Etale Cohomology Course Notes

Etale cohomology course notes

Last update: April 22, 2014 at 10:27pm 2 Contents

1 Introduction5 1.1 Prerequisites...... 5 1.2 References...... 5 1.3 History of ´etalecohomology...... 6

2 Category theory7 1 Categories and functors...... 7 1.1 Categories...... 7 1.2 Functors...... 8 2 Natural transformations and equivalences of categories...... 9 2.1 Natural transformations...... 9 2.2 Equivalence of categories...... 10 3 Representable functors, Yoneda’s Lemma, and universal properties 10 3.1 Yoneda’s Lemma...... 10 3.2 Representable functors...... 11 4 Limits and colimits...... 12 5 Special limits and colimits...... 12 A Correspondences and adjunctions...... 14 B The adjoint functor theorem...... 15

3 Sheaves and the fundamental group 19 6 The category of sheaves on a topological space...... 20 6.1 Examples...... 20 6.2 The espace étalé...... 20 7 Etale spaces and sheaves...... 20 7.1 The equivalence of categories...... 20 7.2 Sheafification...... 23 8 Operations on sheaves...... 24 8.1 Pushforward and pullback...... 24 10 Further operations on sheaves...... 25 10.1 Limits and colimits...... 25 11 Even further operations on sheaves...... 26 11.1 Restriction...... 26 11.2 Stalks...... 26

3 4 CONTENTS

11.3 Extension by the void...... 26 11.4 Set theory...... 26 12 Locally constant sheaves and path lifting...... 27 12.1 Constant sheaves and locally constant sheaves...... 27 12.2 The homotopy lifting property...... 28 14 Uniform spaces...... 30 14.1 Filters and uniformities...... 30 14.2 The topology of a uniform space...... 31 14.3 Complete uniform spaces...... 31 14.4 Completion...... 32 15 Categorical Galois theory...... 33 15.1 Uniform groups...... 33 15.2 Infinite Galois theory...... 34 16 Pseudo-locally constant sheaves...... 35 18 Finite Galois theory...... 37 19 The fundamental group...... 39 19.1 Functoriality of the fundamental group...... 39 19.2 The universal cover...... 39 19.3 The Hawai’ian earring...... 40 A Sheaves of groups and torsors...... 42 B Classification of torsors under locally constant groups...... 44 19.1 Crossed homomorphisms and semidirect products..... 44 19.2 Group objects and group actions...... 44 19.3 Classification of torsors under pseudo-locally constant groups 46 C Fiber functors...... 47 D The espace étalévia the adjoint functor theorem...... 47

4 Commutative algebra 49 21 Affine schemes...... 50 21.1 Limits of schemes...... 50 21.2 Some important schemes...... 50 21.3 Topological rings...... 51 21.4 The Zariski topology...... 51 21.5 Why étalemorphisms?...... 52 22 Smooth and étalemorphisms...... 53 22.1 The functorial perspective...... 53 22.2 The differential perspective...... 54 23 Homology of commutative rings...... 58 24 Homology of modules...... 61 25 Flatness...... 64 26 The equational criterion for flatness...... 65 30 Local criteria for flatness...... 68 31 Flatness of étalemaps...... 70 A Extending étalemaps...... 73 B Completions of rings...... 77 C Zariski’s “Main Theorem”...... 79 CONTENTS 5

D More perspectives on ´etalemaps...... 81 31.1 The analytic perspective...... 81 31.2 The algebraic perspective...... 81 31.3 Equivalence of the deﬁnitions...... 82 E Homology of modules...... 83 31.1 Exact sequences...... 85 F Cohomology of modules...... 86

5 The étaletopology 89 32 Grothendieck topologies...... 89 32.1 Examples...... 90 32.2 Sheaves on Grothendieck topologies...... 91 32.3 More examples of Grothendieck topologies...... 92 33 Sheafification...... 92 33.1 Topological generators...... 92 33.2 Descent data...... 93 34 Fpqc descent...... 94 35 A rapid review of scheme theory...... 96 35.1 A heuristic introduction...... 96 35.2 Schemes as functors...... 98 36 The étaletopology...... 101 37 Henselization...... 101 38 The étalefundamental group...... 101 38.1 Covering spaces...... 101 38.2 Locally constant sheaves...... 102 38.3 The topology on the étalefundamental group...... 102

6 Abelian categories and derived functors 103 39 Abelian categories...... 103 40 Resolution and derived functors...... 104 40.1 Injective and projective objects...... 104 40.2 Complexes...... 104 41 Spectral sequences...... 104

7 Cohomology of sheaves 105 42 Acyclic resolutions...... 105 42.1 Injective resolution...... 105 42.2 Flaccid (ﬂasque) resolution...... 105 42.3 Soft resolution...... 105 42.4 Partitions of unity and de Rham cohomology...... 106 43 Cechˇ cohomology...... 106 44 Compactly supported cohomology...... 107 44.1 The compactly supported cohomology of the real line.. 107 45 Proper base change...... 107 46 Leray spectral sequence...... 108 46.1 K¨unnethformula...... 108 6 CONTENTS

46.2 Homotopy invariance of cohomology...... 108 46.3 The cohomology of spheres...... 109 46.4 The cohomology of complex projective space...... 109 47 Poincar´eduality...... 110 48 Lefschetz ﬁxed point theorem...... 110

8 Grothendieck topologies 111 49 Sieves, functors, and sheaves...... 111 50 Morphisms of sites...... 112 51 Group cohomology...... 113 52 (*) Cohomology in other algebraic categories...... 113 52.1 Groups...... 113 52.2 Rings...... 113 52.3 Commutative rings...... 113 52.4 Hyper-Cechˇ cohomology...... 113 53 Fibered categories and stacks...... 114

9 Schemes 115 54 Solution sets as functors...... 115 55 Solution sets as spaces...... 115 56 Quasi-coherent modules...... 115

10 Properties of schemes 117 57 Flatness...... 117 58 Smooth, unramified, and étalemorphisms...... 117 59 (*) Weakly étalemorphisms...... 117

11 Curves 119 60 Riemann surfaces...... 119 61 Riemann–Roch...... 119 62 Serre duality...... 119 63 The Jacobian...... 119

12 Abelian varieties 121 64 Lattices in complex vector spaces...... 121 65 The dual abelian variety...... 121 66 Geometric class ﬁeld theory...... 121

13 Topologies on schemes 123 67 Faithfully flat descent...... 123 68 The étaletopology...... 123 69 Other topologies...... 123 69.1 The Zariski topology...... 123 69.2 (*) The pro-étaletopology...... 123 69.3 (*) The infinitesimal site...... 123 CONTENTS 7

14 Etale cohomology 125 70 Constructible sheaves and `-adic cohomology...... 125 71 Etale cohomology in low degrees...... 125 72 Etale cohomology and colimits...... 125 73 Cup product...... 126

15 Etale cohomology of points 127 74 Group cohomology...... 127 75 Hilbert’s theorem 90...... 127 76 The Brauer group...... 127 77 Tsen’s theorem...... 127

16 Etale cohomology of curves 131 78 Calculation...... 131 79 Poincar´eduality for curves...... 131

17 Base change theorems 133 80 Smooth base change...... 133 80.1 Auslander–Buchsbaum formula...... 133 80.2 Purity of the branch locus...... 134 8 CONTENTS Chapter 1

Introduction

1.1 Prerequisites

The essential prerequisites for this course are comfort with point set topology and commutative algebra. Here is a partial list of commutative algebra concepts we will use without review:

1. commutative rings,

2. modules under commutative rings,

3. localization,

4. polynomial rings,

5. tensor product,

6. kernel, cokernel, and image.

We will use a lot of homological algebra, but we will review what we use; studying homological algebra concurrently might work well. Comfort with the theory of schemes will be assumed as little as possible. I’ll do my best to review what we use, but there are bound to be some places where background in algebraic geometry is necessary.

1.2 References

Here are a few references I plan to consult as I prepare the class.

Deligne, P. Séminairede GéométrieAlgébriquedu Bois-Marie (SGA 1 4 2 ). Iversen B. Cohomology of sheaves.

9 10 CHAPTER 1. INTRODUCTION

Freitag, E. and Kiehl, R. Etale cohomology and the Weil conjecture.

Weibel, C. An introduction to homological algebra.

Serre, J.-P. Géométriealgebrique et géométrieanalytique.

Mumford, D. Abelian varieties.

Serre, J.-P. Cohomologie galoisienne.

1.3 History of ´etalecohomology Chapter 2

Category theory

Reading and references

Grothendieck, A. Introduction au langage fonctoriel. Facult´edes Sciences d’Alger, Seminaires 1965–66.

Mac Lane, S. Categories for the working mathematician. Springer- Verlag. §§1.2–1.4

Kashiwara, S. and Schapira, P. Categories and sheaves. Chapters 1–3.

For the concept of a correspondence, the only reference I know is Lurie’s Higher topos theory: Lurie, J. Higher topos theory. Section 2.3.1.

1 Categories and functors 1.1 Categories Deﬁnition 1.1. A pre-category C is

PC1 a collection of objects C0 = Ob(C ),

PC2 for each pair of objects X,Y ∈ Ob(C ), a set of morphisms C1(X,Y ) = HomC (X,Y ), and PC3 for each triple of objects X,Y,Z ∈ Ob(C ) and triple of morphisms f ∈ Hom(X,Y ), g ∈ Hom(Y,Z), and h ∈ Hom(X,Z) a property, called commutativity, of the triple (f, g, h).1

A pre-category is called small if Ob(C ) is a set.

1Properly speaking, we should include X, Y , and Z in the notation here.

11 12 CHAPTER 2. CATEGORY THEORY

Deﬁnition 1.2. A pre-category C is a category if it satisﬁes the following conditions:

CAT1 (Composition law) If f ∈ Hom(X,Y ) and g ∈ Hom(Y,Z) then there is a unique g ◦ f ∈ Hom(X,Z) making the triangle (f, g, g ◦ f) commutative. We also write gf = g ◦ f.

CAT2 (Identity elements) Each object X ∈ Ob(C ) has an identity morphism idX ∈ Hom(X,X) satisfying idX ◦ f = f for all f ∈ Hom(Y,X) and g ◦ idX = g for all g ∈ Hom(X,Y ).

CAT3 (Associativity) We have f ◦ (g ◦ h) = (f ◦ g) ◦ h for all f ∈ Hom(Y,Z), g ∈ Hom(X,Y ), and h ∈ Hom(W, X).

Usually we abuse notation and write X ∈ C instead of X ∈ Ob(C ). The opposite category C ◦ of C is formed by deﬁning

HomC ◦ (X,Y ) = HomC (Y,X)

with the expected deﬁnition of composition:

f ◦ g = g ◦ f. C ◦ C

Example 1.3. Any partially ordered set can be viewed as a category.

Example 1.4. For each n, let ∆n be the simplex category whose objects are the non-negative integers {0, 1, . . . , n} and for which Hom∆n (i, j) consists of a single element for i ≤ j and is empty for i > j. This is a special case of the previous example, applied to a totally ordered set with n + 1 elements.

1.2 Functors

Deﬁnition 1.5. Suppose that C and D are pre-categories. A functor F : C → D is a triple of maps

F : Ob(C ) → Ob(D)

F : HomC (X,Y ) → HomD (FX,FY )

such that (F (f),F (g),F (h)) is commutative whenever (f, g, h) is. A functor between categories is a functor between the underlying pre-categories.

Exercise 1.6. Objects of a pre-category C are in bijection with functors ∆0 → C ; morphisms are in bijection with functors ∆1 → C ; commutative triangles are in bijection with functors ∆2 → C . 2. NATURAL TRANSFORMATIONS AND EQUIVALENCES OF CATEGORIES13

2 Natural transformations and equivalences of categories 2.1 Natural transformations Deﬁnition 2.1. Suppose C and D are pre-categories. The product pre-category C × D has as objects Ob(C ) × Ob(D) and

HomC ×D ((X,Y ), (Z,W )) = HomC (X,Z) × HomD (Y,W ).

A triangle ((f, f 0), (g, g0), (h, h0)) is commutative if both (f, g, h) and (f 0, g0, h0) are.

Exercise 2.2. If C and D are categories then C × D is a category.

Deﬁnition 2.3. Let C and D be pre-categories and F,G : C → D functors. A natural transformation from F to G is a functor h : C × ∆1 → D such that if we restrict h0(X) = h(X, 0) and h1(X) = h(X, 1) then

h0 = F

h1 = G. for X and object or morphism of C . A commutative triangle of natural transformations α is a functor α : C × ∆2 → D.

Exercise 2.4. Suppose that C is a small pre-category and D is a category. Then there is a category Hom(C , D) whose objects are the functors from C to D, whose morphisms are natural transformations, and whose commuting triangles are the commuting triangles of natural transformations.

A more concrete way of describing a natural transformation from F to G is as a system of morphisms ϕX : F (X) → G(X) in D, for each X ∈ C , such that for any morphism f : X → Y in C , the diagram

ϕX F (X) / G(X)

F (f) G(f)

ϕY F (Y ) / G(Y ) is commutative. That is, G(f) ◦ ϕX = ϕY ◦ F (f). We write ϕ : F → G. The composition of a natural transformation ϕ : F → G and a natural transformation ψ : G → H is given by the formula (ψ ◦ ϕ)X = ψX ◦ ϕX for all X ∈ C . This category is denoted Hom(C , D). Deﬁnition 2.5. A natural transformation with an inverse is called a natural isomorphism. 14 CHAPTER 2. CATEGORY THEORY

Exercise 2.6. Let F,G : C → D be functors and ϕ : F → G a natural transformation. Then ϕ is a natural isomorphism if and only if ϕX : F (X) → G(X) is an isomorphism for all X ∈ Ob(C ). In particular, if D = Sets then ϕ is a natural isomorphism if and only if ϕX is bijective for all X ∈ Ob(C ).

2.2 Equivalence of categories Deﬁnition 2.7. Categories C and D are said to be equivalent if there are functors F : C → D and G : D → C such that F ◦ G is naturally isomorphic to 2 idD and G ◦ F is naturally isomorphic to idC . Deﬁnition 2.8. A functor F : C → D is said, respectively, to be (i) faithful, F (ii) full, (iii) fully faithful for all X,Y ∈ Ob(C ), the map HomC (X,Y ) −→ HomD (FX,FY ) is (i) injective, (ii) surjective, (iii) bijective. If every object of D is isomorphic to FX for some X ∈ Ob(C ) then F is said to be essentially surjective. Proposition 2.9. A functor is an equivalence of categories if and only if it is fully faithful and essentially surjective.

3 Representable functors, Yoneda’s Lemma, and universal properties 3.1 Yoneda’s Lemma Let C be category. In the last section we saw that Cˆ = Hom(C ◦, Sets) is a category whose objects are functors and whose morphisms are natural transformations. There is a functor ˆ C → C : X 7→ hX where hX is the functor hX (Y ) = HomC (Y,X). If f : Z → Y is a morphism in C then hX (f) is the function sending g : Y → X to g ◦ f : Z → X. Proposition 3.1 (Yoneda’s Lemma). Let X be an object of C and F : C ◦ →

Sets a functor. There is a unique function u : HomCˆ(hX ,F ) → F (X) that is natural in F and sends idhX to idX when F = hX .

Proof. Let ϕ : hX → F be a natural transformation. Then by naturality in F we have idh / ϕ _X _

idX / ϕ(idX ).

2 Here idC and idD denote the identity functors of C and D. 3. REPRESENTABLE FUNCTORS, YONEDA’S LEMMA, AND UNIVERSAL PROPERTIES15

Therefore if u exists then we have u(ϕ) = ϕ(idX ). Now we verify that u(ϕ) = ϕX is a bijection. To see that it is injective, suppose that u(ϕ) = u(ψ), i.e., that ϕX = ψX . Consider any map f : Y → X. Then we have a commutative diagram

ϕX hX (X) / F (X) idX / ϕ(idX )

hX (f) F (f)

ϕY hX (Y ) / F (Y ) f / ϕ(f).

This implies that ϕY (f) = F f(ϕ(idX )) = F f(ψ(idX )) = ψY (f), i.e., that ϕ = ψ. Finally we show that u is surjective. Suppose that α ∈ F (X). Let f ∈ hX (Y ). Set ϕY (f) = F f(α). To check this is natural in Y , suppose we have g : Z → Y . We must check that the diagram below is commutative:

ϕY hX (Y ) / F (Y ) f / F f(α) f

hX (g) F (g)

ϕZ hX (Z) / F (Z) F g(F f(α)) fg / F (fg)(α).

But F is a (contravariant) functor, so F (fg) = F g ◦ F f.

Corollary 3.1.1. The functor X 7→ hX is fully faithful.

Proof. Yoneda’s lemma gives a bijection Hom(hX , hY ) → Hom(X,Y ). It is easy to check that this is inverse to the function h− : Hom(X,Y ) → Hom(hX , hY ).

3.2 Representable functors Deﬁnition 3.2. By Yoneda’s lemma, an X ∈ C and ϕ ∈ F (X) gives a map hX → F . If this map is an isomorphism then we say (X, ϕ) represents F . Sometimes we abuse language and say X represents F when ϕ is clear from context.

We write hX : C ◦ → Hom(C , Sets) for the functor with hX (Y ) = Hom(X,Y ). Note that this functor is contravariant. We have a natural isomorphism Hom(hX ,F ) ' F (X) for covariant functors into Sets, just as in Yoneda’s lemma. We employ the same language about representable functors.

Example 3.3. Let C be a category containing objects Xi, i ∈ I. Let Y F (Y ) = Hom(Y,Xi). i∈I

Then an object representing F is called a product of the Xi. 16 CHAPTER 2. CATEGORY THEORY

Q Example 3.4. Notation as in the last example. Let F (Y ) = i∈I Hom(Xi,Y ). then an object representing F is called a coproduct of the Xi. Exercises: Co- product in the category of sets is disjoint union; coproduct in the category of groups is free product; coproduct in the category of abelian groups is direct sum; coproduct in the category of commutative rings is tensor product; coproduct in the category of topological spaces is disjoint union.

4 Limits and colimits

Definition 4.1. Let C be a category. By a diagram in C we will mean a functor F : I → C for some pre-category I. Suppose that I is a pre-category. Let I. be the pre-category with Ob(I.) = Ob(I) q {1} and Hom(X, 1) = 1 for all X ∈ Ob(I.). Every triangle in I. with final vertex 1 is commutative. Effectively, I. is the diagram obtained by adjoining a final object to I.

Deﬁnition 4.2. Let C be a category, I a diagram, F : I → C a functor. Deﬁne a functor G : C → Sets by ( )

G(X) = F 0 : I. → F 0 = F and F 0(1) = X . C I

An object of C representing G is called the colimit of F . It may be denoted lim F , lim F (X), colim F , or colim F (X). −→ −→X∈I X∈I Dually (i.e., a colimit in C ◦), we have the notion of a limit. It is denoted lim F , lim F (X), etc. ←− ←−X∈I Exercise 4.3. Construct limits and colimits in Sets and familiar categories, like groups, abelian groups, commutative rings, topological spaces, ...

5 Special limits and colimits

Exercise 5.1. Suppose that S , i ∈ I is a small diagram of sets. Then lim S i ←−i∈I i can be constructed as n o Y (xi)i∈I ∈ Si u(xi) = xj for all u : i → j in I . i∈I

Exercise 5.2. Suppose that Sj, j ∈ J is a ﬁltered (small) diagram of sets. Then lim S can be constructed as the set ` S /R where R is the equivalence −→j∈J j j∈J j relation that has s ∼ t whenever s ∈ Si, t ∈ Sj, and u(s) = v(t) for some u : i → k and v : j → k.

Exercise 5.3. Construct small colimits in the category of sets. 5. SPECIAL LIMITS AND COLIMITS 17

There are some diagrams shapes that have special behavior with respect to limits and colimits:

(i) A sequential colimit is one indexed by N.

(ii) A sequential limit is one indexed by N◦.

A non-empty partially ordered set is called filtered (resp. cofiltered) if every pair of elements (equivalently, every finite subset) has an upper bound (resp. lower bound). More generally, a pre-category I is called filtered if, for every finite pre-category J and every functor J → I, there is an extension to J. → I.A pre-category is called cofiltered if its opposite is filtered.

(iii)A ﬁltered colimit is a colimit indexed by a ﬁltered partially ordered set.

(iv)A ﬁltered limit is a limit indexed by a coﬁltered partially ordered set.3

(v)A ﬁnite limit or colimit is one indexed by a ﬁnite pre-category.

◦ Proposition 5.4. Let Xi, i ∈ N be a sequential diagram of sets such that the maps u : X → X are all surjective. Set X = lim X . Then the structural ij i j ←− i maps X → Xi are all surjective.

Proof. We can identify X with the set of tuples (xi)i∈N such that uij(xi) = xj. Fix i ∈ N and x ∈ Xi. Select xj = uij(x) for j ≤ i. Select the remaining values of xj recursively: Assume xj has been deﬁned already for j ≤ N. Since uN+1,N is surjective, we can ﬁnd xN+1 ∈ XN+1 with uN+1,N (xN+1) = xN . This gives an element of X that maps to xi under the map X → Xi.

Exercise 5.5. Show that the analogue of Proposition 5.4 is false for ﬁltered limits. (Hint: See [HS] or [BS, Remark 3.1.6].)

Proposition 5.6. Let I be a ﬁltered diagram and J a ﬁnite pre-category. For any diagram of sets Sij depending on i ∈ I and j ∈ J, the natural map

lim lim S → lim lim S −→ ←− ij ←− −→ ij i∈I j∈J j∈J i∈I is a bijection.

Proof. First we show injectivity. Suppose that x, y ∈ lim lim S . There- −→i∈I ←−j∈J ij 0 fore there are indices i, i ∈ I such that x ∈ lim S and y ∈ lim S 0 . ←−j∈J ij ←−j∈J i j Since I is ﬁltered, we can ﬁnd a single index ≥ i, i0 and therefore we can assume i = i0. Suppose that x and y have the same image in lim lim S . Therefore ←−j∈J −→i∈I ij x and y have the same image in lim S for all j. Therefore, for each j ∈ J, −→i∈I ij there is some i(j) ≥ i such that x and y have the same image in Si(j),j. Choose

3The more accurate cofiltered limit is also common. 18 CHAPTER 2. CATEGORY THEORY i0 ≥ i(j) for all of the (finitely many) j ∈ J. Then x and y have the same image in S 0 . Therefore x and y represent the same element of lim lim S . i j −→i∈I ←−j∈J ij Now we show surjectivity. Let x be an element of lim lim S . There- ←−j∈J −→i∈I ij fore, for each j ∈ J we have an index i(j) ∈ I and xj ∈ Si(j),j representing x. Since J is finite, we can assume that all of the i(j) are equal, say to i. Enlarging i further if necessary, we can assume that the xj represent an element of lim S . This induces an element of lim lim S whose image ←−j∈J ij −→i∈I ←−j∈J ij in lim lim S is x. ←−j∈J −→i∈I ij

Corollary 5.6.1. Suppose that Ai, i ∈ I is a filtered system of sets equipped with an algebraic structure “defined by finite inverse limits”. Then the set A = lim A −→ i can be given the same algebraic structure in a unique way so that A is the colimit of the Ai in the category of objects with that algebraic structure.

In particular, a ﬁltered colimit of groups has a canonical group structure, a ﬁltered colimit of rings has a canonical ring structure, etc.

Exercise 5.7. Show by example that ﬁltered limits do not always commute with ﬁnite colimits.

Exercise 5.8. Let p be a prime number. Show that

lim lim pnZ/pmZ → lim lim pnZ/pmZ −→ ←− ←− −→ n∈Z m∈Z m∈Z n∈Z is a bijection and that it has a natural commutative ring structure under which it is a ﬁeld. This is called the ﬁeld of p-adics.

A Correspondences and adjunctions

Deﬁnition 5.1. A correspondence from a category C to a category D is a functor F : C ◦ × D → Sets. We will say that F is (left) representable by a functor G : C → D if there is a natural isomorphism

F (X,Y ) ' HomD (GX, Y ).

We will say it is (right) representable by H : D → C if there is a natural isomorphism

F (X,Y ) ' HomC (X,HY ). If F is both left and right representable by G and H, respectively, then we say (G, H) is an adjoint pair of functors. The natural equivalence

HomC (X,HY ) ' HomD (GX, Y ) is called the adjunction. B. THE ADJOINT FUNCTOR THEOREM 19

Example 5.2. Let f : A → B be a homomorphism of commutative rings. For an A-module M and a B-module N, deﬁne F (M,N) to be the set of additive functions ϕ : M → N such that

ϕ(ax) = f(a)ϕ(x) for a ∈ A and x ∈ M. Then F is right representable by the functor H that gives a B-module N the A-module structure induced by f. It is left representable by the functor G(M) = B ⊗A M. Therefore (G, H) are an adjoint pair of functors.

Proposition 5.3. To specify an adjunction between G : C → D and H : D → C it is equivalent to give natural transformations ϕ : GH → idD and ψ : idD → HG such that the compositions

G(ψ ) ϕ GX −−−−→X GHGX −−−→GX GX ψ H(ϕ ) HY −−−→HY HGHY −−−−→Y HY coincide with the identity maps of GX and HY , respectively, for all X ∈ Ob(C ) and Y ∈ Ob(D).

4 ← 4

Proposition 5.4. Suppose that F : C → D is a functor.

(i) If F has a right adjoint then F preserves colimits.

(ii) If F has a left adjoint then F preserves limits.

Proof. Let Xi, i ∈ I be a diagram in C with colimit X. Let G be a right adjoint of F . We have bijections (natural in Y )

Hom(FX,Y ) = Hom(X, GY ) = Hom(lim X , GY ) −→ i = lim Hom(X , GY ) = lim Hom(FX ,Y ). ←− i ←− i Therefore FX represents the functor Y 7→ lim Hom(FX ,Y ). ←− i

We will see in the next section that this proposition has a partial converse.

B The adjoint functor theorem

Proposition 5.1. Let C be a category that admits arbitrary small colimits. Assume that there is a set of objects C0 ⊂ C such that every object of C admits a map to some object of C0. Then C has a ﬁnal object.

4todo: proof 20 CHAPTER 2. CATEGORY THEORY

Proof. Taking the colimit of the full subcategory C0 gives an object X of C and we only need to show it is final. Certainly every object of C has a map to X, as it has a map to some Y ∈ C0 and Y has a map to X by definition of the colimit. We therefore only need to check that each object of C has at most one map to X. Before proving the uniqueness, we consider a map f : X → Y with Y ∈ C0 (which is guaranteed to exist by assumption). There is a tautological map g : Y → X, by the definition of a colimit. Then we obtain gf : X → X. I claim that gf = idX . Indeed, it suffices to show that if Z ∈ C0 and h : Z → X is the tautological morphism of the colimit, then gfh = h. But fh : Z → Y is a morphism of C0, so that gfh = h, by definition of a colimit. Suppose that we have two maps u, v : Y → X. Then we can form the coequalizer of u and v and call it W . There is a map W → V for some V ∈ C0 so that u and v are coequalized by a map t : X → V . On the other hand, there is a map s : V → X because V ∈ C0. But by the considerations of the last paragraph, we have st = idX , since V ∈ C0. Therefore

u = stu = stv = v.

Corollary 5.1.1. Let F : C ◦ → Sets be a functor. Assume that C possesses a small collection of objects C0 such that, whenever ξ ∈ F (X) there is an object ∗ X0 ∈ C0, an element ξ0 ∈ F (X0), and a map u : X → X0 such that u ξ0 = ξ. Then F is representable.

Proof. The representability of F is equivalent to the existence of a ﬁnal object for the category C /F . The hypothesis of the corollary implies that C /F satisﬁes the hypothesis of the proposition.

Proposition 5.2. Suppose that C contains a set of objects C0 such that, for every object Y ∈ C , the tautological morphism

lim X → Y −→ X∈C0/Y is an isomorphism. Assume also that C has arbitrary small colimits and the isomorphism classes of quotients of each object form a set. Then C has a ﬁnal object.

Proof. Let Z = lim X and let W = lim Z0, the colimit taken over all quotients −→C0 −→ of Z. Note that W is a quotient of Z. We show that there is a unique map Y → W for any Y ∈ C . First of all, a map exists since we can present Y as lim X for some diagram of X drawn −→ i i from C0. Then each Xi comes with a tautological map to Z and all of these are compatible, by deﬁnition of Z. By the universal property of the colimit, we obtain a map Y → Z, and by composition, a map Y → W .. B. THE ADJOINT FUNCTOR THEOREM 21

To show that this map is unique, we consider two maps Y ⇒ W . The coequalizer W 0 is a quotient of W (it represents a subfunctor of hW ) so that it is also a quotient of Z. Hence it is equal to W , since hW is the intersection of all representable subfunctors of hZ .

