Etale cohomology course notes - Department of Mathematicsmath.colorado.edu/~jonathan.wise/teaching/math8174-spring-2014/notes.pdfMac Lane, S. Categories for the working mathematician.

Etale cohomology course notes

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

2

Contents

1 Introduction 51.1 Prerequisites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.3 History of etale cohomology . . . . . . . . . . . . . . . . . . . . . 6

2 Category theory 71 Categories and functors . . . . . . . . . . . . . . . . . . . . . . . 7

1.1 Categories . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.2 Functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2 Natural transformations and equivalences of categories . . . . . . 92.1 Natural transformations . . . . . . . . . . . . . . . . . . . 92.2 Equivalence of categories . . . . . . . . . . . . . . . . . . 10

3 Representable functors, Yoneda’s Lemma, and universal properties 103.1 Yoneda’s Lemma . . . . . . . . . . . . . . . . . . . . . . . 103.2 Representable functors . . . . . . . . . . . . . . . . . . . . 11

4 Limits and colimits . . . . . . . . . . . . . . . . . . . . . . . . . . 125 Special limits and colimits . . . . . . . . . . . . . . . . . . . . . . 12A Correspondences and adjunctions . . . . . . . . . . . . . . . . . . 14B The adjoint functor theorem . . . . . . . . . . . . . . . . . . . . . 15

3 Sheaves and the fundamental group 196 The category of sheaves on a topological space . . . . . . . . . . 20

6.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . 206.2 The espace etale . . . . . . . . . . . . . . . . . . . . . . . 20

7 Etale spaces and sheaves . . . . . . . . . . . . . . . . . . . . . . . 207.1 The equivalence of categories . . . . . . . . . . . . . . . . 207.2 Sheafification . . . . . . . . . . . . . . . . . . . . . . . . . 23

8 Operations on sheaves . . . . . . . . . . . . . . . . . . . . . . . . 248.1 Pushforward and pullback . . . . . . . . . . . . . . . . . . 24

10 Further operations on sheaves . . . . . . . . . . . . . . . . . . . . 2510.1 Limits and colimits . . . . . . . . . . . . . . . . . . . . . . 25

11 Even further operations on sheaves . . . . . . . . . . . . . . . . . 2611.1 Restriction . . . . . . . . . . . . . . . . . . . . . . . . . . 2611.2 Stalks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3

4 CONTENTS

11.3 Extension by the void . . . . . . . . . . . . . . . . . . . . 2611.4 Set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

12 Locally constant sheaves and path lifting . . . . . . . . . . . . . . 2712.1 Constant sheaves and locally constant sheaves . . . . . . . 2712.2 The homotopy lifting property . . . . . . . . . . . . . . . 28

14 Uniform spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3014.1 Filters and uniformities . . . . . . . . . . . . . . . . . . . 3014.2 The topology of a uniform space . . . . . . . . . . . . . . 3114.3 Complete uniform spaces . . . . . . . . . . . . . . . . . . 3114.4 Completion . . . . . . . . . . . . . . . . . . . . . . . . . . 32

15 Categorical Galois theory . . . . . . . . . . . . . . . . . . . . . . 3315.1 Uniform groups . . . . . . . . . . . . . . . . . . . . . . . . 3315.2 Infinite Galois theory . . . . . . . . . . . . . . . . . . . . 34

16 Pseudo-locally constant sheaves . . . . . . . . . . . . . . . . . . . 3518 Finite Galois theory . . . . . . . . . . . . . . . . . . . . . . . . . 3719 The fundamental group . . . . . . . . . . . . . . . . . . . . . . . 39

19.1 Functoriality of the fundamental group . . . . . . . . . . . 3919.2 The universal cover . . . . . . . . . . . . . . . . . . . . . . 3919.3 The Hawai’ian earring . . . . . . . . . . . . . . . . . . . . 40

A Sheaves of groups and torsors . . . . . . . . . . . . . . . . . . . . 42B Classification of torsors under locally constant groups . . . . . . 44

19.1 Crossed homomorphisms and semidirect products . . . . . 4419.2 Group objects and group actions . . . . . . . . . . . . . . 4419.3 Classification of torsors under pseudo-locally constant groups 46

C Fiber functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47D The espace etale via the adjoint functor theorem . . . . . . . . . 47

4 Commutative algebra 4921 Affine schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

21.1 Limits of schemes . . . . . . . . . . . . . . . . . . . . . . . 5021.2 Some important schemes . . . . . . . . . . . . . . . . . . . 5021.3 Topological rings . . . . . . . . . . . . . . . . . . . . . . . 5121.4 The Zariski topology . . . . . . . . . . . . . . . . . . . . . 5121.5 Why etale morphisms? . . . . . . . . . . . . . . . . . . . . 52

22 Smooth and etale morphisms . . . . . . . . . . . . . . . . . . . . 5322.1 The functorial perspective . . . . . . . . . . . . . . . . . . 5322.2 The differential perspective . . . . . . . . . . . . . . . . . 54

23 Homology of commutative rings . . . . . . . . . . . . . . . . . . . 5824 Homology of modules . . . . . . . . . . . . . . . . . . . . . . . . 6125 Flatness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6426 The equational criterion for flatness . . . . . . . . . . . . . . . . 6530 Local criteria for flatness . . . . . . . . . . . . . . . . . . . . . . . 6831 Flatness of etale maps . . . . . . . . . . . . . . . . . . . . . . . . 70A Extending etale maps . . . . . . . . . . . . . . . . . . . . . . . . 73B Completions of rings . . . . . . . . . . . . . . . . . . . . . . . . . 77C Zariski’s “Main Theorem” . . . . . . . . . . . . . . . . . . . . . . 79

CONTENTS 5

D More perspectives on etale maps . . . . . . . . . . . . . . . . . . 8131.1 The analytic perspective . . . . . . . . . . . . . . . . . . . 8131.2 The algebraic perspective . . . . . . . . . . . . . . . . . . 8131.3 Equivalence of the definitions . . . . . . . . . . . . . . . . 82

E Homology of modules . . . . . . . . . . . . . . . . . . . . . . . . 8331.1 Exact sequences . . . . . . . . . . . . . . . . . . . . . . . 85

F Cohomology of modules . . . . . . . . . . . . . . . . . . . . . . . 86

5 The etale topology 8932 Grothendieck topologies . . . . . . . . . . . . . . . . . . . . . . . 89

32.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . 9032.2 Sheaves on Grothendieck topologies . . . . . . . . . . . . 9132.3 More examples of Grothendieck topologies . . . . . . . . . 92

33 Sheafification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9233.1 Topological generators . . . . . . . . . . . . . . . . . . . . 9233.2 Descent data . . . . . . . . . . . . . . . . . . . . . . . . . 93

34 Fpqc descent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9435 A rapid review of scheme theory . . . . . . . . . . . . . . . . . . 96

35.1 A heuristic introduction . . . . . . . . . . . . . . . . . . . 9635.2 Schemes as functors . . . . . . . . . . . . . . . . . . . . . 98

36 The etale topology . . . . . . . . . . . . . . . . . . . . . . . . . . 10137 Henselization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10138 The etale fundamental group . . . . . . . . . . . . . . . . . . . . 101

38.1 Covering spaces . . . . . . . . . . . . . . . . . . . . . . . . 10138.2 Locally constant sheaves . . . . . . . . . . . . . . . . . . . 10238.3 The topology on the etale fundamental group . . . . . . . 102

6 Abelian categories and derived functors 10339 Abelian categories . . . . . . . . . . . . . . . . . . . . . . . . . . 10340 Resolution and derived functors . . . . . . . . . . . . . . . . . . . 104

40.1 Injective and projective objects . . . . . . . . . . . . . . . 10440.2 Complexes . . . . . . . . . . . . . . . . . . . . . . . . . . 104

41 Spectral sequences . . . . . . . . . . . . . . . . . . . . . . . . . . 104

7 Cohomology of sheaves 10542 Acyclic resolutions . . . . . . . . . . . . . . . . . . . . . . . . . . 105

42.1 Injective resolution . . . . . . . . . . . . . . . . . . . . . . 10542.2 Flaccid (flasque) resolution . . . . . . . . . . . . . . . . . 10542.3 Soft resolution . . . . . . . . . . . . . . . . . . . . . . . . 10542.4 Partitions of unity and de Rham cohomology . . . . . . . 106

43 Cech cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . 10644 Compactly supported cohomology . . . . . . . . . . . . . . . . . 107

44.1 The compactly supported cohomology of the real line . . 10745 Proper base change . . . . . . . . . . . . . . . . . . . . . . . . . . 10746 Leray spectral sequence . . . . . . . . . . . . . . . . . . . . . . . 108

46.1 Kunneth formula . . . . . . . . . . . . . . . . . . . . . . . 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 Poincare duality . . . . . . . . . . . . . . . . . . . . . . . . . . . 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-Cech 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 etale morphisms . . . . . . . . . . . . . 117

59 (*) Weakly etale morphisms . . . . . . . . . . . . . . . . . . . . . 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 etale topology . . . . . . . . . . . . . . . . . . . . . . . . . . 123

69 Other topologies . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

69.1 The Zariski topology . . . . . . . . . . . . . . . . . . . . . 123

69.2 (*) The pro-etale topology . . . . . . . . . . . . . . . . . . 123

69.3 (*) The infinitesimal site . . . . . . . . . . . . . . . . . . . 123

CONTENTS 7

14 Etale cohomology 12570 Constructible sheaves and `-adic cohomology . . . . . . . . . . . 12571 Etale cohomology in low degrees . . . . . . . . . . . . . . . . . . 12572 Etale cohomology and colimits . . . . . . . . . . . . . . . . . . . 12573 Cup product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

15 Etale cohomology of points 12774 Group cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . 12775 Hilbert’s theorem 90 . . . . . . . . . . . . . . . . . . . . . . . . . 12776 The Brauer group . . . . . . . . . . . . . . . . . . . . . . . . . . 12777 Tsen’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

16 Etale cohomology of curves 13178 Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13179 Poincare duality for curves . . . . . . . . . . . . . . . . . . . . . . 131

17 Base change theorems 13380 Smooth base change . . . . . . . . . . . . . . . . . . . . . . . . . 133

80.1 Auslander–Buchsbaum formula . . . . . . . . . . . . . . . 13380.2 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 topologyand commutative algebra. Here is a partial list of commutative algebra conceptswe 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; study-ing homological algebra concurrently might work well.

Comfort with the theory of schemes will be assumed as little as possible. I’lldo my best to review what we use, but there are bound to be some places wherebackground in algebraic geometry is necessary.

1.2 References

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

Deligne, P. Seminaire de Geometrie Algebrique du Bois-Marie (SGA4 1

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. Geometrie algebrique et geometrie analytique.

Mumford, D. Abelian varieties.

Serre, J.-P. Cohomologie galoisienne.

1.3 History of etale cohomology

Chapter 2

Category theory

Reading and references

Grothendieck, A. Introduction au langage fonctoriel. Faculte desSciences 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. Chapters1–3.

For the concept of a correspondence, the only reference I know is Lurie’sHigher 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 com-mutativity, 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 followingconditions:

CAT1 (Composition law) If f ∈ Hom(X,Y ) and g ∈ Hom(Y,Z) then there is aunique 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 morphismidX ∈ Hom(X,X) satisfying idX f = f for all f ∈ Hom(Y,X) andg 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 Cg = g

Cf.

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 arethe non-negative integers 0, 1, . . . , n and for which Hom∆n(i, j) consists of asingle element for i ≤ j and is empty for i > j. This is a special case of theprevious 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 functorbetween 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 trianglesare in bijection with functors ∆2 → C .

2. NATURAL TRANSFORMATIONS AND EQUIVALENCES OF CATEGORIES13

2 Natural transformations and equivalences ofcategories

2.1 Natural transformations

Definition 2.1. Suppose C and D are pre-categories. The product pre-categoryC ×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 ′), (g, g′), (h, h′)) is commutative if both (f, g, h) and (f ′, g′, h′)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. Anatural transformation from F to G is a functor h : C ×∆1 → D such that ifwe 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 Cto D , whose morphisms are natural transformations, and whose commutingtriangles are the commuting triangles of natural transformations.

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

F (X)ϕX //

F (f)

G(X)

G(f)

F (Y )

ϕY // 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 allX ∈ C . This category is denoted Hom(C ,D).

Definition 2.5. A natural transformation with an inverse is called a naturalisomorphism.


Exercise 2.6. Let F,G : C → D be functors and ϕ : F → G a naturaltransformation. 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 ϕXis 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 arefunctors F : C → D and G : D → C such that F G is naturally isomorphic toidD and G F is naturally isomorphic to idC .2

Definition 2.8. A functor F : C → D is said, respectively, to be (i) faithful,

(ii) full, (iii) fully faithful for all X,Y ∈ Ob(C ), the map HomC (X,Y )F−→

HomD(FX,FY ) is (i) injective, (ii) surjective, (iii) bijective. If every object ofD is isomorphic to FX for some X ∈ Ob(C ) then F is said to be essentiallysurjective.

Proposition 2.9. A functor is an equivalence of categories if and only if it isfully faithful and essentially surjective.

3 Representable functors, Yoneda’s Lemma, anduniversal properties

3.1 Yoneda’s Lemma

Let C be category. In the last section we saw that C = Hom(C ,Sets) is acategory whose objects are functors and whose morphisms are natural transfor-mations. There is a functor

C → C : X 7→ hX

where hX is the functor hX(Y ) = HomC (Y,X). If f : Z → Y is a morphism inC 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 : HomC (hX , F ) → F (X) that isnatural in F and sends idhX to idX when F = hX .

Proof. Let ϕ : hX → F be a natural transformation. Then by naturality in Fwe have

idhX //

_

ϕ_

idX

// ϕ(idX).

2Here 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

hX(X)ϕX //

hX(f)

F (X)

F (f)

hX(Y )

ϕY // F (Y )

idX //

ϕ(idX)

f // ϕ(f).

This implies that ϕY (f) = Ff(ϕ(idX)) = Ff(ψ(idX)) = ψY (f), i.e., thatϕ = ψ.

Finally we show that u is surjective. Suppose that α ∈ F (X). Let f ∈hX(Y ). Set ϕY (f) = Ff(α). To check this is natural in Y , suppose we haveg : Z → Y . We must check that the diagram below is commutative:

hX(Y )

hX(g)

ϕY // F (Y )

F (g)

hX(Z)

ϕZ // F (Z)

f // Ff(α)

Fg(Ff(α))

f

fg // F (fg)(α).

But F is a (contravariant) functor, so F (fg) = Fg Ff .

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 easyto 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 maphX → 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 fromcontext.

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 employthe same language about representable functors.

Example 3.3. Let C be a category containing objects Xi, i ∈ I. Let

F (Y ) =∏i∈I

Hom(Y,Xi).

Then an object representing F is called a product of the Xi.


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 ofgroups is free product; coproduct in the category of abelian groups is direct sum;coproduct in the category of commutative rings is tensor product; coproduct inthe 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 functorF : 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 byadjoining a final object to I.

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

G(X) =

F ′ : I.→ C

∣∣∣∣∣ F ′∣∣I

= F and F ′(1) = X

.

An object of C representing G is called the colimit of F . It may be denotedlim−→F , lim−→X∈I F (X), colimF , or colimX∈I F (X).

Dually (i.e., a colimit in C ), we have the notion of a limit. It is denotedlim←−F , lim←−X∈I F (X), etc.

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 Si, i ∈ I is a small diagram of sets. Then lim←−i∈I Sican be constructed as

(xi)i∈I ∈∏i∈I

Si

∣∣∣u(xi) = xj for all u : i→ j in I.

Exercise 5.2. Suppose that Sj , j ∈ J is a filtered (small) diagram of sets. Thenlim−→j∈J Sj can be constructed as the set

∐j∈J Sj/R where R is the equivalence

relation that has s ∼ t whenever s ∈ Si, t ∈ Sj , and u(s) = v(t) for someu : 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 tolimits 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 pairof elements (equivalently, every finite subset) has an upper bound (resp. lowerbound). More generally, a pre-category I is called filtered if, for every finitepre-category J and every functor J → I, there is an extension to J. → I. Apre-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 themaps uij : Xi → Xj are all surjective. Set X = lim←−Xi. Then the structuralmaps 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 valuesof 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 givesan 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 filteredlimits. (Hint: See [HS] or [BS, Remark 3.1.6].)

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

lim−→i∈I

lim←−j∈J

Sij → lim←−j∈J

lim−→i∈I

Sij

is a bijection.

Proof. First we show injectivity. Suppose that x, y ∈ lim−→i∈I lim←−j∈J Sij . There-

fore there are indices i, i′ ∈ I such that x ∈ lim←−j∈J Sij and y ∈ lim←−j∈J Si′j .Since I is filtered, we can find a single index ≥ i, i′ and therefore we can assumei = i′.

Suppose that x and y have the same image in lim←−j∈J lim−→i∈I Sij . Therefore

x and y have the same image in lim−→i∈I Sij for all j. Therefore, for each j ∈ J ,

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.


i′ ≥ i(j) for all of the (finitely many) j ∈ J . Then x and y have the same imagein Si′j . Therefore x and y represent the same element of lim−→i∈I lim←−j∈J Sij .

Now we show surjectivity. Let x be an element of lim←−j∈J lim−→i∈I Sij . There-

fore, for each j ∈ J we have an index i(j) ∈ I and xj ∈ Si(j),j representingx. 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 ele-ment of lim←−j∈J Sij . This induces an element of lim−→i∈I lim←−j∈J Sij whose image

in lim←−j∈J lim−→i∈I Sij is x.

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

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

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

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

lim−→n∈Z

lim←−m∈Z

pnZ/pmZ→ lim←−m∈Z

lim−→n∈Z

pnZ/pmZ

is a bijection and that it has a natural commutative ring structure under whichit 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 afunctor F : C × D → Sets. We will say that F is (left) representable by afunctor 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 naturalisomorphism

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. Foran A-module M and a B-module N , define F (M,N) to be the set of additivefunctions 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 givesa B-module N the A-module structure induced by f . It is left representable bythe functor G(M) = B⊗AM . 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 andψ : idD → HG such that the compositions

GXG(ψX)−−−−→ GHGX

ϕGX−−−→ GX

HYψHY−−−→ HGHY

H(ϕY )−−−−→ 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 adjointof F . We have bijections (natural in Y )

Hom(FX, Y ) = Hom(X,GY ) = Hom(lim−→Xi, GY )

= lim←−Hom(Xi, GY ) = lim←−Hom(FXi, Y ).

Therefore FX represents the functor Y 7→ lim←−Hom(FXi, Y ).

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 admitsa map to some object of C0. Then C has a final object.

