The Brauer–Manin Obstruction on Curves

R. van Dobben de Bruyn

The Brauer–Manin obstruction on curves

Mémoire, July 12, 2013

Supervisor: Prof. J.-L. Colliot-Thélène

Université Paris-Sud XI

Preface

This work aims to deﬁne the Brauer–Manin obstruction and the ﬁnite descent obstructions of [22], at a pace suitable for graduate students. We will give an almost complete proof (following [loc. cit.]) that the abelian descent obstruction (and a fortiori the Brauer–Manin obstruction) is the only one on curves that map non-trivially into an abelian variety of algebraic rank 0 such that the Tate– Shafarevich group contains no nonzero divisible elements. On the way, we will develop all the theory necessary, including Selmer groups, étale cohomology, torsors, and Brauer groups of schemes.

Contents

1 Algebraic geometry8 1.1 Étale morphisms...... 8 1.2 Two results on proper varieties ...... 11 1.3 Adelic points ...... 17

2 Group schemes 20 2.1 Group schemes...... 20 2.2 Abelian varieties ...... 24 2.3 Selmer groups...... 26 2.4 Adelic points of abelian varieties ...... 32 2.5 Jacobians ...... 39

3 Torsors 44 3.1 First cohomology groups...... 44 3.2 Nonabelian cohomology ...... 45 3.3 Torsors ...... 47 3.4 Descent data ...... 53 3.5 Hilbert’s theorem 90...... 55

4 Brauer groups 58 4.1 Azumaya algebras ...... 58 4.2 The Skolem–Noether theorem...... 62 4.3 Brauer groups of Henselian rings ...... 67 4.4 Cohomological Brauer group ...... 69

5 Obstructions for the existence of rational points 74 5.1 Descent obstructions...... 74 5.2 The Brauer–Manin obstruction...... 77 5.3 Obstructions on abelian varieties...... 80 5.4 Obstructions on curves...... 82

A Category theory 86 A.1 Representable functors...... 86 A.2 Limits...... 89 A.3 Functors on limits ...... 91 A.4 Groups in categories...... 94

B Étale cohomology 100 B.1 Sites and sheaves...... 100 B.2 Čech cohomology...... 103 B.3 Sheafification ...... 112 B.4 The étale site...... 116 B.5 Change of site...... 119 B.6 Cohomology...... 125 B.7 Examples of sheaves ...... 127 B.8 The étale site of a field...... 131

References 134

Introduction

It is in general a difficult problem to decide whether a variety X over a number field K has any rational points. On the other hand, finding Kv-points for the various completions of K is relatively easy. Through the Hasse principle, for certain classes of varieties the question of whether XpKq is nonempty has become equivalent to the question of whether XpKvq is nonempty for each completion of K.

However, there are many classes of varieties known for which there is no Hasse principle, i.e. that have points everywhere locally, but not globally. The proof that they have no global point usually requires some cohomological argument. One of the constructions one can carry out is the formation of the Brauer–Manin set Br X XpAK q . Br X It is a subset of XpAK q containing XpKq, and if one can show that XpAK q “ ∅, then in particular X has no rational points.

One of the aims of this thesis is to deﬁne the Brauer–Manin set and show some of its main properties. At the same time, we will provide certain other obstructions to the existence of rational points. The main theorem (Corollary 5.4.6) is that the Brauer–Manin obstruction is the only obstruction for the existence of rational points on curves C that map non-trivially into an abelian variety A of algebraic rank 0 whose Tate–Shafarevich group contains no nontrivial divisible elements.

We develop most of the theory needed to deﬁne all the obstructions involved. In particular, we have a lengthy and almost self-contained appendix on étale cohomology. We assume the reader has familiarity with the language of schemes, to a level equivalent to chapters II and III of Hartshorne [10]. Moreover, the reader is assumed to have some knowledge of algebraic number theory, including Galois cohomology. We will at one point use a theorem of global class ﬁeld theory (Theorem 5.2.5). The language of category theory will be used freely, but we have included an appendix stating some (but possibly not all) of the results we need.

This thesis is for a large part based on an article by M. Stoll [22]. We aim at a pace suitable for graduate students in arithmetic geometry, assuming no knowledge of étale cohomology. The treatment of étale cohomology in Appendix B is mostly based on [15]. We tried to minimise the number of external results needed, but sometimes giving the full proof takes us too far aﬁeld.

6 Notation

Throughout this text, K will denote a ﬁeld, with separable closure K¯ and absolute Galois group ΓK . We will sometimes, by abuse of notation, write K for the scheme Spec K.

If K is a number field, then ΩK will denote its set of places. It consists of the f 8 set of finite places ΩK and the set of infinite places ΩK . If S Ď ΩK is a finite subset containing the infinite places, then AK,S will denote the S-adèles, i.e.

AK,S “ Kv ˆ Ov. vPS vRS ź ź The ring of adèles of K is denoted AK : colim A K , AK “ ÝÑ K,S Ď v S ﬁnite vPΩ źK 8 where the limit is taken over increasing ﬁnite sets S containing ΩK . If S Ď ΩK S is any subset, then AK denotes the adèles with support in S:

S AK “ AK X Kv, vPS ź K S where the intersection is the one taken in vPΩK v. If is the set of finite or f 8 S the set of infinite places, then we will writeśAK and AK respectively for AK . All rings are assumed Noetherian, and all schemes are assumed to be locally Noetherian. We will tacitly assume that morphisms of schemes are locally of finite type, except in the cases where this is obviously false (most notably, a ¯ morphism Spec Kv Ñ X for a completion Kv of K, or the map Spec K Ñ Spec K).

A variety X over a ﬁeld K is a geometrically reduced, separated scheme of ﬁnite ¯ ¯ ¯ type over K. The scheme X ˆK K is denoted X; it is a variety over K.

Recall that a point x P X is nonsingular (or regular) if OX,x is a regular local ring, and X is smooth at x if pΩX{K qx is free of rank dimpOX,xq. When K is algebraically closed, the two are equivalent, but this is not in general true. Note that x P X is smooth if and only if the corresponding point x¯ P X¯ is nonsingular.

A curve over a ﬁeld K will be a smooth, proper, and geometrically connected (hence geometrically integral) variety over K of dimension 1. A standard result shows that it is in fact projective.

If A is an abelian group, then Adiv denotes the subgroup of divisible elements. This need not be a divisible group, as we show in Remark 2.3.16.

7 1 Algebraic geometry

We will assume basic familiarity with the language of schemes, for instance following Hartshorne [10]. In this chapter we will prove some additional results that we will need later on. In the ﬁnal section of this chapter, we introduce some notions that are useful for comparing the K-rational and adelic points for varieties over number ﬁelds.

1.1 Étale morphisms

Definition 1.1.1. Let f : X Ñ Y be a morphism of schemes that is locally of finite type. Then f is unramified at x P X if, for y “ fpxq, the ideal in Ox generated by my is mx, and the field extension kpyq Ñ kpxq is separable. If f is unramified at all x P X, then f is unramified.

Lemma 1.1.2. Let f : X Ñ Y be a morphism of schemes that is locally of ﬁnite type. Then f is unramiﬁed if and only if ΩX{Y “ 0.

Proof. Note that ΩX{Y “ 0 if and only if pΩX{Y qx “ 0 for all x P X. Let x P X be given, and set y “ fpxq.

Let V – Spec A be an aﬃne open neighbourhood of y, and let U – Spec B be an aﬃne open neighbourhood of x contained in f ´1V . Firstly, note that

pΩX{Y q U – pΩB{Aq˜ ˇ pΩ q (by Hartshorne [10], Remark II.8.9.2).ˇ Hence, X{Y x is none other than pΩB{Aqmx . By Matsumura [13], Exercise 25.4, this is the same as ΩBx{Ay . By [loc. cit.], it holds that

ΩBx{Ay bAy kpyq “ ΩpBx{my Bxq{kpyq.

Moreover, we know that ΩBx{Ay is a ﬁnitely generated Bx-module (Hartshorne [10], Corollary II.8.5). Hence, by Nakayama’s lemma, it is zero if and only if

ΩpBx{my Bxq{kpyq “ 0.

But it is a standard result that a k-algebra of finite type R satisfies ΩR{k “ 0 if and only if R is a finite product of finite separable field extensions of k. Since Bx{myBx is also a local ring, this can only be the case if myBx “ mx and kpxq Ñ kpyq is separable.

Hence, f is unramiﬁed at x if and only if pΩX{Y qx “ 0. The result follows by considering these conditions for all x P X.

Definition 1.1.3. Let f : X Ñ Y be a morphism of schemes. Then f is étale if f is locally of finite type, flat, and unramified.

Remark 1.1.4. If Y “ Spec K is a point, then X Ñ Y is étale if and only if X is a (possibly infinite) disjoint union of spectra of finite separable field extensions L{K.

8 Lemma 1.1.5. Open immersions are étale.

Proof. Open immersions are clearly flat, unramified, and locally of finite type.

Lemma 1.1.6. Let f : X Ñ Y and g : Y Ñ Z be étale. Then the composite morphism g ˝ f is étale.

Proof. Clearly, the composition of morphisms that are locally of finite type is locally of finite type, and the composition of flat morphisms is flat. Moreover, we have an exact sequence of sheaves on X:

˚ f ΩY {Z Ñ ΩX{Z Ñ ΩX{Y Ñ 0

(see Hartshorne [10], Prop. II.8.11). Since the ﬁrst and third terms vanish, so does the middle term.

Lemma 1.1.7. Let f : X Ñ Y be an étale morphism, and let Y 1 Ñ Y be any morphism. Then the base change f 1 : X1 Ñ Y 1 of f along Y 1 Ñ Y is étale.

Proof. It is once again clear that the base change of a flat morphism that is locally of finite type is flat and locally of finite type. Moreover, by Hartshorne [10], Prop. II.8.10, we have

˚ ΩX1{Y 1 “ g ΩX{Y ,

1 1 where g : X Ñ X is the base change of Y Ñ Y along f. Hence, ΩX1{Y 1 is zero, since ΩX{Y is.

Lemma 1.1.8. Let f : X Ñ Y be locally of ﬁnite type. Then ΩX{Y “ 0 if and only if the diagonal morphism ∆: X Ñ X ˆY X is an open immersion.

Proof. Recall from Hartshorne [10], Section II.8 that the diagonal morphism factors as X Ñ W Ñ X ˆY X, where W Ď X ˆY X is an open subscheme and ˚ 2 X Ñ W is a closed immersion. Then ΩX{Y is the sheaf ∆ pI {I q, where I is the sheaf of ideals corresponding to the closed immersion X Ñ W .

Hence, it is clear that if X Ñ X ˆY X is an open immersion, then the restriction of I to ∆pXq is zero, hence ΩX{Y “ 0.

2 Conversely, if ΩX{Y “ 0, then Ix{Ix “ 0 for all x P X. But Ix Ď mx, since I is the ideal deﬁning X Ď W . Hence, Ix “ 0 by Nakayama’s lemma.

Hence, X is inside the open set V “ W z Supp I . Since X is deﬁned by I and

I V “ 0, this makes X Ñ V both an open and closed immersion, hence an isomorphism onto its image. Hence, X Ñ V Ñ W Ñ X ˆY X is a composition ˇ ofˇ open immersions, hence an open immersion.

Corollary 1.1.9. Let f : X Ñ Y be any morphism, and let g : Y Ñ Z be unramiﬁed. If gf is étale, then so is f.

9 Proof. Since g is unramiﬁed, the diagonal Y Ñ Y ˆZ Y is an open immersion. One easily sees that the square

1ˆf X X ˆZ Y

f f ˆ1

∆Y Y Y ˆZ Y is a pullback. Hence, 1ˆf is an open immersion, so in particular it is étale. But by deﬁnition the square π2 X ˆZ Y Y

π1 g g ˝ f X Z is a pullback. Hence, π2 is étale, since g ˝ f is. Hence, f “ π2 ˝ p1ˆfq is étale, since it is the composition of two étale morphisms.

Proposition 1.1.10. Let f : X Ñ Y be a closed immersion that is ﬂat (hence étale). Then f is an open immersion.

Proof. Since flat morphisms are open (Hartshorne [10], Exercise III.9.1), we can assume that f is surjective. If V Ď Y is an affine open (say V – Spec A), then U “ f ´1pV q is Spec A{I for some ideal I Ď A. Now Spec A{I Ñ Spec A is surjective, so by Atiyah–MacDonald [4], Exercise 3.16, aec “ a for all ideals a Ď A. In particular, setting a “ 0 we find that A Ñ A{I is injective, i.e. I “ 0.

Hence, f|U : U Ñ V is an isomorphism. Since V was arbitrary, this shows that f is an isomorphism.

Corollary 1.1.11. Let f : X Ñ Y be étale and separated, and suppose Y is connected. Then any section s of f is an isomorphism onto an open connected component.

Proof. The base change of ∆X : X Ñ X ˆY X along s is the map

1ˆs: Y ÝÑ Y ˆY X, where the structure morphism of Y (on the right hand side) as Y -scheme is via fs, which is just 1Y since s is a section. Hence, the second projection π2 : Y ˆY X Ñ X is an isomorphism, which identiﬁes 1ˆs with the map

s: Y Ñ X.

Since f is separated, ∆X is a closed immersion, hence so is 1ˆs “ s. Hence, s is unramiﬁed. Since f and fs “ 1Y are étale, Corollary 1.1.9 shows that s is étale. Hence, by the proposition above it follows that s is an isomorphism onto a clopen set. This set must be a connected component since Y is connected.

10 Corollary 1.1.12. Let f : X Ñ Y be étale and separated, and suppose Y is connected. Then two sections s1, s2 of f for which the topological maps agree on a point must be the same.

Proof. Both s1 and s2 are isomorphisms onto an open connected component, and since their topological maps agree on a point, this must be the same connected component U. But then both s1 and s2 are two-sided inverses of f|U , hence they are the same.

Corollary 1.1.13. Let f, g : X Ñ Y be S-morphisms, where X is a connected S-scheme and Y {S is étale and separated. Suppose x P X is such that fpxq “ gpxq “ y, and that the maps kpyq Ñ kpxq induced by f and g are the same. Then f “ g.

Proof. The maps

1ˆf : X ÝÑ X ˆS Y

1ˆg : X ÝÑ X ˆS Y are sections to the ﬁrst projection π1 : X ˆS Y Ñ X. Moreover, π1 is étale and separated since Y Ñ S is. Since fpxq “ gpxq “ y, the compositions

ÝÑ txu Ñ X ÝÑ X ˆS Y (1.1) induced by f and g both factor as

ÝÑ txu ÝÑ txu ˆS tyu Ñ X ˆS Y, (1.2) and the assumption on the maps kpyq Ñ kpxq implies that the maps in (1.2) coincide for f and g. Hence, so do the compositions in (1.1), so x maps to the same point under 1ˆf and 1ˆg. The result now follows from the preceding corollary.

1.2 Two results on proper varieties

We will prove two well-known theorems about proper varieties (Theorem 1.2.7 and Theorem 1.2.14 below).

Lemma 1.2.1. Let f : X Ñ Y be surjective, and let Y 1 Ñ Y be any morphism. Then the base change X1 Ñ Y 1 of f along Y 1 Ñ Y is surjective.

Proof. Let y1 P Y 1 be given, and let y P Y be its image. There is a commutative cube X1 X 1 Xy1 Xy Y 1 Y. y1 y

11 The left, right and back squares are pullbacks, hence so is the front square. Moreover, Xy is nonempty since X Ñ Y is surjective.

Hence, if U Ď Xy is some nonempty affine open, say U – Spec A, then the 1 1 inverse image of U in Xy1 is SpecpA bkpyq kpy qq. Since U is nonempty, A is 1 not the zero ring. Hence, A bkpyq kpy q is not the zero ring either, since field 1 extensions are faithfully flat. Hence, Xy1 contains a nonempty open subset, hence is nonempty.

Since y1 was arbitrary, we see that all ﬁbres of X1 Ñ Y 1 are nonempty. Hence, X1 Ñ Y 1 is surjective.

Proposition 1.2.2. Let f : X Ñ Y and g : Y Ñ Z be morphisms of schemes. If g is separated and gf proper, then f is proper. If moreover f is surjective and g is of ﬁnite type, then g is proper.

Proof. The ﬁrst statement is Hartshorne [10], Corollary II.4.8(e). For the second, we only have to show that g is universally closed.

1 1 1 1 Let Z be any scheme over Z. Write X “ X ˆZ Z and Y “ Y ˆZ Z . Then we have maps f 1 g1 X1 ÝÑ Y 1 ÝÑ Z1, and f 1 and g1f 1 are closed. Moreover, f 1 is surjective, since surjectivity is stable under base change (by the lemma above).

But if V Ď Y 1 is closed, then f 1pf 1´1pV qq “ V by surjectivity of f 1. Hence, the image of V under g1 is the image of the closed set f 1´1pV q under g1f 1, which is closed.

Corollary 1.2.3. Let f : X Ñ Y be a morphism of separated schemes of ﬁnite type over a base scheme S. Let V be a closed subscheme of X that is proper over S, and let Z be its scheme theoretic image in Y . Then Z is proper over S.

Proof. Since Z Ñ Y is a closed immersion, it is separated. Since Y Ñ S is separated as well, so is Z Ñ S. By the ﬁrst part of the proposition above, the morphism V Ñ Z is proper. Moreover, by deﬁnition of the scheme theoretic image, the morphism V Ñ Z is dominant, hence it is surjective. The result now follows from the second part of the proposition.

Lemma 1.2.4. Let φ: A Ñ B be an injective ring homomorphism, and assume that the induced morphism Spec B Ñ Spec A is closed. Then φ´1pBˆq “ Aˆ.

Proof. The morphism Spec B Ñ Spec A is dominant since φ is injective. Since it is also closed, it is surjective. We clearly have Aˆ Ď φ´1pBˆq.

Now let a P φ´1pBˆq. If a P p for some prime ideal p Ď A, let q be a prime of B such that φ´1pqq “ p. Then φpaq P q, contradicting the assumption that a P φ´1pBˆq. Hence, a is not in any prime ideal, so it is invertible.

12 Lemma 1.2.5. Let φ: A Ñ B be an injective ring homomorphism. Write ψ for the homomorphism ArT s Ñ BrT s, and suppose that the induced morphism 1 1 AB Ñ AA is closed. Then φ is integral.

1 Proof. Let b P B be given, and consider the map BrT s Ñ Br b s. Let C be the image of the composition

1 ArT s Ñ BrT s Ñ B b . 1 “ ‰ 1 1 Then the morphism Spec Br b s Ñ Spec C is the restriction of AB Ñ AA to certain closed subschemes, hence is also a closed map. Moreover, the ring ho- 1 momorphism C Ñ Br b s is injective.

Hence, by the lemma above, the image of T in C is invertible, since the image 1 of T in Br b s is invertible. Hence, b P C, so we can write

n 1 i b “ a i b i“0 ÿ ˆ ˙ n`1 n´i 1 for certain a0, . . . , an P A. Then b ´ aib “ 0 in Br b s, so there exists m P Zě0 such that n ř m n`1 n´i b b ´ aib “ 0 ˜ i“0 ¸ ÿ in B. Hence, b is integral over A.

Corollary 1.2.6. Let f : Spec B Ñ Spec A be a proper morphism of aﬃne schemes. Then f is ﬁnite.

Proof. Let I be the kernel of A Ñ B. Then Spec A{I Ñ Spec A is a closed immersion, hence it is finite. Moreover, A{I Ñ B is injective and g : Spec B Ñ Spec A{I is proper, so by the lemma above, g is integral. Since it is also of finite type, it is finite. The composition of two finite morphisms is finite.

Theorem 1.2.7. Let f : X Ñ Y be a morphism of varieties over a ﬁeld K. If X is proper and connected, and is Y aﬃne, then f is constant (i.e. fpXq is a point).

Proof. By our definition of varieties, X and Y are separated and of finite type over K. By Corollary 1.2.3, the scheme theoretic image Z of f is proper over K. Since Y is affine, so is Z, so by the corollary above, Z is finite over K. Finally, Z is connected since X is, so it is a point.

Lemma 1.2.8. Let S be a scheme, and let Spec R be an aﬃne scheme over S, where R is a domain with ﬁeld of fractions K. Let X{S be separated, and let f, g : Spec R Ñ X be two S-morphisms such that the compositions

ÝÑ Spec K ÝÑ Spec R ÝÑ X coincide. Then f “ g.

13 Proof. Write η for the generic point in Spec R, i.e. the image of Spec K Ñ Spec R. Let U “ Spec A be an aﬃne open neighbourhood of fpηq “ gpηq, and let V Ď f ´1pUq X g´1pUq be an aﬃne open neighbourhood of η of the form 1 V “ Dpxq “ Spec Rr x s for some x P R.

1 1 Now Rr x s is also a domain with fraction ﬁeld K, so Rr x s Ñ K is monic. Moreover, f|V and g|V are given by certain ring homomorphisms

ÝÑ 1 A ÝÑ R x , 1 “ ‰ and the compositions with Rr x s Ñ K coincide. This forces f|V “ g|V .

Now V is dense in Spec R since Spec R is irreducible. Since Spec R is reduced and X is separated, this forces f “ g (cf. Hartshorne [10], Exercise II.4.2).

Deﬁnition 1.2.9. A scheme C is called a Dedekind scheme if it is integral, normal, noetherian, and of dimension 1.

Example 1.2.10. Let R be a ring. Then Spec R is a Dedekind scheme if and only if R is a Dedekind domain.

Example 1.2.11. Let K be a ﬁeld, and X a variety over K. Then X is a Dedekind scheme if and only if X is a nonsingular, connected (hence integral) variety of dimension 1. In particular, this holds for all X ˆK L (L{K ﬁnite) when X is a curve.

Proposition 1.2.12. Let C be an S-scheme that is a Dedekind scheme, and let X be a proper S-scheme. Let U Ď C be a nonempty open subset, and f : U Ñ X an S-morphism. Then f extends uniquely to a morphism on C.

Proof. Since C has dimension 1, the complement of U is ﬁnite. By induction, we can assume that it consists of a single point P . If Q is any closed point on C, then OC,Q is a discrete valuation ring, since it is a normal local noetherian domain of dimension 1. Its fraction ﬁeld is OC,η, where η is the generic point of C.

By the valuative criterion of properness, the map Spec OC,η Ñ X coming from U Ñ X extends uniquely to a map Spec OC,Q Ñ X, making commutative the diagram

Spec OC,η X

Spec OC,Q S.

Since X is of ﬁnite type over S, such a morphism factors as

Spec OC,Q Ñ V Ñ X for some open set V Ď C containing Q (each generator in an aﬃne of X maps into some OV , and we take the intersection over ﬁnitely many generators).

14 Similarly, any two factorisations ÝÑ Spec OC,Q Ñ V ÝÑ X have to coincide on some open W Ď V containing Q.

Applying this to Q “ P , we ﬁnd that there is an open V Ď C containing P and a morphism g : V Ñ X inducing the unique map Spec OC,P Ñ X induced by Spec OC,η Ñ X. Moreover, for any Q P U X V , there exists an open W containing Q such that the compositions ÝÑ W Ñ U X V ÝÑ X induced by f and g coincide. Hence, the maps f, g : U X V Ñ X coincide, so they glue uniquely to a morphism U Y V Ñ X. But V contains the sole point outside U, hence U YV “ C. This shows existence, and uniqueness is clear.

Lemma 1.2.13. Let R be a Dedekind domain with fraction ﬁeld K. Let Z be an integral scheme, and let f g Spec K ÝÑ Z ÝÑ Spec R be morphisms such that their composition is the morphism given by R Ñ K. If f is dominant and g is proper and surjective, then g has a section.

Proof. Let ηZ and η be the generic points of Z and Spec R respectively. Since f is dominant, its image is tηZ u, so gpηZ q “ η. Comparing the respective local rings shows that K Ď OZ,ηZ Ď K, so in fact equality holds.

In particular, g is generically ﬁnite. Hence, by Exercise II.3.7 of Hartshorne [10], there exists an open dense subset U Ď Spec R such that g´1pUq Ñ U is ﬁnite. We can take U to be Spec Ra for some a P R. Then Ra is integrally closed since R is, and g´1pUq “ Spec B for some ring B.

Since OZ,η “ K, the ﬁeld of fractions of B is K. Since the composite map Ra Ñ B Ñ K is injective, so is Ra Ñ B. Hence, B is an Ra-subalgebra of K. It is ﬁnite over Ra, hence it equals Ra since Ra is integrally closed. That is, g : g´1pUq ÝÑ„ U. Hence, we have a section h: U Ñ Z on U. Since R is a Dedekind domain and Z Ñ Spec R is proper, the lemma above shows that h extends uniquely to a section h: Spec R Ñ Z of g.

Theorem 1.2.14. Let S be a scheme, and let Spec R be an aﬃne scheme over S, where R is a Dedekind domain with ﬁeld of fractions K. Let X{S be proper. Then any morphism Spec K Ñ X factors uniquely as

Spec K X.

Spec R

15 Proof. Uniqueness is given by Lemma 1.2.8. For existence, it suﬃces to prove the result for the base change X ˆS Spec R, as Spec R-scheme (note that it is still proper, since properness is stable under base change). That is, we will assume that S “ Spec R.

Let Z be the scheme theoretic image of Spec K Ñ X. Since Spec K is reduced, it is just the reduced induced structure on the closure of the image (Hartshorne [10], Exercise II.3.11(d)). Since Z is the closure of a point, it is irreducible, hence integral since it is reduced.

Since X Ñ Spec R is proper, it is a closed map. Hence, the image of the closed set Z is closed in Spec R. Since it contains the generic point, it must be equal to Spec R. That is, the map Z Ñ Spec R is surjective. Since Z Ñ X and X Ñ Spec R are proper, so is Z Ñ Spec R.

Hence, by Lemma 1.2.13, the map Z Ñ Spec R has a section. Then the composition Spec R Ñ Z Ñ X gives the required map.

Corollary 1.2.15. Let R be a Dedekind domain with fraction ﬁeld K with a map Spec R Ñ S. Let X{S be proper. Then

XpKq “ XpRq.

Proof. This is a reformulation of the theorem.

Remark 1.2.16. Similarly, Lemma 1.2.8 says that the map

XpRq Ñ XpKq is an inclusion whenever X is separated over S (for any domain R).

Remark 1.2.17. Theorem 1.2.14 does not hold for a general (not necessarily normal) domain of dimension 1, even if we restrict to projective schemes over S “ Spec R.

For example, if k is a ﬁeld and R “ krT 2,T 3s – krX,Y s{pX3 ´ Y 2q, then K “ kpT q. If we take X “ Spec krT s, then the map

X Ñ Spec R is ﬁnite, hence projective. However, the canonical K-point of X induced by krT s Ñ kpT q can never factor through Spec R, since the ring homomorphism krT s Ñ kpT q does not factor through R “ krT 2,T 3s.

Similarly, the assumption that R has dimension 1 cannot be dropped. For 2 2 instance, the blow-up X of AK in the origin has the same function ﬁeld as AK , 2 yet there is no section of X Ñ AK .

16 1.3 Adelic points

Throughout this section, K will denote a number ﬁeld, and AK its ring of adèles. We state some basic properties about the AK -points of a K-variety X. For a more complete treatment, see [5].

Proposition 1.3.1. Let X{K be a variety.

(1) There exists a ﬁnite set S Ď ΩK containing the inﬁnite places and a scheme XS over OK,S such that

XS ˆOK,S K – X.

(2) If Y is another variety and YS a model over OK,S, then

colim HomO pXT ,YT q “ HomK pX,Y q, ÝÑ K,T

where XT denotes XS ÔK,S OK,T for S Ď T (and similarly for Y ). (3) If X{K is separated, proper, flat, smooth, affine, or finite, then XT {OK,T has the same property for some finite set T containing S. 1 (4) If XS and XS1 are two such models, then they become isomorphic on some finite set T containing S and S1. Moreover, any two such isomorphisms become the same for T large enough.

Proof. This is Theorem 3.4 of [5].

Proposition 1.3.2. Let XS be a model of X over OK,S. There is a natural identiﬁcation

X pAK q “ pxvq P XpKvq xv P XSpOvq for almost all v . # ˇ + vPΩK ˇ ź ˇ ˇ Proof. This follows from Theorem 3.6 ofˇ [5].

Remark 1.3.3. It is clear from Proposition 1.3.1 (4) that the set XSpOvq (in the deﬁnition) does not depend on S or on the model XS chosen.

Definition 1.3.4. Let X{K be a variety. Then we define a topology on XpAK q as the restricted product topology, via the identification of the proposition.

Proposition 1.3.5. Let X{K be a variety.

(1) XpAK q is a locally compact Hausdorﬀ space, (2) If X is isomorphic to the aﬃne line, then the topology on XpAK q is the same as the usual topology on AK , (3) If X – X1 ˆK X2, then the topology on XpAK q “ X1pAK q ˆ X2pAK q is the product topology, (4) If X Ñ Y is a morphism of K-varieties, then the map XpAK q Ñ Y pAK q is continuous, (5) If X Ñ Y is a closed immersion, then XpAK q Ñ Y pAK q is a closed embedding.

17 Proof. This follows from [5], §3.

Remark 1.3.6. Note however that open immersions do not necessarily go to open embeddings. For example, the topology on the idèles IK Ď AK is not the subspace topology of AK .

Proposition 1.3.7. Let X{K be a proper variety, and let XS{OK,S be proper as in Proposition 1.3.1 (1),(3). Then

X pAK q “ XpKvq “ XSpOvq ˆ XpKvq. f 8 vPΩK vPΩ vPΩK ź źK ź

Proof. Immediate from Proposition 1.3.2 and Theorem 1.2.14.

Deﬁnition 1.3.8. Let X{K be a proper variety. Then we write

XpAK q‚ “ XpKvq ˆ π0pXpKvqq, f vPΩ8 vPΩK K ź vźreal where π0pXpKvqq denotes the set of connected components of XpKvq (in the real topology).

Remark 1.3.9. This is the notation occurring in Stoll’s paper [22]. In the paper itself, the set XpAK q‚ is deﬁned as

XpKvq ˆ π0pXpKvqq, f 8 vPΩ vPΩK źK ź i.e. including the π0 of the complex places as well. This is changed to the deﬁnition above in the errata.

Note that if X is connected, then so is XpKvq for any complex place v, by Shafarevich [19], Theorem VII.2.2.1. Hence, in this case the two deﬁnitions coincide.

2 Group schemes

In this chapter, we will cover some results about group schemes. Sections 1 and 2 are rather general, while the last three sections focus on more specialistic results we will need later on, in Chapter5.

2.1 Group schemes

Deﬁnition 2.1.1. Let S be a base scheme. Then a group scheme over S is a group object in the monoidal category pSch{S, ˆSq. That is, it is a scheme G{S together with S-morphisms µ G ˆS G ÝÑ G η S ÝÑ G ι G ÝÑ G such that the diagrams

1ˆµ G ˆS G ˆS G G ˆS G

µˆ1 µ (2.1) µ G ˆS G G,

1ˆη ηˆ1 G ˆS S G ˆS G S ˆS G µ (2.2) π1 π2 G,

∆G 1ˆι G G ˆS G G ˆS G µ (2.3) η S G commute.

In general, we will assume that all group schemes are ﬂat.

Since it is in general hard to check whether a given object G with morphisms µ, η and ι is actually a group scheme, we will often use the following criterion:

Proposition 2.1.2. Let X{S be a scheme. Then X is a group scheme over S if and only if for every scheme T {S, the set

XpT q “ HomSpT,Xq is a group, and for every morphism g : T Ñ T 1 of S-schemes, the natural map XpT 1q Ñ XpT q is a group homomorphism.

20 Proof. This is Corollary A.4.7.

Corollary 2.1.3. Let G{S be a group scheme, and let S1{S be arbitrary. Then 1 1 1 G “ G ˆS S is a group scheme over S .

Proof. Let T {S1 be a scheme. Then

1 1 G pT q “ HomS1 pT,G ˆS S q “ HomSpT,Gq “ GpT q, where the structure map of T as S-scheme is given by the composition T Ñ S1 Ñ S. Hence, the result follows from the proposition above.

Corollary 2.1.4. If G1, G2 are two group schemes over S, then G1 ˆS G2 is a group scheme over S.

Proof. Let T be an S-scheme. Then

pG1 ˆS G2qpT q “ HomSpT,G1 ˆS G2q “ HomSpT,G1q ˆ HomSpT,G2q, and the result follows from the proposition.

Remark 2.1.5. This result also follows from Corollary A.4.16.