Corollary 5.2.1. Suppose that F : C ◦ → Sets is a functor, that C has arbitrary small colimits, that these are preserved by F , and that C is generated under small colimits by a set of objects C0 ⊂ C . Then F is representable. Proof. The hypotheses guarantee that C /F satisﬁes the conditions of the proposition. 22 CHAPTER 2. CATEGORY THEORY Chapter 3

Sheaves and the fundamental group

Reading and references

R. Hartshorne. Algebraic geometry. Section 2.1.

R. Vakil. Foundations of algebraic geometry. Sections 2.1–2.5, 2.7. Available online: math.stanford.edu/~vakil/216blog

M. Kashiwara and P. Schapira. Sheaves on Manifolds. Sections II.1 and II.2.

R. Godement. Topologie alg´ebriqueet th´eoriedes faisceaux. Sections II.1.1 and II.1.2.

The discussion of uniform spaces is drawn mostly from

N. Bourbaki. Topologie g´en´erale. Sections I.6–7, II.1, II.3.

The presentation on the fundamental group is adapted from the following paper:

B. Bhatt and P. Scholze. “The pro-´etaletopology for schemes”. arXiv:1309.1198

The following papers are also relevant, although I have not studied them thoroughly:

D. Biss. “A generalized approach to the fundamental group”. Amer. Math. Monthly 107 (2000) no. 8.

D. Biss. “The topological fundamental group and generalized covering spaces”. Topology Appl. 124 (2000), no. 3.

23 24 CHAPTER 3. SHEAVES AND THE FUNDAMENTAL GROUP

6 The category of sheaves on a topological space

Deﬁnition 6.1. Let X be a topological space, Open(X) its category of open sets. A presheaf on X is a functor F : Open(X)◦ → Sets. For V ⊂ U, we frequently denote the restriction map F (U) → F (V ) by x 7→ x V . A presheaf is said to be a sheaf if it satisﬁes the following two properties:

SH1 If x, y ∈ F (U) and x V = y V for all V in an open cover of U, then x = y. S SH2 If U = V ∈ V and xV ∈ F (V ) for all V ∈ V are such that xV = V V ∩W xW V ∩W then there is some x ∈ F (U) with x V = xV for all V . We write Γ(U, F ) = H0(U, F ) = F (U).

6.1 Examples Continuous maps to a ﬁxed target, continuous sections, more reﬁned maps or sections, the sheaf of orientations of a manifold

Example 6.2. Let f : X → Y be a continuous function. For each U ⊂ Y , let F (U) be the set of all continuous sections of f over U. That is, F (U) is the set of all continuous functions g : U → X such that fg coincides with the inclusion of U in Y . Then F (U) is called the sheaf of sections of X over Y .

6.2 The espace étalé Definition 6.3. A continuous map of topological spaces f : X → Y is called a local homeomorphism if there is a cover of X by open subsets U such that f(U) ⊂

Y is open and f U : U → f(U) is a homeomorphism. Local homeomorphisms are frequently also called ´etalemaps.

Exercise 6.4. (i) Show that a finite limit of étalespaces over X (taken in the category of topological spaces over X) is an étalespace over X.

(ii) Show that the finiteness assumption is essential by producing an infinite limit that is not an étalespace.

Let ét(X) denote the category of all étalemaps f : Y → X. Morphisms in ét(X) are maps commuting with the projection to X.

7 Etale spaces and sheaves 7.1 The equivalence of categories If Y is an ´etalespace over X, let Y sh be the sheaf of sections of Y . This gives a functor: ´et(X) → Sh(X): Y 7→ Y sh. 7. ETALE SPACES AND SHEAVES 25

We will show that this functor is an equivalence of categories. Suppose that F ∈ Psh(X). Construct a diagram whose elements are U × {σ} for each open U ⊂ X and each σ ∈ F (U). We say that U × {σ} ≤ V × {τ} if U ⊂ V and

τ U = σ. Define F ét = lim U × {σ}. −→ U∈Open(X) σ∈F (U) The colimit is taken in the category of topological spaces. There is projection F ét → X that restricts to the projection on the first factor on U ×{σ} (universal property of colimit).

Exercise 7.1. (i) Show that an arbitrary colimit of ´etalespaces over X (taken in the category of topological spaces) is an ´etalespace over X.

(ii) Conclude that the projection F ´et → X constructed above is a local homeomorphism.

Lemma 7.2. (i) The maps U × {σ} → F ´et are open embeddings.

(ii) The U × {σ} ⊂ F ´et form a basis for the topology.

(iii) The intersection (U ×{σ})∩(V ×{τ}) is W ×{ω} where W ⊂ U ∩V is the

largest open subset of U ∩ V on which σ and τ agree and ω = σ W = τ W .

(iv) We have U × {σ} ⊂ V × {τ} if and only if U ⊂ V and τ U = σ.

Proof. (i) Let V be the image of U × {σ} in F ´et. By deﬁnition of a colimit, the maps U ' U × {σ} → V → U

are continuous. As U → V is a surjection, both maps are bijections. As they are continuous, they are inverse homeomorphisms.

(ii) This is evident from the facts that the U × {σ} are an open cover and

every open V ⊂ U corresponds to an open subset V × {σ V } ⊂ U × {σ}. (iii) Suppose that y ∈ (U ×{σ})∩(V ×{τ}). Then there is a sequence of maps

U × {σ} o W1 × {ω1} / ··· o Wn × {ωn} / V × {τ}

T with y ∈ Wi ⊂ U ∩ V . Therefore we have σ W = τ W for some open neighborhood W of y. This holds for every y in the intersection, so we get the desired conclusion.

(iv) A special case of the previous part. 26 CHAPTER 3. SHEAVES AND THE FUNDAMENTAL GROUP

Before stating the next theorem, we construct natural maps

(Y sh)´et → Y F → (F ´et)sh.

To give (Y sh)ét → Y, we must give a map U × {σ} → Y for any open U ⊂ X and any σ ∈ Y sh(U) in a compatible way. But σ ∈ Y sh(U) = Γ(U, Y ) is a map U → Y commuting with the projection to X. The composition U × {σ}' U −→σ Y is the desired map. To give F → (F ét)sh we must give compatible maps F (U) → Γ(U, F ét) for every open U ⊂ X. If σ ∈ F (U) then we get a tautological map U ' U × {σ} → F ét by the construction of F ét.

Exercise 7.3. Verify that these constructions are natural in F and in Y .

Theorem 7.4. The functors Y 7→ Y sh and F 7→ F ´et are inverse equivalences of categories.

Proof. We verify first that if Y is an étalespace over X then the map ϕ : (Y sh)ét → Y is a homeomorphism. The diagram defining (Y sh)ét can be viewed as the collection of all open U ⊂ Y such that p U : U → p(U) is a homeomorphism. As every one of these open subsets must map bijectively onto its image in Y , we deduce that ϕ is open and a local homeomorphism. We only have to show that it is injective. We know that ϕ U is injective for every U that maps injectively to X, so the only way injectivity could fail is if x ∈ U, y ∈ V , and ϕ(x) = ϕ(y). But then x ∈ U ∩ V , so the diagram

U ∩ V / U

V / (Y sh)ét commutes (all of U ∩ V , U, and V are in the diagram whose colimit defined (Y sh)ét. Thus x and y represent the same element of (Y sh)ét. Now we show that F → (F ét)sh is a bijection when F is a sheaf. First we show injectivity. Suppose σ, τ ∈ F (U) for some open U ⊂ X. Then the two maps U ' U × {σ} → F ét and U ' U × {τ} → F ét coincide. Therefore U × {τ} and U × {σ} have the same image in F ét . Therefore by Lemma ?? ??, we have σ = τ. Now suppose that s ∈ (F ét)sh(U) = Γ(U, F ét). We show that s factors as ét ét U ' U × {σ} → F for some σ ∈ Γ(U, F ). Consider the open sets Vσ = 7. ETALE SPACES AND SHEAVES 27

−1 s (U × {σ}) as σ varies in F (U). These cover U and we have σ ∈ F (Vσ) Vσ for all σ. Moreover, σ = τ Vσ ∩Vτ Vσ ∩Vτ

−1 because Vσ ∩ Vτ = s ((U × {σ}) ∩ (V × {τ})) ⊂ W (notation as in Lemma ??), and σ = τ by deﬁnition. As the Vσ cover U, we can glue together the σ W W Vσ to a single section σ ∈ F (U). Then we have

[ U × {σ} = Vσ × {σ }. Vσ σ

Thus, −1 [ s (U × {σ}) = Vσ = U and s is therefore the map U ' U × {σ} → F ´et. Thus s lies in the image of F (U) → (F ´et)sh(U).

7.2 Sheaﬁﬁcation

Note that the espace étaléis defined for a presheaf, not just a sheaf, and that the map F → (F ét)sh is defined even if F is a presheaf. Therefore if F is a presheaf, we have a canonical map from F into a sheaf. We will abbreviate (F ét)sh to F sh.

Proposition 7.5. Let F be a presheaf and G a sheaf. Any morphism of sheaves u : F → G factors uniquely through the map F → F sh constructed above.

Proof. We have a commutative diagram

F / G

o F sh / Gsh

(by the naturality of the transformation F → F sh). But G is a sheaf so that the vertical arrow on the right is an isomorphism.

Corollary 7.5.1. The sheafification functor F 7→ F sh : Psh(X) → Sh(X) is left adjoint to the inclusion of Sh(X) ⊂ Psh(X). In particular, sheafification preserves arbitrary colimits and limits of sheaves may be computed on the underlying presheaves.

Corollary 7.5.2. The category Sh(X) admits arbitrary small limits and colimits. 28 CHAPTER 3. SHEAVES AND THE FUNDAMENTAL GROUP

8 Operations on sheaves

8.1 Pushforward and pullback

Deﬁnition 8.1. Let F be a sheaf on X, and p : X → Y a continuous function. −1 Deﬁne p∗F (U) = F (p U) for U ⊂ Y open. This is called the pushforward of F via p.

Definition 8.2. Let F be an étalespace over Y and p : X → Y a continuous ∗ function. Define p F = F ×Y X. Extend this definition to sheaves by the equivalence between étale spaces and sheaves.

Lemma 8.3. Pullback of ´etalespaces commutes with all small colimits.

Proof. Let X0 → X be a continuous function and let Y be an étalespace over X. Assume that Y = lim Y where each Y is an étalespace over X. On the −→ i i level of underlying sets, Y 0 = Y × X0 may be identified with lim(Y × X0). X −→ i X We have to check the topologies are the same. The space Y has a basis by open subsets that are images of U ⊂ Yi where U projects isomorphically to an 0 open subset of X. Therefore Y ×X X has a basis of open subsets of the form 0 pi(U ×X V ) where U ⊂ Yi projects isomorphically to an open subset of X and 0 0 0 0 V ⊂ X is open and pi : Yi → Y is the tautological projection. 0 0 On the other hand, Y = lim Yi has a basis of open subsets of the form 0 0 −→ pi(W ) where W ⊂ Yi is open and projects isomorphically to an open subset of 0 X . If U and V are of the form described in the last paragraph then U ×X V 0 0 is of this form. Therefore the topology on Y as lim Yi is at least as fine as the 0 −→ topology as Y ×X X . 0 On the other hand, any such open W ⊂ Yi is a union of open subsets of the form U ×X V where U and V are as in the first paragraph, by definition of the 0 0 product topology on Yi ×X X . Therefore the topology on Y ×X X is at least as fine as the one on lim Y 0. −→ i

Proposition 8.4. There is a natural bijection

∗ HomSh(X)(p F,G) ' HomSh(Y )(F, p∗G) for any sheaves G on X and F on Y .

Proof. Write F = lim U sh for a suitable diagram of U , with each U ⊂ Y open. −→ i i i One way to see that such a diagram exists is to remark that F ´et is a union of Ui × {σ} and equivalence of categories respects colimits. 10. FURTHER OPERATIONS ON SHEAVES 29

Now we may compute:

Hom (p∗F,G) = Hom (p∗ lim U sh,G) Sh(X) Sh(X) −→ i = Hom (lim p∗U sh,G) Sh(X) −→ i = lim Hom (p−1U ,G´et) ←− Sh(X) i = lim G(p−1U ) ←− i = lim Hom (U sh, p G) ←− Sh(Y ) i ∗ = Hom (lim U sh, p G) Sh(Y ) −→ i ∗ = HomSh(Y )(F, p∗G)

As the diagram of the Ui can be selected naturally in F , every equality above is natural in F and G.

Corollary 8.4.1. Let p : X → Y be a continuous function. The functor ∗ p : Sh(Y ) → Sh(X) is left adjoint to the functor p∗ : Sh(X) → Sh(Y ). ∗ In particular, p preserves arbitrary colimits and p∗ preserves arbitrary limits.

Corollary 8.4.2. Pullback of sheaves commutes with all colimits and ﬁnite limits.

Proof. The commutation with all colimits comes from the existence of the right adjoint p∗. For the limits, use the fact that a finite limit of étalespaces is an étalespace (Exercise 6.4).

Exercise 8.5. Why doesn’t pullback commute with all limits? After all, pullback is deﬁned as a limit of topological spaces, and limits commute with limits...

10 Further operations on sheaves

10.1 Limits and colimits Exercise 10.1. Let C be a small category. Show that Cˆ admits all limits and colimits and that these are computed objectwise.

Proposition 10.2. The category of sheaves on a topological space X admits all small limits and small colimits.

Proof. For limits, we verify that a limit of sheaves is a sheaf. For colimits, we compute the colimit as a presheaf and then sheafify. Since sheafification has a right adjoint (the inclusion of sheaves in presheaves), it preserves colimits, so this shows that all diagrams of sheaves admit colimits.

Exercise 10.3. Let F = lim F be a colimit of a diagram of sheaves. Show by −→ i example that the map lim F (U) → F (U) is not always an isomorphism. −→ i 30 CHAPTER 3. SHEAVES AND THE FUNDAMENTAL GROUP

11 Even further operations on sheaves 11.1 Restriction Deﬁnition 11.1. Suppose X is a topological space, i : U → X is the inclusion of an open subset, and F is a sheaf on X. Then i∗F is called the restriction of

F to U. It is also denoted F U . Exercise 11.2. Verify that i∗F (V ) = F (V ) for all open V ⊂ U.

11.2 Stalks Let x be a point of a topological space X. Denote the inclusion by i. This is continuous, so we have a functor i∗ : Sh(X) → Sh(x) = Sets. We can compute i∗F explicitly using the espace étaléof F . By definition, F ét = lim U×{σ}. Pullback preserves colimits, so that i∗F ét = lim i−1(U)× −→σ∈F (U) −→σ∈F (U) {σ}. Note, however, that i−1(U) = ∅ if x 6∈ U and i−1(U) is a singleton if x ∈ U. Therefore we get a i∗F ét = lim {σ} = lim {σ} = lim F (U). −→ −→ −→ x∈U x∈U⊂X σ∈F (U) x∈U⊂X U ⊂ X open σ∈F (U) Exercise 11.3. Conclude from the above that i∗F ét = lim F (U) with the col- −→ imit taken over open neighborhoods of x in X. Definition 11.4. The set i∗F is called the stalk of F at x.

11.3 Extension by the void Let f : X → Y be a local isomorphism. If g : W → X is a local isomorphism then f ◦ g : W → Y is a local isomorphism. This determines a functor

f! : ´et(X) → ´et(Y ):(W, g) 7→ (W, f ◦ g). We can also think of this as a functor from Sh(X) to Sh(Y ). ∗ Proposition 11.5. The functor f! is left adjoint to f .

11.4 Set theory Anything you can do with sets you can do with sheaves. However, the results can sometimes be unexpected. Translating a set-theoretic concept to a sheaf- theoretic one is easy: add the word “locally” before every existential quantiﬁer. Less prosaically, we should replace any statement of the form “there exists some x ∈ F (U) with the following properties P (U, x)...” with “there exists a cover of U by open subsets Vi such that for each i there is an xi ∈ F (Vi) with P (Vi, xi)...” 12. LOCALLY CONSTANT SHEAVES AND PATH LIFTING 31

Definition 11.6. A morphism of sheaves ϕ : F → G is called a (i) injection, (ii) surjection, (iii) bijection if the defining property in sets holds locally in ϕ. The definition of injectivity uses no existential quantifiers: for all x and y, if ϕ(x) = ϕ(y) then x = y. Therefore, a morphism of sheaves ϕ : F → G is injective if and only if the morphisms of sets ϕU : F (U) → G(U) are injective for all open sets U. However, the definition of surjectivity does have an existential quantifier: for all y ∈ G(U) there is an x ∈ F (U) such that ϕ(x) = y. We replace this with the following: for all open U ⊂ X and all y ∈ G(U) there exists an open cover of U by subsets Vi and elements xi ∈ F (Vi) such that y = ϕ(xi). Vi Bijectivity is, of course, the conjunction of injectivity and surjectivity and requires no modification beyond the modification already made to the definition of surjectivity. Example 11.7. Let F be the sheaf on S1 associated to the covering space R → S1. Let G(U) = 1 for every open U ⊂ S1. Then G is the final sheaf on S1 so there is a unique map F → G. (On the level of étalespaces, this is simply the map R → S1.) This map is surjective because F (U) 6= ∅ for any U ⊂ S1 other than U = S1 itself. However, F (S1) = ∅ and G(S1) = 1 so F (S1) → G(S1) is not surjective. The connection between sheaves and sets is useful in set theory. It allows one to create models of set theory in which not all of the familiar axioms hold. For example, the axiom of choice may be formulated Definition 11.8. We say that the axiom of choice holds if every surjection p : X → Y has a section.1 Exercise 11.9. Show that the axiom of choice is false for sheaves on S1. Exercise 11.10. Show that injectivity, surjectivity, and bijectivity of a morphism of sheaves can all be verified locally.

12 Locally constant sheaves and path lifting 12.1 Constant sheaves and locally constant sheaves Proposition 12.1. Let F be a sheaf on X with espace étalé Y . The following properties are equivalent: (i) There is a discrete space S such that Y ' X × S. (ii) There is an isomorphism between F and the sheaf of continuous functions valued in a discrete space S. (iii) There is a set S and an isomorphism between F and the presheaf F 0 defined by F 0(U) = S for all open U ⊂ X.

1 Recall that a section of p is a map s : Y → X such that ps = idY . 32 CHAPTER 3. SHEAVES AND THE FUNDAMENTAL GROUP

Proof. Exercise.

Deﬁnition 12.2. A sheaf satisfying the equivalent conditions of Proposition 12.1 is called a constant sheaf.

Proposition 12.3. Let F be a sheaf over X and Y its espace ´etal´e. The following properties are equivalent:

(i) There is an open cover of X by subsets U such that Y ×X U ' U × S for some discrete space S.

(ii) There is an open cover of X by subsets U such that F U is a constant sheaf on U.

Deﬁnition 12.4. Sheaves satisfying the equivalent conditions of Proposition 12.3 are called locally constant. Etale spaces satisfying these conditions are called covering spaces.

12.2 The homotopy lifting property Let I = [0, 1] denote the unit interval.

Definition 12.5. Let F be a sheaf over X. We say that F satisfies the (unique) path lifting property over X if, for every map f : I → X, the sheaf f ∗F is constant on I. We say that Y satisfies the (unique) homotopy lifting property if, for any f : Y × I → X, the sheaf f ∗F is pulled back from the first projection p : Y × I → Y .

Exercise 12.6. Translate the homotopy lifting property into a property of the espace ´etal´e.

Note that the homotopy lifting property implies the path lifting property (take Y to be a point).

Proposition 12.7. Let X be a topological space and F a sheaf on X × I that ∗ is locally pulled back via the projection p : X × I → X. Then p p∗F → F is an isomorphism of sheaves on X × I.

First we prove a lemma:

Lemma 12.7.1. Let G be a sheaf on X and p : X × I → X the projection. ∗ Then G → p∗p G is a bijection.

∗ ∗ Proof. By deﬁnition, p∗p G = p G(U × I). this can be interpreted as the set of commutative diagrams G´et ; f q U × I / X. 12. LOCALLY CONSTANT SHEAVES AND PATH LIFTING 33

−1 Now, f {x}×I : {x} × I → q (x) is a continuous function from {x} × I ' I to the discrete set q−1(x), so it must be constant. Therefore there is a function ét g : U → G such that f = g ◦ p U×I . Note that g is continuous, because a subset V of U is open if and only if p−1(V ) is open (i.e., U has the quotient topology associated to the map p). Thus every map f ∈ p∗G(U ×I) is the image of some g ∈ G(U) via the map G(U) → p∗G(U × I). Thus the map is surjective. On the other hand, g is unique because U × I → U is surjective. Proof of Proposition 12.7. The lemma allows us to show that the proposition is true if F ' p∗G. Consider the sequence of maps ∗ ∗ ∗ ∗ p G → p p∗p G → p G 2 whose composition is idp∗G by formal properties of adjunction. The first of ← 2 ∗ these arrows is the pullback of the isomorphism G → p∗p G, hence is an isomorphism. Therefore the second arrow is an isomorphism as well. Now we consider the general case. Because a morphism of sheaves can be ∗ verified to be an isomorphism on an open cover, and the formation of p and p∗ commutes with restriction to an open subset of X, it is sufficient to prove the proposition after restricting to the open subsets in a suitable open cover of X.

Lemma 12.7.2. There is an open cover of X by subsets U such that F U×I ' p∗G for some sheaf G on U. Proof. Suppose that x ∈ X. By assumption, every (x, t) ∈ X × I possesses an open neighborhood U × (a, b) on which the assertion holds. As I is compact, T finitely many of these Ui ×(ai, bi) suffice to cover {x}×I. Let us take U = Ui ∗ so that there are sheaves Gi on U such that F ' p Gi . U×(ai,bi) U×(ai,bi) It seems easiest now to think in terms of the espace étalé. We can deduce from the considerations above that there is a finite sequence 0 = t0 < t1 < ét ∗ ét ··· < tn = 1 such that F ' p G . Note that this gives U×[ti,ti+1] i U×[ti,ti+1] isomorphisms F ét ' Gét ' F ét U×{ti} i U×{ti+1} so that all the Gi are isomorphic and we can simplify notation by writing Gi = G for all i. Furthermore, we have compatible isomorphisms

F ét / F ét o F ét / ··· o F ét U×{t0} U×[t0,t1] U×{t1} U×{tn}

o o o o

p∗Gét / Gét o Gét / ··· o Gét . U×{t0} U×[t0,t1] U×{t1} U×{tn}

Gluing together the horizontal diagrams we obtain F ´et ' p∗G´et. 2todo: explain these somewhere 34 CHAPTER 3. SHEAVES AND THE FUNDAMENTAL GROUP

Now we can complete the proof of Proposition 12.7. Choose an open cover ∗ of X by U ⊂ X such that F U×I ' p G for some sheaf G on U. Then we have a commutative diagram

∗ ∼ ∗ ∗ p p∗(F U×I ) p p∗p G

∗ o (p p∗F ) U×I

∼ ∗ F U×I p G

∗ from which it follows that the restriction of p p∗F → F to U × I is an isomorphism. This holds for all U in a cover of X, so the U × I for which ∗ p p∗F U×I → F U×I is an isomorphism form an open cover of X ×I. Therefore ∗ p p∗F → F is locally an isomorphism, hence is an isomorphism.

Corollary 12.7.3. Locally constant sheaves satisfy the homotopy lifting property.

14 Uniform spaces

14.1 Filters and uniformities A ﬁlter is a partial replacement for the concept of a sequence:

Deﬁnition 14.1. Let X be a set. A ﬁlter on X is a family of subsets F of X such that

F1 If U ⊂ V and U ∈ F then V ∈ F .

F2 The set X appears in F and if U and V are in F then so is U ∩ V .

F3 The empty set is not in F .

A uniformity on a set X is a way of speaking about relative distance without invoking the real numbers:

Deﬁnition 14.2. Let X be a set and denote by ∆X the diagonal subset of X × X.A uniformity (or uniform structure) on a set X is a ﬁlter Φ on X × X such that

U1 Every U ∈ Φ contains ∆X as a subset.

U2 If U ∈ Φ then U −1 = {(x, y) (y, x) ∈ U} is in Φ. 14. UNIFORM SPACES 35

U3 For any U ∈ Φ there is a V ∈ Φ such that

V ◦ V = {(x, z) ∃(x, y), (y, z) ∈ V }

is contained in U.

The elements of Φ are called entourages.

Example 14.3. Every metric space has a natural uniformity: Let Φ consist of the sets that contain Uλ, for some λ ≥ 0, where

Uλ = {(x, y) ∈ X × X d(x, y) ≤ λ}.

Exercise 14.4. Let X be a set and G the group of bijections from X to itself.

For each subset Y ⊂ X, let UY be the set of (g, h) ∈ G×G such that g Y = h Y . Call a subset T ⊂ G × G an entourage if there is a ﬁnite subset Y ⊂ X such that UY ⊂ T and take Φ to be the set of entourages so deﬁned. Then Φ is a uniform structure on G.

14.2 The topology of a uniform space A uniform space can be given a topology, just like a metric space can. Let (X, Φ) be a uniform space. Call a subset U ⊂ X a neighborhood of x if there is a V ∈ Φ such that {x} × U = V ∩ {x} × X.3 Call a subset W ⊂ X open if, for every x ∈ W , there is a neigbhorhood U of x in X with U ⊂ W .

Exercise 14.5. This is a topology on X.

14.3 Complete uniform spaces Definition 14.6. A filter F on a uniform space (X, Φ) is called a Cauchy filter if, for any U ∈ Φ there is some A ∈ F such that A × A ⊂ U.4

Definition 14.7. Suppose that (X, Φ) is a uniform space and F is a filter on X. We say that x ∈ X is a limit of F if for any (x, x) ∈ V ∈ Φ there is some A ∈ F with A × {x} ⊂ V .5 T Proposition 14.8. Suppose that (X, Φ) is a uniform space such that U∈Φ U = ∆X. Then a Cauchy filter has at most one limit.

Proof. Suppose that x and y are limits of a Cauchy ﬁlter F . For any U ∈ Φ there are A, B ∈ F such that A × {x} ⊂ U and {y} × B ⊂ U. Replacing A and B by their intersection, we can assume A = B. Then A × {x} ⊂ U and {y}×A ⊂ U. We can assume furthermore that A×A ⊂ U by reﬁning A further

3In more familiar language, U consists of all y ∈ X such that d(x, y) ≤ V . 4One should interpret A × A ⊂ U as d(x, y) ≤ U for all x, y ∈ A. The Cauchy filter condition therefore says that F contains arbitrarily small sets [?, §3.1]. 5In other words, d(y, x) ≤ V for all y ∈ A. 36 CHAPTER 3. SHEAVES AND THE FUNDAMENTAL GROUP if necessary. Therefore we have (a, x) ∈ U for some a ∈ A and (y, b) ∈ U for some b ∈ A and (a, b) ∈ U. Therefore (x, y) ∈ U ◦ U ◦ U. This holds for any U ∈ Φ. If V is any element of Φ we may find U ∈ Φ such T that U ◦ U ◦ U ⊂ V and therefore (x, y) ∈ V . Thus (x, y) ∈ U∈Φ U = ∆X. Thus x = y. Definition 14.9. A uniform space (X, Φ) is said to be complete if every Cauchy filter of X has a limit. It is said to be separated if Cauchy filters have at most one limit.

14.4 Completion Every uniform space has a unique completion. We will only prove the uniqueness here. Definition 14.10. Let (X, Φ) and (Y, Ψ) be uniform spaces. A map f : X → Y is uniformly continuous if, for any f −1V ∈ Φ for any V ∈ Ψ.6 Proposition 14.11. Suppose that Y is a uniform space with a dense subset X and that Z is a separated and complete uniform space. Any uniformly continuous function f : X → Z extends to a uniformly continuous function Y → Z. Proof. For each y ∈ Y there is some entourage V of Y and some x ∈ X ∩ V (y). Therefore the collection V consisting of all X ∩ V (y), as V ranges among entourages of Y , does not include the empty set. It is therefore a filter, and in fact a Cauchy filter since if W is an entourage of X and V is a second entourage such that V 2 ⊂ W , then V (y)×V (y) ⊂ W . Since f is continuous, the filter f(V) is Cauchy, so it has a unique limit f(y) ∈ Z, as Z is separated and complete. Therefore f gives a function from Y to Z. To see that f is uniformly continuous, consider an entourage W of Z. Let T be an entourage of Z such that T 3 ⊂ W . Select an entourage V of Y such that (x, x0) ∈ V ∩ (X × X) implies (f(x), f(x0)) ∈ T . Let U be an entourage of Y such that U 3 ⊂ V . We show that if (y, y0) ∈ U then (f(y), f(y0)) ∈ W . We may select x ∈ U(y) ∩ X and x0 ∈ U(y0) ∩ X such that f(x) ∈ T (f(y)) and x0 ∈ T (f(y0)), by the definition of f. Then (x, x0) = (x, y) ◦ (y, y0) ◦ (y0, x0) ∈ U 3 ⊂ V so that (f(x), f(x0)) ∈ T , by the construction of V . But then (f(y), f(y0)) = (f(y), f(x)) ◦ (f(x), f(x0)) ◦ (f(x0), f(y0)) so (f(y), f(y0)) ∈ T 3 ⊂ W . Corollary 14.11.1. Suppose that X is a uniform space and that Y and Z are two separted and complete uniform spaces containing X as a dense subspace. Then there is a unique uniformly continuous bijection Y ' Z restricting to the identity on X.

6In other words, for any V ∈ Ψ there is a U ∈ Φ such that d(x, y) ≤ U implies d(f(x), f(y)) ≤ V . 15. CATEGORICAL GALOIS THEORY 37

15 Categorical Galois theory

This section is adapted from [BS, Section 7].

15.1 Uniform groups Theorem 15.1. Let G be the group of self-bijections of a set X and let Φ be the uniform structure deﬁned in Exercise 14.4. Then G is separated and complete with respect to Φ.