4todo: proof


Proof. Taking the colimit of the full subcategory C0 gives an object X of C andwe 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 thecolimit. We therefore only need to check that each object of C has at most onemap 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 mapg : Y → X, by the definition of a colimit. Then we obtain gf : X → X. I claimthat gf = idX . Indeed, it suffices to show that if Z ∈ C0 and h : Z → X isthe tautological morphism of the colimit, then gfh = h. But fh : Z → Y is amorphism 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 thecoequalizer 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, thereis a map s : V → X because V ∈ C0. But by the considerations of the lastparagraph, 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 asmall collection of objects C0 such that, whenever ξ ∈ F (X) there is an objectX0 ∈ 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 objectfor the category C /F . The hypothesis of the corollary implies that C /F satisfiesthe hypothesis of the proposition.

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

lim−→X∈C0/Y

X → Y

is an isomorphism. Assume also that C has arbitrary small colimits and theisomorphism classes of quotients of each object form a set. Then C has a finalobject.

Proof. Let Z = lim−→C0X and let W = lim−→Z ′, the colimit taken over all quotients

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−→Xi for some diagram of Xi drawnfrom C0. Then each Xi comes with a tautological map to Z and all of theseare compatible, by definition of Z. By the universal property of the colimit, weobtain 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 . Thecoequalizer W ′ is a quotient of W (it represents a subfunctor of hW ) so that itis also a quotient of Z. Hence it is equal to W , since hW is the intersection ofall representable subfunctors of hZ .

Corollary 5.2.1. Suppose that F : C → Sets is a functor, that C has ar-bitrary small colimits, that these are preserved by F , and that C is generatedunder 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 propo-sition.


Chapter 3

Sheaves and thefundamental group


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.1and II.2.

R. Godement. Topologie algebrique et theorie des faisceaux. SectionsII.1.1 and II.1.2.

The discussion of uniform spaces is drawn mostly from

N. Bourbaki. Topologie generale. Sections I.6–7, II.1, II.3.

The presentation on the fundamental group is adapted from the followingpaper:

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

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

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

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

23

http://math.stanford.edu/~vakil/216blog/

http://arxiv.org/abs/1309.1198

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 opensets. A presheaf on X is a functor F : Open(X) → Sets. For V ⊂ U , wefrequently denote the restriction map F (U) → F (V ) by x 7→ x

∣∣V

. A presheafis 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.

SH2 If U =⋃V ∈V V and xV ∈ F (V ) for all V ∈ V are such that xV

∣∣V ∩W =

xW∣∣V ∩W then there is some x ∈ F (U) with x

∣∣V

= xV for all V .

We writeΓ(U,F ) = H0(U,F ) = F (U).

6.1 Examples

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

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

6.2 The espace etale

Definition 6.3. A continuous map of topological spaces f : X → Y is called alocal homeomorphism if there is a cover ofX by open subsets U such that f(U) ⊂Y is open and f

∣∣U

: U → f(U) is a homeomorphism. Local homeomorphismsare frequently also called etale maps.

Exercise 6.4. (i) Show that a finite limit of etale spaces over X (taken inthe category of topological spaces over X) is an etale space over X.

(ii) Show that the finiteness assumption is essential by producing an infinitelimit that is not an etale space.

Let et(X) denote the category of all etale maps f : Y → X. Morphisms inet(X) are maps commuting with the projection to X.

7 Etale spaces and sheaves

7.1 The equivalence of categories

If Y is an etale space over X, let Y sh be the sheaf of sections of Y . This givesa 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 thatF ∈ Psh(X). Construct a diagram whose elements are U × σ for each openU ⊂ X and each σ ∈ F (U). We say that U × σ ≤ V × τ if U ⊂ V andτ∣∣U

= σ. Define

F et = lim−→U∈Open(X)σ∈F (U)

U × σ.

The colimit is taken in the category of topological spaces. There is projectionF et → X that restricts to the projection on the first factor on U×σ (universalproperty of colimit).

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

(ii) Conclude that the projection F et → X constructed above is a local home-omorphism.

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 thelargest 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. Asthey are continuous, they are inverse homeomorphisms.

(ii) This is evident from the facts that the U × σ are an open cover andevery 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 W1 × ω1oo // · · · Wn × ωnoo // V × τ

with y ∈⋂Wi ⊂ U ∩ V . Therefore we have σ

∣∣W

= τ∣∣W

for some openneighborhood W of y. This holds for every y in the intersection, so we getthe desired conclusion.

(iv) A special case of the previous part.


Before stating the next theorem, we construct natural maps

(Y sh)et → Y

F → (F et)sh.

To give

(Y sh)et → 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 ' Uσ−→ Y is the desired

map.To give

F → (F et)sh

we must give compatible maps F (U) → Γ(U,F et) for every open U ⊂ X.If σ ∈ F (U) then we get a tautological map U ' U × σ → F et by theconstruction of F et.

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 equivalencesof categories.

Proof. We verify first that if Y is an etale space over X then the map ϕ :(Y sh)et → Y is a homeomorphism. The diagram defining (Y sh)et can be viewedas the collection of all open U ⊂ Y such that p

∣∣U

: U → p(U) is a homeomor-phism. As every one of these open subsets must map bijectively onto its imagein Y , we deduce that ϕ is open and a local homeomorphism. We only have toshow that it is injective. We know that ϕ

∣∣U

is injective for every U that mapsinjectively to X, so the only way injectivity could fail is if x ∈ U , y ∈ V , andϕ(x) = ϕ(y). But then x ∈ U ∩ V , so the diagram

U ∩ V //

U

V // (Y sh)et

commutes (all of U ∩ V , U , and V are in the diagram whose colimit defined(Y sh)et. Thus x and y represent the same element of (Y sh)et.

Now we show that F → (F et)sh is a bijection when F is a sheaf. First weshow injectivity. Suppose σ, τ ∈ F (U) for some open U ⊂ X. Then the twomaps U ' U ×σ → F et and U ' U ×τ → F et coincide. Therefore U ×τand U ×σ have the same image in F et . Therefore by Lemma ?? ??, we haveσ = τ .

Now suppose that s ∈ (F et)sh(U) = Γ(U,F et). We show that s factors asU ' U × σ → F et for some σ ∈ Γ(U,F et). Consider the open sets Vσ =

7. ETALE SPACES AND SHEAVES 27

s−1(U × σ) as σ varies in F (U). These cover U and we have σ∣∣Vσ∈ F (Vσ)

for all σ. Moreover,

σ∣∣Vσ∩Vτ

= τ∣∣Vσ∩Vτ

because Vσ ∩Vτ = s−1((U ×σ)∩ (V ×τ)) ⊂W (notation as in Lemma ??),and σ

∣∣W

= τ∣∣W

by definition. As the Vσ cover U , we can glue together the σ∣∣Vσ

to a single section σ ∈ F (U). Then we have

U × σ =⋃σ

Vσ × σ∣∣Vσ.

Thus,

s−1(U × σ) =⋃Vσ = U

and s is therefore the map U ' U × σ → F et. Thus s lies in the image ofF (U)→ (F et)sh(U).

7.2 Sheafification

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

Proposition 7.5. Let F be a presheaf and G a sheaf. Any morphism of sheavesu : 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 thatthe 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, sheafifica-tion preserves arbitrary colimits and limits of sheaves may be computed on theunderlying presheaves.

Corollary 7.5.2. The category Sh(X) admits arbitrary small limits and col-imits.


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.Define p∗F (U) = F (p−1U) for U ⊂ Y open. This is called the pushforward ofF via p.

Definition 8.2. Let F be an etale space over Y and p : X → Y a continuousfunction. Define p∗F = F ×Y X. Extend this definition to sheaves by theequivalence between etale spaces and sheaves.

Lemma 8.3. Pullback of etale spaces commutes with all small colimits.

Proof. Let X ′ → X be a continuous function and let Y be an etale space overX. Assume that Y = lim−→Yi where each Yi is an etale space over X. On thelevel of underlying sets, Y ′ = Y ×X X ′ may be identified with lim−→(Yi×X X ′).We have to check the topologies are the same. The space Y has a basis byopen subsets that are images of U ⊂ Yi where U projects isomorphically to anopen subset of X. Therefore Y ×X X ′ has a basis of open subsets of the formp′i(U ×X V ) where U ⊂ Yi projects isomorphically to an open subset of X andV ⊂ X ′ is open and p′i : Y ′i → Y ′ is the tautological projection.

On the other hand, Y ′ = lim−→Y ′i has a basis of open subsets of the formp′i(W ) where W ⊂ Y ′i is open and projects isomorphically to an open subset ofX ′. If U and V are of the form described in the last paragraph then U ×X Vis of this form. Therefore the topology on Y ′ as lim−→Y ′i is at least as fine as thetopology as Y ×X X ′.

On the other hand, any such open W ⊂ Y ′i is a union of open subsets of theform U ×X V where U and V are as in the first paragraph, by definition of theproduct topology on Yi×X X ′. Therefore the topology on Y ×X X ′ is at leastas fine as the one on lim−→Y ′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 shi for a suitable diagram of Ui, with each Ui ⊂ Y open.

One way to see that such a diagram exists is to remark that F et is a union ofUi × σ and equivalence of categories respects colimits.

10. FURTHER OPERATIONS ON SHEAVES 29

Now we may compute:

HomSh(X)(p∗F,G) = HomSh(X)(p

∗ lim−→U shi , G)

= HomSh(X)(lim−→ p∗U shi , G)

= lim←−HomSh(X)(p−1Ui, G

et)

= lim←−G(p−1Ui)

= lim←−HomSh(Y )(Ushi , p∗G)

= HomSh(Y )(lim−→U shi , p∗G)

= HomSh(Y )(F, p∗G)

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

Corollary 8.4.1. Let p : X → Y be a continuous function. The functorp∗ : 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 finitelimits.

Proof. The commutation with all colimits comes from the existence of the rightadjoint p∗. For the limits, use the fact that a finite limit of etale spaces is anetale space (Exercise 6.4).

Exercise 8.5. Why doesn’t pullback commute with all limits? After all, pull-back 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 C admits all limits andcolimits and that these are computed objectwise.

Proposition 10.2. The category of sheaves on a topological space X admits allsmall 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), itpreserves colimits, so this shows that all diagrams of sheaves admit colimits.

Exercise 10.3. Let F = lim−→Fi be a colimit of a diagram of sheaves. Show byexample that the map lim−→Fi(U)→ F (U) is not always an isomorphism.


11 Even further operations on sheaves

11.1 Restriction

Definition 11.1. Suppose X is a topological space, i : U → X is the inclusionof an open subset, and F is a sheaf on X. Then i∗F is called the restriction ofF 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 iscontinuous, so we have a functor

i∗ : Sh(X)→ Sh(x) = Sets.

We can compute i∗F explicitly using the espace etale of F . By definition, F et =lim−→σ∈F (U)

U×σ. Pullback preserves colimits, so that i∗F et = lim−→σ∈F (U)i−1(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

i∗F et = lim−→x∈U

U ⊂ X openσ∈F (U)

σ = lim−→x∈U⊂X

∐σ∈F (U)

σ = lim−→x∈U⊂X

F (U).

Exercise 11.3. Conclude from the above that i∗F et = 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 isomorphismthen 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 resultscan 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 existssome x ∈ F (U) with the following properties P (U, x)...” with “there exists acover of U by open subsets Vi such that for each i there is an xi ∈ F (Vi) withP (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 isinjective if and only if the morphisms of sets ϕU : F (U) → G(U) are injectivefor 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 withthe following: for all open U ⊂ X and all y ∈ G(U) there exists an open coverof U by subsets Vi and elements xi ∈ F (Vi) such that y

∣∣Vi

= ϕ(xi).Bijectivity is, of course, the conjunction of injectivity and surjectivity and

requires no modification beyond the modification already made to the definitionof surjectivity.

Example 11.7. Let F be the sheaf on S1 associated to the covering spaceR→ 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 etale spaces, this is simply themap R→ S1.) This map is surjective because F (U) 6= ∅ for any U ⊂ S1 otherthan U = S1 itself. However, F (S1) = ∅ and G(S1) = 1 so F (S1) → G(S1) isnot surjective.

The connection between sheaves and sets is useful in set theory. It allowsone 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 surjectionp : 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 mor-phism 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 etale Y . The followingproperties 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 functionsvalued in a discrete space S.

(iii) There is a set S and an isomorphism between F and the presheaf F ′ definedby F ′(U) = S for all open U ⊂ X.

1Recall that a section of p is a map s : Y → X such that ps = idY .


Proof. Exercise.

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

Proposition 12.3. Let F be a sheaf over X and Y its espace etale. Thefollowing properties are equivalent:

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

(ii) There is an open cover of X by subsets U such that F∣∣U

is a constantsheaf on U .

Definition 12.4. Sheaves satisfying the equivalent conditions of Proposition 12.3are called locally constant. Etale spaces satisfying these conditions are calledcovering 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 isconstant on I. We say that Y satisfies the (unique) homotopy lifting propertyif, for any f : Y × I → X, the sheaf f∗F is pulled back from the first projectionp : Y × I → Y .

Exercise 12.6. Translate the homotopy lifting property into a property of theespace etale.

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 thatis locally pulled back via the projection p : X × I → X. Then p∗p∗F → F is anisomorphism 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

Get

q

U × I //

f

;;

X.

12. LOCALLY CONSTANT SHEAVES AND PATH LIFTING 33

Now, f∣∣x×I : x × I → q−1(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 functiong : U → Get 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 quotienttopology associated to the map p). Thus every map f ∈ p∗G(U×I) is the imageof 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 → Uis surjective.

Proof of Proposition 12.7. The lemma allows us to show that the proposition istrue if F ' p∗G. Consider the sequence of maps

p∗G→ p∗p∗p∗G→ p∗G

whose composition is idp∗G by formal properties of adjunction.2 The first of ← 2these arrows is the pullback of the isomorphism G → p∗p

∗G, hence is an iso-morphism. Therefore the second arrow is an isomorphism as well.

Now we consider the general case. Because a morphism of sheaves can beverified 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 theproposition 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 anopen neighborhood U × (a, b) on which the assertion holds. As I is compact,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∣∣U×(ai,bi)

' p∗Gi∣∣U×(ai,bi)

.

It seems easiest now to think in terms of the espace etale. We can deducefrom the considerations above that there is a finite sequence 0 = t0 < t1 <· · · < tn = 1 such that F et

∣∣U×[ti,ti+1]

' p∗Geti

∣∣U×[ti,ti+1]

. Note that this gives

isomorphismsF et∣∣U×ti

' Geti ' F et

∣∣U×ti+1

so that all the Gi are isomorphic and we can simplify notation by writing Gi = Gfor all i. Furthermore, we have compatible isomorphisms

F et∣∣U×t0

//

o

F et∣∣U×[t0,t1]

o

F et∣∣U×t1

oo

o

// · · · F et∣∣U×tn

oo

o

p∗Get∣∣U×t0

// Get∣∣U×[t0,t1]

Get∣∣U×t1

oo // · · · Get∣∣U×tn

.oo

Gluing together the horizontal diagrams we obtain F et ' p∗Get.

2todo: explain these somewhere


Now we can complete the proof of Proposition 12.7. Choose an open coverof X by U ⊂ X such that F

∣∣U×I ' p

∗G for some sheaf G on U . Then we havea commutative diagram

p∗p∗(F∣∣U×I)

o

∼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 iso-morphism. This holds for all U in a cover of X, so the U × I for whichp∗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 prop-erty.

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 Xsuch 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 withoutinvoking the real numbers:

Definition 14.2. Let X be a set and denote by ∆X the diagonal subset ofX ×X. A uniformity (or uniform structure) on a set X is a filter Φ on X ×Xsuch 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 ofthe 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 suchthat UY ⊂ T and take Φ to be the set of entourages so defined. Then Φ is auniform 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 isa 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 Uof 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 filterif, 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 onX. We say that x ∈ X is a limit of F if for any (x, x) ∈ V ∈ Φ there is someA ∈ F with A× x ⊂ V .5

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 ∈ Φthere are A,B ∈ F such that A × x ⊂ U and y × B ⊂ U . Replacing Aand B by their intersection, we can assume A = B. Then A × x ⊂ U andy×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.


if necessary. Therefore we have (a, x) ∈ U for some a ∈ A and (y, b) ∈ U forsome 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 suchthat 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 Cauchyfilter of X has a limit. It is said to be separated if Cauchy filters have at mostone limit.

14.4 Completion

Every uniform space has a unique completion. We will only prove the uniquenesshere.

Definition 14.10. Let (X,Φ) and (Y,Ψ) be uniform spaces. A map f : X → Yis 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 Xand that Z is a separated and complete uniform space. Any uniformly continuousfunction 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 amongentourages of Y , does not include the empty set. It is therefore a filter, and infact a Cauchy filter since if W is an entourage of X and V is a second entouragesuch 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. LetT be an entourage of Z such that T 3 ⊂ W . Select an entourage V of Y suchthat (x, x′) ∈ V ∩ (X ×X) implies (f(x), f(x′)) ∈ T . Let U be an entourage ofY such that U3 ⊂ V . We show that if (y, y′) ∈ U then (f(y), f(y′)) ∈W .

We may select x ∈ U(y) ∩X and x′ ∈ U(y′) ∩X such that f(x) ∈ T (f(y))and x′ ∈ T (f(y′)), by the definition of f . Then

(x, x′) = (x, y) (y, y′) (y′, x′) ∈ U3 ⊂ V

so that (f(x), f(x′)) ∈ T , by the construction of V . But then

(f(y), f(y′)) = (f(y), f(x)) (f(x), f(x′)) (f(x′), f(y′))

so (f(y), f(y′)) ∈ T 3 ⊂W .

Corollary 14.11.1. Suppose that X is a uniform space and that Y and Z aretwo separted and complete uniform spaces containing X as a dense subspace.Then there is a unique uniformly continuous bijection Y ' Z restricting to theidentity on X.

6In other words, for any V ∈ Ψ there is a U ∈ Φ such that d(x, y) ≤ U impliesd(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 theuniform structure defined in Exercise 14.4. Then G is separated and completewith 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∈Φ

U ⊂⋃Y⊂Xfinite

UY = ∆G

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. Weconstruct a limit g. For each x ∈ X, there is an A ∈ F with A × A ⊂ Ux.Select h ∈ A and set g(x) = h(x). This is well-defined, for if we chose h′ ∈ Athen h′(x) = h(x) by definition of Ux. On the other hand, if B ∈ F andB × B ⊂ Ux then A ∩ B 6= ∅ so that the definition does not depend on thechoice of A either.

We must verify that this is actually a limit of F . Supposing U ∈ Φ we mustshow that g × A ⊂ U for some A ∈ F . We may assume U = UY for a finitesubset Y ⊂ X, since every U ∈ Φ contains such a subset. As F is a Cauchyfilter, we can find A ∈ F such that A× A ⊂ UY . Note that UY ⊂ Ux for anyx ∈ 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 setof objects C0. Let F : C → Sets be a functor. Either use the previous theoremor 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 setS 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 × Scontaining ∆S are considered to be entourages.)7

If G is a uniform group, let G-Sets be the category of uniformly continuousactions 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 impliesg(s) = h(s).


