Approximations and Endomorphism Algebras of Modules: Volume 1 – Approximations / Volume 2 – Predictions [2 ed.] 9783110218107, 9783110218114

Citation preview

De Gruyter Expositions in Mathematics 41 Editors Victor P. Maslov, Moscow, Russia Walter D. Neumann, New York, USA Markus J. Pflaum, Boulder, USA Dierk Schleicher, Bremen, Germany Raymond O. Wells, Bremen, Germany

Rüdiger Göbel Jan Trlifaj

Approximations and Endomorphism Algebras of Modules Volume 1 – Approximations

2nd Revised and Extended Edition

De Gruyter

Mathematics Subject Classification 2010: Primary: 1602. Secondary: 03Cxx, 03Exx, 13-XX, 14F05, 16-XX, 18E15, 20Kxx. Keywords: Approximations of Modules, Filtration, Cotorsion Pair, Infinite Dimensional Tilting Theory.

ISBN 978-3-11-021810-7 e-ISBN 978-3-11-021811-4 ISSN 0938-6572 Library of Congress Cataloging-in-Publication Data A CIP catalog record for this book has been applied for at the Library of Congress. Bibliographic information published by the Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available in the internet at http://dnb.dnb.de. © 2012 Walter de Gruyter GmbH & Co. KG, Berlin/Boston Typesetting: PTP-Berlin Protago-TEX-Production GmbH, www.ptp-berlin.eu Printing and binding: Hubert & Co. GmbH & Co. KG, Göttingen Printed on acid-free paper Printed in Germany www.degruyter.com

For our wives Heidi and Kateˇrina and children Ines, and Lucie, Justina, Magdalena, Šimon and Daniel

Contents

Volume 1 Introduction

xvii

List of Symbols

xxv

I Some useful classes of modules 1 S-completions 1.1 Support of elements in By – a first step . . . 1.1.1 Norms . . . . . . . . . . . . . . . . 1.2 Uncountable S in completions . . . . . . . 1.3 Modules of cardinality  2ℵ0 . . . . . . . . 1.3.1 Algebraically independent elements

. . . . .

3 10 11 12 14 14

2 Pure-injective modules 2.1 Direct limits, finitely presented modules and pure submodules . . . 2.2 Characterizations of pure-injective modules . . . . . . . . . . . . .

22 22 37

3 Mittag-Leffler modules 3.1 Characterizations of Mittag-Leffler modules . . . . . . . . . 3.2 Flat Mittag-Leffler modules . . . . . . . . . . . . . . . . . . 3.3 Unions of countable pure chains of pure-projective modules 3.4 Locally projective modules . . . . . . . . . . . . . . . . . . 3.5 Open problems . . . . . . . . . . . . . . . . . . . . . . . .

. . . . .

47 47 60 62 65 79

. . . . . .

80 80 87 91 100 104 111

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

4 Slender modules 4.1 Factors of products and slender modules . . . . . . . . . . . . . . 4.1.1 Radicals commuting with products . . . . . . . . . . . . . 4.1.2 Modules with a linear topology . . . . . . . . . . . . . . 4.1.3 Characterizing slender modules by excluding submodules 4.2 Slender modules over Dedekind domains . . . . . . . . . . . . . . 4.3 Open problems . . . . . . . . . . . . . . . . . . . . . . . . . . .

viii

Contents

II Approximations and cotorsion pairs 5 Approximations of modules 5.1 Preenvelopes and precovers . 5.2 Cotorsion pairs and Tor-pairs 5.3 Minimal approximations . . 5.4 Open problems . . . . . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

115 115 120 126 130

6 Complete cotorsion pairs 6.1 Ext and direct limits . . . . . . . . . . . . . 6.2 The abundance of complete cotorsion pairs . 6.3 Ext and inverse limits . . . . . . . . . . . . 6.4 Open problems . . . . . . . . . . . . . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