Proof. For each finite Y ⊂ X, let UY be the set of pairs (g, h) such that g = Y h Y . Then [ [ U ⊂ UY = ∆G U∈Φ Y ⊂X finite since if g and h act the same way on every finite subset of X they are the same. This proves that G is separated. Now we consider the completeness. Let F be a Cauchy filter on X. We construct a limit g. For each x ∈ X, there is an A ∈ F with A × A ⊂ U{x}. Select h ∈ A and set g(x) = h(x). This is well-defined, for if we chose h0 ∈ A 0 then h (x) = h(x) by definition of U{x}. On the other hand, if B ∈ F and B × B ⊂ U{x} then A ∩ B 6= ∅ so that the definition does not depend on the choice of A either. We must verify that this is actually a limit of F . Supposing U ∈ Φ we must show that {g} × A ⊂ U for some A ∈ F . We may assume U = UY for a finite subset Y ⊂ X, since every U ∈ Φ contains such a subset. As F is a Cauchy filter, we can find A ∈ F such that A × A ⊂ UY . Note that UY ⊂ U{x} for any x ∈ Y so that, by definition of g, we have g(x) = h(x) for any h ∈ A. Therefore (g, h) ∈ UY , i.e., {g} × A ⊂ UY . Exercise 15.2. Let C be a category that is generated under colimits by a set of objects C0. Let F : C → Sets be a functor. Either use the previous theorem or adapt its proof to show that Aut(F ) is separated and complete. Definition 15.3. Suppose that G is a uniform group. An action of G on a set S is called a uniformly continuous action if G × S → S is uniformly continuous. (Here S is given the discrete uniform structure in which all subsets of S × S containing ∆S are considered to be entourages.)7 If G is a uniform group, let G-Sets be the category of uniformly continuous actions of G on (discrete) sets. There is a forgetful functor F : G-Sets → Sets by forgetting the G-action. We get a uniformly continuous function G → Aut(F ).

7This means that for any s ∈ S there is some U ∈ Φ such that d(g, h) ≤ U implies g(s) = h(s). 38 CHAPTER 3. SHEAVES AND THE FUNDAMENTAL GROUP

15.2 Infinite Galois theory Definition 15.4. An infinite Galois theory is a category C and a functor F : C → Sets, called a fiber functor, satisfying the following properties:

IG1 All ﬁnite limits and small colimits exist in C and F preserves these.

IG2 Every object of C is a disjoint union of connected objects.

IG3 The isomorphism classes of connected objects of C form a set. IG4 The ﬁber functor is faithful and conservative.

If, in addition, C and F satisfy the following axiom then they are called a tame inﬁnite Galois theory:

IG5 The group Aut(F ) acts transitively on the set F (Y ) for each connected Y in C .

Theorem 15.5 ([BS, Theorem 7.2.5 (3)]). If C is a tame inﬁnite Galois theory with ﬁber functor F and Aut(F ) = G then C ' G-Sets via the functor induced by F .

Proof. Denote the functor C → G-Sets induced by F by the letter Φ. The forgetful functor from G-Sets → Sets is faithful, and the composition

Φ C −→ G-Sets → Sets coincides with F . We may conclude that Φ is faithful. We next show that Φ is full, i.e., that the map

HomC (Y,Z) → HomG-Sets(Y,Z) is surjective for all Y and Z in C . We may assume that Y is connected. Note that in any category possessing fiber products, Hom(Y,Z) may be identified with the set of sections of Y × Z over Y . This applies to both G-Sets and to C . It is therefore sufficient to show that, for any map Z → Y in C , the induced map Γ(Y,Z) → Γ(Φ(Y ), Φ(Z)) is bijective. A σ ∈ Γ(Φ(Y ), Φ(Z)) corresponds to a connected component T ⊂ Φ(Z) such that the induced map to Φ(Y ) is a bijection. Now, Φ preserves disjoint unions and it preserves the property of connectedness, so there is some connected component Z0 ⊂ Z with Φ(Z0) = T . The map Z0 → Y induces an isomorphism Φ(Z0) → Φ(Y ) and the latter is an isomorphism, so the former is as well (because F is conservative). Finally, we check that Φ is essentially surjective. We show that the essential image of Φ is closed under (small) disjoint unions, finite products, and passage to subobjects and quotients. Then we will show that every object of G-Sets can be constructed using these operations. 16. PSEUDO-LOCALLY CONSTANT SHEAVES 39

(i) The essential image is closed under disjoint unions. Disjoint unions are examples of colimits. (ii) The essential image is closed under finite products. Products are examples of finite limits. (iii) The essential image is closed under passage to subobjects. A subobject of Φ(X) is a disjoint union of connected components of Φ(X). But every ` union of connected components of Φ(X) can be written as Φ(Xi) where Xi are connected components of X. (iv) The essential image is closed under passage to quotients. Suppose that Φ(X) = S and S → T is a surjection. This is the quotient by an equivalence relation R = S ×T S ⊂ S × S. Note that S × S = Φ(X × X). But then R ⊂ S × S is a subobject, so R = Φ(Y ) for some Y ⊂ X × X. Let Z be the colimit of the diagram Y ⇒ Z. Then Φ(Z) is the colimit of R ⇒ S, which is T . Now we show that every object of G-Sets is a quotient of a subobject of a product of objects Φ(X) where X is a connected object of C . First note that every object of G-Sets is a disjoint union of connected G-sets. We may identify any connected G-set S with G/U for an open subgroup U ⊂ G since S has the discrete topology (in this case U is the stabilizer of an element of S). As U is open, there are (by definition of the uniformity on G) a finite collection of connected X1,...,Xk ∈ C and xi ∈ F (Xi) such that the open subgroup

V = {f ∈ G ∀i, f(xi) = xi} is contained in U. There is therefore a surjection on G/V → G/U ' S. On Q the other hand, G/V is isomorphic to the connected component of i Φ(Xi) containing (x1, . . . , xk) as V is the stabilizer of (x1, . . . , xk) in G. We now conclude that any G-set is a disjoint union of connected G-sets, that a connected G-set is a quotient of some G/V is a (connected) subobject Q of Φ(Xi), for some ﬁnite collection {Xi} of connected objects of C . Thus any G-set is a disjoint union of objects that are quotients of objects that are subobjects of objects that are products of objects that are in the image of Φ.

Deﬁnition 15.6. When (C ,F ) is an inﬁnite Galois theory, we write π1(C ,F ) = Aut(F ) and call it the fundamental group.

Proposition 15.7. Suppose (C ,F ) is an inﬁnite Galois theory. Then π1(C ,F ) is a separated and complete uniform group.

16 Pseudo-locally constant sheaves

Definition 16.1. We will say that a sheaf F is pseudo-locally constant if it satisfies the homotopy lifting property (Definition 12.5). We call its espace étaléa pseudo-covering space. The full subcategory of pseudo-locally constant sheaves on X is denoted cov(X). 40 CHAPTER 3. SHEAVES AND THE FUNDAMENTAL GROUP

Suppose that p : Y → X is an object of cov(X). If x, x0 ∈ X and γ : I → X is a continuous path from x to x0 then there is an induced function p−1(x) → p−1(x0). Indeed, γ∗Y is constant, hence isomorphic to I × S for some set s. We therefore have functions

−1 ∗ ∗ ∗ −1 0 p (x) = (γ Y )0 ' Γ(I, γ Y ) ' (γ Y )1 = p (x ).

If x = x0 and i denotes the inclusion of x in X then this is a bijection from i∗Y to itself. As this construction is natural in Y , it gives an automorphism of the functor i∗.8 To state the next proposition, we need the following deﬁnition:

Deﬁnition 16.2. A functor F : C → D is called conservative provided a morphism ϕ of C is an isomorphism if and only if F (ϕ) is an isomorphism. Proposition 16.3. Assume that X is connected and locally path connected. (i) The category cov(X) is closed under small colimits and ﬁnite limits. (ii) Every object of cov(X) is a disjoint union of connected objects. (iii) The connected objects of cov(X) are path connected. (iv) The isomorphism classes of connected objects of cov(X) form a set. (v) If i : x → X is the inclusion of a point then i∗ is a faithful and conservative functor. (vi) If i : x → X is the inclusion of a point then Aut(i∗) acts transitively on i∗Y , for any connected Y ∈ cov(X).

Proof. (i) Suppose that Fi is a diagram in cov(X) with colimit F . We must show that for any map f : Y ×I → X we have f ∗F ' p∗G for some G ∈ Sh(Y ) (where ∗ ∗ ∗ p : Y × I → Y is the projection). But f preserves colimits and f Fi ' p Gi for each i. Therefore,

f ∗F ' f ∗ lim F ' lim f ∗F ' lim p∗G . −→ i −→ i −→ i

But every map Fi → Fj is induced from a uniquely determined map Gi → Gj ∗ (since p admits a section) so that the diagram of f Fi is induced from a diagram of the Gi. It is therefore legitimate to write

f ∗F = lim p∗G = p∗ lim G −→ i −→ i so that F ∈ cov(X). The same argument, with colimits replaced by finite limits, shows that lim F ∈ cov(X) for any finite diagram of F . ←− i i (ii) Let Y be an object of cov(X). Note that Y is locally path connected since Y is locally homeomorphic to X. Consider a cofiltered intersection of open

8In fact, this provides an action of the group of homotopy classes of loops based at x (what is usually called the fundamental group) on the set p−1(x). 18. FINITE GALOIS THEORY 41

and closed subsets Zi of Y . Intersect it with a path connected open subset U of Y . Each Zi ∩ U is open and closed in U, hence empty or equal to U itself. T Therefore the chain Zi ∩ U must stabilize. Thus Zi ∩ U is either empty or T equal to U and in particular is both open and closed in U. Hence Zi is open and closed, since these properties can be verified locally on an open cover. Now suppose that y ∈ Y is a point and consider the collection of all open and closed subsets Zi of Y containing y. The Zi form a cofiltered family with non-empty intersection, which by the above is the minimal open and closed subset of Y containing y. Thus Y is the disjoint union of its minimal open and closed subsets. (iii) Since Y is locally isomorphic to X, it is locally path connected. It is connected by assumption, hence is path connected.9 (iv) If Y ∈ cov(X) is connected then we can connect any two points in Y by a path, which is the unique lift of a path from X. This bounds the set of points of Y by the set of points of X and the set of paths in X. On the other hand, there are only a set’s worth of topologies on a fixed set, and only a set’s worth of functions from that set to X. Therefore there is a set’s worth of isomorphism classes of connected objects of cov(X). (v) First we prove faithfulness. Suppose that i∗ϕ = i∗ψ. We argue that ϕ and ψ agree on all stalks. Indeed, connect x0 ∈ X to x ∈ X by a path u : I → X. ∗ ∗ ∗ ∗ ∗ We get u ϕ 0 = u ψ 0. But u Y is constant, so we get u ϕ = u ψ. Therefore ϕ and ψ agree at x0 as well. This holds for all x and x0 so ϕ and ψ agree on all stalks, hence are the same. The same argument shows that if ϕ is a bijection at x then it is a bijection on every stalk, hence is an isomorphism. (vi) This is copied from [BS]: Suppose p : Y → X is in cov(X) and y, y0 ∈ p−1(x). Choose a path γ : I → Y connected y to y0. Then p ◦ γ is a loop in X based at x, hence provides an automorphism of the fiber functor i∗. The effect of p ◦ γ on y is to trasport it to y0.

18 Finite Galois theory

Deﬁnition 18.1. A ﬁnite Galois theory is a pair (C ,F ) where C is a category and F : C → FinSets is a functor satisfying the following properties:

FGT1 C is closed under ﬁnite limits and ﬁnite colimits and F preserves these.

FGT2 Every object of C is a disjoint union of connected objects.

9A connected, locally path connected space is path connected. Suppose x and y are points of such a space. Find a cover by a well-ordered collection of open sets Ui such that x ∈ U0 and for each i, we have Ui ∩ Uj 6= ∅ for some j < i. Choose such a j and call it j(i). Then y appears in some Ui. Then the sequence i, j(i), j2(i), j3(i),...

is ﬁnite. We may then connected y ∈ Ui to an element z1 ∈ Ui ∩ Uj(i) with a path in Ui, connect z1 to z2 ∈ Uj(i) ∩Uj2(i) by a path in Uj(i), etc. This process is ﬁnite, so we eventually arrive at a path from y to x by concatenation. 42 CHAPTER 3. SHEAVES AND THE FUNDAMENTAL GROUP

FGT3 The full subcategory of connected objects of C is essentially small. FGT4 F is faithful and conservative.

An object X of C is said to be Galois if AutC (X) acts transitively on F (X). We call a ﬁnite Galois theory tame if π1(C ,F ) acts transitively on F (X) for all connected X ∈ C .10

Theorem 18.2. Suppose that (C ,F ) is a tame ﬁnite Galois theory. Then the functor C → π1(C ,F )-Sets is an equivalence of categories. Proof. The proof is exactly as in the inﬁnite case.

Theorem 18.3. Let k be a field and let k be an algebraic closure of k. Let C be the category of finite, separable k-algebras.11 For any k-algebra A, define F (A) = Hom(A, k). Then (C ◦,F ) is a finite Galois theory.

Proof. ?? We can see that C has finite limits, since tensor product of fields preserve products, coproducts, kernels, and cokernels, the verification reduces to the case k = k. Then we can remark that finite, separable k-algebras are equivalent as a category to finite sets. ?? Since a field has no quotients, fields correspond to the connected objects of C ◦. The assertion then comes down to showing that every finite, separable k-algebra is a product of fields. We certainly have an embedding in a product of fields n A → A ⊗ k ' k . k Therefore A is a product of finite, separable k-algebras without zero divisors. But a finite extension of a field without zero divisors is a field. ?? All finite (separable) field extensions of k can be embedded in k. ?? It is sufficient to demonstrate the faithfulness and conservativeness for connected objects of C ◦. First we prove faithfulness. Suppose we have two maps of finite separable extensions of k, say u, v : E0 → E and assume that F (u) = F (v). This means that, for every w : E → k, we have w ◦ u = w ◦ v. But w is injective, so this means u = v. Now we prove that F is conservative. Suppose that u : E → E0 is a morphism of finite, separable field extensions of k such that F (u) is a bijection. That is, every embedding E → k extends uniqely to an embedding E0 → k. Choose some x ∈ E0 and let f be its minimal polynomial. For any root ξ of f in k we can extend an embedding v : E → k to an embedding w : E0 → k with w(x) = ξ. Therefore there is a unique root of f in k. But f is separable, so it has no repeated roots. Therefore f must be linear, so x ∈ E. Thus u is an isomorphism.

10In fact, every finite Galois theory is tame. For our purposes it will be easier to treat this as a separate hypothesis. 11 n A finite k-algebra A is separable the k-algebra, A ⊗k k is isomorphic to k for some n. Finite separable k-algebras are also known as finite étale k-algebras. The category of finite étale k-algebras is denoted fét(k). 19. THE FUNDAMENTAL GROUP 43

(Tameness) Consider two embeddings u, v : E → k. By induction, there is an automorphism σ of k that carries u to v. (Build it up by constructing k as a sequence of extensions of E.) But then σ induces an automorphism of F by sending an embedding w to σ ◦ w and we have σ(u) = v. Corollary 18.3.1. The category of finite separable extensions of a field k is ◦ equivalent to the category of continuous π1(fét(k) , hk)-sets.

19 The fundamental group

Theorem 19.1. Let X be a connected, locally path connected topological space and let G be the automorphism group of a ﬁber functor of cov(X). Then cov(X) ' G-Sets. Proof. The theorem follows formally from Theorem 15.5 and Proposition 16.3.

It is therefore reasonable to write π1(X,F ) = Aut(F ) when X is a connected, locally path connected topological space and F is a ﬁber functor. If F is the ﬁber functor associated to a point x ∈ X then we write π1(X, x).

19.1 Functoriality of the fundamental group Let f : X → Y be a continuous map between based, path connected, locally path connected topological spaces. This induces a functor f ∗ : ét(Y ) → ét(X) that restricts to a functor cov(Y ) → cov(X). Indeed, the homotopy lifting property is phrased in terms of maps I × Z → X, every one of which corresponds to a map I × Z → Y after composition with f. Let x and y be basepoints of X and Y with fiber functors F and G. Then G = F ◦ f ∗. Suppose that γ ∈ Aut(F ). That is, γ is a compatible collection of automorphisms γW of F (W ) for all W ∈ cov(X). Set f∗(γ)Z = γf ∗Z for Z ∈ cov(Y ). This gives a homomorphism

π1(X, x) → π1(Y, y).

19.2 The universal cover Proposition 19.2. Suppose that X is connected and locally path connected and has a simply connected covering space (Y, p). Then π1(X, x) ' AutX (Y ). Proof. We argue that Y represents i∗. Indeed, choose any element y ∈ i∗Y . If Z ∈ cov(X) and z ∈ i∗Z then construct a map Y → Z by, for each y0 ∈ Y , selecting a path γ from y to y0. The projection of this path to X is a path from x to p(y0). It lifts, in a unique way, to a path in Z starting at z. Deﬁne f(y0) to be the endpoint of this lift. This construction is well-deﬁned because the path γ is unique up to homotopy (because Y is simply connected). To complete the proof, we only need to verify that the map f constructed above is continuous. 44 CHAPTER 3. SHEAVES AND THE FUNDAMENTAL GROUP

Note that, by construction, q ◦ f = p (where q : Z → X is the projection). For any y0 ∈ Y , select an open neighborhood U that projects isomorphically to its image in X; do the same for f(y0) ⊂ Z and call the neighborhood V . Then the diagram below commutes:

f p−1(q(V )) ∩ U / q−1(p(U)) ∩ V

( v p(U) ∩ q(V )

The diagonal arrows are homeomorphisms so the horizontal arrow is as well. It follows that f is locally continuous, so it is continuous. The above shows that for any z ∈ i∗Z there is a unique map f : Y → Z such that i∗f(y) = z. Therefore Y represents i∗. It now follows that Aut(i∗) = Aut(Y ), by Yoneda’s lemma.

1 1 Corollary 19.2.1. For any basepoint s ∈ S , we have π1(S , s) = Z.

19.3 The Hawai’ian earring The Hawai’ian earring is the standard example of a topological space without a universal cover. It may be constructed as

n n o [ 2 Xn = x ∈ R d x, (1/n, 0) = 1/n k=1 ∞ [ X = Xn. k=1

W∞ 1 Let Y be the infinite wedge n=1 S . There is a continuous bijection f : Y → X, but the topology on X is coarser than that of Y . We wish to relate the categories cov(X) and cov(Y ). First note that ét(X) → ét(Y ) is faithful. Indeed, a map between objects of ét(X) is determined by what it does pointwise, and pullback to Y only changes the topology, not the points.

Proposition 19.3. The functor cov(X) → cov(Y ) is fully faithful.

Proof. Faithfulness was already demonstrated. To demonstrate that it is full, consider σ ∈ Γ(f −1U, f −1Z) for some open U ⊂ X. We have to show that σ is continuous as a function U → Z. It is suﬃcient to verify this in a neighborhood of 0 ∈ U (since Y is homeomorphic to X away from 0). We can therefore assume U is connected (otherwise replace U with the connected component containing 0). Choose an open subset W ⊂ Z, containing σ(0), that projects homeomorphically onto its image in X. Then there is certainly an extension of

σ σ−1W to a continuous section τ of W ⊂ Z over p(W ). 19. THE FUNDAMENTAL GROUP 45

I claim that τ U∩p(W ) = σ U∩p(W ). Indeed, σ and τ agree at 0 by assumption, so by the path lifting property they agree on the path component of 0 in U ∩ p(W ). In particular, they agree on an open neighborhood of 0 in U ∩ p(W ). Therefore σ is continuous at 0.

Since Y has a universal cover (by a tree with a countably infinite collection of branches at each vertex), we can identify π1(Y, 0) as a free group generated by a countably infinite set. Therefore cov(Y ) may be identified with the category of pairs (S, ϕ), where ϕ = (ϕ1, ϕ2,...) is a sequence of bijections from S to itself.

Proposition 19.4. An object of cov(Y ), viewed as a pair (S, ϕ), lies in the essential image of cov(X) if and only if the following properties hold:

1. For any z ∈ S, all but ﬁnitely many of the ϕi ﬁx z.

2. For any z ∈ S, and any sequence ϕn1 , ϕn2 ,... in which each ϕ appears i1 i2 i

only finitely many times, the sequence of ϕik ··· ϕi1 (z) stabilizes. Proof. The first property expresses that Z is étaleover X. The second is the path lifting property.

Proposition 19.5. The map π1(Y, 0) → π1(X, 0) is injective with dense image. Proof. First we prove injectivity. We must prove that for any two distinct words in the ϕi there is a cover of X on which those words act differently. If the first n of the ϕi appear in these words then we can arrange for this by unwinding the first n of the circles in X. To prove the density, we have to show that every open subgroup of π1(X, 0) meets π1(Y, 0). Suppose that U ⊂ π1(X, 0) is open. Then there is a finite subset W0 ⊂ Z0, for some Z ∈ cov(X), such that U contains the stabilizer of W0. Then each of the finitely many w ∈ W0 is fixed by all but finitely many of the ϕi, so that all of W0 is fixed by all but finitely many of the ϕi. Every one of those ϕi lies in π1(Y, 0).

Let G = π1(X, 0) with the topology in which the open subgroups are those generated by all but ﬁnitely many of the ϕi.

Proposition 19.6. The homomorphism G → π1(X, 0) is continuous and a homeomorphism onto its image.

Proof. The topology on G was constructed to make this homomorphism continuous. We have already seen that it is injective. We have to check that the given topology coincides with the induced one. Take the cover in which ϕi acts faithfully and all other ϕj act trivially. Then the stabilizer of a single element is open in the induced topology. The opens in the topology deﬁned above are ﬁnite intersections of these.

Corollary 19.6.1. π1(X, 0) is the completion of G. 46 CHAPTER 3. SHEAVES AND THE FUNDAMENTAL GROUP

12 Proof. This is immediate by the uniqueness of the completion because π1(X, 0) ← 12 13 → is complete by13 , contains G as a dense subgroup,.

A Sheaves of groups and torsors

Deﬁnition 19.1. A sheaf of groups on a topological space X is a sheaf of sets G with a map G × G → G such that, for every open U ⊂ X, the map G(U) × G(U) → G(U) is the multiplication map for a group structure on G(U).

Deﬁnition 19.2. A left action (resp. right action) of a sheaf of groups G on a sheaf of sets F is a map G × F → F such that for every open U ⊂ X the map G(U) × F (U) → F (U) is a left action (resp. right action) of G(U) on F (U). A sheaf of sets equipped with a left (resp. right) G-action is called a sheaf of left (resp. right) G-sets. Unless speciﬁed otherwise, all actions will be assumed to be from the left. If F and F 0 are left (resp. right) G-sets, a G-morphism ϕ : F → F 0 is a morphism of sheaves such that ϕ(gx) = gϕ(x) for each local section g of G and each local section f of F .

Deﬁnition 19.3. A sheaf of left (resp. right) G-sets F is called a left (resp. right) G-torsor if the following conditions are met:

T1 for each open U ⊂ X the action of G(U) on F (U) is faithful and transitive, and

T2 the open U ⊂ X such that F (U) is non-empty form a cover of X.

If F satisﬁes the ﬁrst condition it is called a pseudo-torsor.

Proposition 19.4. (i) A sheaf of left G-sets F is a G-pseudo-torsor if and only if the map

G × F → F × F :(g, f) 7→ (f, gf)

is an isomorphism.

(ii) A psuedo-torsor F over X is a torsor if and only if F ´et → X is surjective.

(iii) A pseudo-torsor F over X is a torsor if and only if the stalk Fx of F at x is non-empty for all x ∈ X.

(iv) If f : X → Y is a continuous map, G is a sheaf of sets on Y , and F is a G-torsor on Y , then ϕ∗F is a ϕ∗G-torsor on X.

(v) Local properties of G are inherited by any G-torsor. E.g., a torsor under a locally constant sheaf of groups is locally constant.

12todo: reference 13todo: reference A. SHEAVES OF GROUPS AND TORSORS 47

Deﬁnition 19.5. Let G be a sheaf of groups on X. The ﬁrst (non-abelian) cohomology group of G is the set of isomorphism classes of left G-torsors on X. It is denoted H1(X,G).

Note that H1(X,G) is a pointed set: it contains a special point corresponding to the trivial G-torsor. Unless G is abelian, it does not generally have the structure of a group.

Example 19.6. 1. R → R/Z ' S1 is the espace étaléof a Z-torsor. Given a continuous section σ over U ⊂ S1 and a locally constant map τ : U → Z, define τ.σ(x) = τ(x) + σ(x). Note that if σ, σ0 ∈ Γ(U, R) then σ − σ0 is a locally constant function U → Z. Therefore R is a pseudo-torsor over S1.14 ← 14

2. Let G be a sheaf of groups acting faithfully on a sheaf of sets F . Let F 0 be the sheaﬁﬁcation of the presheaf quotient G\F (i.e., the colimit of the diagram G × F ⇒ F in the category of sheaves on X). There is a quotient map p : F → F 0. There is a map

δ : H0(U, F 0) → H1(U, G)

0 00 00 −1 sending a section f of F to the torsor F with F (V ) = p (f V ) for all open V ⊂ U. If x ∈ F 00(V ) and g ∈ G(V ) then regarding x as an element of F (V ) via the inclusion of F 00(V ) ⊂ F (V ) we may act on x with g. By deﬁnition gx ∈ F 00(V ) so this gives F 00 the structure of a G-torsor.

Lemma 19.7. (i) Let G, F , and F 0 be as in Example2. Then the sequence

H0(X,F ) → H0(X,F 0) −→δ H1(X,G)

is exact.

(ii) Let G0 be a sheaf of subgroups of a sheaf of groups G and G00 = G0\G the sheaf of right cosets. Then the sequence

1 → H0(X,G0) → H0(X,G) → H0(X,G00) −→δ H1(X,G0) → H1(X,G)

is exact.

(iii) Let G0 be a sheaf of normal subgroups of G and G00 the sheaf of quotient groups. Then the sequence

1 → H0(X,G0) → H0(X,G) → H0(X,G00) −→δ H1(X,G0) → H1(X,G) → H1(X,G00)

is exact and δ is a group homomorphism.

14todo: explain why it’s a torsor 48 CHAPTER 3. SHEAVES AND THE FUNDAMENTAL GROUP

Example 19.8. Let F be the sheaf of sections of R → S1. Let O be the sheaf of continuous maps from S1 to C; let O∗ ⊂ O be the sheaf of continuous maps from S1 to C∗; ﬁnally, let Z be the sheaf of locally constant maps from S1 to Z. Then there is an exact sequence

exp 0 → Z −−→O2πi −−→O∗ → 0.

This gives a long exact sequence

0 → H0(S1, Z) → H0(S1, O) → H0(S1, O∗) → H1(S1, Z) → H1(S1, O) → H1(S1, O∗)

Let f : S1 → C∗ be the map sending x to e2πix. Since this map does not factor exp as a continuous map S1 → C −−→ C∗, it follows that δ(f) is a non-zero element of H1(S1, Z). In fact, by deﬁnition, δ(f) is obtained as the pre-image of the unit circle in C∗ under the exponential map, hence is the torsor associated to the map R → Z. We will see later that H1(S1, Z) ' Z and is generated by δ(f).

B Classiﬁcation of torsors under locally constant groups

19.1 Crossed homomorphisms and semidirect products Deﬁnition 19.1. Suppose the group π acts (on the right) on another group G by homomorphisms g 7→ gσ.A crossed homomorphism from π to G is a function ϕ : π → G such that ϕ(στ) = ϕ(σ)τ ϕ(τ).

Deﬁnition 19.2. Let π act on the right on G. The elements of the semidirect product π n G are products σg with σ ∈ π and g ∈ G, with the multiplication law σgτh = στgτ h.

Exercise 19.3. (i) Show that σg 7→ σ is a homomorphism π n G → π.

(ii) Construct a bijection between the set of crossed homomorphisms π → G and the set of sections of the projection π n G → π.

19.2 Group objects and group actions

Deﬁnition 19.4. Let C be a category. A group object of C is an object G ∈ ◦ Ob(C ), together with a factorization of the functor hG : C → Sets through the forgetful functor Grp → Sets. B. CLASSIFICATION OF TORSORS UNDER LOCALLY CONSTANT GROUPS49

Exercise 19.5. (i) Show that to specify a group object G of C is equivalent to give a group structure on Hom(X,G) for all X ∈ Ob(C ) such that for every f ∈ Hom(X,Y ), the map Hom(f, G) : Hom(Y,G) → Hom(X,G) is a homomorphism.

(ii) Show that to specify a group object G of C is equivalent to giving maps m : G × G → G, i : G → G, and eX : X → G for all X ∈ Ob(C ) such that the following properties hold: (a) (identity) m ◦ (e, f) = m ◦ (f, e) = f for all f : X → G, (b) (associativity) m ◦ (m × id) = m ◦ (id × m), and

(c) (inverses) m ◦ (i, idG) = m ◦ (idG, i) = eG . Definition 19.6. Let G be a group object of C . An action of G on an object X—or a G-object—of C is a morphism a : G × X → X such that for every object Y of C , the induced map hG(Y ) × hX (Y ) → hX (Y ) is a group action. A morphism of G-objects is a morphism of C that induces a morphism of hG(Y )- sets hX (Y ) → hX (Y ) for all objects Y . The category of pairs (X, a) where X is a G-object of C and a is an action of G on X is called the category of G-objects of C and denoted G-C . Exercise 19.7. Translate actions of group objects into diagrams, as in exercise 19.5. Definition 19.8. An action X of a group object G is called a pseudo-torsor if hX (Y ) is a hG(Y )-pseudo-torsor for all objects Y of C . Exercise 19.9. Show that an action a : G × X → X is a pseudo-torsor if and only if (ha, p2): hG × hX → hX × hX is a natural isomorphism. Definition 19.10. Let G be a group object in the category of π-sets. We call a G-object of the category of π-sets a G-set (instead of a G-π-set). A G-set is said to be a G-torsor if it is a G-pseudo-torsor and its underlying set is non-empty. Proposition 19.11. Let G be a group object of the category of right π-sets.