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 tameinfinite Galois theory :

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

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

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

CΦ−→ 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. Notethat in any category possessing fiber products, Hom(Y, Z) may be identifiedwith the set of sections of Y × Z over Y . This applies to both G-Sets and toC . It is therefore sufficient to show that, for any map Z → Y in C , the inducedmap

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 preservesdisjoint unions and it preserves the property of connectedness, so there is someconnected component Z ′ ⊂ Z with Φ(Z ′) = T . The map Z ′ → Y induces anisomorphism Φ(Z ′)→ Φ(Y ) and the latter is an isomorphism, so the former isas well (because F is conservative).

Finally, we check that Φ is essentially surjective. We show that the essentialimage of Φ is closed under (small) disjoint unions, finite products, and passageto subobjects and quotients. Then we will show that every object of G-Setscan be constructed using these operations.

16. PSEUDO-LOCALLY CONSTANT SHEAVES 39

(i) The essential image is closed under disjoint unions. Disjoint unions areexamples of colimits.

(ii) The essential image is closed under finite products. Products are examplesof finite limits.

(iii) The essential image is closed under passage to subobjects. A subobjectof Φ(X) is a disjoint union of connected components of Φ(X). But everyunion 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 thatΦ(X) = S and S → T is a surjection. This is the quotient by an equiva-lence relation R = S×T S ⊂ S × S. Note that S × S = Φ(X ×X). Butthen R ⊂ S ×S is a subobject, so R = Φ(Y ) for some Y ⊂ X ×X. Let Zbe 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 aproduct of objects Φ(X) where X is a connected object of C . First note thatevery object of G-Sets is a disjoint union of connected G-sets. We may identifyany connected G-set S with G/U for an open subgroup U ⊂ G since S has thediscrete topology (in this case U is the stabilizer of an element of S). As Uis open, there are (by definition of the uniformity on G) a finite collection ofconnected 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. Onthe 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) subobjectof∏

Φ(Xi), for some finite collection Xi of connected objects of C . Thusany G-set is a disjoint union of objects that are quotients of objects that aresubobjects 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 itsatisfies the homotopy lifting property (Definition 12.5). We call its espaceetale a pseudo-covering space. The full subcategory of pseudo-locally constantsheaves on X is denoted cov(X).


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

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

If x = x′ and i denotes the inclusion of x in X then this is a bijection from i∗Yto itself. As this construction is natural in Y , it gives an automorphism of thefunctor i∗.8

To state the next proposition, we need the following definition:

Definition 16.2. A functor F : C → D is called conservative provided amorphism 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 conservativefunctor.

(vi) If i : x → X is the inclusion of a point then Aut(i∗) acts transitively oni∗Y , for any connected Y ∈ cov(X).

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

f∗F ' f∗ lim−→Fi ' lim−→ f∗Fi ' lim−→ p∗Gi.

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 diagramof the Gi. It is therefore legitimate to write

f∗F = lim−→ p∗Gi = p∗ lim−→Gi

so that F ∈ cov(X). The same argument, with colimits replaced by finite limits,shows that lim←−Fi ∈ cov(X) for any finite diagram of Fi.

(ii) Let Y be an object of cov(X). Note that Y is locally path connectedsince 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 (whatis 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 Uof Y . Each Zi ∩ U is open and closed in U , hence empty or equal to U itself.Therefore the chain Zi ∩ U must stabilize. Thus

⋂Zi ∩ U is either empty or

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 withnon-empty intersection, which by the above is the minimal open and closedsubset of Y containing y. Thus Y is the disjoint union of its minimal open andclosed subsets.

(iii) Since Y is locally isomorphic to X, it is locally path connected. It isconnected by assumption, hence is path connected.9

(iv) If Y ∈ cov(X) is connected then we can connect any two points in Y bya path, which is the unique lift of a path from X. This bounds the set of pointsof 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 worthof functions from that set to X. Therefore there is a set’s worth of isomorphismclasses of connected objects of cov(X).

(v) First we prove faithfulness. Suppose that i∗ϕ = i∗ψ. We argue that ϕand ψ agree on all stalks. Indeed, connect x′ ∈ 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 x′ as well. This holds for all x and x′ so ϕ and ψ agree on allstalks, hence are the same.

The same argument shows that if ϕ is a bijection at x then it is a bijectionon every stalk, hence is an isomorphism.

(vi) This is copied from [BS]: Suppose p : Y → X is in cov(X) and y, y′ ∈p−1(x). Choose a path γ : I → Y connected y to y′. Then p γ is a loop in Xbased at x, hence provides an automorphism of the fiber functor i∗. The effectof p γ on y is to trasport it to y′.

18 Finite Galois theory

Definition 18.1. A finite Galois theory is a pair (C , F ) where C is a categoryand 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 pointsof 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 yappears 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 eventuallyarrive at a path from y to x by concatenation.


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 allconnected X ∈ C .10

Theorem 18.2. Suppose that (C , F ) is a tame finite Galois theory. Then thefunctor 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. LetC be the category of finite, separable k-algebras.11 For any k-algebra A, defineF (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 fieldspreserve products, coproducts, kernels, and cokernels, the verification reducesto the case k = k. Then we can remark that finite, separable k-algebras areequivalent as a category to finite sets.

?? Since a field has no quotients, fields correspond to the connected objectsof C . The assertion then comes down to showing that every finite, separablek-algebra is a product of fields. We certainly have an embedding in a productof fields

A→ A⊗kk ' kn.

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 twomaps of finite separable extensions of k, say u, v : E′ → E and assume thatF (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 → E′ is a morphismof finite, separable field extensions of k such that F (u) is a bijection. That is,every embedding E → k extends uniqely to an embedding E′ → k. Choosesome x ∈ E′ and let f be its minimal polynomial. For any root ξ of f in kwe can extend an embedding v : E → k to an embedding w : E′ → k withw(x) = ξ. Therefore there is a unique root of f in k. But f is separable, soit has no repeated roots. Therefore f must be linear, so x ∈ E. Thus u is anisomorphism.

10In fact, every finite Galois theory is tame. For our purposes it will be easier to treat thisas a separate hypothesis.

11A finite k-algebra A is separable the k-algebra, A⊗k k is isomorphic to kn

for some n.Finite separable k-algebras are also known as finite etale k-algebras. The category of finiteetale k-algebras is denoted f et(k).

19. THE FUNDAMENTAL GROUP 43

(Tameness) Consider two embeddings u, v : E → k. By induction, there isan automorphism σ of k that carries u to v. (Build it up by constructing k asa sequence of extensions of E.) But then σ induces an automorphism of F bysending 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 isequivalent to the category of continuous π1(f et(k), hk)-sets.

19 The fundamental group

Theorem 19.1. Let X be a connected, locally path connected topological spaceand let G be the automorphism group of a fiber functor of cov(X). Thencov(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 thefiber 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 pathconnected topological spaces. This induces a functor f∗ : et(Y ) → et(X) thatrestricts to a functor cov(Y )→ cov(X). Indeed, the homotopy lifting propertyis phrased in terms of maps I × Z → X, every one of which corresponds to amap I × Z → Y after composition with f .

Let x and y be basepoints of X and Y with fiber functors F and G. ThenG = F f∗. Suppose that γ ∈ Aut(F ). That is, γ is a compatible collectionof automorphisms γW of F (W ) for all W ∈ cov(X). Set f∗(γ)Z = γf∗Z forZ ∈ 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 andhas 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 . IfZ ∈ cov(X) and z ∈ i∗Z then construct a map Y → Z by, for each y′ ∈ Y ,selecting a path γ from y to y′. The projection of this path to X is a path fromx to p(y′). It lifts, in a unique way, to a path in Z starting at z. Define f(y′)to be the endpoint of this lift.

This construction is well-defined because the path γ is unique up to homo-topy (because Y is simply connected). To complete the proof, we only need toverify that the map f constructed above is continuous.


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

p−1(q(V )) ∩ U

((

f // q−1(p(U)) ∩ V

vvp(U) ∩ q(V )

The diagonal arrows are homeomorphisms so the horizontal arrow is as well. Itfollows 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 → Zsuch that i∗f(y) = z. Therefore Y represents i∗. It now follows that Aut(i∗) =Aut(Y ), by Yoneda’s lemma.

Corollary 19.2.1. For any basepoint s ∈ S1, we have π1(S1, s) = Z.

19.3 The Hawai’ian earring

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

Xn =

n⋃k=1

x ∈ R2

∣∣∣ d(x, (1/n, 0))

= 1/n

X =

∞⋃k=1

Xn.

Let Y be the infinite wedge∨∞n=1 S

1. There is a continuous bijection f : Y → X,but the topology on X is coarser than that of Y . We wish to relate the categoriescov(X) and cov(Y ).

First note that et(X)→ et(Y ) is faithful. Indeed, a map between objects ofet(X) is determined by what it does pointwise, and pullback to Y only changesthe 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 iscontinuous as a function U → Z. It is sufficient to verify this in a neighborhoodof 0 ∈ U (since Y is homeomorphic to X away from 0). We can thereforeassume U is connected (otherwise replace U with the connected componentcontaining 0). Choose an open subset W ⊂ Z, containing σ(0), that projectshomeomorphically onto its image in X. Then there is certainly an extension ofσ∣∣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 collectionof branches at each vertex), we can identify π1(Y, 0) as a free group generated bya countably infinite set. Therefore cov(Y ) may be identified with the categoryof pairs (S,ϕ), where ϕ = (ϕ1, ϕ2, . . .) is a sequence of bijections from S toitself.

Proposition 19.4. An object of cov(Y ), viewed as a pair (S,ϕ), lies in theessential 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 ϕn1i1, ϕn2

i2, . . . in which each ϕi appears

only finitely many times, the sequence of ϕik · · ·ϕi1(z) stabilizes.

Proof. The first property expresses that Z is etale over X. The second is thepath 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 wordsin the ϕi there is a cover of X on which those words act differently. If the firstn of the ϕi appear in these words then we can arrange for this by unwindingthe 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 subsetW0 ⊂ Z0, for some Z ∈ cov(X), such that U contains the stabilizer of W0. Theneach of the finitely many w ∈ W0 is fixed by all but finitely many of the ϕi, sothat all of W0 is fixed by all but finitely many of the ϕi. Every one of those ϕilies in π1(Y, 0).

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

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

Proof. The topology on G was constructed to make this homomorphism con-tinuous. 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. Thenthe stabilizer of a single element is open in the induced topology. The opens inthe topology defined above are finite intersections of these.

Corollary 19.6.1. π1(X, 0) is the completion of G.


Proof. This is immediate by the uniqueness of the completion12 because π1(X, 0) ← 12is complete by13 , contains G as a dense subgroup,.13→

A Sheaves of groups and torsors

Definition 19.1. A sheaf of groups on a topological space X is a sheaf ofsets G with a map G × G → G such that, for every open U ⊂ X, the mapG(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 asheaf of sets F is a map G× F → F such that for every open U ⊂ X the mapG(U)× F (U)→ F (U) is a left action (resp. right action) of G(U) on F (U). Asheaf 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 tobe from the left.

If F and F ′ are left (resp. right) G-sets, a G-morphism ϕ : F → F ′ is amorphism of sheaves such that ϕ(gx) = gϕ(x) for each local section g of G andeach 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 andonly 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 atx 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 aG-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 undera locally constant sheaf of groups is locally constant.

12todo: reference13todo: 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 thatH1(X,G) is a pointed set : it contains a special point correspondingto the trivial G-torsor. Unless G is abelian, it does not generally have thestructure of a group.

Example 19.6. 1. R→ R/Z ' S1 is the espace etale of a Z-torsor. Givena 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, σ′ ∈ Γ(U,R) then σ − σ′ isa locally constant function U → Z. Therefore R is a pseudo-torsor overS1.14 ← 14

2. Let G be a sheaf of groups acting faithfully on a sheaf of sets F . Let F ′

be the sheafification of the presheaf quotient G\F (i.e., the colimit of thediagram G×F ⇒ F in the category of sheaves on X). There is a quotientmap p : F → F ′. There is a map

δ : H0(U,F ′)→ H1(U,G)

sending a section f of F ′ to the torsor F ′′ with F ′′(V ) = p−1(f∣∣V

) for allopen V ⊂ U . If x ∈ F ′′(V ) and g ∈ G(V ) then regarding x as an elementof F (V ) via the inclusion of F ′′(V ) ⊂ F (V ) we may act on x with g. Bydefinition gx ∈ F ′′(V ) so this gives F ′′ the structure of a G-torsor.

Lemma 19.7. (i) Let G, F , and F ′ be as in Example 2. Then the sequence

H0(X,F )→ H0(X,F ′)δ−→ H1(X,G)

is exact.

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

1→ H0(X,G′)→ H0(X,G)→ H0(X,G′′)δ−→ H1(X,G′)→ H1(X,G)

is exact.

(iii) Let G′ be a sheaf of normal subgroups of G and G′′ the sheaf of quotientgroups. Then the sequence

1→ H0(X,G′)→ H0(X,G)→ H0(X,G′′)

δ−→ H1(X,G′)→ H1(X,G)→ H1(X,G′′)

is exact and δ is a group homomorphism.

14todo: explain why it’s a torsor


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

0→ Z2πi−−→ O exp−−→ 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

as a continuous map S1 → Cexp−−→ 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 theunit circle in C∗ under the exponential map, hence is the torsor associated tothe map R→ Z.

We will see later that H1(S1,Z) ' Z and is generated by δ(f).

B Classification of torsors under locally constantgroups

19.1 Crossed homomorphisms and semidirect products

Definition 19.1. Suppose the group π acts (on the right) on another group Gby homomorphisms g 7→ gσ. A crossed homomorphism from π to G is a functionϕ : π → G such that

ϕ(στ) = ϕ(σ)τϕ(τ).

Definition 19.2. Let π act on the right on G. The elements of the semidirectproduct π n G are products σg with σ ∈ π and g ∈ G, with the multiplicationlaw

σgτh = στgτh.

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

(ii) Construct a bijection between the set of crossed homomorphisms π → Gand the set of sections of the projection π nG→ 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 throughthe 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 equivalentto give a group structure on Hom(X,G) for all X ∈ Ob(C ) such that forevery f ∈ Hom(X,Y ), the map Hom(f,G) : Hom(Y,G) → Hom(X,G) isa homomorphism.

(ii) Show that to specify a group object G of C is equivalent to giving mapsm : G × G → G, i : G → G, and eX : X → G for all X ∈ Ob(C ) suchthat 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 objectX—or a G-object—of C is a morphism a : G × X → X such that for everyobject Y of C , the induced map hG(Y )×hX(Y )→ hX(Y ) is a group action. Amorphism 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 actionof 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 exer-cise 19.5.

Definition 19.8. An action X of a group object G is called a pseudo-torsor ifhX(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 andonly 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 aG-object of the category of π-sets a G-set (instead of a G-π-set). A G-set is saidto 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 nG-sets.

(ii) The category of right G-torsors is equivalent to the category of πnG-setsY such that, for any y ∈ Y , the stabilizer subgroup of y in π nG projectsisomorphically to π under the map π nG→ 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 nG, set

yσg = yσ.g.

Note that we have

(yσg)τh = (yσ.g)τh = yστ .gτh

yσgτh = yστgτh = yστ .gτh.


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

(y.g)σ = (yg)σ = ygσ = yσgσ

= yσ.gσ

so the action is a morphism of π-sets.

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

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

We use this to understand the category of G-torsors itself. Let C be thefollowing strange category: objects are pairs (Y, y), where Y is a G-torsor andy ∈ Y , but morphisms are simply morphisms of G-torsors, ignoring y. Weanalyze 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 uniquelydetermined g ∈ G. We then have

StabπnG(z)g =α ∈ π nG

∣∣z.α = ϕ(y)

= g StabπnG(y).

So if s, t : π → π n G are the sections corresponding to StabπnG(y) andStabπnG(y) then we have gs = tg. Now, we can write s(σ) = σϕ(σ) andt(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 (rightG-actions in the category of π-sets) is equivalent to the category whose objectsare crossed homomorphisms π → G with

Hom(ϕ,ψ) =g ∈ G

∣∣∣gσϕ(σ) = ψ(σ)g.

19.3 Classification of torsors under pseudo-locally con-stant groups

Let G be a pseudo-locally constant group on a locally path connected topolog-ical space X. Note that cov(X) ' π1(X,x)-Sets so that G corresponds to agroup object G′ of π1(X,x)-Sets and the categories of G-sets and G′-sets areequivalent. 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 pointof X, and G a pseudo-locally constant sheaf of groups over X. There is anequivalence of categories between the category of G-torsors on X and the categoryof crossed homomorphisms π1(X,x)→ G defined in Theorem 19.12.

Proof. The only thing that remains to be proved is that G-torsors correspondunder the Galois equivalence to G′-torsors. Since pseudo-torsors are definedcategorically, G-pseudo-torsors correspond to G′-pseudo-torsors. But amongpseudo-torsors, the G-torsors may be characterized as the ones whose tauto-logical map to X (on the level of etale spaces) is surjective. Since the Galoiscorrespondence preserves surjections, G-torsors correspond exactly to the G′-pseudo-torsors that surject onto the π-set with a single object (note that thisis the π-set corresponding to X). To say that a set surjects onto the 1-elementset means that the set is non-empty, so that G-torsors do indeed correspond toG′-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 rightadjoint.

Proof. One direction is obvious, since ξ is a pullback functor. Consider thecollection of all open U ⊂ X such that ξ(hU ) 6= ∅ (since ξ∗ is left exact, asubfunctor of the final functor must pull back to a subfunctor of the final functorof Sets, hence must be ∅ or 1). This collection is closed under intersection. Ifξ(hU ) 6= ∅ then ξ(hU ) = 1 so ξ(hV ) = 1 for every V containing U . Furthermore,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,(iii) if U, V ∈ C then U ∩ V ∈ C, and (iv) if X =

⋃Ui then at least one Ui

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 closuresince X is covered by open subsets with compact closure. For each point x ∈ Uthere must be some Vx ∈ C with x 6∈ Vx. Finitely many of the Vx suffice tocover U . But then

⋂Vx = ∅ so ∅ ∈ C.

D The espace etale via the adjoint functor the-orem

We construct the espace etale using the adjoint functor theorem. This reducesto showing that, for any sheaf F on a topological space X, the functor on et(X)sending an etale space Y to Φ(Y ) = Hom(Y sh, F ) is representable by a an etalespace over X. By the adjoint functor theorem, it is enough to show that


1. et(X) admits arbitrary colimits,

2. Φ carries colimits to limits, and

3. et(X) is generated under colimits by a set.

For the first assertion, we must show that et(X) admits arbitary colimits andthat Φ preserves them. First consider a diagram of etale spaces Yi over X.Take the colimit in the category of topological spaces. If y ∈ lim−→Yi then we canrepresent y by some y′ in some Yi. Choose an open neighborhood U of y′ inYi that projects homeomorphically to an open subset of X. Then U → lim−→Yiis injective (since the composition U → lim−→Yi → X is injective) so U mapsbijectively to a subset V ⊂ lim−→Yi. Moreover, this is a homeomorphism, sinceU → V → X is a homeomorphism.