Remark 2.1.6. One could also prove the above two results directly, by deﬁning multiplication, unit and inversion, and showing that they satisfy the necessary relations. However, the proofs we give are easier and in some way more intuitive, since in many cases it is more natural to think of a scheme in terms of its T - points (for all schemes T {S).

We recall from scheme theory the following adjunction.

op Lemma 2.1.7. The functor Sch Ñ Ring given by X ÞÑ ΓpX, OX q is the left adjoint of the functor Spec: Ringop Ñ Sch.

Proof. We need to show that, for any scheme X and for any ring R, there is an isomorphism

„ HomSchpX, Spec Rq ÝÑ HomRingpR, ΓpX, OX qq, natural in both X and R. The isomorphism is given by taking global sections (cf. Hartshorne [10], Exercise II.2.4), and naturality is easy to check.

This gives already many examples of group schemes.

Example 2.1.8. If G “ Spec ZrXs, then for any scheme T there is an isomorphism GpT q – HomRingpZrXs, ΓpT, OT qq – ΓpT, OT q, where the ﬁrst isomorphism is given by the lemma, and the second since ZrXs represents the forgetful functor Ring Ñ Set (cf. Example A.1.4).

21 1 Hence, since ΓpT, OT q is a group and each ΓpT , OT 1 q Ñ ΓpT, OT q is a group homomorphism (for T Ñ T 1 a morphism of schemes), this shows that G is a group scheme. It is denoted Ga, for the additive group.

Example 2.1.9. Similarly, if G “ Spec ZrX,X´1s, then there is a natural isomorphism ˆ GpT q “ ΓpT, OT q , since ZrX,X´1s represents the group of units functor Ring Ñ Set. Hence, G is a group scheme, called the multiplicative group. It is denoted Gm.

n ´1 Example 2.1.10. If G “ Spec ZrtXijui,j“1, det s, then there is a natural isomorphism GpT q “ GLnpΓpT, OT qq, n ´1 since ZrtXijui,j“1, det s represents the functor GLn : Ring Ñ Set. Here, det denotes the element n

det “ sgnpσq Xiσpiq. σPS i“1 ÿn ź The group scheme G is called the general linear group scheme, and is denoted GLn.

Example 2.1.11. If G “ Spec ZrXs{pXn ´ 1q, then there is an isomorphism

n GpT q – tx P ΓpT, OT q : x “ 1u.

Then G is called the group of n-th roots of unity, and is denoted µn.

G Example 2.1.12. Let G be a ﬁnite group. Put X “ SpecpZ q “ gPG Spec Z. Then XpT q – Gπ0pT q, š since a morphism T Ñ X is uniquely determined by choosing a connected component of X for each connected component of T . The thus obtained group scheme is called the constant group scheme on the group G, and is denoted G as well.

In particular, for G “ 0, we get the trivial group scheme.

Deﬁnition 2.1.13. Let K be a ﬁeld. Then a group variety over K is a group scheme that is a variety over K.

Example 2.1.14. We get group varieties

Ga,K , Gm,K , GLn,K ,GK pG ﬁniteq over K, by extension of scalars. Note that µn,K is only a group variety over K when charK - n. Indeed, if charK | n, then KrXs{pXn ´ 1q is not reduced, so G “ µn,K is not a variety by our conventions.

Lemma 2.1.15. Let X be a variety over an algebraically closed ﬁeld K. Then there exists a dense open subset U Ď X that is nonsingular.

22 Proof. For X irreducible, this is Hartshorne [10], Cor. II.8.16.

For general X, let X1,...,Xn be the irreducible components of X. Deﬁne

V “ Xz pXi X Xjq i‰j ď as the (open) set of points that are in one component only. Note that V is nonsingular at a point x P V if and only if X is nonsingular at x, since x is in one irreducible component only.

The irreducibility of each Xi and the fact that Xi Ď Xj for i ‰ j force that each V X Xi is nonempty. Hence, V is dense in X, since its closure contains each Xi. Moreover, the irreducible components of V are the Vi “ V X Xi, and each two have empty intersection. That is, V is the disjoint union of the Vi.

Now each Vi contains some (dense) open Ui that is nonsingular. Then clearly the union U of these Ui is open and dense in V , and nonsingular. The result follows since V is open and dense in X.

Lemma 2.1.16. Let G be a group variety over K. Then G is smooth.

Proof. Recall that G is smooth if and only if G¯ is nonsingular. By the lemma above, G¯ has an open dense subset U which is nonsingular. There are transla- tions ¯ 1ˆa ¯ ¯ µ ¯ τa : G ÝÑ G ˆK¯ G ÝÑ G for various a P GpK¯ q. Each translation is an isomorphism, and the translates of U cover G¯. Hence, G¯ is nonsingular.

Remark 2.1.17. Some authors allow for a more general deﬁnition of group variety, in which reducedness is not assumed. Then the lemma above does not hold for G “ µp,Fp . What we call a group variety is then called a smooth group variety, cf. the lemma.

Deﬁnition 2.1.18. Let G be a group variety over K, let P : Spec K¯ Ñ G be a K¯ -point of G. Observe that there is an isomorphism

„ op ψ :ΓK ÝÑ AutSpec K pSpec K¯ q mapping σ : K¯ Ñ K¯ to its associated morphism Spec K¯ Ð Spec K¯ (in the other direction). ¯ Let σ P ΓK . Then we deﬁne σP as the K-point P ˝ ψpσq of G. That is, there is a commutative diagram

Spec K¯ σP ψpσq G. Spec K¯ P

23 ¯ Remark 2.1.19. This makes GpKq into a ΓK -module, as

pστqP “ P ˝ ψpστq “ P ˝ ψpτq ˝ ψpσq “ σpτP q ¯ for all σ, τ P ΓK , P P GpKq. Moreover, this is a discrete ΓK -module, since any K¯ -point of G is actually deﬁned over some ﬁnite extension L{K.

Finally, if L{K is some separable algebraic extension, then GpLq “ GpK¯ qΓL . Indeed, if P P GpK¯ q and U is some aﬃne open neighbourhood of (the image of) n P , then U is closed in some AK . The K-algebra homomorphisms

φ: KrX1,...,Xns Ñ K¯ such that σ ˝ φ “ φ for all σ P ΓL are exactly those with image inside L, so P is an L-point if and only if it is ΓL-stable. ¯ We shall simply write G for the ΓK -module GpKq.

2.2 Abelian varieties

Deﬁnition 2.2.1. An abelian variety over K is a geometrically connected, proper group variety A over K.

Example 2.2.2. As an uninteresting example, the trivial group scheme gives an abelian variety of dimension 0. Since abelian varieties are connected and reduced, and have an identity section Spec K Ñ A, this is the only dimension 0 abelian variety.

Example 2.2.3. Any elliptic curve is an abelian variety of dimension 1. Using the theory of Jacobians, one can also show that these are the only abelian varieties of dimension 1. See Remark 2.5.4

Remark 2.2.4. As we will see, the group law on an abelian variety is indeed commutative, justifying the name. Note however that it is not true that every commutative group variety over a field K is an abelian variety. For instance, Ga,K is clearly commutative, but not proper (since the only proper morphisms of affine schemes over K are finite morphisms, by Corollary 1.2.6).

Theorem 2.2.5. Let X, Y and Z be varieties over K, such that X is proper and X ˆK Y is irreducible. Let α: X ˆK Y Ñ Z be a morphism of varieties over K, and suppose there exist closed points x P X, y P Y and z P Z such that

αpX ˆK tyuq “ tzu “ αptxu ˆK Y q.

Then αpX ˆK Y q “ tzu.

Proof. Let Z0 be an aﬃne open neighbourhood of z, and let V be the inverse ´1 image of the closed set ZzZ0 in X ˆK Y . Let U “ pX ˆK Y qzV “ α pZ0q be its complement.

24 Since X is proper, the second projection π2 : X ˆK Y Ñ Y is closed, so the image W of V is closed in Y . Moreover,

´1 π2 ptyuq “ X ˆK tyu Ď U, as αpX ˆK tyuq “ tzu Ď Z0. Hence, y is not in W “ π2pV q.

1 1 Now for any closed point y P Y zW , the ﬁbre X ˆK ty u is inside U, so

1 αpX ˆK ty uq Ď Z0.

1 1 Since y is closed, it is ﬁnite over K, so X ˆK ty u is proper over K. Then

1 α 1 : X ˆK ty u Ñ Z0 XˆK ty u ˇ is a map from a proper varietyˇ into an aﬃne variety, so this map has to be 1 constant (Theorem 1.2.7). Since αptxu ˆK ty uq “ tzu, in fact this constant value has to be z. That is, for all y1 P Y zW , it holds that

1 αpX ˆK ty uq “ tzu.

Since the closed points of Y zW are dense in it (by the Nullstellensatz), this forces αpX ˆK pY zW qq “ tzu.

Now X ˆK pY zW q is a nonempty open subset of an irreducible variety, hence it is dense. This gives αpX ˆK Y q “ tzu, as α is continuous and tzu is closed.

Corollary 2.2.6. Let f : A Ñ B be a morphism of abelian varieties over K. Then f can be written uniquely as f “ τb ˝g, with τb a right translation by some closed point b P B and g a homomorphism of abelian varieties.

Proof. Let b “ fp0q, and put g “ τ´b ˝ f. Then gp0q “ 0. Now deﬁne

α: A ˆK A ÝÑ B

pa1, a2q ÞÝÑ gpa1 ` a2q ´ gpa1q ´ gpa2q.

That is, α is the composition

pµ,ιπ1,ιπ2q gˆgˆg µˆ1 µ A ˆ A ÝÑ A ˆ A ˆ A ÝÑ B ˆ B ˆ B ÝÑ B ˆ B ÝÑ B, where we drop the subscript from ˆK to ease notation. One easily sees that

αpA ˆK t0uq “ t0u “ αpt0u ˆK Aq, so α is zero by the theorem. Hence, g is a homomorphism of abelian varieties. It is clear that this factorisation is unique.

Corollary 2.2.7. Let A be an abelian variety over K. Then A is commutative.

Proof. The inversion ι: A Ñ A is a morphism ﬁxing 0. Hence, by the corollary above, it is a homomorphism.

25 2.3 Selmer groups

Here and henceforth, A will denote an abelian variety over a number ﬁeld K.

Deﬁnition 2.3.1. Let rns: A Ñ A be the multiplication by n map. Then we write Arns for the kernel of rns. It is a group scheme by Corollary A.4.16.

Remark 2.3.2. Since rns is unramified, it is in fact étale. The base change of rns along the identity section 0 Ñ A is Arns (this holds in any category with finite products). Hence, Arns is étale over K. In fact, it is finite étale over K.

Lemma 2.3.3. There is a short exact sequence

0 Ñ ApKq{nApKq Ñ H1pK,Arnsq Ñ H1pK,Aqrns Ñ 0.

Proof. There is a short exact sequence of ΓK -modules

n 0 Ñ Arns Ñ A Ñ A Ñ 0.

Then the long exact sequence of Galois cohomology groups gives

n n ApKq Ñ ApKq Ñ H1pK,Arnsq Ñ H1pK,Aq Ñ H1pK,Aq, hence the result.

Remark 2.3.4. This induces a commutative diagram

0 ApKq{nApKq H1pK,Arnsq H1pK,Aqrns 0

1 1 0 ApKvq{nApKvq H pKv,Arnsq H pKv,Aqrns 0, vPΩ vPΩ vPΩ źK źK źK with exact rows.

Deﬁnition 2.3.5. The n-Selmer group of A is the kernel

pnq 1 1 Sel pK,Aq “ ker H pK,Arnsq Ñ H pKv,Aqrns ˜ vPΩ ¸ źK of the diagonal of the right hand square of the diagram above.

Deﬁnition 2.3.6. The Tate–Shafarevich group of A is

1 1 XpK,Aq “ ker H pK,Aq Ñ H pKv,Aq . ˜ vPΩ ¸ źK Corollary 2.3.7. There is a short exact sequence

0 Ñ ApKq{nApKq Ñ SelpnqpK,Aq Ñ XpK,Aqrns Ñ 0.

26 Proof. Restrict the short exact sequence of the lemma to the elements that map to zero in the bottom right term

1 H pKv,Aqrns vPΩ źK of the commutative diagram above.

Remark 2.3.8. One can show that SelpnqpK,Aq is ﬁnite. The proof is essentially the same as that of the weak Mordell–Weil theorem. For an elliptic curve, it is given in Theorem X.4.2(b) of Silverman [20].

In particular, XpK,Aqrns is ﬁnite. There is the following conjecture.

Conjecture 2.3.9. (Shafarevich–Tate) The group XpK,Aq is ﬁnite.

Remark 2.3.10. Since XpK,Aq lives inside H1pK,Aq, it is torsion. Therefore, the conjecture breaks up into two statements:

• XpK,Aqrps “ 0 for almost all primes p, • XpK,Aqtpu is ﬁnite for all primes p.

Only in particular cases are we able to compute XpK,Aqtpu, so we are still quite far away from proving the conjecture.

Lemma 2.3.11. Let m | n. Then there is a commutative diagram

0 ApKq{nApKq SelpnqpK,Aq XpK,Aqrns 0 n m 0 ApKq{mApKq SelpmqpK,Aq XpK,Aqrms 0 with exact rows.

Proof. There is a commutative diagram

n 0 Arns A A 0 n m m 0 Arms A A 0 with exact rows. The associated long exact sequence is

n n A A H1pK,Arnsq H1pK,Aq H1pK,Aq n n m m m m A A H1pK,Armsq H1pK,Aq H1pK,Aq.

The result follows by restricting to the respective Selmer groups.

27 Deﬁnition 2.3.12. Write ApKq for the limit

A K lim A K nA K . {p q “ ÐÝ p q{ p q n Also, put { Sel K,A lim Selpnq K,A , p q “ ÐÝ p q n where the limit is taken overx the maps above. Finally, if B is any abelian group, put TB lim B n “ ÐÝ r s n for the (absolute) Tate module of B, where the limit is taken with respect to the maps n Brns ÝÑm Brms for m | n.

Proposition 2.3.13. Let I be a directed set, and let

0 Ñ pAiq Ñ pBiq Ñ pCiq Ñ 0 be a short exact sequence of projective systems. Then there is an exact sequence 0 lim A lim B lim C . Ñ ÐÝ i Ñ ÐÝ i Ñ ÐÝ i i i i

If moreover I is countable and the maps Ai Ñ Aj for j ď i are surjective, then the sequence is exact on the right as well, i.e. lim Bi Ñ lim Ci is surjective. ÐÝ ÐÝ

Proof. Note that the limit of a projective system pMiq is given by

ψ lim M ker M M , ÐÝ i “ j ÝÑ i i ˜jďi i ¸ ź ź where ψ is the map given by

pmjqj,i ÞÝÑ pmj ´ m¯ iq, where m¯ i denotes the image of mi in Mj under the natural map Mi Ñ Mj. Then the limits of pAiq, pBiq and pCiq are the kernels of the vertical maps in the diagram

0 Aj Bj Cj 0 jďi jďi jďi ź ź ź ψA ψB ψC

0 Ai Bi Ci 0, i i i ź ź ź and the ﬁrst statement follows from taking vertical kernels.

Now if I is countable, say I “ ti0,...u, we inductively construct a subset J Ď I of the form tj0,...u such that ik ď jk and the map k ÞÑ jk gives an isomorphism „ of ordered sets Zě0 ÝÑ J.

28 Namely, pick j0 “ i0, and inductively let jk be such that jk´1 ď jk and ik ď jk. Such jk exists since I is directed, and it is clear that J satisﬁes the promised properties. We shall identify Zě0 with J via k ÞÑ jk.

Now J is coﬁnal since ik ď jk for all k P Zě0, so lim M lim M ÐÝ i “ ÐÝ k iPI kPZě0 for any projective system pMiq. Moreover, the order on J is linear, so

lim M ker M ∆ M , ÐÝ k “ k ÝÑ k k ˜ k k ¸ ź ź where ∆ is the map given by

pmkqk ÞÝÑ pmk ´ m¯ k`1qk, where m¯ k`1 denotes the image of mk`1 in Mk under the map Mk`1 Ñ Mk. We now get a commutative diagram with exact rows

0 Ak Bk Ck 0 k k k ź ź ź ∆A ∆B ∆C

0 Ak Bk Ck 0. k k k ź ź ź If pakqk is given, set m0 “ a0 and inductively pick some mk`1 P Ak`1 such that m¯ k`1 “ mk ´ ak. We can do this since Ak`1 Ñ Ak is surjective. Then by deﬁnition ∆A ppmkqkq “ pmk ´ m¯ k`1qk “ pakqk.

Since pakqk was arbitrary, this shows that ∆A is surjective. The result now follows from the snake lemma.

Corollary 2.3.14. There is a short exact sequence

0 Ñ ApKq Ñ SelpK,Aq Ñ T XpK,Aq Ñ 0.

If moreover XpK,Aqdiv{“ 0, thenx ApKq – SelpK,Aq.

Proof. The ﬁrst statement follows from{ thex proposition, as the maps ApKq{nApKq ÝÑ ApKq{mApKq for m | n are surjective.

The second statement follows as TB “ T pBdivq for any abelian group B. Indeed, if

pbnqnPZą0 P TB Ď Brns nP źZą0 then that means exactly that mbmn “ bn for all m, n P Zą0. Hence, each bn is divisible, so pbnqn P T pBdivq.

29 Lemma 2.3.15. Let B be a torsion abelian group such that Brns is ﬁnite for all n P Zą0. Then Bdiv is a divisible group, so in particular it is the maximal divisible subgroup of B.

Proof. Let b P Bdiv be given, then there exist bn P B such that nbn “ b. What we have to prove is that we can take bn to be in Bdiv.

Let d be the order of b. Then any bn with nbn “ b has order nd. Hence, the subsets Cn “ cn P Brdns ncn “ b of Brdns are all nonempty. Moreover, thereˇ are natural( maps ˇ Cmn ÝÑ Cn

cmn ÞÝÑ mcmn for all m, n. Since Brdns is ﬁnite, so is Cn. Since a projective limit of ﬁnite nonempty sets is nonempty, there exists an element pcnq P lim Cn Ď TB such ÐÝ that ncn “ b for all n. Saying that pcnq P lim Cn means that mcmn “ cn for all ÐÝ m, n, hence each cn is divisible. Hence, b is divisible in Bdiv as well.

Remark 2.3.16. The lemma is no longer valid if we drop the assumption that Brns be ﬁnite. Indeed, let B be the quotient of the group

xnpZ{2nZq n à by the subgroup generated by nxn ´ x1. Then B is a torsion group, and x1 is divisible by construction (and it has order 2). However, no other xn is divisible, and in fact one can show that Bdiv “ t0, x1u. This is clearly not a divisible group.

What happens here is that we can ﬁnd xn with nxn “ x1, but we can not do this in a compatible way, i.e. we do not have mxmn “ xn.

Corollary 2.3.17. We have ApKq “ SelpK,Aq if and only if XpK,Aqdiv “ 0.

Proof. One implication was already{ notedx in Corollary 2.3.14. Conversely, if ApKq “ SelpK,Aq, then T XpK,Aq “ 0. The proof of the lemma above shows that if Bdiv ‰ 0, then TB ‰ 0. Since XpK,Aqrns is ﬁnite for each n, we can apply{ thisx to B “ XpK,Aq to obtain the result.

Remark 2.3.18. The article of Stoll [22] uses the notation Bdiv to mean the maximal divisible subgroup of an abelian group B, as opposed to the set of divisible elements. By the lemma above, in the case of XpK,Aq there is no diﬀerence.

Finally, we will compare ApAK q with ApAK q‚ (see section 1.3 for this notation). Proposition 2.3.19. There is a short exact sequence

0 Ñ ApAK qdiv Ñ ApAK q Ñ ApAK q‚ Ñ 0.

30 Proof. There is a short exact sequence

0 0 Ñ ApKvq Ñ ApAK q Ñ ApAK q‚ Ñ 0 vPΩ8 źK induced by the short exact sequences

0 0 Ñ ApKvq Ñ ApKvq Ñ π0pApKvqq Ñ 0

8 0 for v P ΩK . Here, ApKvq denotes the connected component of the origin, which is a (normal) subgroup with quotient π0pApKvqq. Note that since A is connected, the complex places give a trivial π0, so they do not contribute to ApAK q‚ (compare Remark 1.3.9).

0 It remains to prove that ApKvq “ ApKvqdiv for all infinite places v and ApKvqdiv “ 0 for all finite places v. But ApAK q‚ is compact and totally discon- nected, hence profinite. Hence, it contains no divisible elements, so

0 ApKvqdiv Ď ApKvq for all inﬁnite places, and ApKvqdiv “ 0 for all ﬁnite places.

Finally, a standard result on real or complex Lie groups shows that for a compact Lie group, the exponential

exp: LiepApKvqq Ñ ApKvq

0 is a surjection onto ApKvq . Hence, every element of ApKvq is divisible, so

0 ApKvqdiv “ ApKvq , which ﬁnishes the proof.

Corollary 2.3.20. Let n P Zą0. Then

ApAK q{nApAK q “ ApAK q‚{nApAK q‚.

Proof. Take vertical cokernels in the diagram

0 ApAK qdiv ApAK q ApAK q‚ 0 n n n

0 ApAK qdiv ApAK q ApAK q‚ 0.

The result then follows since the left vertical map is surjective.

Corollary 2.3.21. There is a natural identiﬁcation

ApAK q – ApAK q‚.

{ Proof. This follows since ApAK q‚ is its own proﬁnite completion.

31 Lemma 2.3.22. There is a chain of maps

ApKq ãÑ ApKq ãÑ SelpK,Aq Ñ ApAK q‚, inducing isomorphisms { x „ „ ApKqtors Ñ ApKqtors Ñ SelpK,Aqtors.

Proof. By the deﬁnition of the Selmer{ group,x there are maps

pnq Sel pK,Aq Ñ ApKvq{nApKvq. v ź These induce a map SelpK,Aq Ñ ApAK q by taking the limit, and the right hand side is ApAK q‚ by the corollary above. x { By the Mordell–Weil theorem, there is an isomorphism

r ApKq – ∆ ˆ Z for some ﬁnite group ∆ and some r P Zě0. In particular, ApKqdiv “ 0, so the map ApKq Ñ ApKq is injective. Injectivity of ApKq Ñ SelpK,Aq is Corollary 2.3.14. { { x Finally, the explicit description of ApKq also gives

r ApKq – ∆ ˆ Zˆ , so ApKqtors “ ApKqtors. The identiﬁcation{

{ ApKqtors “ SelpK,Aqtors follows from Corollary 2.3.14{, taking intox account that pTBqtors “ 0 for any abelian group B.

The main result of the next section (and indeed one of the main results of this work) is that the map SelpK,Aq Ñ ApAK q‚ is also injective. x

2.4 Adelic points of abelian varieties

This section is essentially section 3 of Stoll’s paper [22]. Its main ingredient is a theorem of Serre (which we will not prove); see Theorem 2.4.2.

Lemma 2.4.1. Let A{K be an abelian variety. Then there is an isomorphism ˆ AutpAtorsq – GL2gpZq of groups.

32 Proof. Over the complex numbers, there is an isomorphism

g ApCq – C {Λ, for some lattice Λ Ď Cg. Since multiplication by n is deﬁned over K, all n- torsion of A is deﬁned over K¯ . That is, ¯ 2g ApKqtors – ApCqtors “ pQ{Zq . 1 Since Q{Z is the colimit of n Z{Z, we get Hom 2g, 2g lim Hom 1 2g, 2g ppQ{Zq pQ{Zq q “ ÐÝ pp n Z{Zq pQ{Zq q n lim Hom 1 2g, 1 2g “ ÐÝ pp n Z{Zq p n Z{Zq q n lim M n M ˆ . “ ÐÝ 2gpZ{ Zq “ 2gpZq n Taking invertible elements gives the result.

Theorem 2.4.2. Let A{K be an abelian variety. Then the image of ΓK Ñ ˆ ˆˆ d GL2gpZq contains a subgroup pZ q of d-th power scalars, for some d P Zą0.

Proof. See [17].

ˆˆ d ˆ ˆˆ Lemma 2.4.3. Let d P Zą0 be even, and let S “ pZ q Ď Z . Let Z – AutpQ{Zq act on Q{Z in the canonical way. Then there exists D P Zą0 killing i H pS, Q{Zq, for i P t0, 1u.

d ˆ Proof. For p prime, put νp “ mintvppa ´ 1q | a P Zp u. Put

D “ pνp , p ź and note that this indeed a ﬁnite product, for instance since, for p ‰ 2,

d νp ď vpp2 ´ 1q, which is zero for almost all p. There is a decomposition

Q{Z – pQ{Zqtpu “ Qp{Zp. p p à à Since any isomorphism has to map the p-primary torsion into the p-primary ˆˆ torsion, the restriction of the action of Z to Qp{Zp is just given by the action of ˆ ˆˆ the subgroup Zp Ď Z . One easily sees that in fact this is just the multiplication ˆ action of Zp on Qp{Zp.

We have a decomposition

S p ˆqd pQ{Zq “ pQp{Zpq Zp . p à 33 By deﬁnition, each term of the right hand side is given by

ˆ d pZp q d ˆ pQp{Zpq “ tx P Qp{Zp | pa ´ 1qx “ 0 for all a P Zp u.

ˆ d d pZp q Hence, if ap is an element such that vppap ´ 1q is minimal, any x P pQp{Zpq d νp S is killed by ap ´ 1, hence also by p . Hence, pQ{Zq is killed by D.

Now the logarithm induces isomorphisms

ˆ „ Z2 ÝÑ t˘1u ˆ Z2, ˆ „ Zp ÝÑ Z{pp ´ 1qZ ˆ Zp pp oddq.

Since d is even, this gives isomorphisms (including for p “ 2)

ˆ d „ pZp q ÝÑ dpZ{pp ´ 1qZ ˆ Zpq.

ˆ d In particular, pZp q has a topological generator α, corresponding to the element

d ¨ p1, 1q P dpZ{pp ´ 1qZ ˆ Zpq.

ˆ d Since α generates pZp q topologically, any continuous 1-cocycle

ˆ d a: pZp q Ñ Qp{Zp is uniquely determined by its value at α. Moreover, it comes from the coboundary σ ÞÑ σb ´ b if and only if aα “ pα ´ 1qb. This gives an injection

1 ˆ d Qp{Zp H ppZp q , Qp, Zpq ãÑ . pα ´ 1qQp{Zp

Since α ´ 1 ‰ 0 and since Qp{Zp is divisible (by elements of Zp, even), the right hand side is 0. Hence, 1 ˆ d H ppZp q , Qp{Zpq “ 0. ˆ Also, the action of q‰p Zq on Qp{Zp is trivial, so

ś ˆ d ˆ d pZp q pZp q 1 ˆ d ˆ d H pZq q , Qp{Zp “ Homcont pZq q , Qp{Zp ˜q‰p ¸ ˜q‰p ¸ ź ź ˆ d ˆ d pZp q “ Homcont pZq q , pQp{Zpq . ˜q‰p ¸ ź p ˆqd The latter is killed by D since pQp{Zpq Zp is. The inf-res sequence gives

ˆ d pZp q 1 ˆ d 1 1 ˆ d 0 Ñ H ppZp q , Qp{Zpq Ñ H pS, Qp{Zpq Ñ H pZq q , Qp{Zp . ˜q‰p ¸ ź The left term is zero, and the right term is killed by D. Hence, the middle term is killed by D as well. Taking the sum over all primes gives the result.

34 Deﬁnition 2.4.4. Let A{K be an abelian variety. Put

Kn “ KpArnsq for the ﬁeld obtained by adjoining (all coordinates of) the n-torsion to K. Also put 8 K8 “ Kn. n“1 ď Remark 2.4.5. Note that Kn{K is a ﬁnite extension. Moreover, if nP “ 0, then nσpP q “ σpnP q “ 0 as well, for σ P ΓK . Hence, Kn{K is Galois. It follows that also K8{K is Galois.

Proposition 2.4.6. Let A{K be an abelian variety. Then there exists m P Zą0 1 killing all H pKn{K,Arnsq.

Proof. Note that K8 is defined to be the fixed field of the kernel of ΓK Ñ ˆ GL2gpZq. Hence, the image of this morphism is isomorphic to G “ GalpK8{Kq. ˆˆ d By Theorem 2.4.2, it contains S “ pZ q for some d P Zą0. By making S smaller if necessary, we can assume that d is even. Then by the lemma above, there exists D P Zą0 killing

i i 2g H pS, Atorsq “ pH pS, Q{Zqq for i P t0, 1u. ˆ Since S is central in GL2gpZq, it is normal in G. Then the inf-res sequence gives

1 S 1 1 0 Ñ H pG{S, Atorsq Ñ H pG, Atorsq Ñ H pS, Atorsq.

S The ﬁrst term is killed by D since Ators is, and the third term is also killed by D. Hence, the middle term is killed by D2.

Now the short exact sequence

0 Ñ Arns Ñ Ators Ñ Ators Ñ 0 of G-modules (note that all torsion is deﬁned over K8) gives a long exact sequence

1 1 ... Ñ ApKqtors Ñ H pK8{K,Arnsq Ñ H pK8{K,Atorsq Ñ ....

The third term is killed by D2, and the first term is finite. Hence, the middle term is killed by 2 m “ #ApKqtors ¨ D . Finally, the inflation map gives an injection

1 1 H pKn{K,Arnsq ãÑ H pK8{K,Arnsq, which gives the result.

35 Corollary 2.4.7. The kernel of

pnq pnq Sel pK,Aq Ñ Sel pKn,Aq is killed by m.

Proof. There is an inf-res sequence

1 1 1 0 Ñ H pKn{K,Arnsq Ñ H pK,Arnsq Ñ H pKn,Arnsq. (2.4)

By deﬁnition, the Selmer groups live inside the second and third term, and the pnq pnq map Sel pK,Aq Ñ Sel pKn,Aq is the one induced by (2.4). Hence, the kernel 1 is killed by m, since H pKn{K,Arnsq is.

Lemma 2.4.8. Let Q P SelpnqpK,Aq, and let α be the image of Q under the map

pnq pnq 1 Sel pK,Aq Ñ Sel pKn,Aq Ď H pKn,Arnsq “ HomcontpΓKn ,Arnsq.

Let L be the ﬁxed ﬁeld of the kernel of α. Let v be a place of K that splits completely in Kn. Then v splits completely in L if and only if the image of Q in ApKvq{nApKvq is zero.

Proof. Let w be a place of Kn above v. Then v splits completely in L iﬀ w does, and the latter is equivalent to

α “ 0. ΓKn,w ˇ We have a commutative diagram ˇ

pnq pnq Sel pK,Aq Sel pKn,Aq HompΓKn ,Arnsq

ApKvq{nApKvq ApKn,wq{nApKn,wq HompΓKn,w ,Arnsq.

The result follows since both horizontal arrows of the right hand square are injections and the bottom arrow of the left hand square is an isomorphism.

Lemma 2.4.9. Let Q P SelpnqpK,Aq, and let d be the order of mQ. Then the density of places v of K such that v splits completely in Kn and the image of Q 1 in ApKvq{nApKvq is trivial is at most . drKn:Ks

Proof. Let α:ΓKn Ñ Arns and L be as in the lemma above. Let e be the order of α.

By Corollary 2.4.7, the kernel of

pnq pnq Sel pK,Aq Ñ Sel pKn,Aq is killed by m. Since eα “ 0, it follows that eQ is in this kernel, so meQ “ 0. Since d is the order of mQ, this forces d | e.

36 On the other hand, one can easily see that the order of α is the exponent of its image. That is, e “ exppGalpL{Knqq.

In particular, it divides rL : Kns. Hence, d | e | rL : Kns, so

rL : Ks “ rL : KnsrKn : Ks ě d ¨ rKn : Ks.

The result now follows from the lemma above and Chebotarev’s Density Theo- rem.

Lemma 2.4.10. Let I be a directed set, and let pBiq be some projective system. Put B “ lim Bi. Let b1, . . . , br P B have inﬁnite order, and let d P Zą0 be given. ÐÝ Then there exists i P I such that the images of b1, . . . , br all have order at least d in Bi.

Proof. The elements pd ´ 1q!bk P B are nonzero, so there exist ik P I such that pd ´ 1q!pbkqik P Bik is nonzero.

Since I is directed, there exists i P I with i ě ik for all k P t1, . . . , ru. Then

pd ´ 1q!pbkqi ‰ 0 P Bi.

Hence, each pbkqi has at least order d.