(i) The category of right G-sets is equivalent to the category of π n G-sets. (ii) The category of right G-torsors is equivalent to the category of π n G-sets Y such that, for any y ∈ Y , the stabilizer subgroup of y in π n G projects isomorphically to π under the map π n G → π. Proof. (i) A right G-set is a π-set Y with a map of π-sets G × Y → Y : (g, y) 7→ y.g satisfying h.(g.y) = (hg).y. For σg ∈ π n G, set yσg = yσ.g. Note that we have (yσg)τh = (yσ.g)τh = yστ .gτ h τ yσgτh = yστg h = yστ .gτ h. 50 CHAPTER 3. SHEAVES AND THE FUNDAMENTAL GROUP

On the other hand, if S is a π n G-set we can extract actions of π and of G from the inclusions π ⊂ π n G and G ⊂ nG. Moreover, we have

σ (y.g)σ = (yg)σ = ygσ = yσg = yσ.gσ

so the action is a morphism of π-sets.

(ii) View a G-torsor Y as a π n G-set. For each σ ∈ π there is a unique g ∈ G such that yσ.g = y. In other words, the stabilizer of y in π n G contains a unique element that projects to σ.

Corollary 19.11.1. The category of G-torsor with a chosen element (and morphisms preserving that element) is equivalent to the set of crossed homomorphisms π → G.

We use this to understand the category of G-torsors itself. Let C be the following strange category: objects are pairs (Y, y), where Y is a G-torsor and y ∈ Y , but morphisms are simply morphisms of G-torsors, ignoring y. We analyze morphisms in C in terms of crossed homomorphisms. Consider a map ϕ :(Y, y) → (Z, z) in C . Then ϕ(y) = z.g for a uniquely determined g ∈ G. We then have StabπnG(z)g = α ∈ π n G z.α = ϕ(y) = g StabπnG(y).

So if s, t : π → π n G are the sections corresponding to StabπnG(y) and StabπnG(y) then we have gs = tg. Now, we can write s(σ) = σϕ(σ) and t(σ) = σψ(σ) where ϕ and ψ are crossed homomorphisms so we get

gs(σ) = gσϕ(σ) = σgσϕ(σ) t(σ)g = σψ(σ)g whence gσϕ(σ) = ψ(σ)g. This proves the following theorem:

Theorem 19.12. Let G be a right π-set. The category of right G-sets (right G-actions in the category of π-sets) is equivalent to the category whose objects are crossed homomorphisms π → G with n o σ Hom(ϕ, ψ) = g ∈ G g ϕ(σ) = ψ(σ)g .

19.3 Classiﬁcation of torsors under pseudo-locally constant groups Let G be a pseudo-locally constant group on a locally path connected topological space X. Note that cov(X) ' π1(X, x)-Sets so that G corresponds to a 0 0 group object G of π1(X, x)-Sets and the categories of G-sets and G -sets are equivalent. As a corollary to Theorem 19.12, we therefore obtain C. FIBER FUNCTORS 51

Theorem 19.13. Let X be a locally path connected topological space, x a point of X, and G a pseudo-locally constant sheaf of groups over X. There is an equivalence of categories between the category of G-torsors on X and the category of crossed homomorphisms π1(X, x) → G defined in Theorem 19.12. Proof. The only thing that remains to be proved is that G-torsors correspond under the Galois equivalence to G0-torsors. Since pseudo-torsors are defined categorically, G-pseudo-torsors correspond to G0-pseudo-torsors. But among pseudo-torsors, the G-torsors may be characterized as the ones whose tautological map to X (on the level of étale spaces) is surjective. Since the Galois correspondence preserves surjections, G-torsors correspond exactly to the G0- pseudo-torsors that surject onto the π-set with a single object (note that this is the π-set corresponding to X). To say that a set surjects onto the 1-element set means that the set is non-empty, so that G-torsors do indeed correspond to G0-torsors.

C Fiber functors

Proposition 19.1. Assume X is locally compact Hausdorﬀ. A functor ξ : Sh(X) → Sets is a ﬁber functor if and only if ξ is left exact and has a right adjoint.

Proof. One direction is obvious, since ξ is a pullback functor. Consider the ∗ collection of all open U ⊂ X such that ξ(hU ) 6= ∅ (since ξ is left exact, a subfunctor of the final functor must pull back to a subfunctor of the final functor of Sets, hence must be ∅ or 1). This collection is closed under intersection. If ξ(hU ) 6= ∅ then ξ(hU ) = 1 so ξ(hV ) = 1 for every V containing U. Furthermore, S since ξ preserves colimits (it has a right adjoint), if X = Ui then ξ(hUi ) = 1 for at least one index i. Let C be a non-empty collection of open subsets of X that is stable under finite intersections such that (i) ∅ 6∈ C, (ii) if U ⊂ V and U ∈ C then V ∈ C, S (iii) if U, V ∈ C then U ∩ V ∈ C, and (iv) if X = Ui then at least one Ui T is in C. By the Hausdorff hypothesis, U∈C U consists of at most one point. Suppose it is empty. Pick U ∈ C. We can assume that U has compact closure since X is covered by open subsets with compact closure. For each point x ∈ U there must be some Vx ∈ C with x 6∈ Vx. Finitely many of the Vx suffice to T cover U. But then Vx = ∅ so ∅ ∈ C.

D The espace ´etal´evia the adjoint functor theorem

We construct the espace étaléusing the adjoint functor theorem. This reduces to showing that, for any sheaf F on a topological space X, the functor on ét(X) sending an étalespace Y to Φ(Y ) = Hom(Y sh,F ) is representable by a an étale space over X. By the adjoint functor theorem, it is enough to show that 52 CHAPTER 3. SHEAVES AND THE FUNDAMENTAL GROUP

1. ét(X) admits arbitrary colimits, 2. Φ carries colimits to limits, and 3. ét(X) is generated under colimits by a set. For the first assertion, we must show that ét(X) admits arbitary colimits and that Φ preserves them. First consider a diagram of étalespaces Yi over X. Take the colimit in the category of topological spaces. If y ∈ lim Yi then we can 0 −→ 0 represent y by some y in some Yi. Choose an open neighborhood U of y in Y that projects homeomorphically to an open subset of X. Then U → lim Y i −→ i is injective (since the composition U → lim Y → X is injective) so U maps −→ i bijectively to a subset V ⊂ lim Y . Moreover, this is a homeomorphism, since −→ i U → V → X is a homeomorphism. Now we show that Φ carries colimits to limits. It is sufficient to treat coproducts and coequalizers. Suppose Yi is a collection of étalespaces and that sh sh we have maps Yi → F . We extend these (in a unique way) to a map Y → F ` where Y = Yi. Suppose σ ∈ Γ(U, Y ). There is a cover of U by disjoint open subsets Ui such that σ U takes values in Yi (since the Yi are an open cover of i Y by disjoint subsets). On each Ui, the restriction σ lies in Γ(Ui,Yi), so we Ui sh obtain an element of F (Ui) from the map Yi → F . Since the Ui are disjoint, these elements agree on the Ui ∩ Uj, hence glue uniquely to an element of F . We must also consider coequalizers. Consider a diagram p, q : Y1 → Y0. Let Y be the coequalizer. Suppose x ∈ X, U is an open neighborhood of x in X, and σ ∈ Γ(U, Y ). Then σ(x) is the projection of some y0 ∈ Y0. Choose an open 0 neighborhood V0 of y0 ∈ Y0 and let U be the image of V0. Then σ U 0 lies in the 0 0 image of Γ(U ,Y0), so we have a candidate for the image of σ U 0 in F (U ). We must verify this is well-defined. Suppose that we had another choice for y1 ∈ Y0 and another neighborhood V1 ⊂ Y0 as above. Then there is a sequence of points z1, . . . , zn with

p(z1) = y0

q(zi) = p(zi+1)

q(zn) = y1.

Each has an open neighborhood Wi 3 zi projecting homeomorphically to an open neighborhood of x in X. Take the intersection of all of these neighborhoods 00 00 and call it U . Then we have a sequence of sections τi ∈ Γ(U ,Y1) that relate

σ0 U 00 to σ1 U 00 . Therefore there is an open neighborhood of x on which σ0 and σ1 agree. Finally, we note that ´et(X) is generated under colimits by Open(X), which is certainly a set. Chapter 4

Commutative algebra

Reading and references

Most of the results presented here can be found in EGA:

A. Grothendieck. Eléments de géométriealgébrique(rédigésavec la collaboration de Jean Dieudonné).Chapter IV.

You can ﬁnd many basic facts about aﬃne schemes in the exercises of

M. F. Atiyah and I. G. MacDonald. Commutative algebra.

The standard reference for the theory of schemes in general is

R. Hartshorne. Algebraic geometry.

There are many good references on the homology and cohomology of commutative rings (also known as Andr´e–Quillenhomology and cohomology). In Section 23, we only use H0 and H1, which are treated very cleanly in

R. G. Swan. N´eron–Popescu desingularization.

I learned Proposition 23.4 from the following reference (although the presentation there is quite diﬀerent):

A. Grothendieck. Catégoriescofibréesadditives et complexe cotan- gent.

It also appears as Théorème(20.6.11) of Chapter 0 of EGA (volume 4-1). The presentation of the local criterion for flatness was adapted from

R. Vakil. Foundations of algebraic geometry, Section 24.6. June, 2013.

53 54 CHAPTER 4. COMMUTATIVE ALGEBRA

21 Aﬃne schemes

Algebraic geometry is the study of the solutions to problems posed about commutative rings. The most basic, and perhaps the most fundamental, of these problems is a system of polynomial equations: Given a collection of polynomials f1, . . . , fn ∈ Z[x1, . . . , xm], one may ask for the set of solutions to this system in any commutative ring A. In fact, if we deﬁne X(A) to be this set of solutions then X is a covariant functor from commutative rings to sets. It is representable by the commutative ring,

A = Z[x1, . . . , xm]/(f1, . . . , fn). By Yoneda’s lemma, studying the functor X is equivalent to studying the ring A. In this chapter we will explore some of the geometric properties of functors like X and their relationships to algebraic properties of the commutative ring A. The finiteness of the number of variables and the number of equations above was only for concreteness. In general, we may study the solutions to an arbitrary set of polynomials in an arbitrary number of variables. As every commutative ring arises as the quotient of some polynomial ring by some ideal, there is a one-to-one (contravariant) correspondence between the functors introduced above and commutative rings. Definition 21.1. An affine scheme is a covariant, representable functor X : ComRng → Sets. The category of affine schemes is denoted Aff. We write Spec A for the affine scheme represented by a commutative ring A. Exercise 21.2. Verify that Aff is equivalent to ComRng◦.

21.1 Limits of schemes The category of commutative rings admits arbitrary small colimits (and limits, but we ignore those for now). The assignment A 7→ Spec A transforms colimits to limits, so this gives a construction of ﬁber products in the category of aﬃne schemes. Exercise 21.3. Suppose that A → B and A → C are homomorphisms of commutative rings and X = Spec A, Y = Spec B, and Z = Spec C. Verify that Y ×X Z ' Spec B ⊗A C.

21.2 Some important schemes n n n We write A = Spec Z[x1, . . . , xn]. Note that A (A) = A for any commutative ring A. Note that Spec Z has the property,

Hom(X, Spec Z) = 1 for any aﬃne scheme X. 21. AFFINE SCHEMES 55

21.3 Topological rings Let X = Spec A be an aﬃne scheme. Every element of A gives a map Z[x] → A, hence a morphism of schemes X → A1. If we evaluate this on a topological ring C (such as R, C, or Qp) then we get a function

X(C) → A1(C).

We may give the set X(C) the coarsest topology such that all of these maps are continuous.

Example 21.4. 1. If A = Z[x, y]/(x2 + y2 − 1) then X(R) is a circle, X(C) is homeomorphic to C.

2. If A = Z[x, y]/(x2+y2+1) then X(R) is empty and X(C) is homeomorphic to C.

3. If A = Z[x, y]/(y2 − x3 − x) then X(R) is homeomorphic to a line; X(C) is homeomorphic to a punctured torus.

4. If A = Z[x, y]/(y2 − x3 + x) then X(R) is homeomorphic to the disjoint union of a circle and a line; X(C) is homeomorphic to a punctured torus.

21.4 The Zariski topology We write | Spec A| for the set of prime ideals of A. This called the prime spectrum of A and is the reason we use the notation Spec. If f : A → B is a homomorphism and p is a prime ideal of B then f −1p is a prime ideal of A. This gives a map |X| → |Y | associated to any morphism of schemes X → Y .

Exercise 21.5. If A → B is an epimorphism then Spec B → Spec A is injective (meaning that | Spec B| → | Spec A| is injective).

Call V ⊂ | Spec A| closed if V = Spec B for some surjection A → B. Note that B is not uniqely determined by V . The map Spec B → Spec A is called a closed embedding.

Exercise 21.6. This is a topological space.

Exercise 21.7. Show that if A is an integral domain then πn(| Spec A|) = 0 for all n. (Hint: the zero ideal is a dense open subset of | Spec A|.)

Say that X = Spec A is irreducible if any decomposition X = V ∪ W with both V and W closed in X has X = V or X = W . 56 CHAPTER 4. COMMUTATIVE ALGEBRA

21.5 Why ´etalemorphisms? Suppose that X is a scheme. Then there are at least two natural ways we might try to deﬁne a fundamental group of X. On one hand, we might look at the set X(C) of complex points of X. This can be given a topology (since the Zariski topology, with respect to which the maps used to glue X are continuous, is coarser than the topology of Cn). If x ∈ X(C) is a point then we can evaluate π1(X(C), x). On the other hand X has an underlying topological space |X|, and given a point x ∈ |X|, we can compute π1(|X| , x).

Proposition 21.8. If X is an irreducible scheme then πn(|X| , x) = 0 for all n and all x. Therefore the Zariski fundamental group of X is usually uninteresting. We may ask if there is an way to compute π1(X(C), x) algebraically (without relying on the analytic topology on C). Of course, we cannot rely on maps from a circle into X, since the circle is not algebraically defined (it has real dimension 1!). However, sheaf theory gives us another approach. Provided we can understand the category of all covering spaces of X, we can define the fundamental group of X to be the automorphism group of a fiber functor. But what do we mean by a covering space of X? The most naive definition—namely a map X0 → X that is a covering space in the Zariski topology—does not suffice. Consider the most basic nontrivial example of a covering space: Example 21.9. Let X = Spec C[t, t−1]. Then X(C) = C∗. There is a 2-to-1 covering space C∗ → C∗ given by squaring: z 7→ z2. This map can be defined algebraically: it is representable by the map C[t, t−1] → C[u, u−1]: t 7→ u2. This should be a covering space, but it is not a covering space in the Zariski topology (consider the generic point). Thus the Zariski topology is much too coarse to see many interesting covering spaces. However, there is another characterization of covering spaces over C that is better behaved. Consider the differential of the map f : X → X constructed above. We have t = u2 so dt = 2udu. That is, infinitesimal motion in the base can be lifted—in exactly one way—to infinitesimal motion in the source. This is how we shall define étalemorphisms. Definition 21.10. A morphism of schemes f : X → Y is called formally étale if infinitesimal motion in Y can be lifted, in a unique way, to infinitesimal motion in X. That is, any commutative diagram of solid lines S / X ? f S0 / Y, 22. SMOOTH AND ETALE´ MORPHISMS 57 in which S ⊂ S0 is an infinitesimal extension, admits a unique lift.

22 Smooth and ´etalemorphisms

“Etale” means “spread out” in French. Etale morphisms are meant to be the local isomorphisms in the category of schemes. We have seen that local isomorphism in the Zariski topology does not include enough of the maps we would like to consider local isomorphisms. However, in a geometric category there are other ways to characterize local isomorphisms. We will find that the following gives a useful characterization: A map f : X → Y is a local isomorphism if, for any point x ∈ X, and any infinitesimal motion y0 away from y = f(x) in Y , there is a unique way of lifting y0 to infinitesimal motion away from x in X.

22.1 The functorial perspective We encode the notion of infinitesimal motion with the following definition: Definition 22.1. A closed embedding of schemes S ⊂ S0 is called a nilpotent immersion or an infinitesimal thickening if the ideal sheaf IS/S0 is nilpotent. Our definition of an étalemap will also involve the following condition Definition 22.2. A morphism of schemes f : X → Y is called locally of finite presentation (resp. locally of finite type) if f can be represented on the level of charts map maps Spec B → Spec A where B is an A-algebra of finite presentation (resp. of finite type). Definition 22.3. A morphism of schemes f : X → Y is called formally smooth if it has the right lifting property with respect to infinitesimal thickenings of affine schemes.1 It is called formally unramified if its diagonal is formally ← 1 smooth. It is called formally étale if it is formally smooth and formally unramified. If it is locally of finite type and formally unramified, it is called unramified (in the functorial sense). If it is locally of finite presentation and formally smooth, it is called smooth (in the functorial sense). If it is smooth in the infinitesimal sense and unramified in the infinitesimal sense it is called étale (in the functorial sense). Lemma 22.4. It is sufficient to consider just square-zero thickenings in the definitions of formally unramified, formally étale,and formally smooth morphisms. Exercise 22.5. A morphism of schemes f : X → Y is formally unramified if and only if its diagonal is formally smooth. Lemma 22.6. If f and g are composable morphisms and g is étalethen gf is étale if and only if f is étale. 1todo: local right lifting property? 58 CHAPTER 4. COMMUTATIVE ALGEBRA

22.2 The differential perspective Definition 22.7. Let B → A be a homomorphism of commutative rings. The module of relative (Kähler)differentials is generated by the symbols df, for all f ∈ A, with the following relations: D1 d(fg) = f dg + g df, and D2 df = 0 if f lies in the image of A.

The module of relative differentials is notated ΩA/B. We also use ΩA to mean ΩA/Z. Definition 22.8. Let B be a finitely generated polynomial algebra over A and let I ⊂ B be a finitely generated ideal, and let C = B/I. Say that C is étale(in 2 the differential sense) over A if the map I/I → C ⊗B ΩB/A is an isomorphism. Proposition 22.9. A finitely presented A-algebra C is étaleover A in the differential sense if and only if it is étalein the infinitesimal sense. 2 Proof. Let δ denote the map I/I → C ⊗B ΩB/A. Suppose that C is étalein the infinitesimal sense. We show d is surjective. We have ΩC/A = coker(δ). Consider the square-zero extension C+ΩC/A (where 2 = 0) of B. The maps x 7→ x + dx x 7→ x + 0 from B to C + ΩC/A coincide on A, hence must be the same. Therefore dx = 0 for all x ∈ C. Now we show that d is injective. Consider the square-zero extension B/I2 → C. By the infinitesimal lifting property, the diagram CC (22.1) O O

~ C0 o A must admit a completion, whenever C0 is a square-zero extension of C. If J is the ideal of C in C0 then C0 ' C + J. Therefore lifts of the diagram CB (22.2) O O

~ C0 o A can be identiﬁed with Hom(C ⊗B ΩB/A,J). On the other hand, any map I/I2 → J induces a square zero extension C0 → C by pushing out the diagram 0 / I/I2 / B/I2 / C / 0

0 / J / C0 / C / 0. 22. SMOOTH AND ETALE´ MORPHISMS 59

2 We deduce that every map I/I → J factors uniquely through C ⊗B ΩB/A. 2 Therefore I/I injects into C ⊗B ΩB/A. Now we prove the converse. Consider a lifting problem in which D0 is a square-zero extension of D with ideal J:

DCo O O

g B O

f Ö ~ D0 o A

The lifts f form a torsor under Hom(C ⊗B ΩB/A,J) (viewing J as a C-module). The induced map I/I2 → J obstructs the existence of a lift g. But the map 2 I/I → C ⊗B ΩB/A is an isomorphism, so that we may choose a lift f such that the lift g exists. 2 Finally, lifts g form a torsor under Hom(ΩC/A,J), which is zero, since I/I → C ⊗B ΩB/A is surjective. Corollary 22.9.1. If A → B is an étalemorphism of commutative rings then there is a finite type Z-algebra A0 and an étale A0-algebra B0 such that B =

B0 ⊗A0 A.

Proof. There is certainly a finite type Z-algebra A0 and an A0-algebra of finite presentation B0 such that B = B0 ⊗A0 A. Indeed, B is obtained by adjoining finitely many variables to A and imposing finitely many relations among them. We may take A0 to be the commutative ring generated by the coefficients in those relations. Note now that A is the filtered colimit of finitely presented A0-algebras Ai. For each i, let Bi = B ⊗A Ai. I claim that for i sufficiently large, Bi is étale over Ai. Indeed, present B0 as C0/I0 for some finitely generated polynomial algebra C0 over A0 and finitely generated ideal I0. Set C = C0 ⊗A0 A and let I 2 be the ideal of B in C. Then consider the maps δi : Ii/Ii → Bi ⊗Ci ΩCi/Ai and 2 δ : I/I → B ⊗C ΩC/A. We claim that δi is an isomorphism for i sufficiently large. Observe first

lim Ci = lim(Ai ⊗ C0) = (lim Ai) ⊗ C0 = A ⊗ C0 = C −→ −→ A0 −→ A0 A0 and I = lim ker(C → B ) = ker(C → B) −→ i i since ﬁltered colimits are exact. As colimits preserve quotients, tensor products, and the formation of ΩCi/Ai , we also have I/I2 = lim I /I2 −→ i i

B ⊗ ΩC/A = lim Bi ⊗ ΩCi/Ai . C −→ Ci 60 CHAPTER 4. COMMUTATIVE ALGEBRA

2 Since B0 ⊗C0 ΩC0/A0 is finitely generated as a B0-module, and I/I → B ⊗C ΩC/A is surjective, every one of the finitely many generators appears in the image of 2 Ii/Ii for i 0. Replacing 0 by i, we may assume that δ0 is surjective. 2 Now choose a splitting σ0 : B0 ⊗C0 ΩC0/A0 → I0/I0 of δ0, which is guaranteed to exist since B0 ⊗C0 ΩC0/A0 is a free B0-module. Set σi = Ai ⊗A0 σ0. Note that σ = lim σ = A ⊗ σ is surjective, so by the same argument as above, σ −→ i A0 0 i is surjective for i 0. Replacing i by this value, we find that σ0 is the inverse of δ0 and therefore that B0 is étaleover A0.

Corollary 22.9.2. Let A → C a morphism of ﬁnite presentation. The following properties are equivalent:

(i) Spec C → Spec A has the unique right lifting property with respect to infinitesimal extensions of affine schemes (i.e., C is an étale A-algebra).

(ii) Spec C → Spec A has the unique right lifting property with respect to local noetherian rings that are complete with respect to the maximal-adic topology and quotients by powers of the maximal ideal.

(iii) Spec C → Spec A has the unique right lifting property with respect to in- ﬁnitsimal extension of artinian rings.

Proof. That (i)= ⇒ (iii) and (ii)= ⇒ (iii) are clear. For (iii)= ⇒ (ii), we consider a lifting problem D/mn o C O O

| D o A in which D is a complete local noetherian ring with maximal ideal m. We realize D as lim D/mk and observe that we get unique maps C → D/mk for all k ≥ 0 ←−k by (iii). Then by the universal property of lim, we get a map C → D. ←− Now we consider (iii)= ⇒ (i). Let B be a polynomial algebra over A and B → C a surjection with finitely generated ideal I. By the differential criterion for étalemorphisms, Condition (i) holds if and only if

2 δ : I/I → C ⊗ ΩB/A B

2 is an isomorphism. This is an isomorphism if and only if k ⊗C I/I → k ⊗C C ⊗B ΩB/A is an isomorphism for any map from C into a ﬁeld k. Indeed, if this holds then Nakayama’s lemma implies δ is surjective. On the other hand, since C ⊗B ΩB/A is free, we can then select a section and verify the surjectivity of the section the same way. Note now that for k ﬁxed,

2 k ⊗ I/I → k ⊗ C ⊗ ΩB/A C C B 22. SMOOTH AND ETALE´ MORPHISMS 61 is an isomorophism if and only if

2 Hom(C ⊗ ΩB/A, k) → Hom(I/I , k) (22.3) B is an isomorphism. But we may interpret Hom(I/I2, k) as the set of square-zero B-algebra extensions of C by k and we may interpret Hom(C ⊗B ΩB/A, k) as the set of A-algebra maps B → C + k. For (22.3) to be bijective therefore means that every B-algebra extension of C by k is isomorphic to C + k as an A-algebra. Now assume Condition (iii) and consider the a B-algebra extension C0 of C by k. Since C is an ´etale A-algebra, every A-algebra extension of C by k is isomorphic to C + k: the diagram

CC O O

~ C0 o A has a lift, splitting the surjection C0 → C. But to give a B-algebra extension of C by k means to give an A-algebra extension C0 by k and an A-algebra map B → C0. As C0 is necessarily isomorphic to C + k, this means that every B-algebra extension of C by k is a homomorphism B → C + k, as desired. Proposition 22.10. Let A be a commutative ring. An A-algebra C is ´etaleif and only if it can be presented as A[x1, . . . , xn]/(f1, . . . , fn), where the image of ∂fi det ∂x in C is a unit. j i,j

Proof. Assume ﬁrst that the determinant is a unit. Let us set B = A[x1, . . . , xn] and I = (f1, . . . , fn), so that C = B/I. 2 2 Consider the map δ : I/I → C ⊗A ΩB/A. Note that I/I is generated by the images of f1, . . . , fn and that C ⊗A ΩB/A is free with basis dx1, . . . , dn. In ∂fi terms of these bases, δ is given by the matrix ∂x . That is, j i,j

X ∂fi f 7→ dx . i ∂x j j j

We have the following commutative diagram:

P ∼ P Cfi / Cdxj o

2 δ I/I / C ⊗A ΩB/A

P The upper horizontal arrow is invertible by assumption, and the map Cfi → 2 2 I/I is surjective since the fi generate I/I . It follows that δ is bijective. 62 CHAPTER 4. COMMUTATIVE ALGEBRA

Now we consider the converse. Suppose that C is ´etaleso that the map 2 δ : I/I → C ⊗A ΩB/A is bijective. As ΩB/A is free with basis dx1, . . . , dxn, so 2 2 is I/I . Let f1, . . . , fn be elements of I whose images form the basis of I/I P ∂fi corresponding to the dxi. Then as before, δ(fi) = ∂x dxj so that the matrix j ∂fi ∂x is the identity, and in particular is invertible. j i,j 2 The only thing left to check is that the fi generate I. We have I ≡ I (mod f1, . . . , fn). But by assumption I is ﬁnitely generated, so by Nakayama’s lemma we deduce that I ≡ 0 (mod f1, . . . , fn), i.e., that I = (f1, . . . , fn).

23 Homology of commutative rings

Proposition 23.1. Let A → B → C be a sequence of homomorphisms of commutative rings. Then the induced sequence of C-modules

C ⊗ ΩB/A → ΩC/A → ΩC/B → 0 B is exact. If C is a free B-algebra then it is exact on the left as well.

Proof. Consider the functors they represent.

Proposition 23.2. Let B → C be a surjection of A-algebras with kernel I. Then there is an exact sequence

2 d I/I −→ C ⊗ ΩB/A → ΩC/A → 0 B where d is the map sending f (mod I2) to df. Moreover, if C is a free A-algebra, d is injective and the sequence is split.

Proof. Consider the functor represented by Ω. If C is a free A-algebra then the map B → C admits a section, so that B ' C × I with ring structure (a, x)(b, y) = (ab, ay+bx+xy). We can then calculate explicitly that a derivation C × I → J consists of a derivation C → J and a homomorphism I/I2 → J, 2 which may be selected independently. Therefore ΩB/A ' ΩC/A × I/I .

Suppose B is an A-algebra. Deﬁne H0(B/A) = ΩB/A. Suppose that B → C is a surjection of A-algebras and B is free. Let I be the kernel. Deﬁne

2 H1(C/B) = ker(I/I → C ⊗ H0(B/A)). B

Proposition 23.3. Up to canonical isomorphism, the deﬁnition of H1(C/A) does not depend on the choice of B.

Proof. Select A-algebra surjections B1 → C and B2 → C with B1 and B2 free and let I1 and I2 be the kernels. We may certainly ﬁnd a free A-algebra that 23. HOMOLOGY OF COMMUTATIVE RINGS 63

surjects onto both B1 and B2, so we may assume that there is a surjection B1 → B2 with ideal J. By Proposition 23.2, we have a split exact sequence

2 0 → J/J → B2 ⊗ H0(B1/A) → H0(B2/A) → 0. B1

Since the sequence is split, it remains exact upon tensoring with C, and we get

2 0 → C ⊗ J/J → C ⊗ H0(B1/A) → C ⊗ H0(B2/A) → 0. B2 B1 B2

Now consider the following commutative diagram with exact rows and columns

0 0

K1 / K2

2 2 2 0 / J/(I1 ∩ J) / I1/I1 / I2/I2 / 0

2 0 / C ⊗B1 J/J / C ⊗B1 H0(B1/A) / C ⊗B2 H0(B2/A) / 0

H0(C/A) H0(C/A)

0 0

Note that K1 and K2 are the two values of H1(C/A) that we get by computing with B1 and B2, respectively. 2 2 We check that J/(I1 ∩ J) → C ⊗B1 J/J is an isomorphism. Note that 2 2 C ⊗B1 J/J = J/(J + I1J) = J/I1J since J ⊂ I1. On the other hand, we can split B1 as B2 × J by choosing a section, under which splitting we have 2 2 2 I1 = I2 × J. Then I1 = I2 × I1J so that I1 ∩ J = I1J. By the snake lemma, K1 → K2 is an isomorphism.

