C∞-Algebraic Geometry with Corners (London Mathematical Society Lecture Note, 490) [1 ed.] 1009400169, 9781009400169

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C ∞-algebraic geometry with corners Kelli Francis-Staite and Dominic Joyce

Cambridge University Press & Assessment 978-1-009-40016-9 — C∞-Algebraic Geometry with Corners Kelli Francis-Staite , Dominic Joyce Copyright information More Information

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Contents

1

Introduction

page 1 ∞

2

Background on C -schemes 2.1 Introduction to category theory 2.2 C ∞ -rings 2.3 Modules and cotangent modules of C ∞ -rings 2.4 Sheaves 2.5 C ∞ -schemes 2.6 Complete C ∞ -rings 2.7 Sheaves of OX -modules on C ∞ -ringed spaces 2.8 Sheaves of OX -modules on C ∞ -schemes 2.9 Applications of C ∞ -rings and C ∞ -schemes

16 17 23 30 33 35 39 41 42 44

3

Background on manifolds with (g-)corners 3.1 Manifolds with corners 3.2 Monoids 3.3 Manifolds with g-corners 3.4 Boundaries, corners, and the corner functor 3.5 Tangent bundles and b-tangent bundles 3.6 Applications of manifolds with (g-)corners

49 50 52 54 58 61 63

4

(Pre) C ∞ -rings with corners 4.1 Categorical pre C ∞ -rings with corners 4.2 Pre C ∞ -rings with corners 4.3 Adjoints and (co)limits for pre C ∞ -rings with corners 4.4 C ∞ -rings with corners 4.5 Generators, relations, and localization 4.6 Local C ∞ -rings with corners 4.7 Special classes of C ∞ -rings with corners

71 72 73 78 82 85 90 98

iii

iv

Contents

5

C ∞ -schemes with corners 5.1 (Local) C ∞ -ringed spaces with corners 5.2 Special classes of local C ∞ -ringed spaces with corners 5.3 The spectrum functor 5.4 C ∞ -schemes with corners 5.5 Semi-complete C ∞ -rings with corners 5.6 Special classes of C ∞ -schemes with corners 5.7 Fibre products of C ∞ -schemes with corners

105 105 109 110 116 119 127 132

6

Boundaries, corners, and the corner functor 6.1 The corner functor for C ∞ -ringed spaces with corners 6.2 The corner functor for C ∞ -schemes with corners 6.3 The corner functor for firm C ∞ -schemes with corners ˇ ex , M ˇ in on C(X) 6.4 The sheaves of monoids M C(X) C(X) 6.5 The boundary ∂X and k-corners Ck (X) 6.6 Fibre products and corner functors ∞ c 6.7 B- and c-transverse fibre products in Mangc in , C Schto

135 135 140 147 150 154 155 159

7

Modules, and sheaves of modules 7.1 Modules over C ∞ -rings with corners, (b-)cotangent modules 7.2 (B-)cotangent modules for manifolds with (g-)corners 7.3 Some computations of b-cotangent modules 7.4 B-cotangent modules of pushouts 7.5 Sheaves of O X -modules, and (b-)cotangent sheaves 7.6 (B-)cotangent sheaves and the corner functor

165 165 167 171 177 179 183

Further generalizations and applications 8.1 Synthetic Differential Geometry with corners 8.2 C ∞ -stacks with corners 8.3 C ∞ -rings and C ∞ -schemes with a-corners 8.4 Derived manifolds and orbifolds with corners

192 192 193 194 198

8

Bibliography

201

Glossary of Notation

206

Index

211

1 Introduction

This book introduces and studies C ∞ -algebraic geometry with corners. It extends the existing theory of C ∞ -algebraic geometry [23, 43, 50, 53, 77], a version of algebraic geometry that generalizes smooth manifolds to a huge class of singular spaces. C ∞ -algebraic geometry with corners is a generalization based on manifolds with corners rather than on manifolds. It is related to log geometry [82], as log structures are a kind of boundary in Algebraic Geometry. The motivation for this book is two-fold. Firstly, it presents an introduction to C ∞ -algebraic geometry with corners, along with many foundational results, in one self-contained volume. Secondly, it provides the foundations needed to extend current work on Derived Differential Geometry [6–8, 10–13, 44, 45, 64, 91–93] to include derived C ∞ -schemes and C ∞ -stacks with corners, and hence derived manifolds and derived orbifolds with corners. Derived orbifolds with corners have important applications in Symplectic Geometry, for example, in the most general versions of Lagrangian Floer cohomology [28, 29], Fukaya categories [3, 87], and Symplectic Field Theory [24]. In these areas one studies moduli spaces M of J-holomorphic curves, which should be derived orbifolds with corners. (Related structures are the ‘Kuranishi spaces with corners’ of Fukaya–Oh–Ohta–Ono [29, 30] and ‘polyfolds with corners’ of Hofer–Wysocki–Zehnder [37, 38], but derived orbifolds with corners have better functoriality and are, we think, more beautiful.) C ∞ -algebraic geometry was originally suggested by William Lawvere in the late 1960’s [60], and our primary reference is the second author’s monograph [50]. In C ∞ -algebraic geometry, rings are replaced with C ∞ -rings, such as the space C ∞ (X) of smooth functions c : X → R of a manifold X. C ∞ rings are R-algebras but with a far richer structure — for each smooth function f : Rn → R there is an n-fold operation Φf : C ∞ (X)n → C ∞ (X) acting by Φf : c1 , . . . , cn 7→ f (c1 , . . . , cn ), satisfying many natural identities. The basic objects of C ∞ -algebraic geometry are C ∞ -schemes, which form 1

2

Introduction

a category C∞ Sch. They are special examples of (local) C ∞ -ringed spaces X = (X, OX ), a topological space X with a sheaf of C ∞ -rings OX . These form categories LC∞ RS ⊂ C∞ RS. As in Algebraic Geometry, there is a spectrum functor Spec : C∞ Ringsop → LC∞ RS, and a C ∞ -scheme is a local C ∞ -ringed space X covered by open subspaces U ⊆ X with U ∼ = Spec C for some C ∈ C∞ Rings. Any smooth manifold X determines a C ∞ -scheme X = Spec C ∞ (X), which is X with its sheaf of smooth functions OX , giving an embedding Man ⊂ C∞ Sch as a full subcategory. C ∞ -schemes are far more general than manifolds, and C∞ Sch contains many singular or infinite-dimensional objects. In addition, the category of C ∞ schemes addresses several shortcomings of the category of smooth manifolds: it is Cartesian closed, and has all finite limits and directed colimits, unlike the category of manifolds. Thus all fibre products (not just transverse ones) exist in C∞ Sch, and many spaces constructed from fibre products of manifolds can be studied. These good properties are some of the reasons C ∞ -algebraic geometry has been applied in the foundations of Synthetic Differential Geometry [23,53, 75–77], and Derived Differential Geometry. Manifolds with corners X are a generalization of manifolds locally modelled on [0, ∞)k × Rm−k rather than on Rm . They occur in many places in Differential Geometry. There are several candidates for morphisms of manifolds with corners, but for reasons explained below we choose the ‘b-maps’ of Melrose [71–74], which we just call ‘smooth maps’, to be the morphisms in the category Manc of manifolds with corners. A manifold with corners X of dimension n has a boundary ∂X, a manifold with corners of dimension n − 1, with a (not necessarily injective) inclusion map ΠX : ∂X → X. The interior is X ◦ = X \ ΠX (∂X), an ordinary nmanifold. A smooth map f : X → Y is called interior if f (X ◦ ) ⊂ Y ◦ . We write Mancin ⊂ Manc for the subcategory with morphisms interior maps. The symmetric group Sk acts freely on ∂ k X, and the quotient ∂ k X/Sk is the k-corners Ck (X), a manifold with corners of dimension n − k, with C0 (X) = X and C1 (X) = ∂X. For example, the square [0, 1]2 has

C0 [0, 1]2 = [0, 1]2 , C2 [0, 1]2 = {0, 1}2 ,



C1 [0, 1]2 = ∂ [0, 1]2 = {0, 1} × [0, 1] q [0, 1] × {0, 1} . `n The corners of X is C(X) = k=0 Ck (X), a manifold with corners of ˇ c . If f : X → Y is a smooth mixed dimension, which form a category Man map of manifolds with corners (possibly of mixed dimension), we can define a natural interior map C(f ) : C(X) → C(Y ), which need not map Ck (X) → ˇ c → Man ˇ c , which is right Ck (Y ). This defines a corner functor C : Man in

Introduction

3

ˇ c ,→ Man ˇ c . This means that not only do adjoint to the inclusion inc : Man in boundaries ∂X and corners Ck (X) combine into a functorial object C(X), but they are canonically determined just by the two notions of smooth map and interior map of manifolds with corners. Manifolds with g-corners Mangc , as in [47] and §3.3, are a generalization of manifolds with corners whose local models XP are more general than [0, ∞)k × Rm−k , and depend on a weakly toric monoid P . They have nicer properties than manifolds with corners in some ways – for example, in the existence of b-transverse fibre products – and come up in some applications, for example, moduli spaces of ‘quilts’ in Symplectic Geometry [69, 70] may be manifolds with g-corners, but not manifolds with corners. To extend C ∞ -algebraic geometry to include manifolds with (g-)corners, we introduce the notion of a C ∞ -ring with corners C = (C, C ex ), which is a pair consisting of a C ∞ -ring C and a commutative monoid C ex with many intertwining relationships. If X is a manifold with corners, the corresponding C ∞ -ring with corners is C ∞ (X) = (C ∞ (X), Ex(X)), where C ∞ (X) is the C ∞ -ring of smooth maps X → R, and Ex(X) is the monoid of smooth (‘exterior’) maps X → [0, ∞), with the monoid operation being multiplication. The monoid holds information about the corners structure of X. The definition of C ∞ -scheme with corners then follows those of schemes in Algebraic Geometry [36, §II], or C ∞ -schemes. We introduce categories LC∞ RSc ⊂ C∞ RSc of (local) C ∞ -ringed spaces with corners X = (X, O X ), which are topological spaces X with sheaves of C ∞ -ringed spaces with corners O X . We construct a spectrum functor Specc : (C∞ Ringsc )op → LC∞ RSc , which is left adjoint to the global sections functor Γc : LC∞ RSc → (C∞ Ringsc )op . A C ∞ -scheme with corners X is then a local C ∞ -ringed space with corners which can be covered by open U ⊆ X with U ∼ = Specc C ∞ for some C -ring with corners C. We define many interesting subcategories of C∞ Schc , as in Figure 1.1, which reproduces a diagram Figure 5.1 of subcategories of the category of C ∞ schemes with corners C∞ Schc from §5.6. For example, firm C ∞ -schemes with corners C∞ Schcfi ⊂ C∞ Schc have only finitely many boundary faces at each point. We define a subcategory C∞ Schcin ⊂ C∞ Schc of interior C ∞ schemes with corners, with interior morphisms corresponding to interior maps of manifolds with (g-)corners. We study categorical properties of C∞ Schc and its subcategories. For example, fibre products and all finite limits exist in C∞ Schcfi , and most of the other subcategories in Figure 1.1. We construct a corner functor C : C∞ Schc → C∞ Schcin , which is right adjoint to inc : C∞ Schcin ,→ C∞ Schc . If X is a firm C ∞ -scheme with

4

Introduction / Mangc in

Mancin

|

| / Mangc

Manc



 / C∞ Schcto

C∞ Schcsi

 |

C∞ Schcsi,ex

C∞ Schcfg

/ C∞ Schcfi,tf

/ C∞ Schcfi,Z

 | | | / C∞ Schcto,ex / C∞ Schcfi,tf ,ex / C∞ Schcfi,Z,ex 

C∞ Schcsa

 |

C∞ Schcsa,ex

 / C∞ Schctf

 | / C∞ Schctf ,ex

/ C∞ Schcfi,in

|  / C∞ Schcfi,in,ex / C∞ Schcfi

 / C∞ SchcZ

 | / C∞ SchcZ,ex

/ C∞ Schcfp

 / C∞ Schcin

 | / C∞ Schcin,ex

 / C∞ Schc .

Figure 1.1 Subcategories of C ∞ -schemes with corners C∞ Schc

`∞ corners we may write C(X) = k=0 Ck (X), where Ck (X) is the k-corners of X, with C0 (X) ∼ = X, and we define the boundary ∂X = C1 (X). There are full and faithful inclusion functors Manc , Mangc ,→ C∞ Schcfi ∞ c and Mancin , Mangc in ,→ C Schfi,in , which commute with corner functors, boundaries ∂X, and k-corners Ck (X), and preserve (b-)transverse fibre products. Thus, manifolds with (g-)corners can be regarded as special examples of C ∞ -schemes with corners, and C ∞ -schemes with corners generalize manifolds with (g-)corners. The geometry of ‘things with corners’ turns out to be surprisingly interesting and complicated, and has received relatively little attention — for example, our observation that corners arise as right adjoints to inclusions like Mancin ,→ Manc appears to be new. We hope our book will inspire further research.

How this book came to be written Beginning from foundational work of Lurie [64, §4.5] and Spivak [91], the second author has since 2009 been working on a theory of Derived Differential Geometry [44, 45], which is the study of derived manifolds and derived orbifolds. Here ‘derived’ is in the sense of Derived Algebraic Geometry [63, 64, 94–96]. This was aimed at applications in Symplectic Geometry, and is further explained in §2.9 and §8.4, though we summarize below. Fukaya–Oh–Ohta–Ono [28–31] were developing theories of Gromov–Witten invariants, Lagrangian Floer cohomology, and Fukaya categories, involving ‘Kuranishi spaces’, a geometric structure they put on moduli spaces of J-holomorphic curves. At the time there were problems with the definition and theory of Kuranishi spaces. The second author’s view was that Kuranishi

Introduction

5

spaces are actually derived orbifolds, and ideas from derived geometry were needed to understand them. Following [64, 91], he defined derived manifolds and derived orbifolds as special kinds of derived C ∞ -schemes and derived C ∞ -stacks. For more advanced applications in Symplectic Geometry [28, 29], it was essential to have a theory of derived orbifolds with corners. To do this properly in the world of C ∞ -algebraic geometry, it was necessary to go right back to the beginning, and introduce notions of C ∞ -ring with corners and C ∞ scheme with corners, before defining derived C ∞ -schemes and C ∞ -stacks with corners. The first attempt at this was the 2014 MSc thesis of Elana Kalashnikov [52], supervised by the second author, which defined notions of C ∞ -rings with corners and C ∞ -schemes with corners. Building on [52], the first author’s 2019 PhD thesis [25] went into the subject in much more detail, again supervised by the second author. One of the goals of the PhD was to find the best foundations for C ∞ algebraic geometry with corners, and its possible future applications, and the authors explored many options before settling on the definitions given here. This book is a rewritten and expanded version of [25]. It is designed in particular to provide the foundations of theories of ‘derived C ∞ -schemes with corners’ and ‘derived manifolds and orbifolds with corners’ in the final version of the second author’s book [45].

Why we chose these definitions In Chapters 3 and 4 we have chosen particular definitions of the notions of smooth map of manifolds with corners, and of C ∞ -ring with corners, which determine the course of the rest of the theory. The reader may wonder whether these choices could have been otherwise, leading to a different way of algebraizing ‘things with corners’, and a different theory of C ∞ -algebraic geometry with corners. Our answer to this comes in two parts. For the first part, knowing C ∞ -algebraic geometry, the rough starting point for a theory of C ∞ -algebraic geometry with corners is obvious. As in §2.2, a categorical C ∞ -ring is a product-preserving functor F : Euc → Sets, where Euc ⊂ Man is the full subcategory of Euclidean spaces Rn , and a C ∞ -ring C is an equivalent way of repackaging the data in F , with C = F (R), and many operations on C. Here Euc is an Algebraic Theory [1], and a (categorical) C ∞ -ring is an algebra over this Algebraic Theory. Thus it is clear that a categorical C ∞ -ring with corners should be a productpreserving functor F : Eucc → Sets, where Eucc ⊂ Manc is the full

6

Introduction

m−k k subcategory of Euclidean corner spaces Rm . Then a C ∞ k = [0, ∞) × R ring with corners C = (C, C ex ) should be an equivalent way of repackaging F , with C = F (R) and C ex = F ([0, ∞)), and many operations on C, C ex . There are possible choices in two places in this definition. Firstly, how should we define morphisms in Manc and Eucc ? (As we will see in §3.1, there are many different classes of ‘smooth maps’ between manifolds with corners.) And secondly, do we want to impose any additional conditions on the functors F ? For the first, in order for ‘product-preserving’ to make sense, the class of morphisms of manifolds with corners X must be closed under products and direct products. Also, for good properties of boundaries ∂X we want the inclusion ΠX : ∂X → X to be a morphism in Manc (this excludes interior morphisms, for example). With these restrictions, there are only really three sensible choices for the morphisms in Manc , which are called ‘weakly smooth’, ‘smooth’ (or ‘b-maps’), and ‘strongly smooth’ in §3.1, with

{weakly smooth} ⊂ {smooth} ⊂ {strongly smooth}. We rejected ‘weakly smooth’ maps, as they would cause C ∞ -schemes with corners not to have well behaved notions of boundary and corners. We found that smooth maps led to a richer and more interesting theory than strongly smooth maps, and the latter did not work for all the applications we had in mind (for example, it does not correctly model manifolds with g-corners in §3.3). Also the strongly smooth theory is strictly contained in the smooth theory. So we decided on smooth maps as the morphisms in Manc and Eucc . For the second, we impose an extra condition for C = (C, C ex ) coming from a product-preserving F : Eucc → Sets to be a C ∞ -ring with corners, that invertible functions in C ex should have logs in C. To motivate this, note that if X is a manifold with corners and f : X → [0, ∞) is smooth with inverse 1/f then f maps X → (0, ∞), and so log f : X → R is well defined and smooth. The second part of our answer is that we have defined not one theory of C ∞ rings and schemes with corners, but many, as in Figure 1.1; the overarching definitions of C ∞ -rings and schemes with corners are the most general, which contain all the rest. General C ∞ -schemes with corners X ∈ C∞ Schc may have ‘corner behaviour’ which is complicated and pathological, for example, X = [0, ∞)N has infinitely many boundary faces at a point. The subcategories of C∞ Schc in Figure 1.1 have ‘corner behaviour’ which is progressively nicer, and more like ordinary manifolds with corners, as we impose more conditions. The smallest categories C∞ Schcsi , C∞ Schcsi,ex , with ‘simple corners’, behave exactly like manifolds with corners, and C∞ Schcto , C∞ Schcto,ex , with ‘toric corners’, behave exactly like manifolds with g-corners. Different subcategories may be appropriate for different applications.

Introduction

7

In the rest of this introduction we summarize the contents of the chapters.

Chapter 2. C ∞ -rings and C ∞ -schemes We begin with background on category theory, and on C ∞ -rings and C ∞ schemes, mostly following the second author [50]. A categorical C ∞ -ring is a product-preserving functor F : Euc → Sets, where Euc ⊂ Man is the full subcategory of the category of smooth manifolds Man with objects the Euclidean spaces Rn for n > 0. These form a category CC∞ Rings, with morphisms natural transformations, which is equivalent to the category C∞ Rings of C ∞ -rings by mapping F 7→ C =
F (R). Here a C ∞ -ring is data C, (Φf )f :Rn →R C ∞ , where C is a set and Φf : n C → C a map for all smooth maps f : Rn → R, satisfying some axioms. Usually we write this as C, leaving the C ∞ -operations Φf implicit. The motivating examples for X a smooth manifold are C = C ∞ (X), the set of smooth functions c : X → R, with operations Φf (c1 , . . . , cn )(x) = f (c1 (x), . . . , cn (x)). We discuss C-modules, which are just modules over C considered as an Ralgebra, and the cotangent module ΩC , generalizing Γ∞ (T ∗ X), the vector space of smooth sections of T ∗ X, as a module over C ∞ (X). We then define the subcategory C∞ RS of C ∞ -ringed spaces X = (X, OX ) where X is a topological space and OX a sheaf of C ∞ -rings on X, and the full subcategory LC∞ RS ⊂ C∞ RS of local C ∞ -ringed spaces for which the stalks OX,x are local C ∞ -rings. The global sections functor Γ : LC∞ RS → C∞ Ringsop has a right adjoint Spec : C∞ Ringsop → LC∞ RS, the spectrum functor. An object X ∈ LC∞ RS is called a C ∞ -scheme if we may cover X with open U ⊆ X with (U, OX |U ) ∼ = Spec C for C ∈ C∞ Rings. These ∞ form a full subcategory C Sch ⊂ LC∞ RS. There is a full embedding Man ,→ C∞ Sch mapping X 7→ X = (X, OX ), where OX is the sheaf of local smooth functions c : X → R. In classical Algebraic Geometry Γ ◦ Spec ∼ = Id : Ringsop → Ringsop , so Ringsop is equivalent to the category ASch of affine schemes. In C ∞ algebraic geometry it is not true that Γ ◦ Spec ∼ = Id, but we do have Spec ◦Γ ◦ Spec ∼ = Spec. Define a C ∞ -ring C to be complete if C ∼ = Γ ◦ Spec D for some ∞ ∞ D. Write C Ringsco ⊂ C Rings for the full subcategory of complete C ∞ -rings. Then (C∞ Ringsco )op is equivalent to the category AC∞ Sch of affine C ∞ -schemes. We define OX -modules on a C ∞ -scheme X, and the cotangent sheaf T ∗ X, which generalizes cotangent bundles of manifolds. We review applications of C ∞ -algebraic geometry in Synthetic Differential Geometry and Derived Differential Geometry.

8

Introduction

Chapter 3. Manifolds with (g-)corners We discuss manifolds with corners, locally modelled on [0, ∞)k × Rm−k . The definition of manifold with corners X, Y is an obvious generalization of the definition of manifold. However, the definition of smooth map f : X → Y is not obvious: what conditions should we impose on f near the boundary and corners of X, Y ? There are several non-equivalent definitions of morphisms of manifolds with corners in the literature. The one we choose is due to Melrose [73, §1.12], [56, §1], who calls them b-maps. This gives a category Manc . A smooth map f : X → Y is called interior if f (X ◦ ) ⊆ Y ◦ , where X ◦ is the interior of X (that is, if X is locally modelled on [0, ∞)k × Rm−k , then X ◦ is locally modelled on (0, ∞)k × Rm−k ). Write Mancin ⊂ Manc for the subcategory with only interior morphisms. A manifold with corners X has a boundary ∂X, a manifold with corners of dimension dim X−1, with a (non-injective) morphism ΠX : ∂X → X. For example, if X = [0, ∞)2 , then ∂X is the disjoint union ({0}×[0, ∞))q([0, ∞)× {0}) with the obvious map ΠX , so two points in ∂X lie over (0, 0) ∈ X. There is a natural, free action of the symmetric group Sk on the k th boundary ∂ k X, permuting the boundary strata. The k-corners is Ck (X) = ∂ k X/Sk , of dimen`dim X sion dim X − k. The corners of X is C(X) = k=0 Ck (X), an object in ˇ c of manifolds with corners of mixed dimension. the category Man We call X a manifold with faces if ΠX |F : F → X is injective for each connected component F of ∂X (called a face). Now boundaries are not functorial: if f : X → Y is smooth, there is generally no natural morphism ∂f : ∂X → ∂Y . However, there is a natural, interior morphism C(f ) : C(X) → C(Y ), giving the corner functor ˇ c or C : Man ˇ c → Man ˇ c . We explain that C is right C : Manc → Man in in ˇ c ,→ Man ˇ c . Thus, from a categorical adjoint to the inclusion inc : Man in point of view, boundaries and corners in Manc are determined uniquely by the inclusion Mancin ,→ Manc , that is, by the comparison between interior and smooth maps. A manifold with corners X has two notions of (co)tangent bundle: the tangent bundle T X and its dual the cotangent bundle T ∗ X, and the b-tangent bundle b T X and its dual the b-cotangent bundle b T ∗ X. Here T X is the obvious notion of tangent bundle. There is a morphism IX : b T X → T X which identifies Γ∞ (b T X) with the subspace of vector fields v ∈ Γ∞ (T X) which are tangent to every boundary stratum of X. Tangent bundles are functorial over smooth maps f : X → Y (that is, f lifts to T f : T X → T Y linear on the fibres). B-tangent bundles are functorial over interior maps f : X → Y . gc We also discuss the categories Mangc of manifolds with generin ⊂ Man

Introduction

9

alized corners, or manifolds with g-corners, introduced by the second author in [47]. These allow more complicated local models than [0, ∞)k × Rm−k . The local models XP are parametrized by weakly toric monoids P , with XP = [0, ∞)k × Rm−k when P = Nk × Zm−k . So we provide an introduction to the theory of monoids. (All monoids in this book are commutative.) The simplest manifold with g-corners which is not a manifold with corners  is X = (w, x, y, z) ∈ [0, ∞)4 : wx = yz . Most of the theory above extends to manifolds with g-corners, except that tangent bundles T X are not well behaved, and we no longer have Ck (X) ∼ = ∂ k X/Sk . We call morphisms g : X → Z, h : Y → Z in Mancin or Mangc in btransverse if b Tx g ⊕ b Ty h : b Tx X ⊕ b Ty Y → b Tz Z is surjective for all x ∈ X, y ∈ Y with g(x) = h(y) = z. Manifolds with g-corners have the nice property that all b-transverse fibre products exist in Mangc in , whereas fibre products only exist in Mancin , Manc under rather restrictive conditions. We can think c of Mangc in as a kind of closure of Manin under b-transverse fibre products. We discuss applications of manifolds with (g-)corners in the literature, including in Topological Quantum Field Theories, analysis of partial differential equations, and moduli problems in Morse homology and Floer theories whose moduli spaces are manifolds with (g-)corners.

Chapter 4. (Pre) C ∞ -rings with corners We introduce C ∞ -rings with corners. To decide on the correct definition, the obvious starting point is categorical C ∞ -rings, i.e. product-preserving functors F : Euc → Sets. We define a categorical pre C ∞ -ring with corners to be a product-preserving functor F : Eucc → Sets, where Eucc ⊂ Manc is the full subcategory with objects [0, ∞)m × Rn for m, n > 0. These form a category CPC∞ Ringsc , with morphisms natural transformations. Then we define the category PC∞ Ringsc of pre C ∞ -rings with corners,
with objects C, C ex , (Φf )f :Rm ×[0,∞)n →R C ∞ , (Ψg )g:Rm ×[0,∞)n →[0,∞) C ∞ for C, C ex sets and Φf : C m × C nex → C, Ψg : C m × C nex → C ex maps, satisfying some axioms. Usually we write this as C = (C, C ex ), leaving the C ∞ -operations Φf , Ψg implicit. There is an equivalence CPC∞ Ringsc → PC∞ Ringsc mapping F 7→ (C, C ex ) = (F (R), F ([0, ∞))). The motivating example, for X a manifold with (g-)corners, is (C, C ex ) = (C ∞ (X), Ex(X)), where C ∞ (X) is the set of smooth maps c : X → R, and Ex(X) the set of exterior (i.e. smooth, but not necessarily interior) maps c0 : X → [0, ∞). We define the category of C ∞ -rings with corners C∞ Ringsc to be the full subcategory of (C, C ex ) in PC∞ Ringsc satisfying an extra condition, that if c0 ∈ C ex is invertible in C ex then c0 = Ψexp (c) for some c ∈ C. In terms of

10

Introduction

spaces X, this says that if c0 : X → [0, ∞) is smooth and invertible (hence positive) then c0 = exp c for some smooth c = log c0 : X → R. Now we could also have started with Mancin . So we write Euccin ⊂ Mancin for the full subcategory with objects [0, ∞)m × Rn , and define the category CPC∞ Ringscin of categorical interior pre C ∞ -rings with corners to be product-preserving functors F : Euccin → Sets, with morphisms natural transformations. We define a subcategory PC∞ Ringscin ⊂ PC∞ Ringsc of interior pre C ∞ -rings with corners, and an equivalence CPC∞ Ringscin → PC∞ Ringscin mapping F 7→ (C, C ex ) = (F (R), F ([0, ∞)) q {0}). The category C∞ Ringscin of interior C ∞ -rings with corners is the intersection PC∞ Ringscin ∩ C∞ Ringsc in PC∞ Ringsc . The motivating example, for X a manifold with (g-)corners, is (C, C ex ) = (C ∞ (X), In(X) q {0}), for In(X) the interior maps c0 : X → [0, ∞). Although a C ∞ -ring with corners C = (C, C ex ) has a huge number of C ∞ operations Φf , Ψg , it is helpful much of the time to focus on a small subset of these, giving C, C ex smaller, more manageable structures. In particular, we often think of C as an R-algebra, with addition and multiplication given by Φf+ , Φf· : C × C → C for f+ , f· : R2 → R given by f+ (x, y) = x + y, f· (x, y) = xy. And we think of C ex as a monoid, with multiplication given by Ψg· : C ex ×C ex → C ex for g· : [0, ∞)2 → [0, ∞) given by g· (x, y) = xy. Note that monoids also control the corner structure of manifolds with g-corners, in the same way. We define various important subcategories of C ∞ -rings with corners by imposing conditions on C ex as a monoid. For example, a C ∞ -ring with corners C = (C, C ex ) is interior if C ex = C in q {0} where C in is a submonoid of C ex , that is, C ex has no zero divisors. In terms of spaces X, we interpret C in as the monoid In(X) of interior maps c0 : X → [0, ∞). If C, D are interior, a morphism φ = (φ, φex ) : C → D is interior if φex : C ex → Dex maps C in → Din . The subcategory C∞ Schcfi ⊂ C∞ Schc of firm C ∞ -rings with corners are those whose sharpening C ]ex is finitely generated.

Chapter 5. C ∞ -schemes with corners We define the category C∞ RSc of C ∞ -ringed spaces with corners X = ex (X, O X ), where X is a topological space and O X = (OX , OX ) a sheaf of ∞ ∞ c C -rings with corners on X, and the subcategory LC RS ⊂ C∞ RSc of local C ∞ -ringed spaces with corners for which the stalks O X,x for x ∈ X are local C ∞ -rings with corners. The global sections functor Γc : LC∞ RSc → (C∞ Ringsc )op has a right adjoint Specc : (C∞ Ringsc )op → LC∞ RSc , the spectrum functor. An object X in LC∞ RSc is called a C ∞ -scheme with

Introduction

11

corners if we may cover X with open U ⊆ X with (U, O X |U ) ∼ = Specc C for ∞ c ∞ c C in C Rings . These form a full subcategory C Sch ⊂ LC∞ RSc . One might expect that to define corresponding subcategories C∞ RScin ⊂ ∞ C RSc , LC∞ RScin ⊂ LC∞ RSc of interior (local) C ∞ -ringed spaces with corners, we should just replace C∞ Ringsc by C∞ Ringscin ⊂ C∞ Ringsc . However, as inc : C∞ Ringscin ,→ C∞ Ringsc does not preserve limits, a sheaf of interior C ∞ -rings with corners is only a presheaf of C ∞ -rings with corners. So we define X = (X, O X ) ∈ C∞ RSc to be interior if O X is the sheafification, as a sheaf valued in C∞ Ringsc , of a sheaf valued in C∞ Ringscin . Then the stalks O X,x of O X lie in C∞ Ringscin , so ex in OX,x = OX,x q {0}. We define Γcin : LC∞ RScin → (C∞ Ringscin )op by Γcin (X) = (Γ(OX ), in in ex Γ(OX ) q {0}), where Γ(OX ) ⊂ Γ(OX ) is the subset of sections s with in ex s|x ∈ OX,x ⊂ OX,x for each x ∈ X. Then Γcin has a right adjoint Speccin : (C∞ Ringscin )op → LC∞ RScin , where Speccin = Specc |(C∞ Ringscin )op . An object X ∈ LC∞ RScin is called an interior C ∞ -scheme with corners if we may cover X with open U ⊆ X with (U, O X |U ) ∼ = Speccin C for ∞ c ∞ c C ∈ C Ringsin . These form a full subcategory C Schin ⊂ LC∞ RScin , with C∞ Schcin ⊂ C∞ Schc . ∞ c We define full embeddings Mangc ,→ C∞ Schc or Mangc in ,→ C Schin in ex q {0})), where OX , )) or X 7→ (X, (OX , OX mapping X 7→ (X, (OX , OX ex in OX , OX are the sheaves of local smooth functions X → R, and exterior functions X → [0, ∞), and interior functions X → [0, ∞), respectively. Thus, manifolds with (g-)corners may be regarded as special examples of C ∞ schemes with corners. Manifolds with (g-)faces map to affine C ∞ -schemes with corners. For C ∞ -schemes, Γ ◦ Spec ∼ 6 Id but Spec ◦Γ ◦ Spec ∼ = = Spec. Thus we can define a full subcategory C∞ Ringsco ⊂ C∞ Rings of complete C ∞ rings, with (C∞ Ringsco )op equivalent to the category AC∞ Sch of affine C ∞ -schemes. For C ∞ -schemes with corners, the situation is worse: we have both Γc ◦ Specc ∼ 6 Id and Specc ◦Γc ◦ Specc ∼ 6 Specc . To see why, note that if X is a = = manifold with corners then smooth functions c : X → R are essentially local objects — they can be glued using partitions of unity. However, smooth functions c0 : X → [0, ∞) have some strictly global behaviour: there is a locally constant function µc0 : ∂X → N q {∞} giving the order of vanishing of c0 along the boundary ∂X. So the behaviour of c0 near distant points x, y in X is linked, if x, y lie in the image of the same connected component of ∂X. This means that smooth functions c0 : X → [0, ∞) cannot always be glued using partitions of unity, and localizing a C ∞ -ring with corners

12

Introduction

C = (C, C ex ) at an R-point x : C → R, as one does to define Specc , does not see only local behaviour around x. Since Specc ◦Γc ◦ Specc 6∼ = Specc , we cannot define a subcategory of ‘complete’ C ∞ -rings with corners equivalent to affine C ∞ -schemes with corners. As a partial substitute, we define semi-complete C ∞ -rings with corners C = (C, C ex ), such that Γc ◦ Specc is an isomorphism on C and injective on C ex . If X, Y ∈ C∞ Schc , a morphism f : X → Y in C∞ Schc is a morphism in LC∞ RSc . Although locally we can write X ∼ = Specc C, X ∼ = Specc D, because of the lack of a good notion of completeness, we do not know that locally we can write f = Specc φ for some φ : D → C in C∞ Ringsc . One problem this causes is that g : X → Z, h : Y → Z are morphisms in C∞ Schc , we do not know that the fibre product X ×g,Z,h Y in LC∞ RSc (which always exists) lies in C∞ Schc , if g, h are not locally of the form Specc φ. So we do not know that all fibre products exist in C∞ Schc . To get around this, we introduce the full subcategory C∞ Schcfi ⊂ C∞ Schc of firm C ∞ -schemes with corners X, which are locally of the form Specc C for C a firm C ∞ -ring with corners. Morphisms f : X → Y in C∞ Schcfi are always locally of the form Specc φ, so we can prove that C∞ Schcfi is closed under fibre products in LC∞ RSc , and thus all fibre products exist in C∞ Schcfi . In general, C∞ Schc contains a huge variety of objects, many of which are very singular and pathological, and do not fit with our intuitions about manifolds with corners. So it can be helpful to restrict to smaller subcategories of better behaved objects in C∞ Schc , such as C∞ Schcfi . For example, for X to be firm means that locally X has only finitely many boundary strata, which seems likely to hold in almost every interesting application. We define many subcategories of C∞ Schcin , C∞ Schc , and prove results such as existence of reflection functors between them, and existence of fibre products and finite limits in them. Two particularly interesting and well behaved examples are the full subcategories C∞ Schcto ⊂ C∞ Schcin , C∞ Schcto,ex ⊂ C∞ Schc of toric C ∞ -schemes with corners, whose corner structure is controlled by toric monoids in the same way that manifolds with g-corners are.

Chapter 6. Boundaries, corners, and the corner functor One of the most important properties of manifolds with corners X is the existence of boundaries ∂X, and clearly we want to generalize this to C ∞ -schemes with corners. Our starting point, as in Chapter 3, is that the boundary ∂X =

Introduction 13 `dim X C1 (X) is part of the corners C(X) = k=0 Ck (X), and the corner functor ˇ c → Man ˇ c is right adjoint to the inclusion inc : Man ˇ c ,→ Man ˇ c. C : Man in in ∞ c ∞ We construct a right adjoint corner functor C : LC RS → LC RScin to the inclusion inc : LC∞ RScin ,→ LC∞ RSc . We prove that the restriction to C∞ Schc maps to C∞ Schcin , giving C : C∞ Schc → C∞ Schcin right adjoint to inc : C∞ Schcin ,→ C∞ Schc . For X in LC∞ RSc , points in ex C(X) are pairs (x, P ) for x ∈ X and P ⊂ OX,x a prime ideal in the monoid ex OX,x of the stalk O X,x of O X at x. This is an analogue, for X ∈ Manc , of a point in C(X) being (x, γ) for x ∈ X and γ a local corner component of X at x. The corner functors for Manc , Mangc and C∞ Schc commute with the embeddings Manc , Mangc ,→ C∞ Schc . ` To get an analogue of the decomposition C(X) = k>0 Ck (X) for C ∞ schemes with corners, we restrict to the subcategories C∞ Schcfi ⊂ C∞ Schc and C∞ Schcfi,in ⊂ C∞ Schcin of firm C ∞ -schemes with corners, where C : ˇ ex C∞ Schcfi → C∞ Schcfi,in . For X firm there is a locally constant sheaf M C(X) 0 ex ex ˇ /[c =1 of finitely generated monoids on C(X), with M | = O ) X,x C(X) (x,P` ex \ P ]. We define a decomposition C(X) = k>0 Ck (X) with if c0 ∈ OX,x Ck (X) open and closed in C(X), by saying that (x, P ) ∈ Ck (X) if the ˇ ex |(x,P ) is k + 1. This maximum length of a chain of prime ideals in M C(X) ` recovers the usual decomposition C(X) = k>0 Ck (X) if X is a manifold with (g-)corners. We define the boundary ∂X = C1 (X). For toric C ∞ -schemes with corners C maps C∞ Schcto,ex → C∞ Schcto , and C preserves fibre products, and all fibre products exist in C∞ Schcto . We use this to give criteria for when fibre products exist in C∞ Schcto,ex . This is an analogue of results in [47] giving criteria for when fibre products exist in Mangc , given that b-transverse fibre products exist in Mangc in .

Chapter 7. Modules, and sheaves of modules If C = (C, C ex ) is a C ∞ -ring with corners, we define a C-module to be a ex module over C as an R-algebra. Similarly, if X = (X, (OX , OX )) is a C ∞ scheme with corners, we consider OX -modules on X, which are just modules on the underlying C ∞ -scheme X = (X, OX ). So the theory of modules over C ∞ -rings with corners and C ∞ -schemes with corners lifts immediately from modules over C ∞ -rings and C ∞ -schemes in [50, §5], with no additional theory required. If X is a manifold with corners then as in Chapter 3 we have two notions of cotangent bundle T ∗ X, b T ∗ X, where b T ∗ X is functorial only under interior morphisms. Similarly, if C = (C, C ex ) is a C ∞ -ring with corners, we have the

14

Introduction

cotangent module ΩC of C from [50, §5]. If C is interior we also define the bcotangent module b ΩC , which uses the corner structure. If X is a manifold with ∞ ∗ b ∞ b ∗ (g-)faces and C = C ∞ in (X) then ΩC = Γ (T X) and ΩC = Γ ( T X). We show that b-cotangent modules are functorial under interior morphisms and have exact sequences for pushouts. If X is a C ∞ -scheme with corners we define the cotangent sheaf T ∗ X, and if X is interior the b-cotangent sheaf b T ∗ X, by sheafifying the (b-)cotangent modules of O X (U ) for open U ⊂ X. If X = FManc (X) for X ∈ Manc these are the sheaves of sections of T ∗ X and b T ∗ X. We show that Cartesian squares in subcategories such as C∞ Schcto , C∞ Schcfi,in ⊂ C∞ Schcin yield exact sequences of b-cotangent sheaves. On the corners C(X) we construct an ˇ ex ⊗N OC(X) . exact sequence relating b T ∗ C(X), Π∗X (b T ∗ X) and M C(X)

Chapter 8. Further generalizations and applications Finally we propose four directions in which this book could be generalized and applied: Synthetic Differential Geometry with corners. Synthetic Differential Geometry is a subject in which one proves theorems about manifolds in Differential Geometry by reasoning using ‘infinitesimals’, as in Kock [53,54]. C ∞ -schemes are used to provide a ‘model’ for Synthetic Differential Geometry, and so show that the axioms of Synthetic Differential Geometry are consistent. In a similar way, one could develop a theory of ‘Synthetic Differential Geometry with corners’, for proving theorems about manifolds with corners using infinitesimals, and C ∞ -schemes with corners could be used to show that it is consistent. C ∞ -stacks with corners. In classical Algebraic Geometry, schemes are generalized to (Deligne–Mumford or Artin) stacks. The second author [50] extended the theory of C ∞ -schemes to C ∞ -stacks, including Deligne–Mumford C ∞ -stacks. This corresponds to generalizing manifolds to orbifolds. We discuss a theory of C ∞ -stacks with corners, including Deligne–Mumford ∞ C -stacks with corners. These generalize orbifolds with (g-)corners. Much of the theory follows from [50] with only cosmetic changes. C ∞ -rings and C ∞ -schemes with a-corners. Our theory starts with the categories Mancin ⊂ Manc of manifolds with corners defined in Chapter 3. The ac second author [48] also defined categories Manac of manifolds in ⊂ Man with analytic corners, or manifolds with a-corners. Even the simplest objects J0, ∞) in Manac and [0, ∞) in Manc have different smooth structures. There is also a category Manc,ac of manifolds with corners and a-corners containing both Manc and Manac .

Introduction

15

Manifolds with a-corners have applications to partial differential equations with boundary conditions of asymptotic type, and to moduli spaces with boundary and corners, such as moduli spaces of Morse flow lines, in which (we argue) manifolds with a-corners give the correct smooth structure. ac This entire book could be rewritten over Manac or Manc,ac ⊂ in ⊂ Man in Manc,ac rather than Mancin ⊂ Manc . We explain the first steps in this. Derived manifolds and derived orbifolds with corners. Classical Algebraic Geometry has been generalized to Derived Algebraic Geometry, which is now a major area of mathematics. As in §2.9, classical Differential Geometry can be generalized to Derived Differential Geometry, the study of derived manifolds and derived orbifolds, regarded as special examples of derived C ∞ -schemes and derived C ∞ -stacks. Some references are [6–8, 10–13, 44, 45, 64, 91–93]. It is desirable to extend the subject to derived manifolds with corners and derived orbifolds with corners, regarded as special examples of derived C ∞ schemes with corners and derived C ∞ -stacks with corners. This will be done by the second author in [45], with this book as its foundations (see also Steffens [92, 93]). Derived orbifolds with corners will have important applications in Symplectic Geometry, as (we argue) they are the correct way to make Fukaya– Ohta–Oh–Ono’s ‘Kuranishi spaces with corners’ [28–31] into well behaved geometric spaces. Acknowledgements: The first author would like to thank The University of Oxford, St John’s College, the Rhodes Trust, the Centre for Quantum Geometry of Moduli Spaces in Aarhus, and her family, for their support while writing her PhD thesis [25], on which this book is based.

2 Background on C ∞ -schemes

One can think of C ∞ -rings as a specific type of commutative R-algebra, where there are not only the addition and multiplication operations, but operations corresponding to every smooth function Rn → R. Alternatively, C ∞ -rings can also be considered as certain product-preserving functors, and we will introduce both definitions in this chapter. C ∞ -rings, along with a spectrum functor, form the building blocks of C ∞ schemes. As in ordinary Algebraic Geometry, the image of a C ∞ -ring under the spectrum functor gives an affine C ∞ -scheme. However, while there will be many similarities to ordinary Algebraic Geometry, C ∞ -algebraic geometry has several differences that may challenge the reader’s intuition. These include: • C ∞ -rings are non-noetherian, so finitely presented C ∞ -rings are not necessarily finitely generated. • The spectrum functor uses maximal ideals with residue field R, not prime ideals. This makes affine C ∞ -schemes Hausdorff and regular. • Affine C ∞ -schemes are very general, enough so that all manifolds can be represented as affine C ∞ -schemes, and study can be restricted to affine C ∞ schemes in many cases. However, manifolds with corners will not always be affine C ∞ -schemes with corners in Chapter 5. • C ∞ -rings are not in 1-1 correspondence with affine C ∞ -schemes, and the spectrum functor is neither full nor faithful. However, the full subcategory of complete C ∞ -rings is in 1-1 correspondence with affine C ∞ -schemes, and the spectrum functor is full and faithful on this subcategory. Our motivating example throughout this chapter will be a manifold X. We will see that its set of smooth functions C ∞ (X) is a complete C ∞ -ring, and applying the spectrum functor returns the affine C ∞ -scheme with underlying topological space X along with its sheaf of smooth functions. Transverse fibre products of manifolds map to fibre products of (affine) C ∞ -schemes. Unlike 16

2.1 Introduction to category theory

17

the category of manifolds, all fibre products of (affine) C ∞ -schemes exist, which is one of the motivating reasons for considering this category. Our main reference for this chapter is the second author [50, §2–§4]. See also [43], Dubuc [23], Moerdijk and Reyes [77], and Kock [53].

2.1 Introduction to category theory We begin with some well known definitions from category theory. All the material of this section can be found in MacLane [66].

2.1.1 Categories and functors Definition 2.1. A category C consists of a proper class of objects Obj(C), and for all X, Y ∈ Obj(C) a set Hom(X, Y ) of morphisms f from X to Y , written f : X → Y , and for all X, Y, Z ∈ Obj(C) a composition map ◦ : Hom(X, Y )×Hom(Y, Z) → Hom(X, Z), written (f, g) 7→ g ◦f . Composition must be associative, that is, if f : W → X, g : X → Y and h : Y → Z are morphisms in C then (h ◦ g) ◦ f = h ◦ (g ◦ f ). For each X ∈ Obj(C) there must exist an identity morphism idX : X → X such that f ◦idX = f = idY ◦f for all f : X → Y in C. A morphism f : X → Y is an isomorphism if there exists f −1 : Y → X with f −1 ◦ f = idX and f ◦ f −1 = idY . If C is a category, the opposite category C op is C with the directions of all morphisms reversed. That is, we define Obj(C op ) = Obj(C), and for all X, Y, Z ∈ Obj(C) we define HomC op (X, Y ) = HomC (Y, X), and for f : X → Y , g : Y → Z in C we define f ◦C op g = g ◦C f , and idC op X = idC X. We call D a subcategory of C if Obj(D) ⊆ Obj(C), and HomD (X, Y ) ⊆ HomC (X, Y ) for all X, Y ∈ Obj(D), and compositions and identities in D agree with those in C. We call D a full subcategory if also HomD (X, Y ) = HomC (X, Y ) for all X, Y in D. Definition 2.2. Let C, D be categories. A (covariant) functor F : C → D gives for all objects X in C an object F (X) in D, and for all morphisms f : X → Y in C a morphism F (f ) : F (X) → F (Y ) in D, such that F (g◦f ) = F (g) ◦ F (f ) for all f : X → Y , g : Y → Z in C, and F (idX ) = idF (X) for all X ∈ Obj(C). A contravariant functor F : C → D is a covariant functor F : C op → D. Functors compose in the obvious way. Each category C has an obvious identity functor idC : C → C with idC (X) = X and idC (f ) = f for all X, f . A functor F : C → D is called full if the maps HomC (X, Y ) →

18

Background on C ∞ -schemes

HomD (F (X), F (Y )), f 7→ F (f ) are surjective for all X, Y ∈ Obj(C), and faithful if the maps HomC (X, Y ) → HomD (F (X), F (Y )) are injective for all X, Y ∈ Obj(C). A functor F : C → D is called essentially surjective if for every object Y ∈ D there exists X ∈ C such that Y ∼ = F (X) in D. Let C, D be categories and F, G : C → D be functors. A natural transformation η : F ⇒ G gives, for all objects X in C, a morphism η(X) : F (X) → G(X) such that if f : X → Y is a morphism in C then η(Y ) ◦ F (f ) = G(f ) ◦ η(X) as a morphism F (X) → G(Y ) in D. We call η a natural isomorphism if η(X) is an isomorphism for all X ∈ Obj(C). An equivalence between categories C, D is a functor F : C → D such that there exists a functor G : D → C and natural isomorphisms η : G ◦ F ⇒ idC and ζ : F ◦ G ⇒ idD . That is, F is invertible up to natural isomorphism. Then we call C, D equivalent categories. A functor F : C → D is an equivalence if and only if it is full, faithful, and essentially surjective. It is a fundamental principle of category theory that equivalent categories C, D should be thought of as being ‘the same’, and naturally isomorphic functors F, G : C → D should be thought of as being ‘the same’. Note that equivalence of categories C, D is much weaker than strict isomorphism: isomorphism classes of objects in C are naturally in bijection with isomorphism classes of objects in D, but there is no relation between the sizes of the isomorphism classes, so that C could have many more objects than D, for instance.

2.1.2 Limits, colimits and fibre products in categories We shall be interested in various kinds of limits and colimits in our categories of spaces. These are objects in the category with a universal property with respect to some class of diagrams. Definition 2.3. Let C be a category. A diagram ∆ in C is a class of objects Si in C for i ∈ I, and a class of morphisms ρj : Sb(j) → Se(j) in C for j ∈ J, where b, e : J → I. The diagram is called small if I, J are sets (rather than something too large to be a set), and finite if I, J are finite sets. A limit of the diagram ∆ is an object L in C and morphisms πi : L → Si for i ∈ I such that ρj ◦ πb(j) = πe(j) for all j ∈ J, with the universal property 0 0 that given L0 ∈ C and πi0 : L0 → Si for i ∈ I with ρj ◦ πb(j) = πe(j) for all j ∈ J, there is a unique morphism λ : L0 → L with πi0 = πi ◦ λ for all i ∈ I. The limit is called small, or finite, if ∆ is small, or finite. Here are some important kinds of limit:

2.1 Introduction to category theory

19

(i) A terminal object is a limit of the empty diagram. (ii) Let X, Y be objects in C. A product X × Y is a limit of the diagram with two objects X, Y and no morphisms. (iii) Let g : X → Z, h : Y → Z be morphisms in C. A fibre product is g h a limit of a diagram X −→ Z ←− Y . The limit object W is often written X ×g,Z,h Y or X ×Z Y . Explicitly, a fibre product is an object W and morphisms e : W → X and f : W → Y in C, such that g ◦ e = h ◦ f , with the universal property that if e0 : W 0 → X and f 0 : W 0 → Y are morphisms in C with g ◦ e0 = h ◦ f 0 then there is a unique morphism b : W 0 → W with e0 = e ◦ b and f 0 = f ◦ b. The commutative diagram W  X

f

e g

/Y  /Z

h

(2.1)

is called a Cartesian square. If Z is a terminal object then X ×Z Y is a product X × Y . A colimit of the diagram ∆ is an object L in C and morphisms λi : Si → L for i ∈ I such that λb(j) = λe(j) ◦ ρj for all j ∈ J, which has the universal property that given L0 ∈ C and λ0i : Si → L0 for i ∈ I with λ0b(j) = λ0e(j) ◦ ρj for all j ∈ J, there is a unique morphism π : L → L0 with λ0i = π ◦ λi for all i ∈ I. Here are some important kinds of colimit: (iv) An initial object is a colimit of the empty diagram. (v) Let X, Y be objects in C. A coproduct X q Y is a colimit of the diagram with two objects X, Y and no morphisms. (vi) Let e : W → X, f : W → Y be morphisms in C. A pushout is a colimit of a e

f

diagram X ←− W −→ Y . The colimit object Z is often written X qe,W,f Y or X qW Y . Explicitly, a pushout is an object Z and morphisms g : X → Z and h : Y → Z in C, such that g ◦ e = h ◦ f , with the universal property that if g 0 : X → Z 0 and h0 : Y → Z 0 are morphisms in C with g 0 ◦ e = h0 ◦ f then there is a unique morphism b : Z → Z 0 with g 0 = b ◦ g and h0 = b ◦ h. The diagram (2.1) is then called a co-Cartesian square. If W is an initial object then X qW Y is a coproduct X q Y . Limits and colimits may not may not exist. If a limit or colimit exists, it is unique up to canonical isomorphism in C. We say that all limits, or all small limits, or all finite limits exist in C, if limits exist for all diagrams, or all small diagrams, or all finite diagrams respectively; and similarly for colimits.

20

Background on C ∞ -schemes

A category C is called complete if all small limits exist in C, and cocomplete if all small colimits exist in C. Limits in C are equivalent to colimits (of the opposite diagram) in the opposite category C op . So for example, fibre products in C are pushouts in C op . A directed colimit is a colimit for which the diagram ∆ is an upward-directed set, that is, ∆ is a preorder (a category in which there is at most one morphism Si → Sj between any two objects) in which every finite subset has an upper bound. Confusingly, directed colimits are also called inductive limits or direct limits, although they are actually colimits. So we can say that all directed colimits exist in C. The dual concept, called codirected limit, will not be used in this book. By category theory general nonsense, one can prove: Proposition 2.4. Suppose a category C has a terminal object, and all fibre products exist in C. Then all finite limits exist in C. Example 2.5. Let C be a category of spaces, for instance, topological spaces Top, manifolds Man, schemes Sch, or C ∞ -schemes C∞ Sch in §2.5. Then: (i) The terminal object is the point ∗, and exists for all sensible C. (ii) Products X ×Y are the usual products of manifolds, topological spaces, . . . . (iii) Fibre products may or may not exist, depending on C. All fibre products  W = X×g,Z,h Y exist in Top, with W = (x, y) ∈ X×Y : g(x) = h(y) , with the subspace topology as a subset of X × Y . All fibre products also exist in Sch, C∞ Sch. Fibre products X ×g,Z,h Y exist in Man if g, h are transverse, but not in general. (iv) The initial object is the empty set ∅, and exists for all sensible C. (v) Coproducts X q Y are disjoint unions of the spaces X, Y . In Man this exists if dim X = dim Y . (vi) General pushouts in categories such as Man, Sch, C∞ Sch, . . . tend not to exist, and have not been a focus of research. Many important constructions in categories of spaces can be expressed as finite limits. For example, the intersection X ∩ Y of submanifolds X, Y ⊂ Z is a fibre product X ×Z Y . By Proposition 2.4, the existence of finite limits reduces to that of fibre products. So in our study of C ∞ -schemes with corners, we will be particularly interested in existence and properties of fibre products. Example 2.6. Let C be a category of (generalized) commutative algebras over a field K, for instance, commutative C-algebras AlgC , or C ∞ -rings C∞ Rings in §2.2 with K = R. Then:

21

2.1 Introduction to category theory

(i) The terminal object is the zero algebra 0. (ii) Products B × C are direct sums B ⊕ C. (iii) For morphismsβ : B → D, γ : C → D, the fibre product B ×β,D,γ C is the subalgebra (b, c) ∈ B ⊕ C : β(b) = γ(c) in B ⊕ C. (iv) The initial object is the field K. (v) Coproducts B q C are (possibly completed) tensor products B ⊗K C. (vi) For morphisms α : A → B, β : A → C, a pushout is a (possibly completed) tensor product B ⊗α,A,β C.

2.1.3 Adjoint functors Definition 2.7. Let C, D be categories. An adjunction (F, G, ϕ) between C and D consists of functors F : C → D and G : D → C and bijections ∼ =

ϕ(X, Y ) : HomD (F (X), Y ) −→ HomC (X, G(Y )) for all objects X ∈ C and Y ∈ D, which are natural in X, Y , that is, if f : X1 → X2 and g : Y1 → Y2 are morphisms in C, D then the following commutes: HomD (F (X2 ), Y1 )

g◦−

ϕ(X ,Y )

2 1  HomC (X2 , G(Y1 ))

/ HomD (F (X2 ), Y2 )

−◦F (f )

/ HomD (F (X1 ), Y2 ) ϕ(X ,Y )

−◦f

1 2  / HomC (X1 , G(Y1 )) G(g)◦− / HomC (X1 , G(Y2 )).

Adjunctions are often written like this: Co

F

/ D.

G

Then we say that F is left adjoint to G, and G is right adjoint to F . We say that F : C → D has a right adjoint if it can be completed to an adjunction (F, G, ϕ). We say that G : D → C has a left adjoint if it can be completed to an adjunction (F, G, ϕ). Suppose C is a category, and D ⊂ C is a full subcategory of C. We say that D is a reflective subcategory of C if the inclusion inc : D ,→ C has a left adjoint. This left adjoint R : C → D is called a reflection functor. Dually, if C ⊂ D is a full subcategory, we say C is a coreflective subcategory of D if the inclusion inc : C ,→ D has a right adjoint. This right adjoint C : D → C is called a coreflection functor. Here are some properties of adjoint functors:

22

Background on C ∞ -schemes

Theorem 2.8. (a) In Definition 2.7 there are natural transformations η : IdC ⇒ G ◦ F, called the unit of the adjunction, and  : F ◦ G ⇒ IdD , called the counit of the adjunction, such that for X ∈ C and Y ∈ D we have η(X) = ϕ(X, F (X))(idF (X) ) : X −→ G(F (X)), (Y ) = ϕ(G(Y ), Y )−1 (idG(Y ) ) : F (G(Y )) −→ Y. (b) F, G are both equivalences of categories if and only if F, G are both full and faithful, if and only if η,  are both natural isomorphisms. (c) If F : C → D has a right adjoint, then the right adjoint G : D → C is determined up to natural isomorphism by F . (d) If G : D → C has a left adjoint, then the left adjoint F : C → D is determined up to natural isomorphism by G. (e) If F : C → D has a right adjoint then it preserves colimits, that is, F maps a colimit in C to the corresponding colimit in D (which is guaranteed to exist in D, if the initial colimit exists in C). (f) If G : D → C has a left adjoint then it preserves limits, that is, G maps a limit in D to the corresponding limit in C (which is guaranteed to exist in C, if the initial limit exists in D). (g) Let D ⊂ C be a reflective subcategory, with reflection functor R : C → D. Suppose some class of colimits (e.g. all small colimits, or all pushouts) exists in C. Then the same class of colimits exists in D. We can obtain the colimit of a diagram in D by taking the colimit in C and then applying R. (h) Let C ⊂ D be a coreflective subcategory, with coreflection functor C : D → C. Suppose some class of limits (e.g. all small limits, or all fibre products) exists in D. Then the same class of limits exists in C. We can obtain the limit of a diagram in C by taking the limit in D and then applying C. Remark 2.9. We will use adjoint functors in two main ways below. Firstly, by Theorem 2.8(e)–(h), we can use them to prove results on existence of (co)limits. The second is more philosophical, and we illustrate it by two examples: (a) In §2.5 below we define an adjunction LC∞ RS o

Γ

/ C∞ Rings

Spec

between the categories C∞ Rings of C ∞ -rings and LC∞ RS of locally C ∞ -ringed spaces. Here the definition of the global sections functor Γ is simple and obvious, but that of the spectrum functor Spec is complicated

2.2 C ∞ -rings

23

and apparently arbitrary. However, Theorem 2.8(c) implies that Spec is determined up to natural isomorphism by Γ and the adjoint property. This justifies the definition of Spec, showing it could not have been otherwise. (b) In §6.2 below we define an adjunction C∞ Schcin o

inc

/ C∞ Schc .

C

Here C∞ Schc is the category of C ∞ -schemes with corners, and C∞ Schcin the non-full category with only interior morphisms, with inclusion inc : C∞ Schcin ,→ C∞ Schc , and C : C∞ Schc → C∞ Schcin is the ‘corner functor’, which encodes the notions of boundary ∂X and k-corners Ck (X) of a (firm) C ∞ -scheme with corners X in a functorial way. Again, the definition of inc is simple and obvious, and that of C is complicated and contrived. But the adjoint property shows that C is determined up to natural isomorphism by inc, and so justifies the definition.

2.2 C ∞ -rings Here are two equivalent definitions of C ∞ -ring. The first definition describes C ∞ -rings as functors while the second definition describes C ∞ -rings as sets in the style of classical algebra. Definition 2.10. Write Man for the category of manifolds, and Euc for the full subcategory of Man with objects the Euclidean spaces Rn . That is, the objects of Euc are Rn for n = 0, 1, 2, . . . , and the morphisms in Euc are smooth maps f : Rm → Rn . Write Sets for the category of sets. In both Euc and Sets we have notions of (finite) products of objects (that is, Rm+n = Rm × Rn , and products S × T of sets S, T ), and products of morphisms. Define a categorical C ∞ -ring to be a product-preserving functor F : Euc → Sets. Here F should also preserve the empty product, that is, it maps R0 in Euc to the terminal object in Sets, the point ∗. If F, G : Euc → Sets are categorical C ∞ -rings, a morphism η : F → G is a natural transformation η : F ⇒ G. We write CC∞ Rings for the category of categorical C ∞ -rings. Categorical C ∞ -rings are an example of an Algebraic Theory in the sense of Ad´amek, Rosick´y and Vitale [1], and many of the basic categorical properties of C ∞ -rings follow from this. Definition 2.11. A C ∞ -ring is a set C together with operations p n copies q

Φf : C n = C × · · · × C −→ C

24

Background on C ∞ -schemes

for all n > 0 and smooth maps f : Rn → R, where by convention when n = 0 we define C 0 to be the single point {∅}. These operations must satisfy the following relations: suppose m, n > 0, and fi : Rn → R for i = 1, . . . , m and g : Rm → R are smooth functions. Define a smooth function h : Rn → R by
h(x1 , . . . , xn ) = g f1 (x1 , . . . , xn ), . . . , fm (x1 . . . , xn ) , for all (x1 , . . . , xn ) ∈ Rn . Then for all (c1 , . . . , cn ) ∈ C n we have
Φh (c1 , . . . , cn ) = Φg Φf1 (c1 , . . . , cn ), . . . , Φfm (c1 , . . . , cn ) . We also require that for all 1 6 j 6 n, defining πj : Rn → R by πj : (x1 , . . . , xn ) 7→ xj , we have Φπj (c1 , . . . , cn ) = cj for all (c1 , . . . , cn ) ∈ C n . Usually we refer to C as the C ∞ -ring, leaving the C ∞ -operations Φf implicit.

n ∞ A morphism of C ∞ -rings C, (Φf )f :Rn →R C ∞ is a
, D, (Ψf )f :R →R C map φ : C → D such that Ψf φ(c1 ), . . . , φ(cn ) = φ ◦ Φf (c1 , . . . , cn ) for all smooth f : Rn → R and c1 , . . . , cn ∈ C. We will write C∞ Rings for the category of C ∞ -rings. Each C ∞ -ring has an underlying commutative R-algebra structure. The addition map f : R2 → R, f : (x, y) 7→ x + y gives addition ‘+’ on C as an Ralgebra by c + d = Φf (c, d) for c, d ∈ C. The multiplication map g : R2 → R, g : (x, y) 7→ xy gives multiplication ‘ · ’ on C by c · d = Φg (c, d). For each λ ∈ R write λ0 : R → R, λ0 : x 7→ λx, and then scalar multiplication is λc = Φλ0 (c). Let 00 , 10 : R0 → R map ∗ to 0, 1. Then 0 = Φ00 and 1 = Φ10 are the zero element and identity element for C. The projection and composition relations show this gives C the structure of a commutative R-algebra. However, an R-algebra only allows for operations corresponding to polynomials, whereas a C ∞ -ring allows for operations corresponding to all smooth functions and so has a richer structure. Proposition 2.12. There is an equivalence C∞ Rings ∼ = CC∞ Rings. This ∞ identifies C Rings with F : Euc → Sets in CC∞ Rings such that
C in n n F R = C for n > 0, and for smooth f : Rn → R then F (f ) is identified with Φf . Proving Proposition 2.12 is straightforward and relies on F being productpreserving. We leave it as an exercise to guide the reader’s intuition. The following example motivates these definitions: Example 2.13. (a) Let X be a manifold. Define a functor FX : Euc → Sets by FX (Rn ) = HomMan (X, Rn ), and FX (g) = g◦ : HomMan (X, Rm ) → HomMan (X, Rn ) for each morphism g : Rm → Rn in Euc. Then FX is a

2.2 C ∞ -rings

25

categorical C ∞ -ring. If f : X → Y is a smooth map of manifolds define a natural transformation Ff : FY ⇒ FX by Ff (Rn ) = ◦f : HomMan (Y, Rn ) → HomMan (X, Rn ). Then Ff is a morphism in CC∞ Rings. Define a functor CC∞ Rings FMan : Man → CC∞ Ringsop to map X 7→ FX and f 7→ Ff . (b) Let X be a manifold. Write C ∞ (X) for the set of smooth functions c : X → R. For non-negative integers n and smooth f : Rn → R, define C ∞ operations Φf : C ∞ (X)n → C ∞ (X) by composition

Φf (c1 , . . . , cn ) (x) = f c1 (x), . . . , cn (x) , (2.2) for all c1 , . . . , cn ∈ C ∞ (X) and x ∈ X. The composition and projection relations follow directly from the definition of Φf , so that C ∞ (X) forms a C ∞ -ring. If we consider the R-algebra structure of C ∞ (X) as a C ∞ -ring, this is the canonical R-algebra structure on C ∞ (X). If f : X → Y is a smooth map of manifolds, then f ∗ : C ∞ (Y ) → C ∞ (X) mapping c 7→ c ◦ f is a morphism of C ∞ -rings. C∞ Rings Define FMan : Man → C∞ Ringsop to map X 7→ C ∞ (X) and f 7→ C∞ Rings f ∗ . Moerdijk and Reyes [77, Th. I.2.8] show that FMan is full and faithful, and takes transverse fibre products in Man to fibre products in C∞ Ringsop . This fact is non-trivial, as it relies on knowing that all manifolds X can be embedded as a closed subspace of Rn for some large n, that C ∞ (X) is a finitely generated C ∞ -ring (as defined later in Proposition 2.17) and that manifolds admit partitions of unity that behave well with respect to smooth maps. There are many more C ∞ -rings than those that come from manifolds. For example, if X is a smooth manifold of positive dimension, then the set C k (X) of k-differentiable maps f : X → R is a C ∞ -ring with operations Φf defined as in (2.2), and each of these C ∞ -rings is different for all k = 0, 1, . . . . Example 2.14. Consider X = ∗ the point, so dim X = 0, then C ∞ (∗) = R and Example 2.13 shows the C ∞ -operations Φf : Rn → R given by Φf (x1 , . . . , xn ) = f (x1 , . . . , xn ) make R into a C ∞ -ring. It is the initial object in C∞ Rings, and the simplest nonzero example of a C ∞ -ring. The zero C ∞ -ring is the set {0} where all C ∞ -operations Φf : {0} → {0} send 0 7→ 0, and this is the final object in C∞ Rings. Definition 2.15. An ideal I in C is an ideal in C when C is considered as a commutative R-algebra. We do not require it to be closed under all C ∞ operations, as this would force I = C. We can make the R-algebra quotient C/I into a C ∞ -ring using Hadamard’s Lemma. That is, if f : Rn → R is smooth, define ΦIf : (C/I)n → C/I by

ΦIf (c1 + I, . . . , cn + I) (x) = Φf c1 (x), . . . , cn (x) + I.

Background on C ∞ -schemes

26

Then Hadamard’s Lemma says for any smooth function f : Rn → R, there exists gi : R2n → R for i = 1, . . . , n, such that f (x1 , . . . , xn ) − f (y1 , . . . , yn ) =

n X (xi − yi )gi (x1 , . . . , xn , y1 , . . . , yn ). i=1

If d1 , . . . , dn are alternative choices for c1 , . . . , cn , then ci − di ∈ I for each i = 1, . . . , n and Φf (c1 , . . . , cn ) − Φf (d1 , . . . , dn ) =

n X

(ci − di )Φf (c1 , . . . , cn , d1 , . . . , dn )

i=1

lies in I, so ΦIf is independent of the choice of representatives c1 , . . . , cn in C and is well defined. The next definition and proposition come from Ad´amek et al. [1, Rem. 11.21, Prop.s 11.26, 11.28, 11.30 & Cor. 11.33]. Definition 2.16. If A is a set then by [1, Rem. 11.21] we can define the free A ∞ C of FA as C ∞ (RA ), where RA =  -ring F generated by A. We may think A (xa )a∈A : xa ∈ R . Explicitly, we define F to be the set of maps c : RA → R which depend smoothly on only finitely many variables xa , and operations Φf are defined as in (2.2). Regarding xa : RA → R as functions for a ∈ A, we have xa ∈ FA , and we call xa the generators of FA . Then FA has the universal property that if C is any C ∞ -ring then a choice of map α : A → C uniquely determines a morphism φ : FA → C with φ(xa ) = α(a) for a ∈ A. When A = {1, . . . , n} we have FA ∼ = C ∞ (Rn ), as in [53, Prop. III.5.1]. Proposition 2.17. (a) Every object C in C∞ Rings admits a surjective morphism φ : FA → C from some free C ∞ -ring FA . We call C finitely generated if this holds with A finite. The kernel of φ, ker(φ), is an ideal in FA and the quotient FA / ker(φ) is isomorphic to C. (b) Every object C in C∞ Rings fits into a coequalizer diagram FB

α

1- FA

φ

/ C,

(2.3)

β

that is, C is the colimit of FB ⇒ FA in C∞ Rings, where φ is automatically surjective. We call C finitely presented if this holds with A, B finite. Actually as any relation f = g in C is equivalent to the relation f − g = 0, we can simplify (2.3) by taking β to map xb 7→ 0 for all b ∈ B. This means that finitely presented is equivalent to requiring ker(φ) from Proposition 2.17(a) to

2.2 C ∞ -rings

27

be finitely generated as an ideal. But the analogue of this for C ∞ -rings with corners in §4.5 will not hold. In addition, C ∞ (Rn ) is not noetherian, so ideals in a finitely generated C ∞ -ring may not be finitely generated. This implies that finitely presented C ∞ -rings are a proper subcategory of finitely generated C ∞ -rings, in contrast to ordinary Algebraic Geometry where they are equal. We now study limits and colimits in C∞ Rings. For the pushout of morphisms φ : C → D, ψ : C → E in C∞ Rings, we write D qφ,C,ψ E or D qC E. In the special case C = R the coproduct D qR E will be written as D ⊗∞ E. Recall that coproduct of R-algebras A, B is the tensor product A ⊗ B, however D ⊗∞ E is usually different from their tensor product D ⊗ E, as discussed for C ∞ (Rn ) and C ∞ (Rm ) above. For example, for m, n > 0, then C ∞ (Rm ) ⊗∞ C ∞ (Rn ) ∼ = C ∞ (Rm+n ) as in [77, p. 22], which contains m n C ∞ (R ) ⊗ C ∞ (R ) but is larger than this, as it includes elements such as exp(f g) for f ∈ C ∞ (Rm ) and g ∈ C ∞ (Rn ). By Moerdijk and Reyes [77, p. 21–22] and Ad´amek et al. [1, Prop.s 1.21, 2.5 & Th. 4.5] we have: Proposition 2.18. The category C∞ Rings of C ∞ -rings has all small limits and all small colimits. The forgetful functor Π : C∞ Rings → Sets preserves limits and directed colimits, and can be used to compute such (co)limits pointwise, however it does not preserve general colimits such as pushouts. The proof of Proposition 2.18 is straightforward, firstly by proving that the small limits and directed colimits in the category of sets inherit their universal properties and a C ∞ -ring structure. Proving separately that coproducts exist follows from the universal property and considering simple cases. This includes that for m, n > 0 then the coproduct of C ∞ (Rn ) and C ∞ (Rm ) is C ∞ (Rn+m ), and that for ideals I, J we have the coproduct


C ∞ (Rn )/I ⊗∞ C ∞ (Rm )/J ∼ = C ∞ (Rn ) ⊗∞ C ∞ (Rm ) /(I, J). A similar result holds for all finitely generated C ∞ -rings. The proof then uses that any C ∞ -ring is a directed colimit of finitely generated C ∞ -rings to deduce the result. We will need local C ∞ -rings and localizations of C ∞ -rings to define local ∞ C -ringed spaces and C ∞ -schemes in §2.5. Definition 2.19. Recall that a local R-algebra is an R-algebra R with a unique maximal ideal m. The residue field of R is the field (isomorphic to) R/m. A C ∞ -ring C is called local if, regarded as an R-algebra, C is a local R-algebra with residue field R. The quotient morphism gives a (necessarily unique) morphism of C ∞ -rings π : C → R with the property that c ∈ C is invertible if

28

Background on C ∞ -schemes

and only if π(c) 6= 0. Equivalently, if such a morphism π : C → R exists with this property, then C is local with maximal ideal mC ∼ = Ker π. Write C∞ Ringslo ⊂ C∞ Rings for the full subcategory of local C ∞ -rings. Usually morphisms of local rings are required to send maximal ideals into maximal ideals. However, if φ : C → D is any morphism of local C ∞ -rings, we see that φ−1 (mD ) = mC as the residue fields in both cases are R, so there is no difference between local morphisms and morphisms for C ∞ -rings. Remark 2.20. We use the term ‘local C ∞ -ring’ following Dubuc [23, Def. 4] and the second author [43]. They are known by different names in other references, such as Archimedean local C ∞ -rings in [75, §3], C ∞ -local rings in Dubuc [23, Def. 2.13], and pointed local C ∞ -rings in [77, §I.3]. Moerdijk and Reyes [75–77] use ‘local C ∞ -ring’ to mean a C ∞ -ring which is a local R-algebra, and require no restriction on its residue field. The next proposition may be found in Moerdijk and Reyes [77, §I.3] and Dubuc [23, Prop. 5]. Proposition 2.21. All finite colimits exist in C∞ Ringslo , and agree with the corresponding colimits in C∞ Rings. In [25], the first author shows how to extend this theorem to small colimits, and how small limits also exist in C∞ Ringslo but usually do not agree with small limits in C∞ Rings. A key fact used in the proof is the existence of bump functions for Rn . Localizations of C ∞ -rings were studied in [22, 23, 50, 75–77]. Definition 2.22. A localization C(s−1 : s ∈ S) = D of a C ∞ -ring C at a subset S ⊂ C is a C ∞ -ring D and a morphism π : C → D such that π(s) is invertible in D (as an R-algebra) for all s ∈ S, which has the universal property that for any morphism of C ∞ -rings φ : C → E such that φ(s) is invertible in E for all s ∈ S, there is a unique morphism ψ : D → E with φ = ψ ◦ π. We call π : C → D the localization morphism for D. By adding an extra generator s−1 and extra relation s · s−1 − 1 = 0 for each s ∈ S to C, it can be shown that localizations C(s−1 : s ∈ S) always exist and are unique up to unique isomorphism. When S = {c} then C(c−1 ) ∼ = (C ⊗∞ C ∞ (R))/I, where I is the ideal generated by ι1 (c) · ι2 (x) − 1, and x is the generator of C ∞ (R), and ι1 , ι2 are the coproduct morphisms ι1 : C → C ⊗∞ C ∞ (R) and ι2 : C ∞ (R) → C ⊗∞ C ∞ (R). An example of this is that if f ∈ C ∞ (Rn ) is a smooth function, and U = f −1 (R\{0}) ⊆ Rn , then using partitions of unity one can show that C ∞ (U ) ∼ = C ∞ (Rn )(f −1 ), as in [77, Prop. I.1.6].