Now we show that Φ carries colimits to limits. It is sufficient to treat co-products and coequalizers. Suppose Yi is a collection of etale spaces and thatwe have maps Y sh

i → F . We extend these (in a unique way) to a map Y sh → Fwhere Y =

∐Yi. Suppose σ ∈ Γ(U, Y ). There is a cover of U by disjoint open

subsets Ui such that σ∣∣Ui

takes values in Yi (since the Yi are an open cover of

Y by disjoint subsets). On each Ui, the restriction σ∣∣Ui

lies in Γ(Ui, Yi), so we

obtain an element of F (Ui) from the map Y shi → 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 openneighborhood V0 of y0 ∈ Y0 and let U ′ be the image of V0. Then σ

∣∣U ′

lies in the

image of Γ(U ′, Y0), so we have a candidate for the image of σ∣∣U ′

in F (U ′). Wemust 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 pointsz1, . . . , zn with

p(z1) = y0

q(zi) = p(zi+1)

q(zn) = y1.

Each has an open neighborhood Wi 3 zi projecting homeomorphically to anopen neighborhood of x in X. Take the intersection of all of these neighborhoodsand call it U ′′. Then we have a sequence of sections τi ∈ Γ(U ′′, Y1) that relateσ0

∣∣U ′′

to σ1

∣∣U ′′

. Therefore there is an open neighborhood of x on which σ0 andσ1 agree.

Finally, we note that et(X) is generated under colimits by Open(X), whichis certainly a set.

Chapter 4

Commutative algebra


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

A. Grothendieck. Elements de geometrie algebrique (rediges avec lacollaboration de Jean Dieudonne). 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 com-mutative rings (also known as Andre–Quillen homology and cohomology). InSection 23, we only use H0 and H1, which are treated very cleanly in

R. G. Swan. Neron–Popescu desingularization.

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

A. Grothendieck. Categories cofibrees additives et complexe cotan-gent.

It also appears as Theoreme (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 com-mutative rings. The most basic, and perhaps the most fundamental, of theseproblems is a system of polynomial equations:

Given a collection of polynomials f1, . . . , fn ∈ Z[x1, . . . , xm], one may askfor the set of solutions to this system in any commutative ring A. In fact, ifwe define X(A) to be this set of solutions then X is a covariant functor fromcommutative 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 ringA. In this chapter we will explore some of the geometric properties of functorslike X and their relationships to algebraic properties of the commutative ringA.

The finiteness of the number of variables and the number of equations abovewas only for concreteness. In general, we may study the solutions to an arbitraryset of polynomials in an arbitrary number of variables. As every commutativering arises as the quotient of some polynomial ring by some ideal, there isa one-to-one (contravariant) correspondence between the functors introducedabove 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 writeSpecA 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→ SpecA transforms colimitsto limits, so this gives a construction of fiber products in the category of affineschemes.

Exercise 21.3. Suppose that A → B and A → C are homomorphisms ofcommutative rings and X = SpecA, Y = SpecB, and Z = SpecC. Verify thatY ×X Z ' SpecB⊗A C.

21.2 Some important schemes

We write An = Spec Z[x1, . . . , xn]. Note that An(A) = An for any commutativering 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 = SpecA 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 ringC (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 arecontinuous.

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. IfA = Z[x, y]/(x2+y2+1) thenX(R) is empty andX(C) is homeomorphicto 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 disjointunion of a circle and a line; X(C) is homeomorphic to a punctured torus.

21.4 The Zariski topology

We write |SpecA| for the set of prime ideals of A. This called the prime spec-trum of A and is the reason we use the notation Spec. If f : A → B is ahomomorphism 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 SpecB → SpecA is injective(meaning that |SpecB| → |SpecA| is injective).

Call V ⊂ |SpecA| closed if V = SpecB for some surjection A → B. Notethat B is not uniqely determined by V . The map SpecB → SpecA is called aclosed embedding.

Exercise 21.6. This is a topological space.

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

Say that X = SpecA is irreducible if any decomposition X = V ∪W withboth V and W closed in X has X = V or X = W .


21.5 Why etale morphisms?

Suppose that X is a scheme. Then there are at least two natural ways we mighttry to define a fundamental group of X. On one hand, we might look at the setX(C) of complex points of X. This can be given a topology (since the Zariskitopology, with respect to which the maps used to glue X are continuous, iscoarser than the topology of Cn). If x ∈ X(C) is a point then we can evaluateπ1(X(C), x).

On the other hand X has an underlying topological space |X|, and given apoint x ∈ |X|, we can compute π1(|X| , x).

Proposition 21.8. If X is an irreducible scheme then πn(|X| , x) = 0 for all nand all x.

Therefore the Zariski fundamental group of X is usually uninteresting. Wemay ask if there is an way to compute π1(X(C), x) algebraically (without relyingon the analytic topology on C). Of course, we cannot rely on maps from a circleinto 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, wecan define the fundamental group of X to be the automorphism group of a fiberfunctor. But what do we mean by a covering space of X?

The most naive definition—namely a map X ′ → X that is a covering spacein the Zariski topology—does not suffice. Consider the most basic nontrivialexample of a covering space:

Example 21.9. Let X = Spec C[t, t−1]. Then X(C) = C∗. There is a 2-to-1covering space C∗ → C∗ given by squaring: z 7→ z2. This map can be definedalgebraically: 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 Zariskitopology (consider the generic point).

Thus the Zariski topology is much too coarse to see many interesting coveringspaces. However, there is another characterization of covering spaces over C thatis better behaved.

Consider the differential of the map f : X → X constructed above. We havet = u2 so dt = 2udu. That is, infinitesimal motion in the base can be lifted—inexactly one way—to infinitesimal motion in the source. This is how we shalldefine etale morphisms.

Definition 21.10. A morphism of schemes f : X → Y is called formally etale ifinfinitesimal motion in Y can be lifted, in a unique way, to infinitesimal motionin X. That is, any commutative diagram of solid lines

S //

X

f

S′

??

// Y,

22. SMOOTH AND ETALE MORPHISMS 57

in which S ⊂ S′ is an infinitesimal extension, admits a unique lift.

22 Smooth and etale morphisms

“Etale” means “spread out” in French. Etale morphisms are meant to be thelocal isomorphisms in the category of schemes. We have seen that local isomor-phism in the Zariski topology does not include enough of the maps we wouldlike to consider local isomorphisms. However, in a geometric category there areother ways to characterize local isomorphisms. We will find that the followinggives a useful characterization:

A map f : X → Y is a local isomorphism if, for any point x ∈ X,and any infinitesimal motion y′ away from y = f(x) in Y , there is aunique way of lifting y′ 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 ⊂ S′ is called a nilpotentimmersion or an infinitesimal thickening if the ideal sheaf IS/S′ is nilpotent.

Our definition of an etale map will also involve the following condition

Definition 22.2. A morphism of schemes f : X → Y is called locally of fi-nite presentation (resp. locally of finite type) if f can be represented on thelevel of charts map maps SpecB → SpecA where B is an A-algebra of finitepresentation (resp. of finite type).

Definition 22.3. A morphism of schemes f : X → Y is called formally smoothif it has the right lifting property with respect to infinitesimal thickenings ofaffine schemes.1 It is called formally unramified if its diagonal is formally ← 1smooth. It is called formally etale if it is formally smooth and formally un-ramified.

If it is locally of finite type and formally unramified, it is called unramified (inthe 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 infinitesimalsense and unramified in the infinitesimal sense it is called etale (in the functorialsense).

Lemma 22.4. It is sufficient to consider just square-zero thickenings in thedefinitions of formally unramified, formally etale, and formally smooth mor-phisms.

Exercise 22.5. A morphism of schemes f : X → Y is formally unramified ifand only if its diagonal is formally smooth.

Lemma 22.6. If f and g are composable morphisms and g is etale then gf isetale if and only if f is etale.

1todo: local right lifting property?


22.2 The differential perspective

Definition 22.7. Let B → A be a homomorphism of commutative rings. Themodule of relative (Kahler) differentials is generated by the symbols df , for allf ∈ 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 meanΩA/Z.

Definition 22.8. Let B be a finitely generated polynomial algebra over A andlet I ⊂ B be a finitely generated ideal, and let C = B/I. Say that C is etale (inthe differential sense) over A if the map I/I2 → C ⊗B ΩB/A is an isomorphism.

Proposition 22.9. A finitely presented A-algebra C is etale over A in thedifferential sense if and only if it is etale in the infinitesimal sense.

Proof. Let δ denote the map I/I2 → C ⊗B ΩB/A.Suppose that C is etale in the infinitesimal sense. We show d is surjective.

We have ΩC/A = coker(δ). Consider the square-zero extension C+εΩC/A (whereε2 = 0) of B. The maps

x 7→ x+ εdx

x 7→ x+ ε0

from B to C+ εΩC/A coincide on A, hence must be the same. Therefore dx = 0for 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

C C

~~C ′

OO

A

OO

oo

(22.1)

must admit a completion, whenever C ′ is a square-zero extension of C. If J isthe ideal of C in C ′ then C ′ ' C + εJ . Therefore lifts of the diagram

C B

~~C ′

OO

A

OO

oo

(22.2)

can be identified with Hom(C ⊗B ΩB/A, J). On the other hand, any mapI/I2 → J induces a square zero extension C ′ → C by pushing out the dia-gram

0 // I/I2 //

B/I2 //

C //

0

0 // J // C ′ // C // 0.


We deduce that every map I/I2 → J factors uniquely through C ⊗B ΩB/A.Therefore I/I2 injects into C ⊗B ΩB/A.

Now we prove the converse. Consider a lifting problem in which D′ is asquare-zero extension of D with ideal J :

D Coo

g

B

OO

f~~D′

OO

A

OO

oo

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 mapI/I2 → C ⊗B ΩB/A is an isomorphism, so that we may choose a lift f such thatthe lift g exists.

Finally, lifts g form a torsor under Hom(ΩC/A, J), which is zero, since I/I2 →C ⊗B ΩB/A is surjective.

Corollary 22.9.1. If A→ B is an etale morphism of commutative rings thenthere is a finite type Z-algebra A0 and an etale A0-algebra B0 such that B =B0⊗A0

A.

Proof. There is certainly a finite type Z-algebra A0 and an A0-algebra of finitepresentation B0 such that B = B0⊗A0

A. Indeed, B is obtained by adjoiningfinitely many variables to A and imposing finitely many relations among them.We may take A0 to be the commutative ring generated by the coefficients inthose relations.

Note now that A is the filtered colimit of finitely presented A0-algebras Ai.For each i, let Bi = B⊗AAi. I claim that for i sufficiently large, Bi is etaleover Ai. Indeed, present B0 as C0/I0 for some finitely generated polynomialalgebra C0 over A0 and finitely generated ideal I0. Set C = C0⊗A0

A and let Ibe the ideal of B in C. Then consider the maps δi : Ii/I

2i → Bi⊗Ci ΩCi/Ai and

δ : I/I2 → B⊗C ΩC/A.We claim that δi is an isomorphism for i sufficiently large. Observe first

lim−→Ci = lim−→(Ai ⊗A0

C0) = (lim−→Ai) ⊗A0

C0 = A ⊗A0

C0 = C

andI = lim−→ ker(Ci → Bi) = ker(C → B)

since filtered colimits are exact. As colimits preserve quotients, tensor products,and the formation of ΩCi/Ai , we also have

I/I2 = lim−→ Ii/I2i

B⊗C

ΩC/A = lim−→Bi ⊗Ci

ΩCi/Ai .


SinceB0⊗C0 ΩC0/A0is finitely generated as aB0-module, and I/I2 → B⊗C ΩC/A

is surjective, every one of the finitely many generators appears in the image ofIi/I

2i for i 0. Replacing 0 by i, we may assume that δ0 is surjective.

Now choose a splitting σ0 : B0⊗C0ΩC0/A0

→ I0/I20 of δ0, which is guaran-

teed to exist since B0⊗C0ΩC0/A0

is a free B0-module. Set σi = Ai⊗A0σ0. Note

that σ = lim−→σi = A⊗A0 σ0 is surjective, so by the same argument as above, σiis surjective for i 0. Replacing i by this value, we find that σ0 is the inverseof δ0 and therefore that B0 is etale over A0.

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

(i) SpecC → SpecA has the unique right lifting property with respect to in-finitesimal extensions of affine schemes (i.e., C is an etale A-algebra).

(ii) SpecC → SpecA has the unique right lifting property with respect to lo-cal noetherian rings that are complete with respect to the maximal-adictopology and quotients by powers of the maximal ideal.

(iii) SpecC → SpecA 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), weconsider a lifting problem

D/mn Coo

||D

OO

Aoo

OO

in which D is a complete local noetherian ring with maximal ideal m. We realizeD as lim←−kD/m

k and observe that we get unique maps C → D/mk for all k ≥ 0

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 andB → C a surjection with finitely generated ideal I. By the differential criterionfor etale morphisms, Condition (i) holds if and only if

δ : I/I2 → C ⊗B

ΩB/A

is an isomorphism. This is an isomorphism if and only if k⊗C I/I2 → k⊗C C ⊗B ΩB/Ais an isomorphism for any map from C into a field k. Indeed, if this holds thenNakayama’s lemma implies δ is surjective. On the other hand, since C ⊗B ΩB/Ais free, we can then select a section and verify the surjectivity of the section thesame way.

Note now that for k fixed,

k⊗CI/I2 → k⊗

CC ⊗B

ΩB/A


is an isomorophism if and only if

Hom(C ⊗B

ΩB/A, k)→ Hom(I/I2, k) (22.3)

is an isomorphism. But we may interpret Hom(I/I2, k) as the set of square-zeroB-algebra extensions of C by k and we may interpret Hom(C ⊗B ΩB/A, k) asthe set of A-algebra maps B → C + εk. For (22.3) to be bijective thereforemeans that every B-algebra extension of C by k is isomorphic to C + εk as anA-algebra.

Now assume Condition (iii) and consider the a B-algebra extension C ′ ofC by k. Since C is an etale A-algebra, every A-algebra extension of C by k isisomorphic to C + εk: the diagram

C C

~~C ′

OO

Aoo

OO

has a lift, splitting the surjection C ′ → C. But to give a B-algebra extensionof C by k means to give an A-algebra extension C ′ by k and an A-algebra mapB → C ′. As C ′ is necessarily isomorphic to C + εk, this means that everyB-algebra extension of C by k is a homomorphism B → C + εk, as desired.

Proposition 22.10. Let A be a commutative ring. An A-algebra C is etale ifand only if it can be presented as A[x1, . . . , xn]/(f1, . . . , fn), where the image of

det(∂fi∂xj

)i,j

in C is a unit.

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.

Consider the map δ : I/I2 → C ⊗A ΩB/A. Note that I/I2 is generated bythe images of f1, . . . , fn and that C ⊗A ΩB/A is free with basis dx1, . . . , dn. In

terms of these bases, δ is given by the matrix(∂fi∂xj

)i,j

. That is,

fi 7→∑j

∂fi∂xj

dxj .

We have the following commutative diagram:∑Cfi

∼ //

∑Cdxj

o

I/I2 δ // C ⊗A ΩB/A

The upper horizontal arrow is invertible by assumption, and the map∑Cfi →

I/I2 is surjective since the fi generate I/I2. It follows that δ is bijective.


Now we consider the converse. Suppose that C is etale so that the mapδ : I/I2 → C ⊗A ΩB/A is bijective. As ΩB/A is free with basis dx1, . . . , dxn, sois I/I2. Let f1, . . . , fn be elements of I whose images form the basis of I/I2

corresponding to the dxi. Then as before, δ(fi) =∑ ∂fi

∂xjdxj so that the matrix(

∂fi∂xj

)i,j

is the identity, and in particular is invertible.

The only thing left to check is that the fi generate I. We have I ≡ I2

(mod f1, . . . , fn). But by assumption I is finitely generated, so by Nakayama’slemma 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 ofcommutative rings. Then the induced sequence of C-modules

C ⊗B

ΩB/A → ΩC/A → ΩC/B → 0

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

I/I2 d−→ C ⊗B

ΩB/A → ΩC/A → 0

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 thenthe 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 derivationC × I → J consists of a derivation C → J and a homomorphism I/I2 → J ,which may be selected independently. Therefore ΩB/A ' ΩC/A × I/I2.

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

H1(C/B) = ker(I/I2 → C ⊗BH0(B/A)).

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 freeand 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 surjectionB1 → B2 with ideal J . By Proposition 23.2, we have a split exact sequence

0→ J/J2 → B2 ⊗B1

H0(B1/A)→ H0(B2/A)→ 0.

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

0→ C ⊗B2

J/J2 → C ⊗B1

H0(B1/A)→ C ⊗B2

H0(B2/A)→ 0.

Now consider the following commutative diagram with exact rows and columns

0

0

K1

//

K2

0 // J/(I2

1 ∩ J) //

I1/I21

//

I2/I22

//

0

0 // C ⊗B1 J/J2 // 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 computingwith B1 and B2, respectively.

We check that J/(I21 ∩ J) → C ⊗B1

J/J2 is an isomorphism. Note thatC ⊗B1

J/J2 = J/(J2 + I1J) = J/I1J since J ⊂ I1. On the other hand, wecan split B1 as B2 × J by choosing a section, under which splitting we haveI1 = I2 × J . Then I2

1 = I22 × I1J so that I2

1 ∩ 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 ofsquare-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 extensionsof C by J . We construct maps in either direction between Hom(H1(C/A), J)

2todo: Explain what isomorphism classes of extensions are


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

0 // I //

B //

C // 0

0 // J // C ′ // C // 0

with the map I → J induced by the universal property of the kernel. But J2 = 0so we get a map I/I2 → 0. Composing with the map H1(C/A)→ I/I2, we geta map H1(C/A)→ J .

Note that this does not depend on the choice of lift B → C ′. Indeed, if wereplace 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) bythe exactness of the sequence

H1(C/A)→ I/I2 → C ⊗B

ΩB/A.

Now we construct a map Hom(H1(C/A), J)→ ExalA(C, J). SinceH1(C/A) ⊂I/I2 and J is injective, any map H1(C/A) → J may be extended to a mapI/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 // C ′ // 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 → C ′ in thediagram above and does not change C ′ as an A-algebra extension of C.

We leave it as an exercise to verify that these constructions are inverse toone another.

Proposition 23.5. Suppose B → C is a homomorphism of A-algebras. Thereis a long exact sequence

C ⊗BH1(B/A)→ H1(C/A)→ H1(C/B)

δ−→ C ⊗BH0(B/A)→ H0(C/A)→ H0(C/B)→ 0.

24. HOMOLOGY OF MODULES 65

Proof. We can verify exactness by showing that the sequence is exact afterapplying 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 Andre–Quillenhomology. When we discuss Grothendieck topologies we will see how to extendthe sequence.