Theorem 2.4.11. Let Z Ď A be a ﬁnite subscheme of A such that ZpKq “ ZpK¯ q. Let P P SelpK,Aq be such that the image Pv of P in ApKvq is inside ZpKvq “ ZpKq for a set of ﬁnite places v of density 1. Then P P ZpKq. x Proof. Let d ą #ZpKq. Write

P ´ ZpKq “ tQ1,...,Qru Ď SelpK,Aq.

Note that the assumption on P is that there is a setx of ﬁnite places v of density 1 such that one of the Qi maps to 0 in ApKvq.

Now suppose that all the Qi have inﬁnite order. Then so do the mQi, so by the pnq lemma above, there exists n P Zą0 such that the image of mQi in Sel pK,Aq has order di ě d for all i P t1, . . . , ru. By Lemma 2.4.9, the set of places of K that split completely in Kn such that the image of at least one of the Qi in ApKvq{nApKvq is trivial is at most

r 1 r 1 ď ă . d rK : Ks drK : Ks rK : Ks i“1 i n n n ÿ Hence, there is a set of places of positive density that split completely, but for which the images of all the Qi in ApKvq{nApKvq are nonzero. This contradicts the assumption on P , so at least one of the Qi must have ﬁnite order. Hence

P P ZpKq ` SelpK,Aqtors “ ZpKq ` ApKqtors Ď ApKq.

x 37 Now pick a ﬁnite place such that Pv P ZpKvq “ ZpKq. Since the map

ApKq Ñ ApKvq is injective and P lands inside ZpKvq “ ZpKq, in fact P itself is in ZpKq.

Theorem 2.4.12. Let A{K be an abelian variety, and let S be a set of places of K of density 1. Then the map

S SelpK,Aq Ñ ApAK q‚ is injective. x

Proof. We can without loss of generality remove the inﬁnite places from S. Then apply the proposition above to the ﬁnite subscheme Z “ t0u.

Corollary 2.4.13. The map

ApKq Ñ ApAK q‚ induces an identiﬁcation between{ApKq and the topological closure ApKq of ApKq in ApAK q‚. { Proof. The map is injective by the above, and it is continuous since it was deﬁned via the quotients

ApKq{nApKq Ñ ApAK q{nApAK q.

It is a closed map since ApKq is compact and ApAK q‚ is Hausdorﬀ, so the topology on ApKq is just the subspace topology of the closed subset ApKq Ď { ApAK q‚. The result follows since ApKq is dense in ApKq. { { Corollary 2.4.14. Let L{K be ﬁnite. Then the map{

SelpK,Aq Ñ SelpL, Aq is injective. x x

f Proof. Let S “ ΩK be the set of ﬁnite places. Then we have a commutative diagram f SelpK,Aq ApAK q‚

x f SelpL, Aq ApALq‚.

The two horizontal maps arex injective by the theorem, and the right vertical f f map is injective since ApAK q‚ is just ApAK q. Hence, the left vertical map is injective as well.

38 Remark 2.4.15. This last corollary can also be proven directly from the def- initions. Together with the theorem, it allows us to think of all the arrows in the diagram

S ZpKq ApKq ApKq SelpK,Aq ApAK q‚

{ x T ZpLq ApLq ApLq SelpL, Aq ApALq‚

f z x as inclusions, whenever S Ď ΩK is a set of ﬁnite places of density 1 and T is the set of places of L above S.

For more ﬂexibility, we remove from Theorem 2.4.11 the restriction that all points of Z are deﬁned over K.

Theorem 2.4.16. Let Z Ď A be a ﬁnite subscheme. Let P P SelpK,Aq be such that the image Pv P ApKvq is inside ZpKvq for a set S of ﬁnite places v of K of density 1. Then P P ZpKq. x

Proof. Since Z is a ﬁnite scheme, there is a ﬁnite Galois extension L{K such that ZpLq “ ZpK¯ q. Theorem 2.4.11 then implies that the image of P in SelpL, Aq is inside ZpLq. x T S But the image of P in ApALq‚ is GalpL{Kq-stable since it comes from ApAK q‚ (where T is the set of places of L above S). Hence,

P P ZpLqGalpL{Kq “ ZpKq.

Remark 2.4.17. The proof uses that ZpLqGalpL{Kq “ ZpKq. I do not know whether the analogous statement for SelpK,Aq is true as well, i.e. whether the S T intersection of SelpL, Aq and ApAK q‚ inside ApALq‚ is SelpK,Aq. x x x 2.5 Jacobians

We will not prove the existence of Jacobians, but we will state the deﬁnitions and main properties, as well as some results we will need later on.

Theorem 2.5.1. Let C be a curve of genus g over a ﬁeld K. Then there exists an abelian variety J of dimension g together with a map

0 Pic pC ˆK Lq Ñ JpLq for all L{K ﬁnite separable (functorial in L) that is an isomorphism whenever CpLq ‰ ∅. Moreover, J is unique up to a unique isomorphism, and is called the Jacobian of C.

Proof. See e.g. [14], Theorem III.1.6.

39 Theorem 2.5.2. Let C be a curve over K, and let L{K be a ﬁnite separable extension such that CpLq ‰ ∅. Then any point P gives rise to a closed immersion P f : C ˆK L ÝÑ J ˆK L, which on M-points (for M{L ﬁnite separable) is given by 0 CpMq ÝÑ Pic pC ˆK Mq – JpMq Q ÞÝÑ rQ ´ P s.

Proof. See [14], Proposition III.2.3.

Remark 2.5.3. Note that the map f P only depends on P up to translation. That is, P 1 P f “ τrP ´P 1s ˝ f , for P,P 1 P CpLq.

Remark 2.5.4. For C “ E an abelian variety of dimension 1, one sees that E is its own Jacobian. Since the dimension of J is the genus of C, this shows that E has genus 1, hence is an elliptic curve.

We will now turn to Jacobians over the real numbers. In what follows, C will be a curve over R such that CpRq ‰ ∅.

Lemma 2.5.5. Let x1, x2 P CpRq be distinct points. Then there exists a function f P RpCq with no real poles, such that fpx1q ‰ fpx2q.

Proof. Since CpCq is a complex manifold, it is also a real manifold of dimension 2. Hence, as CpRq Ď CpCq has real dimension 1, there exist inﬁnitely many points P P CpCq that are not in CpRq. Fix such a P , and consider the divisor D “ npP ` P q ´ x1, for some n, where P is the complex conjugate of P . Note that D is deﬁned over R.

If 2n ´ 1 ą g ` 1, then Riemann–Roch shows that h0pC, L pDqq “ deg D “ 2n ´ 1, and 0 h pC, L pD ´ x2qq “ degpD ´ x2q “ 2n ´ 2. Hence, there exists a function f P RpCq such that div f ` D is eﬀective, but div f ` D ´ x2 is not. Hence, div f ` D contains no terms x2, so div f contains no terms x2.

Then f is an element of RpCq, and f has only poles at the non-real points P and P . Moreover, it has a zero at x1, and neither a zero nor a pole at x2.

Corollary 2.5.6. The set of functions f P RpCq with no real poles is dense in MappCpRq, Rq, with respect to the topology of uniform convergence.

Proof. This is the Stone–Weierstrass theorem, applied to the compact space CpRq.

40 Proposition 2.5.7. Let C1,...,Cr be the connected components of CpRq. Then there exists a function f P RpCq with no real zeroes or poles such that f is negative on C1 and positive on all other Ci.

Proof. We can create a sequence pfnq of functions uniformly converging to the function that is ´1 on C1 and 1 on the other Ci. Hence, eventually fn will be negative on all of C1 and positive on all of Ci. Note that f then automatically has no real zeroes.

Proposition 2.5.8. Suppose that P,Q P CpRq are two real points such that rP ´ Qs is divisible by 2. Then P and Q lie in the same connected component of CpRq.

Proof. Let C1,...,Cr be the connected components of CpRq, in such a way that P P C1. By the proposition above, there exists f P RpCq such that f has neither zeroes nor poles on CpRq and f is negative on C1 and positive on all other Ci. Since all zeroes and poles of f are non-real, they come in pairs, and div f is of the form nipPi ` Piq. i ÿ Now if rP ´ Qs is divisible, there exists a divisor M and an element g P CpCq such that P ´ Q “ 2M ` div g. Hence, fpP q “ fp2Mq ¨ fpdiv gq. fpQq Note that fp2Mq “ fpMq2, which is always nonnegative. Moreover, a standard result shows that fpdiv gq “ gpdiv fq. Since div f is of the form

nipPi ` Piq, i ÿ this gives ni ni gpdiv fq “ pfpPiqfpPiqq “ pfpPiqfpPiqq , i i ź ź which is nonnegative as well. Hence, fpP q “ fpMq2gpdiv fq ě 0. fpQq

Since P P C1, we have fpP q ă 0. Hence, fpQq is negative as well, so Q P C1.

Corollary 2.5.9. Let C be a curve over R, such that CpRq ‰ ∅. Then the map

π0pCpRqq Ñ π0pJpRqq induced by the embedding of C into its Jacobian J is injective.

0 Proof. We saw in the proof of Proposition 2.3.19 that JpRq “ JpRqdiv, i.e. the identity component of the Jacobian is the subgroup of divisible elements. Now if P,Q P CpRq map to the same component of JpRq, then rP ´Qs is in the identity component, hence it is divisible. Hence, P and Q lie in the same component of CpRq.

41 Corollary 2.5.10. Let K be a number ﬁeld, and let C be a curve over K with CpKq ‰ ∅. Then the map

CpAK q‚ Ñ JpAK q‚ induced by the embedding of C into its Jacobian J is injective.

Proof. At the ﬁnite places and the complex places, there is nothing to prove. At the real places, it follows from the corollary above.

3 Torsors

In this chapter, we will look closely at the ﬁrst cohomology of representable sheaves of abelian groups. We will give an ad hoc deﬁnition of Hˇ 1 for sheaves of (not necessarily commutative) groups, and show that its elements correspond to certain geometrical objects.

3.1 First cohomology groups

Lemma 3.1.1. Let G: B Ñ A be a functor with an exact left adjoint F . Then G preserves injectives.

Proof. Let I P ob B be injective. Let 0 Ñ A Ñ B be an injection in A . Then 0 Ñ FA Ñ FB is exact in B. Hence, the map BpFB,Iq Ñ BpF A, Iq is surjective. But this is A pB, GIq Ñ A pA, GIq, by the adjunction.

In order to compute the ﬁrst cohomology, we want to use Čech cohomology. In order to do this, we need some comparison results between Čech cohomology and sheaf cohomology.

Proposition 3.1.2. Let I be an injective presheaf, and let U “ tUi Ñ UuiPI be a covering of some U P ob C . Then Hˇ ipU , I q “ 0 for all i ą 0.

Proof. This is Lemma III.2.4 of [15].

Theorem 3.1.3. The functors Hˇ ipU , ´q: PShpC q Ñ PShpC q are the right derived functors of Hˇ 0pU , ´q.

Proof. Note that for a short exact sequence of presheaves 0 Ñ F Ñ G Ñ H Ñ 0, we get a commutative diagram

0 0 0

0 F pUq iPI F pUiq pi,jqPI2 F pUi Û Ujq ... ś ś 0 G pUq iPI G pUiq pi,jqPI2 G pUi Û Ujq ... ś ś 0 H pUq iPI H pUiq pi,jqPI2 H pUi Û Ujq ..., ś ś 0 0 0

44 with exact columns. That is, we have an exact sequence

0 Ñ Cˇ‚pU , F q Ñ Cˇ‚pU , G q Ñ Cˇ‚pU , H q Ñ 0 of chain complexes.

This gives a long exact cohomology sequence

0 Ñ Hˇ 0pU , F q Ñ Hˇ 0pU , G q Ñ Hˇ 0pU , H q Ñ Hˇ 1pU , F q Ñ ....

The result now follows from the deﬁnition of right derived functor by a routine computation, since any injective sheaf is acyclic by the proposition above.

Proposition 3.1.4. Let F be a sheaf. Then there is an isomorphism

Hˇ1pF q – H 1pF q.

Proof. Note that the inclusion ShpC q Ñ PShpC q preserves injectives by Lemma 3.1.1, since it has an exact left adjoint by Theorem B.3.6 and Proposition B.3.17.

Now let F Ñ I be a monomorphism into an injective sheaf I . Let G be the presheaf cokernel, then G ` is the sheaf cokernel (Corollary B.3.10). Moreover, G is separated by Lemma B.3.14. The long exact sequence of H i gives an exact sequence 0 Ñ F Ñ I Ñ G ` Ñ H 1pF q Ñ 0 (3.1) in PShpC q, since I is injective. On the other hand, since G is separated, we have Hˇ0pG q “ G `, so the short exact sequence of presheaves

0 Ñ F Ñ I Ñ G Ñ 0 gives the long exact Hˇi-sequence

0 Ñ F Ñ I Ñ G ` Ñ Hˇ1pF q Ñ 0, since I is also injective as presheaf. Comparing with (3.1) gives the result.

3.2 Nonabelian cohomology

We give an ad hoc deﬁnition of Hˇ 1pU, G q when G is a sheaf of (not necessarily abelian) groups.

Deﬁnition 3.2.1. Let G be a sheaf of groups on a site C , and let U “ tUi Ñ UuiPI be a covering of some U P ob C . Then a 1-cocycle for U with values in G is a family pgijqpi,jq P G pUijq i,j I2 p źqP satisfying

gij ¨ gjk “ gik , Uijk Uijk Uijk 3 ´ ˇ ¯ ´ ˇ ¯ ˇ for all pi, j, kq P I , whereˇ the productˇ is the groupˇ law of G pUijkq. We will usually write g for the cocycle pgijqpi,jq.

45 Deﬁnition 3.2.2. Let g, h be two 1-cocycles. Then g is cohomologous to h if there exists a family b “ pbiqi P iPI G pUiq such that ś ´1 hij “ bi ¨ gij ¨ bj , Uij Uij ´ ˇ ¯ ´ ˇ ¯ for all pi, jq P I2. This is clearlyˇ an equivalenceˇ relation, and the set of cohomology classes is denoted Hˇ 1pU , G q. It is called the ﬁrst cohomology of G with respect to U .

Remark 3.2.3. If G is a sheaf of abelian groups, then Hˇ 1pU , G q was already deﬁned, namely as the cohomology of the complex

Cˇ0pU , G q Ñ Cˇ1pU , G q Ñ Cˇ2pU , G q.

The map d1 : Cˇ1pU , G q Ñ Cˇ2pU , G q is deﬁned by

pgijqpi,jq ÞÑ gjk ´ gik ` gij , Uijk Uijk Uijk pi,j,kq ´´ ˇ ¯ ´ ˇ ¯ ´ ˇ ¯¯ hence a 1-cocycle is exactly anˇ element ofˇker d1. Theˇ map d0 is given by

pbiqi ÞÑ bj ´ bi , Uij Uij ´ ˇ ¯ ´ ˇ ¯ hence two 1-cocycles g, h are cohomologousˇ if andˇ only if their diﬀerence is in im d0. Hence, the deﬁnition of Hˇ 1pU , G q given here is the same as the one given in section B.2.

Remark 3.2.4. Just like in the abelian case, if V is a reﬁnement of U , there is a natural morphism Hˇ 1pU , G q Ñ Hˇ 1pV , G q. We deﬁne Hˇ 1 U, colim Hˇ 1 , . p G q “ ÝÑ pU G q U PJU

We have already seen in Remark B.2.14 that JU is a directed set, so the colimit is just a direct limit.

Deﬁnition 3.2.5. A sequence

1 Ñ F Ñ G Ñ H Ñ 1 of sheaves of groups is exact if for every U P ob C , the sequence

1 Ñ F pUq Ñ G pUq Ñ H pUq is exact, and for every h P H pUq, there exists a covering U “ tUi Ñ Uu of U such that the restrictions h|Ui come from elements gi P G pUiq.

Remark 3.2.6. For the case where F , G and H are all sheaves of abelian groups, this notion corresponds to the notion of exactness in the abelian category ShpC q, by Corollary B.3.7 and Corollary B.3.13.

46 In general, analogously to Theorem B.3.6 and Corollary B.3.7, one can see that limits in the category of sheaves of groups are just pointwise. This justifies the first part of the definition of exactness. In general one would hope that Corollary B.3.13 generalises to the statement that regular epimorphisms in the category of sheaves of groups are exactly the G Ñ H satisfying the second part of the definition of exactness. We do not prove this, and we will use the ad hoc notion of surjectivity instead.

Proposition 3.2.7. Let 1 Ñ F Ñ G Ñ H Ñ 1 be a short exact sequence, and let U P ob C . Then there is a long exact sequence of pointed sets

1 Ñ F pUq Ñ G pUq Ñ H pUq Ñ Hˇ 1pU, F q Ñ Hˇ 1pU, G q Ñ Hˇ 1pU, H q.

If moreover F is abelian and F Ñ G lands inside the centre (pointwise), then this sequence can be extended to

... Ñ Hˇ 1pU, F q Ñ Hˇ 1pU, G q Ñ Hˇ 1pU, H q Ñ Hˇ 2pU, F q.

Proof. See [7], sections III.3 and IV.3.

3.3 Torsors

Definition 3.3.1. Let X be a scheme, and let G be a group scheme over X. Then a sheaf torsor for G on Xét (or Xfppf ) is a sheaf of sets S together with a right G-action such that there exists a covering tUi Ñ Xu such that each S |Uét

(resp. S |Ufppf ) is isomorphic to G ˆX U with the canonical G-action.

We will later turn to the case where S can be represented by some X-scheme S. However, we will ﬁrstly characterise sheaf torsors.

Deﬁnition 3.3.2. Let S be a sheaf torsor for G. Let U “ tUi Ñ Xu be a covering that trivialises S . Then in particular S pUiq is nonempty for all i, so we can pick si P S pUiq. Since the action of G ˆX U on S pUiq is isomorphic to G ˆX U, there exists a unique gij P GpUijq such that

si gij “ sj . Uij Uij ´ ˇ ¯ ˇ Then ˇ ˇ si gijgjk “ sk “ si gik. Uijk Uijk ´ ˇ ¯ ´ ˇ ¯ Since the action of G onˇS pUijkq is simply transitive,ˇ this forces

gijgjk “ gik, so g “ pgijq is a 1-cocycle. (We have omitted the restrictions to ease notation.)

Lemma 3.3.3. The cohomology class of g does not depend on the choice of U or of the si. Moreover, it depends on S only up to isomorphism (of sheaves with G-action).

47 1 1 Proof. If we choose other si, then there exist unique bi P GpUiq with sibi “ si. Hence, (omitting restrictions to ease notation)

1 1 1 1 sigijbj “ sjbj “ sj “ sigij “ sibigij,

1 ´1 1 so gij “ bigijbj , and g and g are cohomologous.

Independence of the choice of U follows from independence of the choice of the si, taking a common reﬁnement and restricting the chosen si. The last statement is clear.

Deﬁnition 3.3.4. The association of the cocycle g is denoted S ÞÑ gpS q.

Deﬁnition 3.3.5. Conversely, let g P Hˇ 1pX,Gq be a 1-cocycle; say that g P ˇ 1 H pU ,Gq for a covering U “ tUi Ñ Xu. Let Fn be the presheaf deﬁned by

FnpV q “ GpUi0¨¨¨in ˆX V q, i ,...,i In p 1 źnqP for n P t0, 1u, and note that it is in fact a sheaf. Let d: F0 Ñ F1 be the morphism given on an arbitrary V by

GpUi ˆX V q ÝÑ GpUij ˆX V q iPI i,j I2 ź p źqP ´1 phiqi ÞÝÑ phi hjqi,j.

Now, g is by deﬁnition a global section of F1, so it deﬁnes elements

g “ gij P F1pV q. V Uij ˆX V i,jPI2 ˇ ´ ˇ ¯ Then deﬁne S as the presheafˇ inverseˇ image of g. That is, for any V , we have

S pV q “ s P F0pV q dpsq “ g V . ˇ ˇ ( Remark 3.3.6. Note that S is in fact a subsheafˇ ofˇF0: if s P F0pV q is given, and tVi Ñ V u is some covering such that s|Vi P S pViq for all i, then

dpsq “ d s “ g Vi Vi Vi ˇ ´ ˇ ¯ ˇ for all i, hence by the sheaf conditionˇ ofˇ F1, weˇ must have dpsq “ g|V , so s P S pV q.

Deﬁnition 3.3.7. We equip S with a right G-action in the following way: for any V , we deﬁne

S pV q ˆ GpV q ÝÑ S pV q ´1 ppsiqiPI , hq ÞÝÑ h si , UiˆX V ˆ ˙iPI ´ ˇ ¯ where we think of an element s P S pV q asˇ the element psiqiPI P F0pV q. Note that if psiq P S pV q, then (once again omitting restrictions)

´1 ´1 ´1 ´1 ´1 d h si i “ ph siq h sj i,j “ si sj i,j “ d ppsiqiq . `` ˘ ˘ ` ˘ ` ˘ 48 ´1 Hence, ph siqi is in S pV q as well. Clearly, this gives a right GpV q-action to each S pV q, and the maps S pV q Ñ S pV 1q are G-invariant, so it indeed gives a G-action on S .

Lemma 3.3.8. The sheaf S with the given right G-action is a torsor. Up to isomorphism, it depends neither on the choice of representative for the cocycle we started with, nor on the covering U trivialising g.

Proof. For each n P I, we have

´1 gij “ gin gjn . Uijn Uijn Uijn ˇ ´ ˇ ¯ ´ ˇ ¯ pnq pnˇq ´1 ˇ ˇ Hence, setting s “ psi qi “ pgin qi P F0pUnq, the above identity reads:

g “ d spnq . Un ´ ¯ ˇ pnq That is, S |Un has a global sectionˇ s . But then the morphism of sheaves

G|Un Ñ S |Un given on V Ñ Un by

GpV q ÝÑ S pV q ´1 pnq h ÞÝÑ h s V ˇ gives an isomorphism of sheaves of sets with rightˇ G-actions, showing that Un trivialises S . Since tUn Ñ Xu is a covering, this shows that S is a torsor.

1 Now if g is a cohomologous cocycle, then there exists pbiqi P F0pXq such that

1 ´1 gij “ bigijbj . Deﬁne the maps

f0 : F0pV q ÝÑ F0pV q ´1 phiqi ÞÝÑ phibi qi and

f1 : F1pV q ÝÑ F1pV q ´1 paijqi,j ÞÝÑ pbiaijbj qi,j. This gives a commutative diagram of morphisms of sheaves

d F0 F1

f0 „ f1 „ d F0 F1.

1 1 Since f1pgq “ g , this diagram induces an isomorphism S – S of the inverse 1 images of g and g in F0. Since f0 is deﬁned by multiplication on the right, it commutes with the action of G on S and S 1, which is given on the left. Hence, S – S 1 as G-sheaves, hence as torsors.

49 Hence, S does not depend on the cocycle representing our class. If we chose a diﬀerent covering U 1, then restricting our cocycles to a common reﬁnement shows that S also does not depend on U .

Deﬁnition 3.3.9. The association of the sheaf torsor S is denoted g ÞÑ Sg.

Theorem 3.3.10. The maps S Ñ gpS q and g ÞÑ Sg give a bijection between 1 the set of isomorphism classes of sheaf torsors on X´et (or Xfppf ) and Hˇ pX´et,Gq 1 (resp. Hˇ pXfppf ,Gq).

Proof. If g is a 1-cocycle, say g P Hˇ 1pU ,Gq, then the lemma above shows that pnq U trivialises g, and there are sections s P SgpUnq, deﬁned by

pnq ´1 si “ gin . The cocycle condition on g asserts that (omitting restrictions)

´1 ´1 ´1 gnm gin “ gim , which by the deﬁnition of the G-action on S (Deﬁnition 3.3.7) gives

pnq ´1 pnq pmq pmq s gnm “ pgnmq si “ si “ s . i i ´ ¯ ´ ¯ Hence, g is the cocycle gpSgq associated to Sg, by Deﬁnition 3.3.2.

Conversely, let S be a sheaf torsor. We fix U “ tUi Ñ XuiPI trivialising S , and we fix sections si P S pUiq. Then gij is defined by

sigij “ sj. „ Moreover, the isomorphism of sheaves G|Ui ÝÑ S |Ui given on V Ñ Ui by ψ : GpV q Ñ S pV q ´1 g ÞÑ sig induces an isomorphism „ ψ : F0 ÝÑ S0, where S0pV q “ i S pUi ˆX V q. By the deﬁnition of the right action on F0, for any V Ñ X we have ś ´1 ´1 ψppgiqihq “ ψpph giqiq “ psigi hqi, whenever pgiqi P F0pV q and h P GpV q. Hence, the right action on S0 given by

ptiqi h :“ ptihqi

´1 makes ψ into a G-invariant map. Now let t “ ptiqi “ psihi qi P S0pV q be the element corresponding to some h “ phiqi P F0 under the isomorphism ψ.

´1 Then dphq “ g if and only if hi hj “ gij, i.e. ti|Uij “ tj|Uij . Hence, under this identiﬁcation, the inverse image sheaf of g corresponds to the equaliser of

ÝÑ S pUi ˆX V q ÝÑ S pUij ˆX V q. iPI i,j I2 ź p źqP

50 But this equaliser is just S pV q, since S is a sheaf. Hence, SgpS q is isomorphic to S as sheaf. Since ψ is G-invariant, the actions agree as well, so SgpS q is the same torsor as S .

We will now turn to sheaf torsors that are representable.

Deﬁnition 3.3.11. Let X be a scheme, and G a group scheme over X. Then a torsor for G on X´et (resp. Xfppf ) is a scheme S over X, together with a right G-action on S such that the sheaf represented by S becomes a sheaf torsor.

Remark 3.3.12. That is, a torsor is a scheme S over X with a right G-action such that there exists a covering tUi Ñ Xu in X´et (resp. Xfppf ) such that S|Ui is isomorphic to G|Ui , with its canonical G-action.

Definition 3.3.13. The set of G-torsors on Xét (resp. Xfppf ) up to isomorphism is denoted PHSpG{Xétq (resp. PHSpG{Xfppf q). It is short for principal homogeneous spaces, which is another word for torsors.

Corollary 3.3.14. There is an injection

1 PHSpG{X´etq Ñ Hˇ pX´et,Gq.

Proof. Clear from the theorem.

Proposition 3.3.15. Let S be an X-scheme with a right G-action. Then the following are equivalent:

(1) S is a G-torsor on Xfppf , (2) S is faithfully ﬂat and locally of ﬁnite type over X, and the morphism

pπ1,mq S ˆX G ÝÑ S ˆX S

is an isomorphism (where π1 : S ˆX G Ñ S is the ﬁrst projection, and m: S ˆ G Ñ S is the action).

Proof. It is clear that in the second case, the one-object covering tS Ñ Xu trivialises S, hence S is a torsor.

Conversely, suppose that G is a torsor. Then there is a covering tUi Ñ Xu trivialising S. Hence, also U “ i Ui trivialises S. Note that U Ñ X is faithfully flat and locally of finite type. Now G Ñ X is flat (by our assumptions on group schemes), and in fact faithfullyš flat since η is a section. The morphism

pS ˆX Gq U ÝÑ pS ˆX Sq U ˇ ˇ is an isomorphism. Then by descentˇ theory (see EGAˇ 4 [8], Prop. 2.7.1), S ˆX G Ñ S ˆX S is an isomorphism. Since S becomes isomorphic to G after the faithfully ﬂat base change along S Ñ X, another application of descent theory shows that S is faithfully ﬂat over X.

51 Corollary 3.3.16. Suppose G is smooth over X. Then so is any G-torsor S.

Proof. After the faithfully ﬂat base change along S Ñ X, S becomes isomorphic to G. Hence the result follows from descent theory.

Corollary 3.3.17. Suppose G is smooth over X. Then any G-torsor S for the fppf topology is actually a torsor for the étale topology.

Proof. We have to show that there exists an étale covering tUi Ñ Xu trivialising S. We have a smooth covering S Ñ X trivialising S. By EGA 4 [8], Cor. 17.16.3(ii), there exists a surjective étale morphism S1 Ñ X and an X- morphism S1 Ñ S. That is, tS1 Ñ Xu is a reﬁnement of tS Ñ Xu, and it is an étale covering. It trivialises S since tS Ñ Xu does.

Corollary 3.3.18. Suppose G is smooth over X. Let S be an X-scheme with a right G-action. Then the following are equivalent:

(1) S is a G-torsor on Xét, (2) S is faithfully flat and locally of finite type over X, and the morphism

pπ1,mq S ˆX G ÝÑ S ˆX S

is an isomorphism (where π1 : S ˆX G Ñ S is the ﬁrst projection, and m: S ˆ G Ñ S is the action).

Proof. This is a reformulation of the above.

We will use without proof the following theorem.

Theorem 3.3.19. Assume that we are in one of the following situations:

(1) G is aﬃne over X, (2) G is smooth and separated over X, and dim X ď 1; (3) G is smooth and proper over X, has geometrically connected ﬁbres, and G is regular.

1 Then the inclusion PHSpG{Xfppf q Ñ Hˇ pXfppf ,Gq is an isomorphism.

Proof. This is Theorem 4.3 and Corollary 4.7 of [15].

Corollary 3.3.20. If G is smooth and satisﬁes one of the conditions of the theorem, then

1 1 PHSpG{X´etq “ Hˇ pX´et,Gq “ Hˇ pXfppf ,Gq “ PHSpG{Xfppf q.

Proof. By the theorem, any sheaf torsor S for Xfppf is representable by some S. Since G is smooth, S is in fact a torsor over X´et, so S is a sheaf torsor on X´et.

52 Remark 3.3.21. If G is commutative and quasi-projective, then Theorem 3.9 of [15] proves that the canonical maps i i H pX´et,Gq Ñ H pXfppf ,Gq are isomorphisms, which for i “ 1 gives part of the corollary (since Hˇ 1 “ H1).

3.4 Descent data

Deﬁnition 3.4.1. Let A Ñ B be a faithfully ﬂat ring homomorphism. Then a descent datum for B{A is a B-module N with an isomorphism „ φ: N bA B ÝÑ B bA N of B bA B-modules, such that the diagram

φ2 N bA B bA B B bA B bA N

φ3 φ1

B bA N bA B commutes, where φi is obtained by tensoring φ with 1B on the i-th coordinate.

Example 3.4.2. Let M be an A-module, and let N “ B bA M. Then the canonical descent datum pN, canq is given by the isomorphism

can: pB bA Mq bA B ÝÑ B bA pB bA Mq pb b mq b c ÞÝÑ b b pc b mq. One easily veriﬁes that this is indeed a descent datum.

Deﬁnition 3.4.3. Let pN 1, φ1q, pN 2, φ2q be descent data. Then a morphism of descent data is a B-linear map ψ : N 1 Ñ N 2 making commutative the diagram

φ1 1 1 N bA B B bA N

ψ b 1 1 b ψ φ2 2 2 N bA B B bA N .

i i (We used superscript in pN , φ q since φ1 already has a diﬀerent meaning.)

Example 3.4.4. Clearly, if M1 Ñ M2 is a morphism of A-modules, then it induces a morphism pB bA M1, canq Ñ pB bA M2, canq of descent data. This makes the canonical descent datum into a functor.

Deﬁnition 3.4.5. Let pN, φq be a descent datum. Then deﬁne N φ “ ker α, where α is the map

α: N ÝÑ B bA N n ÞÝÑ 1 b n ´ φpn b 1q.

53 It is an A-module, and there is a canonical A-linear map

φ fφ : N bA B ÝÑ N n b b ÞÝÑ bn.

Proposition 3.4.6. Let pN, φq be a descent datum. Then fφ is an isomorphism.

Proof. Write α0 for the map n ÞÑ 1 b n, and α1 for n ÞÑ φpn b 1q, so that α “ α0 ´ α1. Now consider the diagram

α0 b 1 N bA B B bA N bA B α1 b 1

φ φ1 (3.2) 1 d0 b 1 B bA N B bA B bA N, 1 d1 b 1

1 where di is as in Lemma B.7.2. If n P N and b P B are given, then

1 pφ1 ˝ pα0 b 1qq pn b bq “ φ1p1 b n b bq “ 1 b φpn b bq “ pd0 b 1q ˝ φ pn b bq. ` ˘ Moreover, if we write φpn b bq “ i bi b ni for certain bi P B, ni P N, then the deﬁnition of φ2 gives ř 1 pd1 b 1q ˝ φ pn b bq “ bi b 1 b ni “ φ2pn b 1 b bq. i ` ˘ ÿ On the other hand, we have

pφ1 ˝ pα1 b 1qq pn b bq “ φ1pφpn b 1q b bq “ φ1pφ3pn b 1 b bqq, so the descent datum assumption on φ forces

1 φ1 ˝ pα1 b 1q “ pd1 b 1q ˝ φ.