2.2 C ∞ -rings

29

Definition 2.23. A C ∞ -ring morphism x : C → R, where R is regarded as a C ∞ -ring as in Example 2.14, is called an R-point. Note that a map x : C → R is a morphism of C ∞ -rings whenever it is a morphism of the underlying Ralgebras, as in [77, Prop. I.3.6]. We define C x as the localization C x = C(s−1 : s ∈ C, x(s) 6= 0), and denote the projection morphism by πx : C → C x . Importantly, [76, Lem. 1.1] shows C x is a local C ∞ -ring. We will use R-points x : C → R to define our spectrum functor in §2.5. We can describe C x explicitly as in [50, Prop. 2.14]. Proposition 2.24. Let x : C → R be an R-point of a C ∞ -ring C, and consider the projection morphism πx : C → C x . Then C x ∼ = C/ Ker πx . This kernel is Ker πx = I where  I = c ∈ C : there exists d ∈ C with x(d) 6= 0 in R and c·d = 0 in C . (2.4) While this localization morphism πx : C → C x is surjective, general localizations of C ∞ -rings need not have surjective localization morphisms. Here is an important example of localization of C ∞ -rings. Example 2.25. Let Cp∞ (Rn ) be the set of germs of smooth functions c : Rn → R at p ∈ Rn for n > 0 and p ∈ Rn . We give Cp∞ (Rn ) a C ∞ -ring structure by using (2.2) on germs of functions. There are several equivalent definitions: (i) Cp∞ (Rn ) is the set of ∼-equivalence classes [U, c] of pairs (U, c), where p ∈ U ⊆ Rn is open and c : U → R is smooth, and (U, c) ∼ (U 0 , c0 ) if there exists open p ∈ U 00 ⊆ U ∩ U 0 with c|U 00 = c0 |U 00 . (ii) Cp∞ (Rn ) ∼ = C ∞ (Rn )/Ip , where Ip ⊂ C ∞ (Rn ) is the ideal of functions vanishing near p. (iii) Cp∞ (Rn ) ∼ = C ∞ (Rn )(f −1 : f ∈ C ∞ (Rn ), f (p) 6= 0).  n n ∞ ∞ Then Cp (R ) is local, with maximal ideal mp = [U, c] ∈ Cp (R ) : c(x) = 0 . Finally, we prove some facts about exponentials and logs in (local) C ∞ rings. These will be used in defining C ∞ -rings with corners. Proposition 2.26. (a) Let C be a C ∞ -ring. Then the C ∞ -operation Φexp : C → C induced by exp : R → R is injective. (b) Let C be a local C ∞ -ring, with morphism π : C → R. If a ∈ C with π(a) > 0 then there exists b ∈ C with Φexp (b) = a. This b is unique by (a). Proof For (a), let a ∈ C with b = Φexp (a) ∈ C. Then Φexp (−a) is the inverse b−1 of b. The map t 7→ exp(t) − exp(−t) is a diffeomorphism R → R. Let

30

Background on C ∞ -schemes

e : R → R be its inverse. Define smooth f : R2 → R by f (x, y) = e(x − y). Then f (exp t, exp(−t)) = t. Hence in the C ∞ -ring C we have Φf (b, b−1 ) = Φf (Φexp (a), Φexp ◦− (a)) = Φf ◦(exp,exp ◦−) (a) = Φid (a) = a. But b determines b−1 uniquely, so Φf (b, b−1 ) = a implies that b = Φexp (a) determines a uniquely, and Φexp : C → C is injective. For (b), choose smooth g, h : R → R with g(x) = log x for x > 21 π(a) > 0, h(π(a)) > 0, and h(x) = 0 for x 6 12 π(a). Set b = Φg (a) and c = Φh (a). Then c · (Φexp (b) − a) = Φh (a) · (Φexp ◦g (a) − a) = Φh(x)·(exp ◦g(x)−x) (a) = 0, since h(x) · (x − exp ◦g(x)) = 0. Also π(c) = h(π(a)) > 0, so c is invertible. Thus Φexp (b) = a.

2.3 Modules and cotangent modules of C ∞ -rings We discuss modules and cotangent modules for C ∞ -rings, following [50, §5]. Definition 2.27. A module M over a C ∞ -ring C is a module over C as a commutative R-algebra, and morphisms of C-modules are the usual morphisms of R-algebra modules. Denote µM : C × M → M the multiplication map, and write µM (c, m) = c · m for c ∈ C and m ∈ M . The category C-mod of C-modules is an abelian category. If a C-module M fits into an exact sequence C ⊗ Rn → M → 0 in C-mod then it is finitely generated; if it further fits into an exact sequence C ⊗ Rm → C ⊗ Rn → M → 0 it is finitely presented. This second condition is not automatic from the first as C ∞ -rings are generally not Noetherian. For a morphism φ : C → D of C ∞ -rings and M in C-mod we have φ∗ (M ) = M ⊗C D in D-mod, giving a functor φ∗ : C-mod → D-mod. For N in D-mod there is a C-module φ∗ (N ) = N with C-action µφ∗ (N ) (c, n) = µN (φ(c), n). This gives a functor φ∗ : D-mod → C-mod. Example 2.28. Let Γ∞ (E) be the collection of smooth sections e of a vector bundle E → X of a manifold X, so Γ∞ (E) is a vector space and a module over C ∞ (X). If λ : E → F is a morphism of vector bundles over X, then there is a morphism of C ∞ (X)-modules λ∗ : Γ∞ (E) → Γ∞ (F ) where λ∗ : e 7→ λ ◦ e. For each smooth map of manifolds f : X → Y there is a morphism of ∞ C -rings f ∗ : C ∞ (Y ) → C ∞ (X). Each vector bundle E → Y gives a vector bundle f ∗ (E) → X. Using (f ∗ )∗ : C ∞ (Y )-mod → C ∞ (X)-mod from

2.3 Modules and cotangent modules of C ∞ -rings 31
Definition 2.27, then (f ∗ )∗ Γ∞ (E) = Γ∞ (E)⊗C ∞ (Y ) C ∞ (X) is isomorphic
to Γ∞ f ∗ (E) in C ∞ (X)-mod. The definition of C-module only used the commutative R-algebra structure of C, however the cotangent module ΩC of C uses the full C ∞ -ring structure, making it smaller in some sense than the corresponding classical Algebraic Geometry version of K¨ahler differentials from [36, II 8]. Definition 2.29. Take a C ∞ -ring C and M ∈ C-mod, then a C ∞ -derivation is a map d : C → M that satisfies the following: for any smooth f : Rn → R and elements c1 , . . . , cn ∈ C, then dΦf (c1 , . . . , cn ) −

n P i=1

Φ ∂f (c1 , . . . , cn ) · dci = 0.

(2.5)

∂xi

This implies that d is R-linear and is a derivation of C as a commutative Ralgebra, that is, d(c1 c2 ) = c1 · dc2 + c2 · dc1 for all c1 , c2 ∈ C. The pair (M, d) is called a cotangent module for C if it is universal in the sense that for any M 0 ∈ C-mod with C ∞ -derivation d0 : C → M 0 , there exists a unique morphism of C-modules λ : M → M 0 with d0 = λ ◦ d. Then a cotangent module is unique up to unique isomorphism. We can explicitly construct a cotangent module for C by considering the free C-module over the symbols dc for all c ∈ C, and quotienting by all relations (2.5) for smooth f : Rn → R and elements c1 , . . . , cn ∈ C. We call this construction ‘the’ cotangent module of C, and write it as dC : C → ΩC . If we have a morphism of C ∞ -rings C → D then ΩD = φ∗ (ΩD ) can be considered as a C-module with C ∞ -derivation dD ◦φ : C → ΩD . The universal property of ΩC gives a unique morphism Ωφ : ΩC → ΩD of C-modules such that dD ◦ φ = Ωφ ◦ dC . From this we have a morphism of D-modules (Ωφ )∗ : ΩC ⊗C D → ΩD . If we have two morphisms of C ∞ -rings φ : C → D, ψ : D → E then uniqueness implies that Ωψ◦φ = Ωψ ◦ Ωφ : ΩC → ΩE . Here is our motivating example. Example 2.30. As in Example 2.28, if X is a manifold, its cotangent bundle T ∗ X is a vector bundle over X, and its global sections Γ∞ (T ∗ X) form a C ∞ (X)-module, with C ∞ -derivation d : C ∞ (X) → Γ∞ (T ∗ X), d : c 7→ dc the usual exterior derivative and equation (2.5) following from the chain rule. One can show that (Γ∞ (T ∗ X), d) has the universal property in Definition 2.29, and so forms a cotangent module for C ∞ (X). This is stated in [50, Ex. 5.4], and proved in greater generality in Theorem 7.7(a) below. If we have a smooth map of manifolds f : X → Y , then f ∗ (T ∗ Y ), ∗ T X are vector bundles over X, and the derivative df : f ∗ (T ∗ Y ) → T ∗ X

Background on C ∞ -schemes

32

is a vector bundle morphism. This induces a morphism of C ∞ (X)-modules (df )∗ : Γ∞ (f ∗ (T ∗ Y )) → Γ∞ (T ∗ X), which is identified with (Ωf ∗ )∗ from Definition 2.29 using that Γ∞ (f ∗ (T ∗ Y )) ∼ = Γ∞ (T ∗ Y ) ⊗C ∞ (Y ) C ∞ (X). This example shows that Definition 2.29 abstracts the notion of sections of a cotangent bundle of a manifold to a concept that is well defined for any C ∞ ring. Here are further helpful examples that we later generalize for C ∞ -rings with corners. Example 2.31. (a) Let A be a set and FA the free C ∞ -ring from Definition 2.16, with generators xa ∈ FA for a ∈ A. Then there is a natural isomorphism

 Ω FA ∼ = dxa : a ∈ A R ⊗R FA . (b) Suppose C is defined by a coequalizer diagram (2.3) in C∞ Rings. Then writing (xa )a∈A , (˜ xb )b∈B for the generators of FA , FB , we have an exact sequence



 γ /0 / dxa : a ∈ A ⊗R C δ / ΩC d˜ x b : b ∈ B R ⊗R C R
in C-mod, where if α, β in (2.3) map α : x ˜b 7→ fb (xa )a∈A , β : x ˜b 7→
gb (xa )a∈A , for fb , gb depending only on finitely many xa , then γ, δ are given by X  ∂fb
∂gb
 (xa )a∈A − (xa )a∈A dxa , δ(dxa ) = dC ◦φ(xa ). γ(d˜ xb ) = φ ∂xa ∂xa a∈A

Hence as in [50, Prop. 5.6], if C is finitely generated (or finitely presented) in the sense of Proposition 2.17, then ΩC is finitely generated (or finitely presented). Cotangent modules behave well under localization, as in the following proposition from [50, Prop. 5.7]. This proposition is used to prove the theorem that follows it, and we use it to generalize these results to C ∞ -rings with corners. Proposition 2.32. Let C be a C ∞ -ring, S ⊆ C, and D = C(s−1 : s ∈ S) be the localization of C at S with projection π : C → D, as in Definition 2.22. Then (Ωπ )∗ : ΩC ⊗C D → ΩD is an isomorphism of D-modules. Finally, here is [50, Th. 5.8] which shows how pushouts of C ∞ -rings give exact sequences of cotangent modules. Theorem 2.33. Suppose we are given a pushout diagram of C ∞ -rings: C

β α

 D

/E  / F,

δ γ

33

2.4 Sheaves so that F = D qC E. Then the following sequence of F-modules is exact: ΩC ⊗C,γ◦α F

ΩD ⊗D,γ F / ⊕ ΩE ⊗E,δ F

(Ωα )∗ ⊕−(Ωβ )∗

(Ωγ )∗ ⊕(Ωδ )∗

/ ΩF

/ 0. (2.6)

Here (Ωα )∗ : ΩC ⊗C,γ◦α F → ΩD ⊗D,γ F is induced by Ωα : ΩC → ΩD , and so on. Note the sign of −(Ωβ )∗ in (2.6).

2.4 Sheaves In this section we explain presheaves and sheaves with values in a (nice) category A, following Godement [33] and MacLane and Moerdijk [67]. Throughout we suppose A is complete, that is, all small limits exist in A, and cocomplete, that is, all small colimits exist in A. The categories of sets, abelian groups, rings, C ∞ -rings, monoids etc. all satisfy this, as will (interior) C ∞ rings with corners. Sometimes it is helpful to suppose that objects of A are sets with extra structure, so there is a faithful functor A → Sets taking each object to its underlying set. We use presheaves and sheaves and the facts that follow to define and study local C ∞ -ringed spaces and C ∞ -schemes. Definition 2.34. A presheaf E on a topological space X valued in A gives an object E(U ) ∈ A for every open set U ⊆ X, and a morphism ρU V : E(U ) → E(V ) in A called the restriction map for every inclusion V ⊆ U ⊆ X of open sets, satisfying the conditions that (i) ρU U = idE(U ) : E(U ) → E(U ) for all open U ⊆ X; and (ii) ρU W = ρV W ◦ ρU V : E(U ) → E(W ) for all open W ⊆ V ⊆ U ⊆ X. A presheaf E is called a sheaf if for all open covers {Ui }i∈I of U , then Y Y E(U ) → E(Ui ) ⇒ E(Ui ∩ Uj ) i∈I

i,j∈I

forms an equalizer diagram in A. This implies: (iii) E(∅) = 0 where 0 is the final object in A. If there is a faithful functor F : A → Sets taking an object of A to its underlying set that preserves limits, then a presheaf E valued in A on X is a sheaf if it equivalently satisfies the following: (iv) (Uniqueness.) If U ⊆ X is open, {Vi : i ∈ I} is an open cover of U , and s, t ∈ F (E(U )) with F (ρU Vi )(s) = F (ρU Vi )(t) in F (E(Vi )) for all i ∈ I, then s = t in F (E(U )); and

34

Background on C ∞ -schemes

(v) (Gluing.) If U ⊆ X is open, {Vi : i ∈ I} is an open cover of U , and we are given elements si ∈ F (E(Vi )) for all i ∈ I such that F (ρVi (Vi ∩Vj ) )(si ) = F (ρVj (Vi ∩Vj ) )(sj ) in F (E(Vi ∩ Vj )) for all i, j ∈ I, then there exists s ∈ F (E(U )) with F (ρU Vi )(s) = si for all i ∈ I. If s ∈ F (E(U )) and V ⊆ U is open we write s|V = F (ρU V )(s). If E, F are presheaves or sheaves valued in A on X, then a morphism φ : E → F is a morphism φ(U ) : E(U ) → F(U ) in A for all open U ⊆ X such that the following diagram commutes for all open V ⊆ U ⊆ X E(U )

φ(U )

ρ0U V

ρU V

 E(V )

/ F(U )

φ(V )

 / F(V ),

where ρU V is the restriction map for E, and ρ0U V the restriction map for F. We write PreSh(X, A) and Sh(X, A) for the categories of presheaves and sheaves on a topological space X valued in A. Definition 2.35. For E a presheaf valued in A on a topological space X, then we can define the stalk E x ∈ A at a point x ∈ X to be the direct limit of the E(U ) in A for all U ⊆ X with x ∈ U , using the restriction maps ρU V . If there is a faithful functor F : A → Sets taking an object of A to its underlying set that preserves colimits, then explicitly it can be written as a set of equivalence classes of sections s ∈ F (E(U )) for any open U which contains x, where the equivalence relation is such that s1 ∼ s2 for s1 ∈ F (E(U )) and s2 ∈ F (E(V )) with x ∈ U, V if there is an open set W ⊂ V ∩ U with x ∈ W and s1 |W = s2 |W in F (E(W )). The stalk is an object of A, and the restriction morphisms give rise to morphisms ρU,x : E(U ) → E x . A morphism of presheaves φ : E → F induces morphisms φx : E x → F x for all x ∈ X. If E, F are sheaves then φ is an isomorphism if and only if φx is an isomorphism for all x ∈ X. Definition 2.36. There is a sheafification functor PreSh(X, A) → Sh(X, A), which is left adjoint to the inclusion Sh(X, A) ,→ PreSh(X, A). We write Eˆ for the sheafification of a presheaf E. The adjoint property gives a morphism π : E → Eˆ and a universal property: whenever we have a morphism φ : E → F of presheaves on X and F is a sheaf, then there is a unique morphism φˆ : Eˆ → F with φ = φˆ ◦ π. Thus sheafification is unique up to canonical isomorphism. Sheafifications always exist for our categories A, and there are isomorphisms of stalks E x ∼ = Eˆ x for all x ∈ X. If there is a faithful functor F : A → Sets taking an object of A to its underlying set that preserves colimits and limits, it ˆ ) as the subset of all can be constructed (as in [36, Prop. II.1.2]) by defining E(U

2.5 C ∞ -schemes

35

functions t : U → qx∈U E x such that for all x ∈ U , then t(x) = F (ρV,x )(s) ∈ E x for some s ∈ F (E(V )) for open V ⊂ U , x ∈ V . If f : X → Y is a continuous map of topological spaces, we can consider pushforwards and pullbacks of sheaves by f . We will use both of these definitions when defining C ∞ -schemes (with corners). Definition 2.37. If f : X → Y is a continuous map of topological spaces, and E is a sheaf valued in A on X,
then the direct image
(or pushforward) sheaf −1 f∗ (E) on Y is defined by f∗ (E) (U ) = E f (U ) for all
open U ⊆ V . Here,
we have restriction maps ρ0U V = ρf −1 (U )f −1 (V ) : f∗ (E) (U ) → f∗ (E) (V ) for all open V ⊆ U ⊆ Y so that f∗ (E) is a sheaf valued in A on Y . For a morphism A) we can define f∗ (φ) : f∗ (E) →
φ : E → F−1in Sh(X,
f∗ (F) by f∗ (φ) (U ) = φ f (U ) for all open U ⊆ Y . This gives a morphism f∗ (φ) in Sh(Y, A), and a functor f∗ : Sh(X, A) → Sh(Y, A). For two continuous maps of topological spaces, f : X → Y , g : Y → Z, then (g ◦ f )∗ = g∗ ◦ f∗ . Definition 2.38. For a continuous map f : X → Y and a sheaf E valued in A on Y , we define the pullback (inverse image) of E under f to be the sheafification of the presheaf U 7→ limA⊇f (U ) E(A) for open U ⊆ X, where the direct limit is taken over all open A ⊆ Y containing f (U ), using the restriction maps ρAB in E. We write this sheaf as f −1 (E). If φ : E → F is a morphism in Sh(Y, A), there is a pullback morphism f −1 (φ) : f −1 (E) → f −1 (F). Remark 2.39. For a continuous map f : X → Y of topological spaces we have functors f∗ : Sh(X, A) → Sh(Y, A), and f −1 : Sh(Y, A) → Sh(X, A). Hartshorne [36, Ex. II.1.18] gives a natural bijection

HomX f −1 (E), F ∼ (2.7) = HomY E, f∗ (F) for all E ∈ Sh(Y, A) and F ∈ Sh(X, A), so that f∗ is right adjoint to f −1 , as in §2.1.3. This will be important in several proofs below.

2.5 C ∞ -schemes We recall the definition of (local) C ∞ -ringed spaces, following [50]. These allow us to define a spectrum functor and C ∞ -schemes. Definition 2.40. A C ∞ -ringed space X = (X, OX ) is a topological space X with a sheaf OX of C ∞ -rings on X.

36

Background on C ∞ -schemes

A morphism f = (f, f ] ) : (X, OX ) → (Y, OY ) of C ∞ ringed spaces consists of a continuous map f : X → Y and a morphism f ] : f −1 (OY ) → OX of sheaves of C ∞ -rings on X, for f −1 (OY ) the inverse image sheaf as in Definition 2.38. From (2.7), we know f∗ is right adjoint to f −1 , so there is a natural bijection

(2.8) HomX f −1 (OY ), OX ∼ = HomY OY , f∗ (OX ) . We will write f] : OY → f∗ (OX ) for the morphism of sheaves of C ∞ -rings on Y corresponding to the morphism f ] under (2.8), so that f ] : f −1 (OY ) −→ OX

!

f] : OY −→ f∗ (OX ).

(2.9)

Given two C ∞ -ringed space morphisms f : X → Y and g : Y → Z we can compose them to form

g ◦ f = g ◦ f, (g ◦ f )] = g ◦ f, f ] ◦ f −1 (g ] ) . If we consider f] : OY → f∗ (OX ), then the composition is (g ◦ f )] = g∗ (f] ) ◦ g] : OZ −→ (g ◦ f )∗ (OX ) = g∗ ◦ f∗ (OX ). We call X = (X, OX ) a local C ∞ -ringed space if it is C ∞ -ringed space for which the stalks OX,x of OX at x are local C ∞ -rings for all x ∈ X. As in Definition 2.19, since morphisms of local C ∞ -rings are automatically local morphisms, morphisms of local C ∞ -ringed spaces (X, OX ), (Y, OY ) are just morphisms of C ∞ -ringed spaces without any additional locality condition. Local C ∞ -ringed spaces are called Archimedean C ∞ -spaces in Moerdijk, van Quˆe and Reyes [75, §3]. We will follow the notation of [50] and write C∞ RS for the category of ∞ C -ringed spaces, and LC∞ RS for the full subcategory of local C ∞ -ringed spaces. We write underlined upper case letters such as X, Y , Z, . . . to represent C ∞ -ringed spaces (X, OX ), (Y, OY ), (Z, OZ ), . . . , and underlined lower case letters f , g, . . . to represent morphisms of C ∞ -ringed spaces (f, f ] ), (g, g ] ), . . . . When we write ‘x ∈ X’ we mean that X = (X, OX ) and x ∈ X. If we write ‘U is open in X’ we will mean that U = (U, OU ) and X = (X, OX ) with U ⊆ X an open set and OU = OX |U . Here is our motivating example. Example 2.41. For a manifold X, we have a C ∞ -ringed space X = (X, OX ) with topological space X and its sheaf of smooth functions OX (U ) = C ∞ (U ) for each open subset U ⊆ X, with C ∞ (U ) defined in Example 2.13. If V ⊆ U ⊆ X then the restriction morphisms ρU V : C ∞ (U ) → C ∞ (V ) are the usual restriction of a function to an open subset ρU V : c 7→ c|V .

2.5 C ∞ -schemes

37

As the stalks OX,x at x ∈ X are local C ∞ -rings, isomorphic to the ring of germs as in Example 2.25, then using partitions of unity we can show that X is a local C ∞ -ringed space. For a smooth map of manifolds f : X → Y with corresponding local ∞ C -ringed spaces (X, OX ), (Y, OY ) as above we define f] (U ) : OY (U ) = C ∞ (U ) → OX (f −1 (U )) = C ∞ (f −1 (U )) for each open U ⊆ Y by f] (U ) : c 7→ c ◦ f for all c ∈ C ∞ (U ). This gives a morphism f] : OY → f∗ (OX ) of sheaves of C ∞ -rings on Y . Then f = (f, f ] ) : (X, OX ) → (Y, OY ) is a morphism of (local) C ∞ -ringed spaces with f ] : f −1 (OY ) → OX corresponding to f] under (2.9). To define a spectrum functor taking a C ∞ -ring to an element of LC∞ RS we require the following definition. Definition 2.42. Let C be a C ∞ -ring, and write XC for the set of all R-points x of C, as in Definition 2.22. Write TC for the topology on XC that has basis  of open sets Uc = x ∈ XC : x(c) 6= 0 for all c ∈ C. For each c ∈ C define a map c∗ : XC → R such that c∗ : x 7→ x(c). For a morphism φ : C → D of C ∞ -rings, we can define fφ : XD → XC by fφ (x) = x ◦ φ, which is continuous. From [50, Lem. 4.15], this definition implies that TC is the weakest topology on XC such that the c∗ : XC → R are continuous for all c ∈ C. Also (XC , TC ) is a regular, Hausdorff topological space. We now define the spectrum functor. Definition 2.43. For a C ∞ -ring C, we will define the spectrum of C, written Spec C. Here, Spec C is a local C ∞ -ringed space (X, OX ), with X the topological space XC from Definition 2.42. If U ⊆ X is open then OX (U ) is the ` set of functions s : U → x∈U C x , where we write sx for the image of x under s, such that around each point x ∈ U there is an open subset x ∈ W ⊆ U and element c ∈ C with sy = πy (c) ∈ C y for all y ∈ W . This is a C ∞ -ring with the operations Φf on OX (U ) defined using the operations Φf on C x for x ∈ U . For s ∈ OX (U ), the restriction map of functions s 7→ s|V for open V ⊆ U ⊆ X is a morphism of C ∞ -rings, giving the restriction map ρU V : OX (U ) → OX (V ). The stalk OX,x at x ∈ X is isomorphic to C x , which is a local C ∞ -ring. Hence (X, OX ) is a local C ∞ -ringed space. For a morphism φ : C → D of C ∞ -rings, we have an induced morphism of local C ∞ -rings, φx : C fφ (x) → Dx . If we let (X, OX ) = Spec C, (Y, OY ) = Spec D, then for open U ⊆ X define (fφ )] (U ) : OX (U ) → OY (fφ−1 (U )) by (fφ )] (U )s : x 7→ φx (sfφ (x) ). This gives a morphism (fφ )] : OX → (fφ )∗ (OY ) of sheaves of C ∞ -rings on X. Then f φ = (fφ , fφ] ) : (Y, OY ) →

38

Background on C ∞ -schemes

(X, OX ) is a morphism of local C ∞ -ringed spaces, where fφ] corresponds to (fφ )] under (2.9). Then Spec is a functor C∞ Ringsop → LC∞ RS, called the spectrum functor, where Spec φ : Spec D → Spec C is defined by Spec φ = f φ . This definition of spectrum functor is different to the classical Algebraic Geometry version [36, §II.2], as the topological space corresponds only to maximal ideals of the C ∞ -ring, instead of also including all prime ideals. In this sense, it is coarser, however this corresponds to the topology of a manifold as in the following example. Example 2.44. For a manifold X then Spec C ∞ (X) is isomorphic to the local C ∞ -ringed space X constructed in Example 2.41. Here is [50, Lem 4.28] which shows how the spectrum functor behaves with respect to localizations and open sets. A proof is contained in [25, Lem 2.4.6], which relies on the existence of bump functions for Rn . Lemma 2.45. Let C be a C ∞ -ring, set X = Spec C = (X, OX ), and let c ∈ C. If we write Uc = {x ∈ X : x(c) 6= 0} as in Definition 2.42, then Uc ⊆ X is open and U c = (Uc , OX |Uc ) ∼ = Spec C(c−1 ). We now define the global sections functor and describe its relationship to the spectrum functor. Definition 2.46. The global sections functor Γ : LC∞ RS → C∞ Ringsop takes (X, OX ) to OX (X) and morphisms (f, f ] ) : (X, OX ) → (Y, OY ) to Γ : (f, f ] ) 7→ f] (Y ), for f] relating f ] as in (2.9). For each C ∞ -ring C we can define a morphism ΞC : C → Γ ◦ Spec C. ` Here, for c ∈ C then ΞC (c) : XC → x∈XC C x is defined by ΞC (c)x = πx (c) ∈ C x , so ΞC (c) ∈ OXC (XC ) = Γ ◦ Spec C. This ΞC is a C ∞ -ring morphism as it is built from C ∞ -ring morphisms πx : C → C x , and the C ∞ -operations on OXC (XC ) are defined pointwise in the C x . This defines a natural transformation Ξ : IdC∞ Rings ⇒ Γ ◦ Spec of functors C∞ Rings → C∞ Rings. Theorem 2.47. The functor Spec : C∞ Ringsop → LC∞ RS is right adjoint to Γ : LC∞ RS → C∞ Ringsop , and Ξ is the unit of the adjunction. This implies that Spec preserves limits as in §2.1.3. Hence if we have C ∞ ring morphisms φ : F → D, ψ : F → E in C∞ Rings then their pushout C = D qF E has image that is isomorphic to the fibre product Spec C ∼ = Spec D ×Spec F Spec E. We extend this theorem to C ∞ -schemes with corners in §5.3.

2.6 Complete C ∞ -rings

39

Remark 2.48. Our definition of spectrum functor follows [50] and Dubuc [23], and is called the Archimedean spectrum in Moerdijk et al. [75, §3]. They also show it is a right adjoint to the global sections functor as above. Definition 2.49. Objects X ∈ LC∞ RS that are isomorphic to Spec C for some C ∈ C∞ Rings are called affine C ∞ -schemes. Elements X ∈ LC∞ RS that are locally isomorphic to Spec C for some C ∈ C∞ Rings (depending upon the open sets) are called C ∞ -schemes. We define C∞ Sch and AC∞ Sch to be the full subcategories of C ∞ schemes and affine C ∞ -schemes in LC∞ RS respectively. Remark 2.50. (a) Unlike ordinary Algebraic Geometry, affine C ∞ -schemes are very general objects. All manifolds are affine, and all their fibre products are affine. But not all manifolds with corners are affine C ∞ -schemes with corners. (b) (Alternatives to C ∞ -schemes.) We briefly review other generalizations of manifolds similar to C ∞ -schemes. Such generalizations usually fall into a ‘maps out’ (based on maps X → R) or a ‘maps in’ (based on maps Rn → X) approach. C ∞ -algebraic geometry uses ‘maps out’, as do the C ∞ -differentiable spaces of Navarro Gonz´alez and Sancho de Salas [80], which form a subcategory of C ∞ -schemes. Sikorski [88], Spallek [90], Buchner et al. [9] and Gonz´alez and Sancho de Salas [80] describe other ‘maps out’ approaches. ‘Maps in’ approaches include the diffeological spaces of Souriau [89] and Iglesias-Zemmour [39], and the various Chen spaces from Chen [15–18]. Usually ‘maps out’ approaches deal well with finite limits, and ‘maps in’ approaches work well for infinite dimensional spaces and quotient spaces.

2.6 Complete C ∞ -rings In ordinary Algebraic Geometry, if A is a commutative ring then Γ ◦ Spec A ∼ = op A, and Spec : Rings → ASch is an equivalence of categories, with inverse Γ. For C ∞ -rings C, in general Γ ◦ Spec C ∼ 6 C, and Spec : C∞ Ringsop → = ∞ AC Sch is neither full nor faithful. But as in [50, Prop. 4.34], we have: Proposition 2.51. For each C ∞ -ring C, Spec ΞC : Spec ◦Γ ◦ Spec C → Spec C is an isomorphism in LC∞ RS. This motivates the following definition [50, Def. 4.35]: Definition 2.52. A C ∞ -ring C is called complete if ΞC : C → Γ ◦ Spec C is an isomorphism. We define C∞ Ringsco to be the full subcategory in C∞ Rings

40

Background on C ∞ -schemes

of complete C ∞ -rings. By Proposition 2.51 we see that C∞ Ringsco is equivalent to the image of the functor Γ ◦ Spec : C∞ Rings → C∞ Rings, which gives a left adjoint to the inclusion of C∞ Ringsco into C∞ Rings. Write ∞ ∞ this left adjoint as the functor Πco all = Γ ◦ Spec : C Rings → C Ringsco . An example of a non-complete C ∞ -ring is the quotient C = C ∞ (Rn )/Ics of C ∞ (Rn ) for n > 0 by the ideal Ics of compactly supported functions, and ∼ Πco all (C) = 0 6= C. The next theorem comes from [50, Prop. 4.11 & Th. 4.25]. Theorem 2.53. (a) Spec |(C∞ Ringsco )op : (C∞ Ringsco )op → LC∞ RS is full and faithful, and an equivalence (C∞ Ringsco )op → AC∞ Sch. (b) Let X be an affine C ∞ -scheme. Then X ∼ = Spec OX (X), where OX (X) is a complete C ∞ -ring. ∞ ∞ (c) The functor Πco all : C Rings → C Ringsco is left adjoint to the inclu∞ ∞ sion functor inc : C Ringsco ,→ C Rings. That is, Πco all is a reflection functor.

(d) All small colimits exist in C∞ Ringsco , although they may not coincide with the corresponding small colimits in C∞ Rings. (e) Spec |(C∞ Ringsco )op = Spec ◦ inc : (C∞ Ringsco )op → LC∞ RS is ∞ ∞ op right adjoint to Πco all ◦ Γ : LC RS → (C Ringsco ) . Thus Spec |··· takes ∞ op limits in (C Ringsco ) (equivalently, colimits in C∞ Ringsco ) to limits in LC∞ RS. Using (a), that small limits exist in the category of C∞ Rings, and that Γ : LC∞ RS → C∞ Rings is a left adjoint with image in (C∞ Ringsco )op when restricted to AC∞ Sch, then small limits in C∞ Ringsco exist and coincide with small limits in C∞ Rings. As (C∞ Ringsco )op → AC∞ Sch is an equivalence of categories, then AC∞ Sch also has all small colimits and small limits. As Spec is a right adjoint, then limits in AC∞ Sch coincide with limits in C∞ Sch and LC∞ RS, however it is not necessarily true that colimits in AC∞ Sch coincide with colimits in C∞ Sch and LC∞ RS. In the following theorem we summarize results found in Dubuc [23, Th. 16], Moerdijk and Reyes [76, § II. Prop. 1.2], and the second author [50, Cor. 4.27]. ∞

AC Sch Theorem 2.54. There is a full and faithful functor FMan : Man → AC∞ Sch that takes a manifold X to the affine C ∞ -scheme X = (X, OX ), where OX (U ) = C ∞ (U ) is the usual smooth functions on U . Here (X, OX ) ∼ = AC∞ Sch Spec(C ∞ (X)) and hence X is affine. The functor FMan sends transverse fibre products of manifolds to fibre products of C ∞ -schemes.

2.7 Sheaves of OX -modules on C ∞ -ringed spaces

41

2.7 Sheaves of OX -modules on C ∞ -ringed spaces This section follows [50, §5.3], where we give the basics of sheaves of OX modules for C ∞ -ringed spaces. This includes the pullback of a sheaf of modules and the cotangent sheaf. Our definition of OX -module is the usual definition of sheaf of modules on a ringed space as in Hartshorne [36, §II.5] and Grothendieck [35, §0.4.1], using the R-algebra structure on our C ∞ -rings. The cotangent sheaf uses the cotangent modules of §2.3. Definition 2.55. For each C ∞ -ringed space X = (X, OX ) we define a category OX -mod. The objects are sheaves of OX -modules (or simply OX modules) E on X. Here, E is a functor on open sets U ⊆ X such that E : U 7→ E(U ) in OX (U )-mod is a sheaf as in Definition 2.34. This means we have linear restriction maps E U V : E(U ) → E(V ) for each inclusion of open sets V ⊆ U ⊆ X, such that the following commutes / E(U )

OX (U ) × E(U ) ρ

×E

EUV

 UV UV OX (V ) × E(V )

 / E(V ),

where the horizontal arrows are module multiplication. Morphisms in OX -mod are sheaf morphisms φ : E → F commuting with the OX -actions. An OX module E is called a vector bundle if it is locally free, that is, around every point there is an open set U ⊆ X with E|U ∼ = OX |U ⊗R Rn . Definition 2.56. We define the pullback f ∗ (E) of a sheaf of modules E on Y by a morphism f = (f, f ] ) : X → Y of C ∞ -ringed spaces as f ∗ (E) = f −1 (E) ⊗f −1 (OY ) OX . Here f −1 (E) is as in Definition 2.38, so that f ∗ (E) is a sheaf of modules on X. Morphisms of OY -modules φ : E → F give morphisms of OX -modules f ∗ (φ) = f −1 (φ) ⊗ idOX : f ∗ (E) → f ∗ (F). Definition 2.57. Let X = (X, OX ) be a C ∞ -ringed space. Define a presheaf PT ∗ X of OX -modules on X such that PT ∗ X(U ) is the cotangent module ΩOX (U ) of Definition 2.29, regarded as a module over the C ∞ -ring OX (U ). For open sets V ⊆ U ⊆ X we have restriction morphisms ΩρU V : ΩOX (U ) → ΩOX (V ) associated to the morphisms of C ∞ -rings ρU V : OX (U ) → OX (V ) so that the following commutes: OX (U ) × ΩOX (U )

µOX (U )

ρU V ×Ωρ

UV  OX (V ) × ΩOX (V )

/ ΩOX (U ) ΩρU V

µOX (V )

 / ΩOX (V ) .

42

Background on C ∞ -schemes

Definition 2.29 implies Ωψ◦φ = Ωψ ◦ Ωφ , so this is a well defined presheaf of OX -modules. The cotangent sheaf T ∗ X of X is the sheafification of PT ∗ X. The universal property of sheafification shows that for open U ⊆ X we have an isomorphism of OX |U -modules T ∗ U = T ∗ (U, OX |U ) ∼ = T ∗ X|U . For f : X → Y in C∞ RS we have f ∗ (T ∗ Y ) = f −1 (T ∗ Y ) ⊗f −1 (OY ) OX . The universal properties of sheafification imply that f ∗ (T ∗ Y ) is the sheafification of the presheaf P(f ∗ (T ∗ Y )), where U 7−→ P(f ∗ (T ∗ Y ))(U ) = limV ⊇f (U ) ΩOY (V ) ⊗OY (V ) OX (U ). This gives a presheaf morphism PΩf : P(f ∗ (T ∗ Y )) → PT ∗ X on X, where (PΩf )(U ) = limV ⊇f (U ) (Ωρf −1 (V ) U ◦f] (V ) )∗ . Here, we have morphisms f] (V ) : OY (V ) → OX (f −1 (V )) from f] : OY → f∗ (OX ) corresponding to f ] in f as in (2.9), and ρf −1 (V ) U : OX (f −1 (V )) → OX (U ) in OX so that (Ωρf −1 (V ) U ◦f] (V ) )∗ : ΩOY (V ) ⊗OY (V ) OX (U ) → ΩOX (U ) = (PT ∗ X)(U ) is constructed as in Definition 2.29. Then write Ωf : f ∗ (T ∗ Y ) → T ∗ X for the induced morphism of the associated sheaves. This corresponds to the morphism df : f ∗ (T ∗ Y ) → T ∗ X of vector bundles over a manifold X and smooth map of manifolds f : X → Y as in Example 2.30.

2.8 Sheaves of OX -modules on C ∞ -schemes We define the module spectrum functor MSpec as in [50, Def.s 5.16, 5.17 & 5.25], and its corresponding global sections functor, and recall their properties. Definition 2.58. Let C be a C ∞ -ring and set X = (X, OX ) = Spec C. Let M ∈ C-mod be a C-module. For each open subset U ⊆ X there is a natural morphism C → OX (U ) in C∞ Rings. Using this we make M ⊗C OX (U ) into an OX (U )-module. This assignment U 7→ M ⊗C OX (U ) is naturally a presheaf P MSpec M of OX -modules. Define MSpec M ∈ OX -mod to be its sheafification. A morphism µ : M → N in C-mod induces OX (U )-module morphisms M ⊗C OX (U ) → N ⊗C OX (U ) for all open U ⊆ X, and hence a presheaf morphism, which descends to a morphism MSpec µ : MSpec M → MSpec N in OX -mod. This defines a functor MSpec : C-mod → OX -mod. It is an exact functor of abelian categories. There is also a global sections functor Γ : OX -mod → C-mod mapping

2.8 Sheaves of OX -modules on C ∞ -schemes

43

Γ : E 7→ E(X), where the OX (X)-module E(X) is viewed as a C-module via the natural morphism C → OX (X). For any M ∈ C-mod there is a natural morphism ΞM : M → Γ ◦ MSpec M in C-mod, by composing M → M ⊗C OX (X) = P MSpec M (X) with the sheafification morphism P MSpec M (X) → MSpec M (X) = Γ ◦ MSpec M . Generalizing Definition 2.52, we call M complete if ΞM is an isomorphism. Write C-modco ⊆ C-mod for the full subcategory of complete C-modules. Here is [50, Th.s 5.19, Prop. 5.20, Th. 5.26, & Prop. 5.31]: Theorem 2.59. (a) In Definition 2.58, MSpec : C-mod → OX -mod is left adjoint to Γ : OX -mod → C-mod, generalizing Theorem 2.47. (b) There is a natural isomorphism MSpec ◦Γ ⇒ IdOX -mod . This gives a natural isomorphism MSpec ◦Γ ◦ MSpec ⇒ MSpec, generalizing Proposition 2.51. (c) MSpec |C-modco : C-modco → OX -mod is an equivalence of categories, generalizing Theorem 2.53(a). (d) The functor Πco all = Γ ◦ MSpec : C-mod → C-modco is left adjoint to the inclusion functor inc : C-modco ,→ C-mod, generalizing Theorem 2.53(c). That is, Πco all is a reflection functor. (e) There is a natural isomorphism T ∗ X ∼ = MSpec ΩC in OX -mod. Remark 2.60. (a) In [50, §5.4], following conventional Algebraic Geometry as in Hartshorne [36, §II.5], the first author defined a notion of quasicoherent sheaf E on a C ∞ -scheme X, which is that we may cover X by open U ⊆ X with U ∼ = Spec C and E|U ∼ = MSpec M for C ∈ C∞ Rings and M ∈ C-mod. But then [50, Cor. 5.22] uses Theorem 2.59(c) to show that every OX module is quasicoherent, that is, qcoh(X) = OX -mod, which is not true in conventional Algebraic Geometry. So here we will not bother with the language of quasicoherent sheaves. (b) In conventional Algebraic Geometry one also defines coherent sheaves [36, §II.5] to be OX -modules E locally modelled on MSpec M for M a finitely generated C-module. However, as in [50, Rem. 5.23(b)], coherent sheaves are only well-behaved on noetherian C ∞ -schemes, and most interesting C ∞ rings, such as C ∞ (Rn ) for n > 0, are not noetherian. So coherent sheaves do not seem to be a useful idea in C ∞ -algebraic geometry. For example, coh(X) is not closed under kernels in OX -mod, and is not an abelian category. Here is [50, Th. 5.32], where part (b) is deduced from Theorem 2.33:

Background on C ∞ -schemes

44

Theorem 2.61. (a) Let f : X → Y and g : Y → Z be morphisms of C ∞ schemes. Then in OX -mod we have Ωg◦f = Ωf ◦ f ∗ (Ωg ) : (g ◦ f )∗ (T ∗ Z) −→ T ∗ X. (b) Suppose we are given a Cartesian square in C∞ Sch: W

f

e

 X

/Y h

g

 / Z,

so that W = X ×Z Y . Then the following is exact in OW -mod : (g◦e)∗ (T ∗ Z)

e∗ (Ωg )⊕−f ∗ (Ωh )

/ e∗ (T ∗ X)⊕f ∗ (T ∗ Y )

Ωe ⊕Ωf

/ T ∗W

/ 0.

2.9 Applications of C ∞ -rings and C ∞ -schemes Since the work of Grothendieck, the theory of schemes in Algebraic Geometry has become an enormously powerful tool, and the language in which most modern Algebraic Geometry is written. As a result, Algebraic Geometers are far better at dealing with singular spaces than Differential Geometers are. It seems desirable to have a theory of schemes in Differential Geometry — C ∞ -schemes, or something similar — that could in future be used for the same purposes as schemes in Algebraic Geometry. For example, it seems very likely that many moduli spaces M of differential-geometric objects are naturally C ∞ -schemes, as well as topological spaces. As another example, suppose (X, g) is a Riemannian manifold, and we are interested in the moduli space M of some class of special embedded submanifolds Y ⊂ X, e.g. minimal, or calibrated. We could imagine trying to define a compactification M of M by regarding submanifolds Y ⊂ X as C ∞ subschemes of X, and taking the closure of M in the space of C ∞ -subschemes. For the present, to the authors’ knowledge, applications of C ∞ -algebraic geometry in the literature are confined to two areas: Synthetic Differential Geometry and Derived Differential Geometry, which we now discuss.

2.9.1 Synthetic Differential Geometry Synthetic Differential Geometry is a subject in which one proves theorems about manifolds in Differential Geometry using ‘infinitesimals’. It was used non-rigorously in the 19th century by authors such as Sophus Lie. In the 1960’s

2.9 Applications of C ∞ -rings and C ∞ -schemes

45

William Lawvere [60] suggested a way to make it rigorous, and the subject has since been developed in detail by Anders Kock [53, 54] and others. One supposes that the real numbers R can be enlarged to a ‘number line’ R ⊃ R, a ring containing nonzero ‘infinitesimal’ elements x ∈ R with xn = 0 for some n > 1. An important rˆole in the theory is played by the ‘double point’  D = x ∈ R : x2 = 0 .

(2.10)

One assumes smooth functions f : R → R, and manifolds X, can all be enlarged by infinitesimals in this way. Here are examples of how these are used: (a) If f : R → R is smooth, we can define the derivative 2 f (x) + y df dx for y ∈ R with y = 0.

df dx

by f (x + y) =

(b) If X is a manifold, the tangent bundle T X is the mapping space X D = MapC ∞ (D, X). (c) A vector field on a manifold X can be defined to be a smooth map v : X × D → X with v|X×{0} = idX . The theory is developed axiomatically, with axioms on the properties of infinitesimals, and theorems about manifolds (including classical results not involving infinitesimals) are proved from them. The logic is unusual: since one treats infinitesimals as ordinary points of the ‘set’ R, though it turns out that R is not really an honest set, then only constructive logic is allowed, and the law of the excluded middle may not be used. For the enterprise to be at all credible, we need to know that the axioms of Synthetic Differential Geometry are consistent, as otherwise one could prove any statement from them, true or false. Consistency is proved by constructing a ‘model’ for Synthetic Differential Geometry, that is, a category (in fact, a topos) of spaces C which includes Man as a full subcategory, and contains other ‘infinitesimal’ objects such as the double point D, such that the axioms can be interpreted as true statements in C, and thus proofs in Synthetic Differential Geometry can be reinterpreted as reasoning (with ordinary logic) in C. The connection to C ∞ -algebraic geometry is that, as in Dubuc [23] and Moerdijk and Reyes [77], this category C may be taken
to be∞the category ∞ ∞ 2 C Sch of C -schemes, with D = Spec R[x]/(x ) , as a C -subscheme of R. Most early work on C ∞ -schemes was directed towards proving properties of C∞ Sch needed to verify consistency of various sets of axioms in Synthetic Differential Geometry. Knowing their axioms were consistent, Synthetic Differential Geometers had little reason to study C ∞ -schemes further, so the subject became inactive.

46

Background on C ∞ -schemes

2.9.2 Derived Differential Geometry Derived Algebraic Geometry is a generalization of classical Algebraic Geometry, in which schemes (and stacks) X are replaced by derived schemes (and derived stacks) X, which have a richer geometric structure. A derived scheme X has a classical truncation X = t0 (X), an ordinary scheme. The foundations were developed in the 2000’s by Bertrand To¨en and Gabriele Vezzosi [94–96] and Jacob Lurie [63, 64], and it has now become a major area in Algebraic Geometry. To¨en [94, 95] gives accessible surveys. Quasi-smooth derived schemes are a class of derived schemes X that behave in some ways like smooth schemes, although their classical truncations X = t0 (X) may be very singular. This is known as the ‘hidden smoothness’ philosophy of Kontsevich [55]. A proper quasi-smooth derived C-scheme X has a dimension dimC X, and a virtual class [X]virt in homology H2 dimC X (X, Z), which is the analogue of the fundamental class of a compact complex manifold. Quasi-smooth derived schemes have important applications in enumerative geometry (as does the older notion of scheme with obstruction theory, which turns out to be a semi-classical truncation of a quasi-smooth derived scheme). Various moduli problems, such as moduli of stable coherent sheaves on a projective surface, have derived moduli schemes which are quasi-smooth, and the virtual class is used to define invariants ‘counting’ such moduli spaces. Two characteristic features of Derived Algebraic Geometry are: (i) Derived Algebraic Geometry is always done in ∞-categories, not ordinary categories, as truncating to ordinary categories loses too much information. For example, the structure sheaf O X of a derived scheme is an ∞-sheaf (homotopy sheaf), but truncating to ordinary categories loses the sheaf property. (ii) To pass from smooth algebraic geometry to Derived Algebraic Geometry, we replace vector bundles by perfect complexes of coherent sheaves. So a smooth scheme X has tangent and cotangent bundles T X, T ∗ X, but a derived scheme X has a tangent complex TX and cotangent complex LX , which are concentrated in degrees [0, 1], [−1, 0] if X is quasi-smooth. We can now ask whether there is an analogous ‘derived’ version of Differential Geometry. In particular, can one define ‘derived manifolds’ and ‘derived orbifolds’ X which would be C ∞ analogues of quasi-smooth derived schemes? We might hope that a compact, oriented derived manifold or orbifold X would have a well-defined dimension and virtual class in homology, and could be applied to enumerative invariant problems in Differential Geometry. In the last paragraph of [64, §4.5], Jacob Lurie outlined how to use his huge framework to define an ∞-category of derived C ∞ -schemes, including derived

2.9 Applications of C ∞ -rings and C ∞ -schemes

47

manifolds. In 2008 Lurie’s student David Spivak [91] worked out the details of this, defining an ∞-category of derived C ∞ -schemes X = (X, O X ) which are topological spaces X with an ∞-sheaf O X of simplicial C ∞ -rings, and a full ∞-subcategory of derived manifolds, which are derived C ∞ -schemes locally modelled on fibre products X ×Z Y for X, Y, Z manifolds. Spivak also gave a list of axioms for an ∞-category of ‘derived manifolds’ to satisfy, and showed they hold for his ∞-category. Some years before the invention of Derived Algebraic Geometry, Fukaya– Oh–Ohta–Ono [28–31] were working on theories of Gromov–Witten invariants and Lagrangian Floer theory in Symplectic Geometry. Their theories involved giving moduli spaces M of J-holomorphic curves in a symplectic manifold (X, ω) the structure of a Kuranishi space M, and defining a virtual class/chain [M]virt in homology. In the 2000’s there were still significant problems with the definition and theory of Kuranishi spaces, and the subject was under dispute. When the second author read Spivak’s thesis [91], he realized that Kuranishi spaces are really derived orbifolds. This explained the problems in the theory: vital ideas from Derived Algebraic Geometry were missing, especially the need for higher categories, as these were unknown when Kuranishi spaces were invented. The second author then developed theories of derived manifolds and derived orbifolds [43–46, 49–51] with a view to applications in Symplectic Geometry, as a substitute for Fukaya–Oh–Ohta–Ono’s Kuranishi spaces. The second author found Spivak’s ∞-category far too complicated to work with. So he defined a simplified version, ‘d-manifolds’ and ‘d-orbifolds’ [44, 45], which form 2-categories dMan, dOrb rather than ∞-categories. (Although this would not work in Derived Algebraic Geometry, it turns out that 2-categories are sufficient in C ∞ geometry because of the existence of partitions of unity.) As part of the foundations of this, he developed C ∞ -algebraic geometry in new directions [43,50], in particular OX -modules and C ∞ -stacks. Later, the second author [46, 49, 51] found a definition of Kuranishi spaces using an atlas of charts in the style of Fukaya–Oh–Ohta–Ono, which yielded a 2-category Kur equivalent to dOrb, fixing the problems with the original definition. This gives two different models for Derived Differential Geometry, one starting from derived C ∞ -schemes, and one from Kuranishi spaces. To understand the relationship, observe that there are two ways to define manifolds: (A) A manifold is a Hausdorff, second countable topological space X equipped with a sheaf OX of R-algebras (or C ∞ -rings) such that (X, OX ) is locally isomorphic to Rn with its sheaf of smooth functions ORn . (B) A manifold is a Hausdorff, second countable topological space X equipped with a maximal atlas of charts {(Ui , φi ) : i ∈ I}.

48

Background on C ∞ -schemes

If we try to define derived manifolds by generalizing approach (A), we get some kind of derived C ∞ -scheme, as in [6–8, 10–13, 44, 45, 64, 91–93]; if we try to generalize (B), we get something like Kuranishi spaces in [28–31, 46, 49, 51]. Derived manifolds and orbifolds are interesting for many reasons, including: (a) Much of classical Differential Geometry extends nicely to the derived case. (b) Many mathematical objects are naturally derived manifolds, for example: (i) The solution set of f1 (x1 , . . . , xn ) = · · · = fk (x1 , . . . , xn ) = 0, where x1 , . . . , xn are real variables and f1 , . . . , fk are smooth functions. (ii) (Non-transverse) intersections X ∩ Y of submanifolds X, Y ⊂ Z. (iii) Moduli spaces M of solutions of nonlinear elliptic equations on compact manifolds. Also, if we consider moduli spaces M for nonlinear equations which are elliptic modulo symmetries, and restrict to objects with finite automorphism groups, then M is a derived orbifold. (c) A compact, oriented derived manifold (or orbifold) X has a virtual class ˇ [X]virt in (Steenrod/Cech) homology Hvdim X (X, Z) (or Hvdim X (X, Q)), with deformation invariance properties. Combining this with (b)(iii), we can use derived orbifolds as tools in enumerative invariant theories such as Gromov–Witten invariants in Symplectic Geometry. Now for applications in Symplectic Geometry, especially Lagrangian Floer theory [28, 29] and Fukaya categories [3, 87], it is important to have a theory of derived orbifolds with corners. (Some applications involving ‘quilts’ also require derived orbifolds with g-corners.) To get satisfactory notions of derived manifold or orbifold with corners in the derived C ∞ -scheme approach, it is necessary to go right back to the beginning, and introduce C ∞ -rings and C ∞ schemes with corners. The second author pursued these ideas with his students Elana Kalashnikov [52] and the first author [25], which led to this book. For further references on Derived Differential Geometry see Behrend–Liao– Xu [6], Borisov [7], Borisov–Noel [8], Carchedi [10], Carchedi–Roytenberg [11, 12], Carchedi–Steffens [13], and Steffens [92, 93].

3 Background on manifolds with (g-)corners

Next we discuss manifolds with corners, following Melrose [71–74] and the second author [42], [47, §2], as well as a generalization of them, manifolds with g-corners, introduced in [47]. While the spaces underlying a manifold with corners are generally agreed m−k k ∼ as based on Rm , there is more than one notion of smooth k = [0, ∞) × R map between manifolds with corners. The one we choose was introduced by Melrose [71–74], who calls them b-maps. It is stricter than requiring that all partial derivatives exist and are continuous. We call this latter definition of smooth weakly smooth, while b-maps are called smooth in this text. Smooth maps f : X → Y are compatible with the corners stratifications of X, Y , whereas weakly smooth maps need not be. We call a smooth map f : X → Y interior if f (X ◦ ) ⊆ Y ◦ . The theory of manifolds with corners is generally similar to that of manifolds, including the existence of partitions of unity. However, there are subtle differences. For example, while manifolds with corners admit partitions of unity [74, Lem. 1.6.1], constructions involving smooth maps and partitions of unity may return weakly smooth but not smooth maps. In this sense, the geometry of manifolds with corners is somewhat more global. A key definition in this chapter is a manifold with faces, which is a manifold with corners where the local and the global behaviours coincide. Manifolds with corners can be considered as C ∞ -schemes, and there is a faithful embedding of manifolds with corners into the category of affine C ∞ schemes. However, this embedding is not full, as morphisms of C ∞ -schemes with corners correspond to weakly smooth maps of manifolds with corners. Manifolds with g-corners are a generalization of manifolds with corners which allow more general local models than Rm k . These local models XP depend on a weakly toric monoid P , and we give an introduction to monoids in §3.2, before proceeding with the definition of manifolds with g-corners in 49

50

Background on manifolds with (g-)corners

§3.3. We discuss boundaries and corners for manifolds with (g-)corners in §3.4. We will see that the corners C(X) of a manifold with (g-)corners X behave functorially, so that smooth maps f : X → Y of manifolds with (g-)corners lift to interior maps C(f ) : C(X) → C(Y ) of the corners. In §3.5 we discuss (co)tangent bundles and b-(co)tangent bundles of manifolds with (g-)corners.

3.1 Manifolds with corners We define manifolds with corners, following the second author [47, §2]. m−k k Definition 3.1. Use the notation Rm for 0 6 k 6 m, and k = [0, ∞) × R write points of Rm k as u = (u1 , . . . , um ) for u1 , . . . , uk ∈ [0, ∞), uk+1 , . . . , n um ∈ R. Let U ⊆ Rm k and V ⊆ Rl be open, and f = (f1 , . . . , fn ) : U → V be a continuous map, so that fj = fj (u1 , . . . , um ) maps U → [0, ∞) for j = 1, . . . , l and U → R for j = l + 1, . . . , n. Then we say: a1 +···+am

∂ (a) f is weakly smooth if all derivatives ∂u a1 am fj (u1 , . . . , um ) : U → R 1 ···∂um exist and are continuous for all j = 1, . . . , n and a1 , . . . , am > 0, including one-sided derivatives where ui = 0 for i = 1, . . . , k. By Seeley’s Extension Theorem, this is equivalent to requiring fj to extend to a smooth function fj0 : U 0 → R on open neighbourhood U 0 of U in Rm . (b) f is smooth if it is weakly smooth and every u = (u1 , . . . , um ) ∈ U has an ˜ in U such that for each j = 1, . . . , l, either: open neighbourhood U a

a

(i) we may uniquely write fj (˜ u1 , . . . , u ˜m ) = Fj (˜ u1 , . . . , u ˜m )· u ˜1 1,j · · · u ˜kk,j ˜ , where Fj : U ˜ → (0, ∞) is weakly smooth and for all (˜ u1 , . . . , u ˜m ) ∈ U a1,j , . . . , ak,j ∈ N = {0, 1, 2, . . .}, with ai,j = 0 if ui 6= 0; or (ii) fj |U˜ = 0. (c) f is interior if it is smooth, and case (b)(ii) does not occur. (d) f is strongly smooth if it is smooth, and in case (b)(i), for each j = 1, . . . , l we have ai,j = 1 for at most one i = 1, . . . , k, and ai,j = 0 otherwise. (e) f is a diffeomorphism if it is a smooth bijection with smooth inverse. Definition 3.2. Let X be a second countable Hausdorff topological space. An m-dimensional chart on X is a pair (U, φ), where U ⊆ Rm k is open for some 0 6 k 6 m, and φ : U → X is a homeomorphism with an open set φ(U ) ⊆ X. Let (U, φ), (V, ψ) be m-dimensional charts
on X. We call (U, φ)
and (V, ψ) compatible if ψ −1 ◦φ : φ−1 φ(U )∩ψ(V ) → ψ −1 φ(U )∩ψ(V ) is a diffeom morphism between open subsets of Rm k , Rl , in the sense of Definition 3.1(e). An m-dimensional atlas for X is a system {(Ua , φa ) : a ∈ A} of pairwise S compatible m-dimensional charts on X with X = a∈A φa (Ua ). We call

3.1 Manifolds with corners

51

such an atlas maximal if it is not a proper subset of any other atlas. Any atlas {(Ua , φa ) : a ∈ A} is contained in a unique maximal atlas, the set of all charts (U, φ) of this type on X which are compatible with (Ua , φa ) for all a ∈ A. An m-dimensional manifold with corners is a second countable Hausdorff topological space X equipped with a maximal m-dimensional atlas. Usually we refer to X as the manifold, leaving the atlas implicit, and by a chart (U, φ) on X, we mean an element of the maximal atlas. Now let X, Y be manifolds with corners of dimensions m, n, and f : X → Y a continuous map. We call f weakly smooth, or smooth, or interior, or strongly smooth, if whenever (U, φ), (V, ψ) are charts on X, Y with U ⊆ Rm k , V ⊆ Rnl open, then ψ −1 ◦ f ◦ φ : (f ◦ φ)−1 (ψ(V )) → V is weakly smooth, or smooth, or interior, or strongly smooth, respectively, as maps between open n subsets of Rm k , Rl in the sense of Definition 3.1. We call f : X → Y a diffeomorphism if it is a bijection and f, f −1 are smooth. Write Mancin , Mancst ⊂ Manc ⊂ Mancwe for the categories with objects manifolds with corners, and morphisms interior maps, and strongly smooth maps, and smooth maps, and weakly smooth maps, respectively. We will be interested almost exclusively in the categories Mancin ⊂ Manc . ` ˇ c for the category with objects disjoint unions ∞ Xm , We write Man m=0 where Xm is a manifold with corners of dimension m (we call these manifolds with corners of mixed dimension), allowing Xm = ∅, and morphisms contin`∞ `∞ uous maps f : m=0 Xm → n=0 Yn , such that fmn := f |Xm ∩f −1 (Yn ) : Xm ∩ f −1 (Yn ) → Yn is a smooth map of manifolds with corners for all ˇ c ⊂ Man ˇ c for the category with the same objects, m, n > 0. We write Man in and with morphisms f such that fmn is interior for all m, n > 0. There are ˇ c , Manc ⊂ Man ˇ c . obvious full and faithful embeddings Manc ⊂ Man in in Remark 3.3. Some references on manifolds with corners are Cerf [14], Douady [21], Gillam and Molcho [32, §6.7], Kottke and Melrose [56], MargalefRoig and Outerelo Dominguez [68], Melrose [71–74], Monthubert [78], and the second author [42, 47]. Just as objects, without considering morphisms, most authors define manifolds with corners X as in Definition 3.2. However, Melrose [71–74] and authors who follow him impose an extra condition, that X should be a manifold with faces in the sense of Definition 3.13 below. There is no general agreement in the literature on how to define smooth maps, or morphisms, of manifolds with corners: (i) Our ‘smooth maps’ in Definitions 3.1–3.2 are due to Melrose [73, §1.12], [56, §1], who calls them b-maps. ‘Interior maps’ are also due to Melrose. (ii) The author [42] defined and studied ‘strongly smooth maps’ above (which were just called ‘smooth maps’ in [42]).

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(iii) Gillam and Molcho’s morphisms of manifolds with corners [32, §6.7] coincide with our ‘interior maps’. (iv) Most other authors, such as Cerf [14, §I.1.2], define smooth maps of manifolds with corners to be weakly smooth maps, in our notation. We will base our theory of (interior) C ∞ -rings and C ∞ -schemes with corners on the categories Mancin ⊂ Manc . Section 8.3 will discuss theories based on other categories of ‘manifolds with corners’ including the category Manac of manifolds with analytic corners, or manifolds with a-corners, defined by the second author [48], which are different to the categories above. Definition 3.4. Let X be a manifold with corners (or a manifold with g-corners in §3.3). Smooth maps g : X → [0, ∞) will be called exterior maps, to contrast them with interior maps. We write C ∞ (X) for the set of smooth maps f : X → R, and In(X) for the set of interior maps g : X → [0, ∞), and Ex(X) for the set of exterior maps g : X → [0, ∞). Thus, we have three sets: (a) C ∞ (X) of smooth maps f : X → R; (b) In(X) of interior maps g : X → [0, ∞); and (c) Ex(X) of exterior (smooth) maps g : X → [0, ∞), with In(X) ⊆ Ex(X). Much of the book will be concerned with algebraic structures on these three sets, and generalizations to other spaces X. In Chapter 4 we give (C ∞ (X), In(X) q {0}) and (C ∞ (X), Ex(X)) the (large and complicated) structure of ‘(pre) C ∞ -rings with corners’. But a lot of the time, it will be enough that C ∞ (X) is an R-algebra, and In(X), Ex(X) are monoids under multiplication, as in §3.2.

3.2 Monoids We will use monoids to define manifolds with g-corners in §3.3, and in the study of C ∞ -rings with corners in Chapter 4. Here we recall some facts about monoids, in the style of log geometry. A good reference is Ogus [82, §I]. Definition 3.5. A (commutative) monoid is a set P equipped with an associative commutative binary operation + : P × P → P that has an identity element 0. All monoids in this book will be commutative. A morphism of monoids P → Q is a map of sets that respects the binary operations and sends the identity to the identity. Write Mon for the category of monoids. For any p n copies q

n ∈ N and p ∈ P we will write np = n · p = p + · · · + p, and set 0 · p = 0. The above is additive notation for monoids. We will also very often use

3.2 Monoids

53

multiplicative notation, in which the binary operation is written · : P ×P → P , thought of as multiplication, and the identity element is written 1, and we write pn rather than np, with p0 = 1. If P is a monoid written in multiplicative notation, a zero element is 0 ∈ P with 0 · p = 0 for all p ∈ P . Zero elements need not exist, but are unique if they do. One should not confuse zero elements with identities. The rest of this definition will use additive notation. A submonoid Q of a monoid P is a subset that is closed under the binary operation and contains the identity element. We can form the quotient monoid P/Q which is the set of all ∼-equivalence classes [p] of p ∈ P such that p ∼ p0 if there are q, q 0 ∈ Q with p + q = p0 + q 0 ∈ P . It has an induced monoid structure from the monoid P . There is a morphism π : P → P/Q. This quotient satisfies the following universal property: it is a monoid P/Q with a morphism π : P → P/Q such that π(Q) = {0} and if µ : P → R is a monoid morphism with µ(Q) = {0} then µ = ν ◦ π for a unique morphism ν : P/Q → R. A unit in P is an element p ∈ P that has a (necessarily unique) inverse under the binary operation, p0 , so that p0 + p = 0. Write P × for the set of all units of P . It is a submonoid of P , and an abelian group. A monoid P is an abelian group if and only if P = P × . An ideal I in a monoid P is a nonempty proper subset ∅ = 6 I ( P such that if p ∈ P and i ∈ I then i + p ∈ I, so it is necessarily closed under P ’s binary operation. It must not contain any units. An ideal I is called prime if whenever a + b ∈ I for a, b ∈ P then either a or b is in I. We say the complement P \ I of a prime ideal I is a face which is automatically a submonoid of P . If we have elements pj ∈ P for j in some indexing set J then we can consider the ideal generated by the pj , which we write as hpj ij∈J . It consists of all elements in P of the form a + pj for any a ∈ P and any j ∈ J. Note that if any of the pj are units then the ‘ideal’ generated by these pj is a misnomer, as hpj ij∈J is not an ideal and instead equal to P . For any monoid P there is an associated abelian group P gp called the groupification of P , with a monoid morphism π gp : P → P gp . This has the universal property that any morphism from P to an abelian group factors through π gp , so P gp is unique up to canonical isomorphism. It can be shown to be isomorphic to the quotient monoid (P × P )/∆P , where ∆P = {(p, p) : p ∈ P } is the diagonal submonoid of P × P , and π gp : p 7→ [p, 0]. For a monoid P we make the following definitions: (i) If there is a surjective morphism Nk → P for some k > 0, we call P finitely generated. This morphism can be uniquely written as (n1 , . . . , nk ) 7→ n1 p1

54

(ii) (iii)

(iv) (v) (vi)

(vii)

Background on manifolds with (g-)corners

+ · · · + nk pk for some p1 , . . . , pk ∈ P , which we call the generators of P . This implies that P gp is finitely generated. If there is an isomorphism P ∼ = Nk for k > 0) then P is called free. = NA for some set A (e.g. if P ∼ If P × = {0} we call P sharp. Any monoid has an associated sharpening P ] which is the sharp quotient monoid P/P × with surjection π ] : P → P ] . If π gp : P → P gp is injective we call P integral or cancellative. This occurs if and only if p + p0 = p + p00 implies p0 = p00 for all p, p0 , p00 ∈ P . Then P is isomorphic to its image under π gp , so we consider it a subset of P gp . If P is integral and whenever p ∈ P gp with np ∈ P ⊂ P gp for some n > 1 implies p ∈ P then we call P saturated. If P gp is a torsion-free group, then we call P torsion-free. That is, if there is n > 0 and p ∈ P gp such that np = 0 then p = 0. If P is finitely generated, integral, saturated, and torsion-free then it is called weakly toric. It has rank rank P = dimR (P ⊗N R). For a weakly toric P there is an isomorphism P × ∼ = Zl and P ] is a toric monoid (defined below). × The exact sequence 0 → P → P → P ] → 0 splits, so that P ∼ = P ] × Zl . Then the rank of P is equal to rank P = rank P gp = rank P ] + l. If P is a weakly toric monoid and is also sharp we call P toric (note that saturated and sharp together imply torsion-free). For a toric monoid P its associated group P gp is a finitely generated, torsion-free abelian group, so P gp ∼ = Zk for k > 0. Then the rank of P is rank P = k.

Parts (vi)–(vii) are not universally agreed in the literature. For example, Ogus [82, p. 13], calls our weakly toric monoids toric monoids, and our toric monoids sharp toric monoids. Example 3.6. (a) The most basic toric monoid is Nk under addition for k = 0, 1, . . . , with (Nk )gp ∼ = Zk . (b) Zk under addition for k > 0 is weakly toric, but not toric.
(c) [0, ∞), ·, 1 under multiplication is a monoid that is not finitely generated. It has identity 1 and zero element 0. We have [0, ∞)gp = {1}, so [0, ∞) is not integral, and [0, ∞)× = (0, ∞), so [0, ∞) is not sharp.

3.3 Manifolds with g-corners The second author [47] defined the category Mangc of manifolds with generalized corners, or manifolds with g-corners. These extend manifolds with corners m−k k , X in §3.1, but rather than being locally modelled on Rm k = [0, ∞) × R they are modelled on spaces XP for P a weakly toric monoid in §3.2.