Definition 23.7. An A-algebra B of finite type is called unramified (in thehomological sense) if H0(B/A) = 0. It is called smooth (in the homologicalsense) if H1(B/A) = 0 and it is of finite presentation. It is called etale (in thehomological sense) if it is smooth and unramified.

Proposition 23.8. An A-algebra B is etale in the homological sense if andonly if it is etale in the differential sense.

Proof. Choose a finitely generated free A-algebra C and an A-algebra surjectionC → B with finitely generated ideal I. Then we have a long exact sequence

B⊗CH1(C/A)→ H1(B/A)→ H1(B/C)→ B⊗

CH0(C/A)→ H0(B/A)→ H0(B/C).

Note that H1(C/A) = 0 because C if free and H0(C/B) = 0 because C surjectsonto B. Substituting H1(B/C) = I/I2 and H0(C/A) = ΩC/A, we get

0→ H1(B/A)→ I/I2 δ−→ B⊗C

ΩC/A → H0(B/A)→ 0

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 Mand P/Q where P is a free A-module and Q is a submodule. Define

TorA1 (M,N) = ker(N ⊗AP → N ⊗

AQ).

It is not obvious that this definition is well posed. However, if we had anotherpresentation of M as P ′/Q′ then we could find a commutative diagram:

0 // Q′ //

P ′ //

M // 0

0 // Q // P // M // 0


We may assume without loss of generality that P ′ → P (and therefore alsoQ′ → Q) is surjective. The kernels of Q′ → Q and P ′ → P are isomorphic, sowe write K for the common value. Then we get a commutative diagram withexact rows and columns:

0

N ⊗AK

N ⊗AK

0 // R′ //

N ⊗AQ′ //

N ⊗A P ′ //

N ⊗AM // 0

0 // R // N ⊗AQ //

N ⊗A P

// N ⊗AM // 0

0 0

The map N ⊗AK → N ⊗A P ′ is injective because P ′ → P can be split. Apply-ing the snake lemma shows that R′ → R is an isomorphism, so the definition ofTor1(M,N) does not depend (up to canonical isomorphism) on the choice of Pand Q.

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

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

0→ N ′ → N → N ′′ → 0

is an exact sequence of A-modules. Then there is an exact sequence

TorA1 (M,N ′)→ TorA1 (M,N)→ TorA1 (M,N ′′)

→M ⊗AN ′ →M ⊗

AN →M ⊗

AN ′′ → 0.

Proof. Choose a commutative diagram with exact rows and colums and P ′, P ,

24. HOMOLOGY OF MODULES 67

and P ′′ free:0

0

0

0 // Q′ //

Q //

Q′′ //

0

0 // P ′ //

P //

P ′′ //

0

0 // N ′ //

N //

N ′′ //

0

0 0 0

Tensor with M to get

0

0

0

Tor1(M,N ′) //

Tor1(M,N) //

Tor1(M,N ′′)

0 // Q′⊗AN //

Q⊗AN //

Q′′⊗AN //

0

0 // P ′⊗AN //

P ⊗AN //

P ′′⊗AN //

0

M ′⊗AN //

M ⊗AN //

M ′′⊗AN //

0

0 0 0

Now apply the snake lemma.

Corollary 24.2.1. Suppose that

0→ N ′ → N → N ′′ → 0

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


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

0 //

Tor1(N,M)

Q⊗AK //

P ⊗AK //

M ⊗AK //

0

0 // Q⊗A L //

P ⊗A L //

M ⊗A L //

0

Tor1(M,N) // Q⊗AN //

P ⊗AN //

M ⊗AN //

0

0 0 0

Apply the snake lemma. (Note that the left exactness of the middle row andcolumn 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 chartsSpecB → SpecA, the ring B is a flat A-algebra.33→

Proposition 25.2. An A-module M is flat if and only if Tor1(M,N) = 0 forall 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 andonly if J ⊗AM → A⊗AM 'M is injective for all ideals J of A.

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

0→ Ni → Ni+1 → A/xiA→ 0.

But xiA⊗AM → 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 thatTor1(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 ⊂ Aan ideal, and M a finitely generated A-module. For any submodule M ′ of M ,there is an integer k such that Ik+nM ′ = In(M ′ ∩ IkM) for n ≥ 0.

Proof. This proof is adapted from [Wik].Form the Rees algebra A[tI] =

∑∞n=0 t

nIn. As A is noetherian, I is finitelygenerated, so that A[tI] is a finitely generated A-algebra, hence is noetherianas well. We can also form the A[tI]-module A[tI]M =

∑∞n=0 t

nInM . This isgenerated by M = t0M as an A[tI]-module, so it is finitely generated becauseM is.

Consider the submodule A[t]M ′ ∩ A[tI]M =∑∞n=0 t

n(M ′ ∩ InM). This isan A[tI]-submodule of A[tI]M , hence is finitely generated. Suppose that thegenerators of A[t]M ′ ∩A[tI]M lie in degrees ≤ k. Then looking in degree n+ k,with n ≥ 0, we discover

M ′ ∩ In+kM = In(M ′ ∩ IkM).

26 The equational criterion for flatness

Let M be an A-module. A relation among elements x1, . . . , xn ∈M is a choiceof 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

An⊗AM

(f1,...,fn)⊗A idM−−−−−−−−−−−→ A⊗AM = M.

If K is the kernel of(f1, . . . , fn) : An → A

then the elements of K may be thought of as relations in A. Every relation inA induced a relation in M by the map

K ⊗AM → An⊗

AM.

Let M and N be a A-modules. An M -linear relation in N is an expression∑xi⊗ yi = 0

inM ⊗AN . More specifically, we call this anM -linear relation among y1, . . . , yn ∈N . Equivalently, anM -linear relation among y1, . . . , yn is an element (x1, . . . , xn)of the kernel of

Mn = M ⊗AAn

idM ⊗ y−−−−−→M ⊗AN.

One can use A-linear relations in N to construct M -linear relations. Indeed, anA-linear relation on y1, . . . , yn ∈ N is a tuple (f1, . . . , fn) such that∑

fiyi = 0.


Then for any x ∈M , ∑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 . Thatis, if F = (fij), i = 1, . . . , n, j = 1, . . . ,m is a family of A-linear relationson y1, . . . , yn, and x1, . . . , xm are elements of M then F (x1, . . . , xm) gives anM -linear relation on y:

n∑j=1

n∑i=1

fijxi⊗ yj .

All of this can be said more efficiently in the following way: The M -linearrelations on y ∈ N are the kernel of idM ⊗ y : M ⊗AAn → M ⊗AN . TheA-linear relations on y are the kernel of y : An → N . Then there is a mapM ⊗A ker(y)→ ker(idM ⊗ y).

We say that every M -linear relation on y ∈ Nn is induced from A-linearrelations on N if the map M ⊗A ker(y) → ker(idM ⊗ y) is bijective. Note thatif M is a flat A-module then the sequence

0→M ⊗A

ker(y)→M ⊗AAn →M ⊗

AN

is exact so that every M -linear relation on y ∈ Nn is induced from an A-linearrelation on N . This proves half of the following theorem:

Theorem 26.1 (Equational criterion for flatness). An A-module M is flat ifand only if, for every A-module N , and for every y ∈ Nn, every M -linearrelation 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 →M ⊗AA = M can be expressed as

∑xi⊗ yi = 0 where xi ∈ M and yi ∈ J .

Let J ′ =∑yiA be the ideal generated by yi. Then

∑xi⊗ yi also lies in the

kernel of M ⊗A J ′ → M . It will be sufficient to show that∑xi⊗ yi is zero in

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 ⊗AK →M ⊗

AAn →M ⊗

AA→M ⊗

AA/J → 0

is exact (exactness at M ⊗AA and M ⊗AA/J is the right exactness of tensorproduct; exactness at the left two terms is from the assumption on M). On theother hand, the right exactness of tensor product also gives us

M ⊗AK →M ⊗

AAn →M ⊗

AJ → 0

so that the image of M ⊗AAn → M ⊗AA = 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 ′ → N → N ′′

and

M ⊗AN ′ →M ⊗

AN →M ⊗

AN ′′

to be flat.

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

Proof. If M is faithfully flat and M ⊗AN = 0 then 0→ M ⊗AN → 0 is exactso 0→ N → 0 is exact so N = 0.

Now suppose M ⊗AN = 0 implies N = 0 and consider a sequence4 ← 4

0→ N ′ → N → N ′′.

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

ker(N ′ → K) = coker(N ′ → K) = 0.

This is equivalent to

M ⊗A

ker(N ′ → K) = M ⊗A

coker(N ′ → K) = 0.

But:

M ⊗A

coker(N ′ → K) = coker(M ⊗AN ′ →M ⊗

AK)

M ⊗A

ker(N ′ → K) = ker(M ⊗AN ′ →M ⊗

AK)

The first line is always true; the second line is because M is flat. Reversing whatwe did above, we deduce that these equalities hold if and only if M ⊗AN ′ →M ⊗AK is an isomorphism. But

M ⊗AK = ker(M ⊗

AN →M ⊗

AN ′′)

so this is the same as

0→M ⊗AN ′ →M ⊗

AN →M ⊗

AN ′′

being exact.

4todo: explain why it’s enough to consider sequences like this


30 Local criteria for flatness

Definition 30.1. An A-module M is said to be ideally separated if, for anyideal J ⊂ A, ⋂

In(J ⊗AM) = 0.

Proposition 30.2. Suppose that u : A→ B is a local homomorphism of noethe-rian 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 ⊗AM . Note that MB

is finitely generated. Then⋂In(J ⊗

AM) =

⋂InBMB .

Set N =⋂InBMB . Then IBNB = NB : Artin–Rees guarantees that there is a

k such that N = N ∩ Ik+1B MB = IB(N ∩ IkBMB) ⊂ 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 aNoetherian ring), it follows that N = 0.

Theorem 30.3 (Infinitesimal criterion for flatness). Let M be a module of finitetype over a noetherian local ring A whose maximal ideal is denoted I. Assumethat M is ideally separated. It is then equivalent for M/InM to be flat overA/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 Ais noetherian. It will be sufficient to show that for any such ideal, the map ofA-modules

ϕ : J ⊗AM →M

is injective.55→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) ⊗A/In

M/InM →M/InM.

The commutativity of the diagram

J ⊗AMϕ //

pn

M

J/(J ∩ In)⊗A/InM/InM // M/InM

shows thatker(ϕ) ⊂

⋂n

ker(pn).

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

qn : J ⊗AM

pn+k−−−→ J/(J ∩ In+k) ⊗A/In

M/InM ' J/In(J ∩ Ik) ⊗A/In

M/InM

→ J/InJ ⊗A/In

M/InM ' A/In⊗AJ ⊗AM.

Therefore ⋂n

ker(pn) ⊂⋂n

ker(qn).

On the other hand, ker(qn) = In(J ⊗AM). By assumption, M is ideally sepa-rated, so

⋂In(J ⊗AM) = 0. Thus,

ker(ϕ) ⊂⋂

ker(pn) ⊂⋂

ker(qn) =⋂In(J ⊗

AM) = 0.

Theorem 30.4 (Local criterion for flatness). Suppose that A is a noetherianlocal ring with maximal ideal I. If M is an ideally separated A-module then Mis flat over A if and only if Tor1(M,A/I) = 0.

Proof. The proof is similar to the infinitesimal criterion. Suppose that J ⊂ Ais an ideal and consider the map ϕ : J ⊗AM → M . We may fit this into adiagram with exact rows:

J ⊗AMϕ //

pn

M

Tor1(A/(In + J),M) // J/(In ∩ J)⊗AM // A/In⊗AM // A/(In + J)⊗AM // 0

The lower row is obtained by tensoring the exact sequence

0→ J/(In ∩ J)→ A/In → A/(In + J)→ 0

with M . But A/(In +J) has finite length, so Tor1(A/(In +J),M) = 0. There-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

qn : J ⊗AM → J/InJ

and deduce that ker(pn) ⊂ ker(qn). But ker(qn) = In(J ⊗AM). Therefore,

ker(ϕ) ⊂⋂

ker(pn) ⊂⋂

ker(qn) =⋂In(J ⊗

AM),

which is zero because M is ideally separated.


31 Flatness of etale maps

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

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 derivationL→ K, together with elements δ(xi) such that if fi is the minimal polynomialsatisfied by xi then δ(fi) = 0. Of course, δ(fi) = f ′i(xi)δ(xi). In particular,we can see that any derivation of L into K extends to a derivation of K intoK by setting δ(xi) = f ′i(xi)

−1δ(fi) if f ′i(xi) 6= 0 and choosing δ(xi) = 0 iff ′i(xi) = 0. If K is unramified over k then there is a unique derivation of K intoitself (namely zero), so we must have L = k.

Even more precisely, we can see that if f ′i(xi) = 0 then δ(xi) may be chosenarbitrarily and therefore there is more than one element of Derk(K,K). Wededuce that f ′i(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 etale algebra over a field is a finite product of separable fieldextensions.

Proof. Let k be a field and C an etale k-algebra. Let m be a maximal ideal ofC and K = C/m the quotient. Note that ΩK/k is a quotient of ΩC/k = 0 soΩK/k = 0. Therefore K is unramified over k, so it is a finite separable extension.In particular, it is etale over k.

This implies that C → K is etale. Recall that Hom(m/m2, J) may beidentified with the isomorphism classes of C-algebra extensions of K by J . AC-algebra extension of K by J is a commutative diagram:

K Coo

~~K ′

OO

k

OO

oo

But there is only one such: Up to unique isomorphism, there is only one k-algebra extension K ′ of K by J because K is etale over k. Once K ′ has beenfixed, there is a unique diagram as above because C is etale over k.

It follows that m/m2 = 0. By Nakayama’s lemma, mCm = 0 so there issome y ∈ C r m such that ym = 0 (because m is finitely generated: C isnoetherian because it is of finite type over a field). Therefore yC ⊂ C projectsisomorphically to K.

Let e be the (unique) element of yC that projects to 1 ∈ K. Then e2 = ebecause e and e2 both project to 1 ∈ K. That is, e is idempotent. Then1 − e is also idempotent and we have C = (1 − e)C × eC ' (1 − e)C ×K andeC = yC ' K via the projection C → K.

6This is the uniqueness part of the definition of an etale morphism. Equivalently ΩK/k = 0.

31. FLATNESS OF ETALE MAPS 75

In particular, C ′ = (1−e)C is also an etale k-algebra. If C ′ 6= 0 then we canrepeat the argument on a maximal ideal m′ of C ′, we find that C ′ = K ′ × C ′′.Note that the ideals K ⊂ K×K ′ ⊂ K×K ′×K ′′ ⊂ · · · form an ascending chainin C. Therefore this process must terminate after a finite number of steps andwe find that C is a product of fields. Each of these fields is finite and separableover k by the previous lemma.

Theorem 31.3. An etale morphism of commutative rings is flat.

Let A → C be an etale morphism. There is a noetherian ring A0 and anetale map A0 → C0 such that C = A⊗A0

C0. It is therefore sufficient to assumethat 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 andonly if Mp is zero for all prime ideals p of A.7

Proof. For each x ∈ M define Ix to be the set of all f ∈ A such that fnx = 0for some n ≥ 0. Then I =

⋂x∈M Ix.

If p is a prime then Mp = 0. By definition, this means that for each x ∈Mthere is some f 6∈ p and some n ≥ 0 such that fnx = 0. That is Ix is notcontained in p. This holds for all primes p, so Ix is not contained in any maximalideal. Hence Ix = A. In particular, 1nx = 0 for some n, so x = 0. This holdsfor all x ∈M , so M = 0.

Corollary 31.4.1. Let A be a commutative ring. The localizations Ap of Aform 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 ofa commutative ring B is called an M -regular sequence if for all 1 ≤ i ≤ n, theelement 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. Anypermutation of an M -regular sequence is also M -regular.

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

7In fact, maximal ideals suffice.


that IN = N (where I is the maximal ideal of B) so N = 0 by Nakayama’slemma. 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 somev ∈ M and we have xz = xyv. Rearranging gives x(z − yv) = 0 and since x isnot 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 flatbase extension.8 Since A is a field, any field extension A′ ⊃ A is a faithfullyflat extension. We select an algebraic closure of A for A′. We replace B withB′ = B⊗AA′ and C with C ′ = C ⊗AA′. We may now assume that A is analgebraically closed field.

Since C is etale over the algebraically closed field A, it must be a productof copies of A.9 The image of the map SpecC → SpecB therefore consists9→of a finite collection of maximal ideals. To verify that (f1, . . . , fn) is a regularsequence in B, it is sufficient to show that (f1, . . . , fn) is a regular sequencein Bq for every prime ideal q in the image of SpecC. (It is trivially a regularsequence at the primes that are not in the image.)

We have Cq ' A for each of these primes. Denoting by ξi the image of xiunder 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 reg-ular 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 thisway, 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 Mbe a flat A-module. If f ∈ B is a non-zero divisor in M/I then M/fM is a flatA-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

0→Mf−→M →M/fM → 0

giving rise to the exact sequence below:

0→ Tor1(M/fM,A/I)→M/IMf−→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 divi-sor in B/(I + (f1, . . . , fi−1)) so by induction B/(f1, . . . , fi−1) is flat over A forall i. In particular, C is flat over A.

A Extending etale maps

Proposition 31.1. Let10 A be a commutative ring and C an etale A-algebra. ← 10Suppose that A′ → A is a nilpotent extension. Then there is an extension of Cto an etale A′-algebra and this extension is unique up to a unique isomorphismthat commutes with the projection to C.

Proof. First we consider the uniqueness. Suppose C ′ and C ′′ are two extensions.We can fit them into a commutative diagram:

C C ′′oo

f

C ′

OO

A′

OO

oo

A unique dashed arrow rendering the whole diagram commutative is guaranteedto exist by the definition of a formally etale morphism of commutative rings.Similarly, there is a unique map g : C ′′ → C ′. The uniqueness in the definitionof etale maps shows furthermore that fg = idC′′ and gf = idC′ .

Now we consider the existence. We may present C as B/I where B =A[x1, . . . , xn] and I = (f1, . . . , fn) and det

(∂fi∂xj

)i,j

is a unit in C.11 Lift f1, . . . , fn ← 11

to polynomials f ′1, . . . , f′n ∈ A′[x1, . . . , xn] and set C ′ = A′[x1, . . . , xn]/(f ′1, . . . , f

′n).

Then( ∂f ′i∂xj

)i,j

reduces to a unit in C. But C ′ is a nilpotent extension of C, so

an element of C ′ is a unit if and only if its image in C is a unit.12 Therefore C ′

is etale as an A′-algebra by13 ← 13

10todo: include ref to EGA; maybe just delete the prop11todo: reference12Suppose α reduces to a unit in C = C′/J and J is nilpotent. Let β be an element that

lifts the inverse of the image of α in C to C′. Then αβ ≡ 1 (mod J). Therefore αβ = 1 − γfor some γ ∈ J . But (1− γ)−1 = 1 + γ + γ2 + · · · and this series is finite because γ ∈ J andJ is nilpotent. Therefore β(1− γ)−1 is the inverse of α.