Hence, the squares for the top and bottom arrows of the horizontal pairs in (3.2) commute. Since φ and φ1 are isomorphisms, this implies that the equalisers of the pairs of horizontal arrows are isomorphic.

φ But since A Ñ B is faithfully ﬂat, the equaliser of the top pair is just N bA B, by deﬁnition of N φ. On the other hand, the equaliser of the bottom pair is N, φ by Lemma B.7.2. Hence, N bA B – N.

φ φ Finally, the map N bA B Ñ N bA B ÝÑ B bA N is given by

n b b ÞÝÑ φpn b bq “ p1 b bqφpn b 1q “ p1 b bqp1 b nq “ p1 b bnq,

φ which is the image of fφpnbbq in BbAN. Hence, the isomorphism N bAB – N given above is given by fφ.

Theorem 3.4.7. The canonical descent datum functor gives an equivalence between the category of A-modules and the category of descent data.

54 Proof. Let M be an A-module, and let N “ BbAM. Then one sees immediately from the deﬁnitions that the sequence α 0 Ñ M Ñ N ÝÑ B bA N is isomorphic to the one from Lemma B.7.2. Hence, it is exact, so N can – M. Conversely, if pN, φq is a descent datum, then the proposition above shows that φ fφ is an isomorphism. Moreover, for all n P N , b, c P B it holds that φpbn b cq “ φppb b cqpn b 1qq “ pb b cqφpn b 1q “ pb b cqp1 b nq “ b b nc.

Hence, the diagram

φ can φ pN bA Bq bA B B bA pN bA Bq

fφ b 1 1 b fφ φ N bA B B bA N commutes, so fφ is an isomorphism of descent data.

3.5 Hilbert’s theorem 90

We will prove a generalisation of Hilbert’s theorem 90 for étale cohomology, ˇ 1 giving a concrete description of H pX´et, GLnq. Note that since GLn is smooth ˇ 1 and aﬃne over X, the above shows that this is the same as H pXfppf , GLnq, so will compute the latter instead.

1 Proposition 3.5.1. Let X “ Spec A be aﬃne. Then Hˇ pXfppf , GLnq is the set of locally free A-modules of rank n, up to isomorphism.

ˇ 1 Proof. If g P H pXfppf , GLnq, then g is trivialised by some covering U “ tUi Ñ Xu. By refining, we can assume that all the Ui are affine. Since each Ui Ñ X is flat and locally of finite type, it is open, and since X is compact we only need finitely many Ui to cover X.

Now define U “ i Ui. Since g is trivialised by U , it is also trivialised by the one-object covering U Ñ X, since this is a refinement of U . Now U is affine š since each Ui is; say U “ Spec B.

ˇ 1 Then g is a cocycle in H ptU Ñ Xu, GLnq. Setting I for the index set t˚u, we get

gij ¨ gjk “ gik , Uijk Uijk Uijk i, j, k ´ ˇ i,¯ j ´ kˇ ¯ ˇ for all P t˚u. Althoughˇ and ˇ will alwaysˇ be equal to each other, it is still useful to keep separate indices, to clarify which restriction maps we are talking about.

Now gij is an element of GLnpUijq “ GLnpB bA Bq. We will view it as an isomorphism n „ n gij : B bA B ÝÑ B bA B.

55 n We will write N “ B . Then gij|Uijk is an isomorphism B bA N bA B Ñ N bA B bA B, and likewise for gjk and gik. We get a commutative diagram

gik B bA B bA N N bA B bA B,

gjk gij

B bA N bA B

´1 hence φ “ g˚˚ makes N into a descent datum. Conversely, starting with this descent datum, it is clear that we can recover g.

By Theorem 3.4.7, descent data of this form correspond to A-modules M. For n such an M, we have M bA B – B , so standard descent theory shows that M is locally free of rank n (see EGA 4 [8], Prop. 2.5.2).

Corollary 3.5.2. Let X “ Spec A be aﬃne. Then the canonical map

1 1 Hˇ pXZar, GLnq Ñ Hˇ pXfppf , GLnq is an isomorphism.

ˇ 1 Proof. It is a standard result that H pXZar, GLnq characterises locally free A- modules of rank n. Moreover, the proof above essentially shows that any fppf- locally free A-module of rank n is already trivialised by a Zariski covering. This proves the result.

Theorem 3.5.3. (Hilbert’s theorem 90) Let X be a scheme. Then the canonical map 1 1 Hˇ pXZar, GLnq Ñ Hˇ pXfppf , GLnq is an isomorphism.

Proof. Let g be trivialised by some tUi Ñ Xu. The images Xi of the Ui are open, and by reﬁning we can assume that the Xi are aﬃne. Then g|Xi is trivialised by the one object cover tUi Ñ Xiu, and by the corollary above this shows that it is trivialised by some Zariski cover. But then g itself is trivialised by a Zariski cover.

We state a number of immediate corollaries, each of which can be referred to as Hilbert’s theorem 90.

Corollary 3.5.4. Let X be a scheme. Then the canonical maps

1 1 1 PicpXq “ H pXZar, Gmq Ñ H pX´et, Gmq Ñ H pXfppf , Gmq are isomorphisms.

Proof. The second isomorphism follows from Remark 3.3.21, and the ﬁrst from the theorem, setting n “ 1, and observing that Hˇ 1 “ H1 by Proposition 3.1.4.

56 Corollary 3.5.5. Let K be a ﬁeld. Then

1 H pK, GLnq “ 0.

Proof. Follows since étale cohomology is Galois cohomology by Corollary B.8.7, using the theorem above. Here, H1 denotes the ad hoc definition given in [18] of non-abelian first cohomology (one checks that it still corresponds to the ad hoc definition of étale Hˇ 1).

Corollary 3.5.6. (Classical Hilbert’s theorem 90) Let K be a ﬁeld. Then

H1pK, K¯ ˆq “ 0.

Proof. Follows from either of the corollaries above.

57 4 Brauer groups

In this chapter, we will treat the Azumaya Brauer group and the cohomological Brauer group of a scheme. In the case of Spec K, they are the classical Brauer group of K, which we will assume familiar. Note however that for a general scheme, the two diﬀerent Brauer groups need not coincide.

4.1 Azumaya algebras

Definition 4.1.1. Let X be a scheme. An OX -algebra A is called an Azumaya algebra over X if it is a locally free OX -module of finite rank, and if moreover the canonical map op A bOX A ÝÑ End OX pAq given on any affine (Zariski) open U Ď X by

op ApUq bOX pUq ApUq ÝÑ EndOX pUqpApUqq a b b ÞÝÑ px ÞÑ axbq is an isomorphism.

Remark 4.1.2. If X “ Spec R is aﬃne, then an Azumaya algebra is just an R-algebra A that is a projective module of ﬁnite type such that the map op A bR A ÝÑ EndRpAq a b b ÞÝÑ px ÞÑ axbq is an isomorphism. In this case, we also call A an Azumaya algebra over R.

op ˜ ˜op Note that the isomorphism pAbRA q˜– AbOX A is automatic (cf. Hartshorne [10], Prop. II.5.2(b)), but for the identiﬁcation ˜ pEndRpAqq˜– End OX pAq we use that A is coherent and X noetherian.

Lemma 4.1.3. Let A be an Azumaya algebra over a ring R. Then the centre of A is R.

op Proof. Note that R Ñ A is injective, since R Ñ A bR A “ EndRpAq is, as A is projective. Also, it is clear that the image of R lands inside the centre ZpAq of A. Let C be the R-module ZpAq{R.

Firstly, suppose that R is local, with maximal ideal m and residue ﬁeld k. By Nakayama’s lemma, any lift of a k-basis of A bR k generates A, hence is a basis since A is free. Hence, we can pick a basis a1, . . . , an of A such that a1 “ 1. Let φ: A Ñ A be deﬁned by 1 i “ 1, φpa q “ i 0 i ‰ 1. "

58 op Now if a P ZpAq, then a b 1 is central in A bR A , so px ÞÑ axq is central in EndRpAq. In particular, it commutes with φ, so a “ a ¨ φp1q “ φpa ¨ 1q.

Since the image of φ is in R, this gives a P R, so ZpAq “ R.

Now turn to the general case. By exactness of localisation, any prime p Ď R gives a short exact sequence

0 Ñ Rp Ñ ZpAqp Ñ Cp Ñ 0.

It is clear that the subset ZpAqp Ď Ap is actually inside ZpApq. Hence, Rp Ď ZpAqp Ď ZpApq, and by the local case they are all equal. Hence, Cp “ 0, and we are done since p is arbitrary.

Lemma 4.1.4. Let R be a local ring, and A an Azumaya algebra over R. Then the ideals I Ď A correspond bijectively to the ideals J Ď R via

I ÞÑ I X R, JA Ðß J.

Proof. Note that if φ P EndRpAq, then φ is some sum of endomorphisms of the form x ÞÑ axb. Hence, if I Ď A is an ideal, then φpxq P I whenever x P I. Hence, φpIq Ď I.

Now let a1, . . . , an be a basis for A with a1 “ 1, as above. Let φ1, . . . , φn be the elements of EndRpAq deﬁned by 1 i “ j, φ pa q “ δ “ i j ij 0 i ‰ j, " where δij P R is viewed as an element of A. Note that φipAq Ď R for all i.

If I Ď A is some ideal, and a P I, then n a “ φipaqai. i“1 ÿ Since φipIq Ď I, each term φipaq is in I X R. Hence, a P pI X RqA.

Conversely, if J Ď R is some ideal, and a P JA X R, then we can write n a “ riai i“1 ÿ for certain ri P J. As a P R we must have ri “ 0 for i ą 1, so a P J.

Hence, the maps I ÞÑ I X R and J ÞÑ JA are each others inverse.

Lemma 4.1.5. Let K be a ﬁeld, and let A be a ﬁnite K-algebra. Then A is an Azumaya algebra over X “ Spec K if and only if it is a central simple algebra over K.

59 Proof. If A is a central simple algebra, then so is Aop, and a standard result op then shows that also A bK A is a central simple algebra. Therefore, the map op A bK A ÝÑ EndK pAq is injective, so by dimension reasons it is an isomorphism. Since A is clearly free of ﬁnite rank, this shows that A is an Azumaya algebra.

Conversely, if A is an Azumaya algebra, then A is central by Lemma 4.1.3, and simple by Lemma 4.1.4.

Lemma 4.1.6. Let R be a local ring with maximal ideal m and residue ﬁeld k. Let A be an R-algebra that is a free module of ﬁnite rank. Then A is an Azumaya algebra over R if and only if A bR k is a central simple algebra over k.

Proof. We write f and fk respectively for the morphisms op A bR A ÝÑ EndRpAq op pA bR kq bk pA bR kq ÝÑ EndkpA bR kq. There is a commutative diagram

f b 1 op pA bR A q bR k EndRpAq bR k „ „

op fk pA bR kq bk pA bR kq EndkpA bR kq.

Therefore, if f is an isomorphism, so is fk.

Conversely, if fk is an isomorphism, then so is f b 1. But since A is free of ﬁnite op rank, so are M “ A bR A and N “ EndRpAq. Then both the kernel K and the cokernel C of f are ﬁnitely generated, since R is noetherian.

By right exactness of the tensor product, we have C bR k “ 0. By Nakayama’s lemma, this forces C “ 0, so f is surjective. But then we get a long exact Tor-sequence: R Tor1 pN, kq Ñ K bR k Ñ M bR k Ñ N bR k Ñ 0. Since N is free, the ﬁrst term vanishes. Since f b1 is an isomorphism, this forces K bR k “ 0, hence K “ 0 by Nakayama’s lemma. That is, f is an isomorphism, and the result now follows from the previous lemma.

Lemma 4.1.7. Let A be an OX -algebra that is a locally free OX -module of ﬁnite rank. Then A is an Azumaya algebra over X if and only if Ax is an Azumaya algebra over OX,x for all x P X.

Proof. Let x P X be a point. Since A is coherent, there is a natural isomorphism op op pA bOX A qx – Ax bOX,x Ax . Moreover, since A is locally free, there is also a natural isomorphism

pEnd OX pAqqx – EndOX,x pAxq.

60 Hence, the map op A bOX A ÝÑ End OX pAq is an isomorphism if and only if for each x P X the map

op Ax bOX,x Ax ÝÑ EndOX,x pAxq is.

Proposition 4.1.8. Let A be an OX -algebra that is locally free of ﬁnite rank as OX -module. Then the following are equivalent: (1) A is an Azumaya algebra; (2) Ax is an Azumaya algebra over OX,x for every x P X;

(3) A bOX kpxq is a central simple algebra over kpxq for every x P X.

Proof. Clear from Lemma 4.1.5, 4.1.6, and 4.1.7.

Corollary 4.1.9. Let F be a locally free OX -module of ﬁnite rank n. Then

EndOX pF q is an Azumaya algebra over X.

Proof. The module A “ End OX pF q is locally free of ﬁnite rank since F is, and it is an Azumaya algebra since A bOx kpxq – Mnpkpxqq is a central simple algebra over kpxq, for each x P X.

Corollary 4.1.10. If A and B are Azumaya algebras over X, then so is AbOX B.

Proof. It is locally free of ﬁnite rank since A and B are, and the result now follows from part (3) of the proposition, using the analogous result on central simple algebras.

Remark 4.1.11. We could also prove this directly, by constructing a morphism

End OX pAq bOX End OX pBq ÝÑ End OX pA bOX Bq φ b ψ ÞÝÑ φ b ψ, where the ﬁrst φ b ψ indicates the formal element of the tensor product, and the second indicates the map

φ b ψ : A bOX B ÝÑ A bOX B a b b ÞÝÑ φpaq b ψpbq.

Locally, one shows this to be an isomorphism by choosing bases for A and B.

Deﬁnition 4.1.12. Let A and B be two Azumaya algebras over X. Then A and B are called similar if there exist locally free OX -modules E , F of ﬁnite rank such that

A bOX End OX pE q – B bOX End OX pF q. This is clearly an equivalence relation, and the equivalence class of A is denoted rAs.

61 Definition 4.1.13. The Azumaya-Brauer group BrApXq of a scheme X is the set of equivalence classes of Azumaya algebras on X. Given two Azumaya algebras A, B over X, we define rAs ¨ rBs to be rA bOX Bs. This is well-defined since

End OX pE q bOX End OX pF q – End OX pE bOX F q, for any two locally free OX -modules of ﬁnite rank E , F (similar to the preceding remark).

Remark 4.1.14. The multiplication on BrApXq indeed makes it into a group, op with unit element OX . The inverse of an element A is given by A , by the very deﬁnition of an Azumaya algebra!

Remark 4.1.15. If X “ Spec k is the spectrum of a ﬁeld k and E is a locally ˜n free Ox-module of rank n, then E – k , and

End OX pE q – pMnpkqq˜.

Hence, Lemma 4.1.5 gives BrApXq “ Brpkq.

Deﬁnition 4.1.16. Let f : X Ñ Y be a morphism of schemes. Then an Azu- maya algebra A on Y gives rise to the Azumaya algebra f ˚A on Y . This induces a map ˚ f : BrApY q Ñ BrApXq, op making BrA into a functor Sch Ñ Ab.

4.2 The Skolem–Noether theorem

The aim of this section is to prove a generalisation of the Skolem–Noether theorem for Azumaya algebras over schemes. The treatment is largely based on [12].

Deﬁnition 4.2.1. Let R be a ring, and let A be an Azumaya algebra over R. Let α, β be R-algebra automorphisms of A. Then we write αAβ for the A-bimodule A whose left action is given by α and whose right action is given by β.

If α “ 1, we will just write Aβ for αAβ. In particular, we write A1 for the usual A-bimodule structure on A.

Given an A-bimodule M, we will denote by M A the R-submodule

M A “ tm P M | am “ ma for all a P Au.

For any R-module M, we will equip A bR M with the A-bimodule structure given by xpa b mqy :“ pxayq b m, for x, y P A, a P A and m P M.

62 Proposition 4.2.2. Let α be an R-algebra automorphism of A. Then the natural map

A ψ : A bR pAαq ÝÑ Aα a b b ÞÝÑ ab is an A-bimodule isomorphism.

Proof. This follows from III.5.1 of [12]. The proof uses a descent argument.

Deﬁnition 4.2.3. If α is an R-algebra automorphism of A, then we write Iα A for the R-module pAαq , as above.

Remark 4.2.4. Since A1 bR Iα – Aα, this forces Iα to be projective and faithfully ﬂat, since both A1 and Aα are. By a dimension argument, it must be a line bundle, so it gives an element of PicpRq.

Lemma 4.2.5. The map

AutR– algpAq Ñ PicpRq

α ÞÑ Iα is a group homomorphism.

Proof. If α, β P AutR– algpAq are given, then the map

ψ : Aα bA Aβ ÝÑ Aαβ (4.1) a b b ÞÝÑ aαpbq is pA, Aq-linear, since ψpxpa b bqyq “ ψpxa b bβpyqq “ xaαpbqαpβpyqq for all a P Aα, b P Aβ and x, y P A. It is an isomorphism since the element a b b equals the element aαpbq b 1, for all a P Aα, b P Aβ. Similarly, the map

Iα bR Iβ ÝÑ Aαβ (4.2) a b b ÞÝÑ aαpbq is pA, Aq-linear. The image is inside Iαβ since xaαpbq “ aαpxbq “ aαpbqαpβpxqq for all a P Aα, b P Aβ, x P A. Combining the proposition above (for α, β and αβ) with the isomorphism (4.1), we ﬁnd that the map

pA1 bR Iαq bA pA1 bR Iβq Ñ A1 bR Iαβ induced by (4.2) is an isomorphism. Since A is faithfully ﬂat, this shows that (4.2) itself is an isomorphism.

Theorem 4.2.6. (Rosenberg–Zelinsky sequence) Let A be an Azumaya algebra. Then the sequence

ˆ ˆ 0 Ñ R Ñ A Ñ AutR– algpAq Ñ PicpRq is exact.

63 ˆ ˆ ˆ Proof. The map R Ñ A is the obvious one, and the map A Ñ AutR– algpAq is given by u ÞÝÑ px ÞÑ uxu´1q. Injectivity of Rˆ Ñ Aˆ and exactness at Aˆ follow since the centre of A is R. It remains to show that Iα – R (as R-modules) if and only if α is inner.

´1 ´1 Suppose α is inner, say αpxq “ uxu for all x P A. Then Iα “ ZpAqu Ď A, which is isomorphic to R since ZpAq “ R and right multiplication by u´1 is R-linear.

Conversely, suppose Iα – R. Then the proposition above gives an isomorphism „ ψ : A1 ÝÑ Aα. Let u “ ψp1q´1. By pA, Aq-linearity of ψ, we have ψpaxbq “ aψpxqαpbq for all a, b, x P A. Setting x “ 1 and a “ b´1, this gives u´1 “ b´1u´1αpbq, or αpbq “ ubu´1, for all b P A. Hence, α is inner.

Corollary 4.2.7. Let A be an Azumaya algebra over a ring R with trivial Picard group. Then every automorphism of A is inner.

ˆ Proof. The map A Ñ AutR– algpAq is surjective.

Corollary 4.2.8. (Skolem–Noether for local rings) Let A be an Azumaya algebra over a local ring R. Then every automorphism of A is inner.

Proof. Over a local ring, every projective module is free, hence PicpRq “ 0.

Remark 4.2.9. We will see later that, for A “ MnpRq, the exact sequence from the theorem is part of a (non-abelian) long exact cohomology sequence.

Theorem 4.2.10. (Skolem–Noether for schemes) Let X be a scheme, and let A be an Azumaya algebra over X. Let α be an OX -algebra automorphism of A. Then there exists a Zariski covering tUiu of X and elements ui P ApUiq such that

´1 α pvq “ uivu Vi i ˇ for all Vi Ď Ui open and all v P Aˇ pViq. That is, α is Zariski-locally given by an inner automorphism.

Proof. Let x P X be a point. Then Ax is an Azumaya algebra over OX,x, so by ´1 ˆ the corollary above, αx is given by a ÞÑ uau for some u P Ax .

Let V be an open neighbourhood of x on which u is defined, and let a P ApV q. ´1 Then αpaq ´ uau vanishes at x, hence in some open neighbourhood Ua of x. Taking the intersection of these Ua for some finite set of generators taiu, we find that α is given by a ÞÑ uau´1 on some open neighbourhood of x. Since x is arbitrary, this gives the result.

64 Deﬁnition 4.2.11. Let PGLn be the sheaf of sets on Xfppf represented by Spec S0, where S0 is the degree 0 part of the graded ring

n ´1 S “ ZrtXijui,j“1, det s, where all the Xij have degree 1, and det is the element

det “ sgnpσq Xiσpiq. σPS i“1 ÿn ź Then PGLn is called the projective general linear group scheme. We will prove that it is indeed a group scheme.

Lemma 4.2.12. Let R be a ring such that Pic R “ 0. Then

ˆ PGLnpRq “ GLnpRq{R .

Proof. We know that giving a morphism from an arbitrary scheme X into Pm m`1 is the same as giving a surjection of OX -modules OX Ñ L , where L is a m`1 1 line bundle. Moreover, two such maps OX Ñ L , OX Ñ L give rise to the same morphism X Ñ Pm if and only if there exists an isomorphism φ: L Ñ L 1 making commutative the diagram

m`1 φ OX L 1.

Now since R has trivial Picard group, we ﬁnd that

m m`1 HomSchpSpec R, P q “ tpa0, . . . , anq P R | pa0q ` ... ` panq “ Ru{ „, where two n`1-tuples pa0, . . . , anq, pb0, . . . , bnq are equivalent if and only if there ˆ exists λ P R such that ai “ λbi for all i P t0, . . . , nu. We write ra0 : ... : ans for the equivalence class of pa0, . . . , anq.

Now let S be the graded ring from the deﬁnition of PGLn. Then Spec S0 is a 2 standard open inside Proj S – Pn ´1. Hence,

n n HomSchpSpec R, Proj Sq “ paijqi,j“1 paijq “ R {„, # ˇ + ˇ i,j“1 ˇ ÿ ˇ and ˇ n n ˆ HomSchpSpec R, Spec S0q “ paijqi,j“1 paijq “ R, detpaijq P R {„ . # ˇ + ˇ i,j“1 ˇ ÿ ˆ ˇ n But the condition detpaijq P R forces ˇ i,j“1paijq “ R, hence the right hand ˆ side is GLnpRq{R . By deﬁnition, the left hand side is PGLnpRq, which gives ř the result.

65 Corollary 4.2.13. Let R be a ring such that Pic R “ 0 (e.g. R is a local ring). Then PGLnpRq “ AutR– algpMnpRqq.

Proof. Immediate from Lemma 4.2.12 and Theorem 4.2.6.

Proposition 4.2.14. Let X be a scheme. Then

PGLnpXq “ AutOX – algpMnpOX qq.

Proof. Let F be the Zariski presheaf

U ÞÝÑ AutOU – algpMnpOU qq

on X. It is a subpresheaf of the sheaf End OX pMnpOX qq of local OU -module homomorphisms. For a local OU -module homomorphism, being an automorphism is a local condition. Similarly for being an OU -algebra morphism. Hence, F is a sheaf of groups for the Zariski topology.

Let U Ď X be open, and let φ: U Ñ Spec S0 be a morphism. The discussion in m`1 Lemma 4.2.12 shows that φ is given by a surjection OU Ñ L for some line bundle L on U. If we let tUiu be a Zariski covering of U trivialising L , then

φi “ φ|Ui is given by some matrix Ai P GLnpUiq.

ˆ Moreover, for any i, j, there exists λij P ΓpUi X Uj, O q such that Ai “ λijAj. Hence, the inner automorphisms of MnpOq on Ui and Uj deﬁned by Ai and Aj respectively coincide on Ui X Uj. Hence, since F is a sheaf, this gives a well-deﬁned element of F pUq.

This clearly deﬁnes a morphism of sheaves

PGLn Ñ F , which is an isomorphism since it is so locally, cf. Corollary 4.2.8 and 4.2.13.

Theorem 4.2.15. The projective general linear group PGLn gives a sheaf of groups on the Zariski, étale and fppf sites. Moreover, there is a short exact sequence 1 Ñ Gm Ñ GLn Ñ PGLn Ñ 1 on any of these sites.

Proof. The first statement is immediate from the proposition above. The sequence is clearly exact at Gm and at GLn. Exactness on the right follows from Skolem–Noether for schemes and the definition of an exact sequence of sheaves of groups (Definition 3.2.5).

Remark 4.2.16. The Rosenberg–Zelinsky sequence (Theorem 4.2.6) for Mn is now just a nonabelian long exact cohomology sequence, cf. Proposition 3.2.7.

66 4.3 Brauer groups of Henselian rings

In this section, R will be a local ring with maximal ideal m and residue ﬁeld k.

Deﬁnition 4.3.1. A local ring R is called Henselian if, given a monic polynomial f P RrXs and two coprime monic polynomials g0, h0 P krXs such that ¯ ¯ f “ g0h0, there exist g, h P RrXs such that g¯ “ g0 and h “ h0 and f “ gh.

Example 4.3.2. By (the classical version of) Hensel’s lemma, any complete DVR is Henselian. This is the motivation for the term.

Theorem 4.3.3. Let x be the closed point in X “ Spec R. Then the following are equivalent:

(1) R is Henselian, (2) any ﬁnite R-algebra is isomorphic to a product of local rings, (3) any étale map f : Y Ñ X such that Y has a point y with fpyq “ x and kpyq “ kpxq admits a section s: X Ñ Y .

Proof. See [15], Theorem I.4.2.

Corollary 4.3.4. If R is Henselian and R1 is a ﬁnite local R-algebra, then R1 is Henselian.

Proof. This follows from p1q ô p2q of the theorem.

Remark 4.3.5. Note that if R1 is ﬁnite and local, then the morphism R Ñ R1 is automatically a local ring homomorphism, by the going-up theorem.

Lemma 4.3.6. Let R be Henselian, and let B,C be ﬁnite étale local R-algebras. Then the map

HomR– algpB,Cq ÝÑ Homk– algpB bR k, C bR kq is injective.

Proof. Put X “ Spec C, Y “ Spec B and S “ Spec R. Let x, y, s be the unique closed points. By the going-up theorem, the fibre above s in X (resp. Y ) is txu (resp. tyu). In particular, if f : X Ñ Y is an S-morphism, then fpxq “ y. Note also that B bR k is finite étale over k, and local since B is. Hence, it is a field, so it is the residue field of B.

Now if f, g : B Ñ C are two R-algebra homomorphisms inducing the same map B bR k Ñ C bR k, then Corollary 1.1.13 asserts that f “ g.

Lemma 4.3.7. Let R be Henselian, and let B,C be ﬁnite étale local R-algebras. Then the map

HomR– algpB,Cq ÝÑ Homk– algpB bR k, C bR kq is surjective.

67 Proof. Let g : B bR k Ñ C bR k be an R-algebra homomorphism. This deﬁnes a surjective homomorphism

ψ : C bR B Ñ C bR k c b b ÞÑ cg¯ p¯bq.

This corresponds to a kpxq-point z of X ˆS Y , and since the kernel of C Ñ C bR B Ñ C bR k is the maximal ideal of C, the image of z under the morphism

f : X ˆS Y Ñ X is x. Now by Corollary 4.3.4, C is Henselian, and by Theorem 4.3.3 (3), we get a section s: X Ñ Y of f. That is, we get a map

C bR B Ñ C such that the composition C Ñ C bR B Ñ C is the identity. The composition B Ñ C bR B Ñ C now gives a map inducing the map B bR k Ñ C bR k we started with.

Lemma 4.3.8. Let R be Henselian, and let l be a local étale k-algebra. Then there exists a local étale R-algebra B such that B bR k “ l.

Proof. Since l is finite étale and local, it must be a finite separable field extension of k. Hence, by the theorem of the primitive element, there exists a separable monic irreducible polynomial f0 P krXs such that

l “ krXs{pf0q.

1 Set B “ RrXs{pfq for any monic lift f P RrXs of f0. Then f0pXq is invertible in l, hence f 1pXq is not in m, hence invertible in R. This shows that f is separable as well, so B is étale over R. Clearly B is ﬁnite over R and local (since f is irreducible), and B bR k “ l.

Proposition 4.3.9. If R is Henselian, then the functor B ÞÑ B bR k gives an equivalence between the category of ﬁnite étale R-algebras and the category of ﬁnite étale k-algebras.

Proof. By Theorem 4.3.3 (2) (and since ´bR k commutes with ﬁnite products), we only need to consider local R-algebras. But on the categories of local ﬁnite étale algebras, we have seen that the functor ´ bR k is full (Lemma 4.3.7), faithful (Lemma 4.3.6) and essentially surjective (Lemma 4.3.8).

Theorem 4.3.10. If R is Henselian, then the map

BrApRq ÝÑ BrApkq is injective.

Proof. Let A be an Azumaya algebra over R, and let φ be an isomorphism

„ A bR k ÝÑ Mnpkq.

68 Let ε P A bR k correspond to the matrix e1,1 with coeﬃcient 1 on the upper left entry and 0 elsewhere. Note that ε is an idempotent.

Let a P A be any lift of ε. Then Rras is a ﬁnite commutative R-algebra, so by Theorem 4.3.3 (2), it is a product of local R-algebras. It must be a product of exactly two local R-algebras, since Rras bR k “ krεs is isomorphic to 2 krXs{pX ´ Xq “ k ˆ k. Then Rras – B1 ˆ B2, and one of the elements p1, 0q, p0, 1q gives a nontrivial idempotent e in Rras mapping to ε in krεs.

„ There is an isomorphism of R-modules A ÝÑ Ae ‘ Ap1 ´ eq. Hence, Ae is a ﬁnitely generated projective R-module, hence free of ﬁnite rank since R is local. Now consider the R-algebra homomorphism

ψ : A Ñ EndRpAeq b ÞÑ pc ÞÑ bcq.

Let I be the kernel of ψ. Then I XR “ 0 since Ae is a free module. Hence, I “ 0 ¯ by Lemma 4.1.4. Similarly, the map ψ : A bR k Ñ EndkppA bR kqεq induced by ψ is injective as well. By dimension reasons, ψ¯ is an isomorphism.

Now write C for the cokernel of ψ. Since ψ¯ is an isomorphism and ψ is injective, by right exactness of the tensor product we get C bR k “ 0. Hence, by Nakayama’s lemma, C “ 0. Hence, ψ is an isomorphism, so A is a matrix algebra.

Corollary 4.3.11. If R is Henselian and k is either ﬁnite or separably closed, then BrApRq “ 0.

Proof. In this case, BrApkq “ Brpkq is zero.

Corollary 4.3.12. If R is Henselian and A is an Azumaya algebra over R, then there exists a ﬁnite étale local R-algebra R1 such that

1 1 A bR R – MnpR q.

Proof. If R “ k is a field, this says that there is a finite separable field extension l{k splitting A, cf. [6], Proposition 2.2.5. The general result now follows from Proposition 4.3.9 and Theorem 4.3.10.

4.4 Cohomological Brauer group

Definition 4.4.1. Let x P X be a point, and write x¯ for Spec kpxq. Then we define O colim Γ U, O , X,x¯ “ ÝÑ p U q U where the limit is taken over all étale maps U Ñ X with a factorisation x¯ Ñ U Ñ X. It is a Henselian local ring whose residue field is kpxq “ kpx¯q (see [15], section I.4).

69 Proposition 4.4.2. Let X be a scheme, and let A be an OX -algebra that is of ﬁnite type as OX -module. Then A is an Azumaya algebra if and only if there exists an étale covering tUi Ñ Xu such that each A|Ui is isomorphic to

Mni pOUi q.

Proof. Let A be an Azumaya algebra. If x is a point, then OX,x¯ is a Henselian local ring with separably closed residue ﬁeld, hence

A bOX OX,x¯ – MnpOX,x¯q for some n, by Corollary 4.3.11. But this isomorphism is already deﬁned over some U Ñ X through which x¯ Ñ X factors. In particular, U Ñ X is étale, A|U – MnpOU q, and the image of U in X contains x. Since x was arbitrary, this proves the assertion.