3.3 Manifolds with g-corners

55

Definition 3.7. Let P be a weakly toric monoid. As in [47, §3.2] we define XP = Hom(P, [0, ∞)) to be the set of monoid morphisms x : P → [0, ∞), where the target is considered as a monoid under multiplication as in Example 3.6(c). The interior of XP is defined to be XP◦ = Hom(P, (0, ∞)) where (0, ∞) is a submonoid of [0, ∞), so that XP◦ ⊂ XP . For p ∈ P there is a corresponding function λp : XP → [0, ∞) such that λp (x) = x(p). If p, q ∈ P then λp+q = λp · λq , and λ0 = 1. Define a topology on XP to be the weakest topology such that each λp is continuous. Then XP is locally compact and Hausdorff and XP◦ is an open subset of XP . The interior U ◦ of an open set U ⊂ XP is defined to be U ∩ XP◦ . As P is weakly toric we can take a presentation for P with generators p1 , . . . , pm and relations aj1 p1 + · · · + ajm pm = bj1 p1 + · · · + bjm pm

in P for j = 1, . . . , k,

for aji , bji ∈ N, i = 1, . . . , m, j = 1, . . . , k. Then we have a continuous function λp1 × · · · × λpm : XP → [0, ∞)m that is a homeomorphism onto its image  aj bj aj bj XP0 = (x1 , . . . , xm ) ∈ [0, ∞)m : x1 1 · · · xmm = x11 · · · xmm , j = 1, . . . , k , which is a closed subset of [0, ∞)m . Let U be an open subset of XP , and U 0 = λp1 × · · · × λpm (U ) ⊂ XP0 . Then we say a continuous function f : U → R or f : U → [0, ∞) is smooth if there exists an open neighbourhood W 0 of U 0 in [0, ∞)m and a smooth function g : W 0 → R or g : W 0 → [0, ∞) that is smooth in the sense above, such that f = g ◦ λp1 × · · · × λpm . As in [47, Prop. 3.14], this is independent of the choice of generators p1 , . . . , pm for P . Suppose Q is another weakly toric monoid, and consider open V ⊆ XQ and a continuous function f : U → V . Then f is smooth if λq ◦ f : U → [0, ∞) is smooth for all q ∈ Q in the sense above. We call smooth f interior if f (U ◦ ) ⊆ V ◦ , and a diffeomorphism if it is bijective with smooth inverse. Example 3.8. If P = Nk × Zm−k then P is weakly toric. We can take generators p1 = (1, 0, . . . , 0), p2 = (0, 1, 0, . . . , 0), . . . , pm = (0, . . . , 0, 1), pm+1 = (0, . . . , 0, −1, . . . , −1) with pm+1 having −1 in the k + 1 to m + 1 entries, so the only relation is pk+1 + · · · + pm+1 = 0. Then XP is homeomorphic to  XP0 = (x1 , . . . , xm+1 ) ∈ [0, ∞)m+1 : xk+1 · · · xm+1 = 1 . This means that for (x1 , . . . , xm+1 ) ∈ XP0 we have xk+1 , . . . , xm+1 > 0 m −1 with xm+1 = x−1 k+1 · · · xm . So there is a homeomorphism from XP → Rk mapping (x1 , . . . , xm+1 ) 7→ (x1 , . . . , xk , log(xk+1 ), . . . , log(xm )). In [47,

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Ex. 3.15] we show that this identification XP ∼ = Rm k identifies the topology, and the notion of smooth maps to R, [0, ∞) and between open subsets of XP ∼ = Rm k n m ∼ and XQ = Rl , in Definition 3.1 for Rk and above for XP . Thus, the spaces XP generalize the spaces Rm k used as local models for manifolds with corners. Following [47, §3.3] we define manifolds with g-corners: Definition 3.9. Let X be a topological space. Define a g-chart on X to be a triple (P, U, φ), where P is a weakly toric monoid, U ⊂ XP is open and φ : U → X is a homeomorphism with an open subset φ(U ) ⊂ X. If rank P = m we call (P, U, φ) m-dimensional. For set theory reasons (to ensure a maximal atlas is a set not a class) we suppose P is a submonoid of Zk for some k > 0. We call m-dimensional g-charts
(P, U, φ) and (Q, V, ψ)
on X compatible if ψ −1 ◦ φ : φ−1 φ(U ) ∩ ψ(V ) → ψ −1 φ(U ) ∩ ψ(V ) is a diffeomorphismbetween open subsets of XP and XQ . A g-atlas on X is a family A = (Pi , Ui , φi ) : i ∈ I of pairwise compatible g-charts (Pi , Ui , φi ) on S X with the same dimension m, with X = i∈I φi (Ui ). We call A maximal if it is not a proper subset of any other g-atlas. We define a manifold with gcorners (X, A) to be a Hausdorff, second countable topological space X with a maximal g-atlas A. We define smooth maps, and interior maps, f : X → Y between manifolds with g-corners X, Y as in Definition 3.2. We write Mangc for the category gc of manifolds with g-corners and smooth maps, and Mangc for the in ⊂ Man subcategory of manifolds with g-corners and interior maps. As in Example 3.8, if P ∼ = Nk × Zm−k we may identify XP ∼ = Rm k . So as in [47, Def. 3.22], we may identify Manc ⊂ Mangc and Mancin ⊂ Mangc in as full subcategories, where a manifold with g-corners (X, A) is a manifold with corners if A has a g-subatlas of g-charts (Pi , Ui , φi ) with Pi ∼ = Nk × Zm−k . `∞ gc ˇ We write Man for the category with objects disjoint unions m=0 Xm , where Xm is a manifold with g-corners of dimension m (we call these manifolds with g-corners of mixed dimension), allowing Xm = ∅, and morphisms `∞ `∞ continuous f : m=0 Xm → n=0 Yn , such that fmn := f |Xm ∩f −1 (Yn ) : Xm ∩ f −1 (Yn ) → Yn is a smooth map of manifolds with g-corners for all ˇ c ⊂ Man ˇ c for the category with the same objects, m, n > 0. We write Man in and with morphisms f such that fmn is interior for all m, n > 0. There are ˇ gc and Mangc ⊂ Man ˇ gc . obvious full embeddings Mangc ⊂ Man in in Remark 3.10. Any weakly toric monoid P is isomorphic to P ] ×Zl , where P ] is toric and l > 0. Then XP ∼ = XP ] × XZl ∼ = XP ] × Rl . Hence manifolds with g-corners have local models XQ × Rl for toric monoids Q and l > 0, where XNk ∼ = [0, ∞)k . Each toric monoid Q has a natural point δ0 ∈ XQ called the

57

3.3 Manifolds with g-corners

vertex of XQ , which acts by taking 0 ∈ Q to 1 ∈ [0, ∞) and all non-zero q ∈ Q to zero. Given a manifold with g-corners X and a point x ∈ X, there is a toric monoid Q such that X near x is modelled on XQ × Rl near (δ0 , 0) ∈ XQ × Rl , where rank Q + l = dim X. From [47, Ex. 3.23] we have the simplest example of a manifold with gcorners that is not a manifold with corners. Example 3.11. Let P be the weakly toric monoid of rank 3 with  P = (a, b, c) ∈ Z3 : a > 0, b > 0, a + b > c > 0 . This has generators p1 = (1, 0, 0), p2 = (0, 1, 1), p3 = (0, 1, 0), and p4 = (1, 0, 1) and one relation p1 + p2 = p3 + p4 . The local model it induces is  XP ∼ = XP0 = (x1 , x2 , x3 , x4 ) ∈ [0, ∞)4 : x1 x2 = x3 x4 .

(3.1)

• (0, 0, 0, 0) ∼ = δ0

(x1 , 0, 0, 0)

•

(x1 , 0, x3 , 0) •

(x1 , 0, 0, x4 ) • (0, 0, 0, x4 )

•

•

(0, 0, x3 , 0)

• (0, x2 , x3 , 0) • (0, x2 , 0, x4 )

•

(0, x2 , 0, 0)

Figure 3.1 3-manifold with g-corners XP0 ∼ = XP in (3.1)

Figure 3.1 is a sketch of XP0 as a square-based 3-dimensional infinite pyramid. As in Remark 3.10, XP0 has vertex (0, 0, 0, 0) corresponding to δ0 ∈ XP . It has 1-dimensional edges of points (x1 , 0, 0, 0), (0, x2 , 0, 0), (0, 0, x3 , 0), (0, 0, 0, x4 ), and 2-dimensional faces of points (x1 , 0, x3 , 0), (x1 , 0, 0, x4 ), (0, x2 , x3 , 0), (0, x2 , 0, x4 ). Its interior XP0◦ ∼ = R3 consists of points (x1 , x2 , x3 , x4 ) with x1 , . . . , x4 non-zero and x1 x2 = x3 x4 . Then XP \ {δ0 } is a 3-manifold with corners, but XP is not a manifold with corners near δ0 , as it is not locally isomorphic to Rm k . We give more material on manifolds with g-corners in §3.4, §3.5, and §6.7.1.

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3.4 Boundaries, corners, and the corner functor This section follows the second author [42] and [47, §2.2 & §3.4]. Definition 3.12. Let X be a manifold with corners, or g-corners, of dimension m. For each x ∈ X we define the depth depthX (x) ∈ {0, 1, . . . , m} as follows: if X is a manifold with corners then depthX (x) = k if X near x is locally modelled on Rm k near 0. And if X is a manifold with g-corners then depthX (x) = k if X near x is locally modelled on XP × Rm−k near (δ0 , 0), using Remark 3.10, where P is a toric monoid of rank k. Write S k (X) = {x ∈ X : depthX (x) = k} for k = 0, . . . , m, the codi`m mension k boundary stratum. This defines a stratification X = k=0 S k (X) called the depth stratification. Then S k (X) has a natural structure of a manifold `m without boundary of dimension m − k. Its closure is S k (X) = l=k S l (X). The interior of X is X ◦ := S 0 (X). Define a local k-corner component γ of X at x ∈ X to be a local choice of connected component of S k (X) near x. That is, for any sufficiently small open neighbourhood V of x in X, then γ assigns a choice of connected component W of V ∩ S k (X) with x ∈ W , and if V 0 , W 0 are alternative choices then x ∈ W ∩ W 0 . The local 1-corner components are called local boundary components of X. As sets, we define the boundary and k-corners of X for k = 0, . . . , m by  ∂X = (x, β) : x ∈ X, β is a local boundary component of X at x ,  (3.2) Ck (X) = (x, γ) : x ∈ X, γ is a local k-corner component of X at x , for k = 0, 1, . . . , m. This implies that ∂X = C1 (X) and C0 (X) ∼ = X. In [47, §2.2 & §3.4] we define natural structures of a manifold with corners, or gcorners (depending on which X is) on ∂X and Ck (X), so that dim ∂X = m−1 and dim Ck (X) = m − k. The interiors are (∂X)◦ ∼ = S 1 (X) and Ck (X)◦ ∼ = S k (X). We define smooth maps ΠX : ∂X → X, ΠX : Ck (X) → X by ΠX : (x, β) 7→ x and ΠX : (x, γ) 7→ x. These are generally neither interior, not injective. The next definition will be important in Chapter 5, as a manifold with corners corresponds to an affine C ∞ -scheme with corners if it is a manifold with faces. We extend Definition 3.13 to manifolds with g-corners in Definition 4.37. Definition 3.13. A manifold with corners X is called a manifold with faces if ΠX |F : F → X is injective for each connected component F of ∂X. The faces of X are the components of ∂X, regarded as subsets of X. Melrose [71–74] assumes his manifolds with corners are manifolds with faces. Write

3.4 Boundaries, corners, and the corner functor

59

Manfin ⊂ Mancin , Manf ⊂ Manc for the full subcategories of manifolds with faces.  Example 3.14. Define the teardrop to be the subset T = (x, y) ∈ R2 : x > 0, y 2 6 x2 − x4 , as in [47, Ex. 2.8]. As shown in Figure 3.2, T is a manifold with corners of dimension 2. The teardrop is not a manifold with faces as in Definition 3.13, since the boundary is diffeomorphic to [0, 1], and so is connected, but the map ΠT : ∂T → T is not injective. Oy

o

•

x/

 Figure 3.2 The teardrop T

Remark 3.15. Let X be a manifold with (g-)corners. Then smooth functions f : X → R are local objects: we can combine them using partitions of unity. In contrast, exterior functions g : X → [0, ∞) as in Definition 3.4 cannot be combined using partitions of unity, and have some non-local, global features. To see this, observe that if g : X → [0, ∞) is an exterior function, we may define a multiplicity function µg : ∂X → N ∪ {∞} giving the order of vanishing of g on the boundary faces of X. If x ∈ X and (x1 , . . . , xm ) ∈ Rm k are local coordinates on X near x with x = (0, . . . , 0) then either g = xa1 1 · · · xakk F (x1 , . . . , xm ) near x for F > 0 smooth, and we set µg (x, {xi = 0}) = ai for i = 1, . . . , k, or g = 0 near x, when we write µg (x, {xi = 0}) = ∞. Then µg : ∂X → N ∪ {∞} is locally constant, and hence constant on each connected component of ∂X. Also g interior implies µg maps to N ⊂ N∪{∞}. If two points x1 , x2 ∈ X lie in the image of the same connected component of ∂X then the behaviour of g near x1 , x2 is linked even if x1 , x2 are far away in X. Interior functions {gi : i ∈ I} can be combined with a partition of unity P {ηi : i ∈ I} to give interior i∈I ηi gi if the µgi for i ∈ I are all equal. The boundary ∂X and corners Ck (X) are not functorial under smooth or interior maps: smooth f : X → Y generally do not lift to smooth ∂f : ∂X → ∂Y . However, as in the second author [42, §4] and [47, §2.2 & §3.4], the corners `dim X C(X) = k=0 Ck (X) do behave functorially.

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Definition 3.16. Let X be a manifold with corners, or with g-corners. The corners of X is the manifold with (g-)corners of mixed dimension `dim X C(X) = k=0 Ck (X), ˇ c or Man ˇ gc in Definitions 3.2 and 3.9. By (3.2) we have as an object of Man  C(X) = (x, γ) : x ∈ X, γ is a local k-corner component of X at x, k > 0 . ˇ c Define morphisms ΠX : C(X) → X and ιX : X → C(X) in Man gc ◦ ˇ or Man by ΠX : (x, γ) 7→ x and ιX : x 7→ (x, X ), that is, ιX : ∼ = X −→ C 0 (X) ⊂ C(X). Here ιX is interior, but ΠX generally is not. Now let f : X → Y be a morphism in Manc or Mangc . Suppose γ is a local k-corner component of X at x ∈ X. For each small open neighbourhood V of x in X, γ gives a connected component W of V ∩S k (X) with x ∈ W . As ` ` f preserves the stratifications X = k>0 S k (X), Y = l>0 S l (Y ), we have f (W ) ⊆ S l (Y ) for some l > 0. Since f is continuous, f (W ) is connected, and f (x) ∈ f (W ). Thus there is a unique l-corner component f∗ (γ) of Y at f (x), such that if V˜ is a sufficiently small open neighbourhood of f (x) in Y , then the ˜ of V˜ ∩ S l (Y ) given by f∗ (γ) has W ˜ ∩ f (W ) 6= ∅. connected component W This f∗ (γ) is independent of the choice of sufficiently small V, V˜ , so is welldefined. Define a map C(f ) : C(X) → C(Y ) by C(f ) : (x, γ) 7→ (f (x), f∗ (γ)). As in [47, §2.2 & §3.4], this C(f ) is smooth, and interior, and so is a morphism ˇ gc . Also C(g ◦ f ) = C(g) ◦ C(f ) and C(idX ) = idC(X) . Thus we in Man in ˇ gc , ˇ c and C : Mangc → Man have defined functors C : Manc → Man in in c ˇ ˇ c which we call the corner functors. We extend them to C : Man → Man in ` ` ˇ gc → Man ˇ gc by C( and C : Man m>0 Xm ) = m>0 C(Xm ). in ˇ c ,→ Man ˇ c and Consider the inclusions of subcategories inc : Man in gc gc ˇ ˇ inc : Manin ,→ Man . The morphisms ΠX : C(X) → X give a natuˇ c or Man ˇ gc . The morphisms ral transformation Π : inc ◦C ⇒ Id in Man ˇ c or ιX : X → C(X) give a natural transformation ι : Id ⇒ C ◦ inc in Man in gc ˇ Man in . ˇ c or Man ˇ gc , and Y an object in Remark 3.17. If X is an object in Man gc ˇ c or Man ˇ Man in in , it is easy to check that we have inverse 1-1 correspondences c HomMan ˇ or (inc X, Y ) o gc ˇ Man

f 7−→C(f )◦ιX g7−→ΠY ◦g

/ Hom c (X, C(Y )). ˇ Man in or gc ˇ Man in

ˇ c → Man ˇ c and C : Man ˇ gc → Man ˇ gc are right adjoint Therefore C : Man in in ˇ c ,→ Man ˇ c and inc : Man ˇ gc ,→ Man ˇ gc , and Π, ι are the units to inc : Man in in of the adjunction. (This is a new observation, not in [42, 47].)

3.5 Tangent bundles and b-tangent bundles

61

This implies that the corner functors C, and hence the boundary ∂X and corners Ck (X) of a manifold with (g-)corners X, are not arbitrary constructions, but are determined up to natural isomorphism by the inclusions of subcategories gc Mancin ⊂ Manc and Mangc in ⊂ Man .

3.5 Tangent bundles and b-tangent bundles Finally we briefly discuss (co)tangent bundles of manifolds with (g-)corners. For a manifold with corners X there are two kinds: the (ordinary) tangent bundle T X, the obvious generalization of tangent bundles of manifolds, and the b-tangent bundle b T X introduced by Melrose [73, §2.2], [74, §I.10], [72, §2]. The duals of the tangent bundle and b-tangent bundle are the cotangent bundle T ∗ X and b-cotangent bundle b T ∗ X. If X is a manifold with g-corners, then b T X is well defined, but in general T X is not. We follow [47, §2.3 & §3.5]. Remark 3.18. (a) Let X be a manifold with corners of dimension m. The tangent bundle T X → X, cotangent bundle T ∗ X → X, b-tangent bundle b T X → X, and b-cotangent bundle b T ∗ X → X, are natural rank m vector bundles on X, defined in detail in [47, §2.3]. In local coordinates, if (x1 , . . . , xk , xk+1 , . . . , xm ) ∈ Rm k are local coordinates on an open set b ∗ φ(U ) ⊂ X, from a chart (U, φ) with U ⊂ Rm k open, then T X, . . . , T X are given on φ(U ) by the bases of sections: D ∂ ∂ ∂ ∂ E ,..., , ,..., , T X|φ(U ) = ∂x1 ∂xk ∂xk+1 ∂xm R

 T ∗ X|φ(U ) = dx1 , . . . , dxk , dxk+1 , . . . , dxm R , D ∂ ∂ E ∂ ∂ b T X|φ(U ) = x1 , . . . , xk , ,..., , ∂x1 ∂xk ∂xk+1 ∂xm R

 −1 b ∗ T X|φ(U ) = x−1 1 dx1 , . . . , xk dxk , dxk+1 , . . . , dxm R . ∂ Note that “xi ∂x ” is a nonzero section of b T X even where xi = 0 for i = i −1 1, . . . , k, and “xi dxi ” is a well defined section of b T ∗ X even where xi = 0. These are formal symbols, but are useful as they determine the transition functions for b T X, b T ∗ X under change of coordinates (x1 , . . . , xm ) (˜ x1 , . . . , x ˜m ), etc. There is a natural morphism of vector bundles IX : b T X → T X mapping ∂ ∂ ∂ ∂ xi ∂x 7→ xi · ∂x for i = 1, . . . , k and ∂x 7→ ∂x for i = k + 1, . . . , m. It is i i i i ◦ an isomorphism over X , and has kernel of rank k over S k (X).

(b) Tangent bundles T X in Manc are functorial under smooth maps. That is,

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for any morphism f : X → Y in Manc there is a corresponding morphism T f : T X → T Y defined in [47, Def. 2.14], functorial in f and linear on the fibres of T X, T Y , in a commuting diagram TX

/ TY

Tf

π

π

 X

 / Y.

f

Equivalently, we may write T f as a vector bundle morphism df : T X → f ∗ (T Y ) on X. It has a dual morphism df ∗ : f ∗ (T ∗ Y ) → T ∗ X on cotangent bundles. Similarly, b-tangent bundles b T X in Manc are functorial under interior maps. That is, for any morphism f : X → Y in Mancin there is a corresponding morphism b T f : b T X → b T Y defined in [47, Def. 2.15], functorial in f and linear on the fibres of b T X, b T Y , in a commuting diagram b

TX

b

IX

' π

/ bT Y Tf

TX

IY

&

/ TY

Tf π

π

  X

f

π

  / Y.

Equivalently, we have a vector bundle morphism b df : b T X → f ∗ (b T Y ) on X, with a dual morphism b df ∗ : f ∗ (b T ∗ Y ) → b T ∗ X on b-cotangent bundles. (c) As in [47, §3.5], tangent bundles T X → X do not extend to manifolds with g-corners X. That is, for each x ∈ X we can define a tangent space Tx X with the usual functorial properties of tangent spaces, but dim Tx X need not be locally constant on X, so Tx X, x ∈ X are not the fibres of a vector bundle. However, b-(co)tangent bundles b T X → X, b T ∗ X → X do extend nicely to manifolds with g-corners. If X is locally modelled on XP for a weakly toric monoid P , as in §3.3, then b T X is locally trivial with fibre HomMon (P, R), and b T ∗ X is locally trivial with fibre P ⊗N R. B-tangent bundles are functorial under interior maps, as in (b). So if f : X → Y is a morphism in Mangc in , as b b b in [47, Def. 3.43] we have a morphism T f : T X → T Y , and vector bundle morphisms b df : b T X → f ∗ (b T Y ) and b df ∗ : f ∗ (b T ∗ Y ) → b T ∗ X on X. (d) Let X be a manifold with (g-)corners. Then [47, §3.6] constructs a natural exact sequence of vector bundles of mixed rank on C(X): 0

/ b NC(X)

b

iT

/ Π∗ (b T X) X

b

πT

/ b T (C(X))

/ 0. (3.3)

Here b NC(X) is called the b-normal bundle of C(X) in X. It has rank k on

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3.6 Applications of manifolds with (g-)corners

Ck (X) for k > 0. Note that b πT is not b dΠX from (c) for ΠX : C(X) → X; b dΠX is undefined as ΠX is not interior, and would go the opposite way to b πT . ∗ Here is the dual complex to (3.3), with b NC(X) the b-conormal bundle: 0

/ b T ∗ (C(X))

b

∗ πT

b ∗ iT

/ Π∗ (b T ∗ X) X

/ bN ∗ C(X)

/ 0.

(3.4)

In [47, §3.6] we define the monoid bundle MC(X) → C(X), which is a locally constant bundle of toric monoids with an embedding MC(X) ⊂ b NC(X) such that b NC(X) ∼ = MC(X) ⊗N R. The monoid fibres have rank k on Ck (X). If X is a manifold with corners then MC(X) has fibre Nk on Ck (X), and the embedding MC(X) ⊂ b NC(X) is locally modelled on Nk ⊂ Rk . We also ∨ ∨ have a dual comonoid bundle MC(X) → C(X), with an embedding MC(X) ⊂ b ∗ NC(X) . If (x, γ) ∈ C(X)◦ , then X near x is locally diffeomorphic to XP × Rl for P a toric monoid, such that γ is identified with {δ0 } × Rl for δ0 the vertex in XP . ∗ ∨ Then near (x, γ) in C(X), the fibres of b NC(X) , b NC(X) , MC(X) , MC(X) are ∨ HomMon (P, R), P ⊗N R, P , P , respectively. Example 3.19. Let X be a manifold with corners, and (x, γ) ∈ Ck (X)◦ ⊂ C(X). We will explain the ideas of Remark 3.18(d) near (x, γ). We can choose local coordinates (x1 , . . . , xm ) ∈ Rm k near x in X, such that the local boundary component γ of X at x is x1 = · · · = xk = 0 in coordinates. Then (xk+1 , . . . , xm ) in Rm−k are local coordinates on Ck (X)◦ near (x, γ), and (3.3)–(3.4) become 0

/ b NC(X)

b

/

∂ ∂ , . . . , xk ∂x x1 ∂x 1 k R



0

/ b T ∗ (C(X))

/ Π∗X (b T X)

iT



∗ πT

1

∂

,...,

dxk+1 , . . . , dxm

 R

/ Π∗X (b T ∗ X)

−1 −1 / x1 dx1 , . . . , xk dxk ,

dxk+1 , . . . , dxm

Under these identifications we have

 ∂ MC(X) = x1 ∂x , . . . , xk ∂x∂ k N , 1

/

k

∂ ∂xm R





/ b T (C(X))

/0

πT

∂ , . . . , xk ∂x∂ , x1 ∂x

∂xk+1

b

b

b ∗ iT



∂ , . . . , ∂x∂ R , ∂xk+1 m

/ bN ∗ C(X)



/0

 −1 / x−1 1 dx1 , . . . , xk dxk R .

R

 −1 ∨ MC(X) = x−1 1 dx1 , . . . , xk dxk N .

3.6 Applications of manifolds with (g-)corners Manifolds with corners appear in many places in the literature. Here, with no attempt at completeness, we review a selection of areas involving manifolds with

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corners. Those in §3.6.4–§3.6.6 are also relevant to C ∞ -schemes with corners and derived manifolds with corners, as in §8.4. We also point out where our particular definition of smooth map in §3.1 is important, and a few applications of manifolds with g-corners.

3.6.1 Extended Topological Quantum Field Theories As in Atiyah [2] and Freed [27], an (unextended, oriented, unitary) Topological Quantum Field Theory (TQFT) Z of dimension n assigns the following data:

(a) To each compact, oriented (n − 1)-manifold N , a complex Hilbert space Z(N ). We require that Z(∅) = C, and Z(−N ) = Z(N ), where −N is N ˆ C Z(N2 ). with the opposite orientation, and Z(N1 q N2 ) ∼ = Z(N1 )⊗ (b) To each compact, oriented n-manifold with boundary M , an element Z(M ) ˆ C Z(N2 ) in Z(∂M ). If ∂M12 = −N1 q N2 we interpret Z(M12 ) ∈ Z(N1 )⊗ as a bounded linear map Z(M12 ) : Z(N1 ) → Z(N2 ) using the Hilbert inner product on N1 . We require that if ∂M12 = −N1 q N2 and ∂M23 = −N2 q N3 , and M13 = M12 qN2 M23 is obtained by gluing M12 , M23 at their common boundary component N2 , then Z(M13 ) = Z(M23 )◦Z(M12 ). This can be written as a functor Bord+ hn−1,ni → HilbC from an ‘oriented bor+ dism category’ Bordhn−1,ni to the category of complex Hilbert spaces HilbC . TQFTs arose from Physics, but are also important in Mathematics, for example Donaldson invariants of 4-manifolds [20] and instanton Floer homology of 3-manifolds [19] can be encoded as a kind of 4-dimensional TQFT. As in Freed [26], extended TQFTs generalize this idea to manifolds with corners. The details are complicated. For 0 < d 6 n one can define a symmetric monoidal d-category Bord+ hn−d,ni in which the k-morphisms for 0 6 k 6 d are compact, oriented (n+k−d)-manifolds with corners Nk , such that ∂ l Nk = ∅ for l > k. A d-extended TQFT of dimension n is then a d-functor Z : Bord+ hn−d,ni → C, where C is a suitable C-linear symmetric monoidal dcategory generalizing HilbC . We call Z fully extended if d = n. The Baez–Dolan cobordism hypothesis [5,26] says fully extended TQFTs Z should be classified by the object Z(∗) ∈ C, which should be ‘fully dualizable’, where ∗ is the point, as a 0-manifold. A proof was outlined by Jacob Lurie [65]. This is an active area.

3.6 Applications of manifolds with (g-)corners

65

3.6.2 Manifolds with corners in asymptotic geometry and analysis Manifolds with corners are used extensively in geometric analysis by Richard Melrose [71–74], in his ‘b-calculus’, and by his many mathematical followers. Grieser [34] gives a good introduction. A key idea in Melrose’s philosophy is this: suppose you have a noncompact manifold X and a smooth function f : X → R (or a Riemannian metric g on X, etc.) and you want to describe the asymptotic behaviour of f (or g) at infinity in X. Then you should compactify X to a compact manifold with faces ¯ with interior X ¯ ◦ = X, such that ∂ X ¯ = Y1 q · · · q Yk with Yi connected. X ¯ Choose a smooth function hi : X → [0, ∞) for i = 1, . . . , k which vanishes ¯ \ Yi . Then we should describe the to order 1 along Yi and is positive on X ¯ in X) ¯ by asymptotic behaviour of f (or g) near infinity in X (that is, near ∂ X Qk αi bounding it by functions of h1 , . . . , hk , such as i=1 hi for α1 , . . . , αk ∈ R. ¯ should depend on f (or g). For this to work the compactification X As a typical application, suppose we are given a (possibly singular) elliptic operator P : Γ∞ (E) → Γ∞ (F ) on a manifold X, for vector bundles E, F → X, and we wish to solve the equation P u = v. In good situations we may write Z u(x) = Q(x, y)v(y)dy, y∈X

where Q ∈ Γ(E  F ∗ ) = Γ(Π∗X (E) ⊗ Π∗Y (F )) is a Schwartz kernel (Green’s function) for P . Here Q is defined on a dense open subset U ⊂ X × X, e.g. U = (X × X) \ ∆X , and may have a pole along the diagonal ∆X . Many important properties of P depend only on the asymptotic behaviour of Q, which ¯ as above. can be described by extending U to a compact manifold with faces U ¯ The construction of U could involve taking a real blow up of X × X along the diagonal ∆X . The ‘b-calculus’ provides a tool-kit for understanding the asymptotic behaviour of Schwartz kernels of pseudo-differential operators. An important rˆole is played by smooth maps f : X → Y of manifolds with corners (for example, real blow ups), in the sense of §3.1 (Melrose calls these ‘b-maps’), and by special classes of these (e.g. ‘b-fibrations’). There are theorems [72] which describe how the asymptotic behaviour of Schwartz kernels transform under pullbacks f ∗ and pushforwards f∗ along such maps f . The b-tangent bundle b T X → X is also important in the theory. There is also a place in the Melrose theory for manifolds with g-corners, as in §3.3: these are basically the ‘interior binomial varieties’ of Kottke–Melrose [56], on which the second author’s definition of manifolds with g-corners [47] is based. For example, given a Schwartz kernel Q defined on U ⊂ X × X, the ¯ of U on which Q has a good asymptotic descripminimal compactification U

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Background on manifolds with (g-)corners

¯ into a manifold with tion might be a manifold with g-corners. We can turn U ¯. ordinary corners by taking further real blow ups of corner strata of U

3.6.3 Partial differential equations on manifolds with corners A second way in which manifolds with corners appear in geometry and analysis is in the study of partial differential equations P u = v whose domain is a manifold with corners X, with some boundary conditions on u at ∂X. There are two different types of such problems: (i) the boundary ∂X is at ‘infinite distance’. In this case, the partial differential equation is really on the interior X ◦ , and the boundary conditions at ∂X prescribe the asymptotic behaviour of u at infinity in X ◦ . (ii) the boundary ∂X is at ‘finite distance’, for example Dirichlet or Neumann boundary conditions for Laplace’s equation. Type (i) is closely related to the ideas of §3.6.2, and has been studied by Richard Melrose and his school, and others. See for example Melrose [73] on type (i) analysis for X a manifold with boundary, and Melrose’s student Loya [61, 62] on Fredholm properties of elliptic operators on manifolds X with corners in codimension 2. Problems of type (ii) occur, for example, in Lagrangian Floer theory in Symplectic Geometry [28, 29]. Let (X, ω) be a symplectic manifold, J an almost complex structure on X compatible with ω, and L1 , L2 ⊂ X be embedded, transversely intersecting Lagrangians. To define the Lagrangian Floer cohomology HF ∗ (L1 , L2 ), one must consider moduli spaces of J-holomorphic maps u : Σ → X, where Σ is a complex disc, with the boundary condition that u(∂Σ) ⊂ L1 ∪ L2 . A typical such disc is shown in Figure 3.3.


L1 • L2

Figure 3.3 J-holomorphic disc Σ with boundary in L1 ∪ L2

Really one should regard Σ as a compact 2-manifold with corners, where components of ∂Σ are mapped to L1 or L2 , and codimension 2 corners of Σ are mapped to L1 ∩ L2 . There are similar, analytically well-behaved problems on manifolds with corners of higher dimension, although they are not much studied. For example, if X is a Calabi–Yau m-fold and D ⊂ X is a divisor with simple normal crossings, one could consider special Lagrangian submanifolds L ⊂ X with corners, with ∂L ⊂ D.

3.6 Applications of manifolds with (g-)corners

67

In [48] the second author argues that for problems of type (i), the correct smooth structure on X is actually that of a manifold with a-corners, to be discussed in §8.3.

3.6.4 Moduli spaces of flow lines in Morse theory In §3.6.4–§3.6.6 we discuss moduli problems in Differential Geometry in which the moduli space M is a manifold with corners. Often this happens when we start with a moduli space M of nonsingular objects, which is a manifold, and compactify it by adding a boundary M \ M of singular objects, such that M is a compact manifold with corners, with interior M = M◦ ⊂ M. A well known example of this is moduli spaces of flow lines in Morse theory, as in Schwarz [86] or Austin and Braam [4]. Let X be a compact manifold, and f : X → R be smooth. We call f Morse if the critical locus Crit(f ) consists of isolated points x ∈ X with Hessian Hess f = ∇2 f ∈ S 2 Tx∗ X nondegenerate. For x ∈ Crit(f ) we define µ(x) to be the number of negative eigenvalues of Hess f |x . Fix a Riemannian metric g on X, and let ∇f ∈ Γ∞ (T X) be the associated gradient vector field of f . For x, y ∈ Crit(f ), consider the moduli space M(x, y) of ∼-equivalence d γ(t) = classes [γ] of smooth flow-lines γ : R → X of −∇f (that is, dt −∇f |γ(t) for t ∈ R) with limt→−∞ γ(t) = x and limt→∞ γ(t) = y, with equivalence relation γ ∼ γ 0 if γ(t) = γ 0 (t + c) for some c ∈ R and all t. Next, enlarge M(x, y) to a compactification M(x, y) by including ‘broken flow-lines’, which are sequences ([γ1 ], . . . , [γk ]) for [γi ] ∈ M(zi−1 , zi ) with z0 = x and zk = y. As discussed by Wehrheim [97, §1], there is a widelybelieved folklore result, which we state (without claiming it is true): ‘Folklore Theorem’. Let X be a compact manifold, f : X → R a Morse function, and g a generic Riemannian metric on X. Then M(x, y) has the canonical structure of a compact manifold with corners of dimension µ(x) − µ(y) − 1 for all x, y ∈ Crit(f ), with interior M(x, y)◦ = M(x, y), where M(x, y) = ∅ if µ(x) 6 µ(y). There is a canonical diffeomorphism ` ∂M(x, y) ∼ = z∈Crit(f ) M(x, z) × M(z, y). Moduli spaces M(x, y) of dimensions 0 and 1 are used to construct the Morse homology groups H∗Mo (X, Z) of X, as in [4, 86]. The higher-dimensional moduli spaces contain information on the homotopy type of X. Remark 3.20. (a) Actually proving the ‘Folklore Theorem’, and in particular, describing the smooth structure on M(x, y) near ∂M(x, y), seems to be surprisingly difficult. References such as [4, Lem. 2.5], [86, Th. 3, p. 69] show

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only that M(x, y) is a manifold, and then indicate how to glue the strata of M(x, y) together topologically, but without smooth structures. In [48, §6.2.2] the second author argues that the correct smooth structure on M(x, y) is that of a manifold with a-corners, to be discussed in §8.3. (b) For M(x, y) to be a manifold with corners, it is essential that g should be generic, so that the deformation theory of flow-lines of ∇f is unobstructed. For non-generic g, M(x, y) should be singular, and we expect M(x, y) to be a C ∞ -scheme with corners, or a derived manifold with corners in the sense of §8.4. This is our first example of the idea that C ∞ -algebraic geometry with corners is relevant to the study of moduli spaces, see also §3.6.5–§3.6.6.

3.6.5 Moduli spaces of stable Riemann surfaces with boundary This section follows Fukaya et al. [29, §2]. A Riemann surface with boundary Σ is a complex 1-manifold with real boundary ∂Σ. A prestable or nodal Riemann surface with boundary Σ is a singular complex 1-manifold with boundary ∂Σ, whose only singularities are nodes, of two kinds:  (a) Σ has an interior node p in Σ◦ locally modelled on (0, 0) in (x, y) ∈ C2 : xy = 0 . Such a node can occur as the limit ofa family of nonsingular Riemann surfaces with boundary Σ modelled on (x, y) ∈ C2 : xy =  , for  ∈ C \ {0}.  (b) Σ has a boundary node q locally modelled on (0, 0) in (x, y) ∈ C2 : xy = 0, Im x > 0, Im y > 0 . Such occur as the limit of a family of  a node can 2 nonsingular Σ modelled on (x, y) ∈ C : xy = −, Im x > 0, Im y > 0 , for  > 0. If Σ is a prestable Riemann surface with boundary, the smoothing Σ0 of Σ is the oriented 2-manifold with boundary obtained by smoothing the nodes of Σ, using the families Σ above. The genus and number of boundary components of Σ are the genus and number of boundary components of the smoothing Σ0 . Fix g, h, m, n > 0. Consider triples (Σ, y, z), where: (i) Σ is a compact, connected, prestable Riemann surface with boundary, of genus g, with h boundary components. (ii) y = (y1 , . . . , ym ) for y1 , . . . , ym distinct nonsingular points of Σ◦ , called interior marked points. (iii) z = (z1 , . . . , zn ) for z1 , . . . , zn distinct nonsingular points of ∂Σ, called boundary marked points. We could also require the zi to lie on prescribed boundary components, or to have a prescribed cyclic order around a boundary component, but we will not bother.

3.6 Applications of manifolds with (g-)corners

69

Such a triple (Σ, y, z) is called stable if Aut(Σ, y, z) is finite. Write Mg,h,m,n for the set of isomorphism classes [Σ, y, z] of stable triples (Σ, y, z). Then Mg,h,m,n has the natural structure of a compact orbifold with corners, with dim Mg,h,m,n = 6g + 3h + 2m + n − 6. The orbifold group of a point [Σ, y, z] is Aut(Σ, y, z). The codimension k boundary stratum S k (Mg,h,m,n ), in the sense of Definition 3.12, is the subset of [Σ, y, z] such that Σ has k boundary nodes. Points of the boundary ∂Mg,h,m,n are isomorphism classes [Σ, y, z, q] of a triple (Σ, y, z) together with a chosen boundary node q ∈ Σ. By regarding Σ with chosen boundary node q as being obtained by gluing two other Σ0 , Σ00 at boundary marked points z 0 , z 00 by identifying z 0 = z 00 to make q, we can identify ∂Mg,h,m,n with a disjoint union of products of other Mg0 ,h0 ,m0 ,n0 . If h = 0, which forces n = 0, then Mg,0,m,0 is a compact complex orbifold, without boundary. One can also study maps between moduli spaces, for example the map Mg,h,m,n+1 −→ Mg,h,m,n which acts on (Σ, y, z) by forgetting the last boundary marked point zn+1 , and then stabilizing (Σ, y, (z1 , . . . , zn )) if necessary. Such maps tend to be smooth (Melrose’s ‘b-maps’) in the sense of §3.1, but not strongly smooth, for example, they may be locally modelled on [0, ∞)2 × Rk → [0, ∞) × Rk , (x, y, z) 7→ (xy, z). This is one reason for our choice of definition of smooth maps in §3.1. Quilted discs are generalizations of prestable holomorphic discs Σ (the case g = 0, h = 1 above), which have a real circle C in Σ tangent to a point of ∂Σ. They are used in Symplectic Geometry, for example, to construct functors between Fukaya categories FL : F (X1 , ω1 ) → F (X2 , ω2 ) from a Lagrangian correspondence L ⊂ (X1 × X2 , −ω1
ω2 ). Ma’u and Woodward [69, 70] define moduli spaces Mn,1 of ‘stable n-marked quilted discs’. As in [70, §6], for n > 4 these are not manifolds with corners, but have an exotic corner structure; in the language of §3.3, the Mn,1 are manifolds with g-corners.

3.6.6 Moduli spaces with corners in Floer theories Floer theories are infinite-dimensional generalizations of Morse theory, as in §3.6.4, which are used to construct exotic cohomology theories in Symplectic Geometry and low-dimensional differential topology. They involve moduli spaces M of geometric objects, such as J-holomorphic curves or instantons. In good cases M is a manifold with corners. Often the boundaries ∂M may be written as (fibre) products of other moduli

70

Background on manifolds with (g-)corners

spaces M0 , M00 , and the relations this implies between fundamental or virtual chains [M]virt are used to prove identities in the theory. For example, often one constructs a complex (CF ∗ , d) in which d involves virtual counts #[M]virt for vdim M = 0, and boundary relations for moduli spaces M with vdim M = 1 are used to prove that d2 = 0, so that the cohomology HF ∗ of (CF ∗ , d) is well defined. Thus, the corner structure is essential to the theory. Some examples of Floer theories are: (a) Instanton Floer homology IH∗ (Y ) of a compact, oriented 3-manifold Y , as in Donaldson [19]. This involves studying moduli spaces M of instantons (anti-self-dual connections on a principal SU(2)-bundle) on Y × R. (b) Seiberg–Witten Floer homology HM∗ (Y ) of a compact, oriented 3-manifold Y , as in Kronheimer–Mrowka [57]. This involves studying moduli spaces M of Seiberg–Witten monopoles on Y × R. (c) Lagrangian Floer cohomology HF ∗ (L1 , L2 ) of transversely intersecting Lagrangians L1 , L2 in a symplectic manifold (X, ω), as in Fukaya–Oh– Ohta–Ono [28, 29]. This involves choosing an almost complex structure J on X compatible with ω, and studying moduli spaces M of J-holomorphic maps u : Σ → X with u(Σ) ⊂ L1 ∪ L2 for Σ a prestable holomorphic disc, as in §3.6.3 and §3.6.5. Then HF 0 (L1 , L2 ) is the morphisms Hom(L1 , L2 ) in the Fukaya category F (X, ω) of X. In each of (a)–(c), in good cases the moduli spaces M have the structure of manifolds with corners, but otherwise can be singular, and could be modelled as C ∞ -schemes with corners or C ∞ -stacks with corners, or as derived manifolds or orbifolds with corners in the sense of §8.4. Thus, the ideas of this book could have applications in Floer theories. As in §3.6.5, natural morphisms between moduli spaces of J-holomorphic maps in Symplectic Geometry, for example, maps forgetting a boundary marked point, tend to be smooth in the sense of §3.1, but not strongly smooth. This is one reason for our choice of definition of smooth maps in §3.1. As for quilted discs in §3.6.5, there are also moduli spaces M used in Floer theories which in good cases have the structure of manifolds with g-corners, and in general should be derived orbifolds with g-corners, as in §8.4. Ma’u, Wehrheim and Woodward [69, 70, 98–100] study moduli spaces M of pseudoholomorphic quilts, which are J-holomorphic maps whose domain is a quilted disc, and these should have g-corners rather than ordinary corners. Pardon [84, Rem. 2.16] uses moduli spaces of J-holomorphic curves with g-corners to define contact homology of Legendrian submanifolds. Work of Chris Kottke (private communication) suggests that natural compactifications of SU(2) magnetic monopole spaces may have the structure of manifolds with g-corners.

4 (Pre) C ∞ -rings with corners

This chapter defines and studies (pre-)C ∞ -rings with corners. As in §2.2, a categorical C ∞ -ring is a product-preserving functor F : Euc → Sets, where Euc ⊂ Man is the full subcategory of Euclidean spaces Rn , and a C ∞ -ring C is an equivalent way of repackaging the data in F , with C = F (R), and many operations on C. Here Euc is an Algebraic Theory [1], and a (categorical) C ∞ -ring is an algebra over this Algebraic Theory. Similarly, we define a categorical pre-C ∞ -ring with corners to be a productpreserving functor F : Eucc → Sets, where Eucc ⊂ Manc is the full m−k k subcategory of Euclidean corner spaces Rm . A pre-C ∞ k = [0, ∞) × R ring with corners C = (C, C ex ) is an equivalent way of repackaging F , with C = F (R) and C ex = F ([0, ∞)), and many operations on C, C ex . These were introduced by Kalashnikov [52], who called them C ∞ -rings with corners. Again, Eucc is an Algebraic Theory, and a (categorical) pre-C ∞ -ring with corners is an algebra over this Algebraic Theory, so we can quote results from Algebraic Theories on existence of (co)limits of pre-C ∞ -rings with corners. We define C ∞ -rings with corners in §4.4 to be the full subcategory of pre C -rings with corners satisfying an extra condition, that invertible functions in C ex should have logs in C. To motivate this, note that if X is a manifold with corners and f : X → [0, ∞) is smooth with an inverse 1/f then f maps X → (0, ∞), and so log f : X → R is well defined and smooth. This condition makes C ∞ -rings and C ∞ -schemes with corners better behaved. ∞

Sections 4.5–4.7 develop the theory of C ∞ -rings with corners. This includes studying localizations and local C ∞ -rings with corners, which are used to define a spectrum functor in §5.3. We also introduce special classes of C ∞ rings with corners that are useful in later chapters. Throughout the chapter, we will return to our motivating example of manifolds with (g-)corners, and show how they can be represented as (pre-)C ∞ -rings with corners. 71

(Pre) C ∞ -rings with corners

72

4.1 Categorical pre C ∞ -rings with corners Here is the obvious generalization of Definition 2.10. Definition 4.1. Write Eucc , Euccin for the full subcategories of Manc and m−k k Mancin with objects the Euclidean spaces with corners Rm k = [0, ∞) ×R for 0 6 k 6 m. We have inclusions of subcategories Euc ⊂ Euccin ⊂ Eucc .

(4.1)

Define a categorical pre C ∞ -ring with corners to be a product-preserving functor F : Eucc → Sets. Here F should also preserve the trivial product, i.e. it maps R0 × [0, ∞)0 = {∅} in Eucc to the terminal object in Sets, the point ∗. If F, G : Eucc → Sets are categorical pre C ∞ -rings with corners, a morphism η : F → G is a natural transformation η : F ⇒ G. Such natural transformations are automatically product-preserving. We write CPC∞ Ringsc for the category of categorical pre C ∞ -rings with corners. Define a categorical interior pre C ∞ -ring with corners to be a product-preserving functor F : Euccin → Sets. They form a category CPC∞ Ringscin , with morphisms natural transformations. Define a commutative triangle of functors CPC∞ Ringsc CPC∞ Ringsc

ΠCPC∞ Ringsin c

/ CC∞ Rings 4

∞

Rings ΠCC CPC∞ Ringsc

* CPC∞ Ringscin

∞

Rings ΠCC CPC∞ Ringsc

(4.2)

in

CPC∞ Ringsc

by restriction to subcategories in (4.1), so that for example ΠCPC∞ Ringsin c maps F : Eucc → Sets to F |Euccin : Euccin → Sets. Here is the motivating example: Example 4.2. (a) Let X be a manifold with corners. Define a categorical pre C ∞ -ring with corners F : Eucc → Sets by F = HomManc (X, −). That is, for objects Rm × [0, ∞)n in Eucc ⊂ Manc we have

F Rm × [0, ∞)n = HomManc X, Rm × [0, ∞)n , 0

0

and for morphisms g : Rm × [0, ∞)n → Rm × [0, ∞)n in Eucc we have
0 0
F (g) = g◦ : HomManc X, Rm ×[0, ∞)n −→ HomManc X, Rm ×[0, ∞)n mapping F (g) : h 7→ g ◦ h. Let f : X → Y be a smooth map of manifolds

4.2 Pre C ∞ -rings with corners

73

with corners, and F, G : Eucc → Sets the functors corresponding to X, Y . Define a natural transformation η : G ⇒ F by


η Rm ×[0, ∞)n = ◦f : Hom Y, Rm ×[0, ∞)n −→ Hom X, Rm ×[0, ∞)n mapping η : h 7→ h ◦ f . CPC∞ Ringsc Define a functor FMan : Manc → (CPC∞ Ringsc )op to map c X 7→ F on objects, and f 7→ η on morphisms, for X, Y, F, G, f, η as above. All of this also works if X, Y are manifolds with g-corners, as in §3.3, giving CPC∞ Ringsc a functor FMan : Mangc → (CPC∞ Ringsc )op . gc (b) Similarly, if X is a manifold with (g-)corners, define a categorical interior pre C ∞ -ring with corners F : Euccin → Sets by F = HomMancin (X, −). CPC∞ Ringscin

This gives functors FManc ∞

CPC

FMangc in

Ringscin

in

: Mancin → (CPC∞ Ringscin )op and

∞ c op : Mangc in → (CPC Ringsin ) .

In the language of Algebraic Theories, as in Ad´amek, Rosick´y and Vitale [1], Euc, Eucc , Euccin are algebraic theories (that is, small categories with finite products), and CC∞ Rings, CPC∞ Ringsc , CPC∞ Ringscin are the corresponding categories of algebras. Also the inclusions of subcategories (4.1) are morphisms of algebraic theories, and the functors (4.2) the corresponding morphisms. So, as for Proposition 2.18, Ad´amek et al. [1, Prop.s 1.21, 2.5, 9.3 & Th. 4.5] give important results on their categorical properties: Theorem 4.3. (a) All small limits and directed colimits exist in the categories CPC∞ Ringsc , CPC∞ Ringscin , and they may be computed objectwise in Eucc , Euccin by taking the corresponding small limits/directed colimits in Sets. (b) All small colimits exist in CPC∞ Ringsc , CPC∞ Ringscin , though in general they are not computed objectwise in Eucc , Euccin as colimits in Sets. ∞

CPC∞ Ringsc

Rings in (c) The functors ΠCC CPC∞ Ringsc , . . . , ΠCPC∞ Ringsc in (4.2) have left adjoints, so they preserve limits.

4.2 Pre C ∞ -rings with corners The next definition relates to Definition 4.1 in the same way that Definition 2.11 relates to Definition 2.10. Definition 4.4. A pre C ∞ -ring with corners C assigns the data: (a) Two sets C and C ex .

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(Pre) C ∞ -rings with corners

(b) Operations Φf : C m ×C nex → C for all smooth maps f : Rm ×[0, ∞)n → R. (c) Operations Ψg : C m × C nex → C ex for all exterior g : Rm × [0, ∞)n → [0, ∞). Here we allow one or both of m, n to be zero, and consider S 0 to be the single point {∅} for any set S. These operations must satisfy the following relations: (i) Suppose k, l, m, n > 0, and ei : Rk × [0, ∞)l → R is smooth for i = 1, . . . , m, and fj : Rk × [0, ∞)l → [0, ∞) is exterior for j = 1, . . . , n, and g : Rm × [0, ∞)n → R is smooth. Define smooth h : Rk × [0, ∞)l → R by h(x1 , . . . , xk , y1 , . . . , yl ) = g e1 (x1 , . . . , yl ), . . . , em (x1 , . . . , yl ),
(4.3) f1 (x1 , . . . , yl ), . . . , fn (x1 , . . . , yl ) . Then for all (c1 , . . . , ck , c01 , . . . , c0l ) ∈ C k × C lex we have Φh (c1 , . . . , ck , c01 , . . . , c0l ) = Φg Φe1 (c1 , . . . , c0l ), . . . , Φem (c1 , . . . , c0l ),
Ψf1 (c1 , . . . , c0l ), . . . , Ψfn (c1 , . . . , c0l ) . (ii) Suppose k, l, m, n > 0, and ei : Rk × [0, ∞)l → R is smooth for i = 1, . . . , m, and fj : Rk × [0, ∞)l → [0, ∞) is exterior for j = 1, . . . , n, and g : Rm × [0, ∞)n → [0, ∞) is exterior. Define exterior h : Rk × [0, ∞)l → [0, ∞) by (4.3). Then for all (c1 , . . . , ck , c01 , . . . , c0l ) ∈ C k × C lex we have Ψh (c1 , . . . , ck , c01 , . . . , c0l ) = Ψg Φe1 (c1 , . . . , c0l ), . . . , Φem (c1 , . . . , c0l ),
Ψf1 (c1 , . . . , c0l ), . . . , Ψfn (c1 , . . . , c0l ) . (iii) Write πi : Rm × [0, ∞)n → R for projection to the ith coordinate of Rm for i = 1, . . . , m, and πj0 : Rm × [0, ∞)n → [0, ∞) for projection to the j th coordinate of [0, ∞)n for j = 1, . . . , n. Then for all (c1 , . . . , cm , c01 , . . . , c0n ) in C m × C nex and all i = 1, . . . , m, j = 1, . . . , n we have Φπi (c1 , . . . , cm , c01 , . . . , c0n ) = ci , Ψπj0 (c1 , . . . , cm , c01 , . . . , c0n ) = c0j . We will refer to the operations Φf , Ψg as the C ∞ -operations, and we often write a pre C ∞ -ring with corners as a pair C = (C, C ex ), leaving the C ∞ operations implicit. Let C = (C, C ex ) and D = (D, Dex ) be pre C ∞ -rings with corners. A morphism φ : C → D is a pair φ = (φ, φex ) of maps φ : C → D and φex : C ex → Dex , which commute with all the operations Φf , Ψg on C, D. Write PC∞ Ringsc for the category of pre C ∞ -rings with corners. As for Proposition 2.12, we have:

4.2 Pre C ∞ -rings with corners

75

Proposition 4.5. There is an equivalence of categories from CPC∞ Ringsc ∞ to PC∞ Ringsc , which identifies F : Eucc → Sets in CPC Ringsc
m ∞ c n with C = (C, C ex ) in PC Rings such that F R × [0, ∞) = C m × C nex for m, n > 0. Under this equivalence, for a smooth function f : Rm k → R, we identify F (f ) with Φf , and for an exterior function g : Rm → [0, ∞), we identify F (g) k with Ψg . The proof of the proposition then follows from F being a productpreserving functor, and the definition of pre C ∞ -ring with corners. It is often helpful to work with a small subset of C ∞ -operations. The next definition explains this small subset. Monoids were discussed in §3.2. Definition 4.6. Let C = (C, C ex ) be a pre C ∞ -ring with corners. Then (as in Definition 4.11 below) C is a C ∞ -ring, and thus a commutative R-algebra. The R-algebra structure makes C into a monoid in two ways: under multiplication ‘·’ with identity 1, and under addition ‘+’ with identity 0. Define g : [0, ∞)2 → [0, ∞) by g(x, y) = xy. Then g induces Ψg : C ex × C ex → C ex . Define multiplication · : C ex × C ex → C ex by c0 · c00 = Ψg (c0 , c00 ). The map 1 : R0 → [0, ∞) gives an operation Ψ1 : {∅} → C ex . The identity in C ex is 1C ex = Ψ1 (∅). Then (C ex , ·, 1C ex ) is a monoid. The map 0 : R0 → [0, ∞) gives an operation Ψ0 : {∅} → C ex . This gives a distinguished element 0C ex = Ψ0 (∅) in C ex , which satisfies c0 · 0C ex = 0C ex for all c0 ∈ C ex , that is, 0C ex is a zero element in the monoid (C ex , ·, 1C ex ). We have: (a) C is a commutative R-algebra. (b) C ex is a monoid, in multiplicative notation, with zero element 0C ex ∈ C ex . We write C × ex for the group of invertible elements in C ex . (c) Φi : C ex → C is a monoid morphism, for i : [0, ∞) ,→ R the inclusion and C a monoid under multiplication. (d) Ψexp : C → C ex is a monoid morphism, for exp : R → [0, ∞) and C a monoid under addition. (e) Φexp = Φi ◦ Ψexp : C → C for exp : R → R and Ψexp in (d). Many of our definitions will use only the structures (a)–(e). When we write Φi , Ψexp , Φexp without further explanation, we mean those in (c)–(e). Proposition 4.7. Let C = (C, C ex ) be a pre C ∞ -ring with corners, and suppose c0 lies in the group C × ex of invertible elements in the monoid C ex . Then there exists a unique c ∈ C such that Φexp (c) = Φi (c0 ) in C. 0−1 Proof Since c0 ∈ C × ∈ C× ex we have a unique inverse c ex . As in the proof of Proposition 2.26(a), let e : R → R be the inverse of t 7→ exp(t) − exp(−t).

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(Pre) C ∞ -rings with corners

Define smooth g : [0, ∞)2 → R by g(x, y) = e(x − y). Observe that if (x, y) ∈ [0, ∞)2 with xy = 1 then x = exp t, y = exp(−t) for t = log x, and so exp ◦g(x, y) = exp ◦g(exp t, exp(−t)) = exp ◦e(exp t − exp(−t)) = exp(t) = x. Therefore there is a unique smooth function h : [0, ∞)2 → R with exp ◦g(x, y) − x = h(x, y)(xy − 1).

(4.4)

We have operations Φg , Φh : C 2ex → C. Define c = Φg (c0 , c0−1 ). Then Φexp (c) − Φi (c0 ) = Φexp ◦g(x,y)−x (c0 , c0−1 ) = Φh(x,y)(xy−1) (c0 , c0−1 ) = Φh (c0 , c0−1 ) · (Φi (c0 · c0−1 ) − 1C ) = 0, using Definition 4.1(i) in the first and third steps, and (4.4) in the second. Hence Φexp (c) = Φi (c0 ). Uniqueness of c follows from Proposition 2.26(a). We could define ‘interior pre C ∞ -rings with corners’ following Definition 4.4, but replacing exterior maps by interior maps throughout. Instead we will do something equivalent but more complicated, and define interior pre C ∞ -rings with corners as special examples of pre C ∞ -rings with corners, and interior morphisms as special morphisms between (interior) pre C ∞ -rings with corners. The advantage of this is that we can work with both interior and non-interior pre C ∞ -rings with corners and their morphisms in a single theory. Definition 4.8. Let C = (C, C ex ) be a pre C ∞ -ring with corners. Then C ex is a monoid and 0C ex ∈ C ex with c0 · 0C ex = 0C ex for all c0 ∈ C ex . We call C an interior pre C ∞ -ring with corners if 0C ex 6= 1C ex , and there do not exist c0 , c00 ∈ C ex with c0 6= 0C ex 6= c00 and c0 · c00 = 0C ex . That is, C ex should have no zero divisors. Write C in = C ex \ {0C ex }. Then C ex = C in q {0C ex }, where q is the disjoint union. Since C ex has no zero divisors, C in is closed under multiplication, and 1C ex ∈ C in as 0C ex 6= 1C ex . Thus C in is a submonoid of C ex . We write 1C in = 1C ex . Let C, D be interior pre C ∞ -rings with corners, and φ = (φ, φex ) : C → D be a morphism in PC∞ Ringsc . We call φ interior if φex (C in ) ⊆ Din . Then we write φin = φex |C in : C in → Din . Interior morphisms are closed under composition and include the identity morphisms. Write PC∞ Ringscin for the (non-full) subcategory of PC∞ Ringsc with objects interior pre C ∞ -rings with corners, and morphisms interior morphisms. Lemma 4.9. Let C = (C, C ex ) be an interior pre C ∞ -ring with corners. Then

4.2 Pre C ∞ -rings with corners

77

m n m n C× ex ⊆ C in . If g : R × [0, ∞) → [0, ∞) is interior, then Ψg : C × C ex → m n C ex maps C × C in → C in . × Proof Clearly 0C ex is not invertible, so 0C ex ∈ / C× ex , and C ex ⊆ C ex \{0C ex } = C in . As g is interior we may write

g(x1 , . . . , xm , y1 , . . . , yn ) = y1a1 · · · ynan · exp ◦h(x1 , . . . , yn ),

(4.5)

for a1 , . . . , an ∈ N and h : Rm ×[0, ∞)n → R smooth. Then for c1 , . . . , cm ∈ C and c01 , . . . , c0n ∈ C in we have

 0an 0 0 1 Ψg c1 , . . . , cm , c01 , . . . , c0n = c0a 1 · · · cn ·Ψexp Φh c1 , . . . , cm , c1 , . . . , cn . 0an 1 Here c0a 1 · · · cn ∈ C in as C in is a submonoid of C ex , and Ψexp [·
· · ] ∈ C in as 0 0 Ψexp maps to C × ex ⊆ C in ⊆ C ex . Thus Ψg c1 , . . . , cm , c1 , . . . , cn ∈ C in .

Here is the analogue of Proposition 4.5: Proposition 4.10. There is an equivalence CPC∞ Ringscin ∼ = PC∞ Ringscin , which identifies F : Euccin → Sets in CPC
∞ Ringscin with C = (C, C ex ) in PC∞ Ringscin such that F Rm × [0, ∞)n = C m × C nin for m, n > 0. Proof Let F : Euccin → Sets be a categorical interior pre C ∞ -ring with corners. Define sets C = F (R), C in = F ([0, ∞)), and
C ex = C in q {0C ex }, where q is the disjoint union. Then F Rm × [0, ∞)n = C m × C nin , as F is product-preserving. Let f : Rm × [0, ∞)n → R and g : Rm × [0, ∞)n → [0, ∞) be smooth. We must define maps Φf : C m × C nex → C and Ψg : C m × C nex → C ex . Let c1 , . . . , cm ∈ C and c01 , . . . , c0n ∈ C ex . Then some of c01 , . . . , c0n lie in C in and the rest in {0C ex }. For simplicity suppose that c01 , . . . , c0k ∈ C in and c0k+1 = · · · = c0n = 0C ex for 0 6 k 6 n. Define smooth d : Rm × [0, ∞)k → R, e : Rm × [0, ∞)k → [0, ∞) by d(x1 , . . . , xm , y1 , . . . , yk ) = f (x1 , . . . , xm , y1 , . . . , yk , 0, . . . , 0), e(x1 , . . . , xm , y1 , . . . , yk ) = g(x1 , . . . , xm , y1 , . . . , yk , 0, . . . , 0).

(4.6)

Then F (d) maps C m × C kin → C. Set Φf (c1 , . . . , cm , c01 , . . . , c0k , 0C ex , . . . , 0C ex ) = F (d)(c1 , . . . , cm , c01 , . . . , c0k ). Either e : Rm × [0, ∞)k → [0, ∞) is interior, or e = 0. If e is interior define Ψg (c1 , . . . , cm , c01 , . . . , c0k , 0C ex , . . . , 0C ex ) = F (e)(c1 , . . . , cm , c01 , . . . , c0k ).

(Pre) C ∞ -rings with corners

78

If e = 0 set Ψg (c1 , . . . , cm , c01 , . . . , c0k , 0C ex , . . . , 0C ex ) = 0C ex . This defines Φf , Ψg , and makes C = (C, C ex ) into an interior pre C ∞ -ring with corners. Conversely, let C = (C, C ex ) be an interior pre C ∞ -ring with corners. Then C ex = C in q {0} and we define a product-preserving functor F : Euccin →
m n Sets with F R × [0, ∞) = C m × C nin , using the fact from Lemma 4.9 that Ψg : C m × C nex → C ex maps C m × C nin → C in for g interior. The rest of the proof follows that of Proposition 4.5.

4.3 Adjoints and (co)limits for pre C ∞ -rings with corners The equivalences of categories in Propositions 2.12, 4.5 and 4.10 mean that facts about CPC∞ Ringsc , CPC∞ Ringscin , CC∞ Rings in §4.1 correspond to facts about PC∞ Ringsc , PC∞ Ringscin , C∞ Rings. Hence (4.2) should correspond to a commutative triangle of functors PC∞ Ringsc

∞

PC∞ Ringsc

ΠPC∞ Ringsin c

+

Rings ΠC PC∞ Ringsc

PC∞ Ringscin

/ C∞ Rings. 3 ∞ (4.7) ΠC Rings PC∞ Ringsc in

The explicit definitions, which we give next, follow from the correspondences. ∞

C Rings ∞ Definition 4.11. The functor ΠPC Ringsc → C∞ Rings in ∞ Ringsc : PC (4.7) acts on objects by C = (C, C ex ) 7→ C, where the C ∞ -ring C has C ∞ operations Φf : C m → C from smooth f : Rm → R in Definition 4.4(b) with n = 0, and on morphisms by φ = (φ, φex ) 7→ φ. ∞ Rings ∞ c ∞ Define ΠC PC∞ Ringsc : PC Ringsin → C Rings in (4.7) to be the ∞

in

Rings ∞ c restriction of ΠC PC∞ Ringsc to PC Ringsin . PC∞ Ringsc

∞ To define ΠPC∞ Ringsin -ring with c in (4.7), let C = (C, C ex ) be a pre C ∞ ˜ ex ) ˜ = (C, C corners. We will define an interior pre C -ring with corners C PC∞ Ringscin ˜ ˜ where C ex = C ex q {0C˜ ex }, and set ΠPC∞ Ringsc (C) = C. Here C ex already contains a zero element 0C ex , but we are adding an extra 0C˜ ex with 0C ex 6= 0C˜ ex . Let f : Rm × [0, ∞)n → R and g : Rm × [0, ∞)n → [0, ∞) be smooth, and ˜n → C ˜ f : C m ×C write Φf , Ψg for the operations in C. We must define maps Φ ex m n 0 0 ˜ ˜ ˜ ˜ and Ψg : C × C ex → C ex . Let c1 , . . . , cm ∈ C and c1 , . . . , cn ∈ C ex . Then some of c01 , . . . , c0n lie in C ex and the rest in {0C˜ ex }. For simplicity suppose that c01 , . . . , c0k ∈ C ex and c0k+1 = · · · = c0n = 0C˜ ex for 0 6 k 6 n. Define smooth d : Rm × [0, ∞)k → R, e : Rm × [0, ∞)k → [0, ∞) by (4.6). Set

˜ f (c1 , . . . , cm , c0 , . . . , c0 , 0 ˜ , . . . , 0 ˜ ) = Φd (c1 , . . . , cm , c0 , . . . , c0 ). Φ 1 k C ex 1 k C ex

4.3 Adjoints and (co)limits for pre C ∞ -rings with corners

79

Either e : Rm × [0, ∞)k → [0, ∞) is interior, or e = 0. If e is interior define ˜ g (c1 , . . . , cm , c01 , . . . , c0k , 0 ˜ , . . . , 0 ˜ ) = Ψe (c1 , . . . , cm , c01 , . . . , c0k ). Ψ C ex C ex ˜ g (c1 , . . . , cm , c0 , . . . , c0 , 0 ˜ , . . . , 0 ˜ ) = 0 ˜ . This deIf e = 0 define Ψ 1 k C ex C ex C ex ˜ ex ) into ˜f, Ψ ˜ g . It is easy to check that these make C ˜ = (C, C fines the maps Φ ∞ an interior pre C -ring with corners. ˜ D ˜ as Now let φ : C → D be a morphism in PC∞ Ringsc , and define C, ˜ ex → D ˜ ex by φ˜ex |C = φex and φ˜ex (0 ˜ ) = 0 ˜ . above. Define φ˜ex : C ex C ex Dex ˜ = (φ, φ˜ex ) : C ˜ → D ˜ is a morphism in PC∞ Ringsc . Define Then φ in PC∞ Ringsc PC∞ Ringsc ΠPC∞ Ringsin : PC∞ Ringsc ,→ PC∞ Ringscin by ΠPC∞ Ringsin : C 7→ c c PC∞ Ringscin ˜ ˜ ˜ C and Π ∞ c : φ 7→ φ, for C, . . . , φ as above. PC

Rings

PC∞ Ringsc

∞ We will show that ΠPC∞ Ringsin Ringscin ,→ c is right adjoint to inc : PC ∞ c ∞ PC Rings . Suppose that C, D are pre C -rings with corners with C interior. Then we can define a 1-1 correspondence
HomPC∞ Ringsc inc(C), D = HomPC∞ Ringsc (C, D) ∞

c

PC Rings ∼ = HomPC∞ Ringscin C, ΠPC∞ Ringsin c (D)), ∞

c

ˆ : C → ΠPC∞ Ringsin identifying φ : inc(C) → D with φ PC Ringsc (D), where φ = ˆ = (φ, φˆex ) with φˆex |C in = φex |C in and φˆex (0C ex ) = 0 ˜ . This (φ, φex ) and φ Dex PC∞ Ringsc

is functorial in C, D, and so proves that ΠPC∞ Ringsin c is right adjoint to inc. ∞ c Also define functors Πsm , Πex : PC Rings → Sets by Πsm : C 7→ C, Πex : C 7→ C ex on objects, and Πsm : φ 7→ φ, Πex : φ 7→ φex on morphisms, where ‘sm’, ‘ex’ are short for ‘smooth’ and ‘exterior’. Define functors Πsm , Πin : PC∞ Ringscin → Sets by Πsm : C 7→ C, Πin : C 7→ C in on objects, and Πsm : φ 7→ φ, Πin : φ 7→ φin on morphisms, where ‘in’ is short for ‘interior’. Combining the equivalences in Propositions 2.12, 4.5 and 4.10 with Theorem 4.3 and Definition 4.11 yields: Theorem 4.12. (a) All small limits and directed colimits exist in the categories PC∞ Ringsc , PC∞ Ringscin . The functors Πsm , Πex : PC∞ Ringsc → Sets and Πsm , Πin : PC∞ Ringscin → Sets preserve limits and directed colimits. (b) All small colimits exist in PC∞ Ringsc , PC∞ Ringscin , though in general they are not preserved by Πsm , Πex and Πsm , Πin . ∞

PC∞ Ringsc

∞

Rings C Rings in (c) The functors ΠC PC∞ Ringsc , ΠPC∞ Ringsc , ΠPC∞ Ringsc in (4.7) have left in

80

(Pre) C ∞ -rings with corners PC∞ Ringsc

adjoints, so they preserve limits. The left adjoint of ΠPC∞ Ringsin c is the inclusion inc : PC∞ Ringscin ,→ PC∞ Ringsc , so inc preserves colimits. Example 4.13. The inclusion inc : PC∞ Ringscin ,→ PC∞ Ringsc in general does not preserve limits, and therefore cannot have a left adjoint. For example, suppose C, D are interior pre C ∞ -rings with corners, and write E = C × D and F = C ×in D for the products in PC∞ Ringsc and PC∞ Ringscin . Then Theorem 4.12(a) implies that E = C × D, Eex = C ex × Dex , F = C × D, Fin = C in × Din . Since C ex = C in q {0C ex }, etc., this gives Eex = (C in ×Din )q(C in ×{0Dex })q({0C ex }×Din )q({0C ex }×{0Dex }), Fex = (C in × Din ) q ({0C ex } × {0Dex }). Thus E ∼ 6 F. Moreover in Eex we have (1C ex , 0Dex ) · (0C ex , 1Dex ) = (0C ex , = 0Dex ), so Eex has zero divisors, and E is not an object in PC∞ Ringscin . ∞

∞

Rings C Rings Proposition 4.14. The functors ΠC PC∞ Ringsc , ΠPC∞ Ringscin in (4.7) have right adjoints, so they preserve colimits.

Proof We will define a functor F>0 : C∞ Rings → PC∞ Ringsc , and ∞ Rings ∞ show it is right adjoint to ΠC PC∞ Ringsc . Let C be a C -ring, and define C >0 to be the subset of elements c ∈ C that satisfy the following condition: for all smooth f : Rn → R such that f |[0,∞)×Rn−1 = 0 and for d1 , . . . , dn−1 ∈ C, then Φf (c, d1 , . . . , dn−1 ) = 0 ∈ C. Note that C >0 is non-empty, as it contains Φexp (C) q {0}. Also, any pre C ∞ -ring with corners (D, Dex ) has Φi (Dex ) ⊆ D>0 . We will make (C, C >0 ) into a pre C ∞ -ring with corners. Take a smooth map m f : Rm k → [0, ∞). Then there is a (non-unique) smooth extension g : R → R such that g|Rm = i ◦ f for i : [0, ∞) → R the inclusion map. We define k Ψf (c01 , . . . , c0k , ck+1 , . . . , cm ) = Φg (c01 , . . . , c0k , ck+1 , . . . , cm ) for c0i ∈ C >0 and ci ∈ C. To show this is well defined, say h is another such extension, then (g − h)|Rm = 0. We must prove Φg−h (c01 , . . . , c0k , ck+1 , . . . , cm ) k 0 = 0 for all ci ∈ C >0 and ci ∈ C. As g − h satisfies the hypothesis of Lemma 4.15 below, we can assume g − h = f1 + · · · + fk such that fi |Ri−1 ×[0,∞)×Rm−i−1 = 0. Then Φg−h (c01 , . . . , c0k , ck+1 , . . . , cm ) = Φf1 +···+fk (c01 , . . . , c0k , ck+1 , . . . , cm ) = Φf1 (c01 , . . . , c0k , ck+1 , . . . , cm )+· · ·+Φfk (c01 , . . . , c0k , ck+1 , . . . , cm ) = 0, as c0i are in C >0 . Therefore Ψf is independent of the choice of extension g.

4.3 Adjoints and (co)limits for pre C ∞ -rings with corners

81

To show that Ψf maps to C >0 ⊆ C, suppose e : Rn → R is smooth with e|[0,∞)×Rn−1 = 0, and let d1 , . . . , dn−1 ∈ C. Then Φe (Ψf (c01 , . . . , c0k , ck+1 , . . . , cm ), d1 , . . . , dn−1 ) = Φe (Φg (c01 , . . . , c0k , ck+1 , . . . , cm ), d1 , . . . , dn−1 ) =

(4.8)

Φe◦(g×idRn−1 ) (c01 , . . . , c0k , ck+1 , . . . , cm , d1 , . . . , dn−1 ).