13todo: reference


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

SymC(I/I2)→ grI(B)

is an isomorphism.

Proof. First consider the commutative diagram

C C

||B/In

OO

Aoo

OO

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. NowI/I2 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

Sym≤nC (I/I2)→ B/In+1 with nilpotent kernel, whence a commutative diagram

B/In+1 Boo

zzSym≤nC (I/I2)

OO

A

OO

oo

in which the dashed arrow again exists because B is an etale A-algebra. Thisinduces a section B/In+1 → Sym≤nC (I/I2). But both Sym≤nC (I/I2) and B/In+1

are generated by I/I2, so that the map B/In+1 → Sym≤nC (I/I2) must be sur-jective. As it is a section it is also injective, so it is bijective. In particular, thismap is an isomorphism in each graded degree, which proves the lemma.

Proposition 31.3. Let A be a commutative ring and C an etale A-algebra.Let A′ be a square-zero extension of A by an ideal J . For any homomorphismJ → K (where K is given the A-module structure indcued by the homomorphismA→ C) there is a compatible A′-algebra extension of C by K.14

Although this proposition is true as stated, we will treat only the case whereA is noetherian and K is of finite type.

14The meaning of compatibility is that the exntesions fit into a commutative diagram

0 // J //

A′ //

A //

0

0 // K // C′ // 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) anddet(∂fi∂xj

)i,j

is a unit in C. We may regard K as a B-module. There is certainly

an extension of B by K as a A′-algebra: first take B′ = A′[x1, . . . , xn], which isan extension of B by B⊗A J and then push out to form B′′:

0 // B⊗A J //

B′ //

B // 0

0 // K // B′′ // B // 0.

Now we consider the question of finding an extension C ′ of C by K fitting intoa commutative diagram

0 // K // B′′ //

B //

0

0 // K // C ′ // C // 0.

Recall that C = B/I where I = (f1, . . . , fn). Choose lifts f ′1, . . . , f′n of the fi to

B′′ and let I ′ ⊂ B′′ be the ideal they generate.I claim that I ′ ∩K = 0. It is sufficient to show that I ′p ∩Kp = 0 for every

prime ideal p of B (note that as sets SpecB → SpecB′′ is a bijection). Ifp ∈ SpecB is not in the image of SpecC then Kp = 0 (as K is a C-module), sothe assertion is trivially true. We therefore only need to consider primes p thatare pre-images of primes of C.

Now suppose that∑aif′i ∈ Kp with ai ∈ B′′p . Reducing modulo K + I ′, we

find that∑aifi = 0 in C. As I ′p/(I

′p

2+ I ′p ∩ K) = Ip/I

2p is freely generated

by the fi, this means that ai ≡ 0 mod I ′p +Kp for all i. That is,∑aif′i lies

in (I ′p + Kp)I ′p = I ′p2

since KpI′p = 0. A fortiori,

∑aif′i lies in I ′p

2 ∩Kp. Butnow we can repeat the argument, using the fact that I2

p/I3p is freely generated

by the fifj (Lemma 31.2), to deduce that I ′p ∩Kp is contained in I ′p3 ∩K. By

induction, we get

I ′p ∩K ⊂⋂n≥0

I ′pn.

But this intersection is zero because I ′p is a proper ideal of a local noetherianring.15 ← 15

We may now define C ′ = B′′/I ′. Since K ∩ I ′ = 0, the ideal of C in C ′ isK, and we have the required extension.

Now we analyze what an infinitesimal extension of an etale map looks like.

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

15todo: Explain why B′′ is noetherian? Because it is a square-zero extension of a noetherianring by a finitely generated module...


f . Suppose that M and N have compatible finite filtrations by submodules Mi

such that C ⊗AMi/Mi−1 → Ni/Ni1 is an isomorphism for all i. Then ϕ is anisomorphism.

Proof. By induction. As a base case, we have C ⊗AM0 ' N0. Then we havean exact sequence

0→Mi−1 →Mi →Mi/Mi−1 → 0

inducing a commutative diagram

0 // C ⊗AMi−1α //

β

C ⊗AMi//

C ⊗AMi/Mi−1//

0

0 // Ni−1γ // Ni // Ni/Ni−1

// 0

The leftmost vertical arrow is an isomorphism by induction and the rightmostis 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 A′ → A be an infinitesimal extenesion of commu-tative noetherian rings with ideal J , let C ′ be an etale A′-algebra, and letC = C ′⊗A′ A. Then the ideal of C in C ′ is isomorphic to J ⊗A′ C ′.

Proof. First we reduce to the case where J2 = 0. We can filter the extensionsA′ → A and C ′ → C as sequences of square-zero extensions

A′ = A′0 → A′1 → · · · → A′n = A

C ′ = C ′0 → C ′1 → · · · → C ′n = C.

This gives filtrations on J = IA/A′ and IC/C′ by the IA′/A′j and the IC′/C′j . By

the square-zero case, we have

IA′j/A′j+1⊗A′C ′ ' IC′j/C′j+1

so Lemma 31.4 implies that J ⊗A′ C ′ → IC/C′ is an isomorphism.Now we consider the square-zero case. We consider the functor represented

by I = IC/C′ on C-modules. Giving a C-module homomorphism I → K inducesan square-zero extension of C by K as an A′′-algebra by pushing out the diagram

0 // I //

C ′ //

C // 0

0 // K // C ′′ // C // 0.

Conversely, if we have any such extension we may consider the diagram

C C ′oo

C ′′

OO

A′oo

OO

B. COMPLETIONS OF RINGS 81

which admits a unique completion because C ′ is etale over A′, hence gives amap I → K. We conclude that I represents the functor sending a C-module Kto the set of square-zero A′-algebra extensions of of C by K.

On the other hand, we may consider the functor represented on C-modulesby J ⊗A C,16 which sends a C-module K to HomA(J,K). Therefore the propo-sition reduces to demonstrating that for any A-module homomorphism J → K,there is a square-zero extension C ′′ of C by K as an A′-algebra such that theinduced map on ideals J → K coincides with the given one.

Theorem 31.6. An etale homomorphism of commutative rings is flat.

Proof. Suppose A→ C is etale. There is a noetherian ring A0 and an etale A0-algebra C0 such that C is induced by base extension via a map A0 → A. It istherefore sufficient to treat the case where the base ring is noetherian. Withoutloss 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 ringin 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 anisomorphism. 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 ′ ⊂M finitely generated A-modules. The I-adic topology on M ′ is inducedfrom the I-adic topology on M .

Corollary 31.1.1. The functor M 7→ lim←−M/InM from finitely generated A-

modules to A-modules is exact.

Proof. Consider an exact sequence:

M ′f−→M

g−→M ′′

Suppose that x ∈ lim←−M/InM has zero image in lim←−M′′/InM ′′. Choose yn ∈M

such that yn ≡ x (mod InM). Then g(yn) ≡ 0 (mod InM ′′). That is, g(yn) ∈InM ′′, or, equivalently, yn ∈M ′+ InM . Choose an element zn ∈M ′ such thatzn − yn ∈ InM .

By construction the zn converge in the I-adic topology on M to x. Thisimplies that they converge I-adically to an element of lim←−M

′/InM ′, since theI-adic topology on M ′ is the same as the one induced from M . The limitof an I-adically convergent sequence is unique so x must lie in the image oflim←−M

′/InM ′ → lim←−M/InM .

16Note that J is an A-module because J2 = 0.


Corollary 31.1.2. Let A be a noetherian ring and I ⊂ A an ideal. Let A =lim←−A/I

n be the completion of A at I. Then A is a flat A-algebra.

Proof. For the flatness, it will be sufficient to show that lim←−M/InM 'M ⊗A Awhen M is a finitely generated A-module. There is certainly a map

J ⊗AA→ lim←− J/I

nJ.

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 considerthe following diagram:

J ⊗AQ //

J ⊗A P //

J ⊗A A //

0

lim←−Q/InQ // lim←−P/I

nP // lim←− J/InJ // 0

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

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

Proof. Let I be the maximal ideal of A. Consider a finitely generated A-moduleM . Then the map M → A⊗AM = lim←−M/InM is is injective. Indeed, letN =

⋂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) = InNso we deduce that IN = N . By Nakayama’s lemma, we get N = 0.

This proves that A⊗AM = 0 if and only if M = 0 when M is finitelygenerated. Suppose that M is a general A-module such that A⊗AM = 0.Then for any finitely generated submodule M ′, we have A⊗AM ′ ⊂ A⊗AMsince A is flat. Therefore A⊗AM ′ = 0 so M ′ = 0. But then every finitelygenerated submodule of M is zero so M = 0.

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

(i) A is noetherian.

(ii) grA 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 quotientof I.

(ii) =⇒ (i). Let J be an ideal of A. Each of the generators of J hasa well-defined image in the associated graded ring, and finitely many of thesesuffice to generate the associated ideal. Let x1, . . . , xn be these generators. Forany yn ∈ J ∩ In there is therefore an expression yn ≡

∑aixi (mod J ∩ In+1).

Select zn =∑aixi such that Then yn+1 = yn − zn ∈ J ∩ In+1. Then select

zn+1 =∑bixi so that we can write yn+1 − zn+1 ∈ J ∩ In+1. Repeating this

procedure, we obtain a convergent series∑zk = y. Thus J is generated by

x1, . . . , xn.(iii) =⇒ (ii). We can construct grA 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 noethe-rian 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 ⊂ Jbecasue A → B is a local homomorphism. Therefore the topology on B is nofiner than the I-adic topology. On the other hand, B/IB is a finite dimensionalvector space over A/I by assumption, so the subspaces Jn(B/IB) must stabilize.But they can’t stabilize anywhere other than zero since

⋂Jn = 0.

Now let x1, . . . , xn be representatives of a basis for B/IB in B. For anyyn ∈ InB, we may find zn =

∑aixi such that yn − zn ∈ In+1B. Applying this

inductively to any y ∈ B we obtain a series∑zn that converges I-adically to

y. But the submodule of B generated by x1, . . . , xn is I-adically complete,18 soit contains y.

C Zariski’s “Main Theorem”

Here we will give a second proof of the flatness of etale maps, based on Zariski’s“Main Theorem”.

Definition 31.1. A homomorphism of commutative rings A → B is calledfaithfully flat if a sequence of A-modules

M ′ →M →M ′′

is exact if and only if the induced sequence

B⊗AM ′′ → B⊗

AM → B⊗

AM ′′

17A homomorphism of local rings f : A → B with maximal ideals I and J is called localif f−1(J) = I. This corresponds to the geometric assertion that the map SpecB → SpecAcarries 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.


is exact.

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

Lemma 31.3. Suppose that f : A → B is a quasi-finite local homomorphismof 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 coefficientsin A:

p(x) = anxn + an−1x

n−1 + · · ·+ a0 = 0

with an = 1. This is a relation in the A-module B = B⊗A A. Since A is flatover A, this relation must be induced from a relation in B. That is, there is afinite collection of polynomials qi with coefficients in A, all satisfied by x, suchthat p is a linear combination of the qi, with coefficients in A.1919→ ∑

ciqi = p

Let bi be the leading coefficient of qi. Then∑cibi = an = 1. That is, the bi

generate the unit ideal in A. But A is faithfully flat over A20 so the bi generate20→the unit ideal in A as well. Let c′i be elements of A such that

∑c′ibi = 1. Then

p′ =∑

c′iqi

is a monic polynomial with coefficients in A that is satisfied by x. Therefore xis 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 ofcommutative noetherian rings. Then f may be factored as

A→ C → B

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

Proof. Let C be the integral closure of A in B. If p is a prime ideal 2121→

19todo: needs to be justified20todo: explain why faithfully flat implies surjective21todo: finish

D. MORE PERSPECTIVES ON ETALE MAPS 85

D More perspectives on etale maps

31.1 The analytic perspective

Definition 31.1. Let A be a commutative ring and I ⊂ A an ideal. The ringA = lim←−A/I

n is called the formal completion of A along I.22

Proposition 31.2. Suppose that A is a commutative ring that is separated andcomplete with respect to the powers of an ideal I. Let B be a formally etaleA-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:

HomA(B,A/I)∼←− HomA(B,A/I2)

∼←− HomA(B,A/I3)∼←− · · ·

On the other hand, we also have the following bijection since A is separated andcomplete:

HomA(B,A)∼−→ lim←−

n

HomA(B,A/In)

31.2 The algebraic perspective

Theorem 31.3. An etale morphism is flat.

Proposition 31.4 ([?, Proposition (18.3.1)]). (i) An A-algebra B is unram-ified if and only if it is of finite type and B is a projective B⊗AB-module.

(ii) An A-algebra B is etale if and only if it is of finite presentation, andprojective, both as an A-module and as a B⊗AB-module.

Proof. Recall that for B to be unramified as an A-algebra means the codiagonalmap B⊗AB → 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. Wehave 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⊗AB, hence is projective.

22This definition is not particularly well behaved unless I is finitely generated.


This suggests the following definition, which is in some ways better behavedthan the etale morphisms we study here:

Definition 31.6 ([BS, Definition 1.2]). A morphism of schemes f : X → Y iscalled weakly etale 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 enoughto 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 finitepresentation.

Lemma 31.7. If f : X → Y is locally of finite type then its diagonal morphismX → 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⊗AB →B is of finite presentation. Indeed, we only need to check that the ideal is finitelygenerated. But if t1, . . . , tn generate B as an A-algebra then ti⊗ 1 − 1⊗ tigenerate the ideal of the quotient B⊗AB → B.

Exercise 31.8. Find an example of a homomorphism of commutative ringsthat is not of finite type but whose diagonal is of finite presentation. (Hint: findone 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) SpecB → SpecA 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 ofA-modules

0→ I → A→ B → 0

and tensor it with B to get

I/I2 → Bid−→ B → 0.

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

fn =

n∑i=1

aif2i .

Thus

fn(1− anfn) =

n−1∑i=1

aif2i .

E. HOMOLOGY OF MODULES 87

Note that we have a canonical morphism of A-algebras, A[(1 − anfn)−1] → Bsince 1− anfn maps to 1 in B. Furthermore, f1, . . . , fn−1 suffice to generate Iover A[(1 − anfn)−1]. Proceeding by induction, we find a localization C of Asuch that IC = 0.

Corollary 31.9.1. All different senses of unramified, etale, and smooth coin-cide 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 forthe above proposition to hold. (Hint: consider the ring obtained by adjoiningall 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 ofα(a,m) and m+m′ instead of σ(m,m′).

Example 31.2. Suppose M and N are A-modules. Then M ×N may be givenan A-premodule structure in which

a.(m,n) ∼ (a.m, n) ∼ (m, a.n)

(m,n) + (m,n′) ∼ (m,n+ n′)

(m,n) + (m′, n) ∼ (m+m′, 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 , whereboth X and M are premodules that satisfies the following condition:

Whenever∑aip(xi) ∼ m for some ai ∈ A, xi ∈ X, and y ∈M there

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 ofM and N .

Exercise 31.4. Show that if M is a module then to give an extension of M asa 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 . Foreach (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:


BIEXT1 There are partially defined addition laws:

σ : X(m,n)×X(m′, n)→ X(m+m′, n)

τ : X(m,n)×X(m,n′)→ X(m,n+ n′)

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,n′)), τ((m′, n), (m′, n′)) = τ(σ((m,n), (m′, n)), σ((m,n′), (m′, n′)).

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 andf : M ′ → M is a homomorphism of premodules. Then X ′ = X ×M M ′ isnaturally equipped with the structure of an extension of M ′.

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.

Suppose that∑aip′(xi,mi) ∼ y in M ′. Then

∑aip(xi) ∼ f(y). Therefore

there is a unique x ∈ X with∑aixi ∼ x and p(x) = f(y). Therefore there is a

unique (x, y) ∈ X ×M M ′ with p(x, y) = y and∑ai(xi,mi) ∼ (x, y). Thus X ′

is an extension of M ′.

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 thatJ 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 ′ is a homomorphismof A-modules then there is a completion of the diagram

0 // J //

X //

M ×N //

0

0 // J ′ // X ′ // M ×N // 0

by an biextension X ′ of M and N by J ′ that is unique up to unique isomorphism.

E. HOMOLOGY OF MODULES 89

Proof. Let X ′ be the quotient of J ′ ×X by the antidiagonal action of J . (It isthe universal map from J ′×X into a simply transitive J ′-set that is equivariantwith respect to the homomorphism J × J → J .) The premodule structure onX ′ is induced from J ′ ×X.

We need to show that X ′ →M×N is a fibration. Consider the commutativediagram

J ′ ×X //

X

X ′ // M ×N.

The right vertical arrow is a fibration and the upper horizontal arrow is a fi-bration because J ′ is a module, so the upper horizontal arrow is a base changeof a fibration. As J ′ ×X → M ×N is therefore a fibration, it suffices to showthat J ′ ×X → X ′ is a fibration. Suppose that

∑ai(ji, xi) has an evaluation in

X ′. We show it has a unique evaluation in J ′ ×X. Since X → M ×N factorsthrough X ′, the sum

∑aixi has an evaluation x that is uniquely determined

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 ′×X lifting the given one in X ′

must be of the form (j, x) and (j, x+ u). But (j, x) and (j, x+ u) only have thesame image in X ′ if u = 0.23 ← 23

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

31.1 Exact sequences

Proposition 31.9. Let

0→M ′ →M →M ′′ → 0.

Then BiextA(M ′′, N ; J) is equivalent to the category of diagrams (of premod-ules)

M ′ ×N

vv 0 // J // X →MtimesN // 0

in which the horizontal row is exact.

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

23todo: clean up; explain fibration somewhere? otherwise write the proof without it


pullback we can construct a diagram

0 // J // X //

M ×N //

0

0 // J // Y // M ′′ ×N // 0

The composition M ′×N →M×N →M ′′×N sends (m,n) to (0, n). Thereforewe have X(m,n) = X(0, n) for all m ∈ M ′. On the other hand, X(0, n) hasa distinguished element: since 0.(0, n) ∼ (0, n) there is a unique element ξ ofX(0, n) such that 0.x = ξ and ξ projects to (0, n) in M × N . In view of theidentification X(m,n) = X(0, n) for all m ∈ M ′, this gives a section of X overM ′ ×N . 2424→

F Cohomology of modules

Definition 31.1. Let M and J be A-modules. An extension of M by J is anexact 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 // X ′ // 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

0 // J ′ // X ′ // M ′ // 0.

(31.2)

We write ExtA(M,J) or Ext1(M,J) for the set of isomorphism classes inExt(M,J).

Exercise 31.2. Show that every morphism in ExtA(M,J) is an isomorphism.

Exercise 31.3. (i) Construct a functor ExtA → (A-Mod)2 that sends anextension (31.1) to (M,J).

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

24todo: verify it’s a homomorphism

F. COHOMOLOGY OF MODULES 91

(iii) Show that if J → J ′ is a homomorphism ofA-modules andX ∈ 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 inparticular that ExtA(M,J) is a covariant functor in J .

