280 regular, 280 Separation Lemma, 29

Subject Index separation of disjoint sets, 106 separator in a category, 13 set basic constructible, 17 clopen, 14 closed constructible, 16 in spectrum of bounded distributive lattice, 82 in Zariski spectrum of ring, 69 Stone representation, 87 closed constructible, Stone representation, 85 constructible, 16 in spectrum of bounded distributive lattice, 82 constructibly closed, 16 constructibly open, 16 diagonal, 59 interior of, 121 inversely closed, 23 inversely open, 23 irreducible characterization via ideals in O(X), 6 if specialization total order, 36 in topological space, 4 locally closed in a topological space, 132 partially ordered = poset, 579 patch open, 16 principal, 81 proconstructible, 16 quasi-compact, 3 quasi-compact open, 3 in definition of spectral maps, 11 in definition of spectral spaces, 4 in product space, 52 in real spectrum of ring, 72 in spectrum of bounded distributive lattice, 82 in Zariski spectrum of ring, 69 vs. closed and constructible sets, 19 quasi-compact open, Stone representation, 84 quasi-ordered, 579 well-ordered, 583 Sierpiński space, 8 as co-separator, 13 powers of, 54 simple index on a spring, 480 sober space, 4 points vs. closed irreducible sets, 7 sobrification map, 384

629

sobrification of continuous map, 385 sobrification of topological space, 384 space T0 , 4 T1 , 4 TD , 135 Boolean, 14 = patch space, 19 category, 16 Boolean = (spectral + T1 ), 20 Boolean = spectral Hausdorff, 15 Esakia, 278 hereditarily normal, 292 inverse, 23 Kolmogorov, 4 of Δ-types of a theory, 544 of ϕ-types, 548 of ordinals, 40, 115, 379, 455 of signs, 538 patch, see spectral space, patch, 16 preserving functor, 473 Priestley, 30 sober, 4 sober, points vs. closed irreducible sets, 7 spectral, see spectral space, 4 stably compact, 98 totally order-disconnected, 30 ultrametric, 125 with indeterminates, 473 category of, 473 spatial frame, 324 spatial localic space, 324 Spec has a section over C, 469 specialization, 3 -closed, 3 characterization if total order, 36 characterization of closed and constructible sets, 29 characterization of closed sets, 29 characterization of quasi-compact open sets, 29 fiber product, 63 generic point, 7 immediate, 580 in inverse space, 24 in prime filter spectrum, 90 in prime ideal spectrum, 90 in product space, 52 in real spectrum of ring, 73 in spectrum of bounded distributive lattice, 81

630

Subject Index

in Zariski spectrum of ring, 69 is spectral order on Xcon , 31 order, 3 partial order in T0 -spaces, 4 proper, 580 quasi-order, 3 set, 579 specialization chain, 103 in spectral space, properties, 111 maximal exists, 103 maximal is proconstructible, 111 specialization order, 579 graph of, 59 property (H), 112 property (H+), 112 spectral coproduct of Boolean spaces, 331 spectral forest, 290 spectral map, 11 characterization via specialization, 33 characterizations, 20, 25 closed embedding, 156 closed, characterization, 153 embedding, 156 embedding = extremal monomorphism, 157 epimorphism, characterization, 148 extensions, existence and uniqueness, 166 extremal epi-mono factorization, 191 going-down property, 152 going-up property, 152 graph of, 60 Heyting, 275 identifying, 188 identifying, characterizations, 189 images and preimages of proconstructible sets, 21 images of proconstructible subsets, 143 images, preimages of spectral subspaces, 51 inversely closed, characterization, 153 lifting property, 193 localic coreflection, 404 monomorphism, characterization, 146 monotone for specialization, 33 open embedding, 156 open, characterization, 154 retraction, 158 section, 158 semi-Heyting, 275 vs. continuous map, 12, 15 with finite domain, 12 yields lattice homomorphism, 11

spectral maps product, 54 spectral morphism real, 505 convexity, 510 fiber, 515 spectral order, 30, 181, 183, 185 total, 41 spectral point, 315 spectral quotient exists, 172 modulo a relation, 170 presentation of colimits, 355 unique, 170 spectral real semigroup, 537 spectral reflection is normal, 398 map of X, 372 of a continuous map, 372 of a topological space, 372 spectral reflector, 372 spectral relation, 183 spectral retraction, 158 spectral root system, 290, 400 spectral reflection, 400 spectral section, 158 spectral space, 4 as projective limit of finite spaces, 63 as subspace of powers of Sierpiński space, 54 axioms (S1–S4), 4 bi-localic, 307 Boolean, 15 category, see category Spec, 11 characterization via quasi-compact open sets, 7 constructible subsets, characterization, 18 dense subsets vs. minimal points, 122 finite = finite T0 -space, 8 finite vs. finite poset, 8 free, 415 Heyting, 267 interior of subsets vs. minimal points, 122 inverse localic, 307 inverse space, 23 inverse topology, 23 inverse topology on maximal points, 123 is dcpo, 110 is Priestley space, 34 Krull dimension, 113 localic, 307

Subject Index localic coreflection, 404 maximal points inversely compact, characterization, 128 maximal points, patch closure, 128 minimal points 0-dimensional, 123 minimal points compact, 127 minimal points, constructible closure, 129 not Dedekind-complete, 110 patch (= constructible) topology, 16 patch space, 16 patch space is Boolean, 17 rank, 116 semi-Heyting, 267 Stone representation, 84, 85, 87 subspace topology on minimal points, 122 total specialization order, 39 spectral spaces coproduct, 329 direct sum, 65 product, 52 wedge construction, 42 spectral subspace, 49 characterization, 50 vs. congruence on lattice, 157 spectral topology, 4 on wedge, 42 on well-ordered set, 40 spectral tree contained in algebraic lattice, 296 spectrum p-adic, of a ring, 572 normal subgroup, 75 of T -expansions of A, 555 of a Boolean ring, 70 of a bounded distributive lattice, 81 of a non-commutative ring, 77 of equivalence relations on a set, 75 of ideals in a ring, 74 of preorders, 499 of quadratic modules, 499 of saturated preorders, 499 of subsets, 74 of valuations, 569 prime filters of lattice, 89 prime ideals of a ring, 67 prime ideals of lattice, 89 prime, of a ring, 67 real, 72, 491 subgroup, 75 Zariski, 67 spring, 476