Remarkably, H1(B/A) comes close to representing a functor: Proposition 23.4 (Grothendieck). Suppose that J is an injective C-module. Then Hom(H1(C/A),J) is isomorphic to the set of isomorphism classes of square-zero A-algebra extensions of C by J, naturally in J.2 ← 2

Proof. Let ExalA(C,J) be the set of isomorphism classes of A-algebra extensions of C by J. We construct maps in either direction between Hom(H1(C/A),J)

2todo: Explain what isomorphism classes of extensions are 64 CHAPTER 4. COMMUTATIVE ALGEBRA

0 and ExalA(C,J). Suppose that C is an A-algebra extension of C by J. Choose a surjection B → C with B a free A-algebra. The map B → C can be lifted to a map B → C0 since B is free. Then we get a diagram of exact sequences

0 / I / B / C / 0

0 / J / C0 / C / 0 with the map I → J induced by the universal property of the kernel. But J 2 = 0 2 2 so we get a map I/I → 0. Composing with the map H1(C/A) → I/I , we get a map H1(C/A) → J. Note that this does not depend on the choice of lift B → C0. Indeed, if we replace it by another lift, the diﬀerence between them is a derivation B → J, representable by a map C ⊗B ΩB/A → J, hence restricts to 0 on H1(C/A) by the exactness of the sequence

2 H1(C/A) → I/I → C ⊗ ΩB/A. B

Now we construct a map Hom(H1(C/A),J) → ExalA(C,J). Since H1(C/A) ⊂ 2 I/I and J is injective, any map H1(C/A) → J may be extended to a map I/I2 → J. Pushing out the exact sequence

0 → I/I2 → B/I2 → C → 0 via this map gives a diagram of exact sequences

0 / I / B / C / 0

0 / I/I2 / B/I2 / C / 0

0 / J / C0 / C / 0.

The bottom row is an element of ExalA(C,J). We must also check that this does not depend on the extension I/I2 → J. Two distinct extensions diﬀer by a map C ⊗B ΩB/A → J, but adding an A- derivation B → J to the map I → J only changes the map B → C0 in the diagram above and does not change C0 as an A-algebra extension of C. We leave it as an exercise to verify that these constructions are inverse to one another. Proposition 23.5. Suppose B → C is a homomorphism of A-algebras. There is a long exact sequence

δ C ⊗ H1(B/A) → H1(C/A) → H1(C/B) −→ C ⊗ H0(B/A) → H0(C/A) → H0(C/B) → 0. B B 24. HOMOLOGY OF MODULES 65

Proof. We can verify exactness by showing that the sequence is exact after applying Hom into any injective C-module J. The sequence becomes

0 → DerB(C,J) → DerA(C,J) → DerA(B,J)

→ ExalB(C,J) → ExalA(C,J) → ExalA(B,J) and it is a straightfoward exercise to verify that this is exact. Remark 23.6. This is the beginning of the long exact sequence of André–Quillen homology. When we discuss Grothendieck topologies we will see how to extend the sequence. Definition 23.7. An A-algebra B of finite type is called unramified (in the homological sense) if H0(B/A) = 0. It is called smooth (in the homological sense) if H1(B/A) = 0 and it is of finite presentation. It is called étale(in the homological sense) if it is smooth and unramified. Proposition 23.8. An A-algebra B is étalein the homological sense if and only if it is étalein the differential sense. Proof. Choose a finitely generated free A-algebra C and an A-algebra surjection C → B with finitely generated ideal I. Then we have a long exact sequence

B ⊗ H1(C/A) → H1(B/A) → H1(B/C) → B ⊗ H0(C/A) → H0(B/A) → H0(B/C). C C

Note that H1(C/A) = 0 because C if free and H0(C/B) = 0 because C surjects 2 onto B. Substituting H1(B/C) = I/I and H0(C/A) = ΩC/A, we get

2 δ 0 → H1(B/A) → I/I −→ B ⊗ ΩC/A → H0(B/A) → 0 C from which we deduce that δ is an isomorphism if and only if H1(B/A) = H0(B/A) = 0.

24 Homology of modules

Deﬁnition 24.1. Let M and N be A-modules. Choose a presentation of M and P/Q where P is a free A-module and Q is a submodule. Deﬁne

A Tor1 (M,N) = ker(N ⊗ P → N ⊗ Q). A A It is not obvious that this deﬁnition is well posed. However, if we had another presentation of M as P 0/Q0 then we could ﬁnd a commutative diagram:

0 / Q0 / P 0 / M / 0

0 / Q / P / M / 0 66 CHAPTER 4. COMMUTATIVE ALGEBRA

We may assume without loss of generality that P 0 → P (and therefore also Q0 → Q) is surjective. The kernels of Q0 → Q and P 0 → P are isomorphic, so we write K for the common value. Then we get a commutative diagram with exact rows and columns:

N ⊗A K N ⊗A K

0 0 0 0 / R / N ⊗A Q / N ⊗A P / N ⊗A M / 0

0 / R / N ⊗A Q / N ⊗A P / N ⊗A M / 0

0 0

0 0 The map N ⊗A K → N ⊗A P is injective because P → P can be split. Apply- ing the snake lemma shows that R0 → R is an isomorphism, so the deﬁnition of Tor1(M,N) does not depend (up to canonical isomorphism) on the choice of P and Q.

The fact that Tor1(M,N) is independent of the resolution of M selected immediately implies that if M is a free module then Tor1(M,N) = 0 for all modules N.

Proposition 24.2. Suppose that M is an A-module and

0 → N 0 → N → N 00 → 0 is an exact sequence of A-modules. Then there is an exact sequence

A 0 A A 00 Tor1 (M,N ) → Tor1 (M,N) → Tor1 (M,N ) → M ⊗ N 0 → M ⊗ N → M ⊗ N 00 → 0. A A A

Proof. Choose a commutative diagram with exact rows and colums and P 0, P , 24. HOMOLOGY OF MODULES 67 and P 00 free: 0 0 0

0 / Q0 / Q / Q00 / 0

0 / P 0 / P / P 00 / 0

0 / N 0 / N / N 00 / 0

0 0 0 Tensor with M to get 0 0 0

0 00 Tor1(M,N ) / Tor1(M,N) / Tor1(M,N )

0 00 0 / Q ⊗A N / Q ⊗A N / Q ⊗A N / 0

0 00 0 / P ⊗A N / P ⊗A N / P ⊗A N / 0

0 00 M ⊗A N / M ⊗A N / M ⊗A N / 0

0 0 0 Now apply the snake lemma. Corollary 24.2.1. Suppose that 0 → N 0 → N → N 00 → 0

0 00 is exact and that Tor1(M,N ) = Tor1(M,N ) = 0. Then Tor1(M,N) = 0.

Proposition 24.3. There is a canonical isomorphism Tor1(M,N) ' Tor1(N,M). Proof. Choose resolutions 0 → Q → P → M → 0 0 → K → L → N → 0 68 CHAPTER 4. COMMUTATIVE ALGEBRA

with P and L free. Form a commutative diagram with exact rows:

0 / Tor1(N,M)

Q ⊗A K / P ⊗A K / M ⊗A K / 0

0 / Q ⊗A L / P ⊗A L / M ⊗A L / 0

Tor1(M,N) / Q ⊗A N / P ⊗A N / M ⊗A N / 0

0 0 0

Apply the snake lemma. (Note that the left exactness of the middle row and column come from the vanishing of Tor1(M,L) and Tor1(N,P ).)

25 Flatness

Definition 25.1. A morphism of schemes f : X → Y is called flat if on charts 3 → Spec B → Spec A, the ring B is a flat A-algebra.3

Proposition 25.2. An A-module M is ﬂat if and only if Tor1(M,N) = 0 for all A-modules N.

Proof. Use the long exact sequence.

Corollary 25.2.1. Let A be a noetherian ring. An A-module M is ﬂat if and only if J ⊗A M → A ⊗A M ' M is injective for all ideals J of A.

Proof. Certainly a flat module has this property, so we assume the property holds and show M is flat. Let N be an arbitrary A-module. We must show that Tor1(N,M) = 0. Since Tor commutes with filtered colimits (since tensor products do), we may assume N is finitely generated, say by x1, . . . , xn. Let Ni be the submodule generated by x1, . . . , xi. Then we have exact sequences

0 → Ni → Ni+1 → A/xiA → 0.

But xiA ⊗A M → M is injective by assumption, so that Tor1(A/xiA, M) = 0. On the other hand, we can assume that Tor1(Ni,M) = 0 by induction, so that Tor1(N,M) = 0 as well.

3todo: deﬁne ﬂat in algebra 26. THE EQUATIONAL CRITERION FOR FLATNESS 69

Lemma 25.3 (Artin–Rees). Let A be a commutative noetherian ring, I ⊂ A an ideal, and M a finitely generated A-module. For any submodule M 0 of M, there is an integer k such that Ik+nM 0 = In(M 0 ∩ IkM) for n ≥ 0. Proof. This proof is adapted from [Wik]. P∞ n n Form the Rees algebra A[tI] = n=0 t I . As A is noetherian, I is finitely generated, so that A[tI] is a finitely generated A-algebra, hence is noetherian P∞ n n as well. We can also form the A[tI]-module A[tI]M = n=0 t I M. This is generated by M = t0M as an A[tI]-module, so it is finitely generated because M is. 0 P∞ n 0 n Consider the submodule A[t]M ∩ A[tI]M = n=0 t (M ∩ I M). This is an A[tI]-submodule of A[tI]M, hence is finitely generated. Suppose that the generators of A[t]M 0 ∩ A[tI]M lie in degrees ≤ k. Then looking in degree n + k, with n ≥ 0, we discover

M 0 ∩ In+kM = In(M 0 ∩ IkM).

26 The equational criterion for ﬂatness

Let M be an A-module. A relation among elements x1, . . . , xn ∈ M is a choice P of f1, . . . , fn ∈ A such that fixi = 0. To give a relation in M is the same as to give an element of the kernel of

(f ,...,f ) ⊗ id An ⊗ M −−−−−−−−−−−→1 n A M A ⊗ M = M. A A If K is the kernel of n (f1, . . . , fn): A → A then the elements of K may be thought of as relations in A. Every relation in A induced a relation in M by the map

K ⊗ M → An ⊗ M. A A Let M and N be a A-modules. An M-linear relation in N is an expression X xi ⊗ yi = 0 in M ⊗A N. More speciﬁcally, we call this an M-linear relation among y1, . . . , yn ∈ N. Equivalently, an M-linear relation among y1, . . . , yn is an element (x1, . . . , xn) of the kernel of id ⊗ y M n = M ⊗ An −−−−−→M M ⊗ N. A A One can use A-linear relations in N to construct M-linear relations. Indeed, an A-linear relation on y1, . . . , yn ∈ N is a tuple (f1, . . . , fn) such that X fiyi = 0. 70 CHAPTER 4. COMMUTATIVE ALGEBRA

Then for any x ∈ M, X fix ⊗ yi = 0 so (f1x, . . . , fnx) is an M-linear relation among y1, . . . , yn. Moreover, any M- linear combination of A-linear relations in N is an M-linear relation in N. That is, if F = (fij), i = 1, . . . , n, j = 1, . . . , m is a family of A-linear relations on y1, . . . , yn, and x1, . . . , xm are elements of M then F (x1, . . . , xm) gives an M-linear relation on y: n n X X fijxi ⊗ yj. j=1 i=1 All of this can be said more efficiently in the following way: The M-linear n relations on y ∈ N are the kernel of idM ⊗ y : M ⊗A A → M ⊗A N. The A-linear relations on y are the kernel of y : An → N. Then there is a map M ⊗A ker(y) → ker(idM ⊗ y). We say that every M-linear relation on y ∈ N n is induced from A-linear relations on N if the map M ⊗A ker(y) → ker(idM ⊗ y) is bijective. Note that if M is a flat A-module then the sequence 0 → M ⊗ ker(y) → M ⊗ An → M ⊗ N A A A is exact so that every M-linear relation on y ∈ N n is induced from an A-linear relation on N. This proves half of the following theorem: Theorem 26.1 (Equational criterion for flatness). An A-module M is flat if and only if, for every A-module N, and for every y ∈ N n, every M-linear relation on y is induced from A-linear relations on N. Proof. What remains is to show that the equational criterion implies flatness. Suppose that J ⊂ A is an ideal. An element of the kernel of M ⊗A J → P M ⊗A A = M can be expressed as xi ⊗ yi = 0 where xi ∈ M and yi ∈ J. 0 P P Let J = yiA be the ideal generated by yi. Then xi ⊗ yi also lies in the 0 P kernel of M ⊗A J → M. It will be sufficient to show that xi ⊗ yi is zero in 0 0 M ⊗A J . We may therefore replace J by J and assume that J = (y1, . . . , yn). Setting K = ker(y), we now have an exact sequence: 0 → K → An → A By assumption, the sequence 0 → M ⊗ K → M ⊗ An → M ⊗ A → M ⊗ A/J → 0 A A A A is exact (exactness at M ⊗A A and M ⊗A A/J is the right exactness of tensor product; exactness at the left two terms is from the assumption on M). On the other hand, the right exactness of tensor product also gives us M ⊗ K → M ⊗ An → M ⊗ J → 0 A A A n so that the image of M ⊗A A → M ⊗A A = M is M ⊗A J. In particular, M ⊗A J → M is injective, whence M is flat. 26. THE EQUATIONAL CRITERION FOR FLATNESS 71

Deﬁnition 26.2. An A-module M is called faithfully ﬂat if it is equivalent for

N 0 → N → N 00 and M ⊗ N 0 → M ⊗ N → M ⊗ N 00 A A A to be ﬂat.

Proposition 26.3. A ﬂat A-module M is faithfully ﬂat if and only if the properties N = 0 and M ⊗A N = 0 are equivalent.

Proof. If M is faithfully ﬂat and M ⊗A N = 0 then 0 → M ⊗A N → 0 is exact so 0 → N → 0 is exact so N = 0. 4 Now suppose M ⊗A N = 0 implies N = 0 and consider a sequence ← 4

0 → N 0 → N → N 00.

Let K = ker(N → N 00). For the sequence above to be exact means N 0 → K is an isomorphism. But this is equivalent to

ker(N 0 → K) = coker(N 0 → K) = 0.

This is equivalent to

M ⊗ ker(N 0 → K) = M ⊗ coker(N 0 → K) = 0. A A

But:

M ⊗ coker(N 0 → K) = coker(M ⊗ N 0 → M ⊗ K) A A A M ⊗ ker(N 0 → K) = ker(M ⊗ N 0 → M ⊗ K) A A A

The ﬁrst line is always true; the second line is because M is ﬂat. Reversing what 0 we did above, we deduce that these equalities hold if and only if M ⊗A N → M ⊗A K is an isomorphism. But

M ⊗ K = ker(M ⊗ N → M ⊗ N 00) A A A so this is the same as

0 → M ⊗ N 0 → M ⊗ N → M ⊗ N 00 A A A being exact.

4todo: explain why it’s enough to consider sequences like this 72 CHAPTER 4. COMMUTATIVE ALGEBRA

30 Local criteria for ﬂatness

Deﬁnition 30.1. An A-module M is said to be ideally separated if, for any ideal J ⊂ A, \ In(J ⊗ M) = 0. A Proposition 30.2. Suppose that u : A → B is a local homomorphism of noetherian local rings and M is a ﬁnite type B-module. Then M is ideally separated.

Proof. Let J be an ideal of A. Set IB = IB and MB = J ⊗A M. Note that MB is ﬁnitely generated. Then

\ n \ n I (J ⊗ M) = IBMB. A T n Set N = IBMB. Then IBNB = NB: Artin–Rees guarantees that there is a k+1 k k such that N = N ∩ IB MB = IB(N ∩ IBMB) ⊂ IBN. If K is the maximal ideal of B then this implies a fortiori that KN = N. By Nakayama’s lemma (which applies since N is a submodule of a finitely generated module over a Noetherian ring), it follows that N = 0. Theorem 30.3 (Infinitesimal criterion for flatness). Let M be a module of finite type over a noetherian local ring A whose maximal ideal is denoted I. Assume that M is ideally separated. It is then equivalent for M/InM to be flat over A/In for all n ≥ 0 and for M to be flat over A. Proof. Let J be an ideal of A, which is necessarily finitely generated since A is noetherian. It will be sufficient to show that for any such ideal, the map of A-modules ϕ : J ⊗ M → M A 5 → is injective.5 Note that J/(J ∩ In) is an ideal of A/In, so that the flatness of M/InM over A/In implies the injectivity of the map

J/(J ∩ In) ⊗ M/InM → M/InM. A/In The commutativity of the diagram

ϕ J ⊗A M / M

n n n J/(J ∩ I ) ⊗A/In M/I M / M/I M

shows that \ ker(ϕ) ⊂ ker(pn). n

5todo: reference to justify this 30. LOCAL CRITERIA FOR FLATNESS 73

By the Artin–Rees lemma 25.3, there is an integer k such that J ∩ In+k = In(J ∩ Ik) ⊂ InJ so that we have a map

pn+k n+k n n k n qn : J ⊗ M −−−→ J/(J ∩ I ) ⊗ M/I M ' J/I (J ∩ I ) ⊗ M/I M A A/In A/In → J/InJ ⊗ M/InM ' A/In ⊗ J ⊗ M. A/In A A Therefore \ \ ker(pn) ⊂ ker(qn). n n n On the other hand, ker(qn) = I (J ⊗A M). By assumption, M is ideally sepa- T n rated, so I (J ⊗A M) = 0. Thus,

\ \ \ n ker(ϕ) ⊂ ker(pn) ⊂ ker(qn) = I (J ⊗ M) = 0. A

Theorem 30.4 (Local criterion for flatness). Suppose that A is a noetherian local ring with maximal ideal I. If M is an ideally separated A-module then M is flat over A if and only if Tor1(M, A/I) = 0. Proof. The proof is similar to the infinitesimal criterion. Suppose that J ⊂ A is an ideal and consider the map ϕ : J ⊗A M → M. We may fit this into a diagram with exact rows:

ϕ J ⊗A M / M

n n n n Tor1(A/(I + J),M) / J/(I ∩ J) ⊗A M / A/I ⊗A M / A/(I + J) ⊗A M / 0 The lower row is obtained by tensoring the exact sequence 0 → J/(In ∩ J) → A/In → A/(In + J) → 0

n n with M. But A/(I + J) has ﬁnite length, so Tor1(A/(I + J),M) = 0. There- T fore ker(ϕ) ⊂ n ker(pn). But by Artin–Rees, there is an index k such that J ∩ In+k = In(J ∩ Ik). Therefore we have maps J → J/(In+k ∩ J) ' J/In(Ik ∩ J) → J/InJ. By tensor product with n, we get maps

n qn : J ⊗ M → J/I J A n and deduce that ker(pn) ⊂ ker(qn). But ker(qn) = I (J ⊗A M). Therefore,

\ \ \ n ker(ϕ) ⊂ ker(pn) ⊂ ker(qn) = I (J ⊗ M), A which is zero because M is ideally separated. 74 CHAPTER 4. COMMUTATIVE ALGEBRA

31 Flatness of ´etalemaps

Lemma 31.1. A field extension is unramified6 if and only if it is finite and separable.

Proof. We compute the vector space of k-derivations from K into itself. Sup- pose that K is algebraic over a purely transcendental extension L. Say K = L(x1, . . . , xn). A derivation of K into itself may be specified by a derivation L → K, together with elements δ(xi) such that if fi is the minimal polynomial 0 satisfied by xi then δ(fi) = 0. Of course, δ(fi) = fi (xi)δ(xi). In particular, we can see that any derivation of L into K extends to a derivation of K into 0 −1 0 K by setting δ(xi) = fi (xi) δ(fi) if fi (xi) 6= 0 and choosing δ(xi) = 0 if 0 fi (xi) = 0. If K is unramified over k then there is a unique derivation of K into itself (namely zero), so we must have L = k. 0 Even more precisely, we can see that if fi (xi) = 0 then δ(xi) may be chosen arbitrarily and therefore there is more than one element of Derk(K,K). We 0 deduce that fi (xi) 6= 0 for all i, so that the fi are all separable polynomials, and K is a separable extension of k.

Lemma 31.2. An étalealgebra over a field is a finite product of separable field extensions.

Proof. Let k be a field and C an étale k-algebra. Let m be a maximal ideal of C and K = C/m the quotient. Note that ΩK/k is a quotient of ΩC/k = 0 so ΩK/k = 0. Therefore K is unramified over k, so it is a finite separable extension. In particular, it is étaleover k. This implies that C → K is étale. Recall that Hom(m/m2,J) may be identified with the isomorphism classes of C-algebra extensions of K by J.A C-algebra extension of K by J is a commutative diagram:

KCo O O

~ K0 o k But there is only one such: Up to unique isomorphism, there is only one k- algebra extension K0 of K by J because K is étaleover k. Once K0 has been fixed, there is a unique diagram as above because C is étaleover k. 2 It follows that m/m = 0. By Nakayama’s lemma, mCm = 0 so there is some y ∈ C r m such that ym = 0 (because m is finitely generated: C is noetherian because it is of finite type over a field). Therefore yC ⊂ C projects isomorphically to K. Let e be the (unique) element of yC that projects to 1 ∈ K. Then e2 = e because e and e2 both project to 1 ∈ K. That is, e is idempotent. Then 1 − e is also idempotent and we have C = (1 − e)C × eC ' (1 − e)C × K and eC = yC ' K via the projection C → K.

6 This is the uniqueness part of the deﬁnition of an ´etalemorphism. Equivalently ΩK/k = 0. 31. FLATNESS OF ETALE´ MAPS 75

In particular, C0 = (1 − e)C is also an étale k-algebra. If C0 6= 0 then we can repeat the argument on a maximal ideal m0 of C0, we find that C0 = K0 × C00. Note that the ideals K ⊂ K ×K0 ⊂ K ×K0 ×K00 ⊂ · · · form an ascending chain in C. Therefore this process must terminate after a finite number of steps and we find that C is a product of fields. Each of these fields is finite and separable over k by the previous lemma.

Theorem 31.3. An ´etalemorphism of commutative rings is ﬂat.

Let A → C be an ´etalemorphism. There is a noetherian ring A0 and an

étalemap A0 → C0 such that C = A ⊗A0 C0. It is therefore sufficient to assume that A is noetherian. Flatness may be verified at the level of local rings:

Lemma 31.4. Let A be a commutative ring. An A-module M is zero if and 7 only if Mp is zero for all prime ideals p of A.

n Proof. For each x ∈ M deﬁne Ix to be the set of all f ∈ A such that f x = 0 T for some n ≥ 0. Then I = x∈M Ix. If p is a prime then Mp = 0. By deﬁnition, this means that for each x ∈ M n there is some f 6∈ p and some n ≥ 0 such that f x = 0. That is Ix is not contained in p. This holds for all primes p, so Ix is not contained in any maximal n ideal. Hence Ix = A. In particular, 1 x = 0 for some n, so x = 0. This holds for all x ∈ M, so M = 0.

Corollary 31.4.1. Let A be a commutative ring. The localizations Ap of A form a faithfully ﬂat collection of A-algebras.

We may now assume that A is local in addition to being noetherian. Present C as B/I where B = A[x1, . . . , xn] and I = (f1, . . . , fn). We will verify that the fi are a regular sequence in B.

Deﬁnition 31.5. Let M be a B-module. A sequence of elements f1, . . . , fn of a commutative ring B is called an M-regular sequence if for all 1 ≤ i ≤ n, the element fi is not a zero divisor of M/(f1, . . . , fi−1)M. Note that the deﬁnition of regularity depends only on the sequence of ideals,

(f1), (f1, f2), (f1, f2, f3),...

Lemma 31.6. Let B be a noetherian local ring and M a ﬁnite B-module. Any permutation of an M-regular sequence is also M-regular.

Proof. It is suﬃcient to treat the case of a regular sequence of length 2. Suppose that x is not a zero divisor in M and y is not a zero divisor in M/xM. Let N ⊂ M be the kernel of multiplication by y. If z ∈ N then z must be a multiple of x since y is not a zero divisor modulo x. Therefore N = xN. This implies

7In fact, maximal ideals suﬃce. 76 CHAPTER 4. COMMUTATIVE ALGEBRA

that IN = N (where I is the maximal ideal of B) so N = 0 by Nakayama’s lemma. This shows y is not a zero divisor in M. Now we show that x is not a zero divisor modulo yM. Suppose that xz ≡ 0 (mod yM). Then xz = yw, so yw ≡ 0 (mod xM). Therefore w = xv for some v ∈ M and we have xz = xyv. Rearranging gives x(z − yv) = 0 and since x is not a zero divisor in M, we deduce z = yv ≡ 0 (mod yM).

Lemma 31.7. If A is a ﬁeld then f1, . . . , fn are a regular sequence in B.

Proof. We can check whether a sequence is regular after making a faithfully flat base extension.8 Since A is a field, any field extension A0 ⊃ A is a faithfully flat extension. We select an algebraic closure of A for A0. We replace B with 0 0 0 0 B = B ⊗A A and C with C = C ⊗A A . We may now assume that A is an algebraically closed field. Since C is étaleover the algebraically closed field A, it must be a product 9 → of copies of A.9 The image of the map Spec C → Spec B therefore consists of a finite collection of maximal ideals. To verify that (f1, . . . , fn) is a regular sequence in B, it is sufficient to show that (f1, . . . , fn) is a regular sequence in Bq for every prime ideal q in the image of Spec C. (It is trivially a regular sequence at the primes that are not in the image.) We have Cq ' A for each of these primes. Denoting by ξi the image of xi under the map

A[x1, . . . , xn] = B → Cq ' A we have an equality of ideals:

(x1 − ξ1, . . . , xn − ξn)Bq = (f1, . . . , fn)Bq

Up to a change of coordinates, we can assume that ξi = 0 for all i. The sequence (x1, . . . , xn) is certainly regular. Note that there is some fi— up to reordering we may assume it is fn—such that (x1, . . . , xn) = (x1, x2, . . . , xn−1, fn). Then fn is not a zero divisor in Bq/(x2, . . . , xn) so (x1, x2, . . . , xn−1, fn) is a regular sequence. But then

(x1, . . . , xn−1)Bq/(fn) = (f1, . . . , fn−1)Bq/(fn).

We can therefore ﬁnd fn−1 such that (x1, . . . , xn−1)Bq/(fn) = (x1, . . . , xn−2, fn−1)Bq/(fn) and deduce that (x1, . . . , xn−2, fn−1, fn) is a regular sequence. Continuing this way, we eventually deduce that (f1, . . . , fn) is a regular sequence, as desired.

Lemma 31.8. Let A be a noetherian local ring with maximal ideal I and let M be a ﬂat A-module. If f ∈ B is a non-zero divisor in M/I then M/fM is a ﬂat A-module.

8Should be justiﬁed. 9todo: justify A. EXTENDING ETALE´ MAPS 77

Proof. By the local criterion, it is suﬃcient to show that Tor1(M/fM, A/I) = 0. Consider the exact sequence

f 0 → M −→ M → M/fM → 0 giving rise to the exact sequence below:

f 0 → Tor1(M/fM, A/I) → M/IM −→ M/IM

By assumption, f is not a zero divisor modulo IM, so we conclude Tor1(M/fM, A/I) = 0.

Proof of Theorem 31.3. Present C as B/J with J = (f1, . . . , fn) and B = (x1, . . . , xn). Let I be the maximal ideal of A. Then fi is not a zero divisor in B/(I + (f1, . . . , fi−1)) so by induction B/(f1, . . . , fi−1) is ﬂat over A for all i. In particular, C is ﬂat over A.

A Extending ´etalemaps

Proposition 31.1. Let10 A be a commutative ring and C an étale A-algebra. ← 10 Suppose that A0 → A is a nilpotent extension. Then there is an extension of C to an étale A0-algebra and this extension is unique up to a unique isomorphism that commutes with the projection to C. Proof. First we consider the uniqueness. Suppose C0 and C00 are two extensions. We can fit them into a commutative diagram:

CCo 00 O O f } C0 o A0 A unique dashed arrow rendering the whole diagram commutative is guaranteed to exist by the definition of a formally étalemorphism of commutative rings. Similarly, there is a unique map g : C00 → C0. The uniqueness in the definition of étalemaps shows furthermore that fg = idC00 and gf = idC0 . Now we consider the existence. We may present C as B/I where B = ∂fi 11 A[x1, . . . , xn] and I = (f1, . . . , fn) and det is a unit in C. Lift f1, . . . , fn ← 11 ∂xj i,j 0 0 0 0 0 0 0 to polynomials f1, . . . , fn ∈ A [x1, . . . , xn] and set C = A [x1, . . . , xn]/(f1, . . . , fn). ∂f 0 Then i reduces to a unit in C. But C0 is a nilpotent extension of C, so ∂xj i,j an element of C0 is a unit if and only if its image in C is a unit.12 Therefore C0 is étaleas an A0-algebra by13 ← 13 10todo: include ref to EGA; maybe just delete the prop 11todo: reference 12Suppose α reduces to a unit in C = C0/J and J is nilpotent. Let β be an element that lifts the inverse of the image of α in C to C0. Then αβ ≡ 1 (mod J). Therefore αβ = 1 − γ for some γ ∈ J. But (1 − γ)−1 = 1 + γ + γ2 + ··· and this series is finite because γ ∈ J and J is nilpotent. Therefore β(1 − γ)−1 is the inverse of α. 13todo: reference 78 CHAPTER 4. COMMUTATIVE ALGEBRA

The following lemma is a special case of [?, Chap. 0, (19.5.4.2)]:

Lemma 31.2. Let A be a commutative ring, B = A[x1, . . . , xn], I = (f1, . . . , fn), such that C = B/I is an ´etale A-algebra. Then the map

2 SymC (I/I ) → grI (B) is an isomorphism. Proof. First consider the commutative diagram

CC O O

| B/In o A in which the dashed arrow exists (and is unique) because C is an ´etale A-algebra. Therefore B/In ' C × I/In. This gives a C-module structure to I/In. Now 2 I/I is a free C-module (since it is isomorphic to the free C-module C ⊗A ΩB/A) so that the map I/In → I/I2 has a section. This extends to a surjective map ≤n 2 n+1 SymC (I/I ) → B/I with nilpotent kernel, whence a commutative diagram

B/In+1 o B O O

z ≤n 2 SymC (I/I ) o A in which the dashed arrow again exists because B is an étale A-algebra. This n+1 ≤n 2 ≤n 2 n+1 induces a section B/I → SymC (I/I ). But both SymC (I/I ) and B/I 2 n+1 ≤n 2 are generated by I/I , so that the map B/I → SymC (I/I ) must be surjective. As it is a section it is also injective, so it is bijective. In particular, this map is an isomorphism in each graded degree, which proves the lemma. Proposition 31.3. Let A be a commutative ring and C an étale A-algebra. Let A0 be a square-zero extension of A by an ideal J. For any homomorphism J → K (where K is given the A-module structure indcued by the homomorphism A → C) there is a compatible A0-algebra extension of C by K.14 Although this proposition is true as stated, we will treat only the case where A is noetherian and K is of finite type.