(iv) Show that if M → M ′ is a homomorphism of A-modules and X ′ ∈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 Mand in particular that ExtA(M,J) is a contravariant functor in M .

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

ExtA(M ′, J) //

ExtA(M ′, J ′)

ExtA(M,J) // ExtA(M,J ′)

is commutative (up to canonical isomorphism).

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

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

(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 andthat 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 ′ →M →M ′′ → 0.

(i) Construct a map Hom(M ′, J) → Ext(M ′′, J). (Hint: Consider the se-quence above as an element of Ext(M ′′,M ′).)

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


(ii) Show that Ext(M ′′, J) is equivalent to the category of commutative dia-grams with exact row,

M ′

0 // J // X // M // 0.

(iii) Conclude that there is an exact sequence

0→ Hom(M ′′, J)→ Hom(M,J)→ Hom(M ′, J)

→ Ext1(M ′′, J)→ Ext1(M,J)→ Ext1(M ′, J).

Chapter 5

The etale topology and theetale fundamental group

32 Grothendieck topologies

Definition 32.1. Let C be a category and X an object of C . A sieve of X isa full subcategory R of C /X with the property that whenever (Y, f) ∈ R is in

R and Zg−→ 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 betweenthe 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 → Yis a morphism then hX(h)(g) = gh ∈ hX(Z). But (Z, gh) ∈ R by definition ofa sieve.

The reverse construction sends F ⊂ hX to the collection of pairs (Y, g) whereY ∈ C and g ∈ F (Y ). This is a sieve since if h : Z → Y is a morphism of Cand (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 exer-cise.

We will usually think of sieves as functors rather than categories, since itmakes 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 ofa class of covering sieves J(X) for each object X of C , subject to the conditionsbelow:

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

TOP2 (local character) If R ⊂ R′ ⊂ X with R covering R′ and R′ covering Xthen R covers X.

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

Note that for R ⊂ R′ to be covering means that R×R′ Y ⊂ Y is covering for allY ∈ R′.

32.1 Examples

Topological spaces

Suppose X is a topological space. Let Open(X) be the category of open subsetsof X. For each open U ⊂ X, let J(U) be the set of all sieves R of U such that⋃

V ∈RV = U.

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

The etale topology on a topological space

Let X be a topological space and let et(X) be the category of all etale spacesover X. Call a family of maps f : V → U in et(X) covering if⋃

(V,f)∈R

f(V ) = U.

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 affinesubschemes of X. For U ∈ zar(C), let J(U) be the set of all sieves R of U suchthat ⋃

V ∈R|V | = |U |.

This is the Zariski topology on X.

32. GROTHENDIECK TOPOLOGIES 95

The etale topology on an affine scheme

Let X be an affine scheme and let et(X) be the category of all affine schemesthat are etale over X. A family of maps f : V → U in et(X) is covering if⋃

(V,f)∈R

f(|V |) = |U |.

This is the etale topology on X.

The fpqc topology

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

n⋃i=1

fi(|Vi|) = |U |.

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 → Setsis called a presheaf on C . We say that F is a sheaf if, for any covering sieveR ⊂ 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 definitionfor a sheaf on a topological space.

When the category C has fiber products, the sheaf condition can be statedin 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 isa sheaf if, for every X ∈ C and every covering family Ui → X, the sequencebelow is exact:

F (X)→∏i

F (Ui) ⇒∏i,j

F (Uij)

This can be stated even more explicitly:

Corollary 32.6.1. A presheaf F is a sheaf if and only if the following conditionshold:


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

∣∣Ui

= η∣∣Ui

for all i.

SH2 For every X ∈ C , every covering family Ui → X, and every collectionξi ∈ F (Ui) such that ξi

∣∣Uij

= ξj∣∣Uij

for all i, j there is a ξ ∈ F (X) such

that ξ∣∣F (Ui)

= ξi for all i.

32.3 More examples of Grothendieck topologies

Sets

The category of sets has a Grothendieck topology in which the covers are thesurjective 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 mapsf : S → T of G-sects covering if ⋃

(S,f)

f(S) = T.

This is another example of a topos.

Commutative rings

Let ComRng be the category of commutative rings. Call a family of mapsAi → A covering if, for each finite collection S of elements of A there is someAi in the family whose image in A contains S.

This definition can be applied to groups, abelian groups, or any algebraicstruccture “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 refinementthat is generated by objects of U .

Put more precisely, if X ∈ C and R ⊂ X is a covering sieve then there is acovering sieve R′ ⊂ R and a collection of objects Ui ∈ U and maps pi : Ui → Xsuch that

R′ =⋃i

pi(hUi).

Definition 33.1. A Grothendieck topology C is called a site if it possesses asmall 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 Rbe a covering sieve of X ∈ C . A descent datum for F with respect to R is anelement of HomC (R,F ).

Exercise 33.3. The intersection of two covering sieves of an object of a Grothendiecktopology is covering.

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

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

F+ = lim−→R∈J(X)

HomC (R,F ).

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.By definition of F+ there is a covering sieve R′ ⊂ hX such that ξ and η are

representable by ξ′, η′ ∈ Hom(R′, F ).Let R′′ ⊂ R′ be the set of all Y → X in R′ such that ξ′(Y ) = η′(Y ) as

elements of F (Y ). By assumption we have ξ(Y ) = η(Y ) as elements of F+(Y )for all Y ∈ R. As ξ

∣∣hY

and η∣∣hY

are represented by ξ′∣∣hY ×hX R′

and η′∣∣hY ×hX R′

in Hom(hY ×hX R′, F ), this means (by definition of F+(Y ) as a direct limit overcovering sieves) that there is a covering sieve RY ⊂ hY ×hX R′ ⊂ hY such thatξ′∣∣RY

= η′∣∣RY

.

We conclude that, for each Y ∈ R we have hY ×hX R′′ ⊂ hY is covering,since it contains the covering sieve RY . But ξ′

∣∣R′′

= η′∣∣R′′

so ξ′ and η′ representthe same element of

F+(X) = lim−→R∈J(X)

Hom(R,F ).

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

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×hX hY

is induced from an element

of F (Y ). (This element is necessarily unique if it exists since F is a separatedpresheaf.)

By definition of F+, for any Y ∈ R, there is a covering sieve RY ⊂ hY suchthat ξ

∣∣hY

: hY → F+ is induced from a map RY → F . In particular, R′×hX hYcontains RY for all Y ∈ R. But RY ⊂ hY and R ⊂ hX are both covering so R′

is covering.


Replacing R′ by R ∩ R′, we find that ξ∣∣R′

factors through F . (This usesthe uniqueness of the factorization on each Y ∈ R′.) By definition of F+, thisprovides 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 ∈ R′ and R′ ⊂ Ris 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 sheafifica-tion.

Proposition 33.7. Sheafification is left adjoint to the inclusion of sheaves inpresheaves.

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 limitsand colimits.

Proof. For limits, calculate the limit as a presheaf and observe it is a sheaf. Forcolmits, calculate the colimit as a presheaf and then sheafify.

34 Fpqc descent

Let A be a commutative ring, M an A-module, and X = SpecA. We constructa presheaf M on fpqc(X) by defining M(SpecB) = B⊗AM .

We write OX = A. Notice that M is a presheaf of OX -modules.

Definition 34.1. We will call a presheaf F of OX -modules quasi-coherent if forany map U = SpecC → V = SpecB we the induced map C ⊗B F (B)→ F (C)is an isomorphism.

Proposition 34.2. M is a sheaf on fpqc(X).

Recall that this means

M(U)∼←− Hom(hU , M)→ Hom(R, M)

is a bijection when R is a covering sieve of U . If U = SpecB and we replaceX with U and M with B⊗AM then we can assume without loss of generalitythat U = X.

Lemma 34.3. Suppose that a covering sieve R of X is generated by U1, . . . , Un.Then

Hom(R,F ) = lim←−(∏

i

F (Ui) ⇒∏i,j

F (Uij))

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 =SpecA and let Y = SpecB be a flat, affine scheme over X. Then for anyquasi-coherent presheaf of OX-modules F on X, the natural map

Hom(R ×hXhY , F )← B⊗

AHom(R,F )

is an isomorphism.

Proof. Compute:

Hom(R ×hXhY , F ) = lim←−

(∏i

F (Ui×XY ) ⇒

∏i,j

F (Uij ×XY ))

= lim←−(∏

i

F (Bi⊗AC) ⇒

∏i,j

F (Bij ⊗AC))

= lim←−(∏

i

C ⊗AF (Bi) ⇒

∏i,j

C ⊗AF (Bij)

)= C ⊗

Alim←−(∏

i

F (Bi) ⇒∏i,j

F (Bij))

= C ⊗A

Hom(R,F )

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

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

Bi⊗AM → Bi⊗

AHom(R, M)

are all isomorphisms since the Bi are faithfully flat.Set Ui = SpecBi. The map above is equivalent to the one below (by the

earlier corollary):

Bi⊗AM → Hom(R ×

hXhUi , M)

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

Corollary 34.3.3. If M is an A-module then M is an fpqc sheaf on SpecA.

Proof. Suppose R ⊂ hX is a covering sieve. Then R contains a sieve R′ that isgenerated by a finite, faithfully flat collection of A-algebras. We have just seenthat M → Hom(R′, M) is an isomorphism. Now we show that Hom(R, M) →Hom(R′, M) is an isomorphism. By definition,

Hom(R, M) = lim←−Y ∈R

M(Y ).


We can also write

Hom(R′, M) = lim←−Y ∈R

Hom(R′×XY, M).

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

M(Y )→ Hom(R′×XY, M)

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 = SpecA. The functorM 7→ M

∣∣R

is an equivalence of categories between A-Mod and QCoh(R) forany fpqc covering sieve R of X.

Proof. The inverse functor sends F to Hom(R,F ). The discussion above showed

that Hom(R, M) = M , which proves that these functors are inverse in onedirection. To go the other way, consider Hom(R,F )∼ → F . Evaluating onan A-algebra B, we get B⊗A Hom(R,F ) → F (B) = B⊗A F (A), which is anisomorphism 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 ofpolynomial equations. The solutions to a system of equations can be encodedin a commutative ring: If xi, i ∈ I is a system of variables and fj , j ∈ J is asystem of equations, then the commutative ring

Z[xi∣∣i ∈ I]/(fj

∣∣j ∈ J)

is the universal commutative ring with solutions to the fj . Every commutativering arises this way, for sufficiently many variables and equations. Thus, forevery commutative ring A, we have a scheme SpecA. These schemes are calledthe affine schemes.

A homomorphism of commutative rings A → B gives a morphism of affineschemes

SpecB → SpecA.

This morphism is called a distinguished open embedding if there is some f ∈ Asuch that B is isomorphic, as an A-algebra, to A[f−1].

A scheme is assembled from affine schemes, glued along open embeddings. Ascheme can thus be described as a collection of affine open subschemes SpecAi,

35. A RAPID REVIEW OF SCHEME THEORY 101

with i drawn from some partiallly ordered indexing set I, along with distin-guished open embeddings

SpecAi → SpecAj

whenever i ≤ j. The one condition is that the diagrams

SpecAi //

%%

SpecAj

yySpecAk

commute whenever i ≤ j ≤ k. The collection of SpecAi is called at atlas forthe scheme.

Points

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

X(k) = lim−→i∈I

Hom(Ai, k).

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

Schematic points

Let SpecA be an affine scheme. Write |SpecA| for the set of prime ideals of A.The assignment SpecA 7→ |SpecA| is a covariant functor from affine schemesto sets. If SpecA → SpecB is an open embedding then |SpecA| → |SpecB| isinjective.

Let SpecAi, i ∈ I be an atlas for a scheme X. Define the set of schematicpoints of X to be

lim−→i

|SpecAi| .

Equivalence of atlases

Let SpecAi, i ∈ I be an atlas for a scheme X and I ′ ⊂ I a subset. ThenSpecAi, i ∈ I ′ is an atlas for a scheme X ′. We say that I ′ is a refinement of Iif the map

|X ′| → |X|

is a bijection.

Refinement gives a partial order among atlases for schemes. We call an atlasmaximal if it is maximal with respect to refinement.


Morphisms of schemes

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

SpecA→ SpecB

is a homomorphism of commutative rings B → A. Thus the category of affineschemes is exactly the opposite of the category of commutative rings.

Suppose X and Y are schemes with maximal atlases SpecAi, i ∈ I andSpecBj , j ∈ J , respectively. A morphism from X to Y is a refinement I ′ of I,a map ϕ : I ′ → J of partially ordered sets, and morphisms of affine schemes

SpecAi → SpecBϕ(i)

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

SpecAi //

SpecAj

SpecBϕ(i)

// SpecBϕ(j).

Finite presentation and finite type

A morphism of commutative rings B → A is said to be of finite type if A isfinitely generated as a B-algebra. It is said to be of finite presentation if theideal of relations among finitely many generators of A is finitely generated. Thatis, 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 SpecA → SpecBwhere A is a B-algebra of finite type (resp. finite presentation). If in addition fis quasi-compact1 then f is said to be of finite type (resp. of finite presentation).1→

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 tocome 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 universalsystem 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 ringB, we simply compute HomComRng(A,B).

Another way to think about the solution set of a system of equations is as afunctor. Let

F : ComRng→ Sets

be the functorF (B) =

ξ ∈ BI

∣∣f(ξ) = 0 for all f ∈ J.

Then the commutative ring A defined above represents the functor F . Wethink of F as the abstract “solution space” of the system of equations J in thevariables I.

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

We write SpecA for the functor represented by a commutative ring A. LetI 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 itV (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 arbitrarycommutative ring B. Exercise: Give an example.

2) Even though U and V are complementary, it is not the case that each canbe 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⊗AA/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) ingeneral.

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 anymap ϕ : hA → X, the pullback ϕ−1(V ) ⊂ hA is closed.


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

(i) Suppose that U = SpecA is covered by open affine subschemes Ui =SpecAi and ξ, η ∈ X(A). If ξ

∣∣Ui

= η∣∣Ui

for every i then ξ = η.

(ii) Suppose that U = SpecA is covered by open affine subschemes Ui =SpecAi and for each pair i, j the intersection Ui ∩ Uj is covered by openaffine subschemes Uijk. If ξi ∈ X(Ai) are such that ξi

∣∣Uijk

= ξj∣∣Uijk

for

every i, j, and k then there is a ξ ∈ X(A) such that ξ∣∣Ui

= ξi for all i.

If X is a scheme then for any field k, the set X(k) has the structure of atopological space. The open sets are the U(k) where U is an open subfunctorof X.

Definition 35.4. The underlying topological space of a scheme is the set |X| ofall isomorphism classes of injective maps hk → X where k is a field. If U ⊂ Xis 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 withthe 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 fieldk = 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 thekernel. We want to show that k = fract(A/p) coincides with K. Consider thering K ⊗kK. Since k → K is an epimorphism, K ⊗kK = K. But we have

dimK K ⊗kK = dimkK

from which we obtain K = k.

By a scheme over a base scheme S we mean a scheme X equipped with amorphism X → S.

Definition 35.6. Let f : X → Y be a morphism of schemes. We say that f isetale if it is locally of finite presentation and the diagram

S //

X

f

S′ //

>>

Y

2This is not standard usage of the term prescheme! What are today called schemes wereoriginally called preschemes. As a consequence, the term “prescheme” has the potential to beconfusing. Don’t tell anyone we’re using it.

36. THE ETALE TOPOLOGY 105

admits a unique lift whenever S ⊂ S′ is an infinitesimal extension of affineschemes.

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

36 The etale topology

37 Henselization

38 The etale fundamental group

38.1 Covering spaces

Let F be an etale sheaf over X. We say that X satisfies the valuative criterionfor properness if, whenevery Y is the spectrum of a valuation ring with closedpoint 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 etalemap W → X that this condition coincides with the familiar valuative criterion.

Definition 38.2. Let X be a locally noetherian scheme. An etale cover of Xis an etale sheaf F on X that satisfies the valuative criterion of properness. Weuse cov(X) (or et-cov(X), if it is necessary to emphasize the etale topology) todenote the category of etale covers of X.

Proposition 38.3. Recall that X is assumed locally noetherian. The categorycov(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 finitechain of specializations and generizations connecting ξ to η.

4. The category cov(X) has all finite limits and all small colimits and theinclusion 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.


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 shoowthat every pair of geometric points of Y is contained in some quasi-compactopen subset of Y . Cover Y by noetherian open subschemes Yi

38.2 Locally constant sheaves

Let X be a scheme and et(X) its etale site. A sheaf F on et(X) is said to belocally constant if the collection of etale maps U → X such that F

∣∣U

is constantforms an etale cover of X. Let lcet(X) be the category of locally constantsheaves on et(X).

Proposition 38.4. 1. lcet(X) is closed under finite limits and colimits.

2. If f : Y → X is a morphism of schemes then f∗ : et(X)→ et(Y ) restrictsto an exact functor lcet(X)→ lcet(Y ).

3. If ξ is a geometric point4 of X then restriction to ξ defines an exact functor4→lcet(X)→ Sets.

4. Coproducts in lcet(X) are disjoint.

Define π1(X, ξ) to be the automorphism group of the restriction functorξ∗ : lcet(X)→ Sets.

Proposition 38.5. A locally constant sheaf satisfies the valuative criterion forproperness.

Proof. We can reduce immediately to the case of a locally constant sheaf over avaluation ring. By etale descent, we can replace the base with a strict henseliza-tion. But then a locally constant sheaf is constant, and the assertion is obviousfor constant sheaves.

38.3 The topology on the etale fundamental group

For each sheaf F ∈ lcet(X) we have an action of π1(X, ξ) on the set ξ∗F . Foreach F ∈ lcet(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 forthe compact-open topology on π1(X, ξ).

3todo: get proof from stacks project4todo: define

Chapter 6

Abelian categories andderived functors


Grothendieck, A. Sur quelques points d’algebre homologique. TohokuMath. 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 prod-ucts and coproducts and the natural map∐

i∈IXi →

∏i∈I

Xi

is an isomorphism for all finite collections of objects Xi of C .

An additive category has a simultaneously initial and final object (the emptycoproduct or product, respectively) called the zero object and notated 0. Ittherefore 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 followingtwo 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 thatX 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

dp : Kp → Kp+1

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

Hp(K•) = ker(dp)/ im(dp−1).

A morphism of complexes f : K• → L• is a homomorphism of graded objectswith graded pieces fp : Kp → Lp satisfying dfp = fp+1d.

1 2 31→2→3→

41 Spectral sequences

1todo: Define internal morphism2todo: Define quasi-isomorphism3todo: 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 topologicalspace 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−→K⊂U F (U), with the colimit taken over all open

subsets of X containing K. Any section of F over K is therefore induced fromsome U ⊃ K and can then be lifted to a section of F over X, by the definitionof a flaccid sheaf.

Lemma 42.3. Suppose F is soft. Then H1K(X,F ) = 0 for any compact K ⊂ X.