Here e◦(g×idRn−1 ) : Rm+n−1 → R restricts to zero on Rkm+n−1 ⊂ Rm+n−1 , since (g × idRn−1 )|Rm+n−1 = f × idRn−1 , which maps to [0, ∞) × Rn−1 , k and e|[0,∞)×Rn−1 = 0. Regarding e ◦ (g × idRn−1 ) as an extension of 0 : Rkm+n−1 → R, the argument above shows (4.8) is independent of extension, so we can replace e ◦ (g × idRn−1 ) by 0 : Rm+n−1 → R, and (4.8) is zero. As this holds for all e, n, we have Ψf (c01 , . . . , cm ) ∈ C >0 , as we want. ∞ Similarly, smooth functions f : Rm k → R give well defined C -operations ∞ Φf . Hence (C, C >0 ) is a pre C -ring with corners. Set F>0 (C) = (C, C >0 ). On morphisms, F>0 sends φ : C → D to (φ, φex ) : (C, C >0 ) → (D, D>0 ), where φex = φ|C >0 . As φ respects the C ∞ -operations, the image of φ(C >0 ) ⊆ D>0 ⊆ D, so φex is well defined. ∞ Rings To show F>0 is right adjoint to ΠC PC∞ Ringsc , we describe the unit and counit of the adjunction. The unit acts on (C, C ex ) ∈ PC∞ Ringsc as (idC , Φi ) : (C, C ex ) → (C, C >0 ). The counit acts on C ∈ C∞ Rings as idC : C → C. The ∞ Rings C∞ Rings compositions F>0 ⇒ F>0 ◦ ΠC PC∞ Ringsc ◦ F>0 ⇒ F>0 and ΠPC∞ Ringsc ⇒ ∞

∞

∞

Rings C Rings C Rings ΠC PC∞ Ringsc ◦ F>0 ◦ ΠPC∞ Ringsc ⇒ ΠPC∞ Ringsc are the identity. ∞

∞

Rings C Rings Since ΠC PC∞ Ringsc = ΠPC∞ Ringsc ◦ inc, it now follows from Definition in

PC∞ Ringsc

∞

C Rings 4.11 that ΠPC∞ Ringsin c ◦ F>0 is right adjoint to ΠPC∞ Ringsc . in

The next lemma was used in the previous proof. Lemma 4.15. If f : Rm → R is smooth such that f |Rm = 0, then there are k smooth fi : Rm → R for i = 1, . . . , k such that fi |Ri−1 ×[0,∞)×Rm−i−1 = 0 for i = 1, . . . , k, and f = f1 + . . . + fk . Proof Let f be as in the lemma. Consider the following open subset U = S k−1 \{(x1 , . . . , xk ) : xi > 0} of the dimension k −1 unit sphere S k−1 ⊂ Rk . Define an open cover U1 , . . . , Uk of U by Ui = {(x1 , . . . , xk ) ∈ U : xi < 0}. Take a partition of unity ρi : U → [0, 1] for i = 1, . . . , k, subordinate to Pk {U1 , . . . , Un }, with ρi having support on Ui and i=1 ρi = 1. Define the fi as follows:    f (x , . . . , x )ρ (x1 ,...,xk ) , if x < 0 for some i = 1, . . . , k, 1 m i |(x1 ,...,xk )| i fi = 0, otherwise,

(Pre) C ∞ -rings with corners

82

where |(x1 , . . . , xk )| is the length of the vector (x1 , . . . , xk ) ∈ Rk . It is clear these are smooth where the ρi are defined. The ρi are not defined in the first quadrant of Rk , where all xi > 0, however approaching the boundary of this quadrant, the ρi are all constant. As f |Rm = 0, then all derivatives of f are k zero in this quadrant, so the fi are smooth and identically zero on Rm k . In addition, fi |Ri−1 ×[0,∞)×Rm−i−1 = 0, as the ρi are zero outside of Ui . Finally, P as ρi = 1 and f |Rm = 0, then f = f1 + . . . + fk as required. k

4.4 C ∞ -rings with corners We can now define C ∞ -rings with corners. Definition 4.16. Let C = (C, C ex ) be a pre C ∞ -ring with corners. We call C a C ∞ -ring with corners if any (hence all) of the following hold: (i) Φi |C × : C× ex → C is injective. ex (ii) Ψexp : C → C × ex is surjective. × (iii) Ψexp : C → C ex is a bijection. Here Proposition 4.7 implies that (i),(ii) are equivalent, and Definition 4.6(e) and Proposition 2.26(a) imply that Ψexp : C → C × ex is injective, so (ii),(iii) are equivalent, and therefore (i)–(iii) are equivalent. Write C∞ Ringsc for the full subcategory of C ∞ -rings with corners in PC∞ Ringsc . We call a C ∞ -ring with corners C interior if it is an interior pre C ∞ -ring with corners. Write C∞ Ringscin ⊂ C∞ Ringsc for the subcategory of interior C ∞ -rings with corners, and interior morphisms between them. Define a commutative triangle of functors C∞ Ringsc C∞ Ringsc

/ ∞ 3 C Rings

∞

ΠC∞ Ringsin c

+

Rings ΠC C∞ Ringsc

C

∞

Ringscin

∞

Rings ΠC C∞ Ringsc

(4.9)

in

by restricting the functors in (4.7) to C∞ Ringsc , C∞ Ringscin . It is clear PC∞ Ringsc ∞ from Definition 4.11 that the restriction of ΠPC∞ Ringsin Ringsc maps c to C PC∞ Ringsc

to C∞ Ringscin . Then ΠPC∞ Ringsin right adjoint to inc in Definition 4.11 c C∞ Ringsc

∞ implies ΠC∞ Ringsin Ringscin ,→ C∞ Ringsc . c is right adjoint to inc : C

Remark 4.17. We can interpret the condition that C = (C, C ex ) be a C ∞ -ring with corners  as follows. Imagine there is some ‘space  with corners’ X, such that C = smooth maps c : X → R , and C ex = exterior maps c0 : X →

4.4 C ∞ -rings with corners

83

[0, ∞) . If c0 ∈ C ex is invertible then c0 should map X → (0, ∞), and we require there to exist smooth c = log c0 : X → R in C with c0 = exp c. This makes C ∞ -rings and C ∞ -schemes with corners behave more like manifolds with corners. Here is the motivating example, following Example 4.2: Example 4.18. (a) Let X be a manifold with corners. Define a C ∞ -ring with corners C = (C, C ex ) by C = C ∞ (X) and C ex = Ex(X), as sets. If f : Rm × [0, ∞)n → R is smooth, define the operation Φf : C m × C nex → C by Φf (c1 , . . . , cm , c01 , . . . , c0n ) :

(4.10)


x 7−→ f c1 (x), . . . , cm (x), c01 (x), . . . , c0n (x) .

If g : Rm × [0, ∞)n → [0, ∞) is exterior, define Ψg : C m × C nex → C ex by Ψg (c1 , . . . , cm , c01 , . . . , c0n ) :

(4.11)


x 7−→ g c1 (x), . . . , cm (x), c01 (x), . . . , c0n (x) .

It is easy to check that C is a C ∞ -ring with corners. We write C ∞ (X) = C. Suppose f : X → Y is a smooth map of manifolds with corners, and let C, D be the C ∞ -rings with corners corresponding to X, Y . Write φ = (φ, φex ), where φ : D → C maps φ(d) = d ◦ f and φex : Dex → C ex maps φ(d0 ) = d0 ◦ f . Then φ : D → C is a morphism in C∞ Ringsc . C∞ Ringsc Define a functor FMan : Manc → (C∞ Ringsc )op to map X 7→ C c on objects, and f 7→ φ on morphisms, for X, Y, C, D, f, φ as above. All the above also works for manifolds with g-corners, as in §3.3, giving a C∞ Ringsc functor FMan : Mangc → (C∞ Ringsc )op . gc (b) Similarly, if X is a manifold with (g-)corners, define an interior C ∞ -ring with corners C = (C, C ex ) by C = C ∞ (X) and C ex = In(X) q {0} as sets, where 0 : X → [0, ∞) is the zero function, with C ∞ -operations as in (4.10)–(4.11). We write C ∞ in (X) = C. Suppose f : X → Y is an interior map of manifolds with (g-)corners, and let C, D be the interior C ∞ -rings with corners corresponding to X, Y . Write φ = (φ, φex ), where φ : D → C maps φ(d) = d ◦ f and φex : Dex → C ex maps φ(d0 ) = d0 ◦ f . Then φ : D → C is a morphism in C∞ Ringscin . C∞ Ringscin C∞ Ringsc Define FManc : Mancin → (C∞ Ringscin )op and FMangc in : in

in

∞ c op Mangc to map X 7→ C on objects, and f 7→ φ on in → (C Ringsin ) morphisms, for X, Y, C, D, f, φ as above.

Recall from §2.1.3 that if D is a category and C ⊂ D a full subcategory, then C is a reflective subcategory if the inclusion inc : C ,→ D has a left adjoint Π : D → C, which is called a reflection functor.

(Pre) C ∞ -rings with corners

84

Proposition 4.19. The inclusion inc : C∞ Ringsc ,→ PC∞ Ringsc has a C∞ Ringsc ∞ left adjoint reflection functor ΠPC Ringscin ∞ Ringsc . Its restriction to PC C∞ Ringsc

∞ in is a left adjoint ΠPC∞ Rings Ringscin ,→ PC∞ Ringscin . c for inc : C in

Proof Let C = (C, C ex ) be a pre C ∞ -ring with corners. We will define a C ∞ ˆ ex ). As a set, define C ˆ ex = C ex /∼, where ∼ is ˆ = (C, C ring with corners C 0 00 the equivalence relation on C ex given by c ∼ c if there exists c000 ∈ C × ex with ˆ ex is the quotient of C ex by the group Φi (c000 ) = 1
and c0 = c00 · c000 . That is, C ˆ ex . Ker Φi |C × ⊆ C × . There is a natural surjective projection π ˆ : C ex → C ex

ex

If f : Rm × [0, ∞)n → R is smooth and g : Rm × [0, ∞)n → [0, ∞) is ˆn → C ˆ f : Cm × C exterior, it is not difficult to show there exist unique maps Φ ex m n ˆ →C ˆ ex making the following diagrams commute: ˆg : C × C and Ψ ex C m × C nex m

/C

Φf

n

 id ׈π ˆn Cm × C ex

C m × C nex m

 id ׈π ˆn Cm × C ex

id

 / C,

ˆf Φ

/ C ex

Ψg

n

π ˆ ˆg Ψ



(4.12)

/C ˆ ex ,

ˆ into a pre C ∞ -ring with corners. Clearly C ˆ satisfies Definiand these make C ∞ ˆ tion 4.16(i). Therefore C is a C -ring with corners. Suppose φ = (φ, φex ) : C → D is a morphism of pre C ∞ -rings with ˆ D ˆ as above. Then by a similar argument to (4.12), we corners, and define C, find that there is a unique map φˆex such that the following commutes: C ex

/ Dex

φex

 πˆ ˆ ex C

 /D ˆ ex , π ˆ

ˆex φ

ˆ = (φ, φˆex ) : C ˆ → D ˆ is a morphism of C ∞ -rings with corners. and then φ C∞ Ringsc Define a functor ΠPC∞ Ringsc : PC∞ Ringsc → C∞ Ringsc to map C 7→ ˆ on morphisms. ˆ on objects and φ 7→ φ C Now let C be a pre C ∞ -ring with corners, D a C ∞ -ring with corners, and ˆ:C ˆ → D, as D ˆ = D, φ : C → D a morphism. Then we have a morphism φ ˆ gives a 1-1 correspondence and φ ↔ φ
HomPC∞ Ringsc C, inc(D) = HomPC∞ Ringsc (C, D)
C∞ Ringsc ∼ = HomC∞ Ringsc ΠPC∞ Ringsc (C), D , ∞

c

Rings which is functorial in C, D. Hence ΠC PC∞ Ringsc is left adjoint to inc. ∞ Ringsc ˆ are interior, so ΠC ∞ ˆ φ If C, φ above are interior then C, c restricts C∞ Ringscin to a functor ΠPC∞ Rings : c in adjoint to inc : C∞ Ringscin

PC

∞

PC Ringscin → ,→ PC∞ Ringscin .

C

∞

Rings c Ringsin , which

is left

4.5 Generators, relations, and localization

85

We extend Theorem 4.12, Example 4.13, Proposition 4.14 to C∞ Ringsc . Theorem 4.20. (a) All small limits exist in C∞ Ringsc , C∞ Ringscin . The functors Πsm , Πex : C∞ Ringsc → Sets and Πsm , Πin : C∞ Ringscin → Sets preserve limits. (b) All small colimits exist in C∞ Ringsc , C∞ Ringscin , though in general they are not preserved by Πsm , Πex and Πsm , Πin . ∞

∞

Rings C Rings (c) The functors ΠC C∞ Ringsc , ΠC∞ Ringscin in (4.9) have left and right adjoints, so they preserve limits and colimits. C∞ Ringsc

∞ (d) The left adjoint of ΠC∞ Ringsin Ringscin ,→ C∞ Ringsc , so c is inc : C inc preserves colimits. However, inc does not preserve limits, and has no left adjoint.

Proof For (a), we claim that C∞ Ringsc , C∞ Ringscin are closed under limits in PC∞ Ringsc , PC∞ Ringscin . To see this, note that Πsm , Πex : PC∞ Ringsc → Sets preserve limits by Theorem 4.12(a), and the conditions Definition 4.16(i)–(iii) are closed under limits of sets C = Πsm (C) and C ex = Πex (C). Thus part (a) follows from Theorem 4.12(a). For (b), let J be a small category and F : J → C∞ Ringsc a functor. Then a colimit C = lim F exists in PC∞ Ringsc by Theorem 4.12(b). As −→ ∞ Ringsc C∞ Ringsc ΠC PC∞ Ringsc is a left adjoint it preserves colimits, so ΠPC∞ Ringsc (C) = ∞

∞

c

c

Rings Rings lim ΠC ◦ F in C∞ Ringsc . But ΠC PC∞ Ringsc is the identity on −→ PC∞ Ringsc ∞ c Rings C∞ Ringsc ∼ C∞ Ringsc ⊂ PC∞ Ringsc , so ΠC PC∞ Ringsc◦F = F , and ΠPC∞ Ringsc (C) = lim F in C∞ Ringsc . The same argument works for C∞ Ringscin . −→ ∞ Rings For (c), ΠC C∞ Ringsc has a left adjoint by Theorem 4.12(c) and Proposition ∞

∞

Rings C Rings 4.19, as ΠC C∞ Ringsc = ΠPC∞ Ringsc ◦ inc. It has a right adjoint as F>0 in the ∞ proof of Proposition 4.14 maps to C Ringsc ⊂ PC∞ Ringsc . The same C∞ Ringscin C∞ Ringscin works for ΠC∞ Ringsc . For (d), ΠC∞ Ringsc has left adjoint inc by Definition in 4.16, and inc does not preserve limits by the analogue of Example 4.13.

Remark 4.21. Although inc : C∞ Ringscin ,→ C∞ Ringsc has no left adjoint, as in Theorem 4.20(d), we will show in §6.2 that inc : C∞ Schcin ,→ C∞ Schc does have a right adjoint, which is the analogous statement for spaces.

4.5 Generators, relations, and localization We generalize Definition 2.16:

(Pre) C ∞ -rings with corners

86

Definition 4.22. If A, Aex are sets then by [1, Rem. 14.12] we can define the ex free C ∞ -ring with corners FA,Aex = (FA,Aex , FA,A ex
) generated by  (A, Aex ). A,Aex A ∞ Aex , where R A = (xa )a∈A : We may think of F as C R × [0, ∞) xa ∈ R and [0, ∞)Aex = (ya0 )a0 ∈Aex : ya0 ∈ [0, ∞) . Explicitly, we define FA,Aex to be the set of maps c : RA × [0, ∞)Aex → R which depend ex smoothly on only finitely many variables xa , ya0 , and FA,A to be the set of ex A 0 Aex maps c : R × [0, ∞) → [0, ∞) which depend smoothly on only finitely many variables xa , ya0 , and operations Φf , Ψg are defined as in (4.10)–(4.11). Regarding xa : RA → R and ya0 : [0, ∞)Aex → [0, ∞) as functions for ex a ∈ A, a0 ∈ Aex , we have xa ∈ FA,Aex and ya0 ∈ FA,A , and we call xa , ya0 ex A,Aex the generators of F . Then FA,Aex has the property that if C = (C, C ex ) is any (pre) C ∞ -ring with corners then a choice of maps α : A → C and αex : Aex → C ex uniquely determine a morphism φ : FA,Aex → C with φ(xa ) = α(a) for a ∈ A and φex (ya0 ) = αex (a0 ) for a0 ∈ Aex . We write FA,Aex = Fm,n when A = {1, . . . , m} and Aex = {1, . . . , n}, and then Fm,n ∼ = C ∞ (Rm × [0, ∞)n ) = m+n ∞ C (Rn ). The analogue of all this also holds in C∞ Ringscin , with the same objects m,n F and FA,Ain , which are interior C ∞ -rings with corners, and the difference that (interior) morphisms Fm,n → C in C∞ Ringscin or PC∞ Ringscin are uniquely determined by elements c1 , . . . , cm ∈ C and c01 , . . . , c0n ∈ C in (rather than c01 , . . . , c0n ∈ C ex ), and similarly for FA,Ain with αin : Ain → C in . As for Proposition 2.17, every object in C∞ Ringsc , C∞ Ringscin can be built out of free C ∞ -rings with corners, in a certain sense. Proposition 4.23. (a) Every object C in C∞ Ringsc admits a surjective morphism φ : FA,Aex → C from some free C ∞ -ring with corners FA,Aex . (b) Every object C in C∞ Ringsc fits into a coequalizer diagram FB,Bex

α

-

1 FA,Aex

φ

/ C,

(4.13)

β

that is, C is the colimit of the diagram FB,Bex ⇒ FA,Aex in C∞ Ringsc , where φ : FA,Aex → C is automatically surjective. The analogues of (a) and (b) also hold in C∞ Ringscin . Proof The analogue in PC∞ Ringsc , PC∞ Ringscin holds by Ad´amek et al. [1, Prop.s 11.26, 11.28, 11.30, Cor. 11.33, & Rem. 14.14], by general facts about algebraic theories. The result for C∞ Ringsc , C∞ Ringscin follows by ∞ C∞ Ringscin Ringsc applying the reflection functors ΠC PC∞ Ringsc , ΠPC∞ Ringscin of Proposition 4.19.

4.5 Generators, relations, and localization

87

Definition 4.24. Let C be a C ∞ -ring with corners, and A, Aex be sets. We will write C(xa : a ∈ A)[ya0 : a0 ∈ Aex ] for the C ∞ -ring with corners obtained by adding extra generators xa for a ∈ A of type R and ya0 for a0 ∈ Aex of type [0, ∞) to C. That is, by definition C(xa : a ∈ A)[ya0 : a0 ∈ Aex ] := C ⊗∞ FA,Aex ,

(4.14)

where FA,Aex is the free C ∞ -ring with corners from Definition 4.22, with generators xa for a ∈ A of type R and ya0 for a0 ∈ Aex of type [0, ∞), and ⊗∞ is the coproduct in C∞ Ringsc . As coproducts are a type of colimit, Theorem 4.20(b) implies that C(xa : a ∈ A)[ya0 : a0 ∈ Aex ] is well defined. Since FA,Aex is interior, if C is interior then (4.14) is a coproduct in both C∞ Ringsc and C∞ Ringscin , so C(xa : a ∈ A)[ya0 : a0 ∈ Aex ] is interior by Theorem 4.20(d). By properties of coproducts and free C ∞ -rings with corners, morphisms φ : C(xa : a ∈ A)[ya0 : a0 ∈ Aex ] → D in C∞ Ringsc are uniquely determined by a morphism ψ : C → D and maps α : A → D, αex : Aex → Dex . If C, D, ψ are interior and αex (Aex ) ⊆ Din then φ is interior. Next suppose B, Bex are sets and fb ∈ C for b ∈ B, gb0 , hb0 ∈ C ex for b0 ∈ Bex . We will write C/(fb = 0 : b ∈ B)[gb0 = hb0 : b0 ∈ Bex ] for the C ∞ -ring with corners obtained by imposing relations fb = 0, b ∈ B in C of type R, and gb0 = hb0 , b0 ∈ Bex in C ex of type [0, ∞). That is, we have a coequalizer diagram FB,Bex

α

2, C

π

/ C/(fb = 0 : b ∈ B)[gb0 = hb0 : b0 ∈ Bex ],

(4.15)

β

where α, β are determined uniquely by α(xb ) = fb , αex (yb0 ) = gb0 , β(xb ) = 0, βex (yb0 ) = hb0 for all b ∈ B and b0 ∈ Bex . As coequalizers are a type of colimit, Theorem 4.20(b) shows that C/(fb = 0 : b ∈ B)[gb0 = hb0 : b0 ∈ Bex ] is well defined. If C is interior and gb0 , hb0 ∈ C in for all b0 ∈ Bex (that is, gb0 , hb0 6= 0C ex ) then (4.14) is also a coequalizer in C∞ Ringscin , so Theorem 4.20(d) implies that C/(fb = 0 : b ∈ B)[gb0 = hb0 : b0 ∈ Bex ] and π are interior. Note that round brackets (· · · ) denote generators or relations of type R, and square brackets [· · · ] generators or relations of type [0, ∞). If we add generators or relations of only one type, we use only these brackets. We construct two explicit examples of quotients in C∞ Ringsc : Example 4.25. (a) Say we wish to quotient a C ∞ -ring with corners (C, C ex ) by an ideal I in C. Quotienting the C ∞ -ring by the ideal will result in additional

88

(Pre) C ∞ -rings with corners

relations on the monoid. While this quotient (D, Dex ) is the coequalizer of a diagram such as (4.15), it is also equivalent to the following construction: The quotient is a C ∞ -ring with corners (D, Dex ) with a morphism π = (π, πex ) : (C, C ex ) → (D, Dex ) such that I is contained in the kernel of π, and is universal with respect to this property. That is, if (E, Eex ) is another C ∞ 0 ring with corners with morphism π 0 = (π 0 , πex ) : (C, C ex ) → (E, Eex ) with I 0 contained in the kernel of π , then there is a unique morphism p : (D, Dex ) → (E, Eex ) such that p ◦ π = π 0 . As a coequalizer is a colimit, by Theorem 4.20(c) we have D ∼ = C/I, the quotient in C ∞ -rings. For the monoid, we require that smooth f : R → [0, ∞) give well defined operations Ψf : D → Dex . This means we require that if a − b ∈ I, then Ψf (a) ∼ Ψf (b) ∈ C ex , and this needs to generate a monoid equivalence relation on C ex , so that a quotient by this relation is well defined. If f : R → [0, ∞) is identically zero, this follows. If f : R → [0, ∞) is non-zero and smooth this means that f is positive, and hence that log f is well defined with f = exp ◦ log f . By Hadamard’s Lemma, if a − b ∈ I, then Φg (a) − Φg (b) ∈ I, for all g : R → R smooth, and therefore Φlog f (a) − Φlog f (b) ∈ I. Hence in C ex we only require that if a − b ∈ I, then Ψexp (a) ∼ Ψexp (b) in C ex . The monoid equivalence relation that this generates is equivalent to c01 ∼I c02 ∈ C ex if there exists d ∈ I such that c01 = Ψexp (d) · c02 . We claim that (C/I, C ex /∼I ) is the required C ∞ -ring with corners. If f : [0, ∞) → R is smooth, and c01 ∼I c02 ∈ C ex , then Φf (c01 ) = Φf (Ψexp (d)c02 ) in C for some d ∈ I. Applying Hadamard’s Lemma twice, we have Φf (c01 ) − Φf (c02 ) = Φ(x−y)g(x,y) (Ψexp (d)c02 , c02 ) = Φi (c02 )(Φexp (d) − 1)Φg (Ψexp (d)c02 , c02 ) = Φi (c02 )(d − 0)Φh(x,y) (d, 0)Φg (Ψexp (d)c02 , c02 ), for smooth maps g, h : R2 → R. As d ∈ I, then Φf (c01 ) − Φf (c02 ) ∈ I, and Φf is well defined. A similar proof shows all the C ∞ -operations are well defined, and so (C/I, C ex /∼I ) is a pre C ∞ -ring with corners. We must show that Ψexp : C/I → C ex /∼I has image equal to (not just contained in) the invertible elements (C ex /∼I )× . Say [c01 ] ∈ (C ex /∼I )× , then there is [c02 ] ∈ (C ex /∼I )× such that [c01 ][c02 ] = [c0 d0 ] = [1]. So there is d ∈ I such that c01 c02 = Ψexp (d). However, Ψexp (d) is invertible, so each of c01 , c02 must be invertible in C ex , and using that Ψexp is surjective onto invertible ele∞ ments in C × ex gives the result. Thus (C/I, C ex /∼I ) is a C -ring with corners. The quotient morphisms C → C/I and C ex → C ex /∼I give the required map π. To show that this satisfies the required universal property of either (4.15) or the above, let (E, Eex ) be another C ∞ -ring with corners with a morphism

4.5 Generators, relations, and localization

89

0 π 0 = (π 0 , πex ) : (C, C ex ) → (E, Eex ) such that I ⊂ Ker π 0 . Then the unique morphism p : (D, Dex ) → (E, Eex ) is defined by p([c], [c0 ]) = [π 0 (c), π 0 (c0 )]. The requirement that (E, Eex ) factors through each diagram shows that this morphism is well defined and unique, giving the result. (b) Say we wish to quotient a C ∞ -ring with corners (C, C ex ) by an ideal P in the monoid C ex . By this we mean quotient C ex by the equivalence relation c01 ∼ c02 if c01 = c02 or c01 , c02 ∈ P . This is known as a Rees quotient of semigroups, see Rees [85, p. 389]. Quotienting the monoid by this ideal will result in additional relations on both the monoid and the C ∞ -ring, which we will now make explicit. While this quotient (D, Dex ) is the coequalizer of a diagram such as (4.15), it is also equivalent to the following construction: The quotient is a C ∞ -ring with corners (D, Dex ) with a morphism π = (π, πex ) : (C, C ex ) → (D, Dex ) such that P is contained in the kernel of πex , and is universal with respect to this property. That is, if (E, Eex ) is another 0 C ∞ -ring with corners with morphism π 0 = (π 0 , πex ) : (C, C ex ) → (E, Eex ) 0 with P contained in the kernel of πex , then there is a unique morphism p : (D, Dex ) → (E, Eex ) such that p ◦ π = π 0 . Similarly to part (a), we begin by quotienting C ex by P , and then require that the C ∞ -operations are well defined. As all smooth f : [0, ∞) → R are equal to fˆ ◦ i : [0, ∞) → R for a smooth function fˆ : R → R, we need only require that if c01 ∼ c02 , then Φi (c01 ) ∼ Φi (c02 ). This generates a C ∞ -ring equivalence relation on the C ∞ -ring C; such a C ∞ -ring equivalence relation is the same data as giving an ideal I ⊂ C such that c1 ∼ c2 ∈ C whenever c1 − c2 ∈ I. Here, this equivalence relation will be given by the ideal hΦi (P )i, that is, the ideal generated by the image of P under Φi . Quotienting C by this ideal generates a further condition on the monoid C ex , as in part (a), that is c01 ∼ c02 if there is d ∈ hΦi (P )i such that c01 = Ψexp (d)c02 .
The claim then is that we may take (D, Dex ) equal to C/hΦi (P )i, C ex /∼P where c01 ∼P c02 if either c01 , c02 ∈ P or there is d ∈ hΦi (P )i such that c01 = Ψexp (d)(c02 ). Similar applications of Hadamard’s Lemma as in (a) show that (D, Dex ) is a pre C ∞ -ring with corners, and similar discussions show it is a C ∞ -ring with corners, and is isomorphic to the quotient. We will use the notation

(D, Dex ) = (C, C ex )/∼P = (C/hΦi (P )i, C ex /∼P ) = (C/∼P , C ex /∼P ) to refer to this quotient later. Definition 4.26. Let C = (C, C ex ) be a C ∞ -ring with corners, and A ⊆ C, Aex ⊆ C ex be subsets. A localization C(a−1 : a ∈ A)[a0−1 : a0 ∈ Aex ] of C at (A, Aex ) is a C ∞ -ring with corners D = C(a−1 : a ∈ A)[a0−1 : a0 ∈ Aex ]

90

(Pre) C ∞ -rings with corners

and a morphism π : C → D such that π(a) is invertible in D for all a ∈ A and πex (a0 ) is invertible in Dex for all a0 ∈ Aex , with the universal property that if E = (E, Eex ) is a C ∞ -ring with corners and φ : C → E a morphism with φ(a) invertible in E for all a ∈ A and φex (a0 ) invertible in Eex for all a0 ∈ Aex , then there is a unique morphism ψ : D → E with φ = ψ ◦ π. Localizations C(a−1 : a ∈ A)[a0−1 : a0 ∈ Aex ] always exist, and are unique up to canonical isomorphism. In the notation of Definition 4.24 we may write C(a−1 : a ∈ A)[a0−1 : a0 ∈ Aex ] = (4.16)

 0  0 0 C(xa : a ∈ A)[ya0 : a ∈ Aex ] a·xa = 1 : a ∈ A a ·ya0 = 1 : a ∈ Aex . That is, we add an extra generator xa of type R and an extra relation a · xa = 1 of type R for each a ∈ A, so that xa = a−1 , and similarly for each a0 ∈ Aex . If C is interior and Aex ⊆ C in then C(a−1 : a ∈ A)[a0−1 : a0 ∈ Aex ] makes sense and exists in C∞ Ringscin as well as in C∞ Ringsc , and Theorem 4.20(d) implies that the two localizations are the same. The next lemma follows from Theorem 4.20(c). Lemma 4.27. Let C = (C, C ex ) be a C ∞ -ring with corners and c ∈ C, and write C(c−1 ) = (D, Dex ). Then D ∼ = C(c−1 ), the localization of the C ∞ -ring.

4.6 Local C ∞ -rings with corners We now define local C ∞ -rings with corners. Definition 4.28. Let C = (C, C ex ) be a C ∞ -ring with corners. We call C local if there exists a C ∞ -ring morphism π : C → R such that each c ∈ C is invertible in C if and only if π(c) 6= 0 in R, and each c0 ∈ C ex is invertible in C ex if and only if π ◦ Φi (c0 ) 6= 0 in R. Note that if C is local then π : C → R is determined uniquely by Ker π = {c ∈ C : c is not invertible}, and C is a local C ∞ -ring. Write C∞ Ringsclo ⊂ C∞ Ringsc and C∞ Ringscin,lo ⊂ C∞ Ringscin for the full subcategories of (interior) local C ∞ -rings with corners. Proposition 4.29. C∞ Ringsclo is closed under colimits in C∞ Ringsc . Thus all small colimits exist in C∞ Ringsclo and C∞ Ringscin,lo . Proof Let J be a category and F = (F, Fex ) : J → C∞ Ringsclo a functor, and suppose a colimit C = (C, C ex ) = lim F exists in C∞ Ringsc , with −→ projections φj : F (j) → C. Then C = lim F in C∞ Rings by Theorem −→ 4.20(c). For each j ∈ J we have a unique C ∞ -ring morphism πj : F (j) → R

4.6 Local C ∞ -rings with corners

91

as F (j) is local, with πj = πk ◦ F (α) for all morphisms α : j → k in J. Hence by properties of colimits there is a unique morphism π : C → R with πj = π ◦ φj for all j ∈ J. Suppose c ∈ C with π(c) 6= 0. Then c = φj (f ) for some j ∈ J and f ∈ F (j), with πj (f ) = π ◦ φj (f ) = π(c) 6= 0. Hence as F (j) is local, we have an inverse f −1 ∈ F (j), and φj (f −1 ) = c−1 is the inverse of c in C. The same argument shows that if c0 ∈ C ex with π ◦Φi (c0 ) 6= 0 then c0 is invertible in C ex . So C is local, and C∞ Ringsclo is closed under colimits in C∞ Ringsc . The last part holds by Theorem 4.20(b),(d). Lemma 4.30. If C = (C, C ex ) is a (pre) C ∞ -ring with corners and x : C → R a C ∞ -ring morphism, then we have a morphism of (pre) C ∞ -rings with corners (x, xex ) : (C, C ex ) → (R, [0, ∞)), where xex (c0 ) = x◦Φi (c0 ) for c0 ∈ C ex . Proof Let C = (C, C ex ) and x : C → R be as in the statement. Take c0 ∈ C ex . To show xex = x ◦ Φi : C ex → [0, ∞) is well defined, assume for a contradiction that x ◦ Φi (c0 ) =  < 0 ∈ R. Let f : R → R be a smooth function such that f is the identity on [0, ∞) and it is zero on (−∞, /2). Then f ◦ i = i for i : [0, ∞) → R the inclusion. So we have 0 >  = x ◦ Φi (c0 ) = x ◦ Φf ◦ Φi (c0 ) = f (x ◦ Φi (c0 )) = f () = 0, and xex = x ◦ Φi : C ex → [0, ∞) is well defined. For (x, xex ) to be a morphism of pre C ∞ -rings with corners, it must respect the C ∞ -operations. For example, let f : [0, ∞) → [0, ∞) be smooth, then there is a smooth function g : R → R that extends f , so that g ◦ i = i ◦ f . Then xex (Ψf (c0 )) = x ◦ Φi (Ψf (c0 )) = Φg (x ◦ Φi (c0 )) = Φg (xex (c0 )) as required. A similar proof holds for the other C ∞ -operations. Definition 4.31. Let C = (C, C ex ) be a C ∞ -ring with corners. An R-point x of C is a C ∞ -ring morphism (or equivalently, an R-algebra morphism) x : C → R. Define C x to be the localization
  C x = C c−1 : c ∈ C, x(c) 6= 0 c0−1 : c0 ∈ C ex , x ◦ Φi (c0 ) 6= 0 , (4.17) with projection π x : C → C x . If C is interior then C x is interior by Definition 4.26. Theorem 4.32(a) below shows C x is local. Theorem 4.32(c) is the analogue of Proposition 2.24. The point of the proof is to give an alternative construction of C x from C by imposing relations, but adding no new generators. Theorem 4.32. Let π x : C → C x be as in Definition 4.31. Then: (a) C x is a local C ∞ -ring with corners.

(Pre) C ∞ -rings with corners

92

(b) C x = (C x , C x,ex ) and π x = (πx , πx,ex ), where πx : C → C x is the local C ∞ -ring associated to x : C → R in Definition 2.22. (c) πx : C → C x and πx,ex : C ex → C x,ex are surjective. Proof Proposition 2.24 says that the local C ∞ -ring C x is C/I, for I ⊂ C the ideal defined in (2.4), with πx : C → C x the projection C → C/I. Define D = C x = C/I

and Dex = C ex /∼,

where ∼ is the equivalence relation on C ex given by c0 ∼ c00 if there exists i ∈ I with c00 = Ψexp (i) · c0 , and φ : C → D = C/I, φex : C ex → Dex = C ex /∼ to be the natural surjective projections. Let f : Rm ×[0, ∞)n → R be smooth and g : Rm × [0, ∞)n → [0, ∞) be exterior, and write Φf , Ψg for the operations in C. Then it is not difficult to show there exist unique maps Φ0f , Ψ0g making the following diagrams commute: C m × C nex φ

m

×φn ex

 Dm × Dnex

Φf Φ0f

/C φ

 / D,

C m × C nex m

φ

×φn ex

 Dm × Dnex

Ψg Ψ0g

/ C ex φex

 / Dex ,

(4.18)

and these Φ0f , Ψ0g make D = (D, Dex ) into a C ∞ -ring with corners, and φ = (φ, φex ) : C → D into a surjective morphism. Suppose that F is a C ∞ -ring with corners and χ = (χ, χex ) : C → F a morphism such that χ(c) is invertible in F for all c ∈ C with x(c) 6= 0. The definition D = C x = C(c−1 : c ∈ C, x(c) 6= 0) in C∞ Rings in Definition 2.22 implies that χ : C → F factorizes uniquely as χ = ξ ◦ φ for ξ : D → F a morphism in C∞ Rings. Hence χ(i) = 0 in F for all i ∈ I, so χex (Ψexp (i)) = 1Fex for all i ∈ I. Thus if c0 , c00 ∈ C ex with c00 = Ψexp (i) · c0 for i ∈ I then χex (c0 ) = χex (c00 ). Hence χex factorizes uniquely as χex = ξex ◦ φex for ξex : Dex → Fex . As χ, φ are morphisms in C∞ Ringsc with φ surjective we see that ξ = (ξ, ξex ) : D → F is a morphism. Therefore χ : C → F factorizes uniquely as χ = ξ ◦ φ. Also φ(c) is invertible in D = C x for all c ∈ C with x(c) 6= 0, by definition of C x in Definition 2.22. Therefore we have a canonical isomorphism
D∼ = C c−1 : c ∈ C, x(c) 6= 0
identifying φ : C → D with the projection C → C c−1 : c ∈ C, x(c) 6= 0 . Note that x : C → R factorizes as x = x ˜ ◦φ for a unique morphism x ˜ : D → R. Next define E = D = Cx

and Eex = Dex / ≈,

4.6 Local C ∞ -rings with corners

93

where ≈ is the monoidal equivalence relation on Dex generated by the conditions that d0 ≈ d00 whenever d0 , d00 ∈ Dex with Φ0i (d0 ) = Φ0i (d00 ) in D and x ˜ ◦ Φ0i (d0 ) 6= 0. Write ψ = id : D → E, and let ψex : Dex → Eex be the natural surjective projection. Suppose f : Rm × [0, ∞)n → R is smooth and g : Rm × [0, ∞)n → [0, ∞) is exterior. Then as for (4.18), we claim there are unique maps Φ00f , Ψ00g making the following diagrams commute: Dm × Dnex ψ

m

n ×ψex

 Em × Enex

Φ0f Φ00 f

/D ψ

 / E,

Dm × Dnex ψ

m

n ×ψex

 Em × Enex

Ψ0g Ψ00 g

/ Dex ψex

 / Eex .

(4.19)

To see that Φ00f in (4.19) is well defined, note that as Φ0i : Dex → D is a monoid morphism and ≈ is a monoidal equivalence relation generated by ˜ 0 ◦ ψex . We d0 ≈ d00 when Φ0i (d0 ) = Φ0i (d00 ), we have a factorization Φ0i = Φ i m+n 0 ˜ may extend f to smooth f : R → R, and then Φf factorizes as Dm × Dnex 0 n idm D ×(Φi )

+

4/ D.

Φ0f

Dm+n

Φ0f˜

˜ 0 ◦ ψex , we see that Φ00 in (4.19) exists and is unique. Using Φ0i = Φ i f 00 For Ψg , if g = 0 then Ψ00g = 0Eex = [0Dex ] in (4.19). Otherwise we may write g using a1 , . . . , an and h : Rm × [0, ∞)n → R as in (4.5), and then Ψg0 (d1 , . . . , dm , d01 , . . . , d0n )   = (d01 )a1 · · · (d0n )an · Ψ0exp Φ0h (d1 , . . . , dm , d01 , . . . , d0n ) . Since ψex : Dex → Eex is a monoid morphism, as it is a quotient by a monoidal equivalence relation, we see from this and the previous argument applied to Φh (d1 , . . . , dm , d01 , . . . , d0n ) that Ψ00g in (4.19) exists and is unique. These Φ00f , Ψ00g make E = (E, Eex ) into a C ∞ -ring with corners, and ψ = (ψ, ψex ) : D → E into a surjective morphism. We will show that there is a canonical isomorphism E ∼ = C x which identifies ψ ◦ φ : C → E with π x : C → C x . Firstly, suppose c ∈ C with x(c) 6= 0. Then ψ ◦ φ(c) is invertible in E = C x by definition of C x in Definition 2.22. Secondly, suppose c0 ∈ C ex with x ◦ Φi (c0 ) 6= 0. Set d0 = φex (c0 ). Then Φ0i (d0 ) = φ ◦ Φi (c0 ) is invertible in D. Now in the proof of Proposition 4.7, we do not actually need c0 to be invertible in C ex , it is enough that Φi (c0 ) is invertible in C. Thus this proof shows that there exists a unique d ∈ D with Φ0i (d0 ) = Φ0exp (d) = Φ0i ◦ Ψ0exp (d). But then d0 ≈ Ψ0exp (d), so ψex (d0 ) = ψex ◦ Ψ0exp (d) = Ψ00exp (ψ(d)). Hence ψex ◦ φex (c0 ) = ψex (d0 ) is invertible in Eex , with inverse Ψ00exp (−ψ(d)).

94

(Pre) C ∞ -rings with corners

Thirdly, suppose that G is a C ∞ -ring with corners and ζ = (ζ, ζex ) : C → G a morphism such that ζ(c) is invertible in G for all c ∈ C with x(c) 6= 0 and ζex (c0 ) is invertible in Gex for all c0 ∈ C ex with x ◦ Φi (c0 ) 6= 0. Then ζ = η ◦ φ for a unique η : D → G, by the universal property of D. Since φex : C ex → Dex is surjective and x = x ˜ ◦ φ we see that ηex (d0 ) is invertible 0 0 0 in Gex for all d ∈ Dex with x ˜ ◦ Φi (d ) 6= 0. ˜ ◦ Φ0i (d0 ) 6= 0, so that Let d0 , d00 ∈ Dex with Φ0i (d0 ) = Φ0i (d00 ) in D and x d0 ≈ d00 . Then ηex (d0 ), ηex (d00 ) are invertible in Gex with Φi ◦ ηex (d0 ) = Φi ◦ ηex (d00 ) in G, so Definition 4.16(i) for G implies that ηex (d0 ) = ηex (d00 ). Since ηex : Dex → Gex is a monoid morphism, and ≈ is a monoidal equivalence relation, and ηex (d0 ) = ηex (d00 ) for the generating relations d0 ≈ d00 , we see that ηex factorizes via Dex / ≈. Thus there exists unique θex : Eex → Fex with ηex = θex ◦ ψex . Set θ = η : E = D → G. Then η = θ ◦ ψ as ψ = idD . As ψ is surjective we see that θ = (θ, θex ) : E → F is a morphism in C∞ Ringsc , with η = θ ◦ ψ, so that ζ = θ ◦ ψ ◦ φ. This proves that ψ ◦ φ : C → E satisfies the universal property of π x : C → C x from the localization (4.17), so E ∼ = C x as we claimed. Parts (b),(c) of the theorem are now immediate, as E = C x and φ, ψ are surjective. For (a), observe that x : C → R factorizes as π ◦ πx for π : C x → R a morphism. If c¯ ∈ C x with π(¯ c) 6= 0 then as πx : C → C x is surjective by (c) we have c¯ = πx (c) with x(c) 6= 0, so c¯ = πx (c) is invertible in C x by (4.17). Similarly, if c¯0 ∈ C x,ex with π ◦ Φi (¯ c0 ) 6= 0 then as πx,ex : C ex → C x,ex is surjective we 0 find that c¯ is invertible in C x,ex . Hence C x is a local C ∞ -ring with corners. We can characterize the equivalence relations that define C x,ex = Eex in the above proof using the following proposition. Proposition 4.33. Let C = (C, C ex ) be a C ∞ -ring with corners and x : C → R an R-point of C. Let π x : C → C x be as in Definition 4.31, and I the ideal defined in (2.4). For any c01 , c02 ∈ C ex , then πx,ex (c01 ) = πx,ex (c02 ) if and only if there are elements a0 , b0 ∈ C ex such that Φi (a0 ) − Φi (b0 ) ∈ I, x ◦ Φi (a0 ) 6= 0, and a0 c01 = b0 c02 . Hence C x = (C/I, C ex /∼) where c01 ∼ c02 ∈ C ex if and only if there are elements a0 , b0 ∈ C ex such that Φi (a0 )−Φi (b0 ) ∈ I, x◦Φi (a0 ) 6= 0, and a0 c01 = b0 c02 . Proof We first show that if πx,ex (c01 ) = πx,ex (c02 ), then there are a0 , b0 satisfying the conditions. In Theorem 4.32, we constructed C ∞ -rings with corners D = (D, Dex ) and E = (E, Eex ) and surjective morphisms φ = (φ, φex ) : C → D, ψ = (ψ, ψex ) : D → E, where E = D = C x = C/I,

Dex = C ex / ∼

and Eex = Dex / ≈,

4.6 Local C ∞ -rings with corners

95

and I ⊂ C is the ideal defined in (2.4), and ∼, ≈ are explicit equivalence relations. Then we showed that there is a unique isomorphism E ∼ = C x identifying ψ ◦ φ : C → E with π x : C → C x . As πx,ex (c01 ) = πx,ex (c02 ) we have ψex ◦ φex (c01 ) = ψex ◦ φex (c02 ) in Eex . Thus φex (c01 ) ≈ φex (c02 ). By definition ≈ is the monoidal equivalence relation on Dex generated by the condition that d0 ≈ d00 whenever d0 , d00 ∈ Dex with Φ0i (d0 ) = Φ0i (d00 ) in D and x ˜ ◦ Φ0i (d0 ) 6= 0, where Φ0i : Dex → D is the C ∞ -operation from the inclusion i : [0, ∞) ,→ R, and x : C → R factorizes as x = x ˜ ◦ φ for a unique 0 0 morphism x ˜ : D → R. Hence φex (c1 ) ≈ φex (c2 ) means that there is a finite sequence φex (c01 ) = d00 , d01 , d02 , . . . , d0n−1 , d0n = φex (c02 ) in Dex , and elements e0i , fi0 , gi0 ∈ Dex such that Φ0i (e0i ) = Φ0i (fi0 ), x ˜ ◦ Φ0i (e0i ) 6= 0, and d0i−1 = e0i gi0 , 0 0 0 di = fi gi in Dex for i = 1, . . . , n. As φex : C ex → Dex is surjective we can choose ei , fi , gi ∈ C ex with e0i = φex (ei ), fi0 = φex (fi ), gi0 = φex (gi ) for i = 1, . . . , n. Then the conditions become Φi (ei ) − Φi (fi ) ∈ I, c01

∼ e1 g1 ,

x ◦ Φi (ei ) 6= 0 ∈ R,

ei+1 gi+1 ∼ fi gi ,

i = 1, . . . , n,

i = 1, . . . , n − 1,

c02 ∼ fn gn ,

since equality in Dex lifts to ∼-equivalence in C ex . By definition of ∼, this means that there exist elements h0 , h1 , . . . , hn in the ideal I ⊂ C in (2.4) such that c01 = Ψexp (h0 )e1 g1 ,

c02 = Ψexp (hn )fn gn ,

ei+1 gi+1 = Ψexp (hi )fi gi ,

and

i = 1, . . . , n − 1.

(4.20)

It is not hard to show that for any element h ∈ I, then the conditions Φi (ei ) − Φi (fi ) ∈ I and x ◦ Φi (ei ) 6= 0 ∈ R hold if and only if Φi (Ψexp (h)ei ) − Φi (fi ) ∈ I and x ◦ Φi (Ψexp (h)ei ) 6= 0 ∈ R hold. So we can remove the hi in (4.20). We have that πx,ex (c01 ) = πx,ex (c02 ) if and only if there are ei , fi , gi ∈ C ex such that c01 = e1 g1 ,

c02 = fn gn , and

ei+1 gi+1 = fi gi ,

x ◦ Φi (ei ) 6= 0 ∈ R,

i = 1, . . . , n − 1,

i = 1, . . . , n.

(4.21)

We define a0 = f1 f2 · · · fn and b0 = e1 e2 · · · en . Then using (4.21), we see that a0 c01 = b0 c02 , Φi (a0 ) − Φi (b0 ) ∈ I and x ◦ Φi (a0 ) 6= 0 as required. For the reverse argument, say we have a0 , b0 ∈ C ex with Φi (a0 ) − Φi (b0 ) ∈ I and x ◦ Φi (a0 ) 6= 0. Let n = 1, e1 = a0 , g1 = 0 and f1 = b0 in (4.20) then we see that πx,ex (a0 ) = πx,ex (b0 ). As x ◦ Φi (a0 ) 6= 0, then πx,ex (a0 ) is invertible in C ex . If we also have that a0 c01 = b0 c02 , then as πx,ex is a morphism, πx,ex (c01 ) = πx,ex (c02 ) and the result follows.

96

(Pre) C ∞ -rings with corners

Remark 4.34. Suppose 0 6= c0 ∈ C ex with πx,ex (c0 ) = 0 for some R-point x : C → R. Then Proposition 4.33 gives a0 ∈ C ex with x ◦ Φi (a0 ) 6= 0 and a0 c0 = 0. Thus a0 , c0 are zero divisors in C ex . Conversely, if C is interior then C ex has no zero divisors, so πx,ex (c0 ) 6= 0 for all 0 6= c0 ∈ C ex . Example 4.35. Let X be a manifold with corners, or a manifold with g-corners, and x ∈ X. Define a C ∞ -ring with corners C x = (C, C ex ) such that C is the set of germs at x of smooth functions c : X → R, and C ex is the set of germs at x of exterior functions c0 : X → [0, ∞). That is, elements of C are ∼-equivalence classes [U, c] of pairs (U, c), where U is an open neighbourhood of x in X and c : U → R is smooth, and (U, c) ∼ ˜ , c˜) if there exists an open neighbourhood U ˆ of x in U ∩ U ˜ with c| ˆ = c˜| ˆ . (U U U Similarly, elements of C ex are equivalence classes [U, c0 ], where U is an open neighbourhood of x in X and c0 : U → [0, ∞) is exterior. The C ∞ -operations Φf , Ψg are defined as in (4.10)–(4.11), but for germs. As the set of germs only depends on the local behaviour, the set of germs at x of exterior functions is equal to the set of germs at x of interior functions and the zero function. Hence C x is an interior C ∞ -ring with corners. There is a morphism π : C → R mapping π : [U, c] 7→ c(x). If π([U, c]) 6= 0 then [U, c] is invertible in C, and if π ◦ Φi ([U, c0 ]) = c0 (x) 6= 0 then [U, c0 ] is invertible in C ex . Thus C x is a local C ∞ -ring with corners. Write C ∞ x (X) = Cx. Example 4.18(a) defines a C ∞ -ring with corners C ∞ (X). The localization π x∗ : (C ∞ (X))x∗ of C ∞ (X) at x∗ : C ∞ (X) → R from Definition 4.31 is also a local C ∞ -ring with corners. There is a morphism π x : 0 0 C ∞ (X) → C ∞ x (X) mapping c 7→ [X, c], c 7→ [X, c ]. By the univer∞ sal property of (C (X))x∗ we have π x = λx ◦ π x∗ for a unique morphism λx : (C ∞ (X))x∗ → C ∞ x (X). The proof of [47, Th. 4.41] implies that the C ∞ -ring morphism λx in λx = (λx , λx,ex ) is always an isomorphism. But λx,ex need not be an isomorphism, as the next proposition shows. Proposition 4.36. In Example 4.35, suppose X is a manifold with faces, as in Definition 3.12. Then λx is an isomorphism for all x ∈ X. If X is a manifold with corners, but not a manifold with faces, then λx is not an isomorphism for some x ∈ X. Proof Let X be a manifold with faces, and x ∈ X, giving λx = (λx , λx,ex ). Then λx is an isomorphism as in Example 4.35. We will show λx,ex is both surjective and injective, so that λx is an isomorphism. ∞ For surjectivity, we need to show that for any [U, c0 ] ∈ Cx,ex (X) there

4.6 Local C ∞ -rings with corners

97

∞ exists c00 ∈ Cex (X) with c0 = c00 near x in X, so that c0 = πx,ex (c00 ) = 00 λx,ex (πx∗ ,ex (c )). If [U, c0 ] = 0 we take c0 = 0, so suppose [U, c0 ] 6= 0, and take c00 to be interior. As in Remark 3.15, such c00 has a locally constant map µc00 : ∂X → N such that c00 vanishes to order µc00 along ∂X locally, and c00 > 0 on X ◦ . We choose µc00 such that µc00 = µc0 on any connected component of ∂X which meets x, and µc00 = 0 on other components. Then local choices of interior c00 on X with prescribed vanishing order µc00 on ∂X can be combined using a partition of unity on X, with the local choice c00 = c0 near x, to get c00 as required. ∞ ∞ For injectivity, let c01 , c02 ∈ Cex (X) with πx,ex (c01 ) = πx,ex (c02 ) in Cx,ex (X), 0 0 which means that c1 = c2 near x in X. We must show that πx∗ ,ex (c01 ) = πx∗ ,ex (c02 ) in Cx∞∗ ,ex (X). By Proposition 4.33, this holds if there are a0 , b0 ∈ ∞ Cex (X) such that a0 = b0 near x in X, and a0 (x) 6= 0, and a0 c01 = b0 c02 . ˜ be the connected component of X containing x. If c0 = 0 near x then Let X 1 0 ˜ and a0 = b0 = 0 on X \ X. ˜ c1 |X˜ = c02 |X˜ , and we can take a0 = b0 = 1 on X 0 0 0 ˜ so µc0 , µc0 map ∂ X ˜ → N. Define a , b0 Otherwise c1 , c2 are interior on X, 1 2 ˜ On X, ˜ define locally constant µa0 , µb0 : ∂ X ˜ → N by to be zero on X \ X. µa0 = max(µc02 − µc01 , 0) and µa0 = max(µc01 − µc02 , 0), so that µa0 + µc01 = ˜ and µa0 , µb0 = 0 near Π−1 (x). Let a0 | ˜ : X ˜ → [0, ∞) be µb0 + µc02 on ∂ X, ˜ X X ˜ Then there is a unique arbitrary interior with order of vanishing µa0 on ∂ X. 0 0 0 0 0 ˜ interior b |X˜ : X → [0, ∞) with a c1 = b c2 , defined by b0 = a0 c01 (c02 )−1 on ˜ ◦ , and extended continuously over X. ˜ These a0 , b0 show λx,ex is injective. X If X is a manifold with corners, but not a manifold with faces, then there exist x ∈ X and distinct x01 , x02 ∈ Π−1 X (x) which lie in the same connected 00 ∞ (X) has µc00 (x01 ) = µc00 (x02 ), component ∂i X of ∂X. Then any c ∈ Cex 0 ∞ but there exists [U, c ] ∈ Cx,ex (X) with µc0 (x01 ) 6= µc0 (x02 ), so [U, c0 ] does not lie in the image of πx,ex = λx,ex ◦ πx∗ ,ex , and hence not in the image of λx,ex , as πx∗ ,ex is surjective. Thus λx,ex is not surjective, and λx not an isomorphism.

The proposition justifies the following definition in the g-corners case: Definition 4.37. A manifold with g-corners X is called a manifold with gfaces if λx : (C ∞ (X))x∗ → C ∞ x (X) in Example 4.35 is an isomorphism for all x ∈ X. In §5.4 we will show that a manifold with g-corners defines an affine C ∞ scheme with corners if and only if it is a manifold with g-faces. Example 4.38. For any weakly toric monoid P , one can show using an argument similar to the proof of Proposition 4.36 that XP in Definition 3.5 is a

98

(Pre) C ∞ -rings with corners

manifold with g-faces. Since every manifold with g-corners X can be covered by open neighbourhoods diffeomorphic to XP , it follows that every manifold with g-corners can be covered by open subsets which are manifolds with gfaces. Remark 4.39. Let us compare Definitions 3.13 and 4.37. For a manifold with g-corners X to be a manifold with g-faces it is necessary, but not sufficient, that ΠX |∂i X : ∂i X → X is injective for each connected component ∂i X of ∂X. This is because not every locally constant map µc0 : ∂X → N can be the vanishing multiplicity map of an interior morphism c0 : X → [0, ∞), as g-corner strata of codimension > 3 can impose extra consistency conditions on µc0 . The first author [25, Ex. 5.5.4] gives an example of a manifold with g-corners with ΠX |∂i X injective for all components ∂i X ⊂ ∂X, which is not a manifold with g-faces.

4.7 Special classes of C ∞ -rings with corners We define and study several classes of C ∞ -rings with corners. Throughout, if C = (C, C ex ) is an (interior) C ∞ -ring with corners, then we regard C ex (and C in ) as monoids under multiplication, as in §3.2. Definition 4.40. Let C be a C ∞ -ring with corners, and see Proposition 4.23. We call C finitely generated if there exists a surjective morphism φ : FA,Aex → C from some free C ∞ -ring with corners FA,Aex with A, Aex finite sets. Equivalently, C is finitely generated if it fits into a coequalizer diagram (4.13) with A, Aex finite. We call C finitely presented if it fits into a coequalizer diagram (4.13) with A, Aex , B, Bex finite. Finitely presented implies finitely generated. Write C∞ Ringscfp ⊂ C∞ Ringscfg ⊂ C∞ Ringsc for the full subcategories of finitely presented, and finitely generated, objects in C∞ Ringsc . Proposition 4.41. Suppose we are given a pushout diagram in C∞ Ringsc : C

β

α

 D

/E  / F,

δ γ

4.7 Special classes of C ∞ -rings with corners

99

so F = D qC E, and C, D, E fit into coequalizer diagrams in C∞ Ringsc : -

1 FA,Aex

φ

/ C,

(4.22)

1 FC,Cex

χ

/ D,

(4.23)

-

ψ

/ E.

(4.24)



FB,Bex

ζ η

-

FD,Dex θ ι

FF,Fex

κ

1 FE,Eex

Then F fits into a coequalizer diagram in C∞ Ringsc : FH,Hex

1 FG,Gex

λ µ

/ F,

ω

(4.25)

with

G = C q E, Gex = Cex q Eex , H = A q D q F, Hex = Aex q Dex q Fex . The analogue holds in C∞ Ringscin rather than C∞ Ringsc . Proof As coproducts exist in C∞ Ringsc , by general properties of colimits we may rewrite the pushout F = D qC E as a coequalizer diagram C

ιD ◦α ιE ◦β

// D ⊗ E ∞

(γ,δ)

/ F.

(4.26)

Now in a diagram H ⇒ G, the coequalizer is the quotient of G by relations from each element of H, Hex . Given a surjective morphism I  H, the coequalizers of H ⇒ G and I ⇒ G are the same, as we quotient G by the same set of relations. Hence using (4.22), in (4.26) we can replace C by FA,Aex . Also using (4.23)–(4.24) gives a coequalizer diagram // Coeq FD,Dex ⇒ FC,Cex
⊗ Coeq FF,Fex ⇒ FE,Eex
/ F. FA,Aex ∞ As coproducts commute with coequalizers, this is equivalent to FA,Aex

// Coeq (FD,Dex ⊗ FF,Fex ) ⇒ (FC,Cex ⊗ FE,Eex )
∞ ∞

/ F.

Since FA,Aex is free, the two morphisms FA,Aex → Coeq(· · · ) factor through the surjective morphism FC,Cex ⊗∞ FE,Eex → Coeq(· · · ). Thus we may combine the two coequalizers into one, giving a coequalizer diagram FA,Aex ⊗∞ FD,Dex ⊗∞ FF,Fex

// FC,Cex ⊗∞ FE,Eex

/ F. (4.27)

But coproducts of free C ∞ -rings with corners are free over the disjoint unions of the generating sets. Thus (4.27) is equivalent to (4.25). The same argument works in C∞ Ringscin .

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(Pre) C ∞ -rings with corners

Proposition 4.42. C∞ Ringscfg and C∞ Ringscfp are closed under finite colimits in C∞ Ringsc . Proof C ∞ (∗) = (R, [0, ∞)) is an initial object in C∞ Ringscfg , C∞ Ringscfp . As C∞ Ringscfg , C∞ Ringscfp have an initial object, finite colimits may be written as iterated pushouts, so it is enough that C∞ Ringscfg , C∞ Ringscfp are closed under pushouts. This follows from Proposition 4.41, noting that if A, Aex , C, Cex , E, Eex are finite then G, Gex are finite, and if A, . . . , Fex are finite then G, . . . , Hex are finite. Definition 4.43. We call a C ∞ -ring with corners C = (C, C ex ) firm if the sharpening C ]ex is a finitely generated monoid. We denote by C∞ Ringscfi the full subcategory of C∞ Ringsc consisting of firm C ∞ -rings with corners. If C is firm, then there are c01 , . . . , c0n in C ex whose images under the quotient C ex → C ]ex generate C ]ex . This implies that each element in C ex can be written 0an 1 as Ψexp (c)c0a 1 · · · cn for some c ∈ C and ai > 0. If there exists a surjective A,Aex morphism φ : F → C with Aex finite then C is firm, as [φex (ya0 )] for a0 ∈ Aex generate C ]ex . Hence C∞ Ringscfp ⊂ C∞ Ringscfg ⊂ C∞ Ringscfi . Proposition 4.44. C∞ Ringscfi is closed under finite colimits in C∞ Ringsc . Proof As in the proof of Proposition 4.42, it is enough to show C∞ Ringscfi is closed under pushouts in C∞ Ringsc . Take C, D, E ∈ C∞ Ringscfi with morphisms C → D and C → E, and consider the pushout D qC E, with its morphisms φ : D → D qC E and ψ : E → D qC E. Then every element of (D qC E)ex is of the form Ψf (φ(d1 ), . . . , φ(dm ), ψ(e1 ), . . . , ψ(en ), φex (d01 ), . . . , φex (d0k ), ψex (e01 ), . . . , ψex (e0l )), for smooth f : Rm+n × [0, ∞)k+l → [0, ∞), where di ∈ D, d0i ∈ Dex , ei ∈ E, e0i ∈ Eex , and d0i are generators of the sharpening of Dex , and e0i generate the sharpening of Eex . If f = 0 this gives 0, otherwise we may write a

k+l F (x1 ,...,xm+n ,y1 ,...,yk+l ) f (x1 , . . . , xm+n , y1 , . . . , yk+l ) = y1a1 . . . yk+l e ,

for F : Rm+n × [0, ∞)k+l → R smooth. In (D qC E)]ex , the above element maps to [φex (d01 )]a1 · · · [φex (d0k )]ak [ψex (e01 )]ak+1 · · · [ψex (e0l )]ak+l . Hence (D qC E)]ex is generated by the images of the generators of D]ex and E]ex , and zero, and so (D qC E)]ex is finitely generated. Thus C∞ Ringscfi is closed under pushouts.

4.7 Special classes of C ∞ -rings with corners

101

Definition 4.45. Suppose C = (C, C ex ) is an interior C ∞ -ring with corners, and let C in ⊆ C ex be the submonoid of §4.2, and C ]in its sharpening. Then: (i) We call C integral if C in is an integral monoid. (ii) We call C torsion-free if C in is an integral, torsion-free monoid. (iii) We call C saturated if it is integral, and C in is a saturated monoid. Note that × ∞ ∼ C× in = C as abelian groups since C is a C -ring with corners, so C in is torsion-free. Therefore C saturated implies that C is torsion-free. (iv) We call C toric if it is saturated and firm. This implies that C ]in is a toric monoid. (v) We call C simple if it is toric and C ]in ∼ = Nk for some k ∈ N. We write C∞ Ringscsi ⊂ C∞ Ringscto ⊂ C∞ Ringscsa ⊂ C∞ Ringsctf ⊂ C∞ RingscZ ⊂ C∞ Ringscin for the full subcategories of simple, toric, saturated, torsion-free, and integral objects in C∞ Ringscin . In the first author [25], simple C ∞ -rings with corners were called ‘simplicial’, but we change to simple to avoid confusion with ‘simplicial C ∞ -rings’ as used in Derived Differential Geometry, as in Spivak [91], see §2.9.2. Example 4.46. Let X be a manifold with corners, and C ∞ in (X) be the interior C ∞ -ring with corners from Example 4.18(b). Let S be the set of connected components of ∂X. For each F ∈ S, we choose an interior map cF : X → [0, ∞) which vanishes to order 1 on F , and to order zero on ∂X \ F , such that cF = 1 outside a small neighbourhood UF of ΠX (F ) in X, where we choose {UF : F ∈ S} to be locally finite in X. Then every interior map Q g : X → [0, ∞) may be written uniquely as g = exp(f ) · F ∈S caFF , for f ∈ C ∞ (X) and aF ∈ N, F ∈ S. Hence as monoids we have In(X) ∼ = C ∞ (X) × NS . Therefore C ∞ in (X) is integral, torsion-free, and saturated, and it is simple and toric if and only if ∂X has finitely many connected components. A more complicated proof shows that if X is a manifold with g-corners then C ∞ in (X) is integral, torsion-free, and saturated, and it is toric if ∂X has finitely many connected components. The functor Πin : C∞ Ringscin → Sets mapping C 7→ C in may be en¯ in : C∞ Ringsc → Mon by regarding C in as a monoid, hanced to a functor Π in ¯ in preserves limits and directed coland then Theorem 4.20(a) implies that Π imits. Write Monsa,tf ,Z ⊂ Montf ,Z ⊂ MonZ ⊂ Mon for the full subcategories of saturated, torsion-free integral, and torsion-free integral, and integral monoids. They are closed under limits and directed colimits in Mon. Thus we deduce: Proposition 4.47. C∞ Ringscsa , C∞ Ringsctf and C∞ RingscZ are closed

(Pre) C ∞ -rings with corners

102

under limits and under directed colimits in C∞ Ringscin . Thus, all small limits and directed colimits exist in C∞ Ringscsa , C∞ Ringsctf , C∞ RingscZ . In a similar way to Proposition 4.19 we prove: sa sa Theorem 4.48. There are reflection functors ΠZin , Πtf Z , Πtf , Πin in a diagram Πsa in

p ∞

C

Ringscsa

o

Πsa tf inc

/ C∞ Ringsctf o

Πtf Z inc

/ C∞ RingscZ o

ΠZin inc

/ C∞ Ringscin ,

sa sa such that each of ΠZin , Πtf Z , Πtf , Πin is left adjoint to the corresponding inclusion functor inc.

Proof Let C be an object in C∞ Ringscin . We will construct an object D = ΠZin (C) in C∞ RingscZ and a projection π : C → D, with the property that if φ : C → E is a morphism in C∞ Ringscin with E ∈ C∞ RingscZ then φ = ψ ◦ π for a unique morphism ψ : D → E. Consider the diagram: π=π 0

C = C0 0

α

/ C1 0

φ=φ

π

1

α

/ C2 1

φ1

α

/ ··· 2 2

φ

-.

/D

(4.28)

ψ

 /- E.

Define C 0 = C and φ0 = φ. By induction on n = 0, 1, . . . , if C n , φn are defined, define an object C n+1 ∈ C∞ Ringscin and morphisms αn : C n → C n+1 , φn+1 : C n+1 → E as follows. We have a monoid C nin , which as in §3.2 has an abelian group (C nin )gp with projection π gp : C nin → (C nin )gp , where C nin , C n are integral if π gp is injective. Using the notation of Definition 4.24, we define   (4.29) C n+1 = C n c0 = c00 if c0 , c00 ∈ C nin with π gp (c0 ) = π gp (c00 ) . Write αn : C n → C n+1 for the natural surjective projection. Then C n+1 , αn are both interior, since the relations c0 = c00 in (4.29) are all interior. We have a morphism φn : C n → E with E integral, so by considering the diagram with bottom morphism injective C nin

π gp

φn

 in  Ein

/ (C nin )gp gp (φn in )

π gp

 / (Ein )gp ,

we see that if c0 , c00 ∈ C nin with π gp (c0 ) = π gp (c00 ) then φnin (c0 ) = φnin (c00 ).

4.7 Special classes of C ∞ -rings with corners

103

Thus by the universal property of (4.29), there is a unique morphism φn+1 : C n+1 → E with φn = φn+1 ◦ αn . This completes the inductive step, so we have defined C n , αn , φn for all n = 0, 1, . . . , where C n , αn are independent of E, φ. n ∞ ∞ c Now define D to be the directed colimit D = lim −→n=0 C in C Ringsin , n n+1 n using the morphisms α : C → C . This exists by Theorem 4.20(a), ∞ c and commutes with Πin : C Ringsin → Sets, where we can think of Πin as mapping to monoids. It has a natural projection π : C → D, and also projections π n : C n → D for all n. By the universal property of colimits, there is a unique morphism ψ in C∞ Ringscin making (4.28) commute. The purpose of the quotient (4.29) is to modify C n to make it integral, since if C n were integral then π gp (c0 ) = π gp (c00 ) implies c0 = c00 . It is not obvious that C n+1 in (4.29) is integral, as the quotient modifies (C nin )gp . However, the direct limit D is integral. To see this, suppose d0 , d00 ∈ Din with π gp (d0 ) = π gp (d00 ) m ∞ in (Din )gp . Since Din = − lim m=0 C in in Mon, for m  0 we may write → n gp gp ∞ m 0 m 00 = lim d0 = πin (c ), d00 = πin (c ) for c0 , c00 ∈ C m in . As (Din ) −→n=0 (C in ) gp 0 gp 00 and π (d ) = π (d ), for some n  m we have n−1 n−1 m 0 m 00 π gp ◦ αin ◦ · · · ◦ αin (c ) = π gp ◦ αin ◦ · · · ◦ αin (c ) in (C nin )gp . n n 0 n m 00 But then (4.29) implies that αin ◦ · · · ◦ αin (c ) = αin ◦ · · · ◦ αin (c ) in C n+1 in , 0 00 gp gp so d = d . Therefore π : Din → (Din ) is injective, and D is integral. Set ΠZin (C) = D. If ξ : C → C 0 is a morphism in C∞ Ringscin , by taking E = ΠZin (C 0 ) and φ = π 0 ◦ ξ in (4.28) we see that there is a unique morphism ΠZin (ξ) in C∞ RingscZ making the following commute:

C

π

ΠZin (ξ)

ξ

 C0

/ ΠZ (C) in

π

0

 / ΠZ (C 0 ). in

This defines the functor ΠZin . For any E ∈ C∞ RingscZ , the correspondence between φ and ψ in (4.28) implies that we have a natural bijection

HomC∞ Ringscin C, inc(E) ∼ = HomC∞ RingscZ ΠZin (C), E . This is functorial in C, E, and so ΠZin is left adjoint to inc : C∞ RingscZ ,→ C∞ Ringscin , as we have to prove. sa tf The constructions of Πtf Z , Πtf are very similar. For ΠZ , if C is an object in ∞ c C RingsZ , the analogue of (4.29) is   C n+1 = C n c0 = c00 if c0 , c00 ∈ C nin with π tf (c0 ) = π tf (c00 ) , where π tf : C nin → (C nin )gp /torsion is the natural projection. For Πsa tf , if C is

104

(Pre) C ∞ -rings with corners

an object in C∞ Ringsctf , the analogue of (4.29) is  C n+1 = C n sc0 : c0 ∈ C nin ⊆ (C nin )gp and there exists c00 ∈ (C nin )gp \ C nin   with c0 = nc0 · c00 , n0c = 2, 3, . . . nc0 · sc0 = c0 , all c0 , nc0 , sc0 . sa tf Z Finally we set Πsa in = Πtf ◦ ΠZ ◦ Πin . This completes the proof.

As for Theorem 4.20(b), we deduce: Corollary 4.49. All small colimits exist in C∞ Ringscsa , C∞ Ringsctf and C∞ RingscZ . In Definition 4.24 we explained how to modify a C ∞ -ring with corners C by adding generators C(xa : a ∈ A)[ya0 : a0 ∈ Aex ] and imposing relations C/(fb = 0 : b ∈ B)[gb0 = hb0 : b0 ∈ Bex ]. This is just notation for certain small colimits in C∞ Ringsc or C∞ Ringscin . Corollary 4.49 implies that we can also add generators and relations in C∞ Ringscsa , C∞ Ringsctf and C∞ RingscZ , provided the relations gb0 = hb0 are interior, i.e. gb0 , hb0 6= 0C ex . Proposition 4.50.C∞ Ringscto is closed under finite colimits in C∞ Ringscsa . Proof Let J be a finite category and F : J → C∞ Ringscto a functor. Write C = lim F for the colimit in C∞ Ringsc . As C∞ Ringscto ⊂ C∞ Ringscfi , −→ Proposition 4.44 says C is firm. Now Πsa F in C∞ Ringscsa , as in (C) = lim −→ sa Πin is a reflection functor. By construction in the proof of Theorem 4.48 we sa see that Πsa in takes firm objects to firm objects, so Πin (C) is firm, and hence toric, as toric means saturated and firm.

5 C ∞ -schemes with corners

We extend the theory of C ∞ -schemes in Chapter 2 to C ∞ -schemes with corners. We start by defining (local) C ∞ -ringed spaces with corners in §5.1, using local C ∞ -rings with corners from §4.6, and discuss some some special classes of local C ∞ -ringed spaces with corners in §5.2. We then construct the spectrum functor Specc : (C∞ Ringsc )op → LC∞ RSc in §5.3, and use it to define (affine) C ∞ -schemes with corners in §5.4. The spectrum functor extends the spectrum functor for C ∞ -rings in a natural way, and is right adjoint to the global sections functor Γc : LC∞ RSc → (C∞ Ringsc )op . In Chapter 2 we saw that Spec : C∞ Ringsop → LC∞ RS restricts to an equivalence from the subcategory (C∞ Ringsco )op ⊂ C∞ Ringsop of complete C ∞ -rings to the subcategory AC∞ Sch ⊂ LC∞ RS of affine C ∞ schemes. Unfortunately, no analogous notion of complete C ∞ -rings with corners exists. But in §5.5 we study semi-complete C ∞ -rings with corners, that have some of the good properties of complete C ∞ -rings, and will be useful in later proofs. We introduce many special classes of C ∞ -schemes with corners in §5.6, and then study fibre products of C ∞ -schemes with corners in §5.7, where we show they exist under mild conditions.