14The meaning of compatibility is that the exntesions ﬁt into a commutative diagram 0 / J / A0 / A / 0

0 / K / C0 / C / 0 in which the maps J → K and A → C are the ones already speciﬁed. A. EXTENDING ETALE´ MAPS 79

Proof. Present C as B/I where B = A[x1, . . . , xn] and I = (f1, . . . , fn) and det ∂fi is a unit in C. We may regard K as a B-module. There is certainly ∂xj i,j 0 0 0 an extension of B by K as a A -algebra: ﬁrst take B = A [x1, . . . , xn], which is 00 an extension of B by B ⊗A J and then push out to form B :

0 0 / B ⊗A J / B / B / 0

0 / K / B00 / B / 0.

Now we consider the question of ﬁnding an extension C0 of C by K ﬁtting into a commutative diagram

0 / K / B00 / B / 0

0 / K / C0 / C / 0.

0 0 Recall that C = B/I where I = (f1, . . . , fn). Choose lifts f1, . . . , fn of the fi to B00 and let I0 ⊂ B00 be the ideal they generate. 0 0 I claim that I ∩ K = 0. It is sufficient to show that Ip ∩ Kp = 0 for every prime ideal p of B (note that as sets Spec B → Spec B00 is a bijection). If p ∈ Spec B is not in the image of Spec C then Kp = 0 (as K is a C-module), so the assertion is trivially true. We therefore only need to consider primes p that are pre-images of primes of C. P 0 00 0 Now suppose that aifi ∈ Kp with ai ∈ Bp . Reducing modulo K + I , we P 0 0 2 0 2 find that aifi = 0 in C. As Ip/(Ip + Ip ∩ K) = Ip/Ip is freely generated 0 P 0 by the fi, this means that ai ≡ 0 mod Ip + Kp for all i. That is, aifi lies 0 0 0 2 0 P 0 0 2 in (Ip + Kp)Ip = Ip since KpIp = 0. A fortiori, aifi lies in Ip ∩ Kp. But 2 3 now we can repeat the argument, using the fact that Ip /Ip is freely generated 0 0 3 by the fifj (Lemma 31.2), to deduce that Ip ∩ Kp is contained in Ip ∩ K. By induction, we get 0 \ 0 n Ip ∩ K ⊂ Ip . n≥0 0 But this intersection is zero because Ip is a proper ideal of a local noetherian ring.15 ← 15 We may now define C0 = B00/I0. Since K ∩ I0 = 0, the ideal of C in C0 is K, and we have the required extension.

Now we analyze what an inﬁnitesimal extension of an ´etalemap looks like.

Lemma 31.4. Let f : A → C be a homomorphism of commutative rings, M an A-module, N a C-module, and ϕ : M → N a homomorphism compatible with

15todo: Explain why B00 is noetherian? Because it is a square-zero extension of a noetherian ring by a ﬁnitely generated module... 80 CHAPTER 4. COMMUTATIVE ALGEBRA

f. Suppose that M and N have compatible ﬁnite ﬁltrations by submodules Mi such that C ⊗A Mi/Mi−1 → Ni/Ni1 is an isomorphism for all i. Then ϕ is an isomorphism.

Proof. By induction. As a base case, we have C ⊗A M0 ' N0. Then we have an exact sequence

0 → Mi−1 → Mi → Mi/Mi−1 → 0 inducing a commutative diagram

α 0 / C ⊗A Mi−1 / C ⊗A Mi / C ⊗A Mi/Mi−1 / 0

β γ 0 / Ni−1 / Ni / Ni/Ni−1 / 0 The leftmost vertical arrow is an isomorphism by induction and the rightmost is by assumption. Note that α is injective because if α(x) = 0 then γ(β(x)) = 0, and β is bijective. We may therefore apply the 5-lemma. Proposition 31.5. Let A0 → A be an infinitesimal extenesion of commutative noetherian rings with ideal J, let C0 be an étale A0-algebra, and let 0 0 0 C = C ⊗A0 A. Then the ideal of C in C is isomorphic to J ⊗A0 C . Proof. First we reduce to the case where J 2 = 0. We can filter the extensions A0 → A and C0 → C as sequences of square-zero extensions 0 0 0 0 A = A0 → A1 → · · · → An = A 0 0 0 0 C = C0 → C1 → · · · → Cn = C.

This gives ﬁltrations on J = I 0 and I 0 by the I 0 0 and the I 0 0 . By A/A C/C A /Aj C /Cj the square-zero case, we have 0 IA0 /A0 ⊗ C ' IC0 /C0 j j+1 A0 j j+1 0 so Lemma 31.4 implies that J ⊗A0 C → IC/C0 is an isomorphism. Now we consider the square-zero case. We consider the functor represented by I = IC/C0 on C-modules. Giving a C-module homomorphism I → K induces an square-zero extension of C by K as an A00-algebra by pushing out the diagram

0 / I / C0 / C / 0

0 / K / C00 / C / 0. Conversely, if we have any such extension we may consider the diagram

CCo 0 O O

} C00 o A0 B. COMPLETIONS OF RINGS 81 which admits a unique completion because C0 is étaleover A0, hence gives a map I → K. We conclude that I represents the functor sending a C-module K to the set of square-zero A0-algebra extensions of of C by K. On the other hand, we may consider the functor represented on C-modules 16 by J ⊗A C, which sends a C-module K to HomA(J, K). Therefore the proposition reduces to demonstrating that for any A-module homomorphism J → K, there is a square-zero extension C00 of C by K as an A0-algebra such that the induced map on ideals J → K coincides with the given one. Theorem 31.6. An étalehomomorphism of commutative rings is flat.

Proof. Suppose A → C is étale.There is a noetherian ring A0 and an étale A0- algebra C0 such that C is induced by base extension via a map A0 → A. It is therefore sufficient to treat the case where the base ring is noetherian. Without loss of generality we replace A with A0 and C with C0. Flatness may be verified locally so we may now assume that A is a local ring in addition to being noetherian. Now we use the formal criterion for flatness. This permits us to assume A is artinian. It is sufficient to show that if I ⊂ A is any ideal, then I ⊗A C → IC is an isomorphism. As every ideal of A is nilpotent, this is the content of Proposi- tion 31.5.

B Completions of rings

This section comes mostly from [?, Section 0.7]. The following is a restatement of the Artin–Rees lemma: Proposition 31.1. Let A be a commutative noetherian ring, I ⊂ A an ideal, and M 0 ⊂ M ﬁnitely generated A-modules. The I-adic topology on M 0 is induced from the I-adic topology on M. Corollary 31.1.1. The functor M 7→ lim M/InM from ﬁnitely generated A- ←− modules to Aˆ-modules is exact. Proof. Consider an exact sequence:

f g M 0 −→ M −→ M 00

n 00 n 00 Suppose that x ∈ lim M/I M has zero image in lim M /I M . Choose yn ∈ M ←− n ←− n 00 such that yn ≡ x (mod I M). Then g(yn) ≡ 0 (mod I M ). That is, g(yn) ∈ n 00 0 n 0 I M , or, equivalently, yn ∈ M + I M. Choose an element zn ∈ M such that n zn − yn ∈ I M. By construction the zn converge in the I-adic topology on M to x. This implies that they converge I-adically to an element of lim M 0/InM 0, since the ←− I-adic topology on M 0 is the same as the one induced from M. The limit of an I-adically convergent sequence is unique so x must lie in the image of lim M 0/InM 0 → lim M/InM. ←− ←− 16Note that J is an A-module because J2 = 0. 82 CHAPTER 4. COMMUTATIVE ALGEBRA

Corollary 31.1.2. Let A be a noetherian ring and I ⊂ A an ideal. Let Aˆ = lim A/In be the completion of A at I. Then Aˆ is a flat A-algebra. ←− Proof. For the flatness, it will be sufficient to show that lim M/InM ' M ⊗ Aˆ ←− A when M is a finitely generated A-module. There is certainly a map

J ⊗ Aˆ → lim J/InJ. A ←− To see that it is an isomorphism, consider a presentation of J:

Q → P → J → 0 in which both P and Q are ﬁnitely generated and free A-modules. Now consider the following diagram:

J ⊗A Q / J ⊗A P / J ⊗A Aˆ / 0

lim Q/InQ lim P/InP lim J/InJ 0 ←− / ←− / ←− /

The lower horizonatal sequence is exact by the exactness of completion; the upper sequence is exact by the right exactness of tensor product. On the other hand, one can easily see that the left two verical arrows are isomorphisms because P and Q are ﬁnitely generated and free. The 5-lemma now completes the proof.

Corollary 31.1.3. Suppose that A is a Noetherian local ring and Aˆ is the completion of A with respect to the maximal ideal. Then Aˆ is faithfully ﬂat over A.

Proof. Let I be the maximal ideal of A. Consider a finitely generated A-module M. Then the map M → Aˆ ⊗ M = lim M/InM is is injective. Indeed, let A ←− N = T InM be the kernel. Then by Artin–Rees, there is some index such that In+kM ∩ N = In(IkM ∩ N). But In+kM ∩ N = N and In(IkM ∩ N) = InN so we deduce that IN = N. By Nakayama’s lemma, we get N = 0. This proves that Aˆ ⊗A M = 0 if and only if M = 0 when M is finitely generated. Suppose that M is a general A-module such that Aˆ ⊗A M = 0. 0 0 Then for any finitely generated submodule M , we have Aˆ ⊗A M ⊂ Aˆ ⊗A M 0 0 since Aˆ is flat. Therefore Aˆ ⊗A M = 0 so M = 0. But then every finitely generated submodule of M is zero so M = 0.

Proposition 31.2. Suppose that A is an I-adically complete ring. The following are equivalent:

(i) A is noetherian.

(ii) gr A is noetherian. C. ZARISKI’S “MAIN THEOREM” 83

(iii) A/I is noetherian and I/I2 is finitely generated over A/I. Proof. (i)= ⇒ (iii). Observe that A/I is a quotient of A and I/I2 is a quotient of I. (ii)= ⇒ (i). Let J be an ideal of A. Each of the generators of J has a well-defined image in the associated graded ring, and finitely many of these suffice to generate the associated ideal. Let x1, . . . , xn be these generators. For n P n+1 any yn ∈ J ∩ I there is therefore an expression yn ≡ aixi (mod J ∩ I ). P n+1 Select zn = aixi such that Then yn+1 = yn − zn ∈ J ∩ I . Then select P n+1 zn+1 = bixi so that we can write yn+1 − zn+1 ∈ J ∩ I . Repeating this P procedure, we obtain a convergent series zk = y. Thus J is generated by x1, . . . , xn. (iii)= ⇒ (ii). We can construct gr A as a quotient of the polynomial ring over A/I generated by a collection of generators for I/I2. Proposition 31.3. Let A → B be a local homomorphism17 of complete noetherian local rings. If B is quasi-finite as an A-algebra then it is finite. Proof. Let I be the maximal ideal of A. Before we begin, note that B is I- adically complete. Indeed, if J denotes the maximal ideal of B then IB ⊂ J becasue A → B is a local homomorphism. Therefore the topology on B is no finer than the I-adic topology. On the other hand, B/IB is a finite dimensional vector space over A/I by assumption, so the subspaces J n(B/IB) must stabilize. But they can’t stabilize anywhere other than zero since T J n = 0. Now let x1, . . . , xn be representatives of a basis for B/IB in B. For any n P n+1 yn ∈ I B, we may find zn = aixi such that yn − zn ∈ I B. Applying this P inductively to any y ∈ B we obtain a series zn that converges I-adically to 18 y. But the submodule of B generated by x1, . . . , xn is I-adically complete, so it contains y.

C Zariski’s “Main Theorem”

Here we will give a second proof of the flatness of étalemaps, based on Zariski’s “Main Theorem”. Definition 31.1. A homomorphism of commutative rings A → B is called faithfully flat if a sequence of A-modules

M 0 → M → M 00 is exact if and only if the induced sequence

B ⊗ M 00 → B ⊗ M → B ⊗ M 00 A A A

17A homomorphism of local rings f : A → B with maximal ideals I and J is called local if f −1(J) = I. This corresponds to the geometric assertion that the map Spec B → Spec A carries the closed point to the closed point. 18A ﬁnitely generated A-module is I-adically complete because it is a quotient of an I- adically complete A-module, An. 84 CHAPTER 4. COMMUTATIVE ALGEBRA

is exact.

Definition 31.2. A homomorphism of commutative rings A → B is called finite if B is a finite A-module. It is called quasi-finite if it is of finite type and B ⊗A k is a finite k-vector space for every field k and every homomorphism A → k. It is called integral if every element of B satisfies a monic polynomial with coefficients in A.

Lemma 31.3. Suppose that f : A → B is a quasi-ﬁnite local homomorphism of noetherian local rings then f is ﬁnite.

Proof. We have already seen that the induced map of completions is finite. Suppose that x ∈ B. Then x satisfies a monic polynomial p(x) with coefficients in Aˆ: n n−1 p(x) = anx + an−1x + ··· + a0 = 0

with an = 1. This is a relation in the Aˆ-module Bˆ = B ⊗A Aˆ. Since Aˆ is flat over A, this relation must be induced from a relation in B. That is, there is a finite collection of polynomials qi with coefficients in A, all satisfied by x, such 19 19 → that p is a linear combination of the qi, with coefficients in Aˆ. X ciqi = p

P Let bi be the leading coeﬃcient of qi. Then cibi = an = 1. That is, the bi 20 20 → generate the unit ideal in Aˆ. But Aˆ is faithfully ﬂat over A so the bi generate 0 P 0 the unit ideal in A as well. Let ci be elements of A such that cibi = 1. Then

0 X 0 p = ciqi

is a monic polynomial with coefficients in A that is satisfied by x. Therefore x is integral over A. We conclude that B is a finite type, integral extension of A, so it is finite.

Theorem 31.4. Suppose that f : A → B is a quasi-ﬁnite homomorphism of commutative noetherian rings. Then f may be factored as

A → C → B

where C is ﬁnite over A and C → B is a localization.

21 → Proof. Let C be the integral closure of A in B. If p is a prime ideal 21

19todo: needs to be justified 20todo: explain why faithfully flat implies surjective 21todo: finish D. MORE PERSPECTIVES ON ETALE´ MAPS 85

D More perspectives on étalemaps 31.1 The analytic perspective Definition 31.1. Let A be a commutative ring and I ⊂ A an ideal. The ring Aˆ = lim A/In is called the formal completion of A along I.22 ←− Proposition 31.2. Suppose that A is a commutative ring that is separated and complete with respect to the powers of an ideal I. Let B be a formally étale A-algebra. Then the map

HomA(B,A) → HomA(B, A/I) is bijective.

Proof. By the formal criterion, we have the following chain of bijections:

∼ 2 ∼ 3 ∼ HomA(B, A/I) ←− HomA(B, A/I ) ←− HomA(B, A/I ) ←−· · ·

On the other hand, we also have the following bijection since A is separated and complete: Hom (B,A) −→∼ lim Hom (B, A/In) A ←− A n

31.2 The algebraic perspective Theorem 31.3. An ´etalemorphism is ﬂat.

Proposition 31.4 ([?, Proposition (18.3.1)]). (i) An A-algebra B is unramified if and only if it is of finite type and B is a projective B ⊗A B-module. (ii) An A-algebra B is étaleif and only if it is of finite presentation, and projective, both as an A-module and as a B ⊗A B-module. Proof. Recall that for B to be unramified as an A-algebra means the codiagonal map B ⊗A B → B is a surjective localization map. Lemma 31.5. A surjective localization map of unital commutative rings ϕ : C → B admits a canonical section as a map of C-modules.

Proof. Write B = C[f −1] for some f ∈ C. Choose g ∈ C such that ϕ(g) = f −1. Then B = C/(1 − fg). Consider the C-module map x 7→ fgx : C → fgC. We have 1 − fg 7→ 0 so this descends to a (surjective) map B = C/(1 − fg) → fgC. This is a section of ϕ so it is also injective, hence an isomorphism.

The lemma shows that B is a direct summand of B ⊗A B, hence is projective.

22This deﬁnition is not particularly well behaved unless I is ﬁnitely generated. 86 CHAPTER 4. COMMUTATIVE ALGEBRA

This suggests the following definition, which is in some ways better behaved than the étalemorphisms we study here: Definition 31.6 ([BS, Definition 1.2]). A morphism of schemes f : X → Y is called weakly étale if it is flat and its diagonal embedding is flat.

31.3 Equivalence of the definitions All of the definitions can be phrased in terms of a chart, so it will be enough to treat the case of a morphism of affine schemes. Throughout this discussion, we will assume that B = A[t1, . . . , tn]/(f1, . . . , fm) is an A-algebra of finite presentation. Lemma 31.7. If f : X → Y is locally of finite type then its diagonal morphism X → X ×Y X is locally of finite presentation.

Proof. First note that if B is an A-algebra of finite type then the map B ⊗A B → B is of finite presentation. Indeed, we only need to check that the ideal is finitely generated. But if t1, . . . , tn generate B as an A-algebra then ti ⊗ 1 − 1 ⊗ ti generate the ideal of the quotient B ⊗A B → B. Exercise 31.8. Find an example of a homomorphism of commutative rings that is not of finite type but whose diagonal is of finite presentation. (Hint: find one whose diagonal is an isomorphism.) Proposition 31.9. Let A be a commutative ring and let B = A/I where I = (f1, . . . , fm) is finitely generated. Then the following properties are equivalent: (i) Spec B → Spec A is an open embedding. (ii) B is flat as an A-algebra. (iii) I/I2 = 0. Proof. Certainly an open embedding is flat. Consider the exact sequence of A-modules 0 → I → A → B → 0 and tensor it with B to get

I/I2 → B −→id B → 0.

Therefore ﬂatness implies I/I2 = 0. Suppose now that I/I2 = 0. Then I = I2 = I3 so we can write

n X 2 fn = aifi . i=1 Thus n−1 X 2 fn(1 − anfn) = aifi . i=1 E. HOMOLOGY OF MODULES 87

−1 Note that we have a canonical morphism of A-algebras, A[(1 − anfn) ] → B since 1 − anfn maps to 1 in B. Furthermore, f1, . . . , fn−1 suffice to generate I −1 over A[(1 − anfn) ]. Proceeding by induction, we find a localization C of A such that IC = 0. Corollary 31.9.1. All different senses of unramified, étale,and smooth coincide for surjective homomorphisms. Corollary 31.9.2. All different senses of unramified agree in general. Exercise 31.10. Give an example to show that I must be finitely generated for the above proposition to hold. (Hint: consider the ring obtained by adjoining all positive rational powers of an indeterminante to a field.)

E Homology of modules

Deﬁnition 31.1. An A-premodule is a set M equipped with relations

α : A × M → M σ : M × M → M subject to the usual module axioms. We generally write a.m or am instead of α(a, m) and m + m0 instead of σ(m, m0). Example 31.2. Suppose M and N are A-modules. Then M × N may be given an A-premodule structure in which

a.(m, n) ∼ (a.m, n) ∼ (m, a.n) (m, n) + (m, n0) ∼ (m, n + n0) (m, n) + (m0, n) ∼ (m + m0, n).

Note that we have written ∼ rather than equal, because the elements (a.m, n) and (m, a.n) usually are not the same. Definition 31.3. An extension of premodules is a function p : X → M, where both X and M are premodules that satisfies the following condition: P Whenever aip(xi) ∼ m for some ai ∈ A, xi ∈ X, and y ∈ M there P is a unique x ∈ X with x ∼ aixi in X and p(x) = m. An extension of the bimodule M × N defined above is called a biextension of M and N. Exercise 31.4. Show that if M is a module then to give an extension of M as a premodule is the same as to give an extension of M as a module. Exercise 31.5. Suppose that X is a biextension of A-modules M and N. For each (m, n) ∈ M ×N, let X(m, n) denote the fiber of the projection X → M ×N. Show that X has the following structures and properties: 88 CHAPTER 4. COMMUTATIVE ALGEBRA

BIEXT1 There are partially deﬁned addition laws:

σ : X(m, n) × X(m0, n) → X(m + m0, n) τ : X(m, n) × X(m, n0) → X(m, n + n0)

BIEXT2 For any a ∈ A there are maps

α(a, −): X(m, n) → X(am, n) β(a, −): X(m, n) → X(m, an).

BIEXT3 The addition laws are assocaitive and the A-actions distribute over them. BIEXT4 The addition laws commute with one another:

σ(τ((m, n), (m, n0)), τ((m0, n), (m0, n0)) = τ(σ((m, n), (m0, n)), σ((m, n0), (m0, n0)).

BIEXT5 The actions of A commute with one another:

α(a, β(b, x)) = β(b, α(a, x)).

BIEXT6 A biextension has a unique zero element 0 and σ(0, x) = τ(0, x) for all x. Proposition 31.6. Suppose that X is an extension of a premodule M and 0 0 0 f : M → M is a homomorphism of premodules. Then X = X ×M M is naturally equipped with the structure of an extension of M 0. Proof. Declare that a(x, m) ∼ (y, n) if ax ∼ y and am ∼ n in X and in M, respectively. Declare that (x, m) + (y, n) ∼ (z, p) if x + y ∼ z and m + n ∼ p. P 0 0 P Suppose that aip (xi, mi) ∼ y in M . Then aip(xi) ∼ f(y). Therefore P there is a unique x ∈ X with aixi ∼ x and p(x) = f(y). Therefore there is a 0 P 0 unique (x, y) ∈ X ×M M with p(x, y) = y and ai(xi, mi) ∼ (x, y). Thus X is an extension of M 0. Suppose that p : X → M × N is a biextension. Then p−1(0, 0) is an A- submodule of X. We call this the kernel and say that there is an exact sequence

0 → J → X → M × N → 0. (31.1)

Exercise 31.7. Suppose that X is a biextension of M and N by J. Show that J acts simply transitively on the underlying set of X with quotient M × N. Proposition 31.8. If (31.1) is a biextension and J → J 0 is a homomorphism of A-modules then there is a completion of the diagram

0 / J / X / M × N / 0

0 / J 0 / X0 / M × N / 0 by an biextension X0 of M and N by J 0 that is unique up to unique isomorphism. E. HOMOLOGY OF MODULES 89

Proof. Let X0 be the quotient of J 0 × X by the antidiagonal action of J. (It is the universal map from J 0 ×X into a simply transitive J 0-set that is equivariant with respect to the homomorphism J × J → J.) The premodule structure on X0 is induced from J 0 × X. We need to show that X0 → M ×N is a ﬁbration. Consider the commutative diagram J 0 × X / X

X0 / M × N. The right vertical arrow is a fibration and the upper horizontal arrow is a fibration because J 0 is a module, so the upper horizontal arrow is a base change of a fibration. As J 0 × X → M × N is therefore a fibration, it suffices to show 0 0 P that J × X → X is a fibration. Suppose that ai(ji, xi) has an evaluation in X0. We show it has a unique evaluation in J 0 × X. Since X → M × N factors 0 P through X , the sum aixi has an evaluation x that is uniquely determined P 0 up to an element u of J. But aiji has a unique evaluation (since J is an A-module). Therefore any two evaluations in J 0 × X lifting the given one in X0 must be of the form (j, x) and (j, x + u). But (j, x) and (j, x + u) only have the same image in X0 if u = 0.23 ← 23

The preceding propositions show that ExtA(M × N; J) is covariant with respect to J and contravariant with respect to M × N. As M × N is covariant with M and N, this shows that BiextA(M,N; J) is contravariant with M and N and covariant with J.

31.1 Exact sequences Proposition 31.9. Let

0 → M 0 → M → M 00 → 0.

00 Then BiextA(M ,N; J) is equivalent to the category of diagrams (of premodules) M 0 × N

v 0 / J / X → MtimesN / 0 in which the horizontal row is exact.

Proof. Suppose ﬁrst that we have a biextension of M 00 and N by J. Then by

23todo: clean up; explain ﬁbration somewhere? otherwise write the proof without it 90 CHAPTER 4. COMMUTATIVE ALGEBRA

pullback we can construct a diagram

0 / J / X / M × N / 0

0 / J / Y / M 00 × N / 0 The composition M 0 ×N → M ×N → M 00 ×N sends (m, n) to (0, n). Therefore we have X(m, n) = X(0, n) for all m ∈ M 0. On the other hand, X(0, n) has a distinguished element: since 0.(0, n) ∼ (0, n) there is a unique element ξ of X(0, n) such that 0.x = ξ and ξ projects to (0, n) in M × N. In view of the identiﬁcation X(m, n) = X(0, n) for all m ∈ M 0, this gives a section of X over 24 → M 0 × N. 24

F Cohomology of modules

Deﬁnition 31.1. Let M and J be A-modules. An extension of M by J is an exact sequence 0 → J → X → M → 0. (31.1) A morphism of extensions of M by J is a commutative diagram

0 / J / X / M / 0

0 / J / X0 / M / 0.

Composition is deﬁned in the evident way so that we obtain a category ExtA(M,J). More generally, we can deﬁne a category ExtA whose objects are exact sequences (31.1) but whose morphisms are commutative diagrams

0 / J / X / M / 0 (31.2) 0 / J 0 / X0 / M 0 / 0.

1 We write ExtA(M,J) or Ext (M,J) for the set of isomorphism classes in Ext(M,J).

Exercise 31.2. Show that every morphism in ExtA(M,J) is an isomorphism. 2 Exercise 31.3. (i) Construct a functor ExtA → (A-Mod) that sends an extension (31.1) to (M,J).

(ii) Show that ExtA(M,J) is the fiber of ExtA over (M,J), where the fiber is defined to be the category of objects projecting to (M,J) and morphisms projecting to id(M,J).

24todo: verify it’s a homomorphism F. COHOMOLOGY OF MODULES 91

0 (iii) Show that if J → J is a homomorphism of A-modules and X ∈ ExtA(M,J) then there is a universal (initial) completion of the diagram (31.2). Con- clude that the categories ExtA(M,J) vary covariantly with J. Deduce in particular that ExtA(M,J) is a covariant functor in J.

(iv) Show that if M → M 0 is a homomorphism of A-modules and X0 ∈ 0 ExtA(M ,J) then there is a universal (ﬁnal) completion of the diagram (31.2). Conclude that the categories ExtA(M,J) vary contravariantly with M and in particular that ExtA(M,J) is a contravariant functor in M.

(v) Show that the constructions from the last two parts are compatible in sense that the diagram

0 0 0 ExtA(M ,J) / ExtA(M ,J )

0 ExtA(M,J) / ExtA(M,J )

is commutative (up to canonical isomorphism).

Exercise 31.4. Using the constructions from the last exercise, show that the functors

0 0 (i) ExtA(M ⊕ M ,J) → ExtA(M,J) × ExtA(M ,J) and

0 0 (ii) ExtA(M,J × J ) → ExtA(M,J) × ExtA(M,J ) are equivalences of categories. Using the A-module homomorphisms

(i) M → M ⊕ M : x 7→ (x, x) and

(ii) J × J → J :(x, y) 7→ x + y deduce two group structures on ExtA(M,J). Verify that these are the same and that they are commutative.25 Show that the zero element of Ext(M,J) is represented by the exact sequence

0 → J → M × J → M → 0.

Exercise 31.5. Consider an exact sequence of A-modules:

0 → M 0 → M → M 00 → 0.

(i) Construct a map Hom(M 0,J) → Ext(M 00,J). (Hint: Consider the sequence above as an element of Ext(M 00,M 0).)

25 In fact, these should be viewed as group structures on the groupoid ExtA(M,J). 92 CHAPTER 4. COMMUTATIVE ALGEBRA

(ii) Show that Ext(M 00,J) is equivalent to the category of commutative diagrams with exact row,

M 0

} 0 / J / X / M / 0.

(iii) Conclude that there is an exact sequence

0 → Hom(M 00,J) → Hom(M,J) → Hom(M 0,J) → Ext1(M 00,J) → Ext1(M,J) → Ext1(M 0,J). Chapter 5

The ´etaletopology and the ´etalefundamental group

32 Grothendieck topologies

Deﬁnition 32.1. Let C be a category and X an object of C .A sieve of X is a full subcategory R of C /X with the property that whenever (Y, f) ∈ R is in g R and Z −→ Y is any morphism of C the composition (Z, fg) ∈ R. Sieves can also be thought of as functors:

Proposition 32.2. Let R ⊂ C /X be a sieve. For each Y ∈ C , deﬁne F (Y ) ⊂ hX (Y ) by

F (Y ) = {g : Y → X (Y, g) ∈ R}.