131 131 135 143 154

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

7 Hill lemma and its applications 7.1 The general version of the Hill Lemma . . . . . . . . 7.2 Kaplansky theorem for cotorsion pairs . . . . . . . . 7.3 C-socle sequences and Filt.C/-precovers . . . . . . . 7.4 Singular compactness for C-filtered modules . . . . . 7.5 Ascending and descending properties of modules . . 7.6 The rank version of the Hill Lemma . . . . . . . . . 7.7 Matlis cotorsion and strongly flat modules . . . . . . 7.7.1 Strongly flat modules over valuation domains 7.8 Open problems . . . . . . . . . . . . . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

155 156 161 164 170 174 180 181 189 195

8 Deconstruction of the roots of Ext 8.1 Approximations by modules of finite homological dimensions 8.2 Closure properties providing for deconstruction . . . . . . . . 8.2.1 The tilting case . . . . . . . . . . . . . . . . . . . . . 8.2.2 The cotilting case . . . . . . . . . . . . . . . . . . . . 8.3 The closure of a cotorsion pair . . . . . . . . . . . . . . . . .

. . . . .

. . . . .

. . . . .

196 196 205 205 210 215

9 Modules of projective dimension one 9.1 Structure of P1 and WI for semiprime Goldie rings . . . . . . . . . 9.2 The class lim P1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3 Open problems . . . . . . . . . . . . . . . . . . . . . . . . . . . .

221 221 224 227

10 Kaplansky classes and abstract elementary classes 10.1 Kaplansky classes and deconstructibility . . . 10.2 Flat Mittag-Leffler modules revisited . . . . . 10.3 Abstract elementary classes of the roots of Ext 10.4 Open problems . . . . . . . . . . . . . . . .

228 228 230 240 251

. . . .

. . . .

. . . .

. . . .

. . . . . . . . .

. . . .

. . . . . . . . .

. . . .

. . . . . . . . .

. . . .

. . . . . . . . .

. . . .

. . . .

. . . .

. . . .

. . . .

ix

Contents

11 Independence results for cotorsion pairs 253 11.1 Completeness of cotorsion pairs under the Diamond Principle . . . . 253 11.2 Uniformisation and cotorsion pairs not generated by a set . . . . . . 258 11.3 Open problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265 12 The lattice of cotorsion pairs 266 12.1 Ultra-cotorsion-free modules and the Strong Black Box . . . . . . . 266 12.2 Rational cotorsion pairs . . . . . . . . . . . . . . . . . . . . . . . . 275 12.3 Embedding posets into the lattice of cotorsion pairs . . . . . . . . . 283

III

Tilting and cotilting approximations

13 Tilting approximations 13.1 Tilting modules . . . . . . . . . . . . . . . . . . . . 13.2 Classes of finite type . . . . . . . . . . . . . . . . . 13.2.1 Deconstruction to countable type . . . . . . . 13.2.2 Definability and the Mittag-Leffler condition 13.2.3 Finite type and resolving subcategories . . . 13.3 Localisation of tilting modules . . . . . . . . . . . . 13.4 Product-completeness of tilting modules . . . . . . . 13.5 Open problems . . . . . . . . . . . . . . . . . . . .

. . . . . . . .

295 295 309 310 315 320 324 325 329

14 1-tilting modules and their applications 14.1 Tilting torsion classes . . . . . . . . . . . . . . . . . . . . . . . . 14.2 The structure of 1-tilting modules and classes over particular rings 14.2.1 1-tilting classes over artin algebras . . . . . . . . . . . . . 14.2.2 Tilting modules and classes over Prüfer domains . . . . . 14.2.3 The case of valuation and Dedekind domains . . . . . . . 14.3 Baer modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.3.1 Solution to the Kaplansky problem . . . . . . . . . . . . . 14.3.2 Baer modules over hereditary artin algebras . . . . . . . . 14.4 Matlis localisations . . . . . . . . . . . . . . . . . . . . . . . . . 14.5 Open problems . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . .

330 330 333 333 336 345 347 350 351 353 363

15 Cotilting classes 15.1 Cotilting classes and the classes of cofinite type . . . 15.2 1-cotilting modules and cotilting torsion-free classes 15.3 Cotilting over Prüfer domains . . . . . . . . . . . . . 15.4 Ext-rigid systems . . . . . . . . . . . . . . . . . . . 15.5 Open problems . . . . . . . . . . . . . . . . . . . .

. . . . .

364 364 372 376 378 381

. . . . . . . .

. . . . .

. . . . . . . .

. . . . .

. . . . . . . .

. . . . .

. . . . . . . .

. . . . .

. . . . . . . .

. . . . .

. . . . . . . .

. . . . .

. . . . . . . .

. . . . .

x

Contents

16 Tilting and cotilting classes over commutative noetherian rings 16.1 Cotilting classes and characteristic sequences . . . . . . . 16.1.1 The one-dimensional case . . . . . . . . . . . . . 16.1.2 The n-dimensional case . . . . . . . . . . . . . . 16.2 Tilting classes over commutative noetherian rings . . . . . 16.3 Tilting and cotilting modules over 1-Gorenstein rings . . . 16.4 Tor-pairs over hereditary rings . . . . . . . . . . . . . . . 16.5 Open problems . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

382 382 382 386 392 395 396 398

17 Tilting approximations and the finitistic dimension conjectures 17.1 Finitistic dimension conjectures and the tilting module Tf . . . 17.2 A formula for the little finitistic dimension of right artinian rings 17.3 Artinian rings with P 0 . . . . . . . . . . . . . . . . . . . 24.7.3 Preparing the predictions on G1 for the Step Lemma . 24.8 Application of the Strong Black Box . . . . . . . . . . . . . . 24.9 Fully rigid systems of ℵk -free R-modules with a prescribed R-algebra . . . . . . . . . . . . . . . . . . . . . . . . . . 24.10 Open problems . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . .

. . . . . . . . . . . . . . . . .

636 . 636 . 641 . 645 . 646 . 648 . 650 . 651 . 656 . 658

. . . . . . . . . . . . . . . . .

659 659 660 665 668 672 673 675 675 676 677 682 693 695 698 699 699 709

. . . 715 . . . 716

xiii

Contents

VI Modules and rings related to algebraic topology 25 Localisations and cellular covers, the general theory for R-modules 25.1 A sketch of the categorical settings . . . . . . . . . . . . . . . . . 25.1.1 Localisations represented by morphisms . . . . . . . . . . 25.2 Localisations not represented by morphisms . . . . . . . . . . . . 25.3 Constructing localisations of R-modules . . . . . . . . . . . . . . 25.4 Cellular covers of R-modules . . . . . . . . . . . . . . . . . . . . 25.5 Some additions and cosmetics on cellular covers and localisations 25.6 Excursion on localisations and cellular covers of non-commutative groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25.6.1 Passing to localisations of groups . . . . . . . . . . . . . 25.6.2 Passing to cellular covers of groups . . . . . . . . . . . . 25.7 Open problems . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . .

719 719 721 723 726 733 741

. . . .

743 743 745 751

26 Tame and wild localisations of size  2@0 26.1 Tame localisations . . . . . . . . . . . . . . . . . . . . . . . . . 26.2 The wild case: classical E.R/-algebras . . . . . . . . . . . . . . . 26.2.1 The definition and its connection to problems in algebraic topology . . . . . . . . . . . . . . . . . . . . . . . . . . 26.3 Characterizations of E.R/-algebras . . . . . . . . . . . . . . . . 26.4 Construction of cotorsion-free E.R/-algebras of rank  2ℵ0 . . . 26.5 E.R/-algebras and uniquely transitive modules . . . . . . . . . . 26.5.1 UT-modules over principal ideal domains . . . . . . . . . 26.5.2 Pure-invertible algebras . . . . . . . . . . . . . . . . . . 26.5.3 The inductive step for the construction of UT-modules . . 26.5.4 The construction of UT-modules . . . . . . . . . . . . . .