Conversely, suppose there exists an étale covering tUi Ñ Xu such that each A|Ui is isomorphic to Mni pOUi q. Let U “ i Ui. Then A|U is locally free, hence by descent theory the same goes for A (since U Ñ X is faithfully ﬂat). š

Moreover, if x P X is given, then there exists some y P Ui over x, and the ﬁeld extension kpxq Ñ kpyq is ﬁnite and separable. The assumption forces that pA|U qy – Mni pOU,yq, which implies that

pA bOX kpxqq bkpxq kpyq – Mni pkpyqq.

Hence, A bOX kpxq is a central simple algebra over kpxq, so A is an Azumaya algebra by Proposition 4.1.8.

Lemma 4.4.3. The set of isomorphism classes of Azumaya algebras of rank n2 1 is isomorphic to Hˇ pX´et, PGLnq.

Proof. By the proposition above, an Azumaya algebra of rank n2 is an étale twist of MnpOU q. By Proposition 4.2.14, we have

AutOU – algpMnq – PGLnpUq for all U P obpEt´ {Xq. Hence, an Azumaya algebra deﬁnes an element g P ˇ 1 H pX´et, PGLnq (compare section 3.3).

Conversely, a 1-cocycle for PGLn “ AutOX – algpMnq deﬁnes in particular a 1- 2 cocycle on AutOX pMnq “ GLn . By Hilbert 90, this corresponds to a locally free module A of rank n.

Moreover, the construction from Deﬁnition 3.4.5 shows that A “ N φ is now the equaliser of two OX -algebra homomorphisms, hence is itself an OX -algebra. Any covering tUi Ñ Xu that trivialises A as OX -module also trivialises it as OX -algebra, hence A is an Azumaya algebra.

Proposition 4.4.4. Let X be connected. Then there is a canonical injective homomorphism ˇ 2 BrApXq Ñ H pX´et, Gmq.

70 Proof. By Theorem 4.2.15, we have a short exact sequence

1 Ñ Gm Ñ GLn Ñ PGLn Ñ 1 of sheaves on X´et. Since Gm lands in the centre of GLn, Proposition 3.2.7 gives an exact sequence of pointed sets

ˇ 1 ˇ 1 δn ˇ 2 ... Ñ H pXét, GLnq Ñ H pXét, PGLnq ÝÑ H pXét, Gmq. ˇ 1 ˇ 1 A direct computation shows that H pXét, GLnq Ñ H pXét, PGLnq maps the locally free OX -module E to the Azumaya algebra End OX pE q. Moreover, a further computation shows that

δn`mpA bOX Bq “ δnpAqδmpBq for any Azumaya algebras A, B of ranks n2 and m2 respectively (see [15], The- orem IV.2.5 for the explicit deﬁnition of δn).

Now since X is connected, the rank of an Azumaya algebra A is constant, so ˇ 1 we have A P H pX´et, PGLnq for some n. We deﬁne the image of A to be δnpAq. The above shows that

δnpAq “ δn`mpA bOX End OX pE qq for any locally free OX -module E of rank m. Hence, if rAs P BrApXq, then the image of rAs under the map above does not depend on the Azumaya algebra A representing rAs. Moreover, the above show that ˇ 2 BrApXq Ñ H pX´et, Gmq is a homomorphism, and that the inverse image of 1 is just the trivial class rOX s.

Corollary 4.4.5. Let X be a compact scheme. Then there is a canonical injective homomorphism ˇ 2 BrApXq Ñ H pX´et, Gmq.

Proof. Let X “ i Xi be the decomposition into connected components. Note ˇ 2 that there are only ﬁnitely many. Now both BrApXq and H pX´et, Gmq break up into a directš product over the connected components, so the result follows from the above.

Deﬁnition 4.4.6. The (cohomological) Brauer group BrpXq of a scheme X is 2 the group H pX´et, Gmq.

Remark 4.4.7. Some authors write BrpXq for the Brauer group in terms of Azumaya algebras, and Br1pXq for the cohomological one. However, since we are mostly interested in the latter, we have introduced the notation above (BrA and Br, respectively).

We will use without proof the following theorem.

71 Theorem 4.4.8. Let X be a compact scheme such that every ﬁnite subset of X is contained in an aﬃne open set. Then there are canonical isomorphisms

i „ i Hˇ pX´et, ´q ÝÑ H pX´et, ´q.

Proof. See [15], Theorem III.2.17. The proof given there uses an article of M. Artin [1].

Theorem 4.4.9. Let X be a scheme satisfying the assumptions of the theorem above. Then there is an injection

BrApXq Ñ BrpXq.

ˇ 2 Proof. By the theorem above, we have BrpXq “ H pX´et, Gmq. The result then follows from the corollary preceding it.

Remark 4.4.10. The theorem remains valid if the assumptions from Theorem 4.4.8 are dropped, but a diﬀerent proof is required. A sketch of the proof is given in [15], Theorem IV.2.5.

Remark 4.4.11. Observe that Br X is functorial in X. Indeed, for any smooth group scheme G over S and any morphism f : X Ñ Y of S-schemes, we have a restriction map i i H pY´et,Gq Ñ H pX´et,Gq. If X Ñ Y is étale, it is given by the restriction

i i H pGqpY´etq Ñ H pGqpX´etq,

i where H is the derived functor of the inclusion ShpY´etq Ñ PShpY´etq.

For a general morphism X Ñ Y , we have to go to the big étale site, and we get the same result (we do not include the details of this argument).

We need two more propositions about Brauer groups, which we will state without proof.

Proposition 4.4.12. Let R be a Henselian local ring. Then the map

BrApXq Ñ BrpXq is an isomorphism.

Proof. See [15], Corollary IV.2.12.

Corollary 4.4.13. Let R be a Henselian local ring, and suppose that its residue ﬁeld k is either ﬁnite or separably closed. Then

BrpXq “ 0.

Proof. Immediate from the proposition above and Corollary 4.3.11.

72 Proposition 4.4.14. If X is a smooth variety over K, then every element α P Br X arises Zariski-locally from an Azumaya algebra.

Proof. See [15], Proposition IV.2.15.

73 5 Obstructions for the existence of rational points

In this chapter, K will be a number ﬁeld, and AK the ring of adèles. We will study the set of K-rational points on a K-variety X.

5.1 Descent obstructions

In this section, X will be a K-variety, G an aﬃne group variety that is étale over K, and f : S Ñ X a torsor under G. We need the following construction.

1 Proposition 5.1.1. Let ξ P H pK,Gq, and let S0 Ñ X be the torsor corre- 1 sponding to the image of ξ in Hˇ pX,Gq. Then there exist a group variety Gξ and a torsor fξ : Sξ Ñ X (depending on S) over Gξ, satisfying the following properties:

(1) Gξ is locally on pSpec Kqfppf isomorphic to G; (2) The map

1 1 Hˇ pX,Gq ÝÑ Hˇ pX,Gξq

S ÞÝÑ Sξ

is a bijection, mapping S0 to the trivial torsor; (3) If G is commutative, then Gξ “ G, and the map S ÞÑ Sξ is given by

S ÞÑ S ´ S0,

where ` is the addition on Hˇ 1pX,Gq induced by the addition on G. (4) Sξ is stable under base change: if Y Ñ X is a morphism of K-varieties, then pS ˆX Y qξ “ Sξ ˆX Y.

Proof. See [21], Lemma 2.2.3 and the examples following.

Deﬁnition 5.1.2. The group Gξ is called the inner form of G, and Sξ is called the twist of S (with respect to ξ).

Deﬁnition 5.1.3. The torsor S deﬁnes a map

1 θS : Xpkq ÝÑ H pK,Gq mapping x: Spec K Ñ X to the element of H1pK,Gq corresponding to the torsor S ˆX K Ñ K induced by x: Spec K Ñ X.

Lemma 5.1.4. Let ξ P H1pK,Gq, and let x: Spec K Ñ X be a K-point of X. Then θSpxq “ ξ if and only if Sξ ˆX K has a K-point.

Proof. Let T “ S ˆX K. Then the torsor Tξ has a K-point if and only if it is the trivial torsor. By Proposition 5.1.1 (2), this is equivalent to T “ ξ. But the class corresponding to T is θSpxq.

74 Corollary 5.1.5. There is a decomposition

XpKq “ fξ pSξpKqq . ξ H1 K,G P žp q

Proof. An element x P XpKq is in fξpSξpKqq if and only if the ﬁbre Sξ ˆX K has a K-rational point. Hence, the result follows from the lemma, since each x P XpKq satisﬁes θSpxq “ ξ for exactly one ξ.

Definition 5.1.6. Let f : S Ñ X be a torsor under an étale affine K-variety G. Then we put f XpAK q‚ “ fξ pSξpAK q‚q . (5.1) ξ H1 K,G P ďp q We write f-cov f-sol f-ab XpAK q‚ ,XpAK q‚ ,XpAK q‚ f for the intersections over XpAK q‚ , where f runs over all torsors under finite, finite soluble, finite abelian K-group varieties, respectively.

Remark 5.1.7. One can also introduce the above notation with XpAK q instead f f of XpAK q‚. Note that XpAK q‚ is the image of XpAK q under the natural f surjection XpAK q Ñ XpAK q‚. Indeed, if prxvsq P XpAK q‚ , then there exist ξ and pryvsq P SξpAK q‚ such that

fξpryvsq “ rxvs.

Since SξpCq ‰ ∅ when X ‰ ∅, we can pick yv P SξpKvq for each complex place, giving an element pyvq P SξpAK q mapping to pryvsq. Then fξpyvq “ xv for all ﬁnite places, and fξpyvq is in the connected component rxvs for all real places. f Hence, pfξpyvqq is an element of XpAK q mapping to prxvsq in XpAK q‚.

Note that we do not necessarily have fξpyvq “ xv for real v, but they lie in the same connected component, which is good enough. In other words, the inverse f f image of XpAK q‚ in XpAK q might a priori be larger than XpAK q .

Lemma 5.1.8. If ψ : X1 Ñ X is a morphism of K-varieties, then

1 f-cov f-cov ψpX pAK q‚ q Ď XpAK q‚ , and similarly for the soluble and abelian versions.

Proof. We will prove the ﬁrst statement; the other two follow similarly. Let 1 f-cov pxvq P X pAK q‚ . Let G be an étale aﬃne K-group, and f : S Ñ X a G-torsor. 1 1 1 1 1 Then f : S ˆX X Ñ X is a G-torsor on X , so there exists ξ P H pK,Gq and 1 1 pzvq P pSξ ˆX X qpAK q‚ with fξppzvqq “ pxvq. Then

1 fξ p1 ˆ ψq pzvq “ ψ fξ pzvq “ ψppxvqq,

` f ` ˘˘ ` ` ˘˘ hence ψppxvqq P XpAK q‚ . Since f was arbitrary, the result follows.

We also need the functoriality with respect to K.

75 Proposition 5.1.9. Let L{K be a ﬁnite ﬁeld extension. Then the image of f-cov f-cov XpAK q‚ in XpALq‚ is contained in XpALq‚ , and similarly for the soluble and abelian versions.

Proof. See [22], Proposition 5.16.

f-cov We will now compare XpKq to XpAK q‚ . We will use the following theorem. Theorem 5.1.10. Suppose X is proper. Then there are only ﬁnitely many twists Sξ such that SξpAK q‚ ‰ ∅.

Proof. This is Proposition 5.3.2 of [21].

f-cov Lemma 5.1.11. Let X be a proper variety. Then XpKq Ď XpAK q‚ , where XpKq denotes the topological closure of XpKq in XpAK q‚.

Proof. Let G be a finite group scheme over K, let S Ñ X be a G-torsor, and let 1 ξ P H pK,Gq. Then Gξ is finite over K by descent theory, since it is locally on pSpec Kqfppf isomorphic to G. Hence, Gξ ˆK X Ñ X is finite, so fξ : Sξ Ñ X is finite since Sξ is fppf-locally isomorphic to Gξ.

In particular, Sξ is a proper variety. Hence,

SξpAK q “ SξpKvq, vPΩ źK which is compact since each SξpKvq is compact. Hence, SξpAK q‚ is compact as well. Since XpAK q‚ is Hausdorff, the image of the compact space SξpAK q‚ f is closed. By the theorem above, the union in (5.1) is finite, hence XpAK q‚ is f-cov closed. Taking the intersection over all f, we find that XpAK q‚ is closed. f-cov The result now follows since XpKq Ď XpAK q‚ , by Corollary 5.1.5. Corollary 5.1.12. Let X be a proper variety. Then we have a chain of inclusions

f-cov f-sol f-ab XpKq Ď XpKq Ď XpAK q‚ Ď XpAK q‚ Ď XpAK q‚ Ď XpAK q‚.

Proof. The second inclusion follows from the lemma, and the others are obvious.

Deﬁnition 5.1.13. Let X be a proper variety. Then X is good with respect f-cov to all coverings (soluble coverings, abelian coverings) if XpKq “ XpAK q‚ f-sol f-ab (resp. XpAK q‚ , XpAK q‚ ).

In the ﬁrst case, we also say that X is good. In the third case, we also say that X is very good.

Deﬁnition 5.1.14. Let X be a proper variety. Then X is excellent with respect f-cov to all coverings (soluble coverings, abelian coverings) if XpKq “ XpAK q‚ f-sol f-ab (resp. XpAK q‚ , XpAK q‚ ).

76 5.2 The Brauer–Manin obstruction

We will use without proof the following addendum to Proposition 1.3.1:

Theorem 5.2.1. Let X{K be a variety, and let XS{OK,S be a model. Let G be a smooth group scheme over XS. Then the canonical maps

colim Hn X ,G Hn X ,G ÝÑ pp T q´et q ÝÑ p ´et q T are isomorphisms for all n P Zě0.

Proof. This is SGA 4 [2], Corollary VII.5.9.

Deﬁnition 5.2.2. If xv : Spec Kv Ñ X is a Kv-point of X, then we denote the restriction map Br X Ñ Br Kv by α ÞÑ αpxvq. If α comes from an Azumaya algebra A, then

αpxvq “ A bOX Kv.

2 Proposition 5.2.3. Let X{K be a variety, let α P H pX´et, Gmq, and let

x “ pxvqv P XpAK q Ď XpKvq v ź 2 be an AK -point of X. Then αpxvq “ 0 P H pKv, Gmq for almost all v.

Proof. Let XS be a model over OK,S, cf. Proposition 1.3.1 (1). By enlarging S if necessary, we can assume that xv P XpOvq for v R S. By Theorem 5.2.1, we can assume that α comes from an element of BrpXSq.

Now for each v R S, the map xv : Spec Kv Ñ XS factors through Spec Ov. Hence, the map Br XS Ñ Br Kv factors through Br Ov. But Br Ov “ 0 by Corollary 4.4.13. Since α P Br X is in the image of Br XS Ñ Br X, this shows that αpxvq “ 0 for all v R S.

Deﬁnition 5.2.4. Let x “ pxvqv P XpAK q be an AK -point. Then we write

αpxq “ pαpxvqqv P Br Kv. vPΩK à It is indeed an element of the direct sum by the proposition above.

Recall from global class ﬁeld theory the following theorem:

Theorem 5.2.5. (Brauer–Hasse–Noether) Let K be a number ﬁeld. Then there is a short exact sequence

invv 0 ÝÑ Br K ÝÑ Br Kv ÝÑ Q{Z ÝÑ 0, ř vPΩK à 77 where invv : Br Kv Ñ Q{Z is an injective map that is an isomorphism for all ﬁnite places v.

This inspires the following deﬁnition.

Deﬁnition 5.2.6. Deﬁne a pairing

x´, ´y: Br X ˆ XpAK q Ñ Q{Z

pα, xq ÞÑ inv αpxq “ invvpαpxvqq. v ÿ It is called the Brauer–Manin pairing.

Theorem 5.2.7. Let x P XpAK q. Then for x to come from a K-rational point, it is necessary that xα, xy “ 0 for all α P Br X.

Proof. If x comes from a K-rational point, then αpxq P Br Kv comes from an element of Br K. Hence, it maps to 0 in Q{Z. À α Deﬁnition 5.2.8. Let α P Br X. Then we denote by XpAK q the set α XpAK q “ x P XpAK q xα, xy “ 0 . Similarly, if B Ď Br X is a subset, then we putˇ ( ˇ B XpAK q “ x P XpAK q xα, xy “ 0 for all α P B . For B “ Br X, this set is called the Brauer–Maninˇ obstruction.( ˇ Corollary 5.2.9. We have Br X XpKq Ď XpAK q .

Proof. This is a reformulation of the theorem above.

Br X Corollary 5.2.10. If XpAK q “ ∅, then XpKq “ ∅.

Proof. Immediate from the preceding corollary.

Br X Remark 5.2.11. If X is such that XpAK q ‰ ∅ but XpAK q “ ∅, then X has points everywhere locally, but not globally. Thus the Brauer–Manin obstruction is an obstruction to the Hasse principle. The ﬁrst counterexamples to the Hasse principle can all be explained by the Brauer–Manin obstruction, but today examples are known now where the failure of the Hasse principle is not explained by the Brauer–Manin obstruction.

Finally, we will adjust the Brauer–Manin obstruction to the set XpAK q‚ instead of XpAK q (see Deﬁnition 1.3.8 for this notation).

Proposition 5.2.12. Let X{K be a smooth variety, and let α P Br X. Then the map

α: XpKvq ÝÑ Br Kv

xv ÞÝÑ αpxvq

78 is locally constant (for the topology on XpKvq induced by the topology on Kv).

Proof. The question is local. Hence, by Proposition 4.4.14, we can assume that α is given by an Azumaya algebra A. Moreover, we can assume that X is ˇ 1 connected, so that A is of pure rank n. Then A P H pX´et, PGLnq corresponds to a PGLn-torsor f : S Ñ X.

ˇ 1 If xv : Kv Ñ X is a Kv-point, then A maps to zero under H pX´et, PGLnq Ñ ˇ 1 H pKv, PGLnq if and only if S ˆX Kv is the trivial torsor, i.e. S ˆX Kv has a Kv-rational point. Hence, a Kv-point of X satisﬁes αpxvq “ 0 if and only if xv is the image of a Kv-point of Y . That is:

´1 α p0q “ fpSpKvqq, where α denotes the map XpKvq Ñ Br Kv. But f : S Ñ X is étale and surjective, hence f : SpKvq Ñ XpKvq is an open map (see e.g. the discussion before Theorem 4.5 of [5]). Hence, ´1 ´1 fpSpKvqq is open, so α p0q is open. By translation, each α pcq is open for c P Br Kv, hence α is locally constant.

Corollary 5.2.13. The Brauer–Manin pairing factors through Br X ˆXpAK q‚.

Proof. If α P Br X is given and v is a real place, then the value αpxvq depends only on the connected component on which xv P XpKvq lies.

Definition 5.2.14. This defines a modified Brauer–Manin pairing

x´, ´y: Br X ˆ XpAK q‚ Ñ Q{Z.

If α P Br X, then we put

α XpAK q‚ “ x P XpAK q‚ xα, xy “ 0 . ˇ ( If B Ď Br X, then we write ˇ B XpAK q‚ “ x P XpAK q‚ xα, xy “ 0 for all α P B . ˇ ( Proposition 5.2.15. Let f : S Ñ X beˇ a torsor on X under PGLn, and let 1 A P Hˇ pX´et, PGLnq be the corresponding Azumaya algebra. Then

f A XpAK q‚ “ XpAK q‚ . (5.2)

Proof. Both sides of (5.2) are the images of the respective sets in XpAK q under the natural surjection XpAK q Ñ XpAK q‚. For the left hand side, this is Remark 5.1.7, and for the right hand side, this is by deﬁnition. Hence, it suﬃces to prove the result for XpAK q instead of XpAK q‚.

1 If θv : XpKvq Ñ H pKv, PGLnq denotes the map associating to a Kv-point xv the pullback S ˆX Kv, then an argument similar to Lemma 5.1.4 shows that θvpxvq “ ξ if and only if S ˆX Kv has a Kv-rational point.

79 Hence, if pxvq is an AK -point of X, then θvpxvq “ ξ for all v if and only if S ˆX AK has an AK -point. That is,

fξpXpAK qq “ pxvq P XpAK q θvpxvq “ ξ for all v . f ˇ ( Then XpAK q is the set of pxvq P XpAK qˇ such that the element pθvpxvqq P 1 H pKv, PGLnq lies in the image of the diagonal map

1 1 ś H pK, PGLnq Ñ H pKv, PGLnq.

A ź On the other hand, XpAK q is the set of points pxvq whose image in Br Kv lands inside the image of Br K. The result follows since we have a commutative diagram ś 1 1 H pK, PGLnq H pKv, PGLnq v „ „ ś

BrpKqrns BrpKvqrns. v ś Corollary 5.2.16. We have

BrApXq f XpAK q‚ “ XpAK q‚ . nPZą0 1 fPHˇ pčX,PGLnq In particular, the Brauer–Manin obstruction is a special case of the descent obstruction.

1 Proof. Follows by taking intersections over all elements of H pX, PGLnq for various n.

5.3 Obstructions on abelian varieties

We will ﬁrstly study the ﬁnite abelian obstruction on abelian varieties, and then use the embedding of a curve into its Jacobian to deduce results about obstructions on curves.

Deﬁnition 5.3.1. For each n P Zą0, give the multiplication by n map A Ñ A the structure of a torsor under Arns via

µ A ˆA Arns ÝÑ A.

That is, on each U P obpXfppf q, it is given by

ApUq ˆ ArnspUq ÝÑ ApUq pa, bq ÞÝÑ a ` b.

Lemma 5.3.2. We have

rns ApAK q‚ “ SelpK,Aq. n č x 80 θn 1 Proof. We have a map ApAK q‚ Ñ v H pKv,Arnsq given by pxvq ÞÑ pθvpxvqq. By a computation, one checks that this is the same map as the left arrow of the bottom row of the diagram ś

0 ApKq{nApKq H1pK,Arnsq H1pK,Aqrns 0

1 1 0 ApAK q‚{nApAK q‚ H pKv,Arnsq H pKv,Aqrns 0 vPΩ vPΩ źK źK rns of Remark 2.3.4. As in the proof of Proposition 5.2.15, we ﬁnd that ApAK q‚ is the set

1 1 pxvq P ApAK q θnpxvq P im H pK,Arnsq Ñ H pKv,Arnsq . (5.3) # ˇ ˜ v ¸+ ˇ ˇ ź ˇ 1 By deﬁnition of theˇ Selmer group, if θnpxvq is in the image of H pK,Arnsq, it is in fact in the image of SelpnqpK,Aq, since it comes from an element of ApAK q‚{nApAK q‚. Now write φn for the map

pnq φn : Sel pK,Aq ÝÑ ApAK q‚{nApAK q‚.

´1 pnq Then the sets Cn “ φn pθnpxvqq Ď Sel pK,Aq are all nonempty. Since the Selmer group is ﬁnite, so is Cn. The projective limit of ﬁnite nonempty sets is nonempty, hence pxvq is in the image of the injection φ: SelpK,Aq Ñ ApAK q‚ induced by the φn (it is an injection by Theorem 2.4.12). This gives x rns ApAK q‚ Ď SelpK,Aq, n č and the other inclusion is obvious from (5.3x).

Proposition 5.3.3. Let A be an abelian variety. Then

f-cov f-ab ApAK q‚ “ ApAK q‚ “ SelpK,Aq.

Proof. We only sketch the proof, since it involves terminologyx and results beyond the scope of this thesis. A standard result on abelian varieties shows that the étale fundamental group of AK¯ is the projective limit lim A K¯ n . ÐÝ p qr s n Hence, the multiplication by n maps rns: A Ñ A form a coﬁnal set inside the f-cov f-ab family of all ﬁnite torsors. In particular, both ApAK q‚ and ApAK q‚ can be computed as rns ApAK q‚ , n č by Lemma 5.7 and 5.8 of [22]. The result follows from the computation above.

Corollary 5.3.4. Let A be an abelian variety over K. Then A is very good if and only if XpK,Aqdiv “ 0, and A is excellent with respect to abelian coverings if and only if ApKq is ﬁnite and XpK,Aqdiv “ 0.

81 Proof. We have ApKq “ ApKq by Corollary 2.4.13, hence ApKq “ SelpK,Aq if and only if XpK,Aqdiv “ 0 by Corollary 2.3.17. Moreover, by Mordell–Weil, we have { x r ApKq “ ∆ ˆ Z for some ﬁnite group ∆ and some r P Zě0, so

r ApKq “ ∆ ˆ Zˆ .

In particular, ApKq “ ApKq if and only if ApKq is ﬁnite.

5.4 Obstructions on curves

Recall that for us, curves are smooth, projective, and geometrically connected K-varieties of dimension 1.

The results in this section are basically section 8 of [22].

Theorem 5.4.1. Let C{K be a curve of genus at least 1, and let Z Ď C f-ab be a ﬁnite subscheme. Then the intersection of CpAK q‚ and the image of ZpAK q‚ Ñ CpAK q‚ is ZpKq.

f-ab Proof. It clearly contains ZpKq. Conversely, let pxvq P CpAK q‚ be in the image of ZpAK q‚. Let L{K be a ﬁnite Galois extension over which there is a morphism ψ : CL Ñ JL, where J “ JacpCq is the Jacobian of C.

Let Ξ be the image of ZL in JL, and note that it is still ﬁnite (over L). Let pyvq f-ab be the image of pxvq in CpALq‚. By Proposition 5.1.9, we have pyvq P CpALq‚ .

f-ab Now ψppyvqq is in JpALq‚ by Lemma 5.1.8, and this set equals SelpL, Aq by Proposition 5.3.3. Since pxvq comes from an element of ZpAK q‚, it is clear that ψppyvqq comes from an element of ΞpALq‚. By Theorem 2.4.16, thisx forces

ψppyvqq P ΞpLq “ ψpZpLqq.

By Corollary 2.5.10, the map

ψ : CpALq‚ Ñ JpALq‚ is injective, hence we conclude that pyvq P ZpLq. Then in fact it must be in the image of ZpKq in CpALq‚, since it is GalpL{Kq-invariant. Note that this does not yet force that pxvq is in ZpKq, since the map CpAK q‚ Ñ CpALq‚ is not in general injective.

Now if ZpKq “ ∅, then the image of ZpKq in CpALq‚ is also empty, which shows that pxvq cannot exist, so we are done. If ZpKq ‰ ∅, then in particular CpKq is nonempty, so ψ is already deﬁned over K. Hence, following the above with L “ K gives pxvq P ZpKq, which completes the proof.

82 Theorem 5.4.2. Let φ: C Ñ X be a non-constant morphism into some K- variety X. If X is excellent with respect to all coverings (soluble coverings, abelian coverings), then so is C.

f-cov Proof. We will prove the result for CpAK q‚ . The other two statements follow similarly.

If C has genus 0, then C is a Severi–Brauer variety of dimension 1, hence it satisﬁes the Hasse principle. That is, CpAK q‚ “ ∅ if and only if CpKq “ ∅. Hence, if CpKq “ ∅, the result is immediate. If CpKq ‰ ∅, then C – P1, and CpKq is dense in CpAK q‚ by weak approximation.

Hence, XpKq contains φpCpAK qq‚. Pick a point x0 P CpKq and a point x1 P CpK¯ q with a different image in XpK¯ q, and let L be a finite Galois extension such that x1 P CpLq. Then L{K is completely split at a set of finite primes of positive density. Now let pxvq P CpAK q‚ be such that xv “ x1 is v is a finite place that is completely split (this makes sense since K Ñ Kv factors through L), and xv “ x0 otherwise. Then ψpxvq is not the same for all v, hence φppxvqq cannot be in XpKq. Hence,

f-cov XpKq Ĺ φpCpAK q‚q Ď XpKq Ď XpAK q‚ , contradicting the assumption on X. This completes the proof for genus 0.

f-cov Now suppose C has genus at least 1. Let pxvq P CpAK q‚ . Then by Lemma f-cov 5.1.8, we have φppxvqq P XpAK q‚ “ XpKq. Let Z be the inverse image scheme of P “ φppxvqq. Then Z is quasi-ﬁnite over K since φ is non-constant, so in particular Z is ﬁnite. Moreover, pxvq is in the image of ZpAK q‚ Ñ CpAK q‚ ´1 since ppxvqq P φ pP q. Hence, by the theorem above, pxvq is in ZpKq. In particular, it is in CpKq, which completes the proof.

Corollary 5.4.3. Let C Ñ A be a non-constant map of C into an abelian variety A{K, such that ApKq is ﬁnite and XpK,Aqdiv “ 0. Then C is excellent with respect to abelian coverings.

Proof. This follows from the theorem, since A is excellent with respect to abelian coverings, by Corollary 5.3.4.

Finally, we will state without proof some consequences. We ﬁrstly need a comparison result:

Theorem 5.4.4. Let X be a smooth, projective and geometrically connected variety. Then Br X f-ab XpAK q‚ Ď XpAK q‚ .

Proof. This is a consequence of Theorem 6.1.1 of [21]. It is also included as Theorem 7.1 in [22].

83 In Stoll’s article [loc. cit.], he even proves the following:

f-ab Br C Theorem 5.4.5. If C is a curve, then CpAK q‚ “ CpAK q‚ .

Proof. This is Corollary 7.3 of [22].

Corollary 5.4.6. If C Ñ A is a non-constant map of C into an abelian variety A{K such that ApKq is ﬁnite and XpK,Aqdiv “ 0, then the Brauer–Manin obstruction is the only obstruction for the existence of rational points on C.

Br C Proof. This means that CpKq “ ∅ if and only if CpAK q‚ “ ∅. It is immediate from the above. Note that we only need Theorem 5.4.4, and not the slightly stronger Theorem 5.4.5.

This leads to the following conjecture.

Conjecture 5.4.7. If C is a curve, then C is very good. In other words, CpKq f-ab is dense in CpAK q‚ .

Remark 5.4.8. We can make the following observations:

• For genus 0, the proof of Theorem 5.4.2 shows that C is very good. • For elliptic curves E, we have seen in Corollary 5.3.4 that E is very good if and only if XpK,Eqdiv “ 0. The Tate–Shafarevich conjecture predicts that in fact XpK,Aq is ﬁnite. • By Faltings’ theorem, for curves of genus at least 2, the ﬁniteness of CpKq implies that C is very good if and only if it is excellent with respect to abelian coverings. • We have proven the result when C maps nontrivially into an abelian variety of algebraic rank 0 whose Tate–Shafarevich group contains no nonzero divisible elements. Br C The conjecture would imply that CpKq “ ∅ if and only if CpAK q‚ “ ∅, i.e. the Brauer–Manin obstruction is the only obstruction for the existence of rational points on a curve C.

A Category theory

We will recall the basic notions of category theory. We will assume familiar the notions of category, functor and natural transformation.

Recall that a category C is locally small if for any two objects A, B P ob C , the collection C pA, Bq of morphisms A Ñ B is a set. If C is locally small and moreover the collection ob C of objects is a set, then C is called small. The occasional remark aside, we ignore set-theoretic issues.

A.1 Representable functors

Lemma A.1.1. Let C be a locally small category, and let A P ob C . Then the association B ÞÑ C pA, Bq deﬁnes a functor C pA, ´q: C Ñ Set.

Proof. Given a morphism f : B Ñ C, we get a natural map

C pA, Bq Ñ C pA, Cq g ÞÑ f ˝ g.

This clearly makes C pA, ´q into a functor.

Deﬁnition A.1.2. Let C be a category, and let F : C Ñ Set be a functor. Then „ F is representable if there exists a natural isomorphism C pA, ´q Ñ F for some object A P ob C .

Example A.1.3. The forgetful functor F : Ab Ñ Set is representable: for each abelian group B, the underlying set FB corresponds bijectively to HompZ,Bq.

Example A.1.4. The forgetful functor F : Ring Ñ Set is representable: for each ring R, the underlying set FR corresponds bijectively to HompZrXs,Rq.

Example A.1.5. Let F : Topop Ñ Set be the functor associating to a topological space pX, T q the set T of open sets of X, and to a continuous map f : X Ñ Y the map f ´1 : T pY q Ñ T pXq mapping an open set to its inverse image.

Then F is representable: for each topological space pX, T q, the open sets of X correspond bijectively with the continuous maps X Ñ A, where A “ t0, 1u with op topology t∅, t1u, t0, 1uu. That is, Top pA, ´q “ Topp´,Aq – F .

Lemma A.1.6. Let C be a locally small category. Then the association A ÞÑ C pA, ´q deﬁnes a functor Y : C op Ñ rC , Sets.

Proof. If f : A Ñ B is a morphism in C , then it is easy to check that the maps

C pB,Cq Ñ C pA, Cq g ÞÑ g ˝ f for all C P ob C form a natural transformation. This makes Y into a functor.