Then F is a subfunctor of hX and the assignment R 7→ F is a bijection between the sieves of X and the subfunctors of hX .

Proof. To see it is a subfunctor, notice that if g ∈ F (Y ) ⊂ hX (Y ) and h : Z → Y is a morphism then hX (h)(g) = gh ∈ hX (Z). But (Z, gh) ∈ R by deﬁnition of a sieve. The reverse construction sends F ⊂ hX to the collection of pairs (Y, g) where Y ∈ C and g ∈ F (Y ). This is a sieve since if h : Z → Y is a morphism of C and (Y, g) ∈ R then gh = F (h)(g) ∈ F (Z) so (Z, gh) ∈ R. Verifying that these two constructions are inverse to one another is an exercise. We will usually think of sieves as functors rather than categories, since it makes other deﬁnitions easier. For example: (i) A morphism of sieves is a morphism of functors (a natural transformation).

(ii) A limit of sieves is a limit of functors. In particular, (R ×S T )(X) = R(X) ×S(X) T (X).

93 94 CHAPTER 5. THE ETALE´ TOPOLOGY

It is also convenient to abuse notation and write X for the sieve hX .

Deﬁnition 32.3. A Grothendieck topology on a category C is the distinction of a class of covering sieves J(X) for each object X of C , subject to the conditions below:

TOP1 (stability under base change) If R ⊂ X is covering and Y → X is any morphism then R ×X Y ⊂ Y is covering.

TOP2 (local character) If R ⊂ R0 ⊂ X with R covering R0 and R0 covering X then R covers X.

TOP3 (nontriviality) The sieve X ⊂ X is covering.

0 Note that for R ⊂ R to be covering means that R ×R0 Y ⊂ Y is covering for all Y ∈ R0.

32.1 Examples Topological spaces Suppose X is a topological space. Let Open(X) be the category of open subsets of X. For each open U ⊂ X, let J(U) be the set of all sieves R of U such that [ V = U. V ∈R

That is J(U) consists of all sieves that actually cover U. Then (Open(X),J) is a Grothendieck topology.

The étaletopology on a topological space Let X be a topological space and let ét(X) be the category of all étalespaces over X. Call a family of maps f : V → U in ét(X) covering if [ f(V ) = U. (V,f)∈R

This is an example of a Grothendieck topology. In fact, this is a topos.

The Zariski topology on an affine scheme Suppose X is an affine scheme. Let zar(C) be the category of open affine subschemes of X. For U ∈ zar(C), let J(U) be the set of all sieves R of U such that [ |V | = |U|. V ∈R This is the Zariski topology on X. 32. GROTHENDIECK TOPOLOGIES 95

The étaletopology on an affine scheme Let X be an affine scheme and let ét(X) be the category of all affine schemes that are étaleover X. A family of maps f : V → U in ét(X) is covering if [ f(|V |) = |U|. (V,f)∈R

This is the ´etaletopology on X.

The fpqc topology Let X be an affine scheme and let fpqc(X) be the category of all affine schemes over X. A family of maps f : V → U in fpqc(X) is called covering if it contains a finite subcollection that surjects onto U schematically. That is, there should be (V1, f1),..., (Vn, fn) in the collection such that

n [ fi(|Vi|) = |U|. i=1 This is the fpqc topology on X.

32.2 Sheaves on Grothendieck topologies Deﬁnition 32.4. Let C be a Grothendieck topology. A functor F : C ◦ → Sets is called a presheaf on C . We say that F is a sheaf if, for any covering sieve R ⊂ hX of C the map

F (X) ' Hom(hX ,F ) → Hom(R,F ) is a bijection.

Exercise 32.5. Verify that this deﬁnition is equivalent to the usual deﬁnition for a sheaf on a topological space.

When the category C has ﬁber products, the sheaf condition can be stated in a more familiar way. Suppose that the sieve R is generated by maps Ui → X. Deﬁne Uij = Ui ×X Uj.

Proposition 32.6. Assume that C has ﬁber products. A presheaf F on C is a sheaf if, for every X ∈ C and every covering family Ui → X, the sequence below is exact: Y Y F (X) → F (Ui) ⇒ F (Uij) i i,j This can be stated even more explicitly:

Corollary 32.6.1. A presheaf F is a sheaf if and only if the following conditions hold: 96 CHAPTER 5. THE ETALE´ TOPOLOGY

SH1 For every X ∈ C , every covering family Ui → X, and every pair ξ, η ∈ F (X) we have ξ = η if and only if ξ = η for all i. Ui Ui

SH2 For every X ∈ C , every covering family Ui → X, and every collection

ξi ∈ F (Ui) such that ξi U = ξj U for all i, j there is a ξ ∈ F (X) such ij ij that ξ = ξi for all i. F (Ui)

32.3 More examples of Grothendieck topologies Sets The category of sets has a Grothendieck topology in which the covers are the surjective families. This is a special case of several of the other examples dis- cussed here.

Group actions Let G be a group. Let G-Sets be the category of G-sets. Call a family of maps f : S → T of G-sects covering if [ f(S) = T. (S,f) This is another example of a topos.

Commutative rings Let ComRng be the category of commutative rings. Call a family of maps Ai → A covering if, for each finite collection S of elements of A there is some Ai in the family whose image in A contains S. This definition can be applied to groups, abelian groups, or any algebraic struccture “defined using inverse limits”.

33 Sheafification 33.1 Topological generators Let C be a site. A collection of objects U ⊂ Ob(C ) is said to generate C topo- logically if every covering sieve of every object of C has a covering refinement that is generated by objects of U . Put more precisely, if X ∈ C and R ⊂ X is a covering sieve then there is a 0 covering sieve R ⊂ R and a collection of objects Ui ∈ U and maps pi : Ui → X such that 0 [ R = pi(hUi ). i Definition 33.1. A Grothendieck topology C is called a site if it possesses a small collection of topological generators. 33. SHEAFIFICATION 97

33.2 Descent data Deﬁnition 33.2. Let F be a presheaf on a Grothendieck topology C . Let R be a covering sieve of X ∈ C .A descent datum for F with respect to R is an element of HomCˆ(R,F ). Exercise 33.3. The intersection of two covering sieves of an object of a Grothendieck topology is covering.

Corollary 33.3.1. The covering sieves of an object of a Grothendieck topology are ﬁltered under inclusion. If C is a site then, for each X ∈ C , the category J(X) (under inclusion) has a small coinitial subcategory.

Let F : C ◦ → Sets be a sheaf on a site. Deﬁne

F + = lim Hom (R,F ). −→ Cˆ R∈J(X)

Lemma 33.4. When F is a presheaf, F + is a separated presheaf.

Proof. To say that F + is a separated presheaf means whenever ξ, η ∈ F +(X) and ξ R = η R for a covering sieve R ⊂ hX , we have ξ = η. + 0 By deﬁnition of F there is a covering sieve R ⊂ hX such that ξ and η are representable by ξ0, η0 ∈ Hom(R0,F ). Let R00 ⊂ R0 be the set of all Y → X in R0 such that ξ0(Y ) = η0(Y ) as elements of F (Y ). By assumption we have ξ(Y ) = η(Y ) as elements of F +(Y ) 0 0 for all Y ∈ R. As ξ h and η h are represented by ξ h × R0 and η h × R0 Y Y Y hX Y hX 0 + in Hom(hY ×hX R ,F ), this means (by deﬁnition of F (Y ) as a direct limit over 0 covering sieves) that there is a covering sieve RY ⊂ hY ×hX R ⊂ hY such that ξ0 = η0 . RY RY 00 We conclude that, for each Y ∈ R we have hY ×hX R ⊂ hY is covering, 0 0 0 0 since it contains the covering sieve RY . But ξ R00 = η R00 so ξ and η represent the same element of F +(X) = lim Hom(R,F ). −→ R∈J(X)

Lemma 33.5. If F is a separated presheaf then F + is a sheaf.

+ 0 Proof. Let R ⊂ hX be covering and let ξ ∈ Hom(R,F ). Consider R ⊂ hX , the collection of all Y → X such that ξ R × h is induced from an element hX Y of F (Y ). (This element is necessarily unique if it exists since F is a separated presheaf.) + By deﬁnition of F , for any Y ∈ R, there is a covering sieve RY ⊂ hY such + 0 that ξ : hY → F is induced from a map RY → F . In particular, R ×hX hY hY 0 contains RY for all Y ∈ R. But RY ⊂ hY and R ⊂ hX are both covering so R is covering. 98 CHAPTER 5. THE ETALE´ TOPOLOGY

0 0 Replacing R by R ∩ R , we ﬁnd that ξ R0 factors through F . (This uses the uniqueness of the factorization on each Y ∈ R0.) By deﬁnition of F +, this provides a map η : hX → F .

We still have to check that η R = ξ. That is, we have to check that η(Y ) = ξ(Y ) for each Y ∈ R. But we know η(Y ) = ξ(Y ) for each Y ∈ R0 and R0 ⊂ R is covering, so we obtain η = ξ because F + is separated. Corollary 33.5.1. If F is a presheaf then F ++ is a sheaf. Definition 33.6. The functor F 7→ F ++ is denoted F sh and called sheafification. Proposition 33.7. Sheafification is left adjoint to the inclusion of sheaves in presheaves. Proof. Observe that F → F + is an isomorphism if F is a sheaf. Proposition 33.8. Sheafification is exact and preserves arbitrary colimits. Proof. It is a filtered direct limit. Proposition 33.9. The category of sheaves on a site admits arbitrary limits and colimits. Proof. For limits, calculate the limit as a presheaf and observe it is a sheaf. For colmits, calculate the colimit as a presheaf and then sheafify.

34 Fpqc descent

Let A be a commutative ring, M an A-module, and X = Spec A. We construct a presheaf Mf on fpqc(X) by deﬁning Mf(Spec B) = B ⊗A M. We write OX = Ae. Notice that Mf is a presheaf of OX -modules.

Deﬁnition 34.1. We will call a presheaf F of OX -modules quasi-coherent if for any map U = Spec C → V = Spec B we the induced map C ⊗B F (B) → F (C) is an isomorphism. Proposition 34.2. Mf is a sheaf on fpqc(X). Recall that this means ∼ Mf(U) ←− Hom(hU , Mf) → Hom(R, Mf) is a bijection when R is a covering sieve of U. If U = Spec B and we replace X with U and M with B ⊗A M then we can assume without loss of generality that U = X.

Lemma 34.3. Suppose that a covering sieve R of X is generated by U1,...,Un. Then Y Y Hom(R,F ) = lim F (U ) F (U ) ←− i ⇒ ij i i,j where Uij = Ui ×X Uj. 34. FPQC DESCENT 99

Corollary 34.3.1. Let R be a finitely generated sieve of an affine scheme X = Spec A and let Y = Spec B be a flat, affine scheme over X. Then for any quasi-coherent presheaf of OX -modules F on X, the natural map

Hom(R × hY ,F ) ← B ⊗ Hom(R,F ) hX A is an isomorphism. Proof. Compute: Y Y Hom(R × hY ,F ) = lim F (Ui × Y ) ⇒ F (Uij × Y ) h ←− X X X i i,j Y Y = lim F (Bi ⊗ C) ⇒ F (Bij ⊗ C) ←− A A i i,j Y Y = lim C ⊗ F (Bi) ⇒ C ⊗ F (Bij) ←− A A i i,j Y Y = C ⊗ lim F (Bi) ⇒ F (Bij) A ←− i i,j = C ⊗ Hom(R,F ) A

Corollary 34.3.2. Let X = Spec A and let M be an A-module. Suppose that R is a sieve of X that is generated by a ﬁnite, faithfully ﬂat collection of A-algebras Bi. Then Hom(R, Mf) = M (via the natural map).

Proof. We have a map M = Hom(hX , Mf) → Hom(R, Mf) that we wish to show is an isomorphism. It is suﬃcient to prove that the maps

Bi ⊗ M → Bi ⊗ Hom(R, Mf) A A are all isomorphisms since the Bi are faithfully ﬂat. Set Ui = Spec Bi. The map above is equivalent to the one below (by the earlier corollary):

Bi ⊗ M → Hom(R × hUi , Mf) A hX

But R ×hX hUi = hUi since R ×hX hUi contains idUi . Therefore the map above is an isomorphism by Yoneda’s lemma.

Corollary 34.3.3. If M is an A-module then Mf is an fpqc sheaf on Spec A. 0 Proof. Suppose R ⊂ hX is a covering sieve. Then R contains a sieve R that is generated by a finite, faithfully flat collection of A-algebras. We have just seen that M → Hom(R0, Mf) is an isomorphism. Now we show that Hom(R, Mf) → Hom(R0, Mf) is an isomorphism. By definition, Hom(R, M) = lim M(Y ). f ←− f Y ∈R 100 CHAPTER 5. THE ETALE´ TOPOLOGY

We can also write

Hom(R0, M) = lim Hom(R0 × Y, M). f ←− f Y ∈R X

0 But R ×X Y is generated by a finite, faithfully flat collection of affine Y - schemes. Therefore the map

Mf(Y ) → Hom(R0 × Y, Mf) X is a bijection. Passing to the limit completes the proof. In fact, a stronger result is true: Theorem 34.4. Let A be a commutative ring and X = Spec A. The functor

M 7→ Mf R is an equivalence of categories between A-Mod and QCoh(R) for any fpqc covering sieve R of X. Proof. The inverse functor sends F to Hom(R,F ). The discussion above showed that Hom(R, Mf) = M, which proves that these functors are inverse in one direction. To go the other way, consider Hom(R,F )∼ → F . Evaluating on an A-algebra B, we get B ⊗A Hom(R,F ) → F (B) = B ⊗A F (A), which is an isomorphism because F (A) → Hom(R,F ) is an isomorphism.

35 A rapid review of scheme theory 35.1 A heuristic introduction Atlases A scheme is a space modelled locally on the set of solutions to a system of polynomial equations. The solutions to a system of equations can be encoded in a commutative ring: If xi, i ∈ I is a system of variables and fj, j ∈ J is a system of equations, then the commutative ring

Z[xi i ∈ I]/(fj j ∈ J) is the universal commutative ring with solutions to the fj. Every commutative ring arises this way, for sufficiently many variables and equations. Thus, for every commutative ring A, we have a scheme Spec A. These schemes are called the affine schemes. A homomorphism of commutative rings A → B gives a morphism of affine schemes Spec B → Spec A. This morphism is called a distinguished open embedding if there is some f ∈ A such that B is isomorphic, as an A-algebra, to A[f −1]. A scheme is assembled from affine schemes, glued along open embeddings. A scheme can thus be described as a collection of affine open subschemes Spec Ai, 35. A RAPID REVIEW OF SCHEME THEORY 101 with i drawn from some partiallly ordered indexing set I, along with distinguished open embeddings

Spec Ai → Spec Aj whenever i ≤ j. The one condition is that the diagrams

Spec Ai / Spec Aj

% y Spec Ak commute whenever i ≤ j ≤ k. The collection of Spec Ai is called at atlas for the scheme.

Points

Suppose X is the scheme above. Let k be a ﬁeld. The k-points of X are

X(k) = lim Hom(A , k). −→ i i∈I

In other words, the k-points of X are obtained by gluing together the k-points of the Spec Ai in the atlas.

Schematic points

Equivalence of atlases

0 Let Spec Ai, i ∈ I be an atlas for a scheme X and I ⊂ I a subset. Then 0 0 0 Spec Ai, i ∈ I is an atlas for a scheme X . We say that I is a refinement of I if the map |X0| → |X| is a bijection. Refinement gives a partial order among atlases for schemes. We call an atlas maximal if it is maximal with respect to refinement. 102 CHAPTER 5. THE ETALE´ TOPOLOGY

Morphisms of schemes Let A and B be commutative rings. A morphism of aﬃne schemes

Spec A → Spec B

is a homomorphism of commutative rings B → A. Thus the category of affine schemes is exactly the opposite of the category of commutative rings. Suppose X and Y are schemes with maximal atlases Spec Ai, i ∈ I and 0 Spec Bj, j ∈ J, respectively. A morphism from X to Y is a refinement I of I, a map ϕ : I0 → J of partially ordered sets, and morphisms of affine schemes

Spec Ai → Spec Bϕ(i)

for each i ∈ I0 subject to the condition that, whenever i ≤ j are elements of I0, the following diagram is commutative:

Spec Ai / Spec Aj

Spec Bϕ(i) / Spec Bϕ(j).

Finite presentation and finite type A morphism of commutative rings B → A is said to be of finite type if A is finitely generated as a B-algebra. It is said to be of finite presentation if the ideal of relations among finitely many generators of A is finitely generated. That is, A is of finite type if there is a surjection

B[t1, . . . , tn] → A.

It is of ﬁnite presentation if there is an isomorphism

∼ B[t1, . . . , tn]/(f1, . . . , fm) −→ A.

A morphism of schemes f : X → Y is said to be locally of finite type (resp. locally of finite presentation) if it has charts by morphisms Spec A → Spec B where A is a B-algebra of finite type (resp. finite presentation). If in addition f 1 → is quasi-compact1 then f is said to be of finite type (resp. of finite presentation).

35.2 Schemes as functors An aﬃne scheme is the space of solutions to a system of polynomial equations. Note that this deﬁnition does not specify where the solutions are supposed to come from. In fact, we consider all solutions in all commutative rings at once.

1todo: deﬁne 35. A RAPID REVIEW OF SCHEME THEORY 103

If J is a system of polynomial equations in variables X then the universal system of solutions to the f ∈ J is the ring

A = Z[X]/J

If we want to extract the solutions to these equations in any commutative ring B, we simply compute HomComRng(A, B). Another way to think about the solution set of a system of equations is as a functor. Let F : ComRng → Sets be the functor I F (B) = ξ ∈ B f(ξ) = 0 for all f ∈ J . Then the commutative ring A deﬁned above represents the functor F . We think of F as the abstract “solution space” of the system of equations J in the variables I.

Deﬁnition 35.1. An aﬃne scheme is a functor ComRng → Sets that is representable by a commutative ring.

We write Spec A for the functor represented by a commutative ring A. Let I be a subset of A and deﬁne U(B) = ϕ ∈ Hom(A, B) ϕ(I)B = B .

By definition, U is a subfunctor of hA. We will denote it U(I). By definition, all such subfunctors are called open. We may also define V (B) = ϕ ∈ Hom(A, B) ϕ(I) = 0 .

Note that V ' hA/I . This is another subfunctor of hA. We often denote it V (I). Subfunctors of this form are called closed. The two subfunctors U and V deﬁned above are said to be complementary, since U(k) ∩ V (k) = ∅ and U(k) ∪ V (k) = hA(k) when k is a ﬁeld. Remark 35.2. 1) It is not true that U(B) ∪ V (B) = hA(B) for an arbitrary commutative ring B. Exercise: Give an example.

2) Even though U and V are complementary, it is not the case that each can be recovered from the other. It is possible to get U from V because U(B) may be identiﬁed with the set of ϕ : A → B such that B ⊗A A/I = 0. Exercise: verify this. However, it is usually not possible to recover V (I) from U(I). Exercise: check that U(I) = U(In) but V (I) 6= V (In) in general. In general, a subfunctor U ⊂ X is called open if, for any map ϕ : hA → X, the pullback ϕ−1(U) ⊂ hA is open. Likewise, V ⊂ X is called closed if, for any map ϕ : hA → X, the pullback ϕ−1(V ) ⊂ hA is closed. 104 CHAPTER 5. THE ETALE´ TOPOLOGY

Definition 35.3. A prescheme2 is a functor ComRng → Sets that has an open cover by affine schemes. A prescheme X is called a scheme if it satisfies the following two conditions:

(i) Suppose that U = Spec A is covered by open aﬃne subschemes Ui =

Spec Ai and ξ, η ∈ X(A). If ξ = η for every i then ξ = η. Ui Ui

(ii) Suppose that U = Spec A is covered by open aﬃne subschemes Ui = Spec Ai and for each pair i, j the intersection Ui ∩ Uj is covered by open

affine subschemes Uijk. If ξi ∈ X(Ai) are such that ξi U = ξj U for ijk ijk every i, j, and k then there is a ξ ∈ X(A) such that ξ = ξi for all i. Ui If X is a scheme then for any field k, the set X(k) has the structure of a topological space. The open sets are the U(k) where U is an open subfunctor of X. Definition 35.4. The underlying topological space of a scheme is the set |X| of all isomorphism classes of injective maps hk → X where k is a field. If U ⊂ X is an open subfunctor then |U| ⊂ |X| is called open in the Zariski topology. Proposition 35.5. Suppose X = hA. The points of |X| are in bijection with the prime ideals of A. Proof. In one direction, this is not hard to prove: Given a prime ideal p ⊂ A, the quotient A/p is an integral domain, hence may be embedded in a field k = fract(A/p). The maps A → A/p → fract(A/p) = k give hk → hA. Moreover A → k is an epimorphism, so hk → hA is an injection (by definition). Conversely, suppose that A → K is an epimorphism. Let p ⊂ A be the kernel. We want to show that k = fract(A/p) coincides with K. Consider the ring K ⊗k K. Since k → K is an epimorphism, K ⊗k K = K. But we have

dimK K ⊗ K = dimk K k from which we obtain K = k. By a scheme over a base scheme S we mean a scheme X equipped with a morphism X → S. Definition 35.6. Let f : X → Y be a morphism of schemes. We say that f is étaleif it is locally of finite presentation and the diagram S / X > f S0 / Y 2This is not standard usage of the term prescheme! What are today called schemes were originally called preschemes. As a consequence, the term “prescheme” has the potential to be confusing. Don’t tell anyone we’re using it. 36. THE ETALE´ TOPOLOGY 105 admits a unique lift whenever S ⊂ S0 is an infinitesimal extension of affine schemes.

Regarding X and Y as functors, this means that whenever A0 → A is a nilpotent extension of commutative rings and we have ξ ∈ X(A) and η0 ∈ Y (A0) having the same image in Y (A) there is a ξ0 ∈ X(A0) whose image in X(A) is ξ and whose image in Y (A0) is η0.

36 The étaletopology 37 Henselization 38 The étalefundamental group 38.1 Covering spaces Let F be an étalesheaf over X. We say that X satisfies the valuative criterion for properness if, whenevery Y is the spectrum of a valuation ring with closed point y, and any map f : Y → X, the map

∗ ∗ Γ(Y, f F ) → Γ(y, f Fy) is bijective.

Exercise 38.1. Verify that when F = W sh is the sheaf associated to an ´etale map W → X that this condition coincides with the familiar valuative criterion.

Definition 38.2. Let X be a locally noetherian scheme. An étalecover of X is an étalesheaf F on X that satisfies the valuative criterion of properness. We use cov(X) (or ét-cov(X), if it is necessary to emphasize the étaletopology) to denote the category of étalecovers of X.

Proposition 38.3. Recall that X is assumed locally noetherian. The category cov(X) has the following properties:

1. Every object Y ∈ cov(X) is representable by a scheme.

2. Every object Y ∈ cov(X) is locally noetherian.

3. For any pair of geometric points ξ and η of Y ∈ cov(X), there is a ﬁnite chain of specializations and generizations connecting ξ to η.

4. The category cov(X) has all ﬁnite limits and all small colimits and the inclusion cov(X) ⊂ ´et(X) preserves these.

5. Every object of cov(X) is a disjoint union of connected objects.

6. The category cov(X) is generated under colimits by a set. 106 CHAPTER 5. THE ETALE´ TOPOLOGY

Proof. 1. 3 ← 3 2. In fact, every Y ∈ ´et(X) is locally noetherian since X is locally noetherian. 3. This certainly holds for any noetherian scheme, so it is suﬃcient to shoow that every pair of geometric points of Y is contained in some quasi-compact open subset of Y . Cover Y by noetherian open subschemes Yi

38.2 Locally constant sheaves Let X be a scheme and ét(X) its étalesite. A sheaf F on ét(X) is said to be

locally constant if the collection of étalemaps U → X such that F U is constant forms an étalecover of X. Let lcét(X) be the category of locally constant sheaves on ét(X). Proposition 38.4. 1. lcét(X) is closed under finite limits and colimits.

2. If f : Y → X is a morphism of schemes then f ∗ : ét(X) → ét(Y ) restricts to an exact functor lcét(X) → lcét(Y ). 4 → 3. If ξ is a geometric point4 of X then restriction to ξ defines an exact functor lcét(X) → Sets. 4. Coproducts in lcét(X) are disjoint.

Define π1(X, ξ) to be the automorphism group of the restriction functor ξ∗ : lcét(X) → Sets. Proposition 38.5. A locally constant sheaf satisfies the valuative criterion for properness.

Proof. We can reduce immediately to the case of a locally constant sheaf over a valuation ring. By ´etaledescent, we can replace the base with a strict henselization. But then a locally constant sheaf is constant, and the assertion is obvious for constant sheaves.

38.3 The topology on the étalefundamental group ∗ For each sheaf F ∈ lcét(X) we have an action of π1(X, ξ) on the set ξ F . For each F ∈ lcét(X), and each pair of finite subsets S and T of ξ∗F , let U(F, S, T ) be the set of σ ∈ π1(X, ξ) such that σ(S) ⊂ T . The U(F, S, T ) are a basis for the compact-open topology on π1(X, ξ).

3todo: get proof from stacks project 4todo: deﬁne Chapter 6

Abelian categories and derived functors

Reading and references

Grothendieck, A. Sur quelques points d’alg`ebrehomologique. Tˆohoku Math. J. (2) 9 1957, §§1.1–1.2

39 Abelian categories

Deﬁnition 39.1. A category C is called additive if it possesses all ﬁnite products and coproducts and the natural map a Y Xi → Xi i∈I i∈I

is an isomorphism for all finite collections of objects Xi of C . An additive category has a simultaneously initial and final object (the empty coproduct or product, respectively) called the zero object and notated 0. It therefore makes sense to ask for a morphism of C to possess a kernel or cokernel. Definition 39.2. An additive category C is called abelian if it has the following two properties:

AB1 Every morphism in C has a kernel and a cokernel. AB2 If f is a morphism in C then the canonical map coker ker f → ker coker f

is an isomorphism. A number of other axioms are convenient to impose on abelian categories:

107 108 CHAPTER 6. ABELIAN CATEGORIES AND DERIVED FUNCTORS

AB3 C possesses arbitrary coproducts. AB5 Finite limits in C commute with ﬁltered colimits.

40 Resolution and derived functors 40.1 Injective and projective objects Deﬁnition 40.1. Let X be an object of an abelian category C . We say that X is projective if the functor

C → Ab : Y 7→ Hom(X,Y )

is exact. We say that X is injective if the functor

C → Ab : Y 7→ Hom(Y,X)

is exact.

40.2 Complexes Deﬁnition 40.2. A complex in an abelian category C is a graded object K• equipped with a diﬀerential

p p+1 dp : K → K

satisfying dp+1 ◦ dp = 0. Usually the index is omitted and the condition is written d2 = 0. The homology of K• is deﬁned to be

p • H (K ) = ker(dp)/ im(dp−1).

A morphism of complexes f : K• → L• is a homomorphism of graded objects p p with graded pieces fp : K → L satisfying dfp = fp+1d. 1 → 123 2 → 3 → 41 Spectral sequences

1todo: Define internal morphism 2todo: Define quasi-isomorphism 3todo: Define chain homotopy Chapter 7

Cohomology of sheaves

42 Acyclic resolutions 42.1 Injective resolution 42.2 Flaccid (flasque) resolution 42.3 Soft resolution Definition 42.1. A sheaf F (X) on a locally compact Hausdorff topological space X is called soft if, for any compact K ⊂ X, the map F (X) → Γ(K, i∗F ) is surjective.

Lemma 42.2. Flaccid sheaves are soft.

Proof. We have Γ(K, i∗F ) = lim F (U), with the colimit taken over all open −→K⊂U subsets of X containing K. Any section of F over K is therefore induced from some U ⊃ K and can then be lifted to a section of F over X, by the deﬁnition of a ﬂaccid sheaf.

1 Lemma 42.3. Suppose F is soft. Then HK (X,F ) = 0 for any compact K ⊂ X.

1 Proof. HK (X,F ) parameterizes F -torsors trivialized over X r K. Choose a cover of K by ﬁnitely many compact sets Li over which the torsor P is trivial and proceed as in the ﬂaccid case.1 ← 1

Lemma 42.4. The restriction of a soft sheaf to a closed subset is soft.

Lemma 42.5. If the sequence

0 → F → G → H → 0 is exact and F and G are both soft then so is H.

1todo: ﬁnish proof

109 110 CHAPTER 7. COHOMOLOGY OF SHEAVES

Proof. We have an exact sequence

0 → Γ(K,F ) → Γ(K,G) → Γ(K,H) → H1(K,F ).

Since F restricts to a soft sheaf on K, we know that H1(K,F ) is zero. Therefore we have surjections

Γ(X,G) → Γ(K,G) → Γ(K,H).

On the other hand, the composition factors through Γ(X,H).

Lemma 42.6. Soft sheaves have vanishing compactly supported higher cohomology.

Proof. We know this already for H1. Embed a soft sheaf F in a ﬂaccid sheaf G. Then Hp(X,F ) = Hp−1(X, G/F ) for p ≥ 2. But G/F is soft so we are done by induction.