. . . . . . . .

754 759 761 764 766 767 768 771

27 Tame cellular covers 27.1 Tame cellular covers of abelian groups . . . . . . 27.1.1 Decompositions of cellular covers . . . . 27.2 Cellular covers of divisible groups . . . . . . . . 27.3 Cellular covers of torsion and mixed groups . . . 27.4 When the only cellular covers are the trivial ones 27.5 Open problems . . . . . . . . . . . . . . . . . .

. . . . . .

773 773 773 775 778 780 781

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

752 . 752 . 754

28 Wild cellular covers 782 28.1 Cellular covers of subgroups of Q . . . . . . . . . . . . . . . . . . 782 28.2 The kernels of rank < 2ℵ0 for cellular covering maps . . . . . . . . 783 28.3 Characterizing kernels of cellular covers of abelian groups . . . . . 787 29 Absolute E-rings 29.1 A very simple example shows the idea of the proof 29.2 Constructing strongly rigid coloured trees . . . . . 29.2.1 A shift map for trees . . . . . . . . . . . . 29.2.2 Strongly rigid trees . . . . . . . . . . . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

792 793 795 796 799

xiv

Contents

29.3 29.4

The construction of E-rings . . . . . . . . . . Invariant principal ideals of R . . . . . . . . 29.4.1 The main theorem and consequences . 29.4.2 Large families of absolute E-rings . . The existence of absolute E-modules . . . . . Open problems . . . . . . . . . . . . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

801 802 808 809 810 812

30 Large kernels of cellular covers and large localisations 30.1 Large kernels of cellular covers . . . . . . . . . . . . . . . . 30.2 Large cotorsion-free E.R/-algebras . . . . . . . . . . . . . 30.3 Localisations of cotorsion-free modules . . . . . . . . . . . 30.3.1 Localisations of free R-modules . . . . . . . . . . . 30.3.2 Localisations of subgroups of Q . . . . . . . . . . . 30.3.3 A report on localisations of finite simple groups . . . 30.3.4 Discussing ℵ1 -free E.R/-algebras of cardinality ℵ1 30.4 Open problems . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

815 815 819 823 823 826 830 830 831

29.5 29.6

VII

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

Large cellular covers, localisations and E.R/-algebras

31 Mixed E(R)-modules over Dedekind domains 832 31.1 The construction of mixed E.R/-modules . . . . . . . . . . . . . . 836 31.1.1 E.R/-modules whose torsion part is a direct summand . . . 836 31.1.2 E.R/-modules whose torsion part is not a direct summand . 837 32 E(R)-modules with cotorsion 840 32.1 The construction of E.R/-algebras with cotorsion . . . . . . . . . . 847 32.2 Open problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 849

VIII

Some useful classes of algebras

33 Generalised E(R)-algebras 33.1 Background, strategy, the basic setting and the main result 33.2 The theory of skeletons . . . . . . . . . . . . . . . . . . . 33.3 Bodies . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33.4 Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33.5 Small cancellation of types . . . . . . . . . . . . . . . . . 33.6 Arbitrarily large free skeletons . . . . . . . . . . . . . . . 33.7 The algebraic structure of free bodies BY . . . . . . . . . 33.8 The Step Lemma . . . . . . . . . . . . . . . . . . . . . . 33.9 The main construction using the diamond principle . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

. . . . . . . . .

853 853 856 861 865 867 877 890 892 904

xv

Contents

33.10 33.11 33.12 33.13 33.14

Proof of the Main Theorem with }κ E . . . . . . . . . The main construction in ZFC . . . . . . . . . . . . . Proof of the Main Theorem with the General Black Box Appendix: rigid systems of generalised E.R/-algebras Open problems . . . . . . . . . . . . . . . . . . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

905 906 910 911 912

34 Some more useful classes of algebras 913 34.1 Leavitt type rings: the discrete case . . . . . . . . . . . . . . . . . . 913 34.2 Algebras with a Hausdorff topology . . . . . . . . . . . . . . . . . 920 34.3 Realising particular algebras as endomorphism algebras . . . . . . . 926 Bibliography Index