631

spring morphism, 482 springs category of, 483 category of indexed, 483 stably compact space, 46, 98 Stone space, 14 Stone–Čech compactification, 123, 287, 399 of Boolean spaces, 331 strong order unit, 532 strongly isomorphism-closed subcategory, 377 structure map of an algebra, 422 sub-dcpo, 587 sub-frame, 301 sub-frame generated by a subset of a frame, 301 sub-poset, 580 subcategory epi-coreflective, 403 epireflective, 377 monoreflective, 377 reflective, 94 strongly isomorphism-closed, 377 sublattice bounded, 2, 6 subset strongly dense, 138 very dense in a topological space, 138 subsets with closed neighborhood basis, 107 subspace bi-localic, 315 Heyting, 278 inverse localic, 315 localic, 315 semi-Heyting, 276 spectral, 49 spectral, characterization, 50 subvariety of R n , R a field, 450 successor, 580 immediate, 580 successor ordinal, 584 super atomic Boolean algebra, 115 super quasi-compact set, 235 support of a cone, 489 of a prime cone, 71 of a valuation, 569 supremum (= join) of Q, 585 , 

T , ,  T -spectrum of a structure, 562

632

Subject Index

T0 -quotient existence, 172 modulo a relation, 170 uniqueness, 170 T0 -reflection, 171 of a topological space, 395 T0 -space, 4 spectral reflection, 394 T1 -space, 4 spectral reflection, 395 TD -space, 135, 138 Tarski–Lindenbaum algebra, 553 ternary semigroup, 537 Theorems Alexander Subbasis Theorem, 112 Baire category, 126 Krull’s Height Theorem, 452 open quantifier elimination for RCFs, 521 Principal Ideal Theorem, 452 quantifier elimination (for RCFs), 520 Real Chevalley, 528 Rudin’s Lemma, 217 Separation Lemma, 29 Tarski, 520 Tychonoff’s Product Theorem, 52 theory L -theory, 542 complete, 542 of a structure, 542 Thomason set, 26 tilde map = tilde operation, 525, 528 analytic, 536 top element, 581 topological space 0-dimensional, 122 Boolean, 14 Cantor–Bendixson rank, 114 coherent, 10 completely normal, 45 completely regular, 122 minspectral, 122 Noetherian, 248 normal, 280 perfect, 114 regular, 280 scattered, 115 specialization set, 579 Stone (= Boolean), 14 totally disconnected (= hereditarily disconnected), 14 Tychonoff, 122

topological spaces direct sum = coproduct, 65 topological sum, 65 irreducible subset, 65 topology Alexandroff = fine lower topology, 589 co-compact = dual, 26 coarse lower, 589 coarse upper, 589 cofinite, 35, 210 constructible, see constructible topology, 16 dual, 46 final, 169 fine lower, 40, 589 fine upper, 589 Harrison on real spectrum of a ring, 72 interval, 589 inverse, 23 of product spaces, 52 Lawson, 220 lower, 589 one-point compactification, 14 patch of product space, 52 patch, see constructible topology, 16 quotient, 169 Scott, 212 spectral, 4 upper, 589 Zariski, 67 total order, 579 total preorder, 489 totally disconnected space, 14 totally order-disconnected space, 30 totally ordered set = chain, 579 prime ideal spectrum, 266 transfer principle for RCFs, 520 tree, 585 tree in a forest, 584 Tychonoff plank, 292 Tychonoff space, 122 type Δ-type a set of sentences, 544 n-type, 548 of a set of sentences, 544 of a tuple in a model, 547 space of n-types, 548 space of a theory T , 544 , 

Subject Index

U , ,  ultrametric space, 125 underlying L -structure of an expansion, 541 Uniformization Lemma, 28 unit belonging to an adjunction, 370 universal algebra, 557 up-directed subset of a poset, 582 up-set generated by a subset Q in a poset, 582 in a poset, 581 principal, 581 upper bound, 581 upper element of a jump, 580 upper topology, 589 , 

V , ,  valuation of a ring, 569 valuation spectrum of ring A, 569 value of a under a valuation, 569 vanishing ideal, 448, 526 variety of algebras, 100 very dense subset in a topological space, 138, 374, 391 von Neumann regular ring, 70, 428 , 

W , ,  way below, 207 wedge of spectral spaces, 42, 228, 256, 353, 364 weight of a topological space, 261

well-filtered, 46 well-founded poset, 583 well-generated Boolean algebra, 264 well-ordered set, 583 spectral topology, 40 well-powered category, 149 word empty, 240 finite, 240 infinite, 240 intitial segment, 240 of length n, 240 prefix, 240 tree, 241 words concatenation, 240 prefix order, 241 , 

Z , ,  Zariski spectrum, 67 = prime spectrum of a ring, 67 closed constructible set, 69 of Boolean ring, 71 of von Neumann regular ring, 70 principal closed subset, 423 principal open subset, 423 quasi-compact open set, 69 specialization, 69 topology, 69 Zariski topology, 67 zero set of a continuous function, 287 zero-dimensional space, 122 Ziegler spectrum, 77, 100

633