86 Deﬁnition A.1.7. The functor Y : C op Ñ rC , Sets deﬁned above is called the Yoneda embedding.

Theorem A.1.8. (The Yoneda Lemma) Let C be a locally small category. Let A P ob C be given, and let F : C Ñ Set be a functor. Then there is an isomorphism

„ ψ : NatpC pA, ´q,F q ÝÑ F A,

α ÞÑ αAp1Aq. Moreover, this isomorphism is natural in both A and F .

Proof. Let x P FA be given. Then for B P ob C , deﬁne the map

θpxqB : C pA, Bq Ñ FB f ÞÑ F fpxq.

Then a straightforward argument shows that θpxq is a natural transformation C pA, ´q Ñ F . Moreover, it is clear that

ψpθpxqq “ θpxqAp1Aq “ F p1Aqpxq “ 1FApxq “ x.

Conversely, given a natural transformation α: C pA, ´q Ñ F . Then by naturality of α, for each morphism f : A Ñ B, we have a commutative diagram

αA C pA, Aq FA

f ˝ ´ F f

αB C pA, Bq FB.

Hence,

θpψpαqqBpfq “ F fpψpαqq “ pF f ˝ αAqp1Aq “ αBpf ˝ 1Aq “ αBpfq.

Hence, as f : A Ñ B was arbitrary, θpψpαqq “ α.

Hence, θ is the inverse of ψ. Naturality in F and A is an easy check.

Corollary A.1.9. The Yoneda embedding is full and faithful.

Proof. Set F “ C pB, ´q, then we have a natural isomorphism NatpC pA, ´q, C pB, ´qq – C pB,Aq.

It is straightforward to check that it is given by Y g Ðß g.

Deﬁnition A.1.10. Let F : C Ñ Set be a functor. Then a pair pA, xq consisting of an object A P ob C and an element x P FA is said to represent F if θpxq: C pA, ´q Ñ F (as above) is a natural isomorphism.

Note that a functor F : C Ñ Set is representable if and only if there exists a pair pA, xq representing F . We will show what it means for the examples given above.

87 Example A.1.11. The forgetful functor F : Ab Ñ Set is represented by the pair pZ, 1q, since for any abelian group B the map θp1qB : HompZ,Bq Ñ B is deﬁned by g ÞÑ gp1q. Note that we could also have chosen pZ, ´1q, since the map g ÞÑ gp´1q also deﬁnes an isomorphism HompZ,Bq Ñ B.

Example A.1.12. The forgetful functor F : Ring Ñ Set is represented by pZrXs,Xq, since for any ring R the map HompZrXs,Rq Ñ R given by g ÞÑ gpXq is an isomorphism.

Here, we could have equally well chosen pZrXs, φpXqq for any automorphism φ of ZrXs (since then Y φ is an automorphism of RingpZrXs, ´q). For example, we could have chosen pZrXs, aX ` bq for a P t˘1u, b P Z.

Example A.1.13. The functor F : Topop Ñ Set described above is represented „ by pA, t1uq, since the isomorphism ToppX,Aq Ñ T pXq is given by g ÞÑ g´1t1u.

Note that in the ﬁrst two examples, there are several possible choices for the pair pA, xq. However, it is almost unique, in the following sense.

Corollary A.1.14. (of the Yoneda Lemma) Let pA, xq and pB, yq be two pairs representing a functor F : C Ñ Set. Then there exists a unique isomorphism f : A Ñ„ B such that F fpxq “ y.

Proof. The elements x P FA and y P FB correspond to isomorphisms „ a: C pA, ´q Ñ F „ b: C pB, ´q Ñ F.

„ Hence, there is a unique isomorphism h: C pB, ´q Ñ C pA, ´q such that a˝h “ b (namely, h “ a´1 ˝b). But by the previous corollary, such an isomorphism comes from a unique isomorphism f : A Ñ B. It is straightforward to check that the condition ah “ b is equivalent to the condition F fpxq “ y.

„ Remark A.1.15. Some authors say that the pair pA, aq (where a: C pA, ´q Ñ F is the natural isomorphism corresponding to x) represents F , instead of the pair pA, xq. By the Yoneda lemma, this distinction is purely a matter of taste. We have included this deﬁnition because we believe it gives some intuition behind the Yoneda lemma; see the examples given above.

For reference purposes, we will state the dual of the Yoneda lemma.

Theorem A.1.16. (Contravariant Yoneda Lemma) Let C be a locally small category. Let A P ob C be given, and let F : C op Ñ Set be a functor. Then there is an isomorphism

„ ψ : NatpC p´,Aq,F q ÝÑ F A,

α ÞÑ αAp1Aq. Moreover, this isomorphism is natural in both A and F .

Proof. This follows from replacing C by C op in the Yoneda lemma.

88 A.2 Limits

Limits and colimits will always be assumed to have a small index category, unless otherwise stated.

Deﬁnition A.2.1. Let C be a category, and let J be a small category. Then a diagram of shape J in C is a functor D : J Ñ C .

Example A.2.2. If J is the category ‚Ñ‚ consisting of two objects with exactly one non-identity morphism, then a diagram of shape J is a pair of objects A, B P ob C together with a morphism f : A Ñ B.

Deﬁnition A.2.3. Let D : J Ñ C be a diagram of shape J . Then a cone over D is a pair pA, tajujPob J q, where A is an object of C , and

aj : A Ñ Dpjq is a morphism in C , for all j P ob J , such that for each morphism α: j Ñ j1 in J the diagram A

aj aj1

Dpjq Dpj1q Dpαq commutes.

Deﬁnition A.2.4. Let D : J Ñ C be a diagram, and let pA, tajujPob J q and pB, tbjujPob J q be two cones over D. Then a morphism of cones f : pA, tajuq Ñ pB, tbjuq is a morphism f : A Ñ B such that for each j P ob J the diagram

f A B

aj bj Dpjq commutes.

Clearly, the above deﬁnitions deﬁne a category of cones over D.

Remark A.2.5. If you will, the category of cones over D is just the comma category p∆ Ó Dq, where ∆: C Ñ rJ , C s is the diagonal embedding mapping A P ob C to the constant functor j ÞÑ A.

Deﬁnition A.2.6. Let D : J Ñ C be a diagram, and let pL, tλjuq be a cone over D. Then pL, tλjuq is called a limit of D if for every other cone pA, tajuq there exists a unique morphism of cones f : pA, tajuq Ñ pL, tλjuq.

Remark A.2.7. That is, pL, tλjuq is terminal in p∆ Ó Dq.

Lemma A.2.8. The limit, if it exists, is unique up to unique isomorphism.

89 Proof. We noted that it is a terminal object in a certain category. Hence, it suﬃces to prove that those are unique up to unique isomorphism.

Let D be a category, and suppose A, B P ob D are both terminal objects. Then there is a unique morphism f : A Ñ B and a unique morphism g : B Ñ A. Moreover, the composition gf is the unique morphism A Ñ A, hence has to be the identity. Similarly, fg “ 1B. Hence, f is an isomorphism. It is clearly unique.

Remark A.2.9. We often drop the maps tλju from the notation, and simply call L a limit for D. By the lemma, we can even say that L is the limit of D. We will denote this by L “ lim Dpjq. jPob J

Remark A.2.10. One deﬁnes cocones under a diagram in a dual way. Then a colimit is an initial object in pD Ó ∆q, and it is denoted by colim Dpjq. jPob J

Example A.2.11. Let J be a discrete category, i.e. the only morphisms in J are the identity morphisms. Then a diagram D : J Ñ C is just an pob J q- indexed set of objects Dj P ob C . Then the limit of D (if it exists) is called the product of the Dj, and it is denoted by L “ Dpjq. jPob J ź Example A.2.12. In Set, the product is just the Cartesian product. Indeed, if Xj for j P J are sets, then every set of functions taj : A Ñ Xju gives a unique function a: A Ñ Dpjq jPJ ź x ÞÑ pajpxqqjPJ , such that πja “ aj, where the product is one in the usual sense, with projections πj : jPJ Dpjq Ñ Dpjq. Hence, it is also a product in our sense.

Theś reader is invited to check that products in categories like Gp, Ab, Ring, Top, Sch{S in the way they are usually deﬁned are indeed products in our sense.

Example A.2.13. More generally, in Set, a limit of an arbitrary diagram D : J Ñ Set is given by

1 L :“ pxjqj ob P Dpjq Dpαqpxjq “ xj1 for all α: j Ñ j . $ P J ˇ , jPob J ˇ & ź ˇ . ˇ % ˇ - We end this section with the followingˇ useful characterisation of a limit.

Lemma A.2.14. Let D : J Ñ C be a diagram. Then an object A P ob C is the limit of D if and only if for every B P ob C there is a natural isomorphism C pB,Aq – lim C pB,Dpjqq, jPob J

90 where the limit on the right hand side is one in Set.

Proof. This follows from the description of limits in Set above: the set

lim C pB,Dpjqq jPob J is the set of series pfj : B Ñ DpjqqjPob J of morphisms in C such that

Dpαq ˝ fj “ C pB,Dpαqqpfjq “ fj1

1 for all α: j Ñ j in J . This is exactly the condition on pB, tfjuq to be a cone over D. The result follows since A is the limit if and only if the existence of a morphism f : B Ñ A is equivalent to B being a cone.

A.3 Functors on limits

Deﬁnition A.3.1. Let J be a small category, and let F : C Ñ D be a functor.

(1) We say that F preserves limits of shape J if for any diagram D : J Ñ C and any limit cone pL, tλjuq of D, the cone pF L, tF λjuq is a limit for FD. (2) We say that F reﬂects limits of shape J if for any diagram D : J Ñ C and any cone pL, tλjuq such that pF L, tF λjuq is a limit of FD, the cone pL, tλjuq is a limit of D. (3) We say that F creates limits of shape J if for any diagram D : J Ñ C and any limit pM, tµjuq of FD, there exists a cone pL, tλjuq over D such that pF L, tF λjuq is isomorphic to pM, tµjuq, and moreover any such pL, λjuq is a limit of D.

Remark A.3.2. Dually, we deﬁne when a functor preserves, reﬂects or creates colimits in the obvious way.

Example A.3.3. The forgetful functor F : Ab Ñ Set preserves products. In- deed, if tAjujPJ is a set of groups, then the underlying set of the product A “ jPJ Aj is just the product set of the Aj. In fact, F preserves all limits.

Exampleś A.3.4. However, the forgetful functor F : Ab Ñ Set does not preserve coproducts. Indeed, if abelian groups tAjujPJ are given, then the coproduct in Ab is the direct sum A “ Aj. jPJ à However, the underlying set of A is not the disjoint union of the Aj, which would be the coproduct in Set.

Lemma A.3.5. Let F : C Ñ D be a functor that creates limits of shape J . Then it also reﬂects them. If moreover D has limits of shape J , then F preserves them.

91 Proof. The ﬁrst statement is immediate from the deﬁnition of creating limits.

Now suppose D has limits of shape J , and let D : J Ñ C be a diagram of shape J .

Suppose pL, tλjuq is a limit for D. Then pF L, tF λjuq is a cone over FD. Since D has limits of shape J , we can form the limit pM, tµjuq of FD. Since F 1 1 1 1 creates limits, there exists a cone pL , tλjuq over D such that pFL , tF λjuq is 1 isomorphic to pM, tµjuq, and moreover L is a limit of D.

1 1 Hence, pL, tλjuq is isomorphic to pL , tλjuq, since limits are unique up to unique isomorphism. But this clearly forces 1 1 pF L, tF λjuq – pFL , tF λjuq – pM, tµjuq, so pF L, tF λjuq is also a limit of FD. Hence, F preserves limits.

We will now prove some results about limits in functor categories.

Deﬁnition A.3.6. Let C be a category. Then the discrete category on C is ι the subcategory C disc Ñ C with the same objects as C , but only identity morphisms.

Lemma A.3.7. Let J be a small category, and D a category that has limits of shape J . Then the functor rC , Ds Ñ rC disc, Ds creates limits of shape J .

Proof. Note that limits in rC disc, Ds are just pointwise limits. Hence rC disc, Ds disc has and evA : rC , Ds Ñ D preserves all limits of shape J , for all A P ob C disc.

Let D : J Ñ rC , Ds be a diagram, and suppose pL, tλpjquq is a limit cone of ιD. Let f : A Ñ B be any morphism in C . Then LpBq is a limit of evB D. For each morphism α: j Ñ j1 in J , naturality of Dpαq gives a commutative diagram LpAq 1 λpjqA λpj qA

DpαqA DpjqpAq Dpj1qpAq

Dpjqpfq Dpj1qpfq DpαqB DpjqpBq Dpj1qpBq.

This shows that pLpAq, tDpjqpfq ˝ λpjqAuq is a cone over evB D. Hence, there is a unique morphism Lpfq: LpAq Ñ LpBq making commutative the prism

Lpfq LpBq LpAq DpjqpBq Dpj1qpBq. DpjqpAq Dpj1qpAq

92 This shows that L can be extended to a functor C Ñ D.

Finally suppose any cone pM, tµpjquq over D mapping to pL, tλpjquq is given. Let pC, tγpjquq be any cone over D. Since ιM is a limit for ιD, for each A P ob C there is a unique morphism F pAq: CpAq Ñ MpAq such that µpjqA ˝ F pAq “ γpjqA for every j P ob J .

Now let f : A Ñ B be any morphism in C . We have a diagram

F pAq CpAq MpAq

Cpfq Mpfq F pBq CpBq MpBq.

It is easily seen that the two cones

pCpAq, tµpjqB ˝ Mpfq ˝ F pAquq,

pCpAq, tµpjqB ˝ F pBq ˝ Cpfquq over evB D are the same cone. Hence, by the universal property of MpBq, we have Mpfq ˝ F pAq “ F pBq ˝ Cpfq. Hence, the diagram above commutes, so F is in fact a natural transformation C Ñ M. Hence, pM, tµpjquq is a limit cone for D.

Remark A.3.8. For D “ Set, the lemma above can also be proved with the Yoneda lemma. However, getting the dual statement in a similar way would require some isomorphism

NatpF,GAq – F pAq for some functor GA depending on A. This can not be done canonically (for instance because the left hand side is contravariant in F , whereas the right hand side is covariant). I am not aware of any way to circumvent this. Since we want to know both the lemma above and its dual (even for D “ Set), we have chosen for the argument given above.

Corollary A.3.9. Suppose D is complete. Let A P ob C . Then the category rC , Ds has and the evaluation functor evA : rC , Ds Ñ D preserves all limits.

Proof. The functor ι: rC , Ds Ñ rC disc, Ds creates limits. In particular, rC , Ds has all limits, and ι preserves them. Hence, evA also preserves limits, since limits in rC disc, Ds are pointwise.

Corollary A.3.10. Suppose D is cocomplete. Let A P ob C . Then the category rC , Ds has and the evaluation functor evA : rC , Ds Ñ D preserves all colimits.

Proof. This follows dually.

This gives the following addendum to the Yoneda lemma.

93 Lemma A.3.11. Let C be locally small. Then the (covariant) Yoneda functor Y : C Ñ rC op, Sets preserves and reﬂects limits.

Proof. Let J be a small category, and let D : J Ñ C be a diagram of shape J . By Lemma A.2.14, an object A P ob C is a limit for D if and only if C pB,Aq is the limit for the functor J Ñ Set j ÞÑ C pB,Dpjqq, for each B P ob C . But this functor is none other than evB YD. Hence, the lemma above asserts that A is the limit for D if and only if YA is the limit for YD, which is exactly what we needed to prove.

A.4 Groups in categories

Deﬁnition A.4.1. Let C be a category with ﬁnite products (in particular, it has a terminal object T ). Then a group object in C is an object G P ob C together with morphisms µ G ˆ G ÝÑ G η T ÝÑ G ι G ÝÑ G such that the diagrams

1ˆµ G ˆ G ˆ G G ˆ G

µˆ1 µ (A.1) µ G ˆ G G,

1ˆη ηˆ1 G ˆ T G ˆ G T ˆ G

µ (A.2) π1 π2 G,

∆G 1ˆι G G ˆ G G ˆ G

µ (A.3) η T G commute. If moreover the diagram

pπ2, π1q G ˆ G G ˆ G (A.4) µ µ G, commutes, then G is commutative (or abelian).

94 Remark A.4.2. The morphisms µ, η and ι are the analogues of multiplication, the unit element and inversion, respectively. We will use these names to refer to these morphisms. Diagram (A.1) is associativity, diagram (A.2) is the neutral property of the unit element and diagram (A.3) is invertibility. Diagram (A.4) is commutativity.

Note that we have only asserted that ι is a right inverse; the similar diagram showing that ι is also a left inverse is a formal consequence of these three diagrams, in the same way that for groups any one-sided inverse is automatically two-sided.

Example A.4.3. If C “ Set, then a group object is just a group, in the ordinary sense. Moreover, G is abelian if and only if it is abelian in the ordinary sense.

Example A.4.4. If C “ Top, then a group object is a group G such that the multiplication and inversion are continuous. That is, G is a topological group. Note that the unit T Ñ G is automatically continuous, since the terminal object in Top is the discrete space t˚u.

Also in this example, G is abelian if and only if it is abelian in the ordinary sense.

Proposition A.4.5. Let C be a category with ﬁnite products. Let Y : C Ñ rC op, Sets be the (covariant) Yoneda embedding. Then G P ob C is a group object in C if and only if YG is a group object in rC op, Sets.

Proof. By Lemma A.3.11, Y preserves limits. Since moreover Y is full and faithful, giving a multiplication µ: G ˆ G Ñ G is equivalent to giving a multiplication µ: Y pG ˆ Gq “ YG ˆ YG ÝÑ YG in rC op, Sets, and similarly for η and ι. Moreover, since Y is faithful, the diagrams (A.1), (A.2) and (A.3) hold for G in C if and only if they hold for YG in rC op, Sets.

Lemma A.4.6. Let C be a category. Then a functor G: C op Ñ Set is a group object if and only if each GA is a group object in Set, and moreover each Gf : GB Ñ GA induced by f : A Ñ B is a group homomorphism.

Proof. Note that products are just pointwise. Hence, giving multiplication, unit and inversion morphisms in rC op, Sets is equivalent to giving them on each GA, such that µ, η and ι are natural. Clearly, they satisfy the diagrams in rC op, Sets if and only if they satisfy the diagrams in Set for each A P ob C .

Also, note that naturality of µ is commutativity of the diagram

µB GB ˆ GB GB

Gf ˆGf Gf

µA GA ˆ GA GA,

95 for any morphism f : A Ñ B in C . This is equivalent to each GB Ñ GA being a group homomorphism.

Hence, if G is a group, then each GA is a group and each Gf : GB Ñ GA is a group homomorphism, for f : A Ñ B. Conversely, if each GA is a group and each Gf is a group homomorphism, then we only need to show naturality of η and ι. But this is just the statement that group homomorphisms preserve the identity and commute with inversion.

Corollary A.4.7. Let C be a category with ﬁnite products. Then G is a group object in C if and only if C pA, Gq is a group for each A P ob C and the map C pB,Gq Ñ C pA, Gq is a group homomorphism for each f : A Ñ B.

Proof. This is immediate from the proposition and the lemma above.

Deﬁnition A.4.8. Let C be a category (not necessarily having ﬁnite products). Then a group object in C is an object G such that C pA, Gq is a group for all A P ob C and the map C pB,Gq Ñ C pA, Gq is a group homomorphism for every f : A Ñ B.

Deﬁnition A.4.9. Let C be a category with ﬁnite products, and let G, H be group objects in C . Then a morphism f : G Ñ H of internal groups is a morphism f : G Ñ H in C such that the diagram

µG G ˆ G G

f ˆf f

µH H ˆ H H, commutes.

Lemma A.4.10. Let C be a category, and let G, H : C op Ñ Set be internal group objects in rC op, Sets. Then a natural transformation f : G Ñ H is a morphism of internal groups if and only if each fA : GA Ñ HA is a group homomorphism.

Proof. This is obvious.

Corollary A.4.11. Let C be a category. Then the category GppC q of internal group objects in rC op, Sets is the category rC op, Gps.

Proof. By Lemma A.4.6, an internal group object in rC op, Sets is exactly an object of rC op, Gps. By Lemma A.4.10, the notion of morphism in both categories coincides as well.

Corollary A.4.12. Let C be a category. Then the category AbpC q of internal abelian group objects in rC op, Sets is the category rC op, Abs.

Proof. This follows from the observation that an internal group object G in rC op, Sets is abelian if and only if GA is abelian for each A P ob C .

96 Corollary A.4.13. Let C be a category with ﬁnite products, and let Y : C Ñ rC op, Sets be the Yoneda embedding. Then the category of internal (abelian) group objects in C is equivalent to the subcategory of rC op, Gps (resp. rC op, Abs) of functors F for which the composition with the forgetful functor Gp Ñ Set (resp. Ab Ñ Set) is representable.

Proof. This is clear from the above.

Lemma A.4.14. Let J be a small category, and let C be a category with limits of shape J . Let G: J Ñ C be a diagram such that Gj is an internal group object for each j P ob J . Then lim Gj is a group object in a canonical way.

Proof. By Lemma A.2.14, for each A P ob C it holds that

C pA, lim Gjq “ lim C pA, Gjq. j j

But in fact, the forgetful functor Gp Ñ Set creates limits. That is, we can uniquely put a group structure on the limit making it the limit in Gp. It is op clear that this makes Y plim Gjq into a group object in rC , Gps, which is by deﬁnition representable.

Corollary A.4.15. Let C be a category with limits of shape J . Then the category of internal group objects has and the forgetful functor GppC q Ñ C preserves and reﬂects limits of shape J .

Proof. This is clear from the lemma.

Corollary A.4.16. Let C be a category with finite limits. Then the category of group objects in C has and the forgetful functor GppC q Ñ C preserves and reflects kernels and finite products.

Proof. Special case of the corollary above.

In other words: if Gi is a ﬁnite set of group objects (i P t1, . . . , nu), then Gi is a group object as well. If f : G Ñ H is a morphism of group objects, then its kernel (the equaliser of f and G Ñ T Ñ H) is a group object as well. ś

Deﬁnition A.4.17. Let C be a category with ﬁnite limits, and G a group object in C . Then a left action of G on an object S is a morphism

m: G ˆ S Ñ S such that the diagrams

1ˆm G ˆ G ˆ S G ˆ S

µˆ1 m m G ˆ S S,

97 ηˆ1 T ˆ S G ˆ S

m π1 S commute.

Proposition A.4.18. Let C be a category with ﬁnite products, and let G be a group object in C . Then giving a left action of G on S is equivalent to giving a left action of YG on YS.

Proof. Similar to Proposition A.4.5.

Lemma A.4.19. Let C be a category, and G a group object in rC op, Sets. Then giving a left action of G on S : C op Ñ Set is equivalent to giving a left action of GA on SA for every A P ob C such that all the maps Gf : GB Ñ GA induced by f : A Ñ B are G-invariant maps.

Proof. Similar to Lemma A.4.6.

Corollary A.4.20. Let C be a category with ﬁnite products, and let G be a group object in C . Then giving an action of G on S is equivalent to giving an action of C pA, Gq on C pA, Sq for every A P ob C , such that the map C pB,Sq Ñ C pA, Sq is G-invariant for all f : A Ñ B.

Proof. Clear from the proposition and the lemma.

Remark A.4.21. One can deﬁne a morphism of G-actions in the obvious way, and prove the analogous results. Also, one can deﬁne the notion of right action and their morphisms, and show that the same properties hold.

B Étale cohomology

This chapter will treat étale cohomology and fppf cohomology. The treatment is based on [15], [2] and [11].

B.1 Sites and sheaves

A Grothendieck topology is a generalisation of a topological space. We will study three such topologies in more detail, namely the Zariski topology, the étale topology and the fppf topology.

Remark B.1.1. Recall that, in the Zariski topology, we deﬁne presheaves on the scheme X as functors ToppXqop Ñ Set, where the category ToppXq has as objects the open sets U Ď X and as morphisms the inclusions U Ď V . Then a sheaf is deﬁned to be a presheaf F with the extra condition that for every open set U P ob ToppXq and for every open covering tUiuiPI of U, the diagram

F pUq ÝÑ F pUiq ÝÑ F pUi X Ujq iPI i,j I2 ź p źqP is an equaliser diagram.

The idea of the étale topology is to replace the category ToppXq by a larger category of which the “open sets” are no longer solely actual open sets, but rather étale morphisms U Ñ X for (abstract) schemes U. In order to be able to talk about coverings, we introduce the following.

Deﬁnition B.1.2. A covering family (or simply covering) of an object U in a fi category C is a family tUi ÝÑ UuiPI of morphisms to U.

In our applications (where C “ Sch{X), we will usually require the maps fi to be jointly surjective, i.e. the union of their images should equal U.

Deﬁnition B.1.3. A Grothendieck pretopology on a category C is a collection fi CovpC q of coverings tUi ÝÑ UuiPI for objects U P ob C , subject to the following conditions:

(0) If U0 Ñ U occurs in some covering of U, and V Ñ U is any morphism in C , then the fibred product U0 Û V exists in C ; (1) If tUi Ñ Uu is a covering of U and V Ñ U is arbitrary, then tUiÛ V Ñ V u is a covering of V ; (2) If tUi Ñ UuiPI is a covering of U, and for each Ui we have a covering

tUij Ñ UiujPJi , then the family of composites tUij Ñ Ui Ñ UuiPI,jPJi is a covering of U; (3) If V Ñ U is an isomorphism, then tV Ñ Uu is a one-object covering.

The collection of all coverings is denoted CovpC q.

Note that condition (0) is only included to assure condition (1) makes sense.

100 Deﬁnition B.1.4. A site is a category C together with a Grothendieck pretopology CovpC q.

Deﬁnition B.1.5. A D-presheaf on a category C is a functor F : C op Ñ D. We write PShD pC q for the category of D-presheaves on C .

Remark B.1.6. We will be mostly interested in the case D “ Ab. However, at certain points we will need the case D “ Set or D “ Gp, so we will develop a slightly more general theory.

Deﬁnition B.1.7. A presheaf of abelian groups on a category C is a functor op F : C Ñ Ab. Equivalently, it is an internal abelian group object in PShSetpC q; see Corollary A.4.12. We simply write PShpC q for the category of presheaves of abelian groups on C .

Deﬁnition B.1.8. Let D be a category with products. Let F be a D-presheaf on a site pC , CovpC qq. Then F is a sheaf (with respect to the Grothendieck pretopology CovpC q) if for any covering tUi Ñ UuiPI , the diagram

F pUq ÝÑ F pUiq ÝÑ F pUi ˆU Ujq iPI i,j I2 ź p źqP is an equaliser diagram. We write ShD pC , CovpC qq for the full subcategory of PShD pC q of sheaves. If no confusion about the chosen Grothendieck pretopology is possible, we will simply write ShD pC q. In the case D “ Ab, we will drop the subscript, and simply write ShpC q.

Remark B.1.9. The pair of parallel arrows in the diagram are given as follows. The projections Ui ˆU Uj Ñ Ui and Uj ˆU Ui Ñ Ui induce maps

πi F pUkq ÝÑÑ F pUiq Ñ F pUi ˆU Ujq, kPI ź πi F pUkq ÝÑÑ F pUiq Ñ F pUj ˆU Uiq. kPI ź The parallel arrows above are given by the respective products over all pi, jq P I2 of these two maps.

Example B.1.10. The (small) Zariski site is the category ToppXq of Zariski open sets of X, together with the Grothendieck pretopology given by open coverings tUi Ď UuiPI of U P ob ToppXq.

Fibred products of open subsets are just intersections (this holds both in the category Sch{X and in ToppXq). Hence, it is easy to check that the Zariski site is indeed a site. Note that, in this case, the deﬁnition of sheaves coincides with the usual one.

Remark B.1.11. Note that the deﬁnition of sheaf depends heavily on the given Grothendieck pretopology. We will see examples of presheaves which are a sheaf for the Zariski topology, but not for the étale topology.

We recall the following results from category theory.

101 Lemma B.1.12. Suppose D is complete. Let A P ob C . Then the category rC , Ds has and the evaluation functor evA : rC , Ds Ñ D preserves all limits.

Proof. This is Corollary A.3.9.

Corollary B.1.13. Suppose D is cocomplete. Let A P ob C . Then the category rC , Ds has and the evaluation functor evA : rC , Ds Ñ D preserves all colimits.

Proof. This follows dually.

Corollary B.1.14. The category PShpC q is both complete and cocomplete, and limits and colimits are pointwise.

Proof. This follows since Ab is both complete and cocomplete.

Corollary B.1.15. Let α: F Ñ G be a morphism in PShpC q. Then α is monic (resp. epic) if and only if αA is injective (resp. surjective) for every A P ob C .

Proof. It is easy to see that any morphism f : B Ñ C in some category D is monic if and only if the diagram

1 B B 1 f f B C is a pullback square. By the above, F is the pullback of α along α if and only if F pAq is the pullback of αA along αA for each A P ob C . The result now follows since monomorphisms in Ab are exactly injective maps. The result about epimorphisms follows dually, since epimorphisms in Ab are exactly surjective maps.

Deﬁnition B.1.16. If a morphism of presheaves α: F Ñ G is a monomorphism (resp. an epimorphism), we will say it is injective (resp. surjective).

By the corollary above, a morphism α: F Ñ G is injective (resp. surjective) if and only if the same holds for each αA : F pAq Ñ GpAq.

Corollary B.1.17. The category PShpC q is balanced.

Proof. Let α: F Ñ G be both monic and epic. Then each αA is both injective and surjective, hence an isomorphism. This forces α to be an isomorphism.

Remark B.1.18. The results of Corollary B.1.14 through B.1.17 also hold for the category of presheaves of sets, since Set is complete, cocomplete and balanced.

102 B.2 Čech cohomology

In the section above, we have seen some of the basic properties of the category of presheaves on a category C . In this section and the next, we will show that a certain adjunction (“sheaﬁﬁcation”) gives similar results about the category of sheaves on a site. On the way, we develop another useful tool, namely Čech cohomology.

Deﬁnition B.2.1. Let C be a site, and let F be a presheaf of abelian groups on C . Let U “ tUi Ñ UuiPI be a covering of some U P ob C . Then we write

Ui0¨¨ïp “ Ui0 Û ... Û Uip

p`1 whenever pi0, . . . , ipq P I . Deﬁne ˇp C pU , F q “ F Ui0¨¨¨ip , p`1 pi0,...,ipqPI ź ` ˘ for all p P Zě0. For each j P t0, . . . , pu, there is a natural map

p resj F pUi0¨¨ïj´1ij`1¨¨ïp q ÝÑ F pUi0¨¨ïp q defined by the projection

Ui0 Û ... Û Uip ÑÑ Ui0 Û ... Û Uij´1 Û Uij`1 Û ... Û Uip . This defines homomorphisms p´1 ˇp´1 ˇp dj : C pU , F q Ñ C pU , F q p p psiqiPI ÞÑ res psi0¨¨ïj´1ij`1¨¨ïp q p`1 , j pi0,¨¨¨ ,ipqPI satisfying the relations ` ˘ p p´1 p p´1 dkdj “ dj dk´1 whenever 0 ď j ă k ď p ` 1. Hence, they define a cosimplicial abelian group ˇ0 ÝÑ ˇ1 ÝÑ ˇ2 C pU , F q ÝÑ C pU , F q ÝÑ C pU , F q ¨ ¨ ¨ .

Deﬁnition B.2.2. The Čech complex associated to F with respect to U is the complex d0 d1 0 ÝÑ Cˇ0pU , F q ÝÑ Cˇ1pU , F q ÝÑ .... associated to the cosimplicial abelian group above. That is, its arrows are given by p p´1 j p´1 d “ p´1q dj . j“0 ÿ It is a complex since it comes from a cosimplicial abelian group. That is,

p`1 p p p´1 k`j p p´1 d d “ p´1q dkdj k“0 j“0 ÿ ÿ k`j p p´1 k`j p p´1 “ p´1q dj dk´1 ` p´1q dkdj . 0ďjăkďp`1 0ďkďjďp ÿ ÿ

103 Setting k1 “ k ´ 1 in the ﬁrst sum gives

p p´1 k1`1`j p p´1 k`j p p´1 d d “ p´1q dj dk ` p´1q dkdj , 0ďjďk1ďp 0ďkďjďp ÿ ÿ which is zero since the two sums are equal exactly up to a factor ´1.