933 965

Introduction

This second, revised and substantially extended edition of Approximations and Endomorphism Algebras of Modules reflects both the depth and the width of recent developments in the area since the first edition appeared in 2006. The new division of the monograph into two volumes roughly corresponds to its two central topics, approximation theory and realisation theorems for modules. It is a widely accepted fact that the category of all modules over a general associative ring is too complex to admit classification. Unless the ring is of finite representation type, we must limit attempts at classification to some restricted subcategories of modules. The wild character of the category of all modules, or of one of its subcategories C, is often indicated by the presence of a realisation theorem, that is, by the fact that any reasonable algebra is isomorphic to the endomorphism algebra of a module from C. This results in the existence of pathological direct sum decompositions, and these are generally viewed as obstacles to classification. Realisation theorems have thus turned into important indicators of the “non-classification theory” of modules. In order to overcome this problem, the approximation theory of modules has been developed over the past few decades. The idea here is to select suitable subcategories C whose modules can be classified, and then to approximate arbitrary modules by those from C. These approximations are neither unique nor functorial in general, but there is a rich supply available appropriate to the requirements of various particular applications. Thus approximation theory has developed into an important part of the classification theory of modules. In this monograph we bring the two theories together. In Part I we set the scene by introducing the main classes of modules relevant here: the S-complete, pure-injective, Mittag-Leffler, and slender ones. Parts II and III develop the key tools of approximation theory. Some of the recent applications to the structure of modules are also presented, notably for tilting, cotilting, Baer, and Mittag-Leffler modules. Parts IV and V introduce further basic instruments, prediction principles and their applications to proving realisation theorems. Further tools are developed in Parts VI and VII for answering problems motivated by algebraic topology. Finally, Part VIII supplies us with the algebras needed, for example, for the applications in Part V. Approximation theory goes back to the discovery of the injective hull of a “group with operators” by Reinhold Baer in 1940. Since the late 1950s, injective envelopes, projec-

xviii

Introduction

tive covers as well as pure-injective envelopes, have successfully been used in module theory of arbitrary rings. Independent research by Auslander, Reiten and Smalø in the finite-dimensional case, and by Enochs and Xu for arbitrary modules, created a general theory of preenvelopes and precovers (or left and right approximations) of modules. The notions of a preenvelope and a precover are dual in the category theoretic sense. In the late 1970s Salce observed that these notions are also tied up by a homological notion of a complete cotorsion pair. The point is that, though there is no duality between the categories of all modules, each cotorsion pair .A; B/ makes it possible to produce special B-preenvelopes, once we know special A-precovers exist and vice versa. Later on Enochs showed the important fact that closed and complete cotorsion pairs provide minimal versions of approximations: envelopes and covers. In the late 1990s a variant of the small object argument discovered by Eklof et al. made it possible to prove that essentially all cotorsion pairs are complete. In particular, the flat cover conjecture was proved, saying that every module has a flat cover and a cotorsion envelope. Similarly one can prove the existence of approximations by various classes of modules of bounded homological dimensions. Moreover, torsion-free covers and Warfield cotorsion envelopes exist and, for domains, so do the Matlis cotorsion envelopes and special strongly flat precovers. The coincidence of the Enochs and Warfield cotorsion pairs characterizes Prüfer domains. Bazzoni and Salce have shown that the coincidence of the Enochs and Matlis cotorsion pairs yields a new interesting class of domains: the almost perfect ones. The completeness of a cotorsion pair .A; B/ is often proved by showing that the class A is deconstructible, that is, there is a cardinal κ, such that each A 2 A is a transfinite extension of < κ-presented modules in A. In order to deal efficiently with transfinite extensions, a sort of infinite-dimensional Jordan–Hölder theory has to be developed. Its key tool is the Hill Lemma, which makes it possible to tailor transfinite extensions to the needs of various particular constructions. Moreover, the Lemma has numerous applications far beyond the original setting of approximation theory. There is also the definition of a Kaplansky class which is similar, but in general weaker than the notion of a deconstructible class. However, for classes closed under direct limits, the two concepts coincide and provide an important example of an abstract elementary class (AEC) in the sense of Shelah. In the finite-dimensional case Auslander and Reiten studied approximations induced by tilting and cotilting modules. The theory has recently been extended to infinitely generated modules. This is essential for applications: for example, all non-trivial tilting and cotilting modules over commutative rings are infinitely generated. Moreover, works of Ringel and Lukas indicate the importance of infinite-dimensional tilting modules, even in the setting of finite-dimensional hereditary algebras.