42.4 Partitions of unity and de Rham cohomology 43 Cechˇ cohomology

Let U be a collection of open subsets of X. Let C be the category of their intersections. If F is a sheaf on X we get an induced presheaf on C . The functor H : Sh(X) → Psh(C ) 2 → is left exact. It has an exact left adjoint,23 so it preserves injectives. We 3 → therefore obtain a spectral sequence:

Theorem 43.1. The spectral sequence

lim(p) q(F ) ⇒ Hp+q(X,F ) ←− H converges.

In the statement, we have written H q(F ) for RqH (F ). It is also convenient to write H q(U, F ) rather than H q(F )(U). Note that H q(U, F ) = RqΓ(U, F ).

Corollary 43.1.1. If RqΓ(U, F ) = 0 for 0 < q < n and all U ∈ U then there is an exact sequence

0 → lim(q) (F ) → Hq(X,F ) → lim q(F ) → lim(q+1) (F ) → Hq+1(X,F ). ←− H ←− H ←− H Corollary 43.1.2. Hp(R, Z) vanishes for p > 0.

2todo: explain what the adjoint is 3todo: add reference 44. COMPACTLY SUPPORTED COHOMOLOGY 111

Proof. By induction on p. We know it is true for p = 1 already. Suppose that it is true for p < n. Then suppose α ∈ Hn(R, Z). Choose an open cover of R by intervals U such that α U = 0 for all U in the cover and the triple intersections of the intervals are empty. By induction we have the 5-term exact sequence. We know that the image of α under the map Hq(X,F ) → lim q(F ) ←− H is zero. Therefore α is induced from lim(q) (F ) and it suﬃces to show that ←− H this vanishes. This is obvious for q ≥ 2 since the triple intersections of the U ∈ U are empty.4 ← 4

44 Compactly supported cohomology 44.1 The compactly supported cohomology of the real line The real line R may be compactiﬁed to a circle. Let j : R → S1 be an open embedding. Consider the exact sequence

0 → j!Z → Z → i∗Z → 0 where i is the inclusion of the complement of the image of j. Taking cohomo- mology we get a long exact sequence

0 1 0 1 1 1 1 0 → H (S , Z) → H (S , i∗Z) → Hc (R, Z) → H (S , Z) → 0. Note the following:

0 0 1 (i) Hc (R, Z) = H (S , j!Z) = 0 because a section must have zero stalk at the image of i,

1 1 (ii) H (S , i∗Z) = 0 since i∗Z is a skyscraper sheaf,

0 1 (iii) H (S , i∗Z) = Z, and (iv) we can easily check that

0 1 0 1 Z = H (S , Z) → H (S , i∗Z) = Z

is the identity map.

1 1 1 We conclude that Hc (R, Z) → H (S , Z) = Z is an isomorphism.

45 Proper base change

Deﬁnition 45.1. A morphism of locally compact Hausdorﬀ spaces is called proper if the pre-image of a compact subset of the codomain is compact. Lemma 45.2. Suppose that f : X → Y is proper. Then for any map g : Z → Y ∗ ∗ and any sheaf F on X, the map g f∗F → f∗g F is an isomorphism.

4todo: cleanup presentation 112 CHAPTER 7. COHOMOLOGY OF SHEAVES

Proof. It is suﬃcient to treat the case where Z is a point; assume that and let y = g(Z). We remark that the open neighborhoods f −1(U), for U ⊂ Y an open neighborhood of y, constitute a basis of open neighborhoods of f −1(y), since f −1(y) is compact. Essentially the same argument works to prove that H1 commutes with base change: Lemma 45.3. Suppose that f : X → Y is proper and g : Z → Y is any map. ∗ 1 1 ∗ Then g R f∗F → R f∗g F for any sheaf of groups F on X.

46 Leray spectral sequence

Let f : X → Y be a continuous map. We can functorially identify Γ(X,F ) = Γ(Y, f∗F ). Since f∗ preserves injectives, we get a spectral sequence: Theorem 46.1. The spectral sequence

p q p+q H (Y,R f∗F ) ⇒ H (X,F )

converges.

46.1 K¨unnethformula 46.2 Homotopy invariance of cohomology

Let h : X × I → Y be a homotopy between f and g. Let i0 and i1 be the inclusions of X ×0 and X ×1 in X ×I. Let q : X ×I → X be the projection. By the K¨unnethformula, the map q∗ : H∗(X,A) → H∗(X, q∗A) is an isomorphism. ∗ ∗ Since q ◦ i0 = q ◦ i1 we deduce that i0 and i1 induce the same isomorphism H∗(X × I,A) → H∗(X,A). Therefore we have

∗ ∗ ∗ ∗ ∗ ∗ f = i0h = i1h = g as maps H∗(Y,A) → H∗(X,A). Theorem 46.2. Homotopic maps induce the same map on cohomology with 5 → locally constant coeﬃcients.5 Corollary 46.2.1. If X is contractible then ( A p = 0 Hp(X,A) = 0 else

for any locally constant coeﬃcient sheaf A. In particular, this computes the cohomology of Rn.

5todo: explain that pullback of locally constant coeﬃcient sheaf from Y is canonically homotopy independent 46. LERAY SPECTRAL SEQUENCE 113

46.3 The cohomology of spheres Consider the two closed discs D,E ⊂ Sn with intersection homeomorphic to Sn−1. Then we have a short exact sequence

0 → Z → iD∗Z × iE∗Z → iD∩E∗Z → 0.

The long exact sequence is

Hp(D, Z)×Hp(E, Z) → Hp(Sn−1, Z) → Hp+1(Sn, Z) → Hp+1(D, Z)×Hp+1(E, Z).

The terms on the end vanish for p > 0 so

Hp(Sn, Z) ' Hp−1(Sn−1, Z)

n 1 n for p > 1. Since π1(S ) = 0 for n > 1 we know H (S , Z) = 0 for n > 1, so this gives ( Z p = 0, n Hp(Sn, Z) = . 0 else

46.4 The cohomology of complex projective space First calculation Consider the inclusion i : CPn−1 → CPn with complement j : Cn → CPn. We get an exact sequence

0 → j!Z → Z → i∗Z → 0 hence a long exact sequence

p n p n p n−1 p+1 n Hc (C , Z) → H (CP , Z) → H (CP , Z) → Hc (C , Z).

p n Since Hc (C , Z) = 0 for p 6= 2n, we get

Hp(CPn, Z) ' Hp(CPn−1, Z) p 6= 2n − 1, 2n.

We also have the exact sequences

2n−1 n 2n−1 n 2n−1 n−1 Hc (C , Z) → H (CP , Z) → H (CP , Z) 2n−1 n−1 2n n 2n n 2n n−1 H (CP , Z) → Hc (C , Z) → H (CP , Z) → H (CP , Z)

The extremal terms in both lines vanish6 so we get ← 6 ( 0 p = 2n − 1 Hp(CPn, Z) = Z p = 2n.

6todo: references, explain 114 CHAPTER 7. COHOMOLOGY OF SHEAVES

Second calculation We have a map f : S2n+1 → CPn. We get a spectral sequence

p n q p+q H (CP ,R f∗Z) ⇒ H (X, Z).

q By proper base change, we know that R f∗Z = 0 for q 6= 0, 1 and f∗Z = Z and 1 n R f∗Z is a locally constant sheaf with ﬁbers isomorphic to Z. But π1(CP ) = 0 so the only such sheaf must be Z. This means the E2 of the spectral sequence must look like

H0(CPn, Z) H1(CPn, Z) H2(CPn, Z) ···

+ H0(CPn, Z) H1(CPn, Z) H2(CPn, Z) + ···

and the sequence degenerates on the E3 page. Since the sequence converges to Hp(S2n+1, Z) and this vanishes for all 0 < p < 2n + 1, we deduce that the maps

Hp(CPn, Z) → Hp+2(CPn, Z)

7 → are isomorphisms for p 6= 2n. 7

47 Poincar´eduality

Let X be a topological space, U ∈ U a collection of open subsets of X. We obtain an exact sequence a a · · · → iU∩V !Z → iU !Z → Z → 0. (U,V )∈U 2 U∈U

Taking compactly supported cohomology gives a spectral sequence converging to the compactly supported cohomology of X.

48 Lefschetz ﬁxed point theorem

7todo: we are done if we can show that Hp(CPn, Z) vanishes for p > 2n. Chapter 8

Grothendieck topologies

49 Sieves, functors, and sheaves

Let C be a category and X an object of C . We may define the functor represented by X: hX : C → Sets : Y 7→ HomC (Y,X). Definition 49.1. Let X be an object of C .A sieve of X is a subfunctor of 1 hX . Definition 49.2. A Grothendieck topology on a category C is a collection of sieves R ⊂ hX —called the covering sieves of the topology—satisfying the following conditions:

T1 For each X, the ﬁnal sieve hX ⊂ hX is covering.

T2 If Y → X is a morphism of C then R ×X Y covers Y .

2 T3 If R ⊂ S ⊂ hX are sieves and R is covering then S is covering. Deﬁnition 49.3. A presheaf on the category C is a functor F : C ◦ → Sets.A presheaf is said to be a sheaf if in a given Grothendieck topology if, whenever R is a covering sieve of X in that topology, the natural map

F (X) → HomPsh(C )(R,F ) is a bijection.3

1Equivalently, it is a ﬁbered subcategory of the ﬁbered category represented by X. 2 We say that R ⊂ S is covering if, for each Y ∈ S, the sieve RY = R ×S Y ⊂ hY is covering. 3Looking forward to stacks, it is somewhat preferable to write this condition this way:

F (X) ← HomPsh(C )(hX ,F ) → HomPsh(C )(R,F ).

Note that the map F (X) ← HomPsh(C )(hX ,F ) is automatically a bijection when F is a presheaf. When F is merely a ﬁbered category, this map is not a bijection but still an equivalence of categories.

115 116 CHAPTER 8. GROTHENDIECK TOPOLOGIES

Local properties Let C be a site with ﬁnal object X. Consider a property P of objects of C such that if Y → Z is a morphism of C then P (Z) implies P (Y ). Thus P is a sieve of C . We say that P holds locally in C if P is a covering sieve of X.

50 Morphisms of sites

A morphism of sites C → D is a pair of functors

f ∗ : D → C

f∗ : C → D

∗ ∗ with f left exact and f∗ right adjoint to f . What kind of information do we need to specify such a morphism? A correspondence from C to D is a functor

F : C ◦ × D → Sets.

Suppose that R is a sieve of Y ∈ D. If X ∈ C and f ∈ F (X,Y ) then we can deﬁne f ∗R to be the sieve S of X consisting of those u : X0 → X such that 0 0 0 ∗ there is some v : Y → Y in R and some g ∈ F (X ,Y ) with v∗g = u f. We can visualize this with a diagram:

g X0 / Y 0 (50.1)

u v f X / Y Take care to remember, though, that X and Y are objects of diﬀerent cate- 5 → gories.45 We say that F is a continuous correspondence if, whenever R is a covering sieve of Y , the pullback f ∗R is a covering sieve of X.

Let X be an object of C . Define a category F X/D (or maybe X/F D?) whose objects are pairs (Y, f) where Y ∈ D and f ∈ F (X,Y ). For varying X we can regard F X/D as a fibered category over C . We say that F has local pro-pushforwards if this fibered category is locally filtered. To say F X/D is locally filtered means that the characteristic properties of a filtered category are required to hold after suitable localization:

LFil1 Let Y,Z be objects of F X/D. Let R ⊂ hX be the sieve consisting of all u : X0 → X such that there is an object W of FX0/D and maps W → u∗Y and W → u∗Z. Then R covers X.

4In fact, we can make diagram (50.1) a legitimate diagram in a category F with Ob(F ) = Ob(C ) q Ob(D) and HomF (X,Y ) = F (X,Y ) for X ∈ C and Y ∈ D. Then the deﬁnition of f ∗R is simply the pullback of the sieve R in the usual sense. 5todo: explain intuition from topological spaces 51. GROUP COHOMOLOGY 117

LFil2 Let f, g : Y → Z be two morphisms of F X/D. Let R ⊂ hX be the sieve consisting of all u : X0 → X such that there is an object W of FX0/D and a map h : W → u∗Y equalizing u∗f and u∗g. Theorem 50.1. Every morphism of sites Sh(C ) → Sh(D) is induced from a continuous correspondence with local pro-pushforwards from C to D.6 ∗ ∗ Proof. Suppose that (f , f∗) is a morpshism of sites. Let F (X,Y ) = HomSh(C )((X), f (Y )).

51 Group cohomology 52 (*) Cohomology in other algebraic categories 52.1 Groups 52.2 Rings 52.3 Commutative rings 52.4 Hyper-Cechˇ cohomology Let C be a site and D a diagram in C , i.e., a functor D : D → C . Giving D the chaotic topology, it is clear that D takes covers to covers. If we assume moreover that D admits a local pro-adjoint, we obtain a morphism of sites Π: Sh(C ) → Sh(D) and with it a Leray spectral sequence p q p+q H (D,R Π∗F ) ⇒ H (C ,F ). Proposition 52.1. Given α ∈ Hp(C ,F ), there is a diagram D as above such that α lies in the image of the edge homomoprhism p p H (D, Π∗F ) → H (C ,F ). 0 p Proof. Consider ﬁrst the image of α in H (D,R Π∗F ). We choose a cover of

the final sheaf of C such that α U = 0 for all U in the cover. Our first guess for D will consist of all products of all objects U in the cover. p 1 p−1 The next step in the filtration of H (C ,F ) is a subquotient of H (D,R Π∗F ). 1 We can realize H (D,G) as a subquotient of G(U1). We can therefore represent p p−1 the image of α in F1H (C ,F ) by a class in H (U1,F ). Replace U1 by an étalecover over which this class is trivial. p By the same argument, we can now represent the image of α in F2H (C ,F ) p−2 by a class in H (U2,F ). Replace U2 by a cover trivializing this class. We can p repeat this argument until we have represented the image of α in FpH (C ,F ) = p H (D, Π∗F ). 6Note that the correspondence is not unique, though it is unique up to a unique isomorphism in a suitable sense. 118 CHAPTER 8. GROTHENDIECK TOPOLOGIES

53 Fibered categories and stacks Chapter 9

Schemes

54 Solution sets as functors 55 Solution sets as spaces 56 Quasi-coherent modules

119 120 CHAPTER 9. SCHEMES Chapter 10

Properties of schemes

57 Flatness 58 Smooth, unramified, and étalemorphisms 59 (*) Weakly étalemorphisms

121 122 CHAPTER 10. PROPERTIES OF SCHEMES Chapter 11

Curves

60 Riemann surfaces 61 Riemann–Roch 62 Serre duality 63 The Jacobian

123 124 CHAPTER 11. CURVES Chapter 12

Abelian varieties

64 Lattices in complex vector spaces 65 The dual abelian variety 66 Geometric class ﬁeld theory

125 126 CHAPTER 12. ABELIAN VARIETIES Chapter 13

Topologies on schemes

67 Faithfully flat descent 68 The étaletopology 69 Other topologies 69.1 The Zariski topology 69.2 (*) The pro-étale topology 69.3 (*) The infinitesimal site

127 128 CHAPTER 13. TOPOLOGIES ON SCHEMES Chapter 14

Etale cohomology

70 Constructible sheaves and `-adic cohomology 71 Etale cohomology in low degrees 72 Etale cohomology and colimits

Consider a ﬁltered system of commutative rings A for i ∈ I. Let A = lim A . i −→ i Suppose given a compatible system of Ai-schemes Xi, with

Xj = Xi ⊗ Aj Ai for all i and j. Then set X = X ⊗ A = lim X . Suppose that F , i ∈ I are a i Ai ←− i i compatible system of étalesheaves on the Xi. Then there is a map lim µ∗Hp(X ,F ) → Hp(X,F ). (72.1) −→ i i i We aim to show it is an isomorphism in reasonable circumstances. Lemma 72.1. If F is representable by an algebraic space locally of finite presentation then (??) is an isomorphism for p = 0. Proof. This is true by definition of local finite presentation. Corollary 72.1.1. If F is constructible then (??) is an isomorphism for p = 0. Lemma 72.2. If F is representable by an algebraic space in groups that is locally of finite presentation then (??) is an isomorphism for p = 1. 1 Proof. Use the interpretation of H (Xi,Fi) as Fi-torsors on Xi. An Fi-torsor is locally isomorphic to Fi, hence is locally of finite presentation. It follows that the underlying space of an F -torsor P is induced from some étalespace Pi over Xi. Similarly, the action of F is induced from the action of some Fi; that P is a torsor is equivalent to the map F × P → P × P being an isomorphism, which can be verified over a sufficiently large i.

129 130 CHAPTER 14. ETALE COHOMOLOGY

Using the ideas highlighted above, we can show that any finite diagram of étaleschemes over X is induced from a finite diagram over Xi for some i. We 1 → know1 that any class in Hp(X,F ) can be represented in Hp( ,F ) for some D D diagram D in ét(X). This diagram is induced from some diagram Di in ét(Xi) p p for some i. A class in H (D,F ) is induced from a class in H (Di,Fi ) for D Di sufficiently large i (essentially uniquely). This gives us the surjectivity of (72.1) for all p. p Now we check injectivity. Consider a class α ∈ H (Xi,Fi). We can represent p ∗ α by a class β in H (Di,F ) for some diagram Di in ét(Xi). Then µ α is Di i represented by µ∗β in Hp( ,F ) where = µ∗ . But the map i D D D i Di Hp( ,F ) → Hp(X,F ) D D ∗ is injective, so µi β = 0. Therefore it is the image of a p − 1 cochain γ of F over D. Since the objects of D are coherent, this is induced (by the p = 0 case) from a cochain γi of Fi over Di for i sufficiently large. Furthermore, dγ = β implies dγi = βi for i sufficiently large. This proves the injectivity of (72.1).

73 Cup product

1todo: reference Chapter 15

Etale cohomology of points

74 Group cohomology

75 Hilbert’s theorem 90

1 We know that H (ét(X), Gm) classifies Gm-torsors on the étalesite of X. But we can identify the stack of Gm-torsors with the stack of invertible sheaves of Oét(X)-modules. We know that quasi-coherent OX -modules satisfy étale descent,1 so that ← 1

1 1 H (´et(X), Gm) = H (zar(X), Gm).

In particular, we have Hilbert’s theorem 90:

1 Theorem 75.1. Let X = Spec k with k a ﬁeld. Then H (´et(X), Gm) = 0.

1 1 Proof. We have H (´et(X), Gm) = H (zar(X), Gm) = 0 since zar(X) is equivalent to the category of sheaves on a point.

76 The Brauer group

77 Tsen’s theorem

Definition 77.1. A field k is said to be C1 or quasi-algebraically closed if, whenever f is a homogeneous polynomial of degree d in n variables with coefficients in k and n > d, there is some x ∈ kn with f(x) = 0.

Theorem 77.2 (Tsen). Suppose k is algebraically closed and K has transcendence degree 1 over k. Then K is C1.

1todo: reference to faithfully ﬂat descent

131 132 CHAPTER 15. ETALE COHOMOLOGY OF POINTS

Proof. Let f be a homogeneous polynomial of degree d in n variables, with coefficients in K. View K as the function field of a smooth curve X. The coefficients of f may be regarded as rational functions on X. They will be regular except at a finite set of points. Choose an effective divisor H so that the coefficients of f lie in OX (H). We will show that we can find a zero (α1, . . . , αn) of f in K with poles only along the support of H. Indeed, consider α1, . . . , αn ∈ OX (eH) for some n positive integer e. The αi are chosen from a vector space Γ(X, OX (eH)) so that we have a polynomial map of k-vector spaces

n ϕ : Γ(X, OX (eH)) → Γ(X, OX ((de + 1)H)) : (α1, . . . , αn) 7→ f(α1, . . . , αn). We calculate the dimensions of these vector spaces. By Riemann–Roch, for large e, we have

n dim Γ(X, OX (eH)) = n dim Γ(X, OX (eH)) = n(1 − g + e deg(H)) dim Γ((de + 1)H) = 1 − g + (de + 1) deg(H). Since n > d, the first line is larger than the second provided e is taken large enough. Viewing ϕ as a polynomial function with coefficients in an algebraically closed field, it has more variables than constraints. Since it has at least one solution—the trivial one, (α1, . . . , αn) = 0—it must have a nontrivial one as well 2 2 → (by the Nullstellensatz!). That is, there is a non-zero choice of α1, . . . , αn ∈ Γ(X, OX (eH)) ⊂ K such that f(α1, . . . , αn) = 0.

2 Corollary 77.2.1. If K is a C1 field then H (K, Gm) = 0. Proof. Let D be a division algebra with center K. The dimension of D is n2. It has a determinant function det : D → K, which we may regard as a poynomial of degree n in n2 variables. If n > 1 then it must have a nontrivial zero. Thus D contains a nonzero element α with det(α) = 0. But det is multiplicative, so α cannot have a multiplicative inverse. We conclude that n = 1, i.e., that D = K. Let f : X → Y be a finite morphism of schemes and F a sheaf of groups on Y . We have a map f ∗ : Hp(Y,F ) → Hp(X, f ∗F ). ∗ On the other hand, we can also construct a map f∗f F → F . For now, we will just construct this in the case of interest, namely where f is finite étale. We will construct the map étalelocally in Y , taking advantage of the fact that pushforward commutes with étalebase change in Y . After an étalebase change 0 0 0 0 0 Y → Y (set X = X ×Y Y ) we can assume that X ' S × Y for some finite ∗ S set S. In that case, f∗f F ' F and we have a summation map

∗ S X f∗f F ' F → F :(xs)s∈S 7→ xs. s∈S

2todo: add reference 77. TSEN’S THEOREM 133

p The same argument also proves that f∗ is exact, so that R f∗ = 0 for p > 0. Putting these together, we have

∗ F → f∗f F → F.

Analyzing this map locally, as before, we obtain

∗ ∗ Lemma 77.3. The induced map Γ(Y, f∗f F ) = Γ(X, f F ) → Γ(Y,F ) is multiplication by n, where n is the degree of f. Corollary 77.3.1. Let f : X → Y be a finite étalemap of degree n and F an m-torsion sheaf on Y . If n and m are coprime then the map Hp(Y,F ) → Hp(X, f ∗F ) is injective. Proposition 77.4. Let K be a field of transcendence degree 1 over an algebraically closed field k. Let ` be a prime. Suppose F is a Z/`nZ-module over ét(K). Then Hp(K,F ) = 0 for p ≥ 2. Proof. From the Kummer sequence

1 → µ` → Gm → Gm → 1 a piece of whose long exact sequence is

1 2 2 H (K, Gm) → H (K, µ`) → H (K, Gm)

2 we deduce that H (K, µ`) = 0. If K ⊂ L is a separable extension with [L : K] prime to ` then H2(K,F ) ⊂ H2(L, F ). Let L0 be the union of all such extensions (within a fixed separable closure of K). Then H2(L0,F ) = lim H2(L, F ) so that H2(K,F ) ⊂ H2(L0,F ). −→L We can therefore replace K with L0 and assume that all polynomials over K of degree prime to ` have degree 1. Now let L be the separable closure of K and denote by f the map Spec L → Spec K. We show first that H2(K,F ) = 0 for all ind-`-power torsion modules F . By compatibility with lim,3 we can assume F is finite. Then F has a ← 3 −→ 4 composition series of simple ét(K)-modules. But the only such is Z/`Z ' µ`. Therefore H2(K,F ) = 0. ← 4 ∗ Now consider the embedding F → f∗f F . The latter is acyclic, since we have p q ∗ p+q ∗ p a spectral sequence H (K,R f∗f F ) ⇒ H (L, f F ). We have H (L, F ) = 0 q for p > 0 since L is separably closed, and, for the same reason, R f∗ = 0 for q > 0. We therefore get

p p−1 ∗ H (K,F ) ' H (K, f∗f F/F )

∗ ∗ for p ≥ 2. But f∗f F is ind-`-power torsion, so the same applies to f∗f F/F . Hence by induction Hp(K,F ) = 0 for p ≥ 2.

3todo: write proof 4todo: needs justiﬁcation 134 CHAPTER 15. ETALE COHOMOLOGY OF POINTS Chapter 16

Etale cohomology of curves

78 Calculation 79 Poincar´eduality for curves

135 136 CHAPTER 16. ETALE COHOMOLOGY OF CURVES Chapter 17

Base change theorems

80 Smooth base change 80.1 Auslander–Buchsbaum formula Let R be a noetherian local ring with maximal ideal m and let M be an R- module of finite type. Definition 80.1. The projective dimension of M is the minimal length of a projective resolution of M. The depth of M is the minimal n such that Extn(R/m, M) 6= 0. Theorem 80.2 (Auslander–Buchsbaum formula). Let M be a finite type R- module. Then pd(M) + depth(M) = depth(R). Lemma 80.3. The formula is true if M is projective. Proof. Since a finite type projective R-module is free, this reduces to the case M = R, where it’s obvious.

Consider a projective resolution P• of M of length e. Let d = depth(R). We have a spectral sequence

p p−q Ext (R/m, Pq) ⇒ Ext (R/m, M).

p n We know that Ext (R/m, Pq) = 0 for p < d and q > e. Therefore Ext (R/m, M) = 0 for n < d − e. Therefore pd(M) ≥ d − e. Since e ≥ pd(M) we get pd(M) + depth(M) ≥ depth(R). To prove the reverse inequality, we have to show that

d d Ext (R/m, Pe) → Ext (R/m, Pe−1) (80.1) is not injective. Since Pe is free, we can study its summands individually. Consider a map R → Rn. If the matrix entries of the map are not all drawn

137 138 CHAPTER 17. BASE CHANGE THEOREMS

from m then the map embeds R as a direct summand of Rn. If this happens for all of the summands of Pe then Pe is a direct summand of Pe−1 so the resolution couldn’t have been minimal. Conclusion: the matrix of ϕ : Pe → Pe−1 has at least one column, all of whose entries are in m. Reducing modulo m (and tensoring with Extd(R/m, R)) gives the matrix of (80.1), which therefore has at least one zero column. Hence (80.1) is not injective.

80.2 Purity of the branch locus Theorem 80.4. The depth of an R-module M is the length of a maximal regular sequence. Proof. Note that the theorem is true if M is m-torsion. In that case the depth is easily seen to be zero and there are no regular elements. Suppose t is an M-regular element of m. Then from the exact sequence

0 → M −→t M → M/tM → 0

we get

Extp(R/m, M) → Extp(R/m, M/tM) → Extp+1(R/m, M) −→t Extp+1(R/m, M).

Note that the last map is zero since t ∈ R/m. Therefore Extp(R/m, M/tM) ' Extp+1(R/m, M) for p < depth(M). We conclude that depth(M/tM) = depth(M)− 1. If M has a maximal regular sequence of length n then M/tM has a maximal regular sequence of length n − 1, so we conclude by induction. Corollary 80.4.1. If R is a regular local ring then depth(R) = dim(R). Corollary 80.4.2. If R is a normal local ring then depth(R) ≥ min{2, dim(R)}. Proof. If dim(R) = 0 then the conclusion is obvious. If R is integral and dim(R) > 0 then R contains a zero non-divisor, so depth(R) ≥ 1. This proves the corollary if dim(R) = 1. If dim(R) = 2, choose a zero non-divisor t. If every element of m/tm is a zero divisor then there is some element s ∈ R such that ms ≡ 0 (mod tR). That is m(s/t) ⊂ R. Also s/t 6∈ R since s 6≡ 0 (mod t). Let m−1 be the set of elements x of the fraction ﬁeld of R such that xm ⊂ R. Then m−1 contains and is not equal to R (last paragraph). Since m is not principal, we cannot have m−1m = R. Therefore m−1m = m. Therefore m−1 stabilizes the ﬁnitely generated submodule m of R. Therefore every element of m−1 is integral over R, whence m−1 = R: contradiction. 1 → 1 Corollary 80.4.3. Let R0 ⊃ R be an integral extension of an integrally closed domain R of dimension 2. Then depth(R0) = 2.

1todo: dim R > 2 80. SMOOTH BASE CHANGE 139

Proof. We know that depth(R0) ≤ depth(R) = 2. On the other hand, we can repeat the proof above. Choose t ∈ R not a zero divisor in R and α ∈ R0 such that mα = 0 in R0/tR0. Then let f(x) be an expression of integral dependence of α. We get mf(α) ≡ f(0) (mod tR0) so mf(0) = 0 in R0/tR0. That is mf(0) ∈ tR0 ∩ R = tR since R is integrally closed. (If tR0 3 u then u/t ∈ R0 is integral over R, hence in R.) Put s = f(0) and conclude as in the last corollary. Corollary 80.4.4 (Purity of the branch locus in dimension 2). Suppose that R is a regular local ring and R0 a ﬁnite, normal, generically ´etale R-algebra. Then the branch locus of Spec R0 → Spec R has pure codimension 1.

0 0 0 Proof. We have depthR(R ) + pdR(R ) = depth(R) = dim(R) = 2. But R is 0 0 0 normal, so depthR(R ) = 2. Therefore pdR(R ) = 0, i.e., R is a free R-module. Now the discriminant (the determinant of the trace pairing on R0) detects the ramiﬁcation locus in the base, so that locus is deﬁned by a single equation, hence is purely of codimension 1. 140 CHAPTER 17. BASE CHANGE THEOREMS Bibliography

[BS] B. Bhatt and P. Scholze. The pro-´etale topology for schemes. math.AG:1309.1198.

[HS] G. Higman and A. H. Stone. On inverse systems with trivial limits. J. London Math. Soc., 29:233–236, 1954. [Wik] Wikipedia. Artin–Rees lemma, 2013. [Online; accessed 22 January 2014].

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