Deﬁnition B.2.3. Let C be a site, F a presheaf of abelian groups on C and U a covering of U P ob C . Then the Čech cohomology of F with respect to U is the cohomology Hˇ ipU , F q “ Hi Cˇ‚pU , F q of the Čech complex. ` ˘

We now want to compare the Čech cohomology with respect to diﬀerent coverings. We ﬁrstly need a way to compare to coverings of U.

fi gj Deﬁnition B.2.4. Let U “ tUi ÝÑ UuiPI and V “ tVj ÝÑ UujPJ be two coverings of U P ob C . Then V is a reﬁnement of U if there exists a map α: J Ñ I and for each j P J a morphism ηj : Vj Ñ Uαpjq such that the diagram

ηj Vj Uαpjq

g j fαpjq U commutes. The pair pα, tηjuq is called a reﬁning morphism from V to U .

Deﬁnition B.2.5. Let pα, tηjujPJ q be a reﬁning morphism from V to U as above. Then the ηj induce maps

ηj0¨¨¨jp : Vj0¨¨¨jp Ñ Uαpj0q¨¨¨αpjpq

p`1 for any pj0, . . . , jpq P J . This is turn deﬁnes a morphism ψp : Cˇpp , q Ñ Cˇpp , q pα,tηj uq U F V F given by

si0¨¨¨ip p`1 ÞÑ resηj ¨¨¨j psαpj q¨¨¨αpj qq . pi0,...,ipqPI 0 p 0 p p`1 pj0,...,jpqPJ ` ˘ ´ ¯ p`1 Remark B.2.6. Note that for all pj0, . . . , jpq P J , k P t0, . . . , pu, the diagram

Vj0¨¨¨jp Vj0¨¨¨jk´1jk`1¨¨¨jp

η ηj0¨¨¨jp j0¨¨¨jk´1jk`1¨¨¨jp

Uαpj0q¨¨¨αpjpq Uαpj0q¨¨¨αpjk´1qαpjk`1q¨¨¨αpjpq

ψp dp commutes. Hence, the maps pα,tηj uq commute with all the k, in the sense that

ψp ˝ dp “ dp ˝ ψp´1 . pα,tηj uq k k pα,tηj uq

104 p It follows that the ψpα,tηj uq commute with the coboundary maps d , hence they deﬁne morphisms ρp : Hˇ pp , q Ñ Hˇ pp , q. pα,tηj uq U F V F

Lemma B.2.7. Let U , V be two coverings of U P ob C . Then any two reﬁning morphisms pα, tηjuq, pβ, tθjuq deﬁne the same map on Čech cohomology.

Proof. We will show that the maps on the Čech complex are chain homotopic. For each k P t0, . . . , pu, the maps ηj and θj deﬁne a morphism

ηj0¨¨¨jk ˆ θjk¨¨¨jp : Vj0¨¨¨jp ÝÑ Uαpj0q¨¨¨αpjkqβpjkq¨¨¨βpjpq. This induces maps p`1 ˇp`1 ˇp χk : C pU , F q ÝÑ C pV , F q deﬁned by

p`2 psiqiPI ÞÝÑ resηj ¨¨¨j ˆθj ¨¨¨j psαpj q¨¨¨αpj qβpj q¨¨¨βpj qq . 0 k k p 0 k k p p`1 pj0,...,jpqPJ ´ ¯ χp`1 ˝ dp ψp We note that 0 0 is none other than pβ,tθj uq, since the diagram

ηj0 ˆ θj0¨¨¨jp Vj0¨¨¨jp Uαpj0qβpj0q¨¨¨βpjpq

θj0¨¨¨jp

Uβpj0q¨¨¨βpjpq

χp`1 ˝ dp ψp commutes. Similarly, p p`1 is just pα,tηj uq. By similar arguments, we ﬁnd the following relations: p`1 p p´1 p χk ˝ dl “ dl´1 ˝ χk, 0 ď k ă l ´ 1 ď p. (B.1) p`1 p p´1 p χk ˝ dl “ dl ˝ χk´1, 0 ď l ă k ď p, (B.2) p`1 p p`1 p χk ˝ dk`1 “ χk`1 ˝ dk`1, 0 ă k ă p ´ 1. (B.3)

p`1 p k p`1 We now put χ “ k“0p´1q χk . Then we get: ř p p`1 p p´1 p`1 p p´1 p k`l p`1 p k`l p´1 p χ d ` d χ “ p´1q χk dl ` p´1q dl χk. (B.4) k“0 l“0 l“0 k“0 ÿ ÿ ÿ ÿ Now the ﬁrst double sum splits into

k`l p`1 p ¨ ` ` ` ` ` ˛ p´1q χk dl . (B.5) k“l“0 lăk l“k l“k`1 ląk`1 k“p ˚ ÿ ÿ kÿ‰0 kÿ‰p ÿ l“ÿp`1‹ ˚ ‹ ˝ ‚ The second double sum in (B.4) splits into

k`l p´1 p ` p´1q dl χk. (B.6) ˜lďk ląk¸ ÿ ÿ

105 ψp Now the first term of (B.5) sum gives pβ,tθj uq. The second term cancels against the first term of (B.6), by (B.2). The third and the fourth term cancel against each other, by (B.3). The fifth term cancels against the second term of (B.6), ´ψp by (B.1). Finally, the sixth term is just pα,tηj uq. Hence, we find that

χp`1dp ` dp´1χp “ ψp ´ ψp , pβ,tθj uq pα,tηj uq which gives the chain homotopy we were looking for.

Definition B.2.8. If V is a refinement of U , we will simply write ρppV , U q for ρp the map pα,tηj uq defined above. By the lemma, it depends only on U , V , and the fact that there exists a refining morphism from V to U . We will usually drop the superscript where this does not lead to confusion.

It automatically follows that

ρpW , V qρpV , U q “ ρpW , U q if V is a reﬁnement of U and W of V .

Definition B.2.9. Let U and V be coverings of U P ob C . We write U ” V if one is a refinement of the other and vice versa. This is clearly an equivalence relation, since we can compose refinement morphisms.

Corollary B.2.10. If U ” V , then ρpV , U q is an isomorphism

„ Hˇ ppU , F q ÝÑ Hˇ ppV , F q for any presheaf F on C .

Proof. If pα, tηjujPJ q denotes a reﬁning morphism from V to U and pβ, tθiuiPI q one from U to V , then both the composite

pα ˝ β, tηβpiq ˝ θiuiPI q and the identity are a reﬁning morphism from U to itself. Hence, they induce the same map on Čech cohomology, so

ρpU , V qρpV , U q “ 1.

Similarly for the other composition, hence ρpU , V q is the inverse of ρpV , U q.

Deﬁnition B.2.11. The set of open covers U “ tUi Ñ UuiPI of U up to the equivalence relation above is denoted JU .

Remark B.2.12. The reader who is interested in such matters may convince himself that in any of the cases we study (Zariski, étale, fppf), the collection JU can indeed be taken to be a (small) set. This is however not the case for any site; most notably the fpqc site (which we do not deﬁne) does not have this property.

106 Corollary B.2.13. If V is a reﬁnement of U , then the map ρpV , U q depends only on the classes of U and V in JU .

Proof. This follows from the lemma and the previous corollary.

Remark B.2.14. Note that the ordering V ď U if V is a reﬁnement of U makes JU into a partially ordered set. Moreover, this set is actually directed, as any two coverings tUi Ñ UuiPI , tVi Ñ UujPJ have a common reﬁnement

tUi ˆU Vj ÝÑ Uupi,jqPIˆJ .

This inspires the following deﬁnition.

Deﬁnition B.2.15. Let C be a site, let U P ob C , and let F be a presheaf on C . Then the (absolute) Čech cohomology groups of F are the groups Hˇ p U, colim Hˇ p , , p F q “ ÝÑ pU F q U PJU with respect to the maps ρpV , U q of above. By the preceding remark, it is just a direct limit in the classical sense.

Remark B.2.16. If f : V Ñ U is any morphism in C , then by the axioms of a site, any covering U “ tUi Ñ UuiPI of U gives rise to a covering

U ˆU V “ tUi ˆU V Ñ V uiPI of V . Let V “ tVi Ñ V uiPI denote this covering. Then

Vi0¨¨¨ip “ Vi0 ˆV ... ˆV Vip

“ pUi0 ˆU V q ˆV ... ˆV pUip ˆU V q

“ Ui0¨¨¨ip ˆU V, and f induces morphisms

fi0,¨¨¨ ,ip : Vi0¨¨¨ip Ñ Ui0¨¨¨ip . This determines a map p ˇp ˇp fU ,V : C pU , F q Ñ C pV , F q

p 1 psiqiPI ` ÞÑ presfi psiqqiPIp`1 . p p p It is clear that fU ,V commutes with all the dk, hence also with d , so it defines a well-defined map p ˇ p ˇ p fU ,V : H pU , F q Ñ H pV , F q. 1 1 1 If U is a refinement of U , then V “ U Û V is a refinement of V . It is straightforward to check commutativity of the diagram

f p U ,V Hˇ ppU , F q Hˇ ppV , F q

ρpU 1, U q ρpV 1, V q f p U 1,V 1 Hˇ ppU 1, F q Hˇ ppV 1, F q.

107 p Hence the maps fU ,V give rise to a map f p : Hˇ ppU, F q Ñ Hˇ ppV, F q, making U ÞÑ Hˇ ppU, F q into a presheaf.

Deﬁnition B.2.17. The presheaf U ÞÑ Hˇ ppU, F q is denoted HˇppF q.

Remark B.2.18. If U is the trivial covering tU Ñ Uu, then each CˇppU , F q p p`1 p is just F pUq. The maps d are just i“0 p´1q ¨ 1FpUq, i.e. 0 if p is even and the identity if p is odd. Hence, ř 0 Hˇ 0pU , F q “ ker F pUq ÝÑ F pUq “ F pUq, ´ ¯ which gives a map U Hˇ 0 , colim Hˇ 0 , Hˇ 0 U, . F p q “ pU F q Ñ ÝÑ pV F q “ p F q V PJU

For f : V Ñ U, the pullback V “ U ˆU V is just the trivial covering tV Ñ V u, hence we have a commutative diagram

F pUq Hˇ 0pU , F q Hˇ 0pU, F q

0 resf 0 fU ,V f

F pV q Hˇ 0pV , F q Hˇ 0pV, F q

That is, we get a morphism of presheaves F Ñ Hˇ0pF q.

Deﬁnition B.2.19. Let F be a presheaf on a site C . Then F is separated if the morphism of presheaves F Ñ Hˇ0pF q is injective.

Lemma B.2.20. Let F be a presheaf. Then F is separated if and only if for each U P ob C and for each covering U “ tUi Ñ UuiPI of U, the natural map

F pUq ÝÑ F pUiq iPI ź is injective.

Proof. Clearly, F is separated if and only if for each U P ob C the map F pUq Ñ Hˇ 0pU, F q is injective. Let U P ob C be given. Since Hˇ 0pU, F q is the direct limit colim Hˇ 0 , , ÝÑ pU F q U PJU an element x P F pUq maps to zero in Hˇ 0pU, F q if and only if there exists some ˇ 0 U P JU such that x maps to zero in H pU , F q.

Hence, the map F pUq Ñ Hˇ 0pU, F q is injective if and only if each F pUq Ñ Hˇ 0pU , F q is injective. The result follows since Hˇ 0pU , F q “ ker d0 is a sub- ˇ0 group of C pU , F q “ iPI F pUiq. ś 108 Remark B.2.21. The usual deﬁnition of a separated presheaf found in the literature is the one of the lemma. The word ‘separated’ refers to the fact that one can distinguish elements of F pUq (“global sections”) by their restrictions to each F pUiq (“local sections”). Note however that, in contrast to the Zariski site, elements of F pUq on an abstract site need not be functions of any sort.

Lemma B.2.22. Let F be a presheaf. Then F is a sheaf if and only if the natural morphism of presheaves F Ñ Hˇ0pF q is an isomorphism.

ˇ 0 Proof. Let U “ tUi Ñ UuiPI be a cover of some U P ob C . Note that H pU , F q is, by deﬁnition, the kernel of

0 0 d0 ´ d1 F pUiq F pUi ˆU Ujq. iPI i,j I2 ź p źqP 0 0 Note also that the kernel of d0 ´ d1 is the same thing as the equaliser of the pair

0 d0 F pUiq F pUi ˆU Ujq. 0 iPI d1 i,j I2 ź p źqP Hence, F is a sheaf if and only if F pUq Ñ Hˇ 0pU , F q is an isomorphism for any U P ob C and any covering U of U. The result now follows since for any direct system tAjujPJ of abelian groups over a poset J containing an initial object j0, the natural map A colim A j0 ÝÑ ÝÑ j jPJ is an isomorphism if and only if each Aj0 Ñ Aj is an isomorphism.

Lemma B.2.23. Let F be a presheaf on C . Then Hˇ0pF q is separated.

Proof. Let U P ob C , and let U “ tUi Ñ UuiPI be a covering of U. We want to show that ˇ 0 ˇ 0 φ: H pU, F q Ñ H pUi, F q iPI ź is injective. Denote by φi the i-th component of this map, for all i P I.

ˇ 0 Suppose x P ker φ, say x P H pV , F q for some covering V “ tVj Ñ UujPJ of U. Put Vi “ V ˆU Ui for all i P I. Then φipxq is the image under ˇ 0 ˇ 0 H pVi, F q Ñ H pUi, F q

f 0 x φ x 0 of the element U ,Vi p q (with the notation deﬁned above). Since ip q “ , the standard properties of direct limits give some covering Wi “ tWik Ñ UiukPKi of Ui reﬁning Vi such that

ρ , f 0 x 0, pWi Viq U ,Vi p q “

ˇ 0 ` ˘ as elements of H pWi, F q.

109 Composing with the inclusion ˇ 0 H pWi, F q Ď F pWikq, iPK źi we ﬁnd that each of the restrictions of x to the Wik must be 0.

But the Wik for ﬁxed i cover Ui, hence by the axioms of a site, the set of composites

tWik Ñ Ui Ñ UuiPI,kPKi is a covering of U, which we will denote W . We have seen that x maps to 0 under the natural map ˇ 0 ˇ 0 ρpW , V q: H pV , F q Ñ H pW , F q Ď F pWikq. iPI kźPKi This says exactly that x “ 0 in the direct limit

Hˇ 0 U, colim Hˇ 0 1, , p F q “ ÝÑ pU F q 1 U PJU i.e. that x “ 0. Hence, φ is injective.

We also have the following:

Proposition B.2.24. Let F be a separated presheaf. Then Hˇ0pF q is a sheaf.

Proof. Let U P ob C be given, and let U “ tUi Ñ UuiPI be a covering of U. We already know that ˇ 0 ˇ 0 H pU, F q ÝÑ H pUi, F q iPI ź is injective, by the lemma above. Also, we know that the image of this map is actually inside Hˇ 0pU , Hˇ0pF qq “ ker d0. Hence, it suﬃces to show that it equals ker d0. Let x P Hˇ 0pU , Hˇ0pF qq be ˇ 0 ˇ 0 given. Write xi for its component in H pUi, F q. Since xi P H pUi, F q, there is ˇ 0 a covering Vi “ tVik Ñ UiukPKi of Ui such that xi P H pVi, F q.

2 Now for pi, jq P I , let πij : Uij Ñ Ui be the ﬁrst projection, where Uij denotes Ui ˆU Uj. To ease notation, we shall identify Uij with Uji. We will also write

Vij “ tVijk Ñ UijukPKi for the covering Vi ˆU Uj “ tVik ˆU Uj Ñ UijukPKi of Uij.

Now let pi, jq P I2. We know that the elements

π 0 x Hˇ 0 , p ijqVi,Vij p iq P pVij F q π 0 x Hˇ 0 , p jiqVj ,Vji p jq P pVji F q

ˇ 0 0 become equal in H pUij, F q, since x P ker d .

110 Since the morphism Viki ˆU Vjkj Ñ U factors through Uij, in particular the ˇ 0 images of xi and xj in H pViki ˆU Vjkj , F q are the same.

Since each Vi covers Ui and the Ui cover U, the axioms of a site imply that

V “ tVik Ñ Ui Ñ UuiPI,kPKi is a covering of U. Since any xi is an element of ˇ 0 ˇ0 H pVi, F q Ď C pVi, F q “ F pVikq, kPK źi we can construct an element ˇ0 z P C pV , F q “ F pVikq iPI kźPKi by setting zik “ pxiqk. We have a commutative diagram

F pVikq F pViki ˆU Vjkj q iPI pi,jqPI2 kPKi ź kiźPKi kj PKj

ˇ 0 ˇ 0 H pVik, F q H pViki ˆU Vjkj , F q. iPI pi,jqPI2 kPKi ź kiźPKi kj PKj induced by the morphism F Ñ Hˇ0pF q of presheaves. Moreover, both vertical arrows are injective since F is separated. But z maps to 0 in the lower right group, hence it is already 0 in the upper right group. Hence, z P Hˇ 0pV , F q.

Hence, z gives an element of Hˇ 0pU, F q. It remains to show that the image of z ˇ 0 in H pUi, F q equals xi for each i P I. By deﬁnition, this image is given by

ˇ 0 zjk P H pV Û Ui, F q “ F pVjk Û Uiq. VjkÛ Ui jPI,kPK j jPI ´ ˇ ¯ kźPKj ˇ Since Hˇ0pF q is separated, the map ˇ 0 ˇ 0 H pUi, F q Ñ H pVik, F q kPK źi ˇ 0 is injective. Now in each H pVik, F q, the image of z equals the image of xi, ˇ 0 since xi and xj become the same in H pUij, F q for all j P I.

Corollary B.2.25. Let F be a presheaf. Then Hˇ0pHˇ0pF qq is a sheaf.

Proof. By Lemma B.2.23, Hˇ0pF q is separated, hence by Proposition B.2.24, Hˇ0pHˇ0pF qq is a sheaf.

111 B.3 Sheaﬁﬁcation

Definition B.3.1. Let F be a presheaf. Then we denote by F ` the sheaf Hˇ0pHˇ0pF qq. It is called the sheaf associated to F , or the sheafification of F .

Remark B.3.2. There is a morphism of presheaves F Ñ F `. It is easy to check that this is natural in F , so sheaﬁﬁcation becomes a functor.

Note also that if F is a sheaf, then F Ñ F ` is an isomorphism. Indeed, F Ñ Hˇ0pF q is an isomorphism, hence Hˇ0pF q is a sheaf so also Hˇ0pF q Ñ F ` is an isomorphism.

We will ﬁrstly prove a couple of easy but useful lemmata.

Lemma B.3.3. Let f : F Ñ G be an injective morphism of presheaves, and let f 1 : Hˇ0pF q Ñ Hˇ0pG q be the induced morphism. Then f 1 is injective.

Proof. Let U P ob C , and let s P Hˇ 0pU, F q such that f 1psq “ 0. Then s is of the form ˇ 0 s “ psiqiPI P H pU , F q Ď F pUiq iPI ź 1 for some covering U “ tUi Ñ UuiPI of U. Then f psq is given by

1 ˇ 0 f psq “ pfpsiqqiPI P H pU , G q Ď G pUiq. iPI ź Hence, each fpsiq is zero, so by injectivity of f, every si is zero, hence s “ 0.

Lemma B.3.4. Let F be a presheaf and G be a sheaf, and let f : F Ñ G be a morphism of presheaves. Let ρ: F Ñ Hˇ0pF q be the natural morphism. Then

ker ρ Ď ker f.

In particular, if f is injective, then F is separated.

Proof. By naturality of F Ñ Hˇ0pF q, we have a commutative diagram

ρ F Hˇ0pF q f G Hˇ0pG q.

The bottom arrow is an isomorphism since G is a sheaf, hence the result follows. The last statement follows since ker ρ “ 0 in that case.

Lemma B.3.5. Let F be a presheaf, and let ρ: F Ñ Hˇ0pF q be the natural morphism. Let G be a sheaf, and g : Hˇ0pF q Ñ G a morphism such that gρ “ 0. Then g “ 0.

112 Proof. Let U P ob C be given, and let s P Hˇ 0pU, F q. Then s is of the form ˇ 0 s “ psiqiPI P H pU , F q Ď F pUiq iPI ź for some covering U “ tUi Ñ UuiPI of U. Then gpρpsiqq “ 0 for all i P I. Now s|Ui is given by ˇ 0 s “ sj P H pU Û Ui, F q. Ui Uj Û Ui jPI ˇ ´ ˇ ¯ By definition of Hˇ 0ˇpU , F q, thisˇ is equal to

si Uj ˆU Ui jPI ´ ¯ ˇ 0 ˇ ˇ 0 which is just the image of si P H pUˇ0, F q under the natural map H pU0, F q Ñ ˇ 0 H pU ˆU Ui, F q, where U0 “ tUi Ñ Uiu is the trivial covering of Ui.

That is, ρpsiq “ s|Ui . Hence, gpsq|Ui “ gpρpsiqq “ 0. Since the Ui cover U and since G is a sheaf, this forces gpsq “ 0.

We now come to one of the main theorems about sites.

Theorem B.3.6. Sheaﬁﬁcation p´q` : PShpC q Ñ ShpC q is a left adjoint of the inclusion ShpC q Ñ PShpC q.

Proof. We have to prove that there is a natural bijection ` HomPShpF , G q – HomShpF , G q, for any presheaf F and any sheaf G . Let f : F Ñ G be a morphism of presheaves. By naturality of F Ñ Hˇ0pF q, we have a commutative diagram

F Hˇ0pF q F ` f G Hˇ0pG q G `, of which the arrows of the bottom row are isomorphisms since G is a sheaf. Hence, every morphism F Ñ G factors through f ` : F ` Ñ G . By applying the lemma above (twice), we ﬁnd that this factorisation is unique.

Naturality in F and G is a formal consequence of naturality of F Ñ F `.

Corollary B.3.7. The category ShpC q is complete, and limits are just pointwise limits.

Proof. We note that the sheaf condition says that something is a kernel. Since limits commute with limits, this shows that the presheaf limit of a diagram D : J Ñ ShpC q is in fact a sheaf, so ShpC q is complete.

On the other hand, the inclusion functor ShpC q Ñ PShpC q is a right adjoint. Hence, it preserves limits, so limits are just pointwise by Corollary B.1.14.

113 Corollary B.3.8. A morphism f : F Ñ G of sheaves is a monomorphism if and only if fpUq: F pUq Ñ G pUq is injective for all U P ob C .

Proof. This follows from the description of a monomorphism as a limit, cf. Corol- lary B.1.15.

Remark B.3.9. Note that similar statements about colimits and epimorphisms do not hold. In particular, for an epimorphism f : F Ñ G of sheaves, the maps fpUq: F pUq Ñ G pUq are in general not surjective! Therefore, it of the greatest importance to indicate in which category we work (PShpC q or ShpC q).

Corollary B.3.10. The category ShpC q is cocomplete. Moreover, for a diagram D : J Ñ ShpC q, the colimit is the sheaﬁﬁcation of the colimit in PShpC q.

Proof. This follows since left adjoints preserve colimits.

Lemma B.3.11. Let f : F Ñ G be a morphism of sheaves. Let H be the presheaf cokernel of f, that is,

H pUq “ G pUq{fpF pUqq for all U P ob C . Then f is an epimorphism in ShpC q if and only if H ` “ 0.

Proof. The sequence F Ñ G Ñ H Ñ 0 is exact in PShpC q. Let K be a sheaf, then left exactness of Homp´, K pUqq for all U P ob C gives a short exact sequence of (not necessarily small) abelian groups

f ˚ 0 Ñ HomPShpH , K q Ñ HomPShpG , K q Ñ HomPShpF , K q.

By the sheaﬁﬁcation adjunction, we can also describe this short exact sequence as ˚ ` f 0 Ñ HomShpH , K q Ñ HomShpG , K q Ñ HomShpF , K q. Now f is epic if and only if f ˚ is injective for any sheaf K . This is equivalent to ` HomShpH , K q “ 0 for all sheaves K , which is in turn equivalent to H ` “ 0.

Remark B.3.12. One would be tempted to just use that f is an epimorphism if and only if its sheaf cokernel is 0. This follows immediately once we know that ShpC q is an abelian category. However, this is exactly what we are trying to prove, which is why a diﬀerent argument is needed.

Corollary B.3.13. Let f : F Ñ G be a morphism of sheaves. Then f is an epimorphism in ShpC q if and only if for each U P ob C and for each s P G pUq, there exists a covering U “ tUi Ñ UuiPI of U such that each s|Ui is in the image of fpUiq.

114 Proof. Let H be the presheaf cokernel of f, as in the lemma. Since Hˇ0pH q is separated, the morphism Hˇ0pH q Ñ H ` is injective. Hence, if H ` “ 0, then Hˇ0pH q “ 0. Since H ` is the sheaﬁﬁcation of Hˇ0pH q, the converse is obvious.

Hence, H ` “ 0 if and only if Hˇ0pH q “ 0. But the latter is exactly equivalent to the property stated.

Lemma B.3.14. Let f : F Ñ G be a monomorphism of sheaves. Then its presheaf cokernel H is separated.

Proof. Let U P ob C be given, and let U “ tUi Ñ UuiPI be a covering of U.

Let s P H pUq be such that s|Ui “ 0 for all i P I. Let t P G pUq represent s. 1 Then each t|Ui is in the image of f, say t|Ui “ ti. For i, j P I, it holds that

1 1 f t ´ t “ ti ´ tj “ 0, i UiÛ Uj j UiÛ Uj UiÛ Uj UiÛ Uj ´ ¯ ˇ ˇ1 ˇ ˇ hence by injectivityˇ of f, the tˇi satisfy the glueingˇ condition.ˇ Hence, there exists 1 1 1 1 t P F pUq with t |Ui “ ti for all i P I. Then fpt q|Ui “ ti for all i P I, hence by uniqueness of the glueing condition, fpt1q “ t. Hence, s “ 0, so the map

H pUq Ñ H pUiq iPI ź is injective. By Lemma B.2.20, this is what we needed to prove.

Corollary B.3.15. The category ShpC q is balanced.

Proof. Let f : F Ñ G be both monic and epic. Since it is monic, we know that each fpUq: F pUq Ñ G pUq is injective. On the other hand, we know that its presheaf cokernel H is separated. Hence, it injects into the sheaﬁﬁcation H `, which is zero by Lemma B.3.11. Hence, H is already 0, so each fpUq is an isomorphism, hence f is an isomorphism.

Theorem B.3.16. The category ShpC q is an abelian category.

Proof. We can clearly enrich ShpC q in abelian groups: if f, g : F Ñ G are two morphisms, we deﬁne f ` g : F Ñ G by

ppf ` gqpUqq psq “ pfpUqq psq ` pgpUqq psq for U P ob C , s P F pUq. One easily checks that composition becomes bilinear, so indeed ShpC q is enriched in (not necessarily small) abelian groups.

Since Ab has a terminal object 0 and limits in ShpC q are pointwise, the constant presheaf F deﬁned by F pUq “ 0 is a sheaf, and it is the terminal object in ShpC q. Since 0 is initial in Ab and the constant presheaf 0 is already a sheaf, it is initial in ShpC q by Corollary B.3.10. Hence, ShpC q has a zero object 0.

Clearly ShpC q has binary products, so since it is enriched in abelian groups, it has binary biproducts.

115 Since ShpC q is complete and cocomplete, every arrow has a kernel and a cokernel. So it remains to prove that every monomorphism is a kernel and every epimorphism is a cokernel.

Firstly, let f : F Ñ G be a monomorphism of sheaves. Then for each U P ob C , the map fpUq: F pUq Ñ G pUq is injective, so we can define the presheaf quotient H by H pUq “ G pUq{F pUq. It is the cokernel of f in PShpC q, since this holds pointwise. Then H ` is the cokernel of f in ShpC q. Let g : G Ñ H be the quotient morphism of presheaves, and h` : H Ñ H ` the sheafification morphism. Let g` “ h` ˝g, then we want to show that F “ ker g`. Note that clearly F “ ker g Ď ker g`.

Conversely, let U P ob C be given, and let s P ker g`pUq. Then gpsq becomes 0 in H `. By injectivity of Hˇ0pH q Ñ H `, in fact gpsq has to become 0 in ˇ0 H pH q. Hence, there exists a covering U “ tUi Ñ UuiPI of U such that gpsq|Ui “ 0 for all i P I. Hence, si :“ s|Ui P F pUiq for all i P I.

By the surjectivity criterion of Corollary B.3.13, this says exactly that F Ñ ker g` is an epimorphism. Hence, since ShpC q is balanced, F Ñ ker g` is an isomorphism, so F is the kernel of its cokernel.

Finally, let f : F Ñ G be an epimorphism of sheaves. Let e: E Ď F be the kernel of f, and let h: F Ñ H be the presheaf cokernel of e, that is,

H pUq “ F pUq{E pUq for all U P ob C . Since fe “ 0, we get a morphism of presheaves g : H Ñ G such that gh “ f. Let g` : H ` Ñ G be the associated morphism of sheaves, and note that H ` is the sheaf cokernel of e. Write h` : F Ñ H ` for the composition h F Ñ H Ñ H `. Now g is injective since E pUq is the kernel of fpUq, by deﬁnition. Hence, by applying Lemma B.3.3 twice, we see that g` is injective. On the other hand, we have f “ g`h`, so g` is epic since f is. Hence, as ShpC q is balanced, g` is an isomorphism, and f is the cokernel of its kernel.

Proposition B.3.17. The functor p´q` : PShpC q Ñ ShpC q is exact.

Proof. It is obviously additive. It is right exact since p´q` is the left adjoint of the inclusion ShpC q Ñ PShpC q. On the other hand, it preserves monomorphisms by Lemma B.3.3 (applied twice). Hence, it is exact.

B.4 The étale site

In this section, we will give the main examples of sites we will study. Besides the étale site, the two main examples are the Zariski site and the fppf site.

116 Deﬁnition B.4.1. Let X be a scheme. Then the category Et´ {X is the full subcategory of Sch{X of schemes f : U Ñ X over X for which the structure morphism f is étale.

Lemma B.4.2. Every morphism in Et´ {X is étale.

Proof. A morphism pU, fq Ñ pV, gq of schemes f : U Ñ X, g : V Ñ X over X is just a morphism φ: U Ñ V such that gφ “ f. Since f is étale and g is unramiﬁed, the result follows from Corollary 1.1.9.

We suggestively introduce the following.

Deﬁnition B.4.3. Let X be a scheme, and let U be a scheme over X. Then a family tUi Ñ UuiPI of morphisms to U (in Sch{X) is called an étale covering of U if all the maps Ui Ñ U are étale, and moreover the union of their images is all of U (we will say that the morphisms Ui Ñ U are jointly surjective).

Remark B.4.4. We want to use this deﬁnition to deﬁne a Grothendieck pre- ´ topology on Et{X. A priori, we need to restrict to all coverings tUi Ñ UuiPI for which the structure morphisms Ui Ñ X are étale. However, they are given by the compositions Ui Ñ U Ñ X, so they are automatically étale over X when U is.

Lemma B.4.5. Let tUi Ñ UuiPI be an étale covering of a scheme U. Let V Ñ U be any morphism. Then

tUi ˆU V Ñ V uiPI is an étale covering of V .

Proof. We write Vi for Ui Û V . Write U8 for i Ui, and V8 for i Vi. Then V8 is the fibred product U8 Û V , by the construction of the fibred product. š š Note that tUi Ñ Uu is a covering if and only if U8 Ñ U is surjective.

By Lemma 1.1.7, each Vi Ñ V is étale. By Lemma 1.2.1, V8 Ñ V is surjective since U8 Ñ U is. Hence, tVi Ñ V u is a covering.

Proposition B.4.6. Let X be a scheme. Then the collections tUi Ñ UuiPI that are étale coverings of U deﬁne a Grothendieck pretopology on Et´ {X.

Proof. If U0 Ñ U is an étale morphism and V Ñ U is any morphism of schemes étale over X, then the fibred product U0 Û V exists in Sch{X. Moreover, by Lemma 1.1.7, the morphism U0 Û V Ñ V is étale. Since the structure morphism V Ñ X was étale by assumption, Lemma 1.1.6 asserts that the composite morphism U0 Û V Ñ V Ñ X is étale as well.

117 But this is the structure morphism of U0 ˆU V , since the diagram

U0 ˆU V V

U0 U

´ commutes. Hence, U0 Û V is an object of Et{X. It is clearly the fibred product ´ of U0 and V along U in this category as well, since the inclusion Et{X Ñ Sch{X is full (and faithful). ´ Now let U “ tUi Ñ UuiPI be a covering of U P obpEt{Xq, and let V Ñ U be ´ a morphism in Et{X. Then tUi Û V Ñ V uiPI is an étale covering of V , by Lemma B.4.5.