Introduction

xix

Though possibly large, tilting modules are always close to the finitely generated setting: all tilting modules T are of finite type. That is, the tilting class T D T ?1 equals S ?1 for a set S consisting of strongly finitely presented modules of bounded projective dimension. In particular, T is a definable class in the sense of CrawleyBoevey. Moreover, cotilting modules C are pure-injective, so the cotilting class ?1 C is also definable. These facts were proved in a series of papers, culminating in the works of Bazzoni and Št’ovíˇcek. The set-theoretic methods needed here were developed earlier in the particular setting of the class P1 of all modules of projective dimension  1 by Eklof, Fuchs, Hill and Shelah. However, even in this classical setting, recent research has led to a breakthrough: Angeleri, Bazzoni and Herbera have solved the Kaplansky problem by showing that all Baer modules over domains are projective. Moreover, there is a cotorsion pair .P1 ; DI / where DI is the class of all divisible modules. The finite type of tilting modules yields their classification only in case we have good knowledge of the category of all strongly finitely presented modules. Fortunately, in the commutative noetherian setting, there are other tools available that make it possible to classify all tilting and cotilting classes, using characteristic sequences of associated prime ideals. The latter classification also reveals interesting connections to some long-standing open problems in commutative algebra. Infinite-dimensional tilting modules can be applied to solve several classical problems that at first sight do not appear to be related to tilting. In projective dimension one, tilting modules are crucial for extending the decomposition theory of Matlis localisations from domains to arbitrary commutative rings. A major application of n-tilting modules concerns another exciting feature of approximation theory, its relation to the Bass’ finitistic dimension conjectures. This was discovered in the classical works of Auslander, Reiten, Smalø and Huisgen-Zimmermann in the setting of finite-dimensional modules over finite-dimensional algebras. However, it is possible to investigate this relation for arbitrary modules over right noetherian rings R. It turns out that the second finitistic dimension conjecture holds for R, if and only if there is an (infinitely generated) tilting module Tf representing in a canonical way the category of all finitely presented modules of finite projective dimension. If R is an artin algebra, then the classical case is exactly that one, where Tf can be taken finitely generated; in this case also the first finitistic dimension conjecture holds for R. Using infinite-dimensional tilting theory, Angeleri et al. proved the first finitistic dimension conjecture for all (non-commutative) Iwanaga-Gorenstein rings. We will now briefly discuss how the contents of the respective chapters of this volume reflect the development of the theory mentioned above. Chapter 5 in Part II deals with basics of the theory of approximations of modules, using cotorsion pairs as the principal tool. Here we prove the classical results: the Salce