5.1 (Local) C ∞ -ringed spaces with corners Definition 5.1. A C ∞ -ringed space with corners X = (X, O X ) is a topological space X with a sheaf O X of C ∞ -rings with corners on X. That is, for ex each open set U ⊆ X, then O X (U ) = (OX (U ), OX (U )) is a C ∞ -ring with corners and O X satisfies the sheaf axioms in §2.4. A morphism f = (f, f ] ) : (X, O X ) → (Y, O Y ) of C ∞ -ringed spaces with ] corners is a continuous map f : X → Y and a morphism f ] = (f ] , fex ) : −1 ∞ −1 f (O Y ) → O X of sheaves of C -rings with corners on X, for f (O Y ) = 105

C ∞ -schemes with corners

106

(f −1 (OY ), f −1 (OYex )) as in Definition 2.38. Note that f ] is adjoint to a morphism f ] : O Y → f∗ (O X ) on Y as in (2.9). A local C ∞ -ringed space with corners X = (X, O X ) is a C ∞ -ringed space ex for which the stalks O X,x = (OX,x , OX,x ) of O X at x are local C ∞ -rings with corners for all x ∈ X. We define morphisms of local C ∞ -ringed spaces with corners (X, O X ), (Y, O Y ) to be morphisms of C ∞ -ringed spaces with corners, without any additional locality condition. Write C∞ RSc for the category of C ∞ -ringed spaces with corners, and write LC∞ RSc for the full subcategory of local C ∞ -ringed spaces with corners. For brevity, we will use the notation that bold upper case letters X, Y , Z, . . . represent C ∞ -ringed spaces with corners (X, O X ), (Y, O Y ), (Z, O Z ), . . . , and bold lower case letters f , g, . . . represent morphisms of C ∞ -ringed spaces with corners (f, f ] ), (g, g ] ), . . . . When we write ‘x ∈ X’ we mean that X = (X, O X ) and x ∈ X. When we write ‘U is open in X’ we mean that U = (U, O U ) and X = (X, O X ) with U ⊆ X an open set and O U = O X |U . ex Let X = (X, OX , OX ) ∈ LC∞ RSc , and let U be open in X. Take ex (U ). Then s and s0 induce functions elements s ∈ OX (U ) and s0 ∈ OX s : U → R, s0 : U → [0, ∞), that at each x ∈ U are the compositions ρX,x

π

ρex X,x

π ex

x ex ex OX (U ) −−−→ OX,x −→ R, and OX (U ) −−−→ OX,x −−x→ [0, ∞).

ex Here, ρX,x , ρex X,x are the restriction morphism to the stalks, and πx , πx are the unique morphisms that exist as OX,x is local for each x ∈ X, as in Definition 4.28 and Lemma 4.30. We denote by s(x) and s0 (x) the values of s : U → R and s0 : U → [0, ∞) respectively at the point x ∈ U . We denote by sx ∈ OX,x ex the values of s and s0 under the restriction morphisms to the and s0x ∈ OX,x stalks ρX,x and ρex X,x respectively.

Definition 5.2. Let X = (X, O X ) be a C ∞ -ringed space with corners. We call X an interior C ∞ -ringed space with corners if one (hence all) of the following conditions hold, which are equivalent by Lemma 5.3: ex (a) For all open U ⊆ X and each s0 ∈ OX (U ), then Us0 = {x ∈ U : s0x 6= 0 ∈ ex OX,x }, which is always closed in U , is open in U , and the stalks O X,x are interior C ∞ -rings with corners. ex ˆs0 = {x ∈ U : (b) For all open U ⊆ X and each s0 ∈ OX (U ), then U \ Us0 = U 0 ex sx = 0 ∈ OX,x }, which is always open in U , is closed in U , and the stalks O X,x are interior C ∞ -rings with corners. ex in in (c) OX is the sheafification of a presheaf of the form OX q {0}, where OX in in is a sheaf of monoids, such that O X (U ) = (OX (U ), OX (U ) q {0}) is an interior C ∞ -ring with corners for each open U ⊆ X.

5.1 (Local) C ∞ -ringed spaces with corners

107

 in in In each case, we can define a sheaf of monoids OX , such that OX (U ) = s0 ∈ ex ex OX (U ) : s0x 6= 0 ∈ OX,x for all x ∈ U . We call X an interior local C ∞ -ringed space with corners if X is both a local C ∞ -ringed space with corners and an interior C ∞ -ringed space with corners. If X, Y are interior (local) C ∞ -ringed spaces with corners, a morphism f : X → Y is called interior if the induced maps on stalks f ]x : O Y,f (x) → O X,x are interior morphisms of interior C ∞ -rings with corners for all x ∈ in X. This gives a morphism of sheaves f −1 (OYin ) → OX . Write C∞ RScin ⊂ ∞ c ∞ c ∞ c C RS (and LC RSin ⊂ LC RS ) for the non-full subcategories of interior (local) C ∞ -ringed spaces with corners and interior morphisms. Lemma 5.3. Parts (a)–(c) in Definition 5.2 are equivalent. ˆs0 is open, as the Proof Parts (a), (b) are equivalent by definition. The set U requirement that an element is zero in the stalk is a local requirement. That is, ex (V ) for some x ∈ V ⊆ U . s0x = 0 if and only if s0 |V = 0 ∈ OX ex ex in for (U ) : s0x 6= 0 ∈ OX,x Suppose (a), (b) hold. Write OX (U ) = {s0 ∈ OX in (U ) then s01,x , s02,x 6= 0 for x ∈ U , and as the stalks all x ∈ U }. If s01 , s02 ∈ OX ex in in are interior s01,x · s02,x 6= 0 ∈ OX,x . So s01 · s02 ∈ OX (U ), and OX is a monoid. ∞ in Thus (OX (U ), OX (U ) q {0}) is a pre-C -ring with corners, where the C ∞ operations come from restriction from O X (U ). As the invertible elements of in (U )q{0}) are the same as those the monoid and the C ∞ -rings of (OX (U ), OX ∞ ˆ ex be the sheafification of from O X (U ), it is a C -ring with corners. Let O X in ex in OX q{0}, which is a subsheaf of OX . Note that OX (U )q{0} already satisfies ˆ ex now satisfies gluing. Then uniqueness, so the sheafification process means O X ∞ ex ˆ ) is a sheaf of C -rings with corners. (OX , O X ex ˆ ex ). This ) → (X, OX , O There is a morphism (id, id] , ι]ex ) : (X, OX , OX X is the identity on X, OX . On the sheaves of monoids, we have an inclusion ˆ ex (U ) → Oex (U ). On stalks, any non-zero element of Oex is an ι]ex (U ) : O X X,x X ex (U ). As (a) holds, we can equivalence class represented by a section s0 ∈ OX in choose U 3 x so that s0y 6= 0 for all y ∈ U . Then s0 ∈ OX (U ), so there is 00 0 ] 00 0 ex ˆ (U ) mapping to s under ιex (U ). So sx 7→ sx , and ι]ex is surjective on s ∈O X ˆ ex is a subsheaf of Oex , ι]ex is injective on stalks. Hence stalks as 0 7→ 0. As O X X ] ] (id, id , ιex ) is an isomorphism, and (c) holds. Thus (a), (b) imply (c). ex ex Now suppose (c) holds. If s0 ∈ OX (U ), where OX is the sheafification in 0 ex of OX q {0}, then if sx 6= 0 ∈ OX,x there is an open x ∈ V ⊆ U and in ex 0 s00 ∈ OX (V ) representing s0 |V , and therefore s0x0 6= 0 ∈ OX,x 0 for all x ∈ V , 0 0 0 ex 0 and the Us0 defined in (a) is open. If s1 , s2 ∈ OX (U ), and s1,x , s2,x 6= 0 ∈ ex in OX,x , then there is an open set open x ∈ V ⊆ U and s001 , s002 ∈ OX (V )

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108

in ∈0, / so the stalk O X,x = representing s01 |V , s02 |V . Then s001,x · s002,x ∈ OX,x in (OX,x , OX,x q {0}) is interior, and (a) holds. Hence (c) implies (a).

Remark 5.4. In §4.1 we defined interior C ∞ -rings with corners C = (C, C ex ) as special examples of C ∞ -rings of corners, with C ex = C in q {0}, so that 0 ∈ C ex plays a special, somewhat artificial rˆole in the interior theory; really one would prefer to write C = (C, C in ) and exclude 0 altogether. One place this artificiality appears is in Definition 5.2(a)–(c). We want interior (local) C ∞ -ringed spaces with corners X = (X, O X ) to be special examples of (local) C ∞ -ringed spaces with corners. At the same time, we na¨ıvely expect that if X is interior then O X should be a sheaf of interior C ∞ -rings with corners. But this is false: O X is never a sheaf valued in C∞ Ringscin in the sense of Definition 2.34. For example, O X (∅) = ({0}, {0}), where ({0}, {0}) is the final object in C∞ Ringsc , and it is not equal to ({0}, {0, 1}), the final object in C∞ Ringscin , contradicting Definition 2.34(iii). ∞ c A sheaf O in X on X valued in C Ringsin is also a presheaf (but not a sh ∞ c valued in sheaf) valued in C Rings , so it has a sheafification (O in X) ∞ c C Rings . Conversely, given a sheaf O X on X valued in C∞ Ringsc , we can ask whether it is the sheafification of a (necessarily unique) sheaf O in X valued in C∞ Ringscin . Definition 5.2(a)–(c) characterize when this holds, and this is the correct notion of ‘sheaf of interior C ∞ -rings with corners’ in ˆs0 closed in U in Definition our context. The extra conditions Us0 open and U 5.2(a),(b) are surprising, but they will be essential in the construction of the corner functor in §6.1. The proof of the next theorem is essentially the same as showing ordinary ringed spaces have all small limits. An explicit proof can be found in the first author [25, Th. 5.1.10]. Existence of small limits in C∞ RSc , C∞ RScin uses Theorem 4.20(b); and LC∞ RSc , LC∞ RScin being closed under limits in C∞ RSc , C∞ RScin uses Proposition 4.29; and LC∞ RScin , C∞ RScin being closed under limits in LC∞ RSc , C∞ RSc uses Theorem 4.20(d). Theorem 5.5. The categories C∞ RSc , LC∞ RSc , C∞ RScin , LC∞ RScin have all small limits. Small limits commute with the inclusion and forgetful functors in the following diagram: LC∞ RScin

/ C∞ RSc

 LC∞ RSc

 / C∞ RSc

in

/ Top.

In contrast to Theorem 4.20(d), in Theorem 6.3 below we show that inc : LC∞ RScin ,→ LC∞ RSc has a right adjoint C. This implies:

5.2 Special classes of local C ∞ -ringed spaces with corners

109

Corollary 5.6. The inclusion inc : LC∞ RScin ,→ LC∞ RSc preserves colimits.

5.2 Special classes of local C ∞ -ringed spaces with corners We extend parts of §4.7 to local C ∞ -ringed spaces with corners. Definition 5.7. Let X = (X, O X ) be an interior local C ∞ -ringed space with corners. We call X integral, torsion-free, or saturated, if for each x ∈ X the stalk O X,x is an integral, torsion-free, or saturated, interior local C ∞ -ring with corners, respectively, in the sense of Definition 4.45. Q in in For each open U ⊆ X, the monoid OX (U ) is a submonoid of x∈U OX,x . in If X is integral then each OX,x is an integral monoid, so it is a submonoid Q in in in of a group (OX,x )gp , and OX (U ) is a submonoid of a group x∈U (OX,x )gp , in ∞ so it is integral, and O X (U ) is an integral C -ring with corners. Conversely, if O in X (U ) is integral for all open U ⊆ X one can show the stalks O X,x are integral, so X is integral. Thus, an alternative definition is that X is integral if O in X (U ) is integral for all open U ⊆ X. The analogues hold for torsion-free and saturated. Write LC∞ RScsa ⊂ LC∞ RSctf ⊂ LC∞ RScZ ⊂ LC∞ RScin for the full subcategories of saturated, torsion-free, and integral objects in LC∞ RScin . Also write LC∞ RScsa,ex ⊂ LC∞ RSctf ,ex ⊂ LC∞ RScZ,ex ⊂ LC∞ RScin,ex ⊂ LC∞ RSc for the full subcategories of saturated, . . . , interior objects in LC∞ RSc , but with exterior (i.e. all) rather than interior morphisms. Here is the analogue of Theorem 4.48: sa sa Theorem 5.8. There are coreflection functors ΠZin , Πtf Z , Πtf , Πin in a diagram

p LC∞ RScsa o

Πsa in Πsa tf inc

/ LC∞ RSctf o

Πtf Z inc

/ LC∞ RScZ o

ΠZin inc

/ LC∞ RScin ,

sa sa such that each of ΠZin , Πtf Z , Πtf , Πin is right adjoint to the corresponding inclusion functor inc. sa sa Proof We explain the functor ΠZin ; the arguments for Πtf Z , Πtf , Πin are the ∞ c same. Let X = (X, O X ) be an object in LC RSin . We will construct an ˜ = (X, O ˜ X ) in LC∞ RSc , and set ΠZ (X) = X. ˜ The topological object X in Z ˜ ˜ spaces X in X, X are the same. Define a presheaf P O X of C ∞ -rings with ˜ X (U ) = ΠZ (O in corners on X by P O X (U )) for each open U ⊆ X, where in

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∞ c Z ∞ c O in X (U ) ∈ C Ringsin is as in Definition 5.2(c), and Πin : C Ringsin → ∞ c C RingsZ is as in Theorem 4.48. For open V ⊆ U ⊆ X, the restriction in ˜ X is ΠZ applied to ρU V : O in ˜ morphism ρU V in P O X (U ) → O X (V ). Let O X in ˜X . be the sheafification of P O ˜ = (X, O ˜ X ) is an object in LC∞ RSc . As Definition 5.2(c) implies that X in stalks are given by direct limits as in Definition 2.35, and ΠZin : C∞ Ringscin → C∞ RingscZ preserves direct limits since it is a left adjoint, we have

˜ X,x ∼ ˜ X,x ∼ O = PO = ΠZin (O X,x )

for all x ∈ X.

˜ is an object of LC∞ RSc by Definition 5.7. Hence X Z ˜ Y˜ are as If f = (f, f ] ) : X → Y is a morphism in LC∞ RScin and X, ] ˜ ˜ ˜ → Y˜ in above, in an obvious way we define a morphism f = (f, f ) : X ∞ c ] ˜ LC RSZ such that if x ∈ X with f (x) = y ∈ Y then f acts on stalks by ˜ Y,y ∼ ˜ X,x ∼ f˜]x ∼ = ΠZin (f ]x ) : O = ΠZin (O Y,y ) −→ O = ΠZin (O X,x ). We define ΠZin (f ) = f˜. This gives a functor ΠZin : LC∞ RScin → LC∞ RScZ . We can deduce that ΠZin is right adjoint to inc : LC∞ RScZ ,→ LC∞ RScin from the fact that ΠZin is left adjoint to inc in Theorem 4.48, applied to stalks O X,x , O Y,y as above for all x ∈ X with f (x) = y in Y . Remark 5.9. The analogue of Theorem 5.8 does not work for the categories LC∞ RScsa,ex ⊂ · · · ⊂ LC∞ RScin,ex . This is because the functors ΠZin , . . . , Πsa in in Theorem 4.48 do not extend to exterior morphisms. Since right adjoints preserve limits, Theorems 5.5 and 5.8 imply: Corollary 5.10. All small limits exist in LC∞ RScsa , LC∞ RSctf , LC∞ RScZ .

5.3 The spectrum functor We now define a spectrum functor for C ∞ -rings with corners, in a similar way to Definition 2.43. Definition 5.11. Let C = (C, C ex ) be a C ∞ -ring with corners, and use the notation of Definition 4.31. As in Definition 2.42, write XC for the set of Rpoints of C with topology TC . For each open U ⊆ XC , define O XC (U ) = ` ex (OXC (U ), OX (U )). Here OXC (U ) is the set of functions s : U → x∈U C x C (where we write sx for its value at the point x ∈ U ) such that sx ∈ C x for all x ∈ U , and such that U may be covered by open W ⊆ U for which there exist ex c ∈ C with sx = πx (c) in C x for all x ∈ W . Similarly, OX (U ) is the set of C

5.3 The spectrum functor

111

` s0 : U → x∈U C x,ex with s0x ∈ C x,ex for all x ∈ U , and such that U may be covered by open W ⊆ U for which there exist c0 ∈ C ex with s0x = πx,ex (c0 ) in C x,ex for all x ∈ W . Define operations Φf and Ψg on O XC (U ) pointwise in x ∈ U using the operations Φf and Ψg on C x . This makes O XC (U ) into a C ∞ -ring with corners. If V ⊆ U ⊆ XC are open, the restriction maps ρU V = (ρU V , ρU V,ex ) : O XC (U ) → O XC (V ) mapping ρU V : s 7→ s|V and ρU V,ex : s0 7→ s0 |V are morphisms of C ∞ -rings with corners. ex The local nature of the definition implies that O XC = (OXC , OX ) is a C ∞ sheaf of C -rings with corners on XC . In fact, OXC is the sheaf of C ∞ -rings in Definition 2.43. By Proposition 5.12 below, the stalk O XC ,x at x ∈ XC is naturally isomorphic to C x , which is a local C ∞ -ring with corners by Theorem 4.32(a). Hence (XC , O XC ) is a local C ∞ -ringed space with corners, which we call the spectrum of C, and write as Specc C. Now let φ = (φ, φex ) : C → D be a morphism of C ∞ -rings with corners. As in Definition 2.43, define the continuous function fφ : XD → XC by fφ (x) = x ◦ φ. For open U ⊆ XC define (f φ )] (U ) : O XC (U ) → O XD (fφ−1 (U )) to act by φx : C fφ (x) → Dx on stalks at each x ∈ fφ−1 (U ), where φx is the induced morphism of local C ∞ -rings with corners. Then (f φ )] : O XC → (fφ )∗ (O XD ) is a morphism of sheaves of C ∞ -rings with corners on XC . Let f ]φ : fφ−1 (O XC ) → O XD be the corresponding morphism of sheaves of C ∞ -rings with corners on XD under (2.9). The stalk map f ]φ,x : O XC ,fφ (x) → O XD ,x of f ]φ at x ∈ XD is identified with φx : C fφ (x) → Dx under the isomorphisms O XC ,fφ (x) ∼ = Dx in Proposition 5.12. = C fφ (x) , O XD ,x ∼ ] Then f φ = (fφ , f φ ) : (XD , O XD ) → (XC , O XC ) is a morphism of local C ∞ -ringed spaces with corners. Define Specc φ : Specc D → Specc C by Specc φ = f φ . Then Specc is a functor (C∞ Ringsc )op → LC∞ RSc , the spectrum functor. Proposition 5.12. In Definition 5.11, the stalk O XC ,x of O XC at x ∈ XC is naturally isomorphic to C x . ex Proof We have O XC ,x = (OXC ,x , OX ) where elements [U, s] ∈ OXC ,x C ,x 0 ex and [U, s ] ∈ OXC ,x are ∼-equivalence classes of pairs (U, s) and (U, s0 ), where U is an open neighbourhood of x in XC and s ∈ OXC (U ), s0 ∈ ex OX (U ), and (U, s) ∼ (V, t), (U, s0 ) ∼ (V, t0 ) if there exists open x ∈ W ⊆ C ex U ∩ V with s|W = t|W in OXC (W ) and s0 |W = t0 |W in OX (W ). Define C ∞ a morphism of C -rings with corners Π = (Π, Πex ) : O XC ,x → C x by Π : [U, s] 7→ sx ∈ C x and Πex : [U, s0 ] 7→ s0x ∈ C x,ex .

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Suppose cx ∈ C x and c0x ∈ C x,ex . Then cx = πx (c) for c ∈ C x and c0x = ` πx,ex (c0 ) for c0 ∈ C x,ex by Theorem 4.32(c). Define s : XC → y∈XC C y ` and s0 : XC → y∈XC C y,ex by sy = πy (c) and s0y = πy,ex (c0 ). Then s ∈ OXC (XC ), so that [XC , s] ∈ OXC ,x with Π([XC , s]) = sx = πx (c) = cx , and ex similarly s0 ∈ OX (XC ) with Πex ([XC , s0 ]) = c0x . Hence Π : OXC ,x → C x C ex and Πex : OXC ,x → C x,ex are surjective. Let [U1 , s1 ], [U2 , s2 ] ∈ OXC ,x with Π([U1 , s1 ]) = s1,x = s2,x = Π([U2 , s2 ]). Then by definition of O XC (U1 ), O XC (U2 ) there exists an open neighbourhood V of x in U1 ∩ U2 and c1 , c2 ∈ C with s1,v = πv (c1 ) and s2,v = πv (c2 ) for all v ∈ V . Thus πx (c1 ) = πx (c2 ) as s1,x = s2,x . Hence c1 − c2 lies in the ideal I in (2.4) by Proposition 2.24. Thus there exists d ∈ C with x(d) 6= 0 ∈ R and d · (c1 − c2 ) = 0 ∈ C. Making V smaller we can suppose that v(d) 6= 0 for all v ∈ V , as this is an open condition. Then πv (c1 ) = πv (c2 ) ∈ C v for v ∈ V , since πv (d) · πv (c1 ) = πv (d) · πv (c2 ) as d · c1 = d · c2 and πv (d) is invertible in C v . Thus s1,v = πv (c1 ) = πv (c2 ) = s2,v for v ∈ V , so s1 |V = s2 |V , and [U1 , s1 ] = [V, s1 |V ] = [V, s2 |V ] = [U2 , s2 ]. Therefore Π : OXC ,x → C x is injective, and an isomorphism. ex with Πex ([U1 , s01 ]) = s01,x = s02,x = Let [U1 , s01 ], [U2 , s02 ] lie in OX C ,x Π([U2 , s02 ]). As above there exist an open neighbourhood V of x in U1 ∩ U2 and c01 , c02 ∈ C ex with s01,v = πv,ex (c01 ) and s02,v = πv,ex (c02 ) for all v ∈ V . At this point we can use Proposition 4.33, which says that πx,ex (x02 ) = πx,ex (c01 ) if and only if there are a, b ∈ C ex such that Φi (a) − Φi (b) ∈ Ix , x ◦ Φi (a) 6= 0 and ac1 = bc2 , where Ix is the ideal in (2.4). The third condition does not depend on x, whereas the first two conditions are open conditions in x, that is, if Φi (a) − Φi (b) ∈ Ix , x ◦ Φi (a) 6= 0, then there is an open neighbourhood of X such that Φi (a) − Φi (b) ∈ Iv , v ◦ Φi (a) 6= 0 for all v in that neighbourhood. Making V above smaller if necessary, we can suppose that these conditions hold in V and thus that πv,ex (c01 ) = πv,ex (c02 ) for all v ∈ V . Hence s01,v = s02,v for all v ∈ V , and s01 |V = s02 |V , so that [U1 , s01 ] = [V, s01 |V ] = [V, s02 |V ] = ex [U2 , s02 ]. Therefore Πex : OX → C x,ex is injective, and an isomorphism. So C ,x Π = (Π, Πex ) : O XC ,x → C x is an isomorphism, as we have to prove. Definition 5.13. As C∞ Ringscin is a subcategory of C∞ Ringsc we can define the functor Speccin by restricting Specc to (C∞ Ringscin )op . Let C = (C, C ex ) be an interior C ∞ -ring with corners, and X = Speccin C = (X, OX ). Then Definition 4.26 implies the localizations C x are interior C ∞ -rings with corners, and C x ∼ = O X,x by Proposition 5.12. 0 ex ex If s ∈ OX (U ) with s0x 6= 0 in OX,x at x ∈ X, then s0x0 = πx0 ,ex (c0 ) in ex 0 ∼ C x0 ,ex = OX,x0 for some c ∈ C ex and all x0 in an open neighbourhood V of

113

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x in U . Then c0 6= 0 in C ex , so c0 is non-zero in every stalk by Remark 4.34 as C is interior, and s0 is non-zero at every point in V . Therefore X is an interior local C ∞ -ringed space with corners by Definition 5.2(a). If φ : C → D is a morphism of interior C ∞ -rings with corners, then Speccin φ = (f, f ] ) has stalk map f ]x = φx : C fφ (x) → Dx . This map fits into the commutative diagram C πf

φ

 φ C fφ (x)

/D πx

(x)

φx

 / Dx .

As φ is interior, and the maps π fφ (x) , π x are interior and surjective, then f ]x is interior. This implies that Speccin φ is an interior morphism of interior local C ∞ -ringed spaces with corners. Hence Speccin : (C∞ Ringscin )op → LC∞ RScin is a well defined functor, which we call the interior spectrum functor. Definition 5.14. The global sections functor Γc : LC∞ RSc −→ (C∞ Ringsc )op takes objects (X, O X ) ∈ LC∞ RSc to O X (X) and takes morphisms (f, f ] ) : (X, O X ) → (Y, O Y ) to Γc : (f, f ] ) 7→ f ] (Y ). Here f ] : O Y → f∗ (O X ) corresponds to f ] under (2.9). The composition Γc ◦ Specc maps (C∞ Ringsc )op → (C∞ Ringsc )op , or equivalently maps C∞ Ringsc → C∞ Ringsc . For each C ∞ -ring with corners C we define a morphism ΞC = (Ξ, Ξex ) : C → Γc ◦ Specc C in ` C∞ Ringsc by Ξ(c) : XC → x∈XC OXC ,x , Ξ(c) : x 7→ πx (c) in C x ∼ = ` ex 0 OXC ,x for c ∈ C, and Ξex (c0 ) : XC → x∈XC OX , Ξ (c ) : x → 7 ex C ,x ex for c0 ∈ C ex . This is functorial in C, so that the πx,ex (c0 ) in C x,ex ∼ = OX C ,x ΞC for all C define a natural transformation Ξ : IdC∞ Ringsc ⇒ Γc ◦ Specc of functors IdC∞ Ringsc , Γc ◦ Specc : C∞ Ringsc → C∞ Ringsc . Here is the analogue of Lemma 2.45. Lemma 5.15. Let C be a C ∞ -ring with corners, and X = Specc C. For any c ∈ C, let Uc ⊆ X be as in Definition 2.42. Then U c ∼ = Specc (C(c−1 )). If C −1 is firm, or interior, then so is C(c ). Proof Write C(c−1 ) = (D, Dex ). By Lemma 4.27 we have D ∼ = C(c−1 ). By Lemma 2.45, we need only show there is an isomorphism of stalks C x,ex → Dxˆ,ex . However, using the universal properties of C x , C(c−1 ) and C(c−1 )xˆ this follows by the same reasoning as Lemma 2.45. If C is firm, then so is

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114

C(c−1 ), as C(c−1 )]ex is generated by the image of C ]ex under the morphism C → C(c−1 ). If C is interior, then C(c−1 ) is also interior, as otherwise zero divisors in C(c−1 )ex would have to come from zero divisors in C ex . Theorem 5.16. The functor Specc : (C∞ Ringsc )op → LC∞ RSc is right adjoint to Γc : LC∞ RSc → (C∞ Ringsc )op . Thus for all C in C∞ Ringsc and all X in LC∞ RSc there are inverse bijections HomC∞ Ringsc (C, Γc (X)) o

LC ,X RC ,X

/ Hom ∞ c (X, Specc C). LC RS

(5.1)

If we let X = Specc C then ΞC = RC,X (idX ), and Ξ is the unit of the adjunction between Γc and Specc . Proof We follow the proof of [50, Th. 4.20]. Take X ∈ LC∞ RSc and C ∈ C∞ Ringsc , and let Y = (Y, O Y ) = Specc C. Define a functor RC,X in (5.1) by taking RC,X (f ) : C → Γc (X) to be the composition ΞC

C

/ Γc ◦ Specc C = Γc (Y )

Γc (f )

/ Γc (X)

for each morphism f : X → Y in LC∞ RSc . If X = Specc C then we have ΞC = RC,X (idX ). We see that RC,X is an extension of the functor RC,X constructed in [50, Th. 4.20] for the adjunction between Spec and Γ. This will also occur for LC,X . Given a morphism φ = (φ, φex ) : C → Γc (X) in C∞ Ringsc we define ] ) where (g, g ] ) = LC,X (φ) with LC,X constructed LC,X (φ) = g = (g, g ] , gex in [50, Th. 4.20]. Here, g acts by x 7→ x∗ ◦ φ where x∗ : OX (X) → R is the composition of the σx : OX (X) → OX,x with the unique morphism π : OX,x → R, as O X,x is a local C ∞ -ring with corners. The morphisms ] are constructed as g ] is in [50, Th. 4.20], and we explain this explicitly g ] , gex now. For x ∈ X and g(x) = y ∈ Y , take the stalk map σ x = (σx , σxex ) : O X (X) → O X,x . This gives the following diagram of C ∞ -rings with corners C

φ

πy

 Cy ∼ = O Y,y

/ Γc (X) (5.2)

σx

φx

 / O X,x

π

/ R.

We know C y ∼ = O Y,y by Proposition 5.12 and π : OX,x → R is the unique local morphism. If we have (c, c0 ) ∈ C with y(c) 6= 0, and y ◦ Φi (c0 ) 6= 0 then σ x ◦ φ(c, c0 ) ∈ O X,x with π(σx ◦ φ(c)) 6= 0 and π(Φi ◦ σxex ◦ φex (c)) = π(σx ◦ φ ◦ Φi (c)) 6= 0.

5.3 The spectrum functor

115

As O X,x is a local C ∞ -ring with corners then σ x ◦ φ(c, c0 ) is invertible in O X,x . The universal property of π y : C → C y gives a unique morphism φx : O Y,y → O X,x that makes (5.2) commute. We define g ] (V ) = (g] (V ), g]ex (V )) : O Y (V ) → g∗ (O X )(V ) = O X (U ) for each open V ⊆ Y with U = g −1 (V ) ⊆ X to act by φx = (φx , φx,ex ) ex on stalks at each x ∈ U . We can identify elements s ∈ OX (U ), s0 ∈ OX (U ) ` ` 0 ex with maps s : U → x∈U OX,x and s : U → x∈U OX,x , such that s, s0 are locally of the form s : x 7→ πx (c) in C x ∼ = OX,x for c ∈ C and s0 : ex x 7→ πx,ex (c0 ) in C x,ex ∼ for c0 ∈ C ex , and similarly for t ∈ OY (V ), = OX,x 0 ex t ∈ OY (V ). If t is locally of the form t : y 7→ πy (d) in Dy ∼ = OY,c near y˜ = g(˜ x) for d ∈ D, then s = g] (V )(t) is locally of the form s : x 7→ πx (c) in Cx ∼ ˜ for c = φ(d) ∈ C, so g] (V ) does map OY (V ) → OX (U ), = OX,x near x ex and similarly g]ex (V ) maps OYex (V ) → OX (U ). Hence g ] (V ) is well defined. This defines a morphism g ] : O Y → g∗ (O X ) of sheaves of C ∞ -rings with corners on Y , and we write g ] : g −1 (O Y ) → O X for the corresponding morphism of sheaves of C ∞ -rings with corners on X under (2.9). At a point x ∈ X with g(x) = y ∈ Y , the stalk map g ]x : O Y,y → O X,x is φx . Then g = (g, g ] ) is a morphism in LC∞ RSc , and LC,X (φ) = g. It remains to show that these define natural bijections, but this follows in a very similar way to [50, Th. 4.20]. Definition 5.17. Define the interior global sections functor Γcin : LC∞ RScin → (C∞ Ringscin )op to act on objects (X, O X ) by Γcin : (X, O X ) 7→ (C, C ex ) ex where C = OX (X) and C ex is the set containing the zero element of OX (X) ex and the elements of OX (X) that are non-zero in every stalk. That is, ex ex C ex = {c0 ∈ OX (X) : c0 = 0 ∈ OX (X) ex or σxex (c0 ) 6= 0 ∈ OX,x ∀x ∈ X},

(5.3)

ex ex → OX,x . This is an interior C ∞ -ring with where σxex is the stalk map σxex : OX ∞ corners, where the C -ring with corners structure is given by restriction from ex (X)). We define Γcin to act on morphisms (f, f ] ) : (X, O X ) → (OX (X), OX (Y, O Y ) by Γcin : (f, f ] ) 7→ f ] (Y )|(C,C ex ) for f ] : O Y → f∗ (O X ) corresponding to f ] under (2.9). As in Definition 5.14, for each interior C ∞ -ring with corners C, we define c c ∞ c a morphism Ξin = (Ξin , Ξin C ` ex ) : C → Γin ◦ Specin C in C Ringsin by Ξin (c) : XC → x∈XC OXC ,x , Ξin (c) : x 7→ πx (c) in C x ∼ = OXC ,x for ` in 0 ex in 0 c ∈ C, and Ξex (c ) : XC → x∈XC OXC ,x , Ξex (c ) : x 7→ πx,ex (c0 ) in ex 0 C x,ex ∼ for c0 ∈ C ex . We need to check Ξin = OX ex (c ) lies in (5.3), but C ,x

116

C ∞ -schemes with corners

this is immediate as π x = (πx , πx,ex ) : C → C x is interior. The Ξin C for all C define a natural transformation Ξin : IdC∞ Ringscin ⇒ Γcin ◦ Speccin of functors IdC∞ Ringscin , Γcin ◦ Speccin : C∞ Ringscin → C∞ Ringscin . Theorem 5.18. The functor Speccin : (C∞ Ringscin )op → LC∞ RScin is right adjoint to Γcin : LC∞ RScin → (C∞ Ringscin )op . Proof This proof is identical to that of Theorem 5.16. We need only check that the definition of Γcin (X), which may not be equal to O X (X), gives well c c defined maps σ in x : Γin (X) → O X,x . As Γin (X) is a subobject of O X (X), these maps are the restriction of the stalk maps σ x : O X (X) → O X,x to Γcin (X). The definition of Γcin (X) implies these maps are interior.

5.4 C ∞ -schemes with corners Definition 5.19. A local C ∞ -ringed space with corners that is isomorphic in LC∞ RSc to Specc C for some C ∞ -ring with corners C is called an affine C ∞ -scheme with corners. We define the category AC∞ Schc to be the full subcategory of affine C ∞ -schemes with corners in LC∞ RSc . Let X = (X, O X ) be a local C ∞ -ringed space with corners. We call X a C ∞ -scheme with corners if X can be covered by open sets U ⊆ X such that (U, O X |U ) is a affine C ∞ -scheme with corners. We define the category C∞ Schc of C ∞ -schemes with corners to be the full subcategory of C ∞ schemes with corners in LC∞ RSc . Then AC∞ Schc is a full subcategory of C∞ Schc . A local C ∞ -ringed space with corners that is isomorphic in LC∞ RScin to Speccin C for some interior C ∞ -ring with corners C is called an interior affine C ∞ -scheme with corners. We define the category AC∞ Schcin of interior affine C ∞ -schemes with corners to be the full subcategory of interior affine C ∞ -schemes with corners in LC∞ RScin , so AC∞ Schcin is a nonfull subcategory of AC∞ Schc . We call an object X ∈ LC∞ RScin an interior C ∞ -scheme with corners if it can be covered by open sets U ⊆ X such that (U, O X |U ) is an interior affine C ∞ -scheme with corners. We define the category C∞ Schcin of interior C ∞ -schemes with corners to be the full subcategory of interior C ∞ -schemes with corners in LC∞ RScin . This implies that AC∞ Schcin is a full subcategory of C∞ Schcin . Clearly C∞ Schcin ⊂ C∞ Schc . In Theorem 5.28(b) below we show that an object X in C∞ Schc lies in ∞ C Schcin if and only if lies in LC∞ RScin . That is, the two natural definitions of ‘interior C ∞ -scheme with corners’ turn out to be equivalent.

5.4 C ∞ -schemes with corners

117

Lemma 5.20. Let X be an (interior) C ∞ -scheme with corners, and U ⊆ X be open. Then U is also an (interior) C ∞ -scheme with corners. Proof This follows from Lemma 5.15 and the fact in Definition 2.42 that the topology on Specc C is generated by subsets Uc . We explain the relation with manifolds with (g-)corners in Chapter 3: ∞

c

C Sch Definition 5.21. Define a functor FMan : Manc → C∞ Schc that acts c ∞ c C Sch on objects X ∈ Manc by FMan (X) = (X, O X ), where O X (U ) = c ∞ ∞ C (U ) = (C (U ), Ex(U )) from Example 4.18(a) for each open subset U ⊆ ∞ X. If V ⊆ U ⊆ X are open we define ρU V = (ρU V , ρex U V ) : C (U ) → ∞ ex 0 0 C (V ) by ρU V : c 7→ c|V and ρU V : c 7→ c |V . It is easy to verify that O X is a sheaf of C ∞ -rings with corners on X, so X = (X, O X ) is a C ∞ -ringed space with corners. We show in Theorem 5.22(b) that X is a C ∞ -scheme with corners, and it is also interior. C∞ Schc Let f : X → Y be a morphism in Manc . Writing FMan (X) = c ∞ c C Sch (X, O X ) and FManc (Y ) = (Y, O Y ), for all open U ⊆ Y , we define

f ] (U ) : O Y (U ) = C ∞ (U ) −→ f∗ (O x )(U ) = O X (f −1 (U )) = C ∞ (f −1 (U )) by f] (U ) : c 7→ c ◦ f for all c ∈ C ∞ (U ) and f]ex (U ) : c0 7→ c0 ◦ f for all c0 ∈ Ex(U ). Then f ] (U ) is a morphism of C ∞ rings with corners, and f ] : O Y → f∗ (O X ) is a morphism of sheaves of C ∞ -rings with corners on Y . Let f ] : C∞ Schc f −1 (O Y ) → O X correspond to f ] under (2.9). Then FMan (f ) = f = c ] ∞ c (f, f ) : (X, O X ) → (Y, O Y ) is a morphism in C Sch . Define a functor C∞ Schc C∞ Schc FManc in : Mancin → C∞ Schcin by restriction of FMan to Mancin . c in All the above generalizes immediately from manifolds with corners to maniC∞ Schc folds with g-corners in §3.3, giving functors FMan : Mangc → C∞ Schc gc ∞ c C Schin gc ∞ c and FMangc : Manin → C Schin . It also generalizes to manifolds with in ˇ c , . . . , Man ˇ gc . (g-)corners of mixed dimension Man ∞

c

C∞ Schc

in

C Sch The functors FMan , . . . , FMangc in map to various subcategories of c in ∞ c ∞ c C Sch , C Schin defined in §5.6, including C∞ Schcto,ex , C∞ Schcto of C∞ Schc

∞

c

C Sch toric C ∞ -schemes with corners. We will write FManc to , . . . for FMan , c . . . considered as mapping to these subcategories. We say that a C ∞ -scheme with corners X is a manifold with corners (or is C∞ Schc a manifold with g-corners) if X ∼ (X) for some X ∈ Manc (or = FMan c ∞ c C Sch gc ∼ X = FMangc (X) for some X ∈ Man ).

Theorem 5.22. Let X be a manifold with corners, or a manifold with gC∞ Schc C∞ Schc corners, and X = FMan (X) or X = FMan (X). Then: c gc (a) If X is a manifold with faces or g-faces, then X is an affine C ∞ -scheme

C ∞ -schemes with corners

118

with corners, and is isomorphic to Specc C ∞ (X). It is also an interior affine C ∞ -scheme with corners, with X ∼ = Speccin C ∞ in (X). If X is a manifold with corners, but not with faces, then X is not affine. (b) In general, X is an interior C ∞ -scheme with corners. ∞

c

C∞ Schcin

C Sch (c) The functors FMan , FManc c in ful.

∞

c

C∞ Schcin

C Sch , FMan , FMangc gc

are fully faith-

in

Proof For (a), let X be a manifold with corners or a manifold with g-corners, and write X = (X, O X ). Note that Γc (X) = O X (X) = C ∞ (X) by the definition of O X . Consider the map LC,X in (5.1). In the notation of Theorem 5.16, if we let C = Γc (X) = C ∞ (X), then LC ∞ (X),X is a bijection LC ∞ (X),X : HomC∞ Ringsc (C ∞ (X), C ∞ (X)) −→ HomLC∞ RSc (X, Specc C ∞ (X)). Write Y = Specc C ∞ (X), and define a morphism g = (g, g ] ) : X → Y in LC∞ RSc by g = LC ∞ (X),X (idC ∞ (X) ). The continuous map g : X → Y is defined in the proof of Theorem 5.16 by g(x) = x∗ ◦ idC ∞ (X) , where x∗ is the evaluation map at the point x ∈ X. This is a homeomorphism of topological spaces as in the proof of [50, Th. 4.41]. On stalks at each x ∈ X we have g ]x = λx : (C ∞ (X))x∗ → C ∞ x (X), where λx is as in Example 4.35. If X has faces then λx is an isomorphism by Proposition 4.36, and if X has g-faces then λx is an isomorphism by Definition 4.37. Hence if X has (g-)faces then g, g ] and g are isomorphisms, and X ∼ = Specc C ∞ (X). Essentially the same proof for the interior case, using Speccin , Γcin adjoint by Theorem 5.18, shows that X ∼ = Speccin C ∞ in (X) if X has (g-)faces. If X has corners, but not faces, then Proposition 4.36 gives x ∈ X such that g ]x = λx is not an isomorphism. The proof shows that λx,ex is not surjective. If X ∼ = Specc C for some C ∞ -ring with corners C then C x ∼ = O X,x = ∞ C x (X), and the localization morphism C → C x is surjective. But C → C x ∼ = λx ∞ ∞ ∞ C x (X) factors as C → C (X) −→ C x (X), and λx,ex is not surjective, a contradiction. Hence X is not affine, completing part (a). For (b), for any point x ∈ X, we can find an open neighbourhood U of x which is a manifold with (g-)faces, as in Example 4.38 in the g-corners case. Then U ∼ = Specc C ∞ (U ) ∼ = Speccin C ∞ in (U ) by (a). So X can be covered by open U ⊆ X which are (interior) affine C ∞ -schemes with corners, and X is an (interior) C ∞ -scheme with corners. C∞ Schc For (c), if f, g : X → Y are morphisms in Manc and (f, f ] ) = FMan (f ) c ∞ c ∞ c C Sch ] C Sch = FManc (g) = (g, g ), we see that f = g, so FManc is faithful. Suppose h = (h, h] ) : X → Y is a morphism in C∞ Schc , and x ∈ X with h(y) = y

5.5 Semi-complete C ∞ -rings with corners

119

in Y . We may choose open y ∈ V ⊆ Y and x ∈ U ⊆ h−1 (V ) ⊆ X such that U = (U, O X |U ) and V = (V, O Y |V ) are affine. Then h] induces a morphism h]U V : O Y (V ) = C ∞ (V ) → O X (U ) = C ∞ (U ) in C∞ Ringsc . By considering restriction to points in U, V we see that h]U V maps C ∞ (V ) ∞ ∞ ∞ → C ∞ (U ) and Cex (V ) → Cex (U ) by c 7→ c◦h|U for c in C ∞ (V ) or Cex (V ). Thus, if c : V → R is smooth, or c : V → [0, ∞) is exterior, then c◦h|U : U → R is smooth, or c ◦ h|U : U → [0, ∞) is exterior. Since V is affine, this implies C∞ Schc that h|U : U → V is smooth, and h|U = FMan (h|U ). As we can cover c C∞ Schc X, Y by such U, V , we see that h is smooth and h = FMan (h). Hence c ∞ c ∞ C Sch C Schc C∞ Schc FMan is full. The argument for FManc in , . . . , FMangc in is essentially c in in the same. Theorem 5.22 shows we can regard Manc , Mangc (or Mancin , Mangc in ) as full subcategories of C∞ Schc (or C∞ Schcin ), and regard C ∞ -schemes with corners as generalizations of manifolds with (g-)corners.

5.5 Semi-complete C ∞ -rings with corners Proposition 2.51 says Spec ΞC : Spec ◦Γ ◦ Spec C → Spec C is an isomorphism for any C ∞ -ring C. This was used in §2.6 to define the category C∞ Ringsco of complete C ∞ -rings (those isomorphic to Γ ◦ Spec C), such that (C∞ Ringsco )op is equivalent to the category of affine C ∞ -schemes AC∞ Sch. This can be used to prove that fibre products and finite limits exist in C∞ Sch. We now consider to what extent these results generalize to the corners case. Firstly, it turns out that Specc ΞC need not be an isomorphism for C ∞ -rings with corners C. The next remark and example explain what can go wrong. Remark 5.23. Let C = (C, C ex ) be a C ∞ -ring with corners. Write X = (X, O X ) = Specc C, an affine C ∞ -scheme with corners. The global sections Γc (X) = O X (X) is a C ∞ -ring with corners, with a morphism ΞC : C → O X (X), so we have a morphism of C ∞ -schemes with corners Specc ΞC = Specc (Ξ, Ξex ) : Specc ◦Γc ◦ Specc C −→ Specc C.

(5.4)

We want to know how close (5.4) is to being an isomorphism. First note that as the underlying C ∞ -scheme of X is X = Spec C, Proposition 2.51 shows (5.4) is an isomorphism on the level of C ∞ -schemes, and hence of topological spaces. Thus (5.4) is an isomorphism if Ξex induces an isomorphism of sheaves of monoids. It is sufficient to check this on stalks. Therefore,

C ∞ -schemes with corners

120

(5.4) is an isomorphism if the following is an isomorphism for all x ∈ X: ex ex Ξx,ex : C x,ex ∼ −→ (OX (X))x∗ . = OX,x

Consider the diagram of monoids for x ∈ X: C ex Ξex

 ex OX (X)

πx,ex ρex X,x

π ˆ x,ex

ex / / C x,ex ∼ = OX,x _ 3 Ξx,ex

 / / (Oex (X))x∗ . X

(5.5)

0 ex ex Here ρex X,x takes an element s ∈ OX (X) to its value in the stalk OX,x . It is not difficult to show that the upper left triangle and the outer rectangle in (5.5) commute, πx,ex , π ˆx,ex are surjective, and Ξx,ex is injective. However, as Example 5.24 shows, Ξx,ex need not be surjective (in which case (5.4) is not an isomorphism), and the bottom right triangle of (5.5) need not commute. ex That is, if two elements of OX (X) agree locally, then while they have the ex same image in the stalk OX,x they do not necessarily have the same value in ex (X) at x∗ . This is because for c0 , d0 ∈ C ex , equality in the localization of OX C ex,x requires a global equality. That is, there need to be a0 , b0 ∈ C ex such that a0 c0 = b0 d0 ∈ C ex with a0 , b0 satisfying additional conditions as in Proposition 4.33. In the C ∞ -ring C, this equality is only a local equality, as the a0 and b0 can come from bump functions. However, in the monoid, bump functions do not necessarily exist, meaning this condition is stronger and harder to satisfy. In the following example, C is interior, and the above discussion also holds with Speccin , Γcin in place of Specc , Γc .

Example 5.24. Define open subsets U, V ⊂ R2 by U = (−1, 1) × (−1, ∞) and V = (−1, 1) × (−∞, 1). The important properties for our purposes are that R2 \ U , R2 \ V are connected, but R2 \ (U ∪ V ) is disconnected, being divided into connected components (−∞, −1] × R and [1, ∞) × R. Choose smooth ϕ, ψ : R2 → R such that ϕ|U > 0, ϕ|R2 \U = 0, ψ|V > 0, ψ|R2 \V = 0, so ϕ, ψ map to [0, ∞). Define a C ∞ -ring with corners C by  C = R(x1 , x2 )[y1 , . . . , y6 ] Φi (y1 ) = Φi (y2 ) = 0, Φi (y3 ) = Φi (y4 )
  = ϕ(x1 , x2 ), Φi (y5 ) = Φi (y6 ) = ψ(x1 , x2 ) y1 y3 = y2 y4 , y1 y5 = y2 y6 , using the notation of Definition 4.24. Let X = (X, O X ) = Specc C, an affine C ∞ -scheme with corners. As a topological space we write  X = (x1 , x2 , y1 , . . . , y6 ) ∈ R2 × [0, ∞)6 : y1 = y2 = 0, y3 = y4 = ϕ(x1 , x2 ), y5 = y6 = ψ(x1 , x2 ) .

5.5 Semi-complete C ∞ -rings with corners

121

The projection X → R2 mapping (x1 , x2 , y1 , . . . , y6 ) 7→ (x1 , x2 ) is a homeomorphism, and we use this to identify X ∼ = R2 . We will show that (5.4) is not an isomorphism, in contrast to Proposition 2.51 for C ∞ -rings and C ∞ -schemes. For each x ∈ X we have a monoid OX,x,ex ∼ = C x,ex by Proposition 5.12, containing elements πx,ex (yj ) for j = 1, . . . , 6. If x ∈ U then using Proposition 4.33 and the relations Φi (y3 ) = Φi (y4 ) = ϕ(x1 , x2 ), y1 y3 = y2 y4 , and ϕ(x) > 0 we see that πx,ex (y1 ) = πx,ex (y2 ). Similarly if x ∈ V then Φi (y5 ) = Φi (y6 ) = ψ(x1 , x2 ), y1 y5 = y2 y6 , and ψ(x) > 0 give πx,ex (y1 ) = πx,ex (y2 ). ex Thus as U ∪ V = (−1, 1) × R we may define an element z ∈ OX (X) by   x 6 −1,  πx,ex (y1 ), z(x1 , x2 ) = πx,ex (y1 ) = πx,ex (y2 ), x ∈ (−1, 1),   π (y ), x > 1. x,ex

2

We will show that for x = (−2, 0) in X there does not exist c0 ∈ C ex ex ex (X))x∗ . Thus Ξx,ex : OX,x → with Ξx,ex ◦ πx,ex (c0 ) = π ˆx,ex (z) in (OX ex (OX (X))x∗ is not surjective. This implies that the bottom right triangle of ex (5.5) does not commute at z ∈ OX (X). Also Specc ΞC in (5.4) is not an isomorphism, as Ξx,ex is part of the action of Specc ΞC on stalks at x, and is not an isomorphism. Suppose for a contradiction that such c0 does exist. Then π ˆx,ex ◦ Ξex (c0 ) = π ˆx,ex (z) as the outer rectangle of (5.5) commutes, so by Proposition 4.33 there ex (X) with a0 (x) = b0 (x) 6= 0 and a0 = b0 near x, such that exist a0 , b0 ∈ OX 0 0 0 a z = b Ξex (c ). Locally on X, a0 , b0 are of the form exp(f (x1 , x2 , y1 , . . . , y6 ))y1a1 . . . y6a6 or zero, where the transitions between different representations of this form (e.g. changing a1 , . . . , a6 ) satisfy strict rules. We will consider a0 , b0 in the regions x1 6 −1, x1 > 1, U \ V , V \ U . In each region, a0 , b0 must admit a global representation exp(f )y1a1 . . . y6a6 . Since a0 (x) = b0 (x) 6= 0, we see that a0 , b0 are of the form exp(f ) in x1 6 −1. On crossing the wall x = −1, y 6 −1 between x 6 −1 and U \ V , y3 , y4 become invertible, so a0 , b0 may be of the form exp(f )y3a3 y4a4 in U \ V . On crossing the wall x = 1, y 6 −1 between U \ V and x > 1, we see that a0 , b0 must be of the form exp(f )y3a3 y4a4 in x1 > 1. Similarly, on crossing the wall x = −1, y > 1 between x 6 −1 and V \ U , y5 , y6 become invertible, so a0 , b0 may be of the form exp(f )y5a5 y6a6 in V \ U . On crossing the wall x = 1, y > 1 we see that a0 , b0 must be of the form exp(f )y5a5 y6a6 in x1 > 1. Comparing this with the previous statement gives a5 = a6 = 0. Hence a0 , b0 are invertible at (2, 0). Now c0 is either zero, or of the form exp(f )y1a1 . . . y6a6 . Specializing a0 z =

122

C ∞ -schemes with corners

b0 Ξex (c0 ) at (−2, 0) and using a0 , b0 invertible there gives c0 6= 0 with a1 = 0 and ai = 0 for i 6= 1. But specializing at (2, 0) and using a0 , b0 invertible there gives a2 = 1 and ai = 0 for i 6= 2, a contradiction. So no such c0 exists. One can prove that C above is toric, and hence also firm, saturated, torsionfree, integral, and interior, in the notation of §4.7. There are simpler examples of C lacking these properties for which (5.4) is not an isomorphism and (5.5) does not commute. We chose this example to show that we cannot easily avoid the problem by restricting to nicer classes of C ∞ -rings with corners. Example 5.24 implies that we cannot generalize complete C ∞ -rings in §2.6 to the corners case, such that the analogue of Theorem 2.53(a) holds. But we will use the following proposition to define semi-complete C ∞ -rings with corners, which have some of the good properties of complete C ∞ -rings. Note that Example 5.24 gives an example where no choice of Dex can make the canonical map (D, Dex ) → Γc ◦ Specc (D, Dex ) surjective on the monoids. Proposition 5.25. Let (C, C ex ) be a C ∞ -ring with corners and X = Specc (C, C ex ) the corresponding C ∞ -scheme with corners. Then there is a C ∞ -ring with corners (D, Dex ) with D ∼ = Γ ◦ Spec C a complete C ∞ -ring, such that c ∼ Spec (D, Dex ) = X and the canonical map (D, Dex ) → Γc ◦ Specc (D, Dex ) is an isomorphism on D, and injective on Dex . If (C, C ex ) is firm, or interior, then (D, Dex ) is firm, or interior. Proof We define (D, Dex ) such that D = Γ ◦ Spec C = OX (X), and ex (X) generated by the invertible elements let Dex be the submonoid of OX Ψexp (D) and the image φex (C ex ). One can check that the C ∞ -operations from ex (X)) restrict to C ∞ -operations on (D, Dex ), and make (D, Dex ) (OX (X), OX ∞ into a C -ring with corners. Let Y = Specc (D, Dex ). If (C, C ex ) is firm, then (D, Dex ) is firm, as the sharpening D]ex is the image of C ]ex under φex , hence the image of the generators generates D]ex . If (C, C ex ) is interior, then (D, Dex ) is interior, as elements in both Ψexp (D) and φex (C ex ) have no zero divisors. Now the canonical morphism (φ, φex ) : (C, C ex ) → Γc ◦ Specc (C, C ex ) gives a morphism (ψ, ψex ) : (C, C ex ) → (D, Dex ) where ψ = φ, and ψex = φex with its image restricted to the submonoid Dex of OX,ex (X). As D is complete, then Spec(D) ∼ = Spec(C) ∼ = (X, OX ), and Specc (ψ, ψex ) : Y ∼ = c c ∼ Spec (D, Dex ) → Spec (C, C ex ) = X is an isomorphism on the topological space and the sheaves of C ∞ -rings. To show that Specc (ψ, ψex ) is an isomorphism on the sheaves of monoids, we show ψex induces an isomorphism on the ex ∼ ex ∼ stalks OX,x = C x,ex and OY,x = Dx,ex , for all R-points x ∈ X. The stalk map corresponds to the morphism ψx,ex : C x,ex → Dx,ex , which is defined by ψx,ex (πx,ex (c0 )) = π ˆx,ex (ψex (c0 )), where πx,ex : C ex → C x,ex and

5.5 Semi-complete C ∞ -rings with corners

123

π ˆx,ex : Dex → Dx,ex are the localization morphisms. Now, as Specc (ψ, ψex ) is an isomorphism on the sheaves of C ∞ -rings, we know that ψx : C x → Dx is an isomorphism, which implies we have an isomorphism ψx,ex |C × : x,ex × C× x,ex → Dx,ex of invertible elements in the monoids. This gives the following commutative diagram of monoids:

C ex

ψex

π

C× x,ex



 x,ex / C x,ex

ψx,ex ∼ =

/ Dex ⊂ Oex (X) X  πˆx,ex / Dx,ex o

?_ × 1 Dx,ex .

To show that ψx,ex is injective, suppose a0x , b0x ∈ C x,ex with ψx,ex (a0x ) = ψx,ex (b0x ). As πx,ex is surjective we have a0x = πx,ex (a0 ), b0x = πx,ex (b0 ) in C x,ex for a0 , b0 ∈ C ex . Then in Dx,ex we have π ˆx,ex (ψex (a0 )) = ψx,ex (πx,ex (a0 )) = ψx,ex (a0x ) = ψx,ex (b0x ) = ψx,ex (πx,ex (b0 )) = π ˆx,ex (ψex (b0 )), so Proposition 4.33 gives e0 , f 0 ∈ Dex such that e0 ψex (a0 ) = f 0 ψex (b0 ) in Dex , with Φi (e0 ) − Φi (f 0 ) ∈ I and x ◦ Φi (e0 ) 6= 0. Hence e0 |x a0x = e0 |x πx,ex (a0 ) = e0 |x ψex (a0 )|x = (e0 ψex (a0 ))|x = (f 0 ψex (b0 ))|x = f 0 |x ψex (b0 )|x = f 0 |x πx,ex (b0 ) = f 0 |x b0x

(5.6)

in C x,ex . Under the maps Φi,x : C x,ex → C x and π : C x → R we have Φi,x (e0 |x ) = Φi,x (f 0 |x ) as Φi (e0 ) − Φi (f 0 ) ∈ I, and π ◦ Φi,x (e0 |x ) = x ◦ Φi (e0 ) 6= 0. Hence e0 |x is invertible in C x,ex , so Φi,x (e0 |x ) = Φi,x (f 0 |x ) and Definition 4.16(i) imply that e0 |x = f 0 |x , and thus (5.6) shows that a0x = b0x , so ψx,ex is injective. To show ψx,ex is surjective, let d0x ∈ Dx,ex , and write d0x = π ˆx,ex (d0 ) for d0 ∈ Dex . As Dex is generated by ψex (C ex ) and Ψexp (D) we have d0 = ψex (c0 ) · e0 for c0 ∈ C and e0 ∈ Ψexp (D) invertible. Thus d0x = ψx,ex ◦πx,ex (c0 )· π ˆx,ex (e0 ), × × × 0 : C x,ex → Dx,ex is an isomorphism where π ˆx,ex (e ) ∈ Dx,ex . As ψx,ex |C × x,ex and πx,ex is surjective, there exists c00 ∈ C ex with ψx,ex ◦πx,ex (c00 ) = π ˆx,ex (e0 ). 0 00 0 Then dx = ψx,ex [πx,ex (c c )], so ψx,ex is surjective. Hence ψx,ex is an isomorphism, and Specc (ψ, ψex ) : Specc (D, Dex ) → Specc (C, C ex ) = X is an isomorphism. Thus there is an isomorphism Γc ◦ Specc (D, Dex ) ∼ = Γc ◦ Specc (C, C ex ), c which identifies the canonical map (D, Dex ) → Γ ◦ Specc (D, Dex ) with the ex inclusion (D, Dex ) ,→ (OX (X), OX (X)) = Γc ◦ Specc (C, C ex ). By definition this inclusion is an isomorphism on D, and injective on Dex . This completes the proof.

124

C ∞ -schemes with corners

Definition 5.26. We call a C ∞ -ring with corners D = (D, Dex ) semi-complete if D is complete, and ΞD : (D, Dex ) → Γc ◦ Specc (D, Dex ) is an isomorphism on D (this is equivalent to D complete) and injective on Dex . Write C∞ Ringscsc ⊂ C∞ Ringsc and C∞ Ringscsc,in ⊂ C∞ Ringscin for the full subcategories of (interior) semi-complete C ∞ -rings with corners. Given a C ∞ -ring with corners C, Proposition 5.25 constructs a semi-complete C ∞ -ring with corners D with a morphism ψ : C → D such that Specc ψ : Specc D → Specc C is an isomorphism. It is easy to show that this map ∞ c ∞ c C 7→ D is functorial, yielding a functor Πsc all : C Rings → C Ringssc c ∼ c sc mapping C 7→ D on objects, with Spec = Spec ◦Πall . Similarly we get sc,in Πsc,in : C∞ Ringscin → C∞ Ringscsc,in with Speccin ∼ = Speccin ◦Πin . in Here is a partial analogue of Theorem 2.53: Theorem 5.27. (a) Specc |··· : (C∞ Ringscsc )op → AC∞ Schc is faithful and essentially surjective, but not full. (b) Let X be an affine C ∞ -scheme with corners. Then X ∼ = Specc D, where D is a semi-complete C ∞ -ring with corners. ∞ c ∞ c (c) The functor Πsc all : C Rings → C Ringssc is left adjoint to the inclu∞ c ∞ c sion inc : C Ringssc ,→ C Rings . That is, Πsc all is a reflection functor.

(d) All small colimits exist in C∞ Ringscsc , although they may not coincide with the corresponding small colimits in C∞ Ringsc . (e) The functor Specc |··· : (C∞ Ringscsc )op → LC∞ RSc is right adjoint c ∞ c ∞ c op to Πsc all ◦ Γ : LC RS → (C Ringssc ) . Thus Spec |··· takes limits in ∞ ∞ c op (C Ringssc ) (or colimits in C Ringscsc ) to limits in LC∞ RSc . The analogues of (a)–(e) hold in the interior case, replacing C∞ Ringscsc , . . . , LC∞ RSc by C∞ Ringscsc,in , . . . , LC∞ RScin . Proof For (a), if φ : D → C is a morphism in C∞ Ringscsc , we have a commutative diagram  / Γc ◦ Specc D D ΞD φ Γc ◦Specc φ (5.7)    ΞC c c / Γ ◦ Spec C. C As the rows are injective by semi-completeness, Γc ◦ Specc φ determines φ, so Specc φ determines φ. Thus Specc |··· is injective on morphisms, that is, it is faithful. Essential surjectivity follows by Proposition 5.25. To see that Specc |··· is not full, let C be as in Example 5.24, and set D = C ∞ ([0, ∞)). Then SpeccC,D : HomC∞ Ringscsc (D, C) −→ HomAC∞ Schc (Specc C, Specc D)

5.5 Semi-complete C ∞ -rings with corners

125

may be identified with the non-surjective map ex Ξex : C ex −→ OX (X).

Part (b) is immediate from Proposition 5.25. Part (c) is easy to check, where the unit of the adjunction is ψ : C → D = inc ◦Πsc all (C), and the counit is idD : Πsc ◦inc(D) = D → D. For (d), given a small colimit in C∞ Ringscsc , all ∞ c the colimit exists in C Rings by Theorem 4.20(b), and applying the reflec∞ c tion functor Πsc all gives the (possibly different) colimit in C Ringssc . Part (e) c c holds as Πsc all , Γ are left adjoint to inc, Spec by (c) and Theorem 5.16. The extension to the interior case is immediate. Theorem 5.28. (a) Suppose C ∈ C∞ Ringsc and X = Specc C lies in LC∞ RScin ⊂ LC∞ RSc . Then X ∼ = Speccin D for some D ∈ C∞ Ringscin . (b) An X ∈ C∞ Schc lies in C∞ Schcin if and only if it lies in LC∞ RScin . Proof For (a), by Theorem 5.27(b) we may take C to be semi-complete, so ex that ΞC : (C, C ex ) → (OX (X), OX (X)) is an isomorphism on C and injective on C ex . As X lies in LC∞ RScin , the stalks O X,x are interior for x ∈ X, and writing  in ex ex OX (X) = f 0 ∈ OX (X) : f 0 |x 6= 0 in OX,x for all x ∈ X , in ∞ then O in X (X) = (OX (X), OX (X) q {0}) is an interior C -subring with ex (X)). Define E = (E, Eex ), where E = corners of O X (X) = (OX (X), OX OX (X) and  ` ex Eex = e0 ∈ OX (X) : there exists a decomposition X = i∈I Xi with

Xi open and closed in X and elements c0i ∈ C ex for i ∈ I with ΞC,ex (c0i )|Xi = e0 |Xi for all i ∈ I .

(5.8)

ex (X), We claim E, Eex are closed under the C ∞ -operations on OX (X), OX ∞ so that E is a C -subring with corners of O X (X). To see this, let g : Rm × [0, ∞)n → [0, ∞) be exterior and e1 , . . . , em ∈ E, e01 , . . . , e0n ∈ Eex . Then ej = ΞC (cj ) for unique cj ∈ C, j = 1, . . . , m as ΞC is an isomorphism. ` Also for each k = 1, . . . , n we get a decomposition X = i∈I Xi in (5.8) for e0k . By intersecting the subsets of the decompositions we can take a common ` decomposition X = i∈I Xi for all k = 1, . . . , n, and elements c0i,k ∈ C ex with ΞC,ex (c0i,k )|Xi = e0k |Xi for all i ∈ I and k = 1, . . . , n. Then e0 = ex Ψg (e1 , . . . , em , e01 , . . . , e0n ) ∈ OX (X) satisfies the conditions of (5.8) with ` the same decomposition X = i∈I Xi , and elements c0i = Ψg (c1 , . . . , cm , c0i,1 , . . . , c0i,n ), so e0 ∈ Eex , and E ∈ C∞ Ringsc .

C ∞ -schemes with corners

126

Observe that ΞC : C → O X (X) factors via the inclusion E ,→ O X (X), by taking the decomposition in (5.8) to be into one set X1 = X. in Next define D = E and Din = OX (X) ∩ Eex , so that D = (D, Din q {0}) ∞ is the intersection of the C -subrings O in X (X) and E in O X (X), and thus lies in ∞ c in C Ringsin , as O X (X) does. We now have morphisms in C∞ Ringsc C

/Eo

ΞC

inc

D,

giving morphisms in LC∞ RSc X = Specc C o

Specc ΞC

Specc inc

/ Specc D = Specc D. in (5.9) We will show that both morphisms in (5.9) are isomorphisms, so that X ∼ = Speccin D, as we have to prove. As C ∼ = D = E, equation (5.9) is isomorphic on the level of C ∞ -schemes. Thus it is sufficient to prove (5.9) induces isomorphisms on the stalks of the sheaves of monoids. That is, for each x ∈ X we must show that the following maps are isomorphisms: C x,ex

ΞC ,x,ex

Specc E

/ Ex,ex o

incx,ex

Dx,ex .

Suppose e00 ∈ Ex,ex . Then e00 = πEx,ex (e0 ) for e0 ∈ Eex . Let Xi , c0i , i ∈ I be as in (5.8) for e0 . Then x ∈ Xi for unique i ∈ I. Define a0 , b0 ∈ Eex by a0 |Xi = 1, a0 |X\Xi = 0 and b0 = a0 . Then a0 e0 = b0 ΞC,ex (c0i ), so Proposition 4.33 gives e00 = πEx,ex (e0 ) = πEx,ex ◦ ΞC,ex (c0i ) = ΞC,x,ex ◦ πC x,ex (c0i ). Thus ΞC,x,ex is surjective. ΞC,x,ex

ex ex As the composition C x,ex −→ Ex,ex → OX (X)x → OX,x is an isoc morphism since X = Spec C, we see that ΞC,x,ex is injective, and thus an isomorphism. ex Suppose 06= e00 ∈ Ex,ex . Then e00 = πEx,ex (e0 ) for e0 ∈ Eex ⊆ OX (X). 0 ex Define Y = y ∈ X : e |y 6= 0 in OX,y . Then Y is open and closed in X by Definition 5.2(a), as X is interior, and x ∈ Y as e00 6= 0. Define d0 ∈ Din by d0 |Y = e0 |Y and d0 |X\Y = 1. Define a0 , b0 ∈ Eex by a0 |Y = 1, a0 |X\Y = 0 and b0 = a0 . Then a0 d0 = b0 e0 with x ◦ Φi (a0 ) = 1, so Proposition 4.33 gives e00 = πEx,ex (e0 ) = πEx,ex ◦ inc(d0 ) = incx,ex ◦πDx,ex (d0 ), and incx,ex is surjective. Let d001 , d002 ∈ Dx,ex with incx,ex (d001 ) = incx,ex (d002 ). Then d00i = πDx,ex (d0i ) for d0i ∈ Din q {0}, i = 1, 2. By Proposition 4.33, incx,ex (d001 ) = incx,ex (d002 ) means that there exist a0 , b0 ∈ Eex with Φi (a0 ) = Φi (b0 ) near x, and x ◦ in then d01 = Φi (a0 ) 6= 0, and a0 d01 = b0 d02 in Eex . If a0 d01 |x = b0 d02 |x = 0 in OX,x 0 00 00 0 0 0 d2 = 0, so d1 =d2 = 0, as we want. Otherwise d1 , d2 6= 0, so d1 , d02 ∈ Din . ex Define Y = y ∈ X : a0 |y 6= 0 in OX,y . Then Y is open and closed

5.6 Special classes of C ∞ -schemes with corners 127  ex in X as above, and Y = y ∈ X : b0 |y 6= 0 in OX,y , since a0 d01 = b0 d02 0 0 00 00 00 0 with d1 , d2 ∈ Din . Define a , b ∈ Din by a |Y = a |Y , a00 |X\Y = d02 |X\Y , b00 |Y = b0 |Y , b00 |X\Y = d01 |X\Y . Then Φi (a00 ) = Φi (b00 ) near x as x ∈ Y and Φi (a0 ) = Φi (b0 ) near x, and x ◦ Φi (a00 ) 6= 0, and a00 d01 = b00 d02 , so Proposition 4.33 says that d001 = πDx,ex (d01 ) = πDx,ex (d02 ) = d002 . Hence incx,ex is injective, and thus an isomorphism. This proves (a). Part (b) follows from (a) by covering X with open affine C ∞ -subschemes.

5.6 Special classes of C ∞ -schemes with corners Definition 5.29. Let X be a C ∞ -scheme with corners. We call X finitely presented, or finitely generated, or firm, or integral, or torsion-free, or saturated, or toric, or simple, if we can cover X by open U ⊆ X with U ∼ = Specc C for C a C ∞ -ring with corners which is firm, integral, torsion-free, saturated, toric, or simple, respectively, in the sense of Definitions 4.40, 4.43, and 4.45. We write C∞ Schcfp ⊂ C∞ Schcfg ⊂ C∞ Schcfi ⊂ C∞ Schc for the full subcategories of finitely presented, finitely generated, and firm C ∞ schemes with corners. We write C∞ Schcsi ⊂ C∞ Schcto ⊂ C∞ Schcsa ⊂ C∞ Schctf ⊂ C∞ SchcZ ⊂ C∞ Schcin for the full subcategories of simple, toric, saturated, torsion-free, and integral objects in C∞ Schcin , respectively. We write C∞ Schcsi,ex ⊂ C∞ Schcto,ex ⊂ C∞ Schcsa,ex ⊂ C∞ Schctf ,ex ⊂ C∞ SchcZ,ex ⊂ C∞ Schcin,ex ⊂ C∞ Schc

(5.10)

for the full subcategories of simple, . . . , interior objects in C∞ Schc , but with exterior (i.e. all) rather than interior morphisms. We write C∞ Schcfi,in ⊂ C∞ Schcin ,

C∞ Schcfi,Z ⊂ C∞ SchcZ ,

C∞ Schcfi,tf ⊂ C∞ Schctf ,

C∞ Schcfi,in,ex ⊂ C∞ Schcin,ex ,

∞

C∞ Schcfi,tf ,ex ⊂ C∞ Schctf ,ex ,

C

Schcfi,Z,ex

⊂C

∞

SchcZ,ex ,

for the full subcategories of objects which are also firm. These live in a diagram of subcategories shown in Figure 5.1.

C ∞ -schemes with corners

128

/ Mangc in

Mancin

|

| / Mangc

Manc



 / C∞ Schcto

C∞ Schcsi

 |

C∞ Schcsi,ex

C∞ Schcfg

/ C∞ Schcfi,tf

/ C∞ Schcfi,Z

/ C∞ Schcfp

/ C∞ Schcfi,in

 |  | | | / C∞ Schcto,ex / C∞ Schcfi,tf ,ex / C∞ Schcfi,Z,ex / C∞ Schcfi,in,ex / C∞ Schcfi 

C∞ Schcsa

 |

C∞ Schcsa,ex

 / C∞ Schctf

 | / C∞ Schctf ,ex

 / C∞ SchcZ

 | / C∞ SchcZ,ex

 / C∞ Schcin

 | / C∞ Schcin,ex

 / C∞ Schc .

Figure 5.1 Subcategories of C ∞ -schemes with corners C∞ Schc

Remark 5.30. (i) General C ∞ -schemes with corners X ∈ C∞ Schc may have ‘corner behaviour’ which is very complicated and pathological. The subcategories of C∞ Schc in Figure 5.1 have ‘corner behaviour’ which is progressively nicer, and more like ordinary manifolds with corners, as we impose more conditions. ∞

c

C Sch (ii) It is easy to see that if X is a manifold with corners then X = FMan (X) c m ∞ ∞ c in Definition 5.21 is an object of C Schsi , since C (Rk ) is a simple C ∞ ring with corners, and similarly if X is a manifold with g-corners then X = C∞ Schc FMan (X) is an object of C∞ Schcto . gc We can think of C∞ Schcsi , C∞ Schcsi,ex , C∞ Schcto , C∞ Schcto,ex as good gc analogues in the world of C ∞ -schemes of Mancin , Manc , Mangc in , Man .

(iii) For a C ∞ -scheme with corners X to be firm basically means that locally X has only finitely many boundary faces. This is a mild condition, which will hold automatically in most interesting applications. An example of a non-firm Q I C ∞ -scheme with corners is [0, ∞) := i∈I [0, ∞) for I an infinite set. We will need to restrict to firm C ∞ -schemes with corners to define fibre products in §5.7, and for parts of the theory of corner functors in Chapter 6. Parts (b),(c) of the next theorem are similar to Theorem 5.28(b). Theorem 5.31. (a) The functor ΠZin : LC∞ RScin → LC∞ RScZ of Theorem 5.8 maps C∞ Schcin → C∞ SchcZ . Restricting gives a functor ΠZin : C∞ Schcin → C∞ SchcZ right adjoint to the inclusion inc : C∞ SchcZ ,→ C∞ Schcin . The analogues also hold for inc : C∞ Schcsa ,→ LC∞ RScsa and

5.6 Special classes of C ∞ -schemes with corners

129

sa sa inc : C∞ Schctf ,→ LC∞ RSctf and functors Πtf Z , Πtf , Πin , giving a diagram

p C∞ Schcsa o

Πsa in Πsa tf inc

/ C∞ Schctf o

Πtf Z inc

/ C∞ SchcZ o

ΠZin inc

/ C∞ Schcin .