Proof. H1K(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 trivialand 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. Thereforewe 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 coho-mology.

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

42.4 Partitions of unity and de Rham cohomology

43 Cech cohomology

Let U be a collection of open subsets of X. Let C be the category of theirintersections. If F is a sheaf on X we get an induced presheaf on C . Thefunctor

H : Sh(X)→ Psh(C )

is left exact. It has an exact left adjoint,2 3 so it preserves injectives. We2→3→ therefore obtain a spectral sequence:

Theorem 43.1. The spectral sequence

lim←−(p)H q(F )⇒ Hp+q(X,F )

converges.

In the statement, we have written H q(F ) for RqH (F ). It is also convenientto 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 thereis an exact sequence

0→ lim←−(q)H (F )→ Hq(X,F )→ lim←−H q(F )→ lim←−

(q+1)H (F )→ Hq+1(X,F ).

Corollary 43.1.2. Hp(R,Z) vanishes for p > 0.

2todo: explain what the adjoint is3todo: add reference

44. COMPACTLY SUPPORTED COHOMOLOGY 111

Proof. By induction on p. We know it is true for p = 1 already. Suppose thatit is true for p < n. Then suppose α ∈ Hn(R,Z). Choose an open coverof R by intervals U such that α

∣∣U

= 0 for all U in the cover and the tripleintersections of the intervals are empty. By induction we have the 5-term exactsequence. We know that the image of α under the map Hq(X,F )→ lim←−H q(F )

is zero. Therefore α is induced from lim←−(q) H (F ) and it suffices to show that

this vanishes. This is obvious for q ≥ 2 since the triple intersections of theU ∈ 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 openembedding. 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→ H0(S1,Z)→ H0(S1, i∗Z)→ H1c (R,Z)→ H1(S1,Z)→ 0.

Note the following:

(i) H0c (R,Z) = H0(S1, j!Z) = 0 because a section must have zero stalk at

the image of i,

(ii) H1(S1, i∗Z) = 0 since i∗Z is a skyscraper sheaf,

(iii) H0(S1, i∗Z) = Z, and

(iv) we can easily check that

Z = H0(S1,Z)→ H0(S1, i∗Z) = Z

is the identity map.

We conclude that H1c (R,Z)→ H1(S1,Z) = Z is an isomorphism.

45 Proper base change

Definition 45.1. A morphism of locally compact Hausdorff spaces is calledproper 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 → Yand any sheaf F on X, the map g∗f∗F → f∗g

∗F is an isomorphism.

4todo: cleanup presentation


Proof. It is sufficient to treat the case where Z is a point; assume that and lety = g(Z). We remark that the open neighborhoods f−1(U), for U ⊂ Y an openneighborhood of y, constitute a basis of open neighborhoods of f−1(y), sincef−1(y) is compact.

Essentially the same argument works to prove that H1 commutes with basechange:

Lemma 45.3. Suppose that f : X → Y is proper and g : Z → Y is any map.Then g∗R1f∗F → R1f∗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

Hp(Y,Rqf∗F )⇒ Hp+q(X,F )

converges.

46.1 Kunneth formula

46.2 Homotopy invariance of cohomology

Let h : X × I → Y be a homotopy between f and g. Let i0 and i1 be theinclusions of X×0 and X×1 in X×I. Let q : X×I → X be the projection. Bythe Kunneth formula, the map q∗ : H∗(X,A)→ H∗(X, q∗A) is an isomorphism.Since q i0 = q i1 we deduce that i∗0 and i∗1 induce the same isomorphismH∗(X × I, A)→ H∗(X,A). Therefore we have

f∗ = i∗0h∗ = i∗1h

∗ = g∗

as maps H∗(Y,A)→ H∗(X,A).

Theorem 46.2. Homotopic maps induce the same map on cohomology withlocally constant coefficients.55→

Corollary 46.2.1. If X is contractible then

Hp(X,A) =

A p = 0

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 canonicallyhomotopy independent

46. LERAY SPECTRAL SEQUENCE 113

46.3 The cohomology of spheres

Consider the two closed discs D,E ⊂ Sn with intersection homeomorphic toSn−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)

for p > 1. Since π1(Sn) = 0 for n > 1 we know H1(Sn,Z) = 0 for n > 1, so thisgives

Hp(Sn,Z) =

Z p = 0, n

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

Hpc (Cn,Z)→ Hp(CPn,Z)→ Hp(CPn−1,Z)→ Hp+1

c (Cn,Z).

Since Hpc (Cn,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

H2n−1c (Cn,Z)→ H2n−1(CPn,Z)→ H2n−1(CPn−1,Z)

H2n−1(CPn−1,Z)→ H2nc (Cn,Z)→ H2n(CPn,Z)→ H2n(CPn−1,Z)

The extremal terms in both lines vanish6 so we get ← 6

Hp(CPn,Z) =

0 p = 2n− 1

Z p = 2n.

6todo: references, explain


Second calculation

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

Hp(CPn, Rqf∗Z)⇒ Hp+q(X,Z).

By proper base change, we know that Rqf∗Z = 0 for q 6= 0, 1 and f∗Z = Z andR1f∗Z is a locally constant sheaf with fibers isomorphic to Z. But π1(CPn) = 0so the only such sheaf must be Z. This means the E2 of the spectral sequencemust 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 toHp(S2n+1,Z) and this vanishes for all 0 < p < 2n+ 1, we deduce that the maps

Hp(CPn,Z)→ Hp+2(CPn,Z)

are isomorphisms for p 6= 2n. 77→

47 Poincare duality

Let X be a topological space, U ∈ U a collection of open subsets of X. Weobtain an exact sequence

· · · →∐

(U,V )∈U 2

iU∩V !Z→∐U∈U

iU !Z→ Z→ 0.

Taking compactly supported cohomology gives a spectral sequence convergingto 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 repre-sented 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 ofhX .1

Definition 49.2. A Grothendieck topology on a category C is a collection ofsieves R ⊂ hX—called the covering sieves of the topology—satisfying the fol-lowing 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 .

T3 If R ⊂ S ⊂ hX are sieves and R is covering then S is covering.2

Definition 49.3. A presheaf on the category C is a functor F : C → Sets. Apresheaf is said to be a sheaf if in a given Grothendieck topology if, wheneverR 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.2We 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 apresheaf. When F is merely a fibered category, this map is not a bijection but still anequivalence 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 suchthat if Y → Z is a morphism of C then P (Z) implies P (Y ). Thus P is a sieveof 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 weneed 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 candefine f∗R to be the sieve S of X consisting of those u : X ′ → X such thatthere is some v : Y ′ → Y in R and some g ∈ F (X ′, Y ′) with v∗g = u∗f . We canvisualize this with a diagram:

X ′g //

u

Y ′

v

X

f // Y

(50.1)

Take care to remember, though, that X and Y are objects of different cate-gories.4 55→

We say that F is a continuous correspondence if, whenever R is a coveringsieve of Y , the pullback f∗R is a covering sieve of X.

Let X be an object of C . Define a category FX/D (or maybe X/F D?)whose objects are pairs (Y, f) where Y ∈ D and f ∈ F (X,Y ). For varying Xwe can regard FX/D as a fibered category over C . We say that F has localpro-pushforwards if this fibered category is locally filtered.

To say FX/D is locally filtered means that the characteristic properties ofa filtered category are required to hold after suitable localization:

LFil1 Let Y,Z be objects of FX/D . Let R ⊂ hX be the sieve consisting of allu : X ′ → X such that there is an object W of FX ′/D and maps W → u∗Yand 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 )qOb(D) and HomF (X,Y ) = F (X,Y ) for X ∈ C and Y ∈ D . Then the definition off∗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 FX/D . Let R ⊂ hX be the sieveconsisting of all u : X ′ → X such that there is an object W of FX ′/Dand 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 acontinuous 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 )(ε(X), f∗ε(Y )).

51 Group cohomology

52 (*) Cohomology in other algebraic categories

52.1 Groups

52.2 Rings

52.3 Commutative rings

52.4 Hyper-Cech cohomology

Let C be a site and D a diagram in C , i.e., a functor D : D → C . GivingD the chaotic topology, it is clear that D takes covers to covers. If we assumemoreover 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

Hp(D , RqΠ∗F )⇒ Hp+q(C , F ).

Proposition 52.1. Given α ∈ Hp(C , F ), there is a diagram D as above suchthat α lies in the image of the edge homomoprhism

Hp(D ,Π∗F )→ Hp(C , F ).

Proof. Consider first the image of α in H0(D , RpΠ∗F ). We choose a cover ofthe final sheaf of C such that α

∣∣U

= 0 for all U in the cover. Our first guess forD will consist of all products of all objects U in the cover.

The next step in the filtration ofHp(C , F ) is a subquotient ofH1(D , Rp−1Π∗F ).We can realize H1(D , G) as a subquotient of G(U1). We can therefore representthe image of α in F1H

p(C , F ) by a class in Hp−1(U1, F ). Replace U1 by anetale cover over which this class is trivial.

By the same argument, we can now represent the image of α in F2Hp(C , F )

by a class in Hp−2(U2, F ). Replace U2 by a cover trivializing this class. We canrepeat this argument until we have represented the image of α in FpH

p(C , F ) =Hp(D ,Π∗F ).

6Note that the correspondence is not unique, though it is unique up to a unique isomor-phism 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 etale morphisms

59 (*) Weakly etale morphisms

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 etale topology

69 Other topologies

69.1 The Zariski topology

69.2 (*) The pro-etale 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 Ai for i ∈ I. Let A = lim−→Ai.Suppose given a compatible system of Ai-schemes Xi, with

Xj = Xi ⊗AiAj

for all i and j. Then set X = Xi⊗Ai A = lim←−Xi. Suppose that Fi, i ∈ I are acompatible system of etale sheaves on the Xi. Then there is a map

lim−→µ∗iHp(Xi, Fi)→ Hp(X,F ). (72.1)

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 pre-sentation 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 islocally of finite presentation then (??) is an isomorphism for p = 1.

Proof. Use the interpretation of H1(Xi, Fi) as Fi-torsors on Xi. An Fi-torsoris locally isomorphic to Fi, hence is locally of finite presentation. It follows thatthe underlying space of an F -torsor P is induced from some etale space Pi overXi. Similarly, the action of F is induced from the action of some Fi; that P isa torsor is equivalent to the map F ×P → P ×P being an isomorphism, whichcan 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 ofetale schemes over X is induced from a finite diagram over Xi for some i. Weknow1 that any class in Hp(X,F ) can be represented in Hp(D , F

∣∣D

) for some1→diagram D in et(X). This diagram is induced from some diagram Di in et(Xi)for some i. A class in Hp(D , F

∣∣D

) is induced from a class in Hp(Di, Fi∣∣Di

) for

sufficiently large i (essentially uniquely). This gives us the surjectivity of (72.1)for all p.

Now we check injectivity. Consider a class α ∈ Hp(Xi, Fi). We can representα by a class β in Hp(Di, F

∣∣Di

) for some diagram Di in et(Xi). Then µ∗iα is

represented by µ∗i β in Hp(D , F∣∣D

) where D = µ∗iDi. But the map

Hp(D , F∣∣D

)→ Hp(X,F )

is injective, so µ∗i β = 0. Therefore it is the image of a p− 1 cochain γ of F overD . Since the objects of D are coherent, this is induced (by the p = 0 case) froma cochain γi of Fi over Di for i sufficiently large. Furthermore, dγ = β impliesdγi = 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

We know that H1(et(X),Gm) classifies Gm-torsors on the etale site of X. Butwe can identify the stack of Gm-torsors with the stack of invertible sheavesof Oet(X)-modules. We know that quasi-coherent OX -modules satisfy etaledescent,1 so that ← 1

H1(et(X),Gm) = H1(zar(X),Gm).

In particular, we have Hilbert’s theorem 90:

Theorem 75.1. Let X = Spec k with k a field. Then H1(et(X),Gm) = 0.

Proof. We have H1(et(X),Gm) = H1(zar(X),Gm) = 0 since zar(X) is equiv-alent 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, when-ever f is a homogeneous polynomial of degree d in n variables with coefficientsin 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 transcen-dence 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, withcoefficients in K.

View K as the function field of a smooth curve X. The coefficients of f maybe regarded as rational functions on X. They will be regular except at a finiteset of points. Choose an effective divisor H so that the coefficients of f lie inOX(H).

We will show that we can find a zero (α1, . . . , αn) of f in K with polesonly along the support of H. Indeed, consider α1, . . . , αn ∈ OX(eH) for somepositive integer e. The αi are chosen from a vector space Γ(X,OX(eH))n sothat we have a polynomial map of k-vector spaces

ϕ : Γ(X,OX(eH))n → Γ(X,OX((de+ 1)H)) : (α1, . . . , αn) 7→ f(α1, . . . , αn).

We calculate the dimensions of these vector spaces. By Riemann–Roch, forlarge e, we have

dim Γ(X,OX(eH))n = ndim Γ(X,OX(eH)) = n(1− g + edeg(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 largeenough. Viewing ϕ as a polynomial function with coefficients in an algebraicallyclosed field, it has more variables than constraints. Since it has at least onesolution—the trivial one, (α1, . . . , αn) = 0—it must have a nontrivial one as well(by the Nullstellensatz!).2 That is, there is a non-zero choice of α1, . . . , αn ∈2→Γ(X,OX(eH)) ⊂ K such that f(α1, . . . , αn) = 0.

Corollary 77.2.1. If K is a C1 field then H2(K,Gm) = 0.

Proof. Let D be a division algebra with center K. The dimension of D is n2. Ithas a determinant function det : D → K, which we may regard as a poynomialof degree n in n2 variables. If n > 1 then it must have a nontrivial zero. ThusD 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., thatD = K.

Let f : X → Y be a finite morphism of schemes and F a sheaf of groups onY . 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 etale.We will construct the map etale locally in Y , taking advantage of the fact thatpushforward commutes with etale base change in Y . After an etale base changeY ′ → Y (set X ′ = X ×Y Y ′) we can assume that X ′ ' S × Y ′ for some finiteset S. In that case, f∗f

∗F ' FS and we have a summation map

f∗f∗F ' FS → F : (xs)s∈S 7→

∑s∈S

xs.

2todo: add reference

77. TSEN’S THEOREM 133

The same argument also proves that f∗ is exact, so that Rpf∗ = 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 mul-

tiplication by n, where n is the degree of f .

Corollary 77.3.1. Let f : X → Y be a finite etale map of degree n and Fan 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 alge-braically closed field k. Let ` be a prime. Suppose F is a Z/`nZ-module overet(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

H1(K,Gm)→ H2(K,µ`)→ H2(K,Gm)

we deduce that H2(K,µ`) = 0.If K ⊂ L is a separable extension with [L : K] prime to ` then H2(K,F ) ⊂

H2(L,F ). Let L′ be the union of all such extensions (within a fixed separableclosure of K). Then H2(L′, F ) = lim−→L

H2(L,F ) so that H2(K,F ) ⊂ H2(L′, F ).

We can therefore replace K with L′ and assume that all polynomials over K ofdegree prime to ` have degree 1.

Now let L be the separable closure of K and denote by f the map SpecL→SpecK. We show first that H2(K,F ) = 0 for all ind-`-power torsion modulesF . By compatibility with lim−→,3 we can assume F is finite. Then F has a ← 3

composition series of simple et(K)-modules. But the only such is Z/`Z ' µ`.4

Therefore H2(K,F ) = 0. ← 4Now consider the embedding F → f∗f

∗F . The latter is acyclic, since we havea spectral sequence Hp(K,Rqf∗f

∗F )⇒ Hp+q(L, f∗F ). We have Hp(L,F ) = 0for p > 0 since L is separably closed, and, for the same reason, Rqf∗ = 0 forq > 0. We therefore get

Hp(K,F ) ' Hp−1(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 proof4todo: needs justification

134 CHAPTER 15. ETALE COHOMOLOGY OF POINTS

Chapter 16

Etale cohomology of curves

78 Calculation

79 Poincare duality 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 ofa projective resolution of M . The depth of M is the minimal n such thatExtn(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 caseM = R, where it’s obvious.

Consider a projective resolution P• of M of length e. Let d = depth(R). Wehave a spectral sequence

Extp(R/m,Pq)⇒ Extp−q(R/m,M).

We know that Extp(R/m,Pq) = 0 for p < d and q > e. Therefore Extn(R/m,M) =0 for n < d − e. Therefore pd(M) ≥ d − e. Since e ≥ pd(M) we getpd(M) + depth(M) ≥ depth(R).

To prove the reverse inequality, we have to show that

Extd(R/m,Pe)→ Extd(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 forall of the summands of Pe then Pe is a direct summand of Pe−1 so the resolutioncouldn’t have been minimal. Conclusion: the matrix of ϕ : Pe → Pe−1 has atleast one column, all of whose entries are in m. Reducing modulo m (andtensoring with Extd(R/m,R)) gives the matrix of (80.1), which therefore hasat 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 regularsequence.

Proof. Note that the theorem is true if M is m-torsion. In that case the depthis 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→Mt−→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 maximalregular 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) ≥ min2,dim(R).

Proof. If dim(R) = 0 then the conclusion is obvious. If R is integral anddim(R) > 0 then R contains a zero non-divisor, so depth(R) ≥ 1. This provesthe corollary if dim(R) = 1.

If dim(R) = 2, choose a zero non-divisor t. If every element of m/tm isa 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 notprincipal, we cannot have m−1m = R. Therefore m−1m = m. Therefore m−1

stabilizes the finitely generated submodule m of R. Therefore every element ofm−1 is integral over R, whence m−1 = R: contradiction.

11→

Corollary 80.4.3. Let R′ ⊃ R be an integral extension of an integrally closeddomain R of dimension 2. Then depth(R′) = 2.

1todo: dimR > 2

80. SMOOTH BASE CHANGE 139

Proof. We know that depth(R′) ≤ depth(R) = 2. On the other hand, we canrepeat the proof above. Choose t ∈ R not a zero divisor in R and α ∈ R′ suchthat mα = 0 in R′/tR′. Then let f(x) be an expression of integral dependenceof α. We get mf(α) ≡ f(0) (mod tR′) so mf(0) = 0 in R′/tR′. That ismf(0) ∈ tR′ ∩ R = tR since R is integrally closed. (If tR′ 3 u then u/t ∈ R′is integral over R, hence in R.) Put s = f(0) and conclude as in the lastcorollary.

Corollary 80.4.4 (Purity of the branch locus in dimension 2). Suppose that Ris a regular local ring and R′ a finite, normal, generically etale R-algebra. Thenthe branch locus of SpecR′ → SpecR has pure codimension 1.

Proof. We have depthR(R′) + pdR(R′) = depth(R) = dim(R) = 2. But R′ isnormal, 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 R′) detects theramification 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].

141

http://arxiv.org/abs/1309.1198

Etale cohomology course notes - Department of Mathematicsmath.colorado.edu/~jonathan.wise/teaching/math8174-spring-2014/notes.pdfMac Lane, S. Categories for the working mathematician.

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