xx

Introduction

and Wakamatsu Lemmas, and the result of Enochs on the existence of minimal approximations. The main general methods are presented in Chapter 6. There we prove that complete cotorsion pairs are abundant: any cotorsion pair generated by a set is complete. Moreover, any cotorsion pair cogenerated by a class of pure-injective modules is closed and complete, hence perfect. In Chapter 7 we introduce a powerful tool for dealing with filtrations of modules: the Hill Lemma. It comes in two versions, the general version for arbitrary modules, and the rank version for modules over domains. We present applications of the general version to proving the Kaplansky Theorem for cotorsion pairs, the existence of C-socle sequences, and the Shelah Singular Compactness Theorem for C-filtered modules. There is also an application to algebraic geometry, extending a classic theorem of Raynaud and Gruson on vector bundles to the restricted Drinfeld vector bundles. The rank version of the Hill Lemma is applied to proving a structure theorem for strongly flat modules over valuation domains. Being equipped with the necessary machinery, we embark on constructing module approximations in Chapter 8. We start with approximations by various classes of modules of finite homological dimensions. Then we consider the harder case of characterizing tilting and cotilting approximations. This case involves some more advanced set-theoretic homological algebra. In Chapter 9 we apply our results to the class P1 of all modules of projective dimension  1. We characterize this class over semiprime Goldie rings, and describe the class lim P1 over domains. ! Chapter 10 starts with an investigation of the relations between Kaplansky and deconstructible classes. Here the important example of the Kaplansky, but not deconstructible class of all flat Mittag-Leffler modules, is studied in detail. We also investigate the recent connection of Kaplansky classes to abstract model theory discovered by Baldwin et al., and prove that the AECs thus arising have finite character. Most of our results on the existence of approximations rely on the completeness of the associated cotorsion pairs. Our principal method of proving completeness, namely deconstruction, is rather complex. In Chapter 11 we therefore address the question of whether completeness and generation by a set can be proved more directly and possibly for all cotorsion pairs. It turns out that the answer depends on the applied extension of set theory. We prove the consistency of the following claims: The completeness of all cotorsion pairs C D .A; B/ cogenerated by a set and such that either A is closed under pure submodules or C is hereditary, and B consists of modules of finite injective dimension (using a weak version of Jensen’s Diamond), (ii) The existence of cotorsion pairs not generated by a set (using Shelah’s Uniformisation Principle).

(i)

Introduction

xxi

However, both (i) and (ii) are independent of ZFC. Thus the problem of generation by a set and completeness is not solvable by purely algebraic means in ZFC. Chapter 12 studies the lattice LExt of all cotorsion pairs in the setting of abelian groups. We prove a result by Göbel et al. showing the extreme complexity of LExt in this case: each poset embeds into LExt . Chapter 13 in Part III contains the key result concerning tilting approximations: all tilting modules are of finite type. This result makes it possible to classify all tilting classes in Mod–R by resolving subcategories of mod–R and paves the way for an explicit classification over particular rings. This is the topic of Chapter 14, where we classify all tilting modules over Prüfer domains. Then we deal in detail with the Baer modules: we present a positive solution to the Kaplansky problem (saying that Baer modules over domains are projective) as well as the recent structure theory for Baer modules over hereditary artin algebras. The chapter ends with an application of tilting theory to the structure of the localisations of commutative rings. Given a multiplicative set S consisting of non-zero-divisors of R, we prove that S 1 R is a Matlis localisation (that is, the R-module S 1 R has projective dimension  1), if and only if S 1 R=R decomposes into a direct sum of countably presented R-modules. The dual setting of cotilting modules and classes is studied in Chapter 15. Since all tilting modules are of finite type, the explicit module duality ./d applied to a tilting right R-module always yields a cotilting left R-module. Such cotilting modules are called of cofinite type. We prove that over one-sided noetherian rings all 1-cotilting modules are of cofinite type, but we present an example of Bazzoni’s showing that in general there exist 1-cotilting modules that are not of cofinite type. The result of Št’ovíˇcek on the pureinjectivity of all cotilting modules is also proved here. Chapter 16 covers the recent classification of tilting and cotilting classes over commutative noetherian rings by Angeleri et al. Here the basic tools for the cotilting case are the associated prime ideals, while the tilting case is treated by means of the Auslander-Bridger transpose. The chapter ends with an application showing that Bass tilting and cotilting modules classify all tilting and cotilting modules in the setting of 1-Gorenstein rings. Chapter 17 deals with applications of tilting approximations to the Bass’ finitistic dimension conjectures. Given a right noetherian ring R, we denote by P and P