(b) An object X in C∞ Schcin lies in C∞ Schcsa , C∞ Schctf or C∞ SchcZ if and only if it lies in LC∞ RScsa , LC∞ RSctf or LC∞ RScZ , respectively. (c) Let X be an object in C∞ Schcfi . Then the following are equivalent: (i) X lies in C∞ Schcfi,in . (ii) X lies in LC∞ RScin . (iii) X may be covered by open U ⊆ X with U ∼ = Speccin C for C a firm, interior C ∞ -ring with corners. Also X lies in C∞ Schcto if and only if it lies in LC∞ RScsa . Proof For (a), consider the diagram of functors

ΓcZ

 (C∞ RingscZ )op o O Specc

Z   LC∞ RScZ o

inc ΠZin inc ΠZin

/ (C∞ Ringsc )op in O Γc

Speccin

in  / LC∞ RSc . in

(5.11)

Here SpeccZ , ΓcZ are the restrictions of Speccin , Γcin to (C∞ RingscZ )op and LC∞ RScZ , and do map to LC∞ RScZ , (C∞ RingscZ )op . The functors ΠZin are from Theorems 4.48 and 5.8, and are right adjoint to the inclusions inc, noting that taking opposite categories converts left to right adjoints. Also Speccin is right adjoint to Γcin by Theorem 5.18, which implies that SpeccZ is right adjoint to ΓcZ . Clearly inc ◦ΓcZ = Γcin ◦ inc. As SpeccZ ◦ΠZin , ΠZin ◦ Speccin are the right adjoints of these we have a natural isomorphism SpeccZ ◦ΠZin ∼ = ΠZin ◦ Speccin . c ∞ c Z ∼ Thus, for any C ∈ C Ringsin we have SpecZ ◦Πin (C) = ΠZin ◦ Speccin C in LC∞ RScZ . Let X ∈ C∞ Schcin . Then X is covered by open U ⊂ X with U ∼ = c ∞ c Z Z Specin C for C ∈ C Ringsin . Hence Πin (X) is covered by open Πin (U ) ⊂ ΠZin (X) with ΠZin (U ) ∼ = ΠZin ◦ Speccin C ∼ = SpeccZ ◦ΠZin (C) for ΠZin (C) ∈ ∞ c Z ∞ c C RingsZ . Thus Πin (X) ∈ C SchZ , and ΠZin : C∞ Schcin → C∞ SchcZ . That the restriction ΠZin : C∞ Schcin → C∞ SchcZ is right adjoint to inc follows from Theorem 5.8. This proves the first part of (a). The rest of (a) follows by the same argument. For (b), the ‘only if’ part is obvious. For the ‘if’ part, suppose X lies in

130

C ∞ -schemes with corners

∞ c sa C∞ Schcin and LC∞ RScsa . Then Πsa in (X) lies in C Schsa by (a). But Πin ∞ c ∞ c sa ∼ is isomorphic to the identity on LC RSsa ⊂ LC RSin , so Πin (X) = X, and X lies in C∞ Schcsa . The arguments for C∞ Schctf , C∞ SchcZ are the same. For (c), clearly (iii) ⇒ (i) ⇒ (ii), so it is enough to show that (ii) ⇒ (iii). Suppose X lies in C∞ Schcfi and LC∞ RScin , and x ∈ X. Then x has an open neighbourhood V ⊂ X with V ∼ = Specc D for D firm. Let d01 , . . . , d0n ∈ Dex 0 0 such that [d1 ], . . . , [dn ] are generators for D]ex . Choose an open neighbourhood U of x in V such that for each i = 1, . . . , n, either πu,ex (d0i ) = 0 in Du,ex ∼ = ex ex OX,x for all u ∈ U , or πe,ex (d0i ) 6= 0 in Du,ex ∼ for all u ∈ U . As the = OX,x topology of V ∼ = Specc D is generated by open sets Ud as in Definition 2.42, making U smaller if necessary we can suppose that U = Ud for d ∈ D. Then Lemma 5.15 says that U ∼ = Specc (D(d−1 )). Define C to be the semi-complete C ∞ -ring with corners constructed from D(d−1 ) in Proposition 5.25, so we have a morphism ψ : D(d−1 ) → C with Specc ψ an isomorphism, implying that U ∼ = Specc C. We will show C is firm and interior. Write d001 , . . . , d00n for the images of d01 , . . . , d0n in D(d−1 )ex . Since [d01 ], . . . , [d0n ] generate D]ex , [d001 ], . . . , [d00n ] generate D(d−1 )]ex . As by construction C ex is generated by ψex (D(d−1 )ex ) and invertibles Ψexp (C), so [ψex (d001 )], . . . , [ψex (d00n )] generate C ]ex , and C is firm. ex By definition the natural map Ξex : C ex → OX (U ) is injective. For each 00 ex i = 1, . . . , n, either Ξex ◦ ψex (di )|u = 0 in OX,u for all u ∈ U , so that ex ψex (d00i ) = 0, or Ξex ◦ ψex (d00i )|u = 0 in OX,u for all u ∈ U , so that Ξex ◦ ex 00 ex are interior, and thus ψex (di ) is not a zero divisor in OX (U ), as the OX,u 00 00 ψex (di ) is not a zero divisor in C ex . Since [ψex (d1 )], . . . , [ψex (d00n )] generate C ]ex , and each ψex (d00i ) is either zero or not a zero divisor, we see that C ex has no zero divisors. Also 0C ex 6= 1C ex as U 6= ∅. So C is interior. Thus (ii) implies (iii), and the proof is complete.

Let f : X → Y be a morphism in C∞ Schc , and x ∈ X with f (x) = y in Y . Then by definition, X, Y are locally isomorphic to Specc C, Specc D near x, y for C ∞ -rings with corners C, D. However, it is not clear that f must be locally isomorphic to Specc φ for some morphism φ : D → C in C∞ Ringsc , and the authors expect this is false in some cases. This is because Specc : (C∞ Ringsc )op → AC∞ Schc is not full, as in Theorem 5.27(a), so morphisms Specc C → Specc D in C∞ Schc need not be of the form Specc φ. We now show that if Y is firm, then f is locally isomorphic to Specc φ. We use this in Theorem 5.33 to give a criterion for existence of fibre products in C∞ Schc . Proposition 5.32. Suppose f : X → Y is a morphism in C∞ Schc with Y

5.6 Special classes of C ∞ -schemes with corners

131

firm. Then for all x ∈ X with f (x) = y ∈ Y , there exists a commutative diagram in C∞ Schc : / U  /X Specc C ∼ = Specc φ

 Specc D

f |U

∼ =

  / V 

f



(5.12)

/Y.

Here U ⊆ X, V ⊆ Y are affine open neighbourhoods of x, y with U ⊆ f −1 (V ), and φ : D → C is a morphism in C∞ Ringscsc with D firm. The analogue also holds with C∞ Schc , C∞ Ringscsc replaced by C∞ Schcin and C∞ Ringscsc,in . Proof As Y is a firm C ∞ -scheme with corners there exists an open neighbourhood V of y in Y and an isomorphism V ∼ = Specc D for D a firm C ∞ ring with corners. By Proposition 5.25 we may take D to be semi-complete. As X is a C ∞ -scheme with corners there exists an affine  open neighbour c ˙ ˙ hood U˙ of x in X with U˙ ∼ Spec C. Since subsets U = x ∈ U˙ : x(c) 6= 0 = c for c ∈ C˙ are a basis for the topology of U˙ by Definition 2.43, there exists c ∈ C˙ ¨ := U˙ c an open neighbourhood of x in U˙ ∩ f −1 (V ). Set C ¨ = C(c ˙ −1 ). with U −1 c ¨ is an open neighbourhood of x in f (V ) ⊆ X by ¨ ∼ Then U = Spec C Lemma 5.15. As D is firm, the sharpening D]ex is finitely generated. Choose d01 , . . . , d0n 0 c c ∼ in Dex whose images generate D]ex . Then Ξex D (di ) lies in (Γ ◦ Spec D)ex = ex 0 c c ¨ ] ¨ ex OY (V ) for i = 1, . . . , n, so fex (U ) ◦ ΞD (di ) lies in (Γ ◦ Spec C)ex ∼ = ex ¨ ex ex OX (U ) for i = 1, . . . , n. By definition of the sheaf OX , any section of OX is ex 0 0 ¨ locally modelled on ΞC ¨ (c ) for some c ∈ Cex . Thus, we may choose an open ¨ ex such that f ] (U ¨) ◦ ¨ and elements c0 , . . . , c0n ∈ C neighbourhood U of x in U ex 1 ex 0 ex 0 ΞD (di )|U = ΞC ¨ (ci )|U for i = 1, . . . , n. Making U smaller if necessary, as ¨ Set C = Πsc (C(c ¨c for c ∈ C. ¨ −1 )). above we may take U of the form U all Then U ∼ = Specc C by Lemma 5.15 and Definition 5.26, for U an affine open neighbourhood of x in X with U ⊆ f −1 (V ), and C a semi-complete C ∞ ring. We will show there is a unique morphism φ = (φ, φex ) : D → C such that (5.12) commutes. To see this, note that there is a unique C ∞ -ring morphism φ : C → D making the C ∞ -scheme part of (5.12) commute by Theorem 2.53(a), as C, D are complete. Let d0 ∈ Dex . Since d01 , . . . , d0n generate D]ex we may write d0 = Ψexp (d)(d01 )a1 · · · (d0n )an for d ∈ D and a1 , . . . , an > 0. We claim that defining φex (d0 ) = Ψexp (φ(d))(c01 )a1 · · · (c0n )an is independent of the presentation d0 = Ψexp (d)(d01 )a1 · · · (d0n )an , and yields a monoid morphism φex : Dex → C ex with all the properties we need. To see this, note

132

C ∞ -schemes with corners

that
0 an 0 ex 0 a1 Ξex C (φex (d )) = ΞC Ψexp (φ(d))(c1 ) · · · (cn )
ex 0
a1 0 an = Ξex · · · (Ξex C ◦ Ψexp ◦ φ(d) ΞC (c1 ) C (cn ))
]
a 1 0 ] 0 an = Ψexp ΞC ◦ φ(d) fex (U ) ◦ Ξex · · · (fex (U ) ◦ Ξex D (d1 ) D (dn ))

a 1 ] 0 ] 0 an = Ψexp f ] (U )◦ΞD (d) fex (U )◦Ξex · · · (fex (U )◦Ξex D (d1 ) D (dn ))
0 an ] ex 0 ] ex 0 a1 = fex (U ) ◦ ΞD (d ). = fex (U ) ◦ ΞD Ψexp (d)(d1 ) · · · (dn ) 0 ex Thus Ξex C (φex (d )) is independent of the presentation, and ΞC is injective as C is semi-complete, so φex (d0 ) is well defined. By identifying C, D with C ∞ subrings with corners of O X (U ), O Y (V ), and φ = (φ, φex ) with the restriction of f ] (U ) : O Y (V ) → O X (U ) to these C ∞ -subrings, in a similar way to (5.7), we see that φ is a morphism in C∞ Ringsc , proving the first part. The same arguments work in the interior case.

5.7 Fibre products of C ∞ -schemes with corners We use Proposition 5.32 to construct fibre products of C ∞ -schemes with corners. Theorem 5.33. Suppose g : X → Z, h : Y → Z are morphisms in C∞ Schc , and Z is firm. Then X ×g,Z,h Y exists in C∞ Schc , and is equal to the fibre product in LC∞ RSc . Similarly, if g : X → Z, h : Y → Z are morphisms in C∞ Schcin , and Z is firm, then the fibre product X ×g,Z,h Y exists in C∞ Schcin , and is equal to the fibre product in LC∞ RScin . The inclusion C∞ Schcin ,→ C∞ Schc preserves fibre products. Proof Let g : X → Z, h : Y → Z be morphisms in C∞ Schc ⊂ LC∞ RSc . By Theorem 5.5 there exists a fibre product W = X ×g,Z,h Y in LC∞ RSc , in a Cartesian square in LC∞ RSc : W

f

e

 X

/Y h

g

 / Z.

(5.13)

We will prove that W is a C ∞ -scheme with corners. Then (5.13) is Cartesian in C∞ Schc ⊂ LC∞ RSc , so W is a fibre product X ×g,Z,h Y in C∞ Schc . Let w ∈ W with e(w) = x ∈ X, f (w) = y ∈ Y and g(x) = h(y) = z ∈ Z. As Z is firm we may pick an affine open neighbourhood V of z in Z with V ∼ = Specc F for a firm C ∞ -ring with corners F. By Proposition 5.25 we may take F to be semi-complete.

5.7 Fibre products of C ∞ -schemes with corners

133

By Proposition 5.32 we may choose affine open neighbourhoods T of x in g −1 (V ) ⊂ X and U of y in h−1 (V ) ⊂ Y , and semi-complete C ∞ -rings with corners D, E with isomorphisms T ∼ = Specc E, and mor= Specc D, U ∼ ∞ c phisms φ : F → D, ψ : F → E in C Ringssc such that Specc φ, Specc ψ are identified with g|T : T → V , h|U : U → V by the isomorphisms T ∼ = Specc D, U ∼ = Specc E and V ∼ = Specc F. −1 −1 Set S = e (T ) ∩ f (U ). Then S is an open neighbourhood of w in W . Write C = D qφ,F,ψ E for the pushout in C∞ Ringscsc , which exists by Theorem 5.27(d), with projections η : D → C, θ : E → C. Consider the diagrams S  T

f |S

/U h|U

e|S g|T

 /V,

Specc C

Specc θ

Specc ψ

Specc η

 Specc D

/ Specc E

Specc φ

 / Specc F,

in LC∞ RSc . The left hand square is Cartesian as (5.13) is, by local properties of fibre products. The right hand square is Cartesian by Theorem 5.27(e), as C = D qφ,F,ψ E. The isomorphisms T ∼ = Specc D, U ∼ = Specc E, V ∼ = c Spec F identify the right and bottom sides of both squares. Hence the two squares are isomorphic. Thus S ∼ = Specc C. So each point w in W has an affine open neighbourhood, and W is a C ∞ -scheme with corners, as we have to prove. The proof for the interior case is essentially the same. Theorem 5.31(c) allows us to write V ∼ = Speccin F for F ∈ C∞ Schcfi,in , and Theorem 5.27 and Propositions 5.25 and 5.32 also work in the interior case. The last part then follows from Theorem 5.5. Theorem 5.34. Consider the family of full subcategories: C∞ Schcfp ⊂ LC∞ RSc ,

C∞ Schcfg ⊂ LC∞ RSc ,

C∞ Schcfi ⊂ LC∞ RSc ,

C∞ Schcfi,in ⊂ LC∞ RScin ,

C∞ Schcfi,Z ⊂ LC∞ RScZ ,

C∞ Schcfi,tf ⊂ LC∞ RSctf ,

(5.14)

C∞ Schcto ⊂ LC∞ RScsa . In each case, the subcategory is closed under finite limits in the larger category. Hence, fibre products and all finite limits exist in C∞ Schcfp , . . . , C∞ Schcto . Proof In the proof of Theorem 5.33, if X, Y , Z are finitely presented we may take D, E, F to be finitely presented, and then C is finitely presented by Proposition 4.42, so W = X ×g,Z,h Y is finitely presented. Hence all fibre products exist in C∞ Schcfp , and agree with fibre products in LC∞ RSc . As C∞ Schcfp has a final object Specc (R, [0, ∞)), and all fibre products in

C ∞ -schemes with corners

134

a category with a final object are (iterated) fibre products, then C∞ Schcfp is closed under finite limits in LC∞ RSc , and all such finite limits exist in C∞ Schcfp by Theorem 5.5. The same argument works for C∞ Schcfg , C∞ Schcfi , C∞ Schcfi,in , using Proposition 4.44 for the latter two. For C∞ Schcfi,Z , C∞ Schcfi,tf , C∞ Schcto we apply the coreflection functors Z sa Πin , Πtf Z , Πtf of Theorem 5.8, noting that the restrictions of these map C∞ Schcfi,in

ΠZin

/ C∞ Schc fi,Z

Πtf Z

/ C∞ Schc

fi,tf

Πsa tf

/ C∞ Schcto ,

by Theorem 5.31 and the fact (clear from the proof of Theorem 5.31) that sa ΠZin , Πtf Z , Πtf preserve firmness. Let J be a finite category and F : J → C∞ Schcfi,Z a functor. Then a limit X = lim F exists in C∞ Schcfi,in from above, which is also a limit in ←− LC∞ RScin . Hence ΠZin (X) is a limit lim ΠZin ◦ F in LC∞ RScZ , since ΠZin : ←− LC∞ RScin → LC∞ RScZ preserves limits as it is a right adjoint. But ΠZin acts as the identity on LC∞ RScZ ⊂ LC∞ RScin , so ΠZin (X) is a limit lim F ←− in LC∞ RScZ . As X ∈ C∞ Schcfi,in we have ΠZin (X) ∈ C∞ Schcfi,Z , so C∞ Schcfi,Z is closed under finite limits in LC∞ RScZ , and all finite limits exist in C∞ Schcfi,Z by Corollary 5.10. The arguments for C∞ Schcfi,tf , C∞ Schcto are the same. We will return to the subject of fibre products in §6.6–§6.7, using ‘corner functors’ from Chapter 6 as a tool to study them.

6 Boundaries, corners, and the corner functor

In §3.4, for manifolds with (g-)corners of mixed dimension, we defined corner ˇ c → Man ˇ c and C : Man ˇ gc → Man ˇ gc , which are functors C : Man in in c c ˇ ˇ ˇ gc ,→ right adjoint to the inclusions inc : Manin ,→ Man and inc : Man in gc ˇ Man . These encode the boundary ∂X and k-corners Ck (X) of a manifold with (g-)corners X in a functorial way. This chapter will show that analogous corner functors C : LC∞ RSc → LC∞ RScin and C : C∞ Schc → C∞ Schcin may be defined for local C ∞ ringed spaces and C ∞ -schemes with corners X, which are right adjoint to the inclusions inc : LC∞ RScin ,→ LC∞ RSc and inc : C∞ Schcin ,→ C∞ Schc . This allows us to define the boundary ∂X and k-corners Ck (X) for firm C ∞ -schemes with corners X. We use corner functors to study existence of fibre products in categories such as C∞ Schcto,ex .

6.1 The corner functor for C ∞ -ringed spaces with corners Definition 6.1. Let X = (X, O X ) be a local C ∞ -ringed space with corners. We will define the corners C(X) in LC∞ RScin . As a set, we define  ex C(X) = (x, P ) : x ∈ X, P is a prime ideal in OX,x , (6.1) where prime ideals in monoids are as in Definition 3.5. Define ΠX : C(X) → X as a map of sets by ΠX : (x, P ) 7→ x. We define a topology on C(X) to be the weakest topology such that ΠX is continuous, so Π−1 X (U ) is open for all U ⊂ X, and such that for all open ex U ⊂ X, for all elements s0 ∈ OX (U ), then  −1 ¨ ˙ U˙ s0 = (x, P ) : x ∈ U, s0x ∈ / P ⊆ Π−1 (6.2) X (U ) and Us0 = ΠX (U )\ Us0 −1 ˙ ¨ ¨ are both open and closed in Π−1 X (U ). Then ΠX (U ) = U1 = U0 , and ∅ = U1 =

135

136

Boundaries, corners, and the corner functor  ex ¨s0 : open U ⊂ X, s0 ∈ Oex (U ) U˙ 0 for 0, 1 ∈ OX (U ). The collection U˙ s0 , U X is a subbase for the topology. −1 ex ex Pull back the sheaves OX , OX by ΠX to get sheaves Π−1 X (OX ), ΠX (OX ) ex on C(X). For each x ∈ X and each prime P in OX,x , we have ideals ex ∼ ex P ⊂ OX,x = (Π−1 X (OX ))(x,P ) ,

hΦi (P )i ⊂ OX,x ∼ = (Π−1 X (OX ))(x,P ) .

ex Here hΦi (P )i is the ideal generated by the image of P under Φi : OX,x → OX,x . We define the sheaf of C ∞ -rings OC(X) on C(X) to be the sheafification of the presheaf of C ∞ -rings Π−1 X (OX )/I on C(X), where for open U ⊂ C(X)

 I(U ) = s ∈ Π−1 X (OX )(U ) : s(x,P ) ∈ hΦi (P )i for all (x, P ) ∈ U . ex ex Let OC(X) be the sheafification of the presheaf U 7→ Π−1 X (OX )(U )/∼ where the equivalence relation ∼ is generated by the relations that s01 ∼ s02 if for each (x, P ) ∈ U either s01,x , s02,x ∈ P , or there is p ∈ hΦi (P )i such that ex s01,x = Ψexp (p)s02,x , for s01 , s02 ∈ Π−1 X (OX )(U ). This is similar to quotienting −1 ∞ ex the C -ring with corners (ΠX (OX )(U ), Π−1 X (OX )(U )) by a prime ideal in −1 ex ΠX (OX )(U ), which we described in Example 4.25(b), and creates a sheaf of ex C ∞ -rings with corners O C(X) = (OC(X) , OC(X) ) on C(X). Lemma 6.2 shows C(X) = (C(X), O C(X) ) is an interior local C ∞ -ringed space with corners, and the stalk of O C(X) at (x, P ) is an interior local C ∞ ring with corners isomorphic to O X,x /∼P , in the notation of Example 4.25(b). ] −1 ex Define sheaf morphisms Π]X : Π−1 X (OX ) → OC(X) and ΠX,ex : ΠX (OX ) −1 −1 ex as the sheafifications of the projections ΠX (OX ) → ΠX (OX )/I → OC(X) ] ] ] −1 −1 ex ex and Π−1 X (OX ) → ΠX (OX )/∼. Then ΠX = (ΠX , ΠX,ex ) : ΠX (O X ) → O C(X) is a morphism of sheaves of C ∞ -rings with corners, and ΠX = (ΠX , Π]X ) is a morphism ΠX : C(X) → X in LC∞ RSc . ] Let f = (f, f ] , fex ) : X → Y be a morphism in LC∞ RSc . We will define a morphism C(f ) : C(X) → C(Y ). On topological spaces, we define C(f ) : ex ] ] : OY,f )−1 (P )) where fex,x C(X) → C(Y ) by (x, P ) 7→ (f (x), (fex,x (x) → ex ] ex OX,x is the stalk map of fex . This is continuous as if t ∈ OY (U ) for some open U ⊂ Y then C(f )−1 (U˙ t ) = (f −1˙(U ))f ex (U )(t) is open, and similarly ] ¨t ) = (f −1¨(U )) ex C(f )−1 (U is open. Also ΠY ◦ C(f ) = f ◦ ΠX . f] (U )(t)

To define the sheaf morphism C(f )] : C(f )−1 (O C(Y ) ) → O C(X) , con-

6.1 The corner functor for C ∞ -ringed spaces with corners

137

sider the diagram of presheaves on C(X): −1 −1 C(f )−1 ◦ Π−1 (OY ) Y (OY ) = ΠX ◦ f

 C(f )−1 (Π−1 Y (OY )/IY )

] Π−1 X (f )∗

sheafify

 C(f )−1 (OC(Y ) )

] Π−1 X (f )

/ Π−1 (OX ) X  / Π−1 (OX )/IX X sheafify

C(f )]

 / OC(X) .

] Here there is a unique morphism Π−1 X (f )∗ making the upper rectangle com−1 ] −1 mute as ΠX (f ) maps C(f ) (IY ) → IX by definition of IY , IX , C(f ). Hence there is a unique morphism C(f )] making the lower rectangle commute by properties of sheafification. ex ] Similarly we define C(f )]ex : C(f )−1 (OC(Y ) ) → OC(X) , and C(f ) = ] ] −1 (C(f ) , C(f )ex ) : C(f ) (O C(Y ) ) → O C(X) is a morphism of sheaves of C ∞ -rings with corners on C(X), so C(f ) = (C(f ), C(f )] ) : C(X) → C(Y ) is a morphism in LC∞ RSc . We see that ΠY ◦ C(f ) = f ◦ ΠX . On the stalks at (x, P ) in C(X), we have

C(f )](x,P ) = (f ]x )∗ : O Y,f (x) /∼(fex,x ] )−1 (P ) −→ O X,x / ∼P . ex For the monoid sheaf, if s0 7→ 0 in the stalk, then s0 = [s00 ] for s00 ∈ OY,f (x) , ] −1 00 00 ] (P ), so s00 ∼(fex,x and fex,(x,P ] ) (s )x ∈ P . Then sf (x) ∈ (fex,x ) )−1 (P ) 0, 0 giving s = 0. Therefore C(f ) is an interior morphism. One can check that C(f ◦ g) = C(f ) ◦ C(g) and C(idX ) = idC(X) , and thus C : LC∞ RSc → LC∞ RScin is a well defined functor. If we now assume X is interior, then {(x, {0}) : x ∈ X} is contained in C(X), and there is an inclusion of sets ιX : X ,→ C(X), ιX : x 7→ (x, {0}). The image of X under ιX : X ,→ C(X) is closed in C(X), since for all open U ⊂ X, \ ιX (U ) = U˙ s0 in (U ) s0 ∈OX

is closed in Π−1 X (U ). Also, ιX is continuous, as the definition of interior  implies ¨ s0 ) = x ∈ U : ˙ s0 ) = x ∈ U : s0x 6= 0 ∈ Oex is open, and ι−1 (U ι−1 ( U X,x X X ex ex (U ). is open for all s0 ∈ OX s0x = 0 in OX,x −1 Now any s ∈ OX (U ) gives equivalence classes in Π−1 X (OX )(ΠX (U )), −1 −1 and Π−1 (U )), and ι−1 X (OX )(ΠX (U ))/∼, and OC(X) (Π X (OC(X) )(U ). So −1 ex there is a map OX (U ) → ιX (OC(X) )(U ), and a similar map OX (U ) → −1 ex ιX (OC(X) )(U ). These respect restriction and give morphisms of sheaves of C ∞ -rings with corners. The stalks of ι−1 X (O C(X) ) are isomorphic to the stalks of O C(X) , which are isomorphic to O X,x /∼{0} ∼ = O X,x by Lemma 6.2.

138

Boundaries, corners, and the corner functor

So there is a canonical isomorphism ι]X : ι−1 X (O C(X) ) → O X , and ιX = (ιX , ι]X ) : X → C(X) is a morphism, which is interior, as the stalk maps are isomorphisms. Thus, for each interior X we have defined a morphism ιX : X → C(X) in LC∞ RScin , which is an inclusion of X as an open and closed C ∞ -subscheme with corners in C(X). One can show that ΠX ◦ ιX = idX . Also, if f : X → Y is a morphism in LC∞ RScin one can show that ιY ◦ f = C(f ) ◦ ιX . Lemma 6.2. In Definition 6.1, C(X) = (C(X), O C(X) ) is an interior local C ∞ -ringed space with corners in the sense of Definition 5.2. The stalk O C(X),(x,P ) of O C(X) at (x, P ) in C(X) is an interior local C ∞ -ring with corners isomorphic to O X,x /∼P , in the notation of Example 4.25(b). Proof We first verify the openness conditions for C(X) to be an interior C ∞ ringed space with corners in Definition 5.2(a),(b). Let U ⊂ C(X) be open and ex s0 ∈ OC(X) (U ), and as in Definition 5.2(a),(b) write  ex ˆs0 = U \ Us0 . Us0 = (x, P ) ∈ U : s0(x,P ) 6= 0 ∈ OC(X),(x,P and U ) Then there is an open cover {Ui }i∈I of U such that s0 |Ui = [s0i ] for s0i ∈ ex )(Ui ), where s0i |(x,P ) ∈ / P if (x, P ) ∈ Us0 ∩ Ui and s0i |(x,P ) ∈ P Π−1 (OX ˆi . The definition of the inverse image sheaf implies there if (x, P ) ∈ Us0 ∩ U is an open cover {Wji }j∈Ji of Ui such that s0i |Wi,j = [s0i,j ] for some s0i,j ∈ ex (Vi,j ) for open Vi,j ⊂ X such that Vi,j ⊃ Π(Wi,j ). Then s0i,j |x ∈ / P for OX 0 ˆ all (x, P ) ∈ Us0 ∩ Wi,j , and si,j |x ∈ P for all (x, P ) ∈ Us0 ∩ Wi,j . Thus ˙ )0 Us0 ∩ Wi,j = Wi,j ∩ (Vi,j s

i,j

ˆs0 ∩ Wi,j = Wi,j ∩ (V¨i,j ) 0 U s

i,j

are both open in C(X) by the definition of the topology on C(X) involv¨s0 in (6.2). Taking the union over i ∈ I, j ∈ Ji , and using that ing U˙ s0 , U S 0 ˆ 0 i,j Wi,j = U , we see that Us , Us are open in U , as we have to prove. ex The stalks of OC(X) , OC(X) at (x, P ) ∈ C(X) are OX,x /hΦi (P )i and ex ex /∼P , where s01 ∼P s02 in OX,x if s01 , s02 ∈ P or there is a p ∈ hΦi (P )i OX,x 0 0 such that Ψexp (p)s1 = s2 , as in Example 4.25(b). To see this, consider that the definitions give us the following diagram, where we will show the arrow t exists and is an isomorphism O X,x 

O X,x /∼P

∼ ˆ] Π x

/ Π−1 (O X )(x,P ) ∼ = t

/ / O C(X),(x,P ) . 55 (6.3)

To see that t exists, we use the universal property of O X,x /∼P as in Example

6.1 The corner functor for C ∞ -ringed spaces with corners

139

ex 4.25(b). Let U be an open set in X and take s0 ∈ OX (U ) such that s0x ∈ P . 0 ex −1 Then s maps to an equivalence class in OC(X) (Π (U )). Consider the open  ¨s0 = (x, P ) ∈ C(X) : x ∈ U , s0x ∈ P , then restricting to this set U ¨s0 ⊂ Π−1 (U ) and so we can restrict open set, we see that our (x, P ) is in U 0 ¨s0 . In this open set however, s0x ∈ P for all the equivalence class of s to U ¨s0 so s0 ∼ 0, and s0 lies in the kernel of the composition of the top (x, P ) ∈ U

row of (6.3). Then the universal property of O X,x /∼P says that t must exist and commute with the diagram. Also, t must be surjective as the top line is surjective. To see that t is injective is straightforward. For example, in the monoid ex ex case, if [s01,x ], [s02,x ] ∈ OX,x /∼P with representatives s01 , s02 ∈ OX (U ), and 0 0 0 0 if tex ([s1,x ]) = tex ([s2,x ]) then s1,x ∼ s2,x for all (x, P ) ∈ V for some open V ⊂ Π−1 (U ). This means at every (x, P ) ∈ V then s01,x ∼P s02,x , so this must ex be true at our (x, P ), so that [s01,x ] = [s02,x ] ∈ OX,x /∼P as required. ex Now O X,x /∼P is interior as the complement of P ∈ OX,x has no zero divisors. It is also local, as the unique morphism OX,x → R must have P ex in its kernel so it factors through the morphism OX,x → OX,x /∼P giving a ex unique morphism OX,x /∼P → R with the correct properties to be local. Thus O C(X),(x,P ) is an interior local C ∞ -ring with corners for all (x, P ) ∈ C(X), so C(X) is an interior local C ∞ -ringed space with corners by Definition 5.2. Theorem 6.3. The corner functor C : LC∞ RSc → LC∞ RScin is right adjoint to the inclusion inc : LC∞ RScin ,→ LC∞ RSc . Thus we have functorial isomorphisms HomLC∞ RScin (C(X), Y ) ∼ = HomLC∞ RSc (X, inc(Y )). As C is a right adjoint, it preserves limits in LC∞ RSc , LC∞ RScin . Proof We describe the unit η : Id ⇒ C ◦ inc and counit  : inc ◦C ⇒ Id of the adjunction. Here X = ΠX for X a C ∞ -scheme with corners, and η X = ιX for X an interior C ∞ -scheme with corners. That  is a natural transformation follows from ΠY ◦C(f ) = f ◦ΠX for morphisms f : X → Y in LC∞ RSc , and that η is a natural transformation follows from ιY ◦ f = C(f ) ◦ ιX for morphisms f : X → Y in LC∞ RScin . Finally, to prove the adjunction, we must show the following compositions are the identity natural transformations: C

η∗idC

+3 C ◦ inc ◦C

idC ∗

+3 C,

inc

idinc ∗η

+3 inc ◦C ◦ inc

η∗idinc

+3 inc .

Both of these follow as ΠX ◦ ιX = idX . Definition 5.7 defined subcategories LC∞ RScsa ⊂ · · · ⊂ LC∞ RScin and LC∞ RScsa,ex ⊂ · · · ⊂ LC∞ RScin,ex ⊂ LC∞ RSc , where the objects X

140

Boundaries, corners, and the corner functor

have stalks O X,x which are saturated, torsion-free, integral, or interior. From the definition of C(X) in Definition 6.1, we see that if the stalks O X,x are saturated, . . . , interior then the stalks O C(X),(x,P ) ∼ = O X,x /∼P are also saturated, . . . , interior, as these properties are preserved by quotients by prime ideals, so C(X) is saturated, . . . , interior, respectively. Hence from Theorem 6.3 we deduce: Theorem 6.4. Restricting C in Theorem 6.3 gives corner functors C : LC∞ RScsa,ex −→ LC∞ RScsa , C : LC∞ RSctf ,ex −→ LC∞ RSctf , C : LC∞ RScZ,ex −→ LC∞ RScZ ,

C : LC∞ RScin,ex −→ LC∞ RScin ,

which are right adjoint to the corresponding inclusions.

6.2 The corner functor for C ∞ -schemes with corners Motivated by Theorem 6.3, na¨ıvely we might expect that there exists C˜ : C∞ Ringsc → C∞ Ringscin left adjoint to C∞ Ringscin ,→ C∞ Ringsc , fitting into a diagram of adjoint functors similar to (5.11):

Γcin

 (C∞ Ringscin )op o O Specc

in   ∞ LC RScin o

inc ˜ C inc C

/ (C∞ Ringsc )op O /

Γc ∞

c



Specc

(6.4)

LC RS .

As in the proof of Theorem 5.31(a) this would imply Speccin ◦C˜ ∼ = C ◦ Specc , ∞ c ∞ c allowing us to deduce that C maps C Sch → C Schin . However, no such left adjoint C˜ for inc exists by Theorem 4.20(d). Surprisingly, there is a ‘corner functor’ C˜ : C∞ Ringsc → C∞ Ringscsc such that Specc ◦C˜ maps to LC∞ RScin ⊂ LC∞ RSc , with Specc ◦C˜ ∼ = C ◦ Specc , as we prove in the next definition and theorem. Definition 6.5. Let C = (C, C ex ) be a C ∞ -ring with corners. Using the nota˜ by tion of Definition 4.24, define a C ∞ -ring with corners C  
˜ = C αc0 , βc0 : c0 ∈ C ex C Φi (αc0 ) + Φi (βc0 ) = 1 ∀c0 ∈ C ex (6.5)   0 00 0 0 α1C ex = 1C ex , αc0 αc00 = αc0 c00 ∀c , c ∈ C ex , c βc0 = αc0 βc0 = 0 ∀c ∈ C ex . That is, we add two [0, ∞)-type generators αc0 , βc0 to C for each c0 ∈ C ex , and impose R-type relations (· · · ) and [0, ∞)-type relations [· · · ] as shown. ˜ D ˜ from Let φ : C → D be a morphism in C∞ Ringsc , and define C,

6.2 The corner functor for C ∞ -schemes with corners

141

C, D as above, writing the additional generators in Dex as α ¯ d0 , β¯d0 . Define a ˜ ˜ ˜ morphism φ : C → D by the commutative diagram   C αc0 , βc0 : c0 ∈ C ex φ[α 0 → 7 α ¯

, β 0→ 7 β¯

φex (c ) c φex (c )  c  0 ¯ 0 0 D α ¯ d , βd : d ∈ Dex 0

//C ˜

project 0

0

˜ φ

, c ∈C ex ]

 / / D. ˜

project

That is, we begin with φ : C → D, and act on the extra generators by mapping ˜ αc0 7→ α ¯ φex (c0 ) , βc0 7→ β¯φex (c0 ) for all c0 ∈ C ex . These map the relations in C ˜ ˜ to the relations in D, and so descend to a unique morphism φ as shown. ˜ takes compositions and identities to compositions Clearly mapping φ 7→ φ ˜ on morphisms ˜ on objects and φ 7→ φ and identities, so mapping C 7→ C ∞ c ∞ c defines a functor C Rings → C Rings . For reasons explained in §6.3, we define C˜ : C∞ Ringsc → C∞ Ringscsc to be the composition of this ∞ c ∞ c functor with Πsc all : C Rings → C Ringssc in Definition 5.26. That is, ∞ c we define the corner functor C˜ : C Rings → C∞ Ringscsc to map C 7→ sc ˜ ˜ ˜ ˜ ˜ Πsc all (C) on objects, and φ 7→ Πall (φ) on morphisms, for C, D, φ, C, D, φ as above. ˜ ˜ C : C → C(C) For each C ∈ C∞ Ringsc , define Π to be the composition  C 

inc

  / C αc0 , βc0 : c0 ∈ C ex

/C ˜

project

ψC ˜

˜ / Πsc (C) ˜ = C(C), all

(6.6)

for ψ C˜ as in Definition 5.26. This is functorial in C, and so defines a natural ˜ : Id ⇒ C˜ of functors C∞ Ringsc → C∞ Ringsc . transformation Π ˜ We will also write αc0 , βc0 for the images of αc0 , βc0 in C(C) ex , and write PrC = {P ⊂ C ex : P is a prime ideal}.

(6.7)

Theorem 6.6. In Definition 6.5, with C : LC∞ RSc → LC∞ RScin as in Definition 6.1, for each C ∈ C∞ Ringsc there is a natural isomorphism ˜ η C : Specc C(C) −→ C(Specc C)

(6.8)

˜ lies in LC∞ RScin ⊂ LC∞ RSc . The following in LC∞ RSc . So Specc C(C) diagram commutes in LC∞ RSc : ˜ Specc C(C) ˜C Specc Π

/ C(Specc C)

+

ηC c

Spec C.

s

ΠSpecc C

(6.9)

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Boundaries, corners, and the corner functor

For each morphism φ : C → D in C∞ Ringsc the following commutes: ˜ Specc C(D)

ηD ∼ =

C(Specc φ)

˜ Specc C(φ)

 ˜ Specc C(C)

/ C(Specc D)

ηC ∼ =

 / C(Specc C),

(6.10)

˜ so Specc C(φ) is a morphism in LC∞ RScin . Thus η : Specc ◦C˜ ⇒ C ◦Specc is a natural isomorphism of functors (C∞ Ringsc )op → LC∞ RScin . ˜ ˜ as in (6.4), so that C(C) Proof Let C ∈ C∞ Ringsc , and define C = sc ˜ c c ˜ As in (6.6), there is a ˜ = Spec C. Πall (C). Write X = Spec C and X ˜ ˜ ˜ → C(C) → such that Specc ψ C˜ : Specc C(C) functorial morphism ψ C˜ : C c ˜ ˜ ˇ Spec C is an isomorphism. Write ΠC : C → C for the composition of the first ˜ C = ψ˜ ◦ Π ˇ C , and set Π ˇ X = Specc Π ˇC : two morphisms in (6.6), so that Π C ˜ → X. We will construct an isomorphism X ˜ −→ C(Specc C) = C(X) ˜ = Specc C ζC : X

(6.11)

in LC∞ RSc , and define η C in (6.8) by η C = ζ C ◦ Specc ψ C˜ . First we define the isomorphism (6.11) at the level of points. A point of ˜ → R. Composing with the projection ˜ is an R-algebra morphism x X ˜ : C ˜ gives a morphism x C[αc0 , βc0 : c0 ∈ C ex ] → C ˆ : C[αc0 , βc0 : c0 ∈ C ex ] → R. Such morphisms are in 1-1 correspondence with data (x, ac0 , bc0 : c0 ∈ C ex ), where x : C → R is an R-algebra morphism (i.e. x ∈ X) and ac0 = x ˆ(αc0 ), bc0 = x ˆ(βc0 ) lie in [0, ∞) ⊆ R for all c0 ∈ C ex . A morphism x ˆ : C[αc0 , βc0 : ˜ → R if and only if x c0 ∈ C ex ] → R descends to a (unique) morphism x ˜:C ˆ maps the relations in (6.5) to relations in R. Hence we may identify  ˜∼ X = (x, ac0 , bc0 : c0 ∈ C ex ) : x ∈ X, ac0 , bc0 ∈ [0, ∞) ∀c0 ∈ C ex , ac0 + bc0 = 1 ∀c0 ∈ C ex , a1C ex = 1, ac0 ac00 = ac0 c00 ∀c0 , c00 ∈ C ex , x ◦ Φi (c0 )bc0 = ac0 bc0 = 0 ∀c0 ∈ C ex . (6.12) We can simplify (6.12). Let (x, ac0 , bc0 : c0 ∈ C ex ) be a point in the right hand side. The equations ac0 + bc0 = 1 and ac0 bc0 = 0 imply that (ac0 , bc0 ) is (1, 0) or (0, 1) for each c0 . Define P = {c0 ∈ C ex : ac0 = 0}. Then a1C ex = 1 implies that 1C ex ∈ / P , and ac0 ac00 = ac0 c00 implies that if c0 ∈ C ex and c00 ∈ P 0 00 then c c ∈ P , so P is an ideal. Also ac0 ac00 = ac0 c00 implies that if c0 c00 ∈ P then c0 ∈ P or c00 ∈ P , so P is a prime ideal. The condition x ◦ Φi (c0 )bc0 = 0 then becomes x ◦ Φi (c0 ) = 0 if c0 ∈ P . Conversely, if P is a prime ideal and x ◦ Φi (c0 ) = 0 for c0 ∈ P , then setting ac0 = 0, bc0 = 1 if c0 ∈ P and ac0 = 1, bc0 = 0 if c0 ∈ C ex \ P then all the

6.2 The corner functor for C ∞ -schemes with corners 143
equations of (6.12) hold. Thus, mapping x, ac0 , bc0 : c0 ∈ C ex 7→ x, P =
{c0 ∈ C ex : ac0 = 0} turns (6.12) into a bijection  ˜∼ (6.13) X = (x, P ) : x ∈ X, P ∈ PrC , x ◦ Φi (c0 ) = 0 if c0 ∈ P . Lemma 6.7. Let x ∈ X, so that x : C → R is an R-point, with localization ex πx : C → C x ∼ . = O X,x giving a morphism πx,ex : C ex → C x,ex ∼ = OX,x Then we have inverse bijective maps 

P ∈ PrC : x◦Φi (c0 ) = 0 ∀c0 ∈ P



o

P 7−→Px =πx,ex (P )

/ Pr ∼ Pr Cx = O X,x . (6.14)

−1 Px 7−→P =πx,ex (Px )

Proof Let P lie in the left hand side of (6.14), and set Px = πx,ex (P ). Then 1 ∈ / Px as x ◦ Φi (c0 ) = 0 for all c0 ∈ P . If c0x ∈ C x,ex and c00x ∈ Px then 0 00 cx cx ∈ Px as P is an ideal and πx,ex is surjective. Thus Px is an ideal. To show Px is prime, suppose c0x , d0x ∈ C ex,x with c0x d0x ∈ Px . As πx,ex is surjective we can choose c0 , d0 ∈ C ex and p0 ∈ P with πx,ex (c0 ) = c0x , πx,ex (d0 ) = d0x and πx,ex (c0 d0 ) = πx,ex (p). Hence Proposition 4.33 gives a0 , b0 ∈ C ex with a0 c0 d0 = b0 p0 and x ◦ Φi (a0 ) 6= 0. But b0 p0 ∈ P as P is an ideal, so a0 c0 d0 ∈ P . Since P is prime, one of a0 , c0 , d0 must lie in P , and a0 ∈ / P as x ◦ Φi (a0 ) 6= 0, 0 0 0 0 so one of c , d must lie in P . Thus one of cx , dx lie in Px , and Px is prime. Hence the top morphism in (6.14) is well defined. −1 Next let Px ∈ PrC x , and set P = πx,ex (Px ). It is easy to check P lies in the left hand side of (6.14). So the bottom morphism in (6.14) is well defined. −1 (πx,ex (P )). Suppose P lies in the left hand side of (6.14). Clearly P ⊆ πx,ex 0 0 −1 If c ∈ πx,ex (πx,ex (P )) then there exists p ∈ P with πx,ex (c0 ) = πx,ex (p0 ). Proposition 4.33 gives a0 , b0 ∈ C ex with a0 c0 = b0 p0 and x ◦ Φi (a0 ) 6= 0, and p0 ∈ P implies a0 c0 = b0 p0 ∈ P , which implies c0 ∈ P as πx,ex (a0 ) 6= 0 so a0 ∈ / −1 −1 P . Hence P = πx,ex (πx,ex (P )). If Px ∈ PrC x then Px = πx,ex (πx,ex (Px )) as πx,ex is surjective. Thus the maps in (6.14) are inverse. Combining equations (6.1) and (6.13) and Lemma 6.7 gives a bijection of ˜ → R corresponds to ˜ → C(X), defined explicitly by, if x sets ζC : X ˜ : C 0 0 (x, ac0 , bc0 : c ∈ C ex ) in (6.25), and P = {c ∈ C ex : ac0 = 0} is the ex , then ζC (˜ x) = associated prime ideal, and Px = πx,ex (P ) in C x,ex ∼ = OX,x (x, Px ) in C(X) in (6.1). Then in an analogue of (6.9), on the level of sets we have ˇ X = ΠX ◦ ζC : X ˜ −→ X, Π (6.15) as both sides map x ˜ 7→ x. ˜ → C(X) is a homeomorphism of topological spaces. Lemma 6.8. ζC : X

144

Boundaries, corners, and the corner functor

˜ in Definition 2.42, the topology Proof By definition of the topology on Spec C ˜ As C ˜ is ˜ ˜ ˜ on X is generated by open sets Uc˜ = {˜ x∈X:x ˜(˜ c) 6= 0} for c˜ ∈ C. 0 ˜c for c ∈ C, a quotient of C[αc0 , βc0 : c ∈ C ex ], the topology is generated by U ˜Φ (α 0 ) , U ˜Φ (β 0 ) for all c0 ∈ C ex and smooth f : [0, ∞) → R. But as and U f f c c ˜ with αc0 + βc0 = 1, the only possibilities αc0 , βc0 take only values 0,1 on X ˜Φ (α 0 ) , U ˜Φ (β 0 ) are V˙ c0 = {˜ ˜ : αc0 |x˜ 6= 0} and V¨c0 = {˜ ˜ : for U x∈X x∈X f f c c ˜ ˜ ˙ αc0 |x˜ 6= 1}. So the topology on X is generated by Uc for all c ∈ C and Vc0 , V¨c0 for all c0 ∈ C ex . For C(X) in Definition 6.1, the topology is generated by open sets Π−1 X (Uc ) ¨s0 for open U ⊂ X and s0 ∈ Oex (U ). As X = for c ∈ C and subsets U˙ s0 , U X ¨s0 , it is enough to take U = X and Specc C, rather than taking all such U˙ s0 , U ¨s0 can be generated from s0 = ΞC,ex (c0 ) for all c0 ∈ C ex , as all other U˙ s0 , U −1 ˙ ¨ ΠX (Uc ) and XΞC,ex (c0 ) , XΞC ,ex (c0 ) . Thus, the topology on C(X) is generated 0 ˙ ¨ by Π−1 X (Uc ) for all c ∈ C and XΞC ,ex (c0 ) , XΞC ,ex (c0 ) for all c ∈ C ex . −1 ˜ ˙ It is easy to check that ζC maps Uc 7→ ΠX (Uc ), Vc0 7→ X˙ ΞC,ex (c0 ) and ¨ Ξ (c0 ) . Therefore ζC is a bijection and maps a basis for the topology V¨c0 7→ X C ,ex ˜ of X to a basis for the topology of C(X), so it is a homeomorphism. ˜ there is a unique Lemma 6.9. In sheaves of C ∞ -rings with corners on X, ] morphism ζ C making the following diagram commute: ζC−1 ◦ Π−1 X (O X ) 

−1 ζC (Π]X )

ζC−1 (O C(X) )

by (6.15) ζ ]C

ˇ −1 (O X ) Π X ˇ] Π X



(6.16)

/ O X˜ .

Furthermore, ζ ]C is an isomorphism. Proof By Definition 6.1, Π]X : Π−1 X (O X ) → O C(X) is the universal surjective quotient of Π−1 (O ) in sheaves of C ∞ -rings with corners, such that X X ex 0 if U ⊆ C(X) is open and s0 ∈ Π−1 X (OX )(U ) with s |(x,Px ) ∈ Px for each ] 0 ex (x, Px ) ∈ U then ΠX,ex (U )(s ) = 0 in OC(X) (U ). As X = Specc C, locally ex on X any section of OX is of the form ΞC,ex (c0 ) for c0 ∈ C ex , so locally on −1 0 C(X) any section s0 of Π−1 X (O X ) is of the form ΠX,ex ◦ ΞC,ex (c ). Thus it is −1 0 enough to verify the universal property above when s = ΠX,ex ◦ ΞC,ex (c0 )|U for c0 ∈ C ex , as it is local in s0 anyway. That is, Π]X : Π−1 X (O X ) → O C(X) is characterized by the property that if c0 ∈ C ex and U ⊂ C(X) ex is open with πx,ex (c0 ) ∈ Px ⊂ C x,ex ∼ for all (x, Px ) ∈ U then = OX,x ] 0 ΠX,ex (C(X)) ◦ ΞC,ex (c )|U = 0. Since ζC is a homeomorphism by Lemma 6.8, the pullback ζC−1 (Π]X ) in (6.16) has the analogous universal property. Let U ⊆ C(X) and c0 ∈ C ex have

6.2 The corner functor for C ∞ -schemes with corners

145

˜ = ζ −1 (U ). Suppose x ˜ with ζC (˜ the properties above, and set U ˜ ∈ U x) = C (x, Px ) in U ⊆ C(X), and let x ˜ correspond to (x, ac00 , bc00 : c00 ∈ C ex ) in (6.12). By construction of ζC we have ac0 = 0, bc0 = 1 as πx,ex (c0 ) ∈ Px . Thus ˜ ex in (6.5) implies that π ˜ x˜,ex ∼ the relation c0 βc0 = 0 in C ˜x˜,ex (c0 ) = 0 in C = ] 0 ex ˜ ˇ ˜ ˜ ∈ U we have Π (X)◦ΞC,ex (c )| ˜ = 0. Hence O . As this holds for all x ˜ x X,˜

X,ex U ] ] −1 the universal property of ζC (ΠX ) gives a unique morphism ζ C making (6.16)

commute. To show ζ ]C is an isomorphism, it is enough to show it is an isomorphism ˜ Let x on stalks at each x ˜ ∈ X. ˜ correspond to (x, ac0 , bc0 : c0 ∈ C ex ) in (6.12), with ζC (˜ x) = (x, Px ), so that ac0 = 0, bc0 = 1 if πx,ex (c0 ) ∈ Px and ac0 = 1, bc0 = 0 otherwise. The stalk ζC−1 (O C(X) )x˜ is O C(X),(x,Px ) , which ˜ ˜ . By the values of ac0 , bc0 , in (6.5) we know is C x /∼Px . The stalk O X,˜ ˜ x is C x that αc0 = 0, βc0 = 1 near x ˜ if πx,ex (c0 ) ∈ Px , and αc0 = 1, βc0 = 0 near x ˜ ˜ ˜ otherwise. Hence the generators αc0 , βc0 add no new generators to C x˜ , and C x˜ ˜ x˜ according is a quotient of C x . The relations in (6.5) turn into relations in C to the local values 0 or 1 of αc0 , βc0 . Therefore we see that   ˜ x˜ ∼ C = C x πx,ex (c0 ) = 0 if c0 ∈ C ex with πx,ex (c0 ) ∈ Px = C x /∼Px , where the second step holds as πx,ex is surjective. So we have an isomorphism ] of stalks ζC−1 (O C(X) )x˜ ∼ = O X,˜ ˜ x , which is the natural map ζ C,˜ x. ˜ → C(X), Lemmas 6.8 and 6.9 show ζ C = (ζC , ζ ]C ) is an isomorphism X c as in (6.11). So η C = ζ C ◦ Spec ψ C˜ is an isomorphism in (6.8). Equations (6.15)–(6.16) imply that (6.9) commutes. This proves the first part of Theorem 6.6. For the second part, consider the diagram ηD

˜ Specc C(D)

Specc ψ D ˜

c ˜ ˜ Specc C(φ)=Spec Πsc all (φ)

 ˜ Specc C(C)

Specc ψ C ˜

/ Specc D ˜

ζD

C(Specc φ)

˜ Specc φ

 / Specc C ˜

. / C(Specc D)

ζC

 / C(Specc C). 0

ηC

The left hand square commutes as ψ : Id ⇒ Πsc all is a natural transformation in Definition 5.26. The right hand square commutes by functoriality of ζ C in (6.11). The semicircles commute by definition of η C , η D . Hence (6.10) commutes, and Theorem 6.6 follows. The next proposition follows from the proof of Theorem 6.6. Here C(X)P is closed as it may be defined by the equations αc0 = 0 for c0 ∈ P , αc0 = 1 for c0 ∈ C ex \ P . Example 6.18 below shows the C(X)P need not be open.

146

Boundaries, corners, and the corner functor

Proposition 6.10. Let C = (C, C ex ) ∈ C∞ Ringsc , and write X = Specc C. With C(X) as in (6.1), for each P ∈ PrC , define  X P = x ∈ X : x ◦ Φi (c0 ) = 0 for all c0 ∈ P , (6.17)  P ex ∼ C(X) = (x, Px ) ∈ C(X) : πx,ex (P ) = Px in C x,ex = OX,x  −1 = (x, Px ) ∈ C(X) : P = πx,ex (Px )  (6.18) = (x, Px ) ∈ C(X) : if c0 ∈ C ex then πx,ex (c0 ) ∈ Px in C x,ex ∼ = Oex if and only if c0 ∈ P ⊆ C(X), X,x

where the three expressions in (6.18) are equivalent. Then ΠX : C(X) → X restricts to a bijection C(X)P → X P . Also C(X)P is closed in C(X), but not necessarily open, and just as sets we have a C(X) = C(X)P . (6.19) P ∈PrC

As an easy consequence of Theorem 6.6, we deduce: Theorem 6.11. The corner functor C : LC∞ RSc → LC∞ RScin of §6.1 maps C∞ Schc → C∞ Schcin . We write C : C∞ Schc → C∞ Schcin for the restriction, and call it a corner functor. It is right adjoint to inc : C∞ Schcin ,→ C∞ Schc . Thus it preserves limits in C∞ Schc , C∞ Schcin . Proof Let X lie in C∞ Schc . Then we can cover X by open U ⊆ X with U∼ = Specc C for C ∈ C∞ Ringsc . Since the definition of ΠX : C(X) → X c ∼ ∼ is local over X, we have Π−1 X (U ) = C(U ) = C(Spec C). Theorem 6.6 gives c c ˜ ∞ ∼ ˜ C(Spec C) = Spec C(C), with C(C) ∈ C Ringsc . ˜ lies in LC∞ RScin ⊂ LC∞ RSc , Theorem 5.28 shows Since Specc C(C) c ˜ that Spec C(C) ∼ = Speccin D for D ∈ C∞ Ringscin . Thus we can cover −1 c ∼ C(X) by open Π−1 X (U ) ⊆ C(X) with ΠX (U ) = Specin D for D ∈ ∞ c ∞ C Ringsin , so C(X) is an interior C -scheme with corners. Therefore C : LC∞ RSc → LC∞ RScin maps C∞ Schc → C∞ Schcin . The rest is immediate from Theorem 6.3. The next proposition is easy to prove by considering local models. ˇ c , Man ˇ gc Proposition 6.12. The corner functors C for Manc , Mangc , Man ∞ c in Definition 3.16, and for C Sch in Theorem 6.11, commute with the funcC∞ Schc C∞ Schc tors FMan , . . . , FMan in Definition 5.21 up to natural isomorphism. c gc ˇ Combining Theorems 5.31(b), 6.4 and 6.11 yields:

6.3 The corner functor for firm C ∞ -schemes with corners

147

Theorem 6.13. Restricting C in Theorem 6.11 gives corner functors C : C∞ Schcsa,ex −→ C∞ Schcsa ,

C : C∞ Schctf ,ex −→ C∞ Schctf ,

C : C∞ SchcZ,ex −→ C∞ SchcZ ,

C : C∞ Schcin,ex −→ C∞ Schcin ,

in the notation of Definition 5.29, which are right adjoint to the corresponding inclusions. As these corner functors are right adjoints, they preserve limits.

6.3 The corner functor for firm C ∞ -schemes with corners ˜ The following proposition is why we included Πsc all in defining C in Definition ˜ ˜ which need not hold 6.5. It forces relations on αc0 , βc0 in C(C) = Πsc ( C), all ˜ in C. ˜ Proposition 6.14. Let C ∈ C∞ Ringsc and write D = C(C) with morphism 0 ˜ ˜ ΠC : C → D, so we have elements αc0 , βc0 , ΠC,ex (c ) in Dex for all c0 ∈ C ex . Write c0 7→ [c0 ], d0 7→ [d0 ] for the projections C ex 7→ C ]ex , Dex 7→ D]ex . Then: (a) If c01 , c02 ∈ C ex with [c01 ] = [c02 ] in C ]ex then αc01 = αc02 , βc01 = βc02 in Dex . (b) If c01 , . . . , c0n ∈ C ex and i1 , . . . , in , j1 , . . . , jn are positive integers then αc0i1 ···c0in n = αc0j1 ···c0j n and βc0i1 ···c0in = βc0j1 ···c0jn in Dex . n n n 1 1 1 1 ] ˜ C,ex (c0 )] for all c0 ∈ C ex . (c) Dex is generated by [αc0 ], [βc0 ], [Π (d) If C is firm then D is firm. Thus C˜ maps C∞ Ringsc → C∞ Ringsc . fi

fi

˜ and ˜ so that D ⊆ O ˜ (X), ˜ be as in (6.5) and X ˜ = Specc C, Proof Let C X ex 0 0 ˜ elements of Dex are global sections of the sheaf OX˜ on X. Hence d1 , d2 ∈ Dex ex ∼ ˜ ˜ have d01 = d02 if and only if d01 |x˜ = d02 |x˜ in OX,˜ ˜ ∈ X. ˜,ex for all x ˜ x = Cx ˜ with ζC (˜ In the proof of Theorem 6.6 we saw that if x ˜∈X x) = (x, Px ) in C(X) and c0 ∈ C ex , and P corresponds to Px under (6.14), then αc0 |x˜ = 0, ex ∼ ˜ 0 βc0 |x˜ = 1 in OX,˜ ˜,ex if c ∈ P , and αc0 |x ˜ = 1, βc0 |x ˜ = 0 otherwise. ˜ x = Cx 0 0 0 0 Thus, if c1 , c2 ∈ C ex satisfy c1 ∈ P if and only if c2 ∈ P for any prime ideal ˜ so αc0 = αc0 , P ⊂ C ex , then αc01 |x˜ = αc02 |x˜ , βc01 |x˜ = βc02 |x˜ for all x ˜ ∈ X, 1 2 βc01 = βc02 in Dex . Parts (a),(b) are consequences of this criterion. For (a), as prime ideals in C ex are preimages of prime ideals in C ]ex , if [c01 ] = [c02 ] in C ]ex then c01 ∈ P if and only if c02 ∈ P for any prime P ⊂ C ex . For (b), if i1 , . . . , jn > 0 the prime 0j1 0in 0jn 1 property of P implies that c0i 1 · · · cn ∈ P if and only if c1 · · · cn ∈ P . ˜ ] is generated by [αc0 ], [βc0 ], [c0 ] for c0 ∈ C ex by (6.5), and ψ ] For (c), C : ex ˜ C,ex ] ] sc ˜ ˜ C ex → Dex is surjective as in the construction of D = Πall (C) in Proposition ˜ ex , so (c) follows. 5.25, Dex is generated in Oex (X) by invertibles and C ˜ X

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For (d), let C be firm, so that C ]ex is finitely generated, and choose c01 , . . . , c0n in C ex such that [c01 ], . . . , [c0n ] generate C ]ex . Then (a),(b) imply that there are at most 2n distinct αc0 and 2n distinct βc0 in Dex , as every αc0 equals αQi∈S c0i for some subset S ⊆ {1, . . . , n}. Hence (c) shows that D]ex is generated by the ˜ C,ex (c0 )] 2n+1 + n elements [αQi∈S c0i ], [βQi∈S c0i ] for S ⊆ {1, . . . , n} and [Π i for i = 1, . . . , m. Therefore D is firm. As for Theorem 6.11, Theorems 5.31(c), 6.11 and Proposition 6.14(d) imply: Theorem 6.15. The corner functor C : C∞ Schc → C∞ Schcin from §6.2 maps C∞ Schcfi → C∞ Schcfi,in . We call the restriction C : C∞ Schcfi → C∞ Schcfi,in a corner functor. It is right adjoint to inc : C∞ Schcfi,in ,→ C∞ Schcfi . As C is a right adjoint, it preserves limits in C∞ Schcfi , C∞ Schcfi,in . Combining Theorems 5.31(c), 6.13, and 6.15 yields: Theorem 6.16. Restricting C in Theorem 6.15 gives corner functors C : C∞ Schcto,ex −→ C∞ Schcto ,

C : C∞ Schcfi,tf ,ex −→ C∞ Schcfi,tf ,

C : C∞ Schcfi,Z,ex −→ C∞ Schcfi,Z ,

C : C∞ Schcfi,in,ex −→ C∞ Schcfi,in ,

in the notation of Definition 5.29, which are right adjoint to the corresponding inclusions. As these corner functors are right adjoints, they preserve limits. For a firm C ∞ -ring with corners C with X = Specc C, we can give an alternative description of C(X): Theorem 6.17. Let C be a firm C ∞ -ring with corners and X = Specc C. Then PrC in (6.7) is finite. Hence in the decomposition (6.19), the subsets C(X)P ⊆ C(X) are open as well as closed, and (6.19) lifts to a disjoint union a C(X) = C(X)P in C∞ Schcfi,in . (6.20) P ∈PrC

For each P ∈ PrC , Example 4.25(b) defines a C ∞ -ring with corners C/∼P with projection π P : C → C/∼P , where C/∼P is firm and interior. Write X P = Speccin (C/∼P ) and ΠP = Specc π P : X P → X. Then
C ΠP ◦ ιX P : X P −→ C(X) (6.21) is an isomorphism with the open and closed C ∞ -subscheme C(X)P ⊆ C(X). ` Thus P ∈PrC X P ∼ = C(X). Proof First note that prime ideals P ⊂ C ex are in 1-1 correspondence with prime ideals P ] in C ]ex , which is finitely generated as C is firm. If [c01 ], . . . , [c0n ]

6.3 The corner functor for firm C ∞ -schemes with corners 149

 are generators of C ]ex then P ] = {[c01 ], . . . , [c0n ]} ∩ P ] , so there are at most 2n such prime ideals P ] . Hence PrC is finite. Thus in (6.19), C(X) is the disjoint union of finitely many closed sets C(X)P , so the C(X)P are also open. Equation (6.20) follows. ˜ Write D = C(C) as in Proposition 6.14 and Y = Specc D, so Theorem 6.6 gives an isomorphism η C : Y → C(X). Fix P ∈ PrC , and write Y P = P P η −1 is open and closed in Y with η C |Y P : Y P → C (C(X) ), so that Y P C(X) an isomorphism. The proof of Theorem 6.6 implies that Y P may be defined as a C ∞ -subscheme of Y by the equations αc0 = 0, βc0 = 1 for c0 ∈ P and αc0 = 1, βc0 = 0 for c0 ∈ C ex \ P , where this is only finitely many equations, as there are only finitely many distinct αc0 in Dex by the proof of Proposition 6.14(d). Thus we see that
YP ∼ = Specc D/[αc0 = 0, βc0 = 1, c0 ∈ P , αc00 = 1, βc00 = 0, c00 ∈ C ex \ P ]
0 00 ˜ = Specc Πsc all (C)/[αc0 = 0, βc0 = 1, c ∈ P , αc00 = 1, βc00 = 0, c ∈ C ex \ P ]
0 00 ∼ ˜ = Specc ◦Πsc all C/[αc0 = 0, βc0 = 1, c ∈ P , αc00 = 1, βc00 = 0, c ∈ C ex \ P ]  
= Specc C αc0 , βc0 : c0 ∈ C ex Φi (αc0 ) + Φi (βc0 ) = 1 ∀c0 ∈ C ex  α1C ex = 1C ex , αc0 αc00 = αc0 c00 ∀c0 , c00 ∈ C ex , c0 βc0 = αc0 βc0 = 0 ∀c0 ∈ C ex 
αc0 = 0, βc0 = 1, c0 ∈ P , αc00 = 1, βc00 = 0, c00 ∈ C ex \ P
∼ (6.22) = Specc C/[c0 = 0, c0 ∈ P ] = Specc (C/∼P ) = X P , where in the third step we use that Πsc all commutes with applying relations c ∼ after composing with Specc , in the fourth that Specc ◦Πsc all = Spec and (6.5), and in the fifth that adding generators αc0 , βc0 and setting them to 0 or 1 is equivalent to adding no generators, and the only remaining effective relations are c0 βc0 = 0 when c0 ∈ P , so that βc0 = 1, giving c0 = 0. Composing the inverse of (6.22) with η C |Y P gives a morphism θ P : X P → C(X) which is an isomorphism with C(X)P ⊂ C(X). As C/∼P is interior, X P is interior, so (6.21) is well defined. One can check, from the actions on points and on stalks O X P ,x , O C(X),(x,Px ) , that (6.21) is θ P . The theorem follows. We explain by example how Theorem 6.17 goes wrong in the non-firm case. ∞

c

C Sch Example 6.18. Write [0, ∞) = FMan ([0, ∞)) = Specc C ∞ ([0, ∞)) c ∞ as an affine C -scheme with corners. Proposition 6.12 gives C([0, ∞)) ∼ = {∗} q [0, ∞), writing ∂[0, ∞) = {∗}, a single point. Q Let S be an infinite set. Consider X = s∈S [0, ∞)s , the product in ∞ c C Sch of a copy [0, ∞)s of [0, ∞) for each s in S. Since Specc preserves Ns∈S limits, we have X = Specc C for C = ∞ C ∞ ([0, ∞))s .

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Boundaries, corners, and the corner functor

As C : C∞ Schc → C∞ Schcin preserves limits by Theorem 6.11 we have
Q
Q C(X) = C s∈S [0, ∞)s ∼ (6.23) = s∈S {∗}s q [0, ∞)s . Just as a set, we may decompose the right hand side of (6.23) as
Q Q ` T ∼` C(X) = T ⊆S t∈T {∗}t × s∈S\T [0, ∞)s =: T ⊆S C(X) . (6.24) One can show that subsets T ⊆ S are in natural 1-1 correspondence with prime ideals P ⊂ C ex , and (6.24) is the decomposition (6.19) for X = Specc C. Since S is an infinite set, the subsets C(X)T ⊂ C(X) in (6.24) are closed, but not open, in (6.24). This is because C(X) has the product topology, as the forgetful functor C∞ Schc → Top preserves limits, and an infinite product of Q topological spaces s∈S Xs has a weaker topology than one might expect: if Q Q Us ⊆ Xs is open for s ∈ S then s∈S Us is generally not open in s∈S Xs , unless Us = Xs for all but finitely many s ∈ S. Although Theorem 6.17 makes sense when C is not firm and PrC is infinite, ` it is false in general, as C(X) ∼ = P X P fails already for topological spaces.

ˇ ex , M ˇ in on C(X) 6.4 The sheaves of monoids M C(X) C(X) As in Remark 3.18(d), for a manifold with g-corners X, in [47, §3.6] we defined ∨ the comonoid bundle MC(X) → C(X), a local system of toric monoids on ∨ C(X) with the property that if (x, γ) ∈ C(X)◦ and MC(X) |(x,γ) = P then X near x is locally diffeomorphic to XP × Rdim X−rank P near (δ0 , 0). If X ∨ is a manifold with corners then MC(X) has fibre Nk over Ck (X). We now ∞ generalize these ideas to firm C -schemes with corners. Definition 6.19. Let X = (X, O X ) be a local C ∞ -ringed space with corners, so that Definition 6.1 defines the corners C(X) = (C(X), O C(X) ) in LC∞ RScin , with a morphism ΠX : C(X) → X. We have a sheaf of −1 ex ex ex monoids Π−1 X (OX ) on C(X), with stalks (ΠX (OX ))(x,P ) = OX,x at (x, P ) −1 ex in C(X), and we have a natural prime ideal P ⊂ (ΠX (OX ))(x,P ) in the stalk at each (x, P ) ∈ C(X). ˇ ex on C(X) by, for open U ⊆ C(X), Define a presheaf of monoids P M C(X)  ˇ ex (U ) = Π−1 (Oex )(U ) s = 1 : s ∈ Π−1 (Oex )(U ), s|(x,P ) ∈ PM /P X X C(X) X X  for all (x, P ) ∈ U . ˇ ex (U ) → For open V ⊆ U ⊆ C(X), the restriction map ρU V : P M C(X) ˇ ex (V ) is induced by ρU V : Π−1 (Oex )(U ) → Π−1 (Oex )(V ). This PM X X X X C(X)

ˇ ex , M ˇ in on C(X) 6.4 The sheaves of monoids M C(X) C(X)

151

ˇ ex valued in Mon. Write M ˇ ex for its gives a well defined presheaf P M C(X) C(X) sheafification. ex ˇ ex The surjective projections Π−1 X (OX )(U ) → P MC(X) (U ) for open U ⊆ ex ˇ ex C(X) induce a surjective morphism ΠMˇ ex : Π−1 X (OX ) → MC(X) . Local C(X)

ex ex sections s of Π−1 X (OX ) lying in OX,x \ P at each (x, P ) have π(s) = 1 in ex ex ˇ ˇ ˇ ex M C(X) , and MC(X) is universal with this property. The stalk of MC(X) at (x, P ) is   ex ex ˇ ex )(x,P ) ∼ (M s = 1 : s ∈ OX,x \P . (6.25) = OX,x C(X)

Now suppose that X is interior. Then as O X,x is interior for x ∈ X, the ex in ex monoid OX,x has no zero divisors, and OX,x = OX,x \ {0} is also a monoid. in ex ˇ ˇ ˇ ex Write MC(X) ⊂ MC(X) for the subsheaf of sections of M C(X) which are in ˇ nonzero at every (x, P ) ∈ C(X). Then MC(X) is also a sheaf of monoids ˇ in )(x,P ) = (M ˇ ex )(x,P ) \ {0} at each (x, P ) ∈ on C(X), with stalk (M C(X) C(X) C(X). Next let f : X → Y be a morphism in LC∞ RSc , so that C(f ) : C(X) → C(Y ) is a morphism in LC∞ RScin . Consider the diagram: ex oo C(f )−1 (OC(Y )) C(f )]ex

 ex oo OC(X)

ex C(f )−1 ◦Π−1 ˇ ex ) Y (OY ) = / / C(f )−1 (M −1 C(Y ) −1 ex ) Π ◦f ◦(O −1 Y X ] C(f ) (Π ˇ ex )

C(f )−1 (ΠY,ex )

M

C(Y )

] Π−1 X (fex )

Π]X,ex

 ex Π−1 (O X) X

ΠM ˇ ex

ˇ ex M C(f )

(6.26) 

ˇ ex . //M C(X)

C(X)

ex −1 ˇ ex Here C(f )−1 (ΠMˇ ex ) : C(f )−1 ◦Π−1 (MC(Y ) ) is the quoY (OY )  C(f ) C(Y )

ex tient sheaf of monoids which is universal for morphisms C(f )−1◦Π−1 Y (OY ) → M , for M a sheaf of monoids on C(X), such that a local section s of C(f )−1 ◦ ex ex Π−1 Y (OY ) is mapped to 1 in M if s|(x,P ) ∈ OY,y \ Q at each (x, P ) ∈ C(X) with C(f )(x, P ) = (y, Q) in C(Y ). ] ˇ ex , the morphism Π ˇ ex ◦ Π−1 (fex Now taking M = M ) : C(f )−1 ◦ MC(X) X C(X) ˇ ex has the property required. This is because C(f ) is inΠ−1 (Oex ) → M Y

Y

C(X)

ex ex terior, so C(f )]ex maps OC(Y ),(y,Q) \ {0} → OC(X),(x,P ) \ {0}, and thus −1 ] ex ex ex ΠX (fex ) maps OY,y \ Q → OX,x \ P , and ΠMˇ ex maps OX,x \ P → 1. C(X) −1 ex −1 ex ˇ ˇ ex Hence there is a unique morphism M : C(f ) ◦ Π (O ) → M C(f )

Y

Y

C(X)

of sheaves of monoids on C(X) making (6.26) commute. If g : Y → Z is another morphism in LC∞ RSc , then composing (6.26) −1 ˇ ex ˇ ex ˇ ex for f , g vertically we see that M (MC(g) ). So the C(g◦f ) = MC(f ) ◦ C(f ) ex ex ˇ ˇ MC(X) , MC(f ) are contravariantly functorial.

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ˇ ex Proposition 6.20. Let X be a firm C ∞ -scheme with corners. Then M C(X) in Definition 6.19 is a locally constant sheaf of monoids on C(X), and the ˇ ex )(x,P ) for (x, P ) ∈ C(X) are finitely generated sharp monoids, stalks (M C(X) with zero elements not equal to the identity. If X is interior, the same holds ˇ in , but without with zero elements. for M C(X) Proof As the proposition is local in X, it is sufficient to prove it in the case that X = Specc C for C a firm C ∞ -ring with corners. Then Theorem 6.17 ` defines an isomorphism P ∈PrC X P → C(X). ˇ ex to the image We will prove that for P ∈ PrC , the restriction of M C(X) C(X)P of X P under (6.21) is the constant sheaf with fibre the monoid     C ex c0 = 1 : c0 ∈ C ex \ P ∼ (6.27) = C ]ex / c00 = 1 : c00 ∈ C ]ex \ P ] . × ] Here C ]ex = C ex /C × ex and P = P/C ex , and the two sides of (6.27) are iso× morphic as C ex ⊆ C ex \ P , so quotienting by C × ex commutes with setting all elements of C ex \ P or C ]ex \ P ] equal to 1. Equation (6.27) is finitely generated as C ]ex is, since C is firm. Also (6.27) is the disjoint union of {1}, and the image of P , which is an ideal in (6.27), and so contains no units. Hence (6.27) is sharp. The image of 0C ex is a zero element not equal to the identity. Suppose P ∈ PrC , and x0 ∈ X P maps to (x, Px ) ∈ C(X) under (6.21), ex = C x,ex is a prime ideal, with Px = πx,ex (P ) ( C x,ex and where Px ⊂ OX,x −1 P = πx,ex (Px ) ( C ex by Lemma 6.7. Consider the diagram of monoids:

C ex

project

πx,ex

 C x,ex

project

/ / C ex

 0  c = 1 : c0 ∈ C ex \ P = (6.27)

(πx,ex )∗  (6.28)   00  ˇ ex )(x,P ) . / / C x,ex c = 1 : c00 ∈ C x,ex \Px = (M x C(X)

−1 (Px ), if c0 ∈ C ex \ P then c00 = πx,ex (c0 ) ∈ C x,ex \ Px , so there As P = πx,ex is a unique, surjective morphism (πx,ex )∗ making (6.28) commute. We claim that (πx,ex )∗ is an isomorphism. To see this, note that as in §4.6, C x,ex may be obtained from C ex by inverting all elements c0 ∈ C ex with x ◦ Φi (c0 ) 6= 0, and setting c00 = 1 for all c00 ∈ Ψexp (I), where I ⊂ C is the ideal vanishing near x. Here c0 ∈ C ex with x ◦ Φi (c0 ) 6= 0 and c00 ∈ Ψexp (I) both lie in the set C ex \ P of elements set to 1 in the right hand of (6.28). Also first inverting c0 and then setting c0 = 1 is equivalent to just setting c0 = 1. ˇ ex )(x,P ) at each Thus (πx,ex )∗ is an isomorphism. So the fibre of (M x C(X) (x, Px ) in the image of X P is naturally isomorphic to (6.27), which is indeˇ ex is constant with fibre (6.27) on pendent of (x, Px ). It easily follows that M C(X) ˇ in when X is interior is immediate. the image X P . The analogue for M C(X)

ˇ ex , M ˇ in on C(X) 6.4 The sheaves of monoids M C(X) C(X)

153

Remark 6.21. If X lies in LC∞ RSc or C∞ Schc , but not in C∞ Schcfi , then ˇ ex in Definition 6.19 may not be locally constant. M C(X) For example, suppose C ∈ C∞ Ringsc is not firm, and C ex has infinitely many prime ideals P , and X = Specc C. Then (6.19) decomposes C(X) into infinitely many disjoint closed subsets C(X)P , which need not be open. The ˇ ex |C(X)P is constant on C(X)P with proof of Proposition 6.20 shows that M C(X) value (6.27) for each P . But if the C(X)P are not open, as happens in Example ˇ ex is not locally constant. 6.18, then in general M C(X) Proposition 6.20 implies that if X is a firm C ∞ -scheme with corners then a C(X) = CM (X), iso. classes [M ] of finitely generated sharp monoids M with zero elements

where CM (X) ⊆ C(X) is the open and closed C ∞ -subscheme of points ˇ ex )(x,P ) ∼ (x, Px ) in C(X) with (M M. x = C(X) c Example 6.22. In Theorem 6.17, let C = C ∞ (Rm k ), so that X = Spec C = ∞ c m C Sch FMan (Rk ). Take P to be the prime ideal P = hx1 , . . . , xk i in C ex , so that c (6.21) maps X P to the corner stratum {x1 = · · · = xk = 0} of Rm k in C(X). ˇ ex on X P is the constant sheaf The proof of Proposition 6.20 shows that M C(X) with fibre (6.27). In this case C is interior with  C in = xa1 1 · · · xakk exp f : ai ∈ N, f ∈ C ∞ (Rm k ) , C ex = C in q {0}. (6.29)  exp f : f ∈ C ∞ (Rm Since C ex \ P = C × ex = k ) , equation (6.27) is ˇ ex has fibre Nk q {0}, and M ˇ in C ex /C × = C ] ∼ = Nk q {0}. So M ex

ex

C(X)

C(X)

has fibre Nk , on X P . C∞ Schc (Y ). Then PropoNow let Y be a manifold with corners, and Y = FMan c ∞ c C Sch ∼ (C(Y )). Let (y, γ) ∈ Ck (Y )◦ . Then sition 6.12 says that C(Y ) = FMan c ˇ k y ∈ S (Y ), and Y near y is locally diffeomorphic to Rm k near 0, such that the local k-corner component γ of Y at y is identified with {x1 = · · · = xk = ex in 0} in Rm k . Hence C(Y ) and MC(Y ) , MC(Y ) near (y, γ) are locally isomor
ˇ ex , M ˇ in phic to C(X), M C(X) C(X) near 0, {x1 = · · · = xk = 0} . Therefore ˇ ex , M ˇ in have fibres Nk q {0}, Nk on Ck (Y )◦ , and hence on Ck (Y ), as M C(Y ) C(Y ) ˇ ex , M ˇ in are locally constant by PropoCk (Y )◦ is dense in Ck (Y ) and M C(Y )

C(Y )

sition 6.20. ∨ As in Remark 3.18(d), in [47, §3.6] we define the comonoid bundle MC(Y ) → C(Y ), which has fibre Nk over Ck (Y ). It is easy to check that the sheaf of ∨ ˇ in continuous sections of MC(Y ) is canonically isomorphic to MC(Y ) . Example 6.23. We can generalize Example 6.22 to manifolds with g-corners.

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Boundaries, corners, and the corner functor

Let Q be a toric monoid and l > 0, and set C = C ∞ (XQ × Rl ), so that C∞ Schc X = Specc C = FMan (XQ × Rl ). Take P to be the prime ideal of gc functions in C ex vanishing on {δ0 } × Rl , so that (6.21) maps X P to the corner stratum {δ0 } × Rl in C(X). The analogue of (6.29) is  C in = λq exp f : q ∈ Q, f ∈ C ∞ (XQ × Rl ) , C ex = C in q {0}, for λq : XQ → [0, ∞) as in Definition 3.7, so C ]in ∼ = Q and C ]ex ∼ = Q q {0}, ex in P ˇ ˇ and M , M have fibres Q q {0}, Q on X . C(X) C(X) C∞ Schc Now let Y be a manifold with g-corners, and Y = FMan (Y ), so that c C∞ Schc C(Y ) ∼ (C(Y )) by Proposition 6.12. Let (y, γ) ∈ C(Y )◦ . Then Y = FMan c ˇ near y is locally diffeomorphic to XQ × Rl near (δ0 , 0) for some toric monoid Q and l > 0, such that the local corner component γ of Y at y is identified with ˇ ex , M ˇ in have fibres Q q {0}, Q at (y, γ). {δ0 } × Rl in XQ × Rl . Hence M C(Y ) C(Y ) ∨ As in Remark 3.18(d), the comonoid bundle MC(Y ) → C(Y ) of [47, §3.6] also has fibre Q at (y, γ). It is easy to check that the sheaf of continuous sections ∨ ˇ in of MC(Y ) is canonically isomorphic to MC(Y ) .

6.5 The boundary ∂X and k-corners Ck (X) If X is a manifold with (g-)corners then the corners C(X) from §3.4 has a `dim X decomposition C(X) = k=0 Ck (X) with C0 (X) ∼ = X and C1 (X) = ∂X. We generalize this to firm (interior) C ∞ -schemes with corners X. Definition 6.24. The dimension dim M in N q {∞} of a monoid M is the maximum length d (or ∞ if there is no maximum) of a chain of prime ideals ∅ ( P1 ( P2 ( · · · ( Pd ( M . If M is finitely generated then dim M < ∞. If M is toric then dim M = dimR (M ⊗N R). Let X be a firm C ∞ -scheme with corners. For each k = 0, 1, . . . , define the k-corners Ck (X) ⊆ C(X) to be the C ∞ -subscheme of (x, P ) ∈ C(X) ˇ ex )(x,P ) = k + 1. Here (x, P ) 7→ dim(M ˇ ex )(x,P ) is a lowith dim(M C(X) C(X) cally constant function C(X) → N by Proposition 6.20, so Ck (X) is open ˇ ex )(x,P ) > 1 as (M ˇ ex )(x,P ) has at and closed in C(X). Also dim(M C(X) C(X) ex ex ˇ ˇ least one prime ideal (M ) \ {1}, since ( M ) (x,P ) is sharp and C(X) (x,P ) `C(X) ex ˇ (MC(X) )(x,P ) \ {1} = 6 ∅ as 0 6= 1. Hence C(X) = k>0 Ck (X). We define the boundary ∂X to be ∂X = C1 (X) ⊂ C(X). ˇ ex )(x,P ) = dim(M ˇ in )(x,P ) + 1, as chains If X is interior then dim(M C(X) C(X) ˇ in )(x,P ) lift to chains in (M ˇ ex )(x,P ) , plus {0}. Hence of ideals in (M C(X) C(X) ˇ in )(x,P ) = k. Ck (X) is the C ∞ -subscheme of (x, P ) ∈ C(X) with dim(M C(X)

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6.6 Fibre products and corner functors

ˇ in )(x,P ) 6= {1} then dim(M ˇ in )(x,P ) > 0 as For X interior, if (M C(X) C(X) in in ˇ ˇ (M ) \{1} is prime. But ( M ) = {1} if and only if P = {0}, C(X) (x,P ) C(X) (x,P ) in ex ˇ as 0 6= p ∈ P descends to [p] 6= 1 in (MC(X) )(x,P ) since OX,x has no zero divisors. Hence C0 (X) = {(x, {0}) : x ∈ X} ⊆ C(X). It is now easy to see that ιX : X → C(X) in Definition 6.1 is an isomorphism ιX : X → C0 (X). ∞

c

C Sch Example 6.25. Let X be a manifold with (g-)corners and X = FMan (X), gc ∞ which is a firm interior C -scheme with corners. As in Proposition 6.12 there C∞ Schc is a natural isomorphism C(X) ∼ (C(X)). It is easy to check that = FMan gc ˇ ∞ c C Sch (Ck (X)) for each k = 0, . . . , dim X. this identifies Ck (X) ∼ = FMan gc

Example 6.26. If X lies in C∞ Schcfi but not C∞ Schcfi,in then C0 (X) can
c contain (x, P ) with
P 6= {0}. For
instance, if X = Spec R[x, y]/[xy = 0] then (0, 0), hxi and (0,
0), hyi lie in C0 (X). Also {0} is not prime in ex OX,(0,0) , so (0, 0), {0} ∈ / C(X). Clearly X ∼ 6 C0 (X), as only one of the = two is interior.

6.6 Fibre products and corner functors The authors’ favourite categories of ‘nice’ C ∞ -schemes with corners gengc eralizing Mangc are C∞ Schcto , C∞ Schcto,ex . Theorem 5.34 says in , Man that fibre products and finite limits exist in C∞ Schcto . Now we address the question of when fibre products exist in C∞ Schcto,ex , or more generally in C∞ Schcfi,in,ex , or in categories like those in (5.10) in which the objects are interior, but the morphisms need not be interior. We use corner functors as a tool to do this. Theorem 6.27. Let g : X → Z, h : Y → Z be morphisms in C∞ Schcto,ex . Consider the question of whether a fibre product W = X ×g,Z,h Y exists in C∞ Schcto,ex , with projections e : W → X, f : W → Y . (a) If W exists then the following is a bijection of sets:  (e, f ) : W −→ (x, y) ∈ X × Y : g(x) = h(y) .

(6.30)

Note that this need not hold for fibre products in C∞ Schcto . (b) If W exists then W is isomorphic to an open and closed C ∞ -subscheme W 0 of C(X) ×C(g),C(Z),C(h) C(Y ), where the fibre product is taken in C∞ Schcto , using C(g), C(h) interior, and exists by Theorem 5.34.

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Boundaries, corners, and the corner functor

(c) Writing C(X)×C(Z) C(Y ) for the topological space of C(X)×C(Z) C(Y ), we have a natural continuous map C(X) ×C(Z) C(Y ) w0 

(ΠC(X) ,ΠC(Y ) )

 / (x, y) ∈ X × Y : g(x) = h(y) , (6.31)
ΠX ×ΠY / (x, Px ), (y, Py )  / (x, y).

If there does not exist an open and closed subset W 0 ⊂ C(X) ×C(Z) C(Y ) such that (6.31) restricts to a bijection on W 0 , then no fibre product W = X ×g,Z,h Y exists in C∞ Schcto,ex . (d) Suppose g, h are interior, and a fibre product W = X ×g,Z,h Y , e, f exists in C∞ Schcto . Then W is also a fibre product in C∞ Schcto,ex if and only if the following diagram is Cartesian in C∞ Schcto : C(W )

C(f )

C(e)

 C(X)

/ C(Y ) C(h)

C(g)

 / C(Z),

(6.32)

that is, if C(W ) ∼ = C(X) ×C(g),C(Z),C(h) C(Y ) in C∞ Schcto . The analogues hold for C∞ Schcfi,in,ex , C∞ Schcfi,Z,ex , C∞ Schcfi,tf ,ex , except that (6.30) is a bijection for fibre products in C∞ Schcfi,in . Proof For (a), suppose W exists, and apply the universal property of fibre products for morphisms from the point ∗ into W , X, Y , Z. There is a 1-1 correspondence between morphisms w : ∗ → W in C∞ Schcto,ex and points w0 = w(∗) ∈ W . Using this, the universal property gives the bijection (6.30). This argument does not work for fibre products in C∞ Schcto , as (necessarily interior) morphisms w : ∗ → W need not correspond to points of W . For example, morphisms w : ∗ → [0, ∞) in C∞ Schcto correspond to points of (0, ∞) not [0, ∞). Equation (6.30) is not a bijection for the fibre products in C∞ Schcto in Examples 6.29–6.30. This also holds for C∞ Schcfi,Z,ex and C∞ Schcfi,tf ,ex . The inclusion inc : C∞ Schcfi,in ,→ C∞ Schcfi,in,ex preserves fibre products by Theorems 5.33 and 5.34. Thus a fibre product in C∞ Schcfi,in is also a fibre product in C∞ Schcfi,in,ex , and (6.30) is a bijection. For (b), suppose W exists. As C : C∞ Schcto,ex → C∞ Schcto preserves limits by Theorem 6.16, we have C(W ) ∼ = C(X) ×C(g),C(Z),C(h) C(Y ). But ιW : W → C(W ) is an isomorphism with an open and closed C ∞ subscheme C0 (W ) ⊆ C(W ) by Definition 6.24. Part (c) follows from (a),(b). For (d), let g, h be interior and W be a fibre product in C∞ Schcto . If W is a

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6.6 Fibre products and corner functors

fibre product in C∞ Schcto,ex then (6.32) is Cartesian as C : C∞ Schcto,ex → C∞ Schcto preserves limits by Theorem 6.16, proving the ‘only if’ part. Suppose (6.32) is Cartesian. Let c : V → X and d : V → Y be morphisms in C∞ Schcto,ex with g ◦ c = h ◦ d. Consider the diagram: in

V in

b

d ΠW in C(c)◦ιV c

a

&

W

+ C(W )

f

ΠX

/. Y u

in

 / C(X) g

/, C(Y )

in C(f )

C(e)

u

e

*  u X

C(d)◦ιV

ΠY

C(g)

in

 / Z.u

h

C(h) in

 / C(Z)

ΠZ

Here morphisms labelled ‘in’ are interior. The morphisms C(c) ◦ ιV : V → C(X), C(d) ◦ ιV : V → C(Y ) are interior with C(g) ◦ C(c) ◦ ιV = C(h) ◦ C(d) ◦ ιV as g ◦ c = h ◦ d. So the Cartesian property of (6.32) gives unique interior b : V → C(W ) with C(e)◦b = C(c)◦ιV and C(f )◦b = C(d)◦ιV . Set a = ΠW ◦b : V → W . Then e ◦ a = e ◦ ΠW ◦ b = ΠX ◦ C(e) ◦ b = ΠX ◦ C(c) ◦ ιV = c ◦ ΠV ◦ ιV = c, and similarly f ◦ a = d. We claim that a : V → W is unique with e ◦ a = c and f ◦ a = d, which shows that W = X ×g,Z,h Y in C∞ Schcto,ex . To see this, note that as C is right adjoint to inc : C∞ Schcto ,→ C∞ Schcto,ex , there is a 1-1 correspondence between morphisms a : V → W in C∞ Schcto,ex and morphisms b : V → C(W ) in C∞ Schcto with a = ΠW ◦ b. But e ◦ a = c, f ◦ a = d imply that C(e) ◦ b = C(c) ◦ ιV , C(f ) ◦ b = C(d) ◦ ιV , and b is unique under these conditions as above. This proves the ‘if’ part. Here are three examples: Example 6.28. Define manifolds with corners X = [0, ∞)2 , Y = ∗ a point, and Z = [0, ∞), and exterior maps g : X → Z, h : Y → Z by g(x1 , x2 ) = x1 x2 and h(∗) = 0. Let X, Y , Z, g, h be the images of X, . . . , h under C∞ Schc ˜ = X ×g,Z,h Y exists FMan . As X, Y , Z are firm, a fibre product W c ∞ c c ˜ in C Schfi , with W = Spec C, where C = R[x1 , x2 ]/[x1 x2 = 0]. Here ˜ are not interior, as the relation x1 x2 = 0 is not of interior type. In C, W

158

Boundaries, corners, and the corner functor

Theorem 6.27 we have C(X) ×C(Z) C(Y ) ∼ = ([0, ∞) × {0}) q ({0} × [0, ∞)) q {(0, 0)},  {(x, y) ∈ X × Y : g(x) = h(y) = (x1 , x2 ) ∈ [0, ∞)2 : x1 x2 = 0 , where the map (6.31) takes (x1 , x2 ) 7→ (x1 , x2 ). Considering a neighbourhood of (0, 0) we see that no such open and closed subset W 0 exists in Theorem 6.27(c). Hence no fibre product W = X ×Z Y exists in C∞ Schcfi,in,ex , or C∞ Schcto,ex , or in any category like those in (5.10) in which the objects are interior, but the morphisms need not be interior. The next two examples are ‘b-transverse’, but not ‘c-transverse’, in the sense of Definition 6.31 below. Example 6.29. Define manifolds with corners X = Y = [0, ∞), Z = [0, ∞)2 , and interior maps g : X → Z, h : Y → Z by g(x) = (x, x) and h(y) = (y, y 2 ). Let X, Y , Z, g, h be the images of X, . . . , h under C∞ Schc FManc in . Then g : X → Z, h : Y → Z are morphisms in C∞ Schcto , in ˜ = X ×g,Z,h Y exists in C∞ Schcto by Theorem 5.34. so a fibre product W ˜ = Specc C, where Using the proof of Theorem 5.34 we can show that W in
2 ∼ ∼ ∞ C = Πsa (6.33) in R[x, y]/[x = y, x = y ] = R[x, y]/[x = y = 1] = C (∗), ˜ is a single ordinary point, which maps to x = 1 in X and y = 1 so W in Y . This is because on applying ΠZin in Theorem 4.48, x = y, x = y 2 force x = y = 1. ˆ = X ×Z Y in C∞ Schc is In contrast, the fibre product W fi,in
ˆ = Speccin R[x, y]/[x = y, x = y 2 ] , W which is two points, an ordinary point at x = y = 1, and another at x = y = 0 ˆ is also with a non-toric corner structure. Theorems 5.33–5.34 imply that W ∞ c the fibre product in C Schfi . For the fibre product W = X ×Z Y in C∞ Schcto,ex , we have C(X) ×C(Z) C(Y ) ∼ = {(1, 1)} q {(0, 0)}

in C∞ Schcto ,

˜ , and which is two ordinary points, (1, 1) from C0 (X) ×C0 (Z) C0 (Y ) ∼ =W (0, 0) from C1 (X) ×C2 (Z) C1 (Y ) ∼ = ∗ ×∗ ∗. (If we had formed the fibre ∞ c product in C Schfi,in there would have been a third, non-toric point over ˆ .) Also (0, 0) in C0 (X) ×C0 (Z) C0 (Y ) ∼ =W  {(x, y) ∈ X × Y : g(x) = h(y) = (1, 1), (0, 0) . Thus W 0 = C(X) ×C(Z) C(Y ) satisfies the condition in Theorem 6.27(c), and

∞ c 6.7 B- and c-transverse fibre products in Mangc in , C Schto

159

in fact W = C(X) ×C(Z) C(Y ) ∼ = {(1, 1)} q {(0, 0)} is a fibre product in C∞ Schcto,ex . Hence in this case fibre products X ×Z Y exist in C∞ Schcto and C∞ Schcto,ex , but are different, so inc : C∞ Schcto ,→ C∞ Schcto,ex does not preserve limits, in contrast to the last part of Theorem 5.33. The same holds for inc : C∞ Schcfi,Z ,→ C∞ Schcfi,Z,ex and inc : C∞ Schcfi,tf ,→ ˜ , W are the same in these categories. C∞ Schcfi,tf ,ex , as the fibre products W Example 6.30. Define manifolds with corners X = [0, ∞) × R, Y = [0, ∞) and Z = [0, ∞)2 , and interior maps g : X → Z, h : Y → Z by g(x1 , x2 ) = (x1 , x1 ex2 ) and h(y) = (y, y). Let X, Y , Z, g, h be the images of X, . . . , h C∞ Schc under FMan . Then g : X → Z, h : Y → Z are morphisms in C∞ Schcto , c ˜ = X ×g,Z,h Y exists in C∞ Schcto by Theorem 5.34. so a fibre product W ˜ = Specc C, where As for (6.33) we find that W in
x2 ∼ C = Πsa in R[x1 , x2 , y]/[x1 = y, x1 e = y] = R[x1 , x2 , y]/[x1 = y, x2 = 0] ∼ = R[y] ∼ = C ∞ ([0, ∞)), ˜ ∼ so W = [0, ∞). Similarly, we have a fibre product in C∞ Schcto   C(X) ×C(Z) C(Y ) ∼ = (y, 0, y) : y ∈ [0, ∞) q (0, x2 , 0) : x2 ∈ R , ˜ , and the second is where the first component is C0 (X) ×C0 (Z) C0 (Y ) ∼ =W C1 (X) ×C2 (Z) C1 (Y ). Also  {(x, y) ∈ X × Y : g(x) = h(y) = (x1 , x2 , y) ∈ [0, ∞) × R × [0, ∞) : either x1 = y, x2 = 0 or x1 = y = 0 . Considering a neighbourhood of (0, 0, 0) we see from Theorem 6.27(c) that no fibre product W = X ×Z Y exists in C∞ Schcto,ex . Thus in this case a fibre product X ×Z Y exists in C∞ Schcto , but not in C∞ Schcto,ex .

∞ c 6.7 B- and c-transverse fibre products in Mangc in , C Schto

6.7.1 Results on b- and c-transverse fibre products from [47] In [47, Def. 4.24] we define transversality for manifolds with g-corners: Definition 6.31. Let g : X → Z and h : Y → Z be morphisms in Mangc in or ˇ gc . Then: Man in (a) We call g, h b-transverse if b Tx g ⊕ b Ty h : b Tx X ⊕ b Ty Y → b Tz Z is surjective for all x ∈ X and y ∈ Y with g(x) = h(y) = z ∈ Z.

160

Boundaries, corners, and the corner functor

(b) We call g, h c-transverse if they are b-transverse, and for all x ∈ X and ˜x g ⊕ b N ˜y h : b N ˜x X ⊕ y ∈ Y with g(x) = h(y) = z ∈ Z, the linear map b N b ˜ b ˜ Ny Y → Nz Z is surjective, and the submonoid  ˜ xX ×M ˜yY : M ˜ x g(λ) = M ˜ y h(µ) in M ˜zZ ⊆ M ˜ xX ×M ˜yY (λ, µ) ∈ M ˜ xX × M ˜ y Y of M ˜ xX × M ˜yY . is not contained in any proper face F ( M Here are the main results on b- and c-transversality [47, Th.s 4.26–4.28]: Theorem 6.32. Let g : X → Z and h : Y → Z be b-transverse (or ctransverse) morphisms in Mangc in . Then C(g) : C(X) → C(Z) and C(h) : ˇ gc . C(Y ) → C(Z) are also b-transverse (or c-transverse) morphisms in Man in Theorem 6.33. Let g : X → Z and h : Y → Z be b-transverse morphisms gc in Mangc in . Then a fibre product W = X ×g,Z,h Y exists in Manin , with dim W = dim X + dim Y − dim Z. Explicitly, we may write  W ◦ = (x, y) ∈ X ◦ × Y ◦ : g(x) = h(y) in Z ◦ , and take W to be the closure W ◦ of W ◦ in X ×Y, and then W is an embedded submanifold of X ×Y, and e : W → X and f : W → Y act by e : (x, y) 7→ x and f : (x, y) 7→ y. Theorem 6.34. Suppose g : X → Z and h : Y → Z are c-transverse morgc phisms in Mangc in . Then a fibre product W = X ×g,Z,h Y exists in Man , with dim W = dim X + dim Y − dim Z. Explicitly, we may write  W = (x, y) ∈ X × Y : g(x) = h(y) in Z , and then W is an embedded submanifold of X × Y, and e : W → X and f : W → Y act by e : (x, y) 7→ x and f : (x, y) 7→ y. This W is also a fibre product in Mangc in , and agrees with that in Theorem 6.33. ˇ gc and Man ˇ gc : Furthermore, the following is Cartesian in both Man in C(W )  C(e) C(X)

C(f ) C(g)

/ C(Y ) C(h)  / C(Z).

(6.34)

Equation (6.34) has a grading-preserving property, in that if (w, β) ∈ Ci (W ) with C(e)(w, β) = (x, γ) ∈ Cj (X), and C(f )(w, β) = (y, δ) ∈ Ck (Y ), and C(g)(x, γ) = C(h)(y, δ) = (z, ) ∈ Cl (Z), then i + l = j + k. Hence ` Ci (W ) ∼ = j,k,l>0:i=j+k−l Cjl (X) ×C(g)|··· ,Cl (Z),C(h)|··· Ckl (Y ), writing Cjl (X) = Cj (X) ∩ C(g)−1 (Cl (Z)) and Ckl (Y ) = Ck (Y ) ∩ C(h)−1

∞ c 6.7 B- and c-transverse fibre products in Mangc in , C Schto

161

(Cl (Z)), which are open and closed in Cj (X), Ck (Y ). When i = 1, this gives a formula for ∂W .

C∞ Schcto

6.7.2 FMangc in

C∞ Schcto,ex

, FMangc

preserve b-, c-transverse fibre products

We generalize the last part of Theorem 2.54 to the corners case. C∞ Schcto

Theorem 6.35. (a) The functor FMangc in

∞ c : Mangc in → C Schto takes b-

∞ c and c-transverse fibre products in Mangc in to fibre products in C Schto . C∞ Schc

(b) The functor FMangc to,ex : Mangc → C∞ Schcto,ex takes c-transverse fibre products in Mangc to fibre products in C∞ Schcto,ex . Proof For (a), suppose g : X → Z and h : Y → Z are b-transverse in gc Mangc in , and let W = X ×g,Z,h Y be the fibre product in Manin given by Theorem 6.33, with projections e : W ∞ → X and f : W → Y . Write W , . . . , h C Schc ˜ = X ×g,Z,h Y be the for the images of W, . . . , h under FMangc to . Let W in ˜ → X, f˜ : W ˜ → Y, ˜: W fibre product in C∞ Schcto , with projections e ˜ which exists by Theorem 5.34. As g ◦ e = h ◦ f , the universal property of W ˜ ˜ gives a unique morphism b : W → W with e = e ˜ ◦ b and f = f ◦ b. We must prove b is an isomorphism. ˜ form a sheaf, this question is First note that as morphisms b : W → W ˜ local in W and hence in X, Y , Z, so it is enough to prove it over small open neighbourhoods of points x ∈ X, y ∈ Y , z ∈ Z with g(x) = h(y) = z. Thus we may replace X, Y, Z by small open neighbourhoods of x, y, z in X, Y, Z, or equivalently, by small open neighbourhoods of (δ0 , 0) in the local models XQ ×Rl for manifolds with g-corners, where Q is a toric monoid, as in Remark 3.10. This also replaces W by a small open neighbourhood of w = (x, y) in W . As small open balls about (δ0 , 0) in XQ × Rl are diffeomorphic to XQ × Rl , this means we can suppose that W, . . . , Z are of the form XQ × Rl . So we can take W ∼ = XP ×Rk ,

X∼ = XQ ×Rl ,

Y ∼ = XR ×Rm ,

Z∼ = XS ×Rn , (6.35)

for toric monoids P, Q, R, S and k, l, m, n > 0, where e, f, g, h : (δ0 , 0) 7→ (δ0 , 0). Then W, X, Y, Z are manifolds with g-faces, so by Theorem 5.22(a) we may take W = Speccin C ∞ in (W ), and similarly for X, Y , Z. Then e = c ∗ ∗ ∗ Specin (e , eex ), where e : C ∞ (X) → C ∞ (W ), e∗ex : In(X) q {0} → In(W ) q {0} are the pullbacks, and similarly for f , g, h. Form a commutative

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Boundaries, corners, and the corner functor

diagram in C∞ Ringscto : 2 C∞ in (Y )

(h∗ ,h∗ ex )

∗ (f ∗ ,fex )

ψ

C∞ in (Z)

)

5C

φ ∗ (g ∗ ,gex )

,

C∞ in (X)

β

) / C∞ (W ), 5 in

(e∗ ,e∗ ex )

∞ c where C is the pushout C ∞ C∞ in (X) qC ∞ in (Y ) in C Ringsto , which exin (Z) ists by Proposition 4.50, and φ, ψ are the projections to the pushout, and β exists by the universal property of pushouts. Since Speccin : (C∞ Ringscto )op → ˜ = Specc C and b = Specc β, C∞ Schcto preserves limits, it follows that W in in and it is enough to prove that β is an isomorphism. The pushout C is equivalent to a coequalizer diagram in C∞ Ringscto :

C∞ in (Z)

∗ (g ∗ ,gex )◦iC ∞ (X) in

∗

(h

,h∗ (Y ) ex )◦iC ∞ in

// C ∞ (X) ⊗to C ∞ (Y ) in in ∞

/C

(6.36)

∞ ∞ c to for C ∞ in (X) ⊗∞ C in (Y ) the coproduct in C Ringsto . Now one can show ∞ ∞ ∞ c that the coproduct C in (X) ⊗∞ C in (Y ) in C Rings is C ∞ in (X × Y ). Since this is toric, it coincides with the coproduct in C∞ Ringscto . Thus we may replace (6.36) by the coequalizer diagram in C∞ Ringscto :

C∞ in (Z)

((g◦πX )∗ ,(g◦πX )∗ ex ) ((h◦πY )∗ ,(h◦πY )∗ ex )

// C ∞ (X × Y ) in

/ C.

(6.37)

We deduce that  
∞ ∗ 0 ∗ 0 0 C∼ = Πsa in C in (X ×Y ) (g◦πX ) (c ) = (h◦πY ) (c ), c ∈ In(Z) . (6.38) ∞ c Here the term C ∞ in (X × Y )/[· · · ] is the coequalizer of (6.37) in C Rings . There is no need to also impose R-type relations (g ◦ πX )∗ (c) = (h ◦ πY )∗ (c) for x ∈ C ∞ (Z), as these are implied by (g ◦ πX )∗ (c0 ) = (h ◦ πY )∗ (c0 ) for c0 = Ψexp (c) in In(X). The proof of Theorem 4.20(b) then implies that ∞ c Πsa in (· · · ) is the coequalizer of (6.37) in C Ringsto . sa ∞ In (6.38), we think of Πin as mapping C Ringscfi,in → C∞ Ringscto . It does not change C ∞ in (X × Y ) as this is already toric, but it can add extra relations. From the construction of Πsa in in the proof of Theorem 4.48, we can prove that  0 0 0 0 0 C∼ = C∞ in (X × Y ) c1 = c2 : c1 , c2 ∈ In(X × Y ), for some c ∈ In(Z)  and n > 1 we have c01 (g ◦ πX )∗ (c0 )n = c02 (h ◦ πY )∗ (c0 )n . (6.39)

∞ c 6.7 B- and c-transverse fibre products in Mangc in , C Schto

163

Here imposing relations c01 = c02 if c01 (g ◦ πX )∗ (c0 ) = c02 (h ◦ πY )∗ (c0 ) for c0 ∈ In(Z) has the effect of making (6.39) integral, that is, it applies ΠZin . (Note that taking c01 = (h ◦ πY )∗ (c0 ) and c02 = (g ◦ πX )∗ (c0 ) recovers the relations in (6.38), but c01 (g ◦ πX )∗ (c0 ) = c02 (h ◦ πY )∗ (c0 ) also includes relations in the integral completion.) Then forcing c01 = c02 when c01 (g ◦ πX )∗ (c0 )n = c02 (h ◦ πY )∗ (c0 )n for n > 1 also makes (6.39) torsion-free and saturated, that tf sa sa tf Z is, it applies Πsa tf ◦ ΠZ , so overall we apply Πin = Πtf ◦ ΠZ ◦ Πin . ∞ Using (6.39), we now show that β = (β, βex ) : C → C in (W ) is an isomorphism, where W is an embedded submanifold in X × Y as in Theorem 6.33. To show that β, βex are surjective, it is enough to show that the restriction maps C ∞ (X × Y ) → C ∞ (W ), In(X × Y ) → In(W ) are surjective. That is, all smooth or interior maps W → R, [0, ∞) should extend to X × Y . Locally near (x, y) in W, X × Y this follows from extension properties of smooth maps on XP × Rl in [47, Prop. 3.14], and globally it follows from (6.35), which ensures there are no obstructions from multiplicity functions µc0 as in Remark 3.15. To show β (or βex ) is injective, we need to show that if c1 , c2 ∈ C ∞ (X × Y ) (or c01 , c02 ∈ In(X × Y )), then c1 |W = c2 |W (or c01 |W = c02 |W ) if and only if c1 − c2 lies in the ideal I in C ∞ (X × Y ) generated by the relations in (6.39) (or c01 ∼ c02 , where ∼ is the equivalence relation on In(X × Y ) generated by the relations in (6.39)). Since W is an embedded submanifold defined as a subset of X × Y by the b-transverse relations (g ◦ πX )∗ (c0 ) = (h ◦ πY )∗ (c0 ) for c0 ∈ In(Z), this is indeed true, though proving it carefully requires a little work on ideals in C ∞ (X × Y ). Therefore β is an isomorphism, completing part (a). For (b), suppose g : X → Z and h : Y → Z are c-transverse in Mangc in , which implies g, h b-transverse, and let W = X ×g,Z,h Y be the fibre product gc in both Mangc given by Theorems 6.33–6.34. Then (6.34) is in and Man gc Cartesian in Manin by Theorem 6.34. Write W , . . . , h for the images of C∞ Schc W, . . . , h under FMangc to . Then part (a), Proposition 6.12, and Theorem 6.16 in imply that W = X ×Z Y in C∞ Schcto , and (6.32) is Cartesian in C∞ Schcto . Hence Theorem 6.27(d) says W = X ×Z Y in C∞ Schcto,ex , and the theorem follows. Example 6.36. Let X = Y = Z = [0, ∞), and g : X → Z, h : Y → Z map g(x) = x2 , h(y) = y 2 . Then g, h are b- and c-transverse, and the fibre product gc W = X ×g,Z,h Y in Mangc is W = [0, ∞) with projections in and Man e : W → X, f : W → Y mapping e(w) = w, f (w) = w. C∞ Schc Let W , . . . , h be the images of W, . . . , h under FMan . Then Theorem c ∞ c ∞ 6.35 implies that W = X ×g,Z,h Y in C Schto and C Schcto,ex . But what about fibre products X ×g,Z,h Y in other categories?

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As X, Y , Z = Speccin C ∞ in ([0, ∞)) are affine, we can show that the fibre ˆ = X ×Z Y in C∞ Schc is Specc C, where C = R[x, y]/[x2 = product W in fi,in y 2 ]. Then C ]in ∼ = Z × Z2 , identifying [x] = (1, 0) and [y] = (1, 1) in Z × Z2 . We see that C is firm and integral, but not torsion-free (as C ]in has torsion Z2 ) ˆ is also firm and or saturated (as x, y ∈ C in with x 6= y but x2 = y 2 ). Thus W ˆ. integral, but not torsion-free or saturated, and W 6∼ =W C∞ Schc C∞ SchcZ gc ∞ c This shows FMangc : Manin → C SchZ and FMangc fi,in : Mangc in in in → C∞ Schcfi,in need not preserve b-transverse fibre products. Example 6.29 ˆ is not integral. One conclusion is that in Theorem above is similar, but W 6.35 it is essential to work in the categories C∞ Schcto , C∞ Schcto,ex of toric C ∞ -schemes with corners, rather than larger categories such as C∞ Schcfi,tf , C∞ Schcfi,Z , C∞ Schcfi,in from §5.6–§5.7. Thus, C∞ Schcto , C∞ Schcto,ex gc may be regarded as natural generalizations of Mangc in , Man .

6.7.3 Restriction to manifolds with corners There is an easy way to restrict Theorem 6.35 to Mancin , Manc : Definition 6.37. Let g : h → Z and h : Y → Z be morphisms in Mancin . Following the second author [49, §2.5.4], we call g, h sb-transverse or sctransverse (short for strictly b-transverse or strictly c-transverse) if they are btransverse or c-transverse in Mangc in , respectively, and the fibre product W = X ×g,Z,h Y in Mangc is a manifold with corners, not just g-corners. In [49, in §2.5.4] we explain the sb- and sc-transverse conditions explicitly. Theorems 6.33–6.34 imply that sb- and sc-transverse fibre products exist in Mancin and Manc , respectively. Then Theorem 6.35 implies: C∞ Schc

Corollary 6.38. (a) The functor FManc to : Mancin → C∞ Schcto takes sbin and sc-transverse fibre products in Mancin to fibre products in C∞ Schcto . C∞ Schc

(b) The functor FManc to,ex : Manc → C∞ Schcto,ex takes sc-transverse fibre products in Manc to fibre products in C∞ Schcto,ex .

7 Modules, and sheaves of modules

In this chapter we discuss modules over C ∞ -rings with corners, sheaves of O X -modules on C ∞ -schemes with corners X, and (b-)cotangent modules and sheaves. This extends the corresponding concepts for C ∞ -schemes from §2.3, §2.7 and §2.8.

7.1 Modules over C ∞ -rings with corners, (b-)cotangent modules In §2.3 we discussed modules over C ∞ -rings. Here is the corners analogue: Definition 7.1. Let C = (C, C ex ) be a C ∞ -ring with corners. A module M over C, or C-module, is a module over C regarded as a commutative Ralgebra as in Definition 2.27, and morphisms of C-modules are morphisms of R-algebra modules. Then C-modules form an abelian category, which we write as C-mod. The basic theory of §2.3 extends trivially to the corners case. So if φ = (φ, φex ) : C → D is a morphism in C∞ Ringsc then using φ : C → D we get functors φ∗ : C-mod → D-mod mapping M 7→ M ⊗C D, and φ∗ : D-mod → C-mod mapping N 7→ N . One might expect that modules over C = (C, C ex ) should also include some kind of monoid module over C ex , but we do not do this. Example 7.2. Let X be a manifold with corners (or g-corners) and E → X be a vector bundle, and write Γ∞ (E) for the vector space of smooth sections e of E. This is a module over C ∞ (X), and hence over the C ∞ -rings with corners C ∞ (X) and C ∞ in (X) from Example 4.18. Section 2.3 discussed cotangent modules of C ∞ -rings, the analogues of 165

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Modules, and sheaves of modules

cotangent bundles of manifolds. As in §3.5 manifolds with corners X have cotangent bundles T ∗ X which are functorial over smooth maps, and b-cotangent bundles b T ∗ X, which are functorial only over interior maps. In a similar way, for a C ∞ -ring with corners C we will define the cotangent module ΩC , and if C is interior we will also define the b-cotangent module b ΩC . Definition 7.3. Let C = (C, C ex ) be a C ∞ -ring with corners. Define the cotangent module ΩC of C to be the cotangent module ΩC of §2.3, regarded as a C-module. If φ = (φ, φex ) : C → D is a morphism in C∞ Ringsc then from Ωφ in §2.3 we get functorial morphisms Ωφ : ΩC → ΩD = φ∗ (ΩD ) in C-mod and (Ωφ )∗ : φ∗ (ΩC ) = ΩC ⊗C D → ΩD in D-mod. Definition 7.4. Let C = (C, C ex ) be an interior C ∞ -ring with corners, so that C ex = C in q {0C ex } with C in a monoid. Let M be a C-module. A b-derivation is a map din : C in → M satisfying the following corners analogue of (2.5): for any interior g : [0, ∞)n → [0, ∞) and elements c01 , . . . , c0n ∈ C in , then n
P din Ψg (c01 , . . . , c0n ) − Φg−1 xi i=1

∂g ∂xi

(c01 , . . . , c0n ) · din c0i = 0.

(7.1)

∂g Here g interior implies that g −1 xi ∂x : [0, ∞)n → R is smooth. i It is easy to show that (7.1) implies:

(i) din : C in → M is a monoid morphism, where M is a monoid over addition. (ii) d = din ◦ Ψexp : C → M is a C ∞ -derivation in the sense of Definition 2.29. (iii) din ◦ Ψexp (Φi (c0 )) = Φi (c0 ) · din c0 for all c0 ∈ C in . As interior g : [0, ∞)n → [0, ∞) are of the form g = xa1 1 · · · xann exp f (x1 , 0 0 0 0an 1 . . . , xn ), with Ψg (c01 , . . . , c0n ) = c0a 1 · · · cn Ψexp (Φf (c1 , . . . , cn )) for c1 , . . . , 0 cn ∈ C in , we can show that (i)–(iii) are equivalent to (7.1). We call such a pair (M, din ) a b-cotangent module for C if it has the universal property that for any b-derivation d0in : C in → M 0 , there exists a unique morphism of C-modules b λ : M → M 0 with d0in = b λ ◦ din . There is a natural construction for a b-cotangent module: we take M to be the quotient of the free C-module with basis of symbols din c0 for c0 ∈ C in by the C-submodule spanned by all expressions of the form (7.1). Thus b-cotangent modules exist, and are unique up to unique isomorphism. When we speak of ‘the’ b-cotangent module, we mean that constructed above, and we write it as dC,in : C in → b ΩC . Since dC,in ◦ Ψexp : C → b ΩC is a C ∞ -derivation, the universal property of ΩC = ΩC in §2.3 implies that there is a unique C-module morphism IC : ΩC → b ΩC with dC,in ◦ Ψexp = IC ◦ dC : C → b ΩC . Let φ : C → D be a morphism in C∞ Ringscin . Then we have a monoid

7.2 (B-)cotangent modules for manifolds with (g-)corners

167

morphism φin : C in → Din . Regarding b ΩD as a C-module using φ : C → D, then dD,in ◦ φin : C in → b ΩD becomes a b-derivation. Thus by the universal property of b ΩC , there exists a unique C-module morphism b Ωφ : b ΩC → b ΩD with dD,in ◦ φin = b Ωφ ◦ dC,in . This then induces a morphism of Dmodules (b Ωφ )∗ : φ∗ (b ΩC ) = b ΩC ⊗C D → b ΩD . If φ : C → D, ψ : D → E are morphisms in C∞ Ringscin then b Ωψ◦φ = b Ωψ ◦ b Ωφ : b ΩC → b ΩE . Remark 7.5. (a) We may think of din c0 above as d(log c0 ). As in Example 7.6(b), if X is a manifold with corners and c0 : X → [0, ∞) is interior, then d(log c0 ) is well defined section of b T ∗ X. This holds even at ∂X where we allow c0 = 0 and log c0 is undefined, as b T ∗ X is spanned by x−1 1 dx1 , . . . , m x−1 dx , dx , . . . , dx in local coordinates (x , . . . , x ) ∈ R k k+1 m 1 m k , and the k −1 −1 0 factors x1 , . . . , xk control the singularities of d log c . (b) In Definition 7.4 we could have omitted the condition that C be interior, and considered monoid morphisms dex : C ex → M such that d = dex ◦ Ψexp : C → M is a C ∞ -derivation and dex ◦ Ψexp (Φi (c0 )) = Φi (c0 )dex c0 for all c0 ∈ C ex . However, since c0 · 0C ex = 0C ex for all c0 ∈ C ex we would have dex c0 + dex 0C ex = dex 0C ex in M , so dex c0 = 0, and this modified definition would give b ΩC = 0 for any C. To get a nontrivial definition we took C to be interior, and defined din only on C in = C ex \ {0C ex }. C∞ Ringsc ∞ If C lies in the image of ΠC∞ Ringsin Ringsc ,→ C∞ Ringscin from c : C Definition 4.11 then b ΩC = 0, since C in then contains a zero element 0C in with c0 · 0C in = 0C in for all c0 ∈ C in . If din : C in → M is a b-derivation then it is a morphism from a monoid to an abelian group, and so factors through π gp : C in → (C in )gp . This suggests that b-cotangent modules may be most interesting for integral C ∞ -rings with corners, as in §4.7, for which π gp : C in → (C in )gp is injective.

7.2 (B-)cotangent modules for manifolds with (g-)corners Example 7.6. (a) Let X be a manifold with corners. Then Γ∞ (T ∗ X) is a module over the R-algebra C ∞ (X), and so over the C ∞ -rings with corners C ∞ (X) and C ∞ in (X) from Example 4.18. The exterior derivative d : C ∞ (X) → Γ∞ (T ∗ X) is a C ∞ -derivation, and there is a unique morphism λ : ΩC ∞ (X) → Γ∞ (T ∗ X) such that d = λ ◦ d. (b) Let X be a manifold with corners (or g-corners), with b-cotangent bundle b ∗ T X as in §3.5. Example 4.18(b) defines a C ∞ -ring with corners C ∞ in (X) = C with C in = In(X), the monoid of interior maps c0 : X → [0, ∞). We have

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a C ∞ (X)-module Γ∞ (b T ∗ X). Define din : In(X) → Γ∞ (b T ∗ X) by din (c0 ) = c0−1 · b dc0 = b d(log c0 ),

(7.2)

∗ where b d = IX ◦ d is the composition of the exterior derivative d : C ∞ (X) → ∞ ∗ ∗ Γ (T X) with the projection IX : T ∗ X → b T ∗ X. Here (7.2) makes sense on the interior X ◦ where c0 > 0, but has a unique smooth extension over X \ X ◦ . We can now show that din : In(X) → Γ∞ (b T ∗ X) is a b-derivation in → the sense of Definition 7.4, so there is a unique morphism b λ : b ΩC ∞ in (X) Γ∞ (b T ∗ X) such that din = b λ ◦ din .

Theorem 7.7. (a) Let X be a manifold with faces, as in Definition 3.13, such that ∂X has finitely many connected components. Then Γ∞ (T ∗ X) is the cotangent module of C ∞ (X). That is, λ from Example 7.6(a) is an isomorphism. (b) Let X be a manifold with faces, or a manifold with g-faces, as in Definition 4.37, such that ∂X has finitely many connected components. Then Γ∞ (b T ∗ X) b is the b-cotangent module of C ∞ in (X). That is, λ from Example 7.6(b) is an isomorphism. Proof The first part of the proof is common to both (a) and (b). Let X be a manifold with faces, or with g-faces, and suppose ∂X has finitely many connected components, which we write ∂1 X, . . . , ∂N X. We will show that the ∞ sharpened monoid Cin (X)] is finitely generated. 0 ∞ ∞ ∞ Write [c ] ∈ Cin (X)] for the Cin (X)× -orbit of c0 ∈ Cin (X). Each such c0 ∞ vanishes to some order ai ∈ N on ∂i X for i = 1, . . . , N . The map Cin (X) → N 0 N , c 7→ (a1 , . . . , aN ) is a monoid morphism, which descends to a morphism ∞ Cin (X)] → NN , [c0 ] 7→ (a1 , . . . , aN ), as NN is sharp. ∞ It is easy to show that this morphism Cin (X)] → NN is injective. If X is a ∞ manifold with faces it is also surjective, so Cin (X)] ∼ = NN . If X is a manifold ∞ ] with g-faces, the image of Cin (X) is the submonoid of NN satisfying some linear equations coming from the higher-dimensional g-corner strata. Therefore ∞ Cin (X)] is a toric monoid, and so finitely generated. ∞ ∞ Let x1 , . . . , xk ∈ Cin (X) whose images [x1 ], . . . , [xk ] generate Cin (X)] . Following the proof of the Whitney Embedding Theorem, we may choose a smooth map (xk+1 , . . . , xl ) : X → Rl−k for some l with l − k > 2 dim X which is a closed embedding (in a weak sense, in which the inclusion [0, ∞) ,→ R counts as an embedding). Then F := (x1 , . . . , xk , xk+1 , . . . , xl ) : X ,→ [0, ∞)k × Rl−k = Rlk is an interior closed embedding in a stronger sense, which embeds the corner strata of X into corner strata of Rlk . Here as X has (g-)faces, by Proposition 4.36 and Definition 4.37 the natural morphism λx : (C ∞ (X))x∗ → C ∞ x (X) is an isomorphism for all x ∈ X,

169

7.2 (B-)cotangent modules for manifolds with (g-)corners

∞ ∞ so as [x1 ], . . . , [xk ] generate Cin (X)] , they also generate Cx,in (X)] for each x ∈ X. Geometrically this means that X near x is nicely embedded in Rlk near F (x), in such a way that the corner strata of X near x all come from corner ˜ = F (X), as a closed, embedded submanifold strata of Rlk near F (x). Write X l of Rk . l ∞ ∞ c Thus we have a morphism F ∗ : C ∞ in (Rk ) → C in (X) in C Ringsin . We claim that F ∗ is surjective. To see this, note that F ∗ is surjective on C ∞ -rings as it is a closed embedding, so every smooth function F (X) → R extends ∗ ] ∞ to a smooth function Rlk → R in the usual way. Also (Fin ) : Cin (Rlk )] ∼ = ak a1 k ∞ ] N → Cin (X) is surjective as it maps (a1 , . . . , ak ) 7→ [x1 · · · xk ], where ∞ ∗ ∞ ∞ [x1 ], . . . , [xk ] generate Cin (X)] . Surjectivity of Fin : Cin (Rlk ) → Cin (X) follows. l ∞ Therefore C ∞ in (X) is isomorphic to the quotient of C in (Rk ) by relations ∗ ˜ as a generating the kernel of Fin , or equivalently, by equations defining X submanifold of Rlk . If X is a manifold with faces then relations of type f = 0 for f ∈ C ∞ (Rlk ) are sufficient, so in the notation of Definition 4.24 we have 
l ∞ ∼ ∞ l C∞ (7.3) ˜ =0 . in (X) = C in (Rk ) f = 0 : f ∈ C (Rk ), f |X

If X is a manifold with g-faces we may also need relations of type g = h for ∞ g, h ∈ Cin (Rlk ), giving 
l ∞ ∼ ∞ l C∞ ˜ =0 in (X) = C in (Rk ) f = 0 : f ∈ C (Rk ), f |X (7.4)   ∞ g = h : g, h ∈ Cin (Rlk ), g|X˜ = h|X˜ . We now prove part (a), so suppose X is a manifold with faces and ∂X has finitely many boundary components, so that (7.3) holds. Consider the commutative diagram of C ∞ (X)-modules with exact rows: 0

0

/ K = Ker(ΩF ∗ ) 

∞ (X) = k ⊗C ∞ (Rk ) C / ΩC ∞ in (Rl ) l

hdx1 , . . . , dxl i⊗R C ∞ (X) ∼ =

µ

/ Γ∞ (Ker(dF ∗ ))

/



/ ΩC ∞ (X) in

ΩF ∗

λ



∞ ∗ Γ (F (T Rkl )) = Γ (dF ) ∞ ∗ Γ (T X) hdx1 , . . . , dxl i⊗R C ∞ (X)

∞

∗

∗

/

/0 (7.5) / 0.

Here as F ∗ : C ∞ (Rkl ) → C ∞ (X) is surjective, ΩF ∗ is surjective, and the top row is exact. As the exact sequence 0 → Ker(dF ∗ ) → F ∗ (T ∗ Rkl ) → T ∗ X → 0 of vector bundles on X splits, the bottom row is exact. It is easy to check that ΩC ∞ = ΩC ∞ (Rkl ) = hdx1 , . . . , dxl i ⊗R C ∞ (Rkl ) satisfies the universal k in (Rl ) property for cotangent modules. Since Γ∞ (T ∗ Rkl ) = hdx1 , . . . , dxl iC ∞ (Rkl ) , we see that the two centre terms are hdx1 , . . . , dxl i ⊗R C ∞ (X), and are iso-

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morphic. Clearly the right hand square commutes. Hence by exactness there is a unique map µ : K → Γ∞ (Ker(dF ∗ )) making the diagram commute. As the right hand square of (7.5) commutes, with the central column an isomorphism, we see that λ is surjective. To prove that λ is injective, and hence ˜ is cut out an isomorphism, it is enough to show that µ is surjective. Since X k l ∞ transversely in Rl by equations f = 0 for f ∈ C (Rk ) with f |X˜ = 0, as in (7.3), we see that the vector bundle Ker(dF ∗ ) → X is spanned over C ∞ (X) by sections F ∗ (df ) for f ∈ C ∞ (Rlk ) with f |X˜ = 0. But then (df ) ⊗ 1 lies in K = Ker(ΩF ∗ ) with F ∗ (df ) = µ((df ) ⊗ 1), so the image of µ contains a spanning set, and µ is surjective. This proves (a). For (b), allow X to have faces or g-faces, so that (7.4) holds, and replace (7.5) by the commutative diagram with exact rows

0

/ b K = Ker(b ΩF ∗ )

bΩ

C ∞ (Rk ) ⊗C ∞ (Rk ) C

dxk+1 , . . . , dxl i⊗R

b



0

∼ =

µ

/ Γ∞ (Ker(b dF ∗ ))

∞ (X)

in l l / = hx−1 −1 1 dx1 , . . . , xk dxk ,

∞

∗ b

Γ (F ( T

∗

C ∞ (X)

b

/ b ΩC ∞ (X) in

b



Rkl ))

ΩF ∗

∞ b

/ = hx−1 dx1 , . . . , x−1 dxkΓ, 1 k

λ

 / Γ∞ (b T ∗ X)

( dF ∗ )

/0

/ 0.

dxk+1 , . . . , dxl i⊗R C ∞ (X)

k −1 −1 ∞ Here b ΩC ∞ k = hx 1 dx1 , . . . , xk dxk , dxk+1 , . . . , dxl i⊗R C (Rl ) satisin (Rl ) fies the universal property for b-cotangent modules. The argument above gives a unique map b µ making the diagram commute, and shows b λ is surjective. To prove b λ is injective, and hence an isomorphism, it is enough to show that b µ is surjective. ˜ is cut out b-transversely in Rk by equations f = 0 for f ∈ C ∞ (Rl ) Since X k l ∞ with f |X˜ = 0, and (if X has g-faces) equations g = h for g, h ∈ Cin (Rlk ) with g|X˜ = h|X˜ , we see that the vector bundle Ker(b dF ∗ ) → X is spanned over C ∞ (X) by sections F ∗ (b d exp(f )) for f ∈ C ∞ (Rlk ) with f |X˜ = 0 and ∞ F ∗ (b dg−b dh) for g, h ∈ Cin (Rlk ) with g|X˜ = h|X˜ . But then b d exp(f )⊗1 and (b dg−b dh)⊗1 lie in b K = Ker(b ΩF ∗ ) with F ∗ (b d exp(f )) = b µ(b d exp(f )⊗ 1) and F ∗ (b dg − b dh) = b µ((b dg − b dh) ⊗ 1), so the image of b µ contains a spanning set, and b µ is surjective. This proves (b).

Remark 7.8. The first author [25, Prop.s 4.7.5 & 4.7.9] proves by a different method that if we allow ∂X to have infinitely many connected components, then λ is still an isomorphism in Theorem 7.7(a), and b λ is surjective in (b).

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7.3 Some computations of b-cotangent modules

7.3 Some computations of b-cotangent modules The next proposition generalizes Example 2.31: Proposition 7.9. (a) Let A, Ain be sets and FA,Ain the free interior C ∞ -ring in with corners from Definition 4.22, with generators xa ∈ FA,Ain , ya0 ∈ FA,A in for a ∈ A, a0 ∈ Ain . Then there is a natural isomorphism

 b ΩFA,Ain ∼ = din Ψexp (xa ), din ya0 : a ∈ A, a0 ∈ Ain R ⊗R FA,Ain . (7.6) (b) Suppose C is defined by a coequalizer diagram (4.13) in C∞ Ringscin . Then writing (xa )a∈A , (ya0 )a0 ∈Ain and (˜ xb )b∈B , (˜ yb0 )b0 ∈Bin for the generators of FA,Ain and FB,Bin , we have an exact sequence



din Ψexp (˜ xb ), b ∈ B, γ / din Ψexp (xa ), a ∈ A, δ / b / 0 (7.7) ΩC din y˜b0 , b0 ∈ Bin R ⊗R C din ya0 , a0 ∈ Ain R ⊗R C in C-mod, where if α = (α, αex ) and β = (β, βex ) in (4.13) map
α:x ˜b 7−→ eb (xa )a∈A , (ya0 )a0 ∈Ain ,
αex : y˜b0 7−→ gb0 (xa )a∈A , (ya0 )a0 ∈Ain ,
β:x ˜b 7−→ fb (xa )a∈A , (ya0 )a0 ∈Ain ,
βex : y˜b0 7−→ hb0 (xa )a∈A , (ya0 )a0 ∈Ain ,

(7.8)

for eb , fb , gb0 , hb0 depending only on finitely many xa , ya0 , then γ, δ are given by X
∂eb (xa )a∈A , (ya0 )a0 ∈Ain γ(din Ψexp (˜ xb )) = φ ∂x a

a∈A ∂fb 0 )a0 ∈A − ∂x (x ) , (y din Ψexp (xa ) a a∈A a in a X
∂eb (xa )a∈A , (ya0 )a0 ∈Ain + φ ya0 ∂y 0 a

a0 ∈Ain ∂fb − ya0 ∂y (xa )a∈A , (ya0 )a0 ∈Ain din ya0 , a0 X
∂gb0 (xa )a∈A , (ya0 )a0 ∈Ain γ(din y˜b0 ) = φ gb−1 (7.9) 0 ∂x a

a∈A −1 ∂hb0 − hb0 ∂xa (xa )a∈A , (ya0 )a0 ∈Ain din Ψexp (xa ) X
∂gb0 + φ ya0 gb−1 (xa )a∈A , (ya0 )a0 ∈Ain 0 ∂y a0

a0 ∈Ain ∂hb0 0 0 0 − ya0 h−1 b0 ∂y 0 (xa )a∈A , (ya )a ∈Ain din ya , a

δ(din Ψexp (xa )) = din ◦ φin (Ψexp (xa )),

δ(din ya0 ) = din ◦ φin (ya0 ).

Hence if C is finitely generated (or finitely presented) in the sense of Definition 4.40, then b ΩC is finitely generated (or finitely presented). Proof Part (a) follows from (b) with B = Bin = ∅, so it is enough to prove

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Modules, and sheaves of modules

(b). For (b), first note that by Definition 7.4, we have 

 b ΩC = din c0 : c0 ∈ C in R ⊗R C hall relations (7.1)i.

(7.10)

Now let S ⊆ C in be a generating subset, that is, every c0 ∈ C in may be written as c0 = Ψg (s1 , . . . , sn ) for some interior g : [0, ∞)n → [0, ∞) and c0

0

s1 , . . . , sn ∈ S. For each c0 ∈ C in \ S, choose interior g c : [0, ∞)n → 0 0 0 0 [0, ∞) and sc1 , . . . , scnc0 ∈ S with c0 = Ψgc0 (sc1 , . . . , scnc0 ), using the Axiom of Choice. Then (7.1) gives 0

din c0 =

c n P

i=1

Φ

0

c0

(g c0 )−1 xi ∂g ∂x

0

0

(sc1 , . . . , scnc0 ) · din sci .

(7.11)

i

Hence b ΩC is spanned over C by din s for s ∈ S, and we may write b ΩC as hdin s : s ∈ SiR ⊗R C/(relations). The relations required are the result of taking all relations (7.1) in (7.10), and using (7.11) to substitute for all terms din c0 for c0 ∈ C in \ S, so that the resulting relation is a finite C-linear combination of din s for s ∈ S. A calculation using Definition 4.4(ii) shows that this relation is of the form (7.1) with c01 , . . . , c0n ∈ S. Hence

 

 b ΩC ∼ = din c0 : c0 ∈ S R ⊗R C relations (7.1) with c01 , . . . , c0n ∈ S . (7.12) Now let C be as in (4.13). Then C has R type generators φ(xa ) for a ∈ A and [0, ∞) type generators φin (ya0 ) for a0 ∈ Ain . But (7.12) only allows for [0, ∞) type generators. To convert R type generators to [0, ∞) type generators observe that, in a similar way to the proof of Proposition 4.7, any smooth function R → R or R → [0, ∞) may be extended to a smooth function [0, ∞)2 → R or [0, ∞)2 → [0, ∞) by mapping t ∈ R to (x, y) = (et , e−t ) ∈ [0, ∞)2 , which then satisfies xy = 1. Thus, we may replace R type generators φ(xa ) ∈ C by pairs of [0, ∞) type generators φin ◦ Ψexp (xa ), φin ◦ Ψexp (−xa ) in C in , satisfying the relation (φin ◦ Ψexp (xa )) · (φin ◦ Ψexp (−xa )) = 1. But applying din to this relation yields  din (φin ◦ Ψexp (xa )) + din (φ  in ◦ Ψexp (−xa )) = 0. Hence by taking S = φin ◦ Ψexp (±xa ) : a ∈ A ∪ φin (ya0 ) : a0 ∈ Ain in (7.12), we see that ΩC ∼ (7.13) =

 din (φin ◦ Ψexp (±xa )) : a ∈ A, din (φin (ya0 )) : a0 ∈ Ain R ⊗R C

. din (φin ◦ Ψexp (xa )) + din (φin ◦ Ψexp (−xa )) = 0, a ∈ A, other relations (7.1) with c0i = φin ◦ Ψexp (±xa ), φin (ya0 )

b

Clearly we can omit the generator din φin ◦ Ψexp (−xa ) and the first relation on the bottom of (7.13) for a ∈ A. We can also omit the ‘φin ’ in

7.3 Some computations of b-cotangent modules

173

din (φin ◦ Ψexp (±xa )), din (φin (ya0 )), as these are only formal symbols anyin way. Furthermore, in FA,A in (4.13), the only relations between the generators in 0 Ψexp (±xa ), a ∈ A, ya0 , a ∈ Ain , are generated by Ψexp (xa )·Ψexp (−xa ) = 1 for a ∈ A. Hence in (7.13), the ‘other relations’ must all come from FB,Bin . in But the relations in FA,A from FB,Bin are generated by αin (Ψexp (±˜ xb )) = in βin (Ψexp (±˜ xb )), b ∈ B, and αin (˜ yb0 ) = βin (˜ yb0 ), b0 ∈ Bin . Putting all this together and using (7.8) yields

b

 din Ψexp (xa ) : a ∈ A, din ya0 : a0 ∈ Ain R ⊗R C

ΩC ∼ . =

din ◦ Ψexp eb (xa )a∈A , (ya0 )a0 ∈Ain

0 0 −din ◦ Ψexp fb (xa )a∈A , (y

a )a ∈Ain = 0, b ∈ B, 0 0 0 din gb (xa )a∈A , (ya )a ∈Ain

 −din hb0 (xa )a∈A , (ya0 )a0 ∈Ain = 0, b0 ∈ Bin

(7.14)

By (7.1) and (7.9), the relations in (7.14) are γ(din Ψexp (˜ xb )) = 0, b ∈ B, 0 b ∼ 0 and γ(din y˜b ) = 0, b ∈ Bin . Hence (7.14) shows that ΩC = Coker γ. Since this isomorphism is realized by mapping din Ψexp (xa ) 7→ din ◦ φin (Ψexp (xa ), din ya0 7→ din ◦ φin (ya0 ), which is δ in (7.9), we see that (7.7) is exact, proving (b). We can compute the b-cotangent module of an interior C ∞ -ring with corners C modified by generators and relations, as in §4.5. Proposition 7.10. Suppose C ∈ C∞ Ringscin , and we are given sets A, Ain , B, Bin , so that C(xa : a ∈ A)[ya0 : a0 ∈ Ain ] ∈ C∞ Ringscin , and elements fb ∈ C(xa : a ∈ A)[ya0 : a0 ∈ Ain ] and gb0 , hb0 ∈ C(xa : a ∈ A)[ya0 : a0 ∈ Ain ]in for b ∈ B and b0 ∈ Bin . Define D ∈ C∞ Ringscin by

  D = C(xa : a ∈ A)[ya0 : a0 ∈ Ain ] fb = 0, b ∈ B gb0 = hb0 , b0 ∈ Bin , (7.15) in the notation of Definition 4.24, with natural morphism π : C → D. Then we have an exact sequence in D-mod:

hdin Ψexp (˜ xb ), b ∈ BiR ⊗R D ⊕hdin (˜ yb0 ), b0 ∈ Bin iR ⊗R D

b

α

ΩC ⊗C D⊕

/ hdin Ψexp (xa ), a ∈ AiR ⊗R D ⊕hdin (ya0 ),

a0 ∈ A

in iR ⊗R D

β

/ b ΩD / 0. (7.16)

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Modules, and sheaves of modules

Here we regard α as mapping to b

ΩC(xa :a∈A)[ya0 :a0 ∈Ain ] ⊗C(xa :a∈A)[ya0 :a0 ∈Ain ] D

= b ΩC⊗∞ FA,Ain ⊗C⊗∞ FA,Ain D ∼ = b ΩC ⊗C (C ⊗∞ FA,Ain ) ⊕ b Ω

FA,Ain


⊗FA,Ain (C ⊗∞ FA,Ain )

⊗C⊗∞ FA,Ain D b ∼ = ΩC ⊗C D ⊕ b ΩFA,Ain ⊗FA,Ain D ∼ = b ΩC ⊗C D ⊕ hpa , a ∈ AiR ⊗R D ⊕ hqa0 , a0 ∈ Ain iR ⊗R D, using Proposition 7.9(a) in the last step, and then α, β act by

 α=

 din Ψexp (˜ xb ) 7−→ din ◦ Ψexp (fb ) , din (˜ yb0 ) 7−→ din (gb0 ) − din (hb0 )

b  ( Ω π )∗ β =  id  . id

(7.17)

Proof Proposition 4.23(b) gives a coequalizer diagram for C in C∞ Ringscin :

FD,Din

γ

0. FC,Cin

/ C.

φ

(7.18)

δ

We can extend this to a coequalizer diagram for D: FBqD,Bin qDin

 ζ

0. FAqC,Ain qCin

ψ

/ D.

(7.19)

Here , ζ act as γ, δ do on the generators from D, Din , and as (˜ xb ) = f˜b , ˜ in (˜ yb0 ) = g˜b0 , ζ(˜ xb ) = 0, ζin (˜ yb0 ) = hb0 on the generators from B, Bin , where ˜ b0 are lifts of fb , gb0 , hb0 along the surjective morphism FAqC,Ain qCin f˜b , g˜b0 , h → C(xa : a ∈ A)[ya0 : a0 ∈ Ain ] induced by φ. Also ψ acts as φ does on the generators from C, Cin , and as the identity on generators xa , ya0 from A, Ain . Thus the effect of enlarging FC,Cin to FAqC,Ain qCin is to enlarge C to C(xa : a ∈ A)[ya0 : a0 ∈ Ain ], and the effect of enlarging FD,Din to FBqD,Bin qDin is to quotient C(xa : a ∈ A)[ya0 : a0 ∈ Ain ] by extra relations (˜ xb ) = ζ(˜ xb ) for b ∈ B, which means fb = 0, and in (˜ yb0 ) = ζin (˜ yb0 ) for b0 ∈ Bin , which means gb0 = hb0 . So the result is D in (7.15).

175

7.3 Some computations of b-cotangent modules Now consider the commutative diagram:

0 id



din Ψexp (˜ vd ), d ∈ D,  din w ˜d0 , d0 ∈ Din R ⊗R D 0 ∗





hdin Ψexp (˜ xb ), b ∈ BiR ⊗R D ⊕hdin (˜ yb0 ), b0 ∈ Bin iR ⊗R D

din Ψexp (˜ xb ), b ∈ B,  0 0 , b ∈ Bin d y ˜ ⊗ D⊕ in b R R /

din Ψexp (˜ vd ), d ∈ D,  din w ˜d0 , d0 ∈ Din R ⊗R D




din Ψexp (xa ), a ∈ A,  din ya0 , a0 ∈ Ain R ⊗R D

din Ψexp (vc ), c ∈ C,  din wc0 , c0 ∈ Cin R ⊗R D 0 ∗ id 0








0








din Ψexp (xa ), a ∈ A,  din ya0 , a0 ∈ Ain R ⊗R D

din Ψexp (vc ), c ∈ C,  din wc0 , c0 ∈ Cin R ⊗R D




b Ω ⊗ C D⊕ α C hdin Ψexp (xa ), a ∈ AiR ⊗R D ⊕hdin (ya0 ), a0 ∈ Ain iR ⊗R D

/

∗0 ∗∗

(∗ ∗) β



/ b ΩD

/0



0.

The middle column is the direct sum of − ⊗C D applied to (7.7) from Proposition 7.9(b) for (7.18) and the identity map on hdin Ψexp (xa ), din ya0 iR ⊗R D, and the right hand column is (7.7) for (7.19). Thus the columns are exact. Some diagram-chasing in exact sequences and the snake lemma then shows the third row is exact. But this is (7.16), so the proposition follows. Here is an analogue of Proposition 2.32: Proposition 7.11. Let C be an interior C ∞ -ring with corners and A ⊆ C, Ain ⊆ C in be subsets, and write D = C(a−1 : a ∈ A)[a0−1 : a0 ∈ Ain ] for the localization in Definition 4.26, with projection π : C → D. Then (b Ωπ )∗ : b ΩC ⊗C D → b ΩD is an isomorphism of D-modules. Proof We may write D by adding generators and relations to C as in (4.16), where we add an R-generator xa and R-relation a · xa = 1 for each a ∈ A, and a [0, ∞)-generator ya0 and [0, ∞)-relation a0 · ya0 = 1 for each a0 ∈ Ain . Then Proposition 7.10 gives an exact sequence (7.16) computing b ΩD . But in this case the contributions from the generators xa , ya0 in the middle of (7.16) are exactly cancelled by the contributions from the relations a · xa = 1, a0 · ya0 = 1 on the left of (7.16). That is, α in (7.16) induces an isomorphism from the x ˜b , y˜b0 terms to the xa , ya0 terms. So by exactness we see that the top component of β in (7.16), which is (b Ωπ )∗ : b ΩC ⊗C D → b ΩD , is an isomorphism. sa sa We show that the reflection functors ΠZin , Πtf Z , Πtf , Πin in Theorem 4.48 do not change b-cotangent modules, in a certain sense.

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Modules, and sheaves of modules

Proposition 7.12. Theorem 4.48 gives ΠZin : C∞ Ringscin → C∞ RingscZ left adjoint to inc : C∞ RingscZ ,→ C∞ Ringscin . For C ∈ C∞ Ringscin , there is a natural surjective π : C → ΠZin (C) =: D in C∞ Ringscin , the unit of the adjunction. Thus Definition 7.4 gives a morphism (b Ωπ )∗ : π ∗ (b ΩC ) = b ΩC ⊗C D → b ΩD in D-mod. Then (b Ωπ )∗ is an isomorphism. sa sa The analogue holds for the other functors Πtf Z , Πtf , Πin in Theorem 4.48. Proof The definition of D = ΠZin (C) in the proof of Theorem 4.48 involves inductively constructing a sequence C = C 0  C 1  C 2  · · · and setting n n+1 n ∞ 0 n D = lim b0 = hb0 , b ∈ Bin ], where the relations −→n=0 C . Here C n = C /[g gp gp are gb0 = hb0 if gb0 , hb0 ∈ C in with π (gb0 ) = π (hb0 ). This is equivalent to the existence of ib0 ∈ C nin with gb0 ib0 = hb0 ib0 . But then in b ΩC n we have din gb0 + din ib0 = din (gb0 ib0 ) = din (hb0 ib0 ) = din hb0 + din ib0 , so din gb0 = din hb0 . Applying Proposition 7.10 with C n in place of C and A = Ain = B = ∅ gives an exact sequence in C n+1 -mod: n hdin (˜ yb0 ), b0 ∈ Bin iR ⊗R D

α

/ b ΩC n ⊗C n C n+1

β

/ b ΩC n+1

/ 0.

Here α maps din (˜ yb0 ) 7→ din gb0 − din hb0 = 0 by (7.17), so α = 0, and β is an isomorphism. Applying −⊗C n+1 D shows that the natural map b ΩC n ⊗C n D → b ΩC n+1 ⊗C n+1 D is an isomorphism. Thus by induction on n and C 0 = C we see that b ΩC ⊗C D ∼ = b ΩC n ⊗C n D for all n > 0. n ∼ ˜ b 0 , b0 ∈ ` ˜ b0 Now D = C/[˜ gb0 = h ˜b0 = h n>0 Bin ], where the relations g n+1 n 0 n are lifts to C of all relations gb0 = hb0 in C = C /[gb0 = hb0 , b ∈ Bin ] b b n n n 0 0 for all n > 0. Since din gb = din hb in ΩC , and hence in ΩC ⊗C D, and b ˜ b0 in b ΩC ⊗C D for all ΩC ⊗ C D ∼ = b ΩC n ⊗C n D, we see that din g˜b0 = din h ` n ˜ such g˜b0 , hb0 . Thus in (7.16) for C, D with A, Ain , B = ∅ and Bin = n>0 Bin b b b we again have α = 0, so β = ( Ωπ )∗ : ΩC ⊗C D → ΩD is an isomorphism. n+1 For Πtf = C n /[gb0 = hb0 , b0 ∈ Z a similar proof works. In this case C n Bin ], with relations gb0 = hb0 if gb0 , hb0 ∈ C nin with π tf (gb0 ) = π tf (hb0 ), which (as C n is integral) is equivalent to gbn0 = hnb0 for some n > 1. But then n din gb0 = din (gbn0 ) = din (hnb0 ) = n din hb0 , so din gb0 = din hb0 , and the rest of the proof is unchanged. n n+1 n n = C n [yb0 : b0 ∈ Bin ]/[yb0b0 = hb0 , b0 ∈ Bin ], for hb0 in For Πsa tf , now C n n gp 00 C in ⊆ (C in ) and nb0 > 2 for which there exists (unique) c ∈ (C nin )gp \ C nin n with nb0 ·c00 = hb0 . That is, we introduce generators yb0 and relations yb0b0 = hb0 nb 0 in pairs. Since yb0 = hb0 implies that nb0 · din yb0 = din hb0 , again (7.16) implies that b ΩC n ⊗C n C n+1 → b ΩC n+1 is an isomorphism, and the rest of the sa tf Z sa argument is similar. As Πsa in = Πtf ◦ ΠZ ◦ Πin , the result for Πin follows.

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7.4 B-cotangent modules of pushouts

7.4 B-cotangent modules of pushouts The next theorem generalizes Theorem 2.33. Theorem 7.13. Suppose we are given a pushout diagram in the categories C∞ Ringscin , C∞ RingscZ , C∞ Ringsctf , or C∞ Ringscsa : C

β

α

 D

/E (7.20)

 / F,

δ γ

so that F = D qC E. Then the following sequence of F-modules is exact: b

b (b Ωα )∗ ⊕−(b Ωβ )∗ ΩD ⊗D,γ F (b Ωγ )∗ ⊕(b Ωδ )∗ / b ΩF / ΩC ⊗C,γ◦α F ⊕b ΩE ⊗E,δ F

/ 0. (7.21)

Here (b Ωα )∗ : b ΩC ⊗C,γ◦α F → b ΩD ⊗D,γ F is induced by b Ωα : b ΩC → ΩD , and so on. Note the sign of −(b Ωβ )∗ in (7.21).

b

Proof First suppose (7.20) is a pushout in C∞ Ringscin . By b Ωψ◦φ = b Ωψ ◦ b Ωφ in Definition 7.4 and commutativity of (7.20) we have b Ωγ ◦ b Ωα = b Ωγ◦α = b Ωδ◦β = b Ωδ ◦ b Ωβ : b ΩC → b ΩF . Tensoring with F then gives b ( Ωγ )∗ ◦ (b Ωα )∗ = (b Ωδ )∗ ◦ (b Ωβ )∗ : b ΩC ⊗C F → b ΩF . As the composition of morphisms in (7.21) is (b Ωγ )∗ ◦ (b Ωα )∗ − (b Ωδ )∗ ◦ (b Ωβ )∗ , this implies (7.21) is a complex. Apply Proposition 4.41 to (7.20), using Proposition 4.23(b) to choose coequalizer presentations (4.22)–(4.24) for C, D, E in C∞ Ringscin , so we have a presentation (4.25) for F. Thus Proposition 7.9(b) gives exact sequences



din Ψexp (˜ sb ), b ∈ B, 1 / din Ψexp (sa ), a ∈ A, ζ1 / b ΩC ⊗ C F din ta0 , a0 ∈ Ain R ⊗R F din t˜b0 , b0 ∈ Bin R ⊗R F



din Ψexp (˜ ud ), d ∈ D, 2 / din Ψexp (uc ), c ∈ C, ζ2 / b ΩD ⊗ D F din v˜b0 , d0 ∈ Din R ⊗R F din vc0 , c0 ∈ Cin R ⊗R F



din Ψexp (w ˜f ), f ∈ F, 3 / din Ψexp (we ), e ∈ E, ζ3 / b ΩE ⊗E F din x ˜f 0 , f 0 ∈ Fin R ⊗R F din xe0 , e0 ∈ Ein R ⊗R F



din Ψexp (˜ yh ), h ∈ H, 4 / din Ψexp (yg ), g ∈ G, ζ4 / b / ΩF din z˜h0 , h0 ∈ Hin R ⊗R F din zg0 , g 0 ∈ Gin R ⊗R F

/ 0, (7.22) / 0, (7.23) / 0, (7.24)

0,

(7.25)

where for (7.22)–(7.24) we have tensored (7.7) for C, D, E with F, and we write s1 , . . . , z˜h0 for the generators in FA,Ain , . . . , FH,Hin as shown.

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Modules, and sheaves of modules

Now consider the diagram

din Ψexp (sa ), a ∈ A,  din ta0 , a0 ∈ Ain R ⊗R F⊕

din Ψexp (˜ ud ), d ∈ D,  din v˜b0 , d0 ∈ Din R ⊗R F⊕

din Ψexp (w ˜f ), f ∈ F,  din x ˜f 0 , f 0 ∈ Fin R ⊗R F bΩ C



(ζ1 0 0)

⊗C F



4 = θ1 −θ2

(b Ωα )∗ −(b Ωβ )∗

!

din Ψexp (uc ), c ∈ C,  0 d / in vc0 , c ∈ Cin R ⊗R F⊕  din Ψexp (we ), e ∈ E, 2 0  0 3 din xe0 , e0 ∈ Ein ⊗ F R R

ζ4

ΩE ⊗E F

(b Ωδ )∗ )

/ 0x (7.26)

idb Ω

  ζ2 0 0 ζ3

 ((b Ωγ )∗ / bb ΩD ⊗D F ⊕

/ b ΩF

/ b ΩF

F

/ 0.

Here the top row is (7.25), except that we have substituted for G, Gin , H, Hin in terms of A, . . . , Fin as in (4.25), and labelled the corresponding generators as in (7.22)–(7.24). Then 4 in (7.25) is written in matrix form in (7.26) as shown, where θ1 , θ2 are induced by the morphism FA,Ain → FC,Cin ⊗∞ FE,Ein ∼ = FCqE,Cin qEin = FG,Gin lifting FA,Ain → D ⊗∞ E chosen during the proof of Proposition 4.41. The bottom line is the complex (7.21). The left hand square of (7.26) commutes as ζ2 ◦ 2 = ζ3 ◦ 3 = 0 by exactness of (7.23)–(7.24) and ζ2 ◦ θ1 = (b Ωα )∗ ◦ ζ1 , ζ3 ◦ θ2 = (b Ωβ )∗ ◦ ζ1 follow from the definition of θ1 , θ2 . The right hand square commutes as ζ4 and (b Ωγ )∗ ◦ ζ2 map din Ψexp (uc ) 7→ din Ψexp ◦ γ ◦ χ(uc ) and din vc0 7→ din γin ◦ χin (vc0 ), and similarly for (b Ωδ )∗ ◦ ζ3 . Hence (7.26) commutes. The columns are surjective since ζ1 , ζ2 , ζ3 are surjective as (7.22)–(7.24) are exact and identities are surjective.
The bottom right morphism (b Ωγ )∗ (b Ωδ )∗ in (7.26) is surjective as ζ4 is and the right hand square commutes. Also surjectivity of the middle
colb b umn implies that it maps Ker ζ4 surjectively onto Ker ( Ωγ )∗ ( Ωδ )∗ . But Ker ζ4 = Im 4 as the top row is exact, so as the left hand square com

mutes we see that (b Ωα )∗ − (b Ωβ )∗ T surjects onto Ker (b Ωγ )∗ (b Ωδ )∗ , and the bottom row of (7.26), which is (7.21), is exact. This proves the theorem for C∞ Ringscin . ˜ = D qC E Next suppose (7.20) is a pushout diagram in C∞ RingscZ . Let F ∞ c be the pushout in C Ringsin , which exists by Theorem 4.20(b), with projec˜ δ˜ : E → F. ˜ Since ΠZ : C∞ Ringsc → C∞ Ringsc in tions γ ˜ : D → F, in Z in Theorem 4.48 preserves colimits as it is a left adjoint, and acts as the identity ˜ The natural projecon C, D, E as they are integral, we see that F ∼ = ΠZin (F). Z ˜ → Π (F) ˜ ∼ tion π : F = F is the morphism induced by (7.20) and the pushout in ˜ ˜ property of F, so that γ = π ◦ γ ˜ and δ = π ◦ δ. ˜ The first part implies that (7.21) with F, γ ˜ , δ˜ in place of F, γ, δ is exact.

7.5 Sheaves of O X -modules, and (b-)cotangent sheaves

179

Applying − ⊗F˜ F to this shows the following is exact: b ΩD ⊗D,γ F (b Ωγ˜ )∗ ⊕(b Ωδ˜)∗ / ⊕b ΩE ⊗E,δ F / bΩ˜ ⊗˜ F F F

(b Ωα )∗ ⊕−(b Ωβ )∗ b

ΩC ⊗C,γ◦α F

/ 0. (7.27)

b Now Proposition 7.12 shows that (b Ωπ )∗ : b ΩF ˜ F → ΩF is an isomor˜ ⊗F b b phism. Combining this with (7.27) and ( Ωγ )∗ = ( Ωπ )∗ ◦ (b Ωγ˜ )∗ , (b Ωγ )∗ = (b Ωπ )∗ ◦ (b Ωγ˜ )∗ as γ = π ◦ γ ˜ , δ = π ◦ δ˜ shows that (7.21) is exact. The ∞ c ∞ sa proofs for C Ringstf , C Ringscsa are the same, using Πtf Z , Πtf in Theorem 4.48.

7.5 Sheaves of O X -modules, and (b-)cotangent sheaves We define sheaves of O X -modules on a C ∞ -ringed space with corners, as in §2.7. Definition 7.14. Let (X, O X ) be a C ∞ -ringed space with corners. A sheaf of O X -modules, or simply an O X -module, E on X, is a sheaf of OX -modules on X, as in Definition 2.55, where OX is the sheaf of C ∞ -rings in O X . A morphism of sheaves of O X -modules is a morphism of sheaves of OX -modules. Then O X -modules form an abelian category, which we write as O X -mod. An O X -module E is called a vector bundle of rank n if we may cover X by open U ⊆ X with E|U ∼ = OX |U ⊗R Rn . Pullback sheaves are defined analogously to Definition 2.56. ] ex Definition 7.15. Let f = (f, f ] , fex ) : (X, OX , OX ) → (Y, OY , OYex ) be a ∞ c morphism in C RS , and E be an O Y -module. Define the pullback f ∗ (E) by f ∗ (E) = f ∗ (E), the pullback in Definition 2.56 for f = (f, f ] ) : (X, OX ) → (Y, OY ). If φ : E → F is a morphism of O Y -modules we have a morphism of O X -modules f ∗ (φ) = f ∗ (φ) = f −1 (φ) ⊗ idOX : f ∗ (E) → f ∗ (F).

Example 7.16. Let X be a manifold with corners, or with g-corners, and E → C∞ Schc X a vector bundle. Write X = FMan (X) for the associated C ∞ -scheme gc with corners. For each open set U ⊆ X, write E(U ) = Γ∞ (E|U ), as a module over OX (U ) = C ∞ (U ), and for open V ⊆ U ⊆ X define ρU V : E(U ) → E(V ) by ρU V (e) = e|V . Then E is the O X -module of smooth sections of E. If F → X is another vector bundle with O X -module F, then vector bundle morphisms E → F are in natural 1-1 correspondence with O X -module morphisms E → F. If f : W → X is a smooth map of manifolds with (g-)corners

180

Modules, and sheaves of modules ∞

c

C Sch and f = FMan : W → X then the vector bundle f ∗ (E) → W corregc sponds naturally to the O W -module f ∗ (E).

Definition 7.17. Let X = (X, O X ) be a C ∞ -ringed space with corners, and X = (X, OX ) the underlying C ∞ -ringed space. Define the cotangent sheaf T ∗ X of X to be the cotangent sheaf T ∗ X of X from Definition 2.57. If U ⊆ X is open then we have an equality of O X |U -modules T ∗ U = T ∗ (U, O X |U ) = T ∗ X|U . ] Let f = (f, f ] , fex ) : X → Y be a morphism of C ∞ -ringed spaces. ∗ Define Ωf : f (T ∗ Y ) → T ∗ X to be a morphism of the cotangent sheaves by Ωf = Ωf , where f = (f, f ] ) : (X, OX ) → (Y, OY ).

Definition 7.18. Let X = (X, O X ) be an interior local C ∞ -ringed space in with corners. For each open U ⊂ X, let dU,in : OX (U ) → b ΩC U be the ∞ b-cotangent module associated to the interior C -ring with corners C U = in in (U ) is the set (OX (U ), OX (U ) q {0}), where q is the disjoint union. Here OX ex ex for all x ∈ U . (U ) that are non-zero in each stalk OX,x of all elements of OX in Note that OX is a sheaf of monoids on X. For each open U ⊆ X, the b-cotangent modules b ΩC U define a presheaf P b T ∗ X of O X -modules, with restriction map b ΩρU V : b ΩC U → b ΩC V defined in Definition 7.4. Denote the sheafification of this presheaf the bcotangent sheaf b T ∗ X of X. Properties of sheafification imply that, for each open set U ⊆ X, there is a canonical morphism b ΩC U → b T ∗ X(U ), and we have an equality of O X |U -modules b T ∗ (U, O X |U ) = b T ∗ X|U . Also, for each x ∈ X, the stalk b T ∗ X|x ∼ = b ΩOX,x , where b ΩOX,x is the b-cotangent module of the interior local C ∞ -ring with corners O X,x . ] For a morphism f = (f, f ] , fex ) : X → Y of interior local C ∞ -ringed spaces, we define the morphism of b-cotangent sheaves b Ωf : f ∗ (b T ∗ Y ) → b ∗ T X by firstly noting that f ∗ (b T ∗ Y ) is the sheafification of the presheaf P(f ∗ (b T ∗ Y )) acting by U 7−→ P(f ∗ (b T ∗ Y ))(U ) = limV ⊇f (U ) b ΩOY (V ) ⊗OY (V ) O X (U ), as in Definition 2.57. Then, following Definition 2.57, define a morphism of presheaves P b Ωf : P(f ∗ (b T ∗ Y )) → PT ∗ X on X by (P b Ωf )(U ) = limV ⊇f (U ) (b Ωρf −1 (V ) U ◦f ] (V ) )∗ , where (b Ωρf −1 (V ) U ◦f ] (V ) )∗ : b ΩOY (V ) ⊗OY (V ) OX (U ) → b ΩOX (U ) = (P b T ∗ X)(U ) is constructed as in Definition 2.29 from the C ∞ -ring with corners morphisms

7.5 Sheaves of O X -modules, and (b-)cotangent sheaves

181

f ] (V ) : O Y (V ) → O X (f −1 (V )) from f ] : O Y → f∗ (O X ) corresponding to f ] in f as in (2.9), and ρf −1 (V ) U : O X (f −1 (V )) → O X (U ) in O X . Define b Ωf : f ∗ (b T ∗ Y ) → b T ∗ X to be the induced morphism of the associated sheaves. We prove an analogue of Theorem 2.59(e): Proposition 7.19. Let C ∈ C∞ Ringscin , and set X = Speccin C. Then there is a canonical isomorphism b T ∗ X ∼ = MSpec b ΩC in O X -mod. Proof By Definitions 2.58 and 7.18, MSpec b ΩC and b T ∗ X are sheafifications of presheaves P MSpec b ΩC , P b T ∗ X, where for open U ⊆ X we have P MSpec b ΩC (U ) = b ΩC ⊗C OX (U ) and

P b T ∗ X(U ) = b ΩOin , X (U )

in writing O in X (U ) = (OX (U ), OX (U ) q {0}). We have morphisms ΞC : C → in ∞ c O X (X) in C Ringsin from Definition 5.14, and restriction ρXU : O in X (X) → O in (U ) from O , and so as in Definition 7.3 a morphism of O X X (U )X b b modules Pρ(U ) := (ρXU ◦ ΞC )∗ : ΩC ⊗C OX (U ) → ΩOin . This X (U ) defines a morphism of presheaves Pρ : P MSpec b ΩC → P b T ∗ X, and so sheafifying induces a morphism ρ : MSpec b ΩC → b T ∗ X. The induced morphism on stalks at x ∈ X is ρx = (π x )∗ : b ΩC ⊗C C x → b ΩC x , where π x : C → C x ∼ = O X,x is projection to the local C ∞ -ring with corners C x . But C x is the localization (4.17), so Proposition 7.11 says (π x )∗ is an isomorphism. Hence ρ : MSpec ΩC → T ∗ X is a sheaf morphism which induces isomorphisms on stalks at all x ∈ X, so ρ is an isomorphism.

The next theorem shows that T ∗ X, b T ∗ X are good analogues of the (b-)cotangent bundles of manifolds with (g-)corners. ∞

c

∞

c

C Sch (X) Theorem 7.20. (a) Let X be a manifold with corners, and X = FMan c ∞ the associated C -scheme with corners. Then there is a natural, functorial isomorphism between the cotangent sheaf T ∗ X of X, and the O X -module T ∗ X associated to the vector bundle T ∗ X → X in Example 7.16. C Sch (X) (b) Let X be a manifold with corners, or g-corners, and X = FMan gc ∞ the associated C -scheme with corners. Then there is a natural, functorial isomorphism between the b-cotangent sheaf b T ∗ X of X, and the O X -module b ∗ T X associated to the vector bundle b T ∗ X → X in Example 7.16.

Proof For (a), every x ∈ X has arbitrarily small open neighbourhoods x ∈ U ⊆ X such that U is a manifold with faces, and ∂U has finitely many boundary components (for example, we can take U ∼ = Rm k ). Then we have PT ∗ X(U ) = ΩOX (U ) = ΩC ∞ (U ) ∼ = Γ∞ (T ∗ X|U ) = T ∗ X(U ).

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Modules, and sheaves of modules

Here in the first step PT ∗ X is the presheaf in Definition 2.57, whose sheafification is T ∗ X = T ∗ X by Definition 7.17. In the second step OX (U ) = C ∞ (U ) by Definition 5.21. The third step holds by Theorem 7.7(a), and the fourth by Example 7.16. Hence we have natural isomorphisms PT ∗ X(U ) ∼ = T ∗ X(U ) for arbitrarily small open neighbourhoods U of each x ∈ X. These isomorphisms are compatible with the restriction morphisms in PT ∗ X and T ∗ X. Thus properties of sheafification give a canonical isomorphism T ∗ X ∼ = T ∗ X. Part (b) is proved in the same way, using Definition 7.18, Example 4.38, and Theorem 7.7(b). Here is a corners analogue of Theorem 2.61. It is proved following the proof of Theorem 2.61 in [50, §5.6], but using as input the identity b Ωψ◦φ = b Ωψ ◦ b Ωφ in Definition 7.4 for (a), and Theorem 7.13 for (b). We restrict to firm C ∞ -schemes with corners so that Proposition 5.32 applies. Theorem 7.21. (a) Let f : X → Y and g : Y → Z be morphisms in C∞ Schcfi,in . Then in O X -mod we have b

Ωg◦f = b Ωf ◦ f ∗ (b Ωg ) : (g ◦ f )∗ (T ∗ Z) −→ T ∗ X.

(b) Suppose we are given a Cartesian square in C∞ Schcfi,in , C∞ Schcfi,Z , C∞ Schcfi,tf or C∞ Schcto : W

f

e

/Y  / Z,

h

 X

g

so that W = X ×Z Y . Then the following is exact in O W -mod : ∗ b

∗

e∗ (b T ∗ X) b Ω ⊕b Ωf / ⊕ f ∗ (b T ∗ Y ) e / bT ∗W

e∗ (b Ωg )⊕−f ∗ (b Ωh )

(g◦e) ( T Z)

/ 0.

(7.28)

Example 7.22. Define morphisms g : X → Z, h : Y → Z in Mancin by X = [0, ∞), Y = ∗, Z = R, g(x) = xn for some n > 0, and h(∗) = 0. C∞ Schc Write X, Y , Z, g, h for the images of X, . . . , h under FManc to . Write W = in X ×g,Z,h Y for the fibre product in C∞ Schcto , which exists by Theorem 5.34 (it is also the fibre product in C∞ Schcfi,tf , C∞ Schcfi,Z and C∞ Schcfi,in ). Now b T ∗ X = hx−1 dxiR , b T ∗ Y = 0 and b T ∗ Z = hdziR , where b Ωg maps dz 7→ nxn · x−1 dx. So by Theorem 7.20(b), b T ∗ X, b T ∗ Z are trivial line bundles, such that b Ωg is multiplication by nxn on X, and b T ∗ Y = 0. Since W is defined by xn = 0 in X, we see that e∗ (b Ωg ) = 0 on W . Thus (7.28) implies that b T ∗ W ∼ = e∗ (b T ∗ X), so b T ∗ W is a rank 1 vector bundle on W . We might have hoped that if W is an interior C ∞ -scheme with corners such

183

7.6 (B-)cotangent sheaves and the corner functor

that b T ∗ W is a vector bundle, and W satisfies a few other obvious necessary conditions (e.g. W should be Hausdorff, second countable, toric, locally finitely presented) then W should be a manifold with g-corners. However, in this example b T ∗ W is a vector bundle, but W is not a manifold with g-corners, and there do not seem to be obvious conditions to impose which exclude W .

7.6 (B-)cotangent sheaves and the corner functor If X is a manifold with (g-)corners, then as in [47, §3.5] and Remark 3.18(d) we have an exact sequence of vector bundles on C(X): / b T ∗ (C(X))

0

b

∗ πT

/ Π∗ (b T ∗ X) X

b ∗ iT

/ bN ∗ C(X)

/ 0.

(7.29)

Note that b πT∗ is not the morphism b dΠ∗X associated to ΠX : C(X) → X: this is not defined as ΠX is not interior, and if it were it would map in the opposite direction. Both b πT∗ , b i∗T are specially constructed using the geometry of corners. We will construct an analogue of (7.29) for (firm, interior) C ∞ -schemes with corners. First we define an analogue for (interior) C ∞ -rings with corners. Definition 7.23. Let C be an interior C ∞ -ring with corners, and P ∈ PrC be a prime ideal in C ex . Then Example 4.25(b) defines a C ∞ -ring with corners D = C/∼P , which is also interior, with (non-interior) projection π P : C → D. Writing Pin = P \ {0} ⊂ C in , define a monoid  0   00 ] ] 0 00 ∼ ] CP in = C in c = 1 : c ∈ C in \Pin = C in / c = 1 : c ∈ C in \Pin , (7.30) with projection λP : C in → C P in . We will define an exact sequence in D-mod: b

b

ΩD

πC ,P

/ b ΩC ⊗ C D

b

iC ,P

/ CP in ⊗N D

/ 0.

(7.31)

Beware of a possible confusion here: in (7.30) we regard C P in as a monoid under multiplication, with identity 1, but in (7.31) it is better to think of C P in as a monoid under addition, so that the identity 1 ∈ C P in is mapped to 0 in the D-module C P in ⊗N D, i.e. 1 ⊗ 1 = 0. This arises as for b-cotangent modules we think of din = d ◦ log, where log maps multiplication to addition and 1 7→ 0. As in Example 4.25(b), Dex = C ex /∼P , where for c01 , c02 ∈ C ex we have 0 c1 ∼P c02 if either c01 , c02 ∈ P or c01 = Ψexp (c)c02 for c in the ideal hΦi (P )i in C generated by Φi (P ). Thus C in \Pin is mapped surjectively to Din = Dex \{0}. Define a map d0in : Din → b ΩC ⊗C D by d0in [c0 ] = (din c0 ) ⊗ 1, where 0 c ∈ C in \ Pin , and [c0 ] ∈ Din is its ∼P -equivalence class, and din c0 ∈ b ΩC ,

184

Modules, and sheaves of modules

so that (din c0 ) ⊗ 1 lies in b ΩC ⊗C D. To show this is well defined, note that if c01 , c02 ∈ C in \ Pin with [c01 ] = [c02 ] then c01 = Ψexp (c)c02 for c ∈ hΦi (P )i. We may write c = a1 Φi (b01 )+· · ·+an Φi (b0n ) for a1 , . . . , an ∈ C and b01 , . . . , b0n ∈ P . Hence
Pn din c01 = Ψexp (c) · din c02 + j=1 c02 Φi (b0j ) din Ψexp (aj ) + aj · din b0j in b ΩC , using Definition 7.4. Applying − ⊗ 1 : b ΩC → b ΩC ⊗C D, we have πP (Ψexp (c)) = 1 as D = C/I with c ∈ I, and πP (c02 Φi (b0j )) = 0 as b0j ∈ P so c02 Φi (b0j ) ∈ hΦi (P )i. Hence din c01 ⊗ 1 = 1 · (din c02 ⊗ 1) + 0 = din c02 ⊗ 1, so d0in is well defined. As din is a b-derivation, d0in is too. Thus by the universal property of b ΩD there is a unique morphism b πC,P : b ΩD → b ΩC ⊗C D in D-mod with d0in = b πC,P ◦ din , as required in (7.31). 00 0 P 0 Next, define a map d00in : C in → C P in ⊗N D by din (c ) = λ (c ) ⊗ 1. Regard P C in ⊗N D as a C-module via πP : C → D, with trivial C-action on the C P in factor, so that c·(¯ c ⊗d) = c¯⊗(πP (c)d) for c ∈ C, d ∈ D and c¯ ∈ C P . We claim in that d00in is a b-derivation. To prove this, let g : [0, ∞)n → [0, ∞) be interior and c01 , . . . , c0n ∈ C in . Write g(x1 , . . . , xn ) = xa1 1 · · · xann exp(f (x1 , . . . , xn )), for a1 , . . . , an > 0 and f : [0, ∞)n → R smooth. Then

0an 1 d00in Ψg (c01 , . . . , c0n ) = d00in c0a 1 · · · cn Ψexp (f (x1 , . . . , xn ))
0an 1 = λP c0a 1 · · · cn Ψexp (f (x1 , . . . , xn )) ⊗ 1
Pn (7.32) = j=1 aj λP (c0j ) ⊗ 1 + λP (Ψexp (f (x1 , . . . , xn )) ⊗ 1 Pn = j=1 aj d00in (c0j ) + 0, using in the third step that λP is a monoid morphism, and in the fourth that λP (Ψexp (f (x1 , . . . , xn ))) = 1 by (7.30) as Ψexp (f (x1 , . . . , xn )) ∈ C in \ Pin , so λP (Ψexp (f (x1 , . . . , xn ))) ⊗ 1 = 0 as − ⊗ 1 maps 1 7→ 0. Now g −1 xj

∂g ∂f = aj + xj (x1 , . . . , xn ). ∂xj ∂xj

Hence Φg−1 xi

∂g ∂xi

(c01 , . . . , c0n ) = aj + Φi (c0j )Φ

∂f ∂xj

(c01 , . . . , c0n ).

(7.33)

Combining (7.32) with πP applied to (7.33) we see that
Pn
d00in Ψg (c01 , . . . , c0n ) = j=1 πP Φg−1 xj ∂g (c01 , . . . , c0n ) · d00in c0j (7.34) ∂xj

Pn 0 − j=1 πP Φi (cj ) πP Φ ∂f (c01 , . . . , c0n ) · d00in c0j . ∂xj

For each j = 1, . . . , n, if c0j ∈ P then Φi (c0j ) ∈ hΦi (P )i so πP (Φi (c0j )) = 0,

185

7.6 (B-)cotangent sheaves and the corner functor

/ P then λP (c0j ) = 1 so d00in c0j = 0. Hence the second line of (7.34) and if c0j ∈ is zero. So d00in satisfies (7.1), and is a b-derivation. Thus there exists a unique morphism b i0C,P : b ΩC → C P in ⊗N D in C-mod with d00in = b i0C,P ◦ din , by the universal property of b ΩC . Applying − ⊗C D to b i0C,P and noting that D ⊗C D ∼ = D as πP : C → D is surjective gives a morphism b iC,P : b ΩC ⊗C D → C P in ⊗N D, as required in (7.31). Proposition 7.24 shows (7.31) is an exact sequence. Now suppose also that φ : C → E is a morphism in C∞ Ringscin , and Q ⊂ Eex is a prime ideal with P = φ−1 ex (Q), and write F = E/∼Q . As φex (P ) ⊆ Q there is a unique morphism ψ : D → F with ψ◦π P = π Q ◦φ : C → F. From the definitions, it is not difficult to show that the following diagram commutes: b

ΩD ⊗ D F

b

πC ,P ⊗id

(b Ωψ )∗

 b ΩF

/ b Ω C ⊗C F b

b

πE,Q

b

/ CP in ⊗N F

iC ,P ⊗id

(7.35)

(φin )∗ ⊗id

Ωφ ⊗id

 / b Ω E ⊗E F

/0

b

iE,Q

/

EQ in

 ⊗N F

/ 0.

Proposition 7.24. Equation (7.31) is an exact sequence. Proof Work in the situation of Definition 7.23. Then b ΩD is generated over D by din [c0 ] for c0 ∈ C in \ Pin . But b

iC,P ◦ b πC,P (din [c0 ]) = b iC,P ◦d0in [c0 ] = b iC,P ((din c0 )⊗1) = b i0C,P (din c0 ) = d00in (c0 ) = λP (c0 ) ⊗ 1 = 1 ⊗ 1 = 0.

Hence b iC,P ◦ b πC,P = 0, so (7.31) is a complex. By Proposition 4.23(b), C fits into a coequalizer diagram in C∞ Ringscin : FB,Bin

α

1 FA,Ain

φ

/ C.

(7.36)

β A,Ain and Define prime ideals Q = φ−1 ex (P ) ⊂ Fex −1 −1 in . R = αex (Q) = (φex ◦αex )−1 (P ) = (φex ◦βex )−1 (P ) = βex (Q) ⊂ FB,B ex in in , FB,B are parametrized by As FA,Ain , FB,Bin are free, prime ideals in FA,A ex ex subsets of Ain , Bin . Thus there are unique subsets A˙ in ⊆ Ain , B˙ in ⊆ Bin with Q = hya0 : a0 ∈ A˙ in i and R = h˜ yb0 : b0 ∈ B˙ in i. Set A¨in = Ain \ A˙ in and ¨ ˙ Bin = Bin \ Bin . Then

¨

FA,Ain /∼Q = FA,Ain /[ya0 = 0 : a0 ∈ A˙ in ] ∼ = FA,Ain , ¨ FB,Bin /∼R = FB,Bin /[˜ yb0 = 0 : b0 ∈ B˙ in ] ∼ = FB,Bin .

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Modules, and sheaves of modules

Thus equation (7.36) descends to a coequalizer diagram for D: γ

¨

-

/ D.

ψ

¨

1 FA,Ain

FB,Bin

(7.37)

δ

Consider the diagram in D-mod:

din Ψexp (˜ xb ), b ∈ B,  ¨in ⊗R D din y˜b0 , b0 ∈ B R



inc



xb ), b ∈ B, / din Ψexp0(˜  din y˜b0 , b ∈ Bin

γ ∗ −δ ∗







project

⊗ D R R

/ din y˜b0 , b0 ∈ B˙ in

⊗R D

α∗ −β ∗



/0

 R

α∗ −β ∗

din Ψexp (xa ), a ∈ A, inc / din Ψexp (xa ), a ∈ A, project / din ya0 , a0 ∈ A˙ in    R ¨in ⊗R D din ya0 , a0 ∈ Ain R ⊗R D din ya0 , a0 ∈ A ⊗R D R ψ∗



b

bΩ D

din ya0 7→[ya0 ]

φ∗ πC,P





/ b ΩC ⊗C D

b

iC,P



0

/0



/ C Pin ⊗N D 

0

0.

/0 (7.38)

Here the first column is (7.7) for (7.37), and so is exact. The maps γ ∗ , δ ∗ , ψ ∗ are defined as in (7.8)–(7.9) using γ, δ, ψ. The second column is − ⊗C D applied to (7.7) for (7.36) in the same way, and so is exact. For the third column of (7.38), equation (7.36) and the definition of Q, R induce a coequalizer diagram in Mon:  0 in FB,B y˜ = 1 : in  B,Bin 0 y˜ ∈ Fin \Rin

(αin )∗ (βin )∗

A,A  . Fin in y 0 = 1 :  0 0 in y ∈ FA,A \ Qin in

(φin )∗

 0 P / C in = C in c =  1: c0 ∈ C in \ Pin .

Applying − ⊗N D, which preserves coequalizers, gives an exact sequence  0  0 in in FB,B y˜ = 1 : (αin )∗ −(βin )∗/ FA,A y = 1 : (φin )∗/ P in in   C in ⊗N D A,Ain in 0 y˜0 ∈ FB,B \R ⊗ D y ∈ F \Qin ⊗N D in N in in

/ 0.

But this is isomorphic to the third column of (7.38), which is therefore exact. The top two squares of (7.38) commute by the relation between (7.36) and (7.37). The bottom two squares commute as they are (7.35) for φ : FA,Ain → C, using (7.6). Thus (7.38) commutes. The columns are all exact, and the top ¨in . The third two rows are obviously exact as Ain = A˙ in q A¨in , Bin = B˙ in q B row is a complex as above. Therefore by some standard diagram-chasing in exact sequences, the third row, which is (7.31), is exact. Example 7.25. Set C = R(x)[y]/[y = y 2 , ex y = y], and let P = hyi ⊂ C ex .

187

7.6 (B-)cotangent sheaves and the corner functor Then D = C/∼P ∼ = R(x). Proposition 7.10 implies that b b

ΩC ∼ =

hdin Ψexp (x), din yiR ⊗R C = 0, din y = 2din y, din Ψexp (x) + din y = din y

ΩD ∼ = hdin Ψexp (x)iR ⊗R D.

P 2 Also (7.30) implies that C P in = {[1], [y]} with [y] = [y], so C in ⊗N D = 0. Thus in this case (7.31) becomes the exact sequence D → 0 → 0 → 0.

Note in particular that in Example 7.25, in contrast to (7.29), if we add ‘0 −→’ to the left of (7.31), it may no longer be exact. However, this is exact if we also assume C is toric. Proposition 7.26. In Definition 7.23, suppose C is toric. Then we may extend (7.31) to an exact sequence in D-mod: / b ΩD

0

b

πC ,P

/ b ΩC ⊗ C D

b

iC ,P

/ 0.

/ CP in ⊗N D

(7.39)

Proof We will explain how to modify the proof of Proposition 7.24. Firstly, as C is toric we may take (7.36) to be a coequalizer diagram in C∞ Ringscto rather than in C∞ Ringscin , which implies Ain , Bin are finite. We choose it so that |Ain | is as small as possible. This implies that we have a 1-1 correspondence  ∼ = φ]in : Ain ∼ = [ya0 ] : a0 ∈ Ain −→  0 (7.40) [c ] ∈ C ]in \ {1} : [c0 ] 6= [c01 ][c02 ] for [c01 ], [c02 ] ∈ C ]in \ {1} , where the second line is the unique minimal set of generators of C ]in . Having fixed Ain , we choose (7.36) so that |Bin | is also as small as possible, that is, we define C using the fewest possible [0, ∞)-type relations in C∞ Ringscto . From (7.36) we may form diagrams N

Bin

=

α]in

in ] (FB,B ) in

] βin

0

/

.

0N

Ain

in ] = (FA,A ) in

QBin = αin −βin / QAin = B,Bin ] A,Ain ] (Fin ) ⊗N Q (Fin ) ⊗N Q ]

]

φ]in

φ]in

/ C] , in

/ C ] ⊗N Q in

/ 0.

(7.41)

(7.42)

Here (7.41) is a coequalizer in the category of toric monoids Monto , as (7.36) is in C∞ Ringscto . This implies (7.42) is exact at the third and fourth terms. Choosing |Bin | as small as possible also forces (7.42) to be exact at the second term. Suppose for a contradiction that 0 6= (yb0 : b0 ∈ Bin ) ∈ ] ] − βin ) in (7.42). By rescaling we can suppose all yb0 lie in Z ⊂ Q. Ker(αin

188

Modules, and sheaves of modules

in in for b0 ∈ Bin . and βin (yb0 ) = hb0 ∈ FA,A Write αin (yb0 ) = gb0 ∈ FA,A in in ] ] A,Ain ] 0 Then (yb0 : b ∈ Bin ) ∈ Ker(αin − βin ) implies that in (Fin ) we have

[gb0 ]yb0

Q b0 ∈B

in :yb0 >0

[hb0 ]yb0

Q

=

[hb0 ]−yb0

Q

b0 ∈Bin :yb0 0

[gb0 ]−yb0 .

Q b0 ∈B

in :yb0 0

Q

−yb0

hb0

b0 ∈Bin :yb0 0

−yb0

gb0

(7.43)

.

b0 ∈Bin :yb0 0, i = 1, . . . , k, (8.1) i    ∂f (x , . . . , x ), i = k + 1, . . . , m. ∂xi

1

m

∂f We say that b ∂f exists if (8.1) is well defined, that is, if ∂x exists on U ∩{xi > i ∂f 0} if i = 1, . . . , k, and ∂xi exists on U if i = k + 1, . . . , m. Nl m We can iterate b-derivatives (if they exist), to get maps b ∂ l f : U → R for l = 0, 1, . . . , by taking b-derivatives of components of b ∂ j f for j = 0, . . . , l − 1.

(i) We say that f is roughly differentiable, or r-differentiable, if b ∂f exists and is a continuous map b ∂f : U → Rm . Nl m (ii) We say that f is roughly smooth, or r-smooth, if b ∂ l f : U → R is r-differentiable for all l = 0, 1, . . . . (iii) We say that f is analytically differentiable, or a-differentiable, if it is rdifferentiable and for any compact subset S ⊆ U and i = 1, . . . , k, there exist positive constants C, α such that b ∂i f (x1 , . . . , xm ) 6 Cxα for all (x1 , . . . , xm ) ∈ S. i Nl m (iv) We say that f is analytically smooth, or a-smooth, if b ∂ l f : U → R is a-differentiable for all l = 0, 1, . . . . One can show that f is a-smooth if for all a1 , . . . , am ∈ N and for any compact subset S ⊆ U , there exist positive constants C, α such that a +···+a Y m ∂ 1 a xiα−ai ∂x 1 · · · ∂xamm f (x1 , . . . , xm ) 6 C 1 i=1,...,k: ai >0

for all (x1 , . . . , xm ) ∈ S with xi > 0 if i = 1, . . . , k with ai > 0, where continuous partial derivatives must exist at the required points. If f, g : U → R are a-smooth (or r-smooth) and λ, µ ∈ R then λf + µg and

196

Further generalizations and applications

f g : U → R are a-smooth (or r-smooth). Thus, the set C ∞ (U ) of a-smooth functions f : U → R is an R-algebra, and in fact a C ∞ -ring. If I ⊆ R is an open interval, such as I = (0, ∞), we say that a map f : U → I is a-smooth, or just smooth, if it is a-smooth as a map f : U → R. Definition 8.2. Let U ⊆ Rk,m and V ⊆ Rl,n be open, and f = (f1 , . . . , fn ) : U → V be a continuous map, so that fj = fj (x1 , . . . , xm ) maps U → J0, ∞) for j = 1, . . . , l and U → R for j = l + 1, . . . , n. Then we say: (a) f is r-smooth if fj : U → R is r-smooth in the sense of Definition 8.1 for j = l + 1, . . . , n, and every u = (x1 , . . . , xm ) ∈ U has an open neighbourhood ˜ in U such that for each j = 1, . . . , l, either: U a

a

(i) we may uniquely write fj (˜ x1 , . . . , x ˜m ) = Fj (˜ x1 , . . . , x ˜m )· x ˜1 1,j · · · x ˜kk,j ˜ , where Fj : U ˜ → (0, ∞) is r-smooth as in for all (˜ x1 , . . . , x ˜m ) ∈ U Definition 8.1, and a1,j , . . . , ak,j ∈ [0, ∞), with ai,j = 0 if xi 6= 0; or (ii) fj |U˜ = 0. (b) f is a-smooth if fj : U → R is a-smooth in the sense of Definition 8.1 for j = l + 1, . . . , n, and every u = (x1 , . . . , xm ) ∈ U has an open neighbourhood ˜ in U such that for each j = 1, . . . , l, either: U a

a

(i) we may uniquely write fj (˜ x1 , . . . , x ˜m ) = Fj (˜ x1 , . . . , x ˜m )· x ˜1 1,j · · · x ˜kk,j ˜ ˜ for all (˜ x1 , . . . , x ˜m ) ∈ U , where Fj : U → (0, ∞) is a-smooth as in Definition 8.1, and a1,j , . . . , ak,j ∈ [0, ∞), with ai,j = 0 if xi 6= 0; or (ii) fj |U˜ = 0. (c) f is interior if it is a-smooth, and case (b)(ii) does not occur. (d) f is an a-diffeomorphism if it is an a-smooth bijection with a-smooth inverse. Definition 8.3. We define the category Manac of manifolds with a-corners X and a-smooth maps f : X → Y following the definition of Manc in Definition 3.2, but using a-charts (U, φ) with U ⊆ Rk,m open, where a-charts (U, φ), (V, ψ) are compatible if ψ −1 ◦ φ is an a-diffeomorphism between open subsets of Rk,m , Rl,m , in the sense of Definition 8.2(d). We also define an asmooth map f : X → Y to be interior if it is locally modelled on interior maps between open subsets of Rk,m , Rl,n , in the sense of Definition 8.2(c), and we ac write Manac for the subcategory with interior morphisms. in ⊂ Man In [48, §3.5] the second author also defines the category Manc,ac of manifolds with corners and a-corners, which are locally modelled on Rl,m = k [0, ∞)k × J0, ∞)l × Rm−k−l , with smooth functions defined as in Definitions 8.1–8.2 for the J0, ∞) factors, and as in Definition 3.1 for the [0, ∞) factors. There are full embeddings Manc ,→ Manc,ac and Manac ,→ Manc,ac .

8.3 C ∞ -rings and C ∞ -schemes with a-corners

197

For X ∈ Manc,ac , the boundary ∂X = ∂ c X q ∂ ac X decomposes into the ordinary boundary ∂ c X and the a-boundary ∂ ac X. Remark 8.4. (a) Manifolds with a-corners are significantly different to manifolds with corners. Even the simplest examples J0, ∞) in Manac and [0, ∞) in Manc have different smooth structures. For example, xα : J0, ∞) → R is a-smooth for all α ∈ [0, ∞), but xα : [0, ∞) → R is only smooth for α ∈ N. (b) Here are two of the reasons for introducing manifolds with a-corners in [48]. Firstly, boundary conditions for partial differential equations are usually of two types: ∂u ∂u (i) Boundary ‘at finite distance’. For example, solve ∂x 2 + ∂y 2 = f on the 2 closed unit disc D2 ⊂ R with boundary condition u|∂D2 = g. (ii) Boundary ‘at infinite distance’, or ‘of asymptotic type’. For example, solve 2 ∂u ∂u 2 2 α ∂x2 + ∂y 2 = f on R with asymptotic condition u = O((x +y ) ), α < 0.

We argue in [48] that it is natural to write type (i) using ordinary manifolds with corners, but type (ii) using manifolds with a-corners. In the example, we ¯ 2 by adding a boundary would compactify R2 to a manifold with a-corners R 1 ¯ 2 → R. circle S at infinity, and then u extends to a-smooth u ¯:R Secondly, and related, several important areas of geometry involve forming moduli spaces M of some geometric objects, such that M is a manifold with corners, or a more general space such as a Kuranishi space with corners in Fukaya et al. [28–31], where the ‘boundary’ ∂M parametrizes singular objects included to make M compact. Examples include moduli spaces of Morse flow lines [4, 86], of J-holomorphic curves with boundary in a Lagrangian [28, 29], and of J-holomorphic curves with cylindrical ends in Symplectic Field Theory [24]. We argue in [48] that the natural smooth structure to put on M in such moduli problems is that of a manifold with a-corners (or Kuranishi space with a-corners, etc.). This resolves a number of technical problems in these theories. We can now write down a new version of our theory, starting instead with ac ac the full subcategories Eucac ⊂ Manac with objects in ⊂ Manin , Euc l,m m−l R = J0, ∞)l × R for 0 6 l 6 m. This yields categories of (pre) C ∞ ∞ rings with a-corners PC Ringsac , C∞ Ringsac and C ∞ -schemes with acorners C∞ Schac . We write C ∈ PC∞ Ringsac as (C, C a-ex ) satisfying the analogue of Definition 4.4, but for the operations Φf , Ψg we take f : Rm × J0, ∞)n → R and g : Rm × J0, ∞)n → J0, ∞) to be a-smooth. Similarly, we could start instead with the full subcategories Eucc,ac ⊂ in l,m c,ac c,ac k l Manc,ac , Euc ⊂ Man with objects R = [0, ∞) × J0, ∞) × in k

198

Further generalizations and applications

Rm−k−l for m > k + l. This yields categories of (pre) C ∞ -rings with corners and a-corners PC∞ Ringsc,ac , C∞ Ringsc,ac and C ∞ -schemes with corners and a-corners C∞ Schc,ac . We write C ∈ PC∞ Ringsc,ac as (C, C ex , C a-ex ), where a functor F : Eucc,ac → Sets corresponds to C = (C, C ex , C a-ex ) with C = F (R), C ex = F ([0, ∞)) and C a-ex = F (J0, ∞)). Much of Chapters 4–7 generalizes immediately to the a-corners case, with only obvious changes. Here are some important differences, though: (a) There should be a category Mangac of ‘manifolds with generalized a-corners’, which relates to Manac as Mangc relates to Manc . But no such category has been written down yet. (b) The special classes of C ∞ -rings with corners C = (C, C ex ) in §4.7, and hence of C ∞ -schemes with corners in §5.6, relied on treating C ex or C in as monoids, and imposing conditions such as C ]ex finitely generated. For (interior) C ∞ -rings with a-corners, C a-ex (and C a-in ) are still monoids. However, requiring C ]a-ex to be finitely generated is not a useful condition. For example, if C = C ∞ (J0, ∞)) then by mapping [xα ] 7→ e−α , [0] 7→ 0 we ] may identify C ]a-ex ∼ = [0, 1], C a-in ∼ = (0, 1] as monoids under multiplication, and [0, 1], (0, 1] are not finitely generated monoids. Instead, we should regard C a-ex and C a-in as [0, ∞)-modules. That is, C a-ex and C a-in are monoids under multiplication, with operations c 7→ cα for c in C a-ex or C a-in and α ∈ [0, ∞) satisfying (cd)α = cα dα , (cα )β = cαβ , and c0 = 1. The operation c 7→ cα is Ψxα : C a-ex → C a-ex . Monoids are to abelian groups as [0, ∞)-modules are to real vector spaces. (c) Continuing (b), we should for example define C ∈ C∞ Ringsac to be firm if C ]a-ex is a finitely generated [0, ∞)-module, that is, there is a surjective [0, ∞)-module morphism [0, ∞)n → C ]a-ex . Key properties of firm C ∞ schemes with corners such as Proposition 5.32, and hence Theorem 5.33, then extend to the a-corners case. (d) Continuing (b), to define the corners C(X) in Definition 6.1 in the a-corners a-ex case, for points (x, P ) ∈ C(X) we should take P to be a prime ideal in OX,x considered as a [0, ∞)-module, not as a monoid. We leave it to the interested reader to work out the details.

8.4 Derived manifolds and orbifolds with corners In §2.9.2 we discussed the subject of Derived Differential Geometry, the study of ‘derived manifolds’ and ‘derived orbifolds’, where ‘derived’ is in the sense

8.4 Derived manifolds and orbifolds with corners

199

of Derived Algebraic Geometry. There are two main approaches to defining ∞categories or 2-categories of derived manifolds and derived orbifolds X, one in which X is a special example of a derived C ∞ -scheme or derived C ∞ -stack, as in [6–8,10–13,44,45,64,91–93], and one in which X is a ‘Kuranishi space’ defined by an atlas of charts, as in [46, 49, 51]. (See also Fukaya–Oh–Ohta– Ono [28–30] for an earlier definition of Kuranishi spaces not using derived geometry, in which morphisms between Kuranishi spaces are not defined.) It is obviously desirable to extend Derived Differential Geometry to theories of derived manifolds with corners, and derived orbifolds with corners. One reason to want to do this is that many moduli spaces used in Morse theory, as in §3.6.4, and in Floer theories, as in §3.6.6, should naturally be derived manifolds or orbifolds with corners (or possibly with a-corners, see §8.3), leading to important applications of Derived Differential Geometry with corners. For example, in the ‘Folklore Theorem’ in §3.6.4 we noted that if f : X → R is a Morse function on a manifold X and g is a generic Riemannian metric on X then the moduli spaces M(x, y) of (broken) flow lines of −∇f are expected to be manifolds with corners. If we drop the assumption that g is generic, so that the deformation theory of flow lines may be obstructed, we should expect M(x, y) to be a derived manifold with corners. The same applies to moduli spaces used in Floer theories. For Floer theories in Symplectic Geometry based on moduli spaces M of J-holomorphic curves, including Lagrangian Floer cohomology [28, 29], Fukaya categories [3, 87], and Symplectic Field Theory [24], having a good theory of derived orbifolds with corners is particularly important, as in the most general case one cannot make M into a manifold or orbifold with corners by a genericity assumption, as M must be singular (and hence ‘derived’) at J-holomorphic curves which are nontrivial branched covers of other J-holomorphic curves. This is why Kuranishi spaces were developed first in Symplectic Geometry [28–30]. Some topics in Symplectic Geometry require a theory of differential geometry of derived orbifolds with corners. For example, the universal curve over a moduli space Mg,h,m,n (X, J) of J-holomorphic curves with marked points in (X, ω) is a forgetful morphism Mg,h,m+1,n (X, J) → Mg,h,m,n (X, J) which forgets an interior marked point, and this is a (1-)morphism of derived orbifolds with corners. This is difficult to model in the non-derived approach of [28–30]. The second author will use the theory of C ∞ -algebraic geometry with corners of this book in the final version of [45] to study 2-categories dManc of derived manifolds with corners and dOrbc of derived orbifolds with corners, defined as special examples of certain types of derived C ∞ -schemes and derived C ∞ -stacks with corners. Pelle Steffens [92, §4.1], [93, §5] also uses this book to define an ∞-category of derived manifolds with corners, using Lurie’s

200

Further generalizations and applications

Derived Algebraic Geometry framework [63,64]. See the second author [46,49] for a parallel treatment using Kuranishi spaces with corners. Although the details of any kind of derived geometry are complex, including corners in some theory of derived manifolds is in principle straightforward. All the discrete aspects of corner structure — for example, the monoids P in the local models XP for manifolds with g-corners in §3.3, and the monoid bundles ˇ ex , M ˇ in MC(X) , M C(X) C(X) in §3.5 and §6.4 — remain classical and do not need to be derived. So the processes of making the geometry derived, and adding corners, are fairly independent. For example, to pass from Kuranishi spaces to Kuranishi spaces with corners in [46, 49], one just replaces manifolds by manifolds with corners, and uses appropriate classes of morphisms in Manc .

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Glossary of Notation

AC∞ Sch category of affine C ∞ -schemes, 39 AC∞ Schc category of affine C ∞ -schemes with corners, 116 AC∞ Schcin category of interior affine C ∞ -schemes with corners, 116 b NC(X) b-normal bundle of C(X) in a manifold with (g-)corners X, 62 b ∗ NC(X) b-conormal bundle of C(X) in a manifold with (g-)corners X, 63 b ˇ NC(X) b-conormal bundle of C(X) in a C ∞ -scheme with corners X, 190 ΩC b-cotangent module of C ∞ -ring with corners C, 166 b TX b-tangent bundle of a manifold with (g-)corners X, 61 b ∗ T X b-cotangent bundle of a manifold with (g-)corners X, 61 b ∗ T X b-cotangent sheaf of C ∞ -ringed space with corners X, 180 C : C∞ Schc → C∞ Schcin corner functor on C ∞ -schemes with corners, 146 C : LC∞ RSc → LC∞ RScin corner functor on local C ∞ -ringed spaces with corners, 137 ˇ c corner functor on manifolds with corners, 60 C : Manc → Man in gc ˇ gc corner functor on manifolds with g-corners, 60 C : Man → Man in C(a−1 : a ∈ A)[a0−1 : a0 ∈ Aex ] localization of C ∞ -ring with corners C, 89 C ∞ (X) C ∞ -ring of smooth maps f : X → R, 52 C(X) corners of a manifold with (g-)corners X, 60 C(X) corners of a C ∞ -ringed space with corners X, 135 CC∞ Rings category of categorical C ∞ -rings, 23 C, D, E, . . . C ∞ -rings, 23 C, D, E, . . . (pre) C ∞ -rings with corners, 73 Ck (X) k-corners of a manifold with corners X, 58 Ck (X) k-corners of a firm C ∞ -scheme with corners X, 154 C-mod category of modules over a C ∞ -ring C, 30 C-modco category of complete C-modules, 43 C-mod category of modules over a C ∞ -ring with corners C, 165 b

206

Glossary of Notation

207

CPC∞ Ringsc category of categorical pre C ∞ -rings with corners, 72 CPC∞ Ringscin category of categorical interior pre C ∞ -rings with corners, 72 ∞ C Rings category of C ∞ -rings, 24 C∞ Ringsco category of complete C ∞ -rings, 39 C∞ Ringslo category of local C ∞ -rings, 28 C∞ Ringsc category of C ∞ -rings with corners, 82 C∞ Ringscfg category of finitely generated C ∞ -rings with corners, 98 C∞ Ringscfi category of firm C ∞ -ring with corners, 100 C∞ Ringscfp category of finitely presented C ∞ -rings with corners, 98 C∞ Ringscin category of interior C ∞ -rings with corners, 82 C∞ Ringscin,lo category of interior local C ∞ -rings with corners, 90 C∞ Ringsclo category of local C ∞ -rings with corners, 90 C∞ Ringscsa category of saturated C ∞ -rings with corners, 101 C∞ Ringscsc category of semi-complete C ∞ -rings with corners, 124 C∞ Ringscsc,in category of interior semi-complete C ∞ -rings with corners, 124 C∞ Ringscsi category of simple C ∞ -rings with corners, 101 C∞ Ringsctf category of torsion-free C ∞ -rings with corners, 101 C∞ Ringscto category of toric C ∞ -rings with corners, 101 C∞ RingscZ category of integral C ∞ -rings with corners, 101 C∞ RS category of C ∞ -ringed spaces, 36 C∞ RSc category of C ∞ -ringed spaces with corners, 106 C∞ RScin category of interior C ∞ -ringed spaces with corners, 107 C∞ Sch category of C ∞ -schemes, 39 C∞ Schc category of C ∞ -schemes with corners, 116 C∞ Schcfg category of finitely generated C ∞ -schemes with corners, 127 C∞ Schcfi category of firm C ∞ -schemes with corners, 127 C∞ Schcfi,in category of firm interior C ∞ -schemes with corners, 127 C∞ Schcfi,in,ex category of firm interior C ∞ -schemes with corners with exterior morphisms, 127 ∞ c C Schfi,tf category of firm torsion-free C ∞ -schemes with corners, 127 C∞ Schcfi,tf ,ex category of firm torsion-free C ∞ -schemes with corners and exterior morphisms, 127 ∞ c C Schfi,Z category of firm integral C ∞ -schemes with corners, 127 C∞ Schcfi,Z,ex category of firm integral C ∞ -schemes with corners and exterior morphisms, 127 ∞ c C Schfp category of finitely presented C ∞ -schemes with corners, 127 C∞ Schcin category of interior C ∞ -schemes with corners, 116

208

Glossary of Notation

C∞ Schcin,ex category of interior C ∞ -schemes with corners and exterior morphisms, 127 C∞ Schcsa category of saturated C ∞ -schemes with corners, 127 C∞ Schcsa,ex category of saturated C ∞ -schemes with corners and exterior morphisms, 127 C∞ Schcsi category of simple C ∞ -schemes with corners, 127 C∞ Schcsi,ex category of simple C ∞ -schemes with corners and exterior morphisms, 127 C∞ Schctf category of torsion-free C ∞ -schemes with corners, 127 C∞ Schctf ,ex category of torsion-free C ∞ -schemes with corners and exterior morphisms, 127 C∞ Schcto category of toric C ∞ -schemes with corners, 127 C∞ Schcto,ex category of toric C ∞ -schemes with corners and exterior morphisms, 127 C∞ SchcZ category of integral C ∞ -schemes with corners, 127 C∞ SchcZ,ex category of integral C ∞ -schemes with corners and exterior morphisms, 127 depthX codimension of boundary stratum in a manifold with corners X, 58 ∂X boundary of a manifold with (g-)corners X, 58 ∂X boundary of a firm C ∞ -scheme with corners X, 154 Euc category of Euclidean spaces Rn , 23 c Euc category of Euclidean spaces with corners Rm k , 72 c Eucin category of Euclidean spaces with corners and interior maps, 72 Ex(X) monoid of exterior maps g : X → [0, ∞), 52 FA free C ∞ -ring generated by a set A, 26 FA,Aex free C ∞ -ring with corners generated by (A, Aex ), 86 FA,Ain free interior C ∞ -ring with corners generated by (A, Ain ), 86 Γ : LC∞ RS → C∞ Ringsop global sections functor, 38 Γc : LC∞ RSc → (C∞ Ringsc )op global sections functor, 113 Γcin : LC∞ RScin → (C∞ Ringscin )op global sections functor, 115 In(X) monoid of interior maps g : X → [0, ∞), 52 ∞ LC RS category of local C ∞ -ringed spaces, 36 LC∞ RSc category of local C ∞ -ringed spaces with corners, 106 LC∞ RScin category of interior local C ∞ -ringed spaces with corners, 107 LC∞ RScsa category of saturated local C ∞ -ringed spaces with corners, 109 LC∞ RScsa,ex category of saturated local C ∞ -ringed spaces with corners with exterior morphisms, 109 ∞ c LC RStf category of torsion-free local C ∞ -ringed spaces with corners, 109

Glossary of Notation

209

LC∞ RSctf ,ex category of torsion-free local C ∞ -ringed spaces with corners with exterior morphisms, 109 LC∞ RScZ category of integral local C ∞ -ringed spaces with corners, 109 LC∞ RScZ,ex category of integral local C ∞ -ringed spaces with corners with exterior morphisms, 109 Man category of manifolds, 23 Manac category of manifolds with a-corners, 196 Manac category of manifolds with a-corners and interior maps, 196 in Manc category of manifolds with corners and smooth maps, 51 ˇ c Man category of manifolds with corners of mixed dimension, 51 Manc,ac category of manifolds with corners and a-corners, 196 Mancin category of manifolds with corners and interior maps, 51 ˇ c Man category of manifolds with corners of mixed dimension and interior in maps, 51 Mancst category of manifolds with corners and strongly smooth maps, 51 Mancwe category of manifolds with corners and weakly smooth maps, 51 Manf category of manifolds with faces, 59 Manfin category of manifolds with faces and interior maps, 59 Mangc category of manifolds with g-corners, 56 ˇ gc category of manifolds with g-corners of mixed dimension, 56 Man Mangc category of manifolds with g-corners and interior maps, 56 in ˇ gc category of manifolds with g-corners of mixed dimension and inteMan in rior maps, 56 MC(X) monoid bundle of C(X) in a manifold with (g-)corners X, 63 ∨ MC(X) comonoid bundle of C(X) in a manifold with (g-)corners X, 63 ex ˇ ˇ M , M in sheaves of monoids on C(X), 151 C(X)

C(X)

Mon category of (commutative) monoids, 52 Monsa,tf ,Z category of saturated, torsion-free, integral monoids, 101 Montf ,Z category of torsion-free, integral monoids, 101 MonZ category of integral monoids, 101 MSpec : C-mod → OX -mod spectrum functor on C-modules, 42 ΩC cotangent module of C ∞ -ring C, 31 ΩC cotangent module of C ∞ -ring with corners C, 166 OX structure sheaf of C ∞ -scheme X, 35 ex O X = (OX , OX ) structure sheaf of C ∞ -scheme with corners X, 105 ] P sharpening of a monoid P , 54 P× abelian group of units in a monoid P , 53 P gp groupification of a monoid P , 53 PC∞ Ringsc category of pre C ∞ -rings with corners, 74

210

Glossary of Notation

PC∞ Ringscin category of interior pre C ∞ -rings with corners, 76 PrC set of prime ideals P ⊂ C ex for a C ∞ -ring with corners C, 141 Sets category of sets, 23 Spec : C∞ Ringsop → LC∞ RS spectrum functor on C ∞ -rings, 38 Specc : (C∞ Ringsc )op → LC∞ RSc spectrum functor on C ∞ -rings with corners, 111 Speccin : (C∞ Ringscin )op → LC∞ RScin spectrum functor on interior C ∞ rings with corners, 113 TX tangent bundle of a manifold with corners X, 61 T ∗X cotangent bundle of a manifold with corners X, 61 T ∗X cotangent sheaf of C ∞ -ringed space with corners X, 180 ◦ X interior of a manifold with corners X, 58 X, Y , Z, . . . C ∞ -ringed spaces or C ∞ -schemes, 36 X, Y , Z, . . . C ∞ -ringed spaces or C ∞ -schemes with corners, 106

Index

adjoint functor, 21–23, 34, 35, 38, 40, 43, 60, 73, 79, 80, 83, 85, 102, 109, 114, 116, 124, 128, 139, 140, 146–148 counit of the adjunction, 22, 81, 125, 139 unit of the adjunction, 22, 38, 81, 114, 125, 139, 176 Algebraic Theory, 5, 71, 73 Axiom of Choice, 172 b-calculus, 65

C ∞ -derivation, 31, 166, 167 C ∞ -differentiable space, 39 C ∞ -ring, 23–30 C ∞ -derivation, 31, 166, 167 categorical, 23 complete, 39–40 cotangent module, 31–33 definition, 23 finitely generated, 26 finitely presented, 26 free, 26 ideal in, 25 local, 27 localization of, 28 module, see module over C ∞ -ring R-point, 29 spectrum functor Spec, 37 C ∞ -ring with a-corners, 194–198 C ∞ -ring with corners, 82–104 adjoint functors, 85 finitely generated, 98, 171 finitely presented, 98, 171 firm, 100, 113 free, 86, 171 generators of, 86 integral, 101

interior, 82, 113 limits and colimits of, 85, 86, 90, 100, 102, 104, 124 local, 90–98 localization of, 89, 113 module, see module over C ∞ -ring with corners of a manifold with corners, 83 pushout, 177–179 R-point of, 91 relations in, 87–89 saturated, 101 semi-complete, 119–127 simple, 101 spectrum functor, 110–116 toric, 101 torsion-free, 101 C ∞ -ringed space, 35–39 cotangent sheaf, 42 local, 36 of a manifold, 36 OX -module on, 41–44 pullback, 41 C ∞ -ringed space with corners, 105–109 b-cotangent sheaf b T ∗ X , 180 and the corner functor, 183–191 of a fibre product, 182 corner functor, 135–140 corners C(X), 135 cotangent sheaf T ∗ X , 180 global sections functor, 113, 115 integral, 109 interior, 106 interior morphism, 107 limits and colimits of, 108, 124, 139 O X -module, 179–183

211

212

Index

pullback, 179 saturated, 109 sheaf of O X -modules, see O X -module torsion-free, 109 C ∞ -scheme, 35–39 affine, 39 cotangent sheaf, 42, 44 definition, 39 of a manifold, 36, 38 OX -module on, 42–44 C ∞ -scheme with a-corners, 194–198 C ∞ -scheme with corners, 116–134 affine, 116 b-cotangent sheaf b T ∗ X , 180 and the corner functor, 183–191 of a fibre product, 182 boundary ∂X , 154 corner functor, 140–150 corners C(X), 135 conormal bundle, 190 cotangent sheaf T ∗ X , 180 definition, 116 fibre product, 132–134, 155–164 finitely generated, 127 finitely presented, 127 firm, 127, 128, 147–155, 189–191 integral, 127 k-corners Ck (X), 154 limits of, 133 of a manifold with (g-)corners, 117 O X -module, 179–183 pullback, 179 saturated, 127 sheaf of O X -modules, see O X -module simple, 127 toric, 127, 164 torsion-free, 127 C ∞ -stack with corners, 193–194 Cartesian square, 19 categorical C ∞ -ring, 23 categorical pre C ∞ -ring with corners, 72–73 interior, 72 limits and colimits of, 73 of a manifold with corners, 72 category, 17–23 adjunction, 21 cocomplete, 20, 33 colimit in, 18–21 complete, 20, 33 coproduct, 19 directed colimit in, 20

equivalence between, 18 fibre product, 19, 132–134, 155–164 functor, 17 faithful, 18 full, 17 natural isomorphism, 18 natural transformation, 18 initial object, 19 limit in, 18–21, 40 finite, 18 small, 18 morphism, 17 opposite, 17 product, 19 pushout, 19 subcategory, 17 full, 17 terminal object, 19, 20 universal property, 18 category of algebras, 73 Chen space, 39 co-Cartesian square, 19 coequalizer diagram, 26, 32, 86, 98, 162, 171 coherent sheaf, 43 coreflection functor, 21, 109, 134 coreflective subcategory, 21 cotangent module of C ∞ -ring, 31 of a pushout of C ∞ -rings, 33 relation to cotangent bundles, 31 Derived Algebraic Geometry, 46 derived C ∞ -scheme, 46 Derived Differential Geometry, 46–48, 198–200 derived manifold, 46–48, 198–200 with corners, 1, 198–200 derived orbifold, 46–48, 198–200 with corners, 1, 198–200 diffeological space, 39 directed colimit, 20 Floer theories, 69–70, 199 instanton Floer homology, 70 Lagrangian Floer cohomology, 1, 4, 47, 66, 70, 199 Seiberg–Witten Floer homology, 70 Fukaya category, 1, 4, 48, 69, 70, 199 functor, 17 adjoint, 21–23, 34, 35, 38, 40, 43, 60, 73, 79, 80, 83, 85, 102, 109, 114, 116, 124, 128, 139, 140, 146–148 essentially surjective, 18, 124 faithful, 18

Index full, 17 natural isomorphism, 18 natural transformation, 18 preserves colimits, 22 preserves limits, 22 reflection, 40, 43, 83, 102, 124, 175 Gromov–Witten invariants, 4, 47, 48 Hadamard’s Lemma, 25, 26, 88, 89 instanton Floer homology, 70 Kuranishi space, 4, 47 with corners, 1, 15, 197 Lagrangian Floer cohomology, 1, 4, 47, 66, 70, 199 local C ∞ -ringed space with corners, 105–110 b-cotangent sheaf b T ∗ X , 180 and the corner functor, 183–191 of a fibre product, 182 corner functor, 135–140 corners C(X), 135 cotangent sheaf T ∗ X , 180 global sections functor, 113, 115 integral, 109 limits and colimits of, 108, 110, 124, 139 O X -module, 179–183 pullback, 179 saturated, 109 sheaf of O X -modules, see O X -module torsion-free, 109 manifold with a-corners, 52, 67, 68, 194–198 manifold with corners, 49–63 applications in analysis, 65–67 b-cotangent bundle, 61–63, 181 b-cotangent module, 167–170 b-map, 51, 69 b-tangent bundle, 61–63 b-transverse fibre product, 159–164 boundary, 58–61 c-transverse fibre product, 159–164 ∨ comonoid bundle MC(X) , 63, 150 corner functor, 58–61 corners C(X), 60 cotangent bundle, 61–63 cotangent module, 167–170 definition, 51 depth stratification, 58 diffeomorphism, 50, 51 exterior map, 52 interior, 58 interior map, 50, 51, 62 k-corners Ck (X), 58–61

213

local k-corner component, 58 local boundary component, 58 moduli spaces are, 67–70 monoid bundle MC(X) , 63 sb-transverse fibre product, 164 sc-transverse fibre product, 164 smooth map, 50, 51 strongly smooth map, 50, 51 tangent bundle, 61–63 weakly smooth map, 50, 51 with faces, see manifold with faces manifold with faces, 8, 51, 58, 59, 65, 96, 117, 168 manifold with g-corners, 54–63, 65 b-cotangent bundle, 181 b-cotangent module, 167–170 b-transverse fibre product, 159–164 c-transverse fibre product, 159–164 ∨ comonoid bundle MC(X) , 63, 150 cotangent module, 167–170 definition, 56 moduli spaces are, 69, 70 monoid bundle MC(X) , 63 of mixed dimension, 56 with g-faces, see manifold with g-faces manifold with g-faces, 97, 168 module over C ∞ -ring, 30–33 complete, 43 cotangent module, 31–33 finitely presented, 30 spectrum, 42 module over C ∞ -ring with corners, 165–167 b-cotangent module, 165–179 definition, 166 of pushout, 177–179 b-derivation, 166 cotangent module, 165–170 definition, 166 monoid, 52–54, 75–76, 100, 101 dimension, 154 face, 53 finitely generated, 53 groupification, 53 ideal prime, 53 ideal in, 53 integral, 54 rank, 54 saturated, 54 sharp, 54 toric, 54

214

Index

torsion-free, 54 weakly toric, 54 Morse theory, 67–68, 199 pre C ∞ -ring with corners, 73–82 adjoint functors, 78–82 categorical, see categorical pre C ∞ -ring with corners interior, 76 limits and colimits of, 78–82 presheaf, 33–35 definition, 33 sheafification, 34 stalk, 34 pushout, 19 quasicoherent sheaf, 43 quilts in Symplectic Geometry, 3, 48, 69, 70 reflection functor, 21, 40, 43, 83, 102, 124, 175 reflective subcategory, 21 Seiberg–Witten Floer homology, 70 sheaf, 33–35 definition, 33 direct image, 35 inverse image, 35 pullback, 35 pushforward, 35 stalk, 34, 111 Symplectic Field Theory, 1, 197, 199 Synthetic Differential Geometry, 2, 44–45 with corners, 192–193 Topological Quantum Field Theory, 64 Whitney Embedding Theorem, 168