If tUi Ñ UuiPI is a covering of U, and for each i P I we have a covering tUij Ñ UiujPJi , then clearly all the maps Uij Ñ U are étale. Since the Uij Ñ Ui are jointly surjective and the Ui Ñ U are, so are the Uij Ñ U. Hence, the family

tUij Ñ Ui Ñ UuiPI,jPJi is a covering of U.

„ Finally, it is clear that the one object family tV Ñ Uu is a covering of U.

Deﬁnition B.4.7. Let X be a scheme. Then the (small) étale site X´et is the category Et´ {X endowed with the Grothendieck pretopology described above.

Remark B.4.8. Despite the name, it is not a small category. The word small is included to distinguish it from the big étale site, which is given by the same Grothendieck topology, but with underlying category Sch{X. We will not study this site in more detail. We do note that the proof that it is indeed a site is the same as the proof above, except that existence of ﬁbred products are automatic.

Definition B.4.9. Let X be a scheme, and let U be a scheme over X. Then a family tUi Ñ UuiPI of morphisms to U (in Sch{X) is called an fppf covering of U if all the maps Ui Ñ U are flat and locally of finite type, and moreover the morphisms Ui Ñ U are jointly surjective.

Remark B.4.10. The term fppf is short for fidèlement plat de présentation finie, which is French for ‘faithfully flat of finite presentation’. The faithful part refers to the fact that the Ui Ñ U are jointly surjective. Since we assume all schemes to be locally Noetherian, a morphism is locally of finite presentation if and only if it is locally of finite type.

Proposition B.4.11. Let X be a scheme. Then the collections tUi Ñ UuiPI that are fppf coverings of U P obpSch{Xq deﬁne a Grothendieck pretopology on Sch{X.

Proof. Analogous to Proposition B.4.6.

118 Deﬁnition B.4.12. The category Sch{X together with the Grothendieck topology given by fppf coverings is called the big fppf site. It is denoted Xfppf .

Remark B.4.13. One does not usually define a small fppf site. One reason for this is that there is no analogue of Corollary B.4.2, since a morphism U Ñ V of schemes that are flat over X is not necessarily flat. For example, if X “ k is 1 (the spectrum of) a field, then both k and Ak are flat over k, but the morphism 1 1 k Ñ Ak mapping the single point to the origin in Ak is not flat.

Some authors write pSch{Xqfppf for the big fppf site, to indicate that its deﬁ- nition is not analogous to the small étale site, but rather to the big étale site, which is usually denoted pSch{Xq´et. Since we will only use the small étale and big fppf site, we will not make this distinction.

Remark B.4.14. Recall that the (small) Zariski site, from Deﬁnition B.1.10, is deﬁned as the small category

ToppXq of open sets on X, together with the Grothendieck topology given by coverings in the classical, topological sense: a covering of U Ď X is a family tUi Ď UuiPI such that the union of all the Ui is U.

We will denote this site by XZar. As opposed to the two other sites we are considering, its underlying category is actually a small category, being a full subcategory of the power set of X, viewed as poset.

Remark B.4.15. Note that the underlying categories of XZar, Xét and Xfppf have fibred products. For XZar, they are given by the intersection (which is also the scheme theoretic fibred product). For Xét, this is proven in Proposition B.4.6, bearing in mind that any morphism in Et´ {X is étale (in order to assert that the fibred product is again étale over X). For Xfppf , it is just the fibred product in Sch{X, which coincides with the fibred product in Sch.

Hence, not only do fibred products exist in the categories XZar, Xét and Xfppf , but they are also preserved and reflected by the inclusion functor to Sch{X. In particular, when writing a fibred product in any of the above categories, it will be understood as the fibred product in the category of schemes.

B.5 Change of site

Deﬁnition B.5.1. Let u: C Ñ D be a functor. Then the functor

PShpDq Ñ PShpC q F ÞÑ F ˝ uop is denoted up. Here, uop : C op Ñ D op denotes the opposite functor of u.

Lemma B.5.2. Let u: C Ñ D be a functor. Then up preserves all limits and colimits.

119 Proof. We will prove the statement about limits; the one about colimits follows similarly.

Let F be a limit of a diagram D : J Ñ PShpDq. Since limits in presheaf 1 categories are pointwise, this implies that F pU q is the limit of evU 1 D for all 1 p U P ob D. Hence, u pF qpUq “ F pupUqq is the limit of evupUq D for all U P ob C , so uppF q is the limit of upD.

Corollary B.5.3. The functor up is exact.

Proof. It preserves ﬁnite limits and colimits.

p In what follows, we will construct a left adjoint up for u , under certain conditions on u.

Deﬁnition B.5.4. Let u: C Ñ D be a functor, and let A P ob D. Then we write IA for the comma category pA Ó uq. That is, an object of IA is an object U P ob C together with a morphism f : A Ñ upUq, and a morphism φ: pU, fq Ñ pV, gq is a morphism φ: U Ñ V making commutative the diagram

A f g

upUq upV q. upφq

Deﬁnition B.5.5. Let F be a presheaf on C . For A P ob D and pU, fq P ob IA, deﬁne DA,F pU, fq “ F pUq.

If φ: pU, fq Ñ pV, gq is a morphism in IA, then deﬁne

DA,F φ: DpV, gq Ñ DpU, fq as the restriction F pV q Ñ F pUq deﬁned by φ. This clearly deﬁnes a functor

op DA,F : IA Ñ Ab.

The colimit of this diagram is denoted uppF qpAq. We will drop the F from the subscript when it causes no confusion.

Remark B.5.6. The careful reader should convince himself that the category IA is equivalent to a small category for each of the sites we study. Hence, the limit can be seen as a small limit, and is thus well-deﬁned.

Deﬁnition B.5.7. If a: A Ñ B is a morphism in D, then there is a functor

IB Ñ IA deﬁned on objects pU, fq P ob IB by pU, faq, and on morphisms φ: pU, fq Ñ pV, gq by φ, viewed as morphism pU, faq Ñ pV, gaq.

120 In particular, for each pU, fq P ob IB, we get a morphism

D U, f u A colim D V, g Bp q ÝÑ ppF qp q “ op Ap q pV,gqPIA by viewing DBpU, fq as DApU, faq.

Lemma B.5.8. The maps DBpU, fq Ñ uppF qpAq make uppF qpAq into a cocone under DB.

Proof. Given a morphism φ: pU, fq Ñ pV, gq in IB, we also have a morphism φ: pU, faq Ñ pV, gaq in IA. Hence, the diagram

resφ DApU, faq DApV, gaq

uppF qpAq commutes.

Corollary B.5.9. There is a unique homomorphism

uppF qpBq Ñ uppF qpAq deﬁned on DBpU, fq by the map DApU, faq Ñ uppF qpAq. Moreover, these maps make uppF q a presheaf on D.

Proof. Only the last statement is new. But functoriality in A follows from the uniqueness statement.

One easily sees that this construction is functorial in F . Hence, we obtain a functor up : PShpC q Ñ PShpDq.

p Theorem B.5.10. The functor up is a left adjoint for u .

Proof. Let F be a presheaf on C . Let U P ob C . Then pU, 1q is an object of IupUq, hence it gives rise to a morphism

p F pUq Ñ uppF qpupUqq “ pu upF qpUq.

A simple inspection shows that these maps are compatible for diﬀerent U P ob C , hence we get a morphism of presheaves

p ηF : F Ñ u upF .

Conversely, let G be a presheaf on D. Let A P ob D, and let pU, fq P ob IA. Then we get a morphism

resf : G pupUqq Ñ G pAq.

121 By the deﬁnition of the category IA, these maps are compatible for diﬀerent pU, fq P ob IA. Hence, they make G pAq into a cocone under DA,upG . Hence, there is a unique morphism

p pupu G qpAq Ñ G pAq given by the resf . One checks that this is natural in A, giving a morphism of presheaves p εG : upu G Ñ G . The constructions above are clearly natural in F and G , so η and ε are natural transformations. In order to check that they are the unit and counit of an adjunction, one needs to check commutativity of the following two diagrams:

u η η p p F p p u G p p upF upu upF u G u upu G

εu p pF u εG 1 1 p upF u G .

We omit the veriﬁcation.

Corollary B.5.11. The functor up is right exact.

Proof. Any left adjoint is right exact.

We want to know in which cases up is also left exact. Since it is deﬁned by a colimit, it is natural to ask whether that colimit is a direct limit.

Lemma B.5.12. Suppose C has and u preserves ﬁnite limits. Let A P ob D. Then IA is coﬁltered.

Proof. Since C has ﬁnite limits, in particular it has a terminal object T . More- over, upT q is terminal in D. Hence, there exists a unique f : A Ñ upT q. Hence, pT, fq is an object of IA, so IA is nonempty.

Now let pU, fq, pV, gq be objects of IA. We can form the product U ˆ V in C , and we know that upU ˆ V q “ upUq ˆ upV q. In particular, we get a morphism

f ˆ g : A ÝÑ upU ˆ V q, with projections π1 : upU ˆ V q Ñ upUq, π2 : upU ˆ V q Ñ upV q such that π1 ˝ pf ˆ gq “ f and π2 ˝ pf ˆ gq “ g.

Hence, we have an object pU ˆ V, f ˆ gq in IA, together with morphisms

π1 : pU ˆ V, f ˆ gq ÝÑ pU, fq

π2 : pU ˆ V, f ˆ gq ÝÑ pV, gq.

Finally, let a, b: pU, fq Ñ pV, gq be a pair of parallel morphisms in IA. Then let w : W Ñ U be the equaliser of a and b in C . Then upW q is the equaliser of upaq and upbq.

122 Since af “ g “ bf, there exists a unique morphism

h: A Ñ upW q such that upwq ˝ h “ f. In particular, we get a morphism w : pW, hq Ñ pU, fq such that aw “ bw.

Hence, IA is coﬁltered.

Corollary B.5.13. Suppose C has and u preserves ﬁbred products and a terminal object. Then up is exact.

Proof. It is a standard result from category theory that all finite limits can be built from fibred products and a terminal object. Hence, C has and u preserves finite limits.

Hence, by the lemma above, for every A P ob D, the category IA is cofiltered. This makes uppF qpAq a filtered colimit, and we know that filtered colimits in Ab are exact. The result follows since limits (and hence exactness) in presheaf categories are pointwise.

Deﬁnition B.5.14. Let C , D be sites. Then a functor u: C Ñ D is continuous if it preserves ﬁbred products that exist in C , and for every covering tUi Ñ UuiPI of some U P ob C , the image tupUiq Ñ upUquiPI is a covering of upUq.

Example B.5.15. If X1 Ñ X is a morphism of schemes and U Ñ X is étale, 1 1 1 then by Lemma 1.1.7, the base change U “ U ˆX X Ñ X is étale as well. Hence, we get a functor

1 u: X´et Ñ X´et 1 U ÞÑ U ˆX X .

Moreover, by Lemma B.4.5, we see that, for any covering tUi Ñ Uu of U P ´ 1 1 1 obpEt{Xq, the associated family tUi ˆU U Ñ U u is also a covering of U . ´ ´ Finally, if U1,U2 P obpEt{Xq are two schemes over a third scheme U P obpEt{Xq, we have isomorphisms

1 1 pU1 Û U2q ˆX X – pU1 Û U2q Û U 1 1 – pU1 Û U q Û 1 pU2 Û U q 1 1 – pU1 ˆX X q Û 1 pU2 ˆX X q.

Hence, u preserves ﬁbred products, so u is continuous.

Example B.5.16. Similarly, if X1 Ñ X is a morphism of schemes, then it deﬁnes continuous functors

1 XZar Ñ XZar 1 Xfppf Ñ Xfppf on the Zariski and fppf sites.

123 Example B.5.17. If X is a scheme, we get inclusion functors XZar Ñ X´et Ñ Xfppf . It is clear that they preserve coverings, and they preserve ﬁbred products by Remark B.4.15. Hence, they are continuous.

In particular, if X1 Ñ X is a morphism of schemes, we also get continuous functors

1 XZar Ñ X´et 1 XZar Ñ Xfppf 1 Z´et Ñ Xfppf ,

1 1 1 obtained by the composition Xτ Ñ Xτ Ñ Xτ 1 , for τ, τ P tZar, ´et, fppfu. One 1 easily sees that it is also given by the composition Xτ Ñ Xτ 1 Ñ Xτ 1 .

Remark B.5.18. Note that in each of the given examples, the site C has and the functor u: C Ñ D preserves ﬁbred products and terminal objects. Hence, Corollary B.5.13 applies, and there is an adjunction

PShpC q ÐÝÝÑ PShpDq of exact functors.

Lemma B.5.19. Let u: C Ñ D be a continuous functor of sites. Let F be a sheaf on D. Then upF is a sheaf on C .

p Proof. Since u is the right adjoint of up, it preserves all limits. In particular, it preserves products and equalisers. Moreover, since u is continuous, it preserves ﬁbred products that exist in C and it preserves coverings. Hence, the sheaf p condition of u F on the covering tUi Ñ Uu of U P ob C is just the sheaf condition of F on the covering tupUiq Ñ upUqu of upUq.

Deﬁnition B.5.20. The restriction of up to ShpDq Ñ ShpC q is denoted us.

Deﬁnition B.5.21. The composite functor

` up p´q ShpC q Ñ PShpC q ÝÑ PShpDq ÝÑ ShpDq is denoted us.

s Theorem B.5.22. The functor us is a left adjoint of u .

Proof. This follows from the chain of adjunctions

PShpC q ÐÝÝÑ PShpDq ÐÝÝÑ ShpDq, noting that the composition from right to left lands inside ShpC q by Lemma B.5.19.

Proposition B.5.23. Suppose C has and u preserves ﬁbred products and a terminal object. Then us is exact.

124 Proof. It is clearly right exact, being a left adjoint. Moreover, the functors ShpC q Ñ PShpC q Ñ PShpDq Ñ ShpDq preserve ﬁnite limits by Theorem B.3.6, Corollary B.5.13 and Proposition B.3.17, respectively.

Deﬁnition B.5.24. A morphism of sites f : D Ñ C is a continuous functor u: C Ñ D such that us is exact.

Remark B.5.25. Note that f and u go in opposite directions. This is to emphasise the geometrical nature, as illustrated by the following examples.

Example B.5.26. Let f : X Ñ Y be a morphism of schemes. Then f defines a continuous functor u: Yét Ñ Xét as above. This gives a morphism of sites Xét Ñ Yét, which we will denote by fét. We will drop the subscript and confusingly write f if the site is understood.

For all the sites we are interested in, we indeed get a morphism of sites, according to Proposition B.5.23 and Remark B.5.18.

Deﬁnition B.5.27. Let f : D Ñ C be a morphism of sites. Then we denote by s f˚ : ShpDq Ñ ShpC q the functor u . It is called the direct image functor.

Deﬁnition B.5.28. Let f : D Ñ C be a morphism of sites. Then we denote by ´1 f : ShpC q Ñ ShpDq the functor us. It is called the inverse image functor.

Theorem B.5.29. Let f : D Ñ C be a morphism of sites. There is a natural isomorphism ´1 HomShpC qpG , f˚F q – HomShpDqpf G , F q. Moreover, f ´1 is exact.

Proof. This is a reformulation of the above.

Remark B.5.30. For the Zariski site, this is just the well-known adjunction from basic sheaf theory (cf. Hartshorne [10], Exercise II.1.18).

B.6 Cohomology

Deﬁnition B.6.1. Let A be an abelian category. Then an object I P A is injective if the functor A p´,Iq is exact.

Deﬁnition B.6.2. Let A be an abelian category. Then A has enough injectives if for each object A P ob A there exists an injective object I P ob A together with a monomorphism A Ñ I.

Deﬁnition B.6.3. Let A be an abelian category. Then an injective resolution of an object A P ob A is an exact sequence 0 Ñ A Ñ I0 Ñ I1 Ñ ..., where each Ii is injective.

125 Remark B.6.4. Suppose A has enough injectives, and let A P ob A . Write A0 “ A, and let I0 “ I. Then inductively also Ai “ Ii´1{Ai´1 injects into some injective object Ii, and we get an injective resolution 0 Ñ A Ñ I0 Ñ I1 Ñ ....

Hence, A has enough injectives if and only if every object has an injective resolution.

We recall the following procedure:

Deﬁnition B.6.5. Let A and B be abelian categories, and let F : A Ñ B be a left exact functor. Assume A has enough injectives. Then the right derived functors RiF : A Ñ B of F are deﬁned as follows:

Let A P ob A . Choose an injective resolution 0 Ñ A Ñ I0 Ñ I1 Ñ I2 Ñ ... of A. Then we get a truncated chain complex 0 Ñ I0 Ñ I1 Ñ I2 Ñ ....

Applying F to the complex, we obtain a chain complex 0 Ñ FI0 Ñ FI1 Ñ FI2 Ñ ... in B. Then we denote by pRiF qA the i-th cohomology of this chain complex.

Remark B.6.6. Since F is left exact, the sequence 0 Ñ FA Ñ FI0 Ñ FI1 is exact. Hence, FA is the kernel of FI0 Ñ FI1, which is the same thing as pR0F qA.

The standard results then show that this deﬁnition depends only on the chosen injective resolution of A up to isomorphism.

Deﬁnition B.6.7. Let A be an abelian category. Then A satisﬁes: • (AB3) if A is cocomplete; • (AB4) if A is cocomplete and direct sums are exact; • (AB5) if A is cocomplete and direct limits are exact.

Dually, A satisﬁes (AB3*), (AB4*) or (AB5*) if the abelian category A op satisﬁes (AB3), (AB4) or (AB5) respectively.

Remark B.6.8. Note that A has (AB3) if and only if it has coproducts, i.e. direct sums. This is since any abelian category has coequalisers, and arbitrary colimits can be constructed from colimits and coequalisers.

Deﬁnition B.6.9. Let A be a category. Then an object U P ob A is a generator if for every monomorphism A Ñ B in A , there exists a morphism U Ñ B that does not factor through A.

126 Theorem B.6.10. Let A be an abelian category. Suppose A satisﬁes (AB5) and (AB3*), and that there exists a generator U P ob A . Then A has enough injectives.

Proof. See Grothendieck’s T¯ohokupaper [9], Théorème 1.10.1. The same proof is also included in the Stacks Project [11], Tag 079H.

Theorem B.6.11. Let C be a site, and suppose C is equivalent to a small category. Then the category ShpC q has enough injectives.

Proof. We know that ShpC q is complete and cocomplete, by Corollary B.3.7 and B.3.10. Hence, it satisfies (AB3) and (AB3*). Moreover, (AB5) follows since colimits in ShpC q are the sheafification of the corresponding colimit in PShpC q, following Corollary B.3.10, and since Ab satisfies (AB5).

It remains to exhibit a generator for ShpC q. This is done in [15], after Lemma III.1.3.

Deﬁnition B.6.12. Let C be a site, such that C is equivalent to a small category. Let U P ob C , and let F be a sheaf on C . Then the cohomology of F on U is HipU, F q “ RiΓpU, F q. The cohomology presheaf H ipF q is deﬁned as the right derived functor of the inclusion ShpC q Ñ PShpC q. Note that

ΓpU, H ipF qq “ HipU, F q,

0 0 since the functor evU : PShpC q Ñ Ab is exact and evU H pF q “ H pU, F q.

B.7 Examples of sheaves

So far, we haven’t seen a single sheaf for the étale or fppf topologies. There is an easy way to check whether a presheaf is a sheaf:

Lemma B.7.1. Let F be a presheaf for the étale (of fppf) topology. Then F ´ is a sheaf if and only if F |UZar is a sheaf on UZar for every U P obpEt{Xq (resp. obpSch{Xq), and for any covering tV Ñ Uu with U and V both aﬃne, the sequence d0 0 Ñ F pUq Ñ F pV q ÝÑ F pV ˆU V q is exact.

Proof. It is clear that the two properties hold when F is a sheaf. Conversely, let F satisfy the two properties above.

If tVi Ñ Uu is a covering, and V “ Vi is the disjoint union, then the first condition asserts that š F pV q “ F pViq. ź 127 Moreover, V Û V is the disjoint union Vi Û Vj. Hence, in the commutative diagram š 0 F pUq F pViq F pVi Û Vjq ś ś 0 F pUq F pV q F pV Û V q, all vertical arrows are isomorphisms.

Hence, if the index set I is ﬁnite, and all the Vi as well as U are aﬃne, our second assumption on F implies that the top row of this diagram is exact, since the bottom row is.

Now let tVi Ñ Uu be an arbitrary covering. By the above, to check the sheaf f condition on tVi Ñ Uu, it suﬃces to check the sheaf condition on tV Ñ Uu, where V “ Vi is the disjoint union.

Now write Uš“ Ui as the union (not necessarily disjoint) of affine schemes Ui, ´1 and cover the inverse image f pUiq with affines Vik. Since f is flat, it is open, Ť so the image of Vik is open. Since Ui is affine, it is compact, hence there are finitely many Vik such that their images cover Ui, so we can assume that there are finitely many Vik for any i.

We have a commutative diagram

0 0 0

0 F pUq F pUiq F pUi Û Ujq i i,j ś ś 0 F pV q F pVikq F pVik Û Vjlq i k i,j k,l ś ś ś ś F pV Û V q F pVik Û Vilq. i k,l ś ś The top two rows are exact since F is a sheaf on the respective Zariski sites, and the middle column is exact since tVik Ñ Uiu is a finite covering of the affine Ui by the affines Vik. Then F pUq Ñ F pV q is injective, so F is separated.

Since F is separated, the right column is exact. If x P F pV q maps to zero in F pV ˆU V q, then a simple diagram chase shows that x must come from some element in F pUq. Hence, the left column is exact, so F is a sheaf.

Lemma B.7.2. Let f : A Ñ B be a faithfully ﬂat ring homomorphism, and let M be an A-module. Then the chain complex

0 1 1bd 1bd b2 0 ÝÑ M ÝÑ M bA B ÝÑ M bA B ÝÑ ... n n i n is exact, where the maps d “ i“0p´1q di are given by n bn bn`1 di : B ÝÑ Bř

b1 b . . . bn ÞÝÑ b0 b ... b bi´1 b 1 b bi`1 b ... b bn.

128 Proof. The standard argument shows that it is a chain complex (compare the Čech complex of Deﬁnition B.2.2).

Assume ﬁrstly that f has a retraction g : B Ñ A (that is, gf “ 1A).

Then we deﬁne hn : Bbn ÝÑ Bbn´1

b1 b ... b bn ÞÝÑ gpb1qb2 b b3 b ... b bn.

Then one easily sees that

n`1 n n´1 n h di`1 ` di h “ 0

n`1 n for all i P t0, . . . , n ´ 1u. Hence, only the term h d0 remains, so

n`1 n n´1 n n`1 n h d ` d h “ h d0 “ 1.

bn bn Hence, h is a contraction for pB q, so the same goes for 1 b h on pM bA B q.

Now in the general case, we tensor everything over A with B. Since B is faithfully ﬂat over A, the sequence of M is exact if and only if the same holds for the sequence of M bA B over B (with respect to the B-algebra B bA B). But the ring homomorphism B Ñ B bA B has a section, given by b1 bb2 ÞÑ b1b2.

Proposition B.7.3. Let f : Spec B Ñ Spec A be a faithfully flat morphism of finite type of affine schemes, and let Z be any scheme. Then the diagram

ÝÑ HompSpec A, Zq Ñ HompSpec B,Zq ÝÑ HompSpec B bA B,Zq is an equaliser diagram (in Set).

Proof. The lemma asserts that the diagram

Ñ A Ñ B Ñ B bA B is an equaliser in ModA, hence also in Set (since ModA Ñ Set preserves limits). Then it is an equaliser in Ring as well: if C Ñ B is a ring homomorphism such Ñ that the compositions C Ñ B Ñ B bA B agree, then it factors set-theoretically through A. Since A is a subring of B, the obtained map C Ñ A has to be a ring homomorphism, since C Ñ B is.

Hence, if Z “ Spec C is aﬃne, the result is true. Now for general Z, we will ﬁrstly prove that the map HompSpec A, Zq Ñ HompSpec B,Zq is injective, i.e. that Spec B Ñ Spec A is an epimorphism. Let g1, g2 : Spec A Ñ Z be such that g1f “ g2f. Since f is surjective, the topological maps g1 and g2 have to coincide.

If x P Spec A is a point, and z “ g1pxq “ g2pxq, let U be an aﬃne open ´1 ´1 neighbourhood of z. Let V Ď g1 pUq “ g2 pUq be an aﬃne open containing x; without loss of generality of the form V “ Spec Aa for some a P A.

129 Note that Spec Ba is faithfully ﬂat over Spec Aa. Since

pg1fq “ pg2fq , Spec Ba Spec Ba ˇ ˇ the above shows that g1|Spec Aˇa “ g2|Spec Aa ,ˇ since U is aﬃne. Since x was arbitrary, this shows g1 “ g2, so the map

HompSpec A, Zq Ñ HompSpec B,Zq is injective.

Now let h: Spec B Ñ Z be such that hπ1 “ hπ2, where πi : Spec B bA B Ñ Spec B is the i-th projection. Let x P Spec A be given, and let y P Spec B be in its fibre. Let z “ hpyq, and let U be an affine open neighbourhood of z. Since f is flat, it is open, so fph´1pUqq is an open neighbourhood of x.

´1 Let a P A such that Spec Aa Ď Spec A contains x and is contained in fph pUqq. Now if y1, y2 P Spec B are two points with fpy1q “ fpy2q, then the ﬁbred product ty1u ˆSpec A ty2u is nonempty, so there exists a point y P Spec B bA B with πipyq “ yi for i P t1, 2u. Hence,

hpy1q “ hπ1pyq “ hπ2pyq “ hpy2q,

´1 ´1 so y1 P h pUq if and only if y2 P h pUq. In particular,

´1 ´1 ´1 ´1 f pSpec Aaq Ď f pfph pUqqq “ h pUq.

´1 But f pSpec Aaq is Spec Ba. Then by the aﬃne case treated above, there exists ga : Spec Aa Ñ U such that

ga ˝ f “ h . Spec Ba Spec Ba ˇ ˇ By the uniqueness statement above,ˇ the restrictionsˇ of ga and ga1 have to coincide on Spec Aaa1 “ Spec AaXSpec Aa1 , so they glue to a morphism g : Spec A Ñ Z satisfying gf “ h.

Theorem B.7.4. Let S be a scheme. Let X be an S-scheme, and let G be a commutative group scheme over S. Then the presheaf F of abelian groups deﬁned by U ÞÑ HomSpU, Gq is a sheaf for the étale and fppf topologies on X.

´ Proof. For every U P obpEt{Xq (resp. obpSch{Xq), the restriction of F to UZar is a sheaf, by ‘glueing morphisms’. Moreover, if tV Ñ Uu is a one-object covering with both U “ Spec A and V “ Spec B aﬃne, then the proposition above shows that the diagram

ÝÑ F pUq Ñ F pV q ÝÑ F pV ˆU V q is an equaliser in Set. That is, the sequence

d0 0 Ñ F pUq Ñ F pV q ÝÑ F pV ˆU V q is exact, so the result follows from Lemma B.7.1.

130 Definition B.7.5. Let X be a scheme. Let F be a sheaf of OX -modules on XZar. Then define the presheaf W pF q on Xét (or Xfppf ) by

W pF qpUq “ ΓpU, f ˚F q, for any f : U Ñ X étale (or any morphism f : U Ñ X, respectively).

Theorem B.7.6. Let F be a quasi-coherent sheaf of OX -modules on XZar. Then W pF q is a sheaf on X´et (or Xfppf ).

´ Proof. Clearly its restriction to UZar is a sheaf for every U P obpEt{Xq. If tf : V Ñ Uu is a one-object covering with U “ Spec A and V “ Spec B both aﬃne, then W pF q|UZar corresponds to an A-module M. If g : U Ñ X denotes the structure map, then

W pF q “ pgfq˚F “ f ˚ W pF q . VZar UZar ˇ ´ ˇ ¯ By Hartshorne [10], Propositionˇ II.5.2(e), the latterˇ is just pM bA Bq˜. The sequence 0 Ñ M Ñ M bA B Ñ M bA B bA B is exact by Lemma B.7.2, hence W pF q is a sheaf by Lemma B.7.1.

B.8 The étale site of a ﬁeld

In this section, we will have X “ Spec K, where K is a field. If F is a sheaf on Xét and L{K is a finite separable extension, then we will simply write F pLq for F pSpec Lq. ¯ We will fix a separable closure K of K, with absolute Galois group ΓK . We denote by x the unique point in X, and we will write x¯ for Spec K¯ .

Deﬁnition B.8.1. Let F be a sheaf on X´et. Then the restriction of the functor

F : pEt´ {Xqop Ñ Ab to the subcategory consisting of Spec L for L Ď K¯ ﬁnite over K gives a functor

F : tL Ď K¯ | L{K ﬁniteu Ñ Ab.

We denote its colimit by AF .

Remark B.8.2. As the category over which the colimit is taken is a directed set, the above is just a direct limit. In particular, AF is the union of the images of F pLq in it. Since F is a sheaf, the maps F pLq Ñ F pMq associated to an extension L Ď M are injective, hence the direct limit is a union

AF “ F pLq. (B.7) LĎK¯ rL:Kďsă8

131 Now each F pLq comes with a ΓK -action, and the actions are compatible as L varies. This deﬁnes a ΓK -module structure on AF . It is a discrete ΓK -module H ¯ H by (B.7), as AF “ F pLq whenever L “ K (where H Ď ΓK denotes an open subgroup). Finally, note that the association

F ÞÝÑ AF is functorial in F , since F pLq Ñ G pLq commutes with the ΓK -actions for every ﬁnite L{K contained in K¯ .

Deﬁnition B.8.3. Let A be a discrete ΓK -module. Then deﬁne the presheaf ´ FA : Et{X Ñ Ab H ¯ H by setting FApSpec Lq “ A when L “ K for H Ď ΓK open, and

FA Spec Li “ FApSpec Liq. ˜iPI ¸ iPI ž ź Lemma B.8.4. Let A be a discrete ΓK -module. Then FA is a sheaf.

Proof. Whenever U Ñ Spec K is étale, we have U “ Spec Li for certain Li{K ¯ ﬁnite and contained in K. Hence, by the second part of the deﬁnition, FA|UZar is a sheaf. š

Now let K Ď L Ď M be a tower of ﬁnite extensions contained in K¯ . Let M 1 be 1 a ﬁnite extension of M such that M {L is Galois with group G “ tσ1, . . . , σnu. We have a commutative diagram

0 FApLq FApMq FApM bL Mq (B.8) 1 1 1 0 FApLq FApM q FApM bL M q.

1 1 Since M {L is Galois, there exists α P M such that σ1pαq, . . . , σnpαq form an L-basis for M 1. This induces an isomorphism of L-algebras

1 1 „ 1n M bL M ÝÑ M , n m m m σ m . 1 b 2 ÞÝÑ 1 ip 2q i“1

1 ÝÑ ` 1 1˘ „ 1n In particular, the compositions M ÝÑ M bL M ÝÑ M are given by m ÞÑ pm, . . . , mq and m ÞÑ pσ1pmq, . . . , σnpmqq. The bottom row of diagram (B.8) is given by 0 Ñ BG Ñ B Ñ Bn, where B “ AΓM1 . The last map is given by

b ÞÑ pb ´ σ1pbq, . . . , b ´ σnpbqq, hence its kernel is exactly BG, so the bottom row of (B.8) is exact. Since all maps in the left hand square are injective, one easily sees that this forces the top row to be exact as well. Hence, the sheaf condition is satisﬁed for the covering tSpec M Ñ Spec Lu.

132 To proceed to the general affine case, observe that if U is étale over K and affine, then (by compactness of affine schemes) U is a finite union of Spec Li. If tV Ñ Uu is a one-element covering with U and V affine, the first argument of the proof of Lemma B.7.1 deduces the sheaf condition of tV Ñ Uu from that of Spec M Ñ Spec L for various L, M. Hence, Lemma B.7.1 gives the result.

Remark B.8.5. Just like the construction F ÞÑ AF , also the construction A ÞÑ FA is functorial.

Theorem B.8.6. The functors ShpX´etq Ø ΓK ´ Mod given by F ÞÑ AF and A ÞÑ FA give an equivalence of the two categories.

Proof. We already remarked that both are indeed functors. It is clear from the deﬁnition that AFA – A for any discrete ΓK -module A, and conversely that

FAF – F for any sheaf F on X´et.

Corollary B.8.7. For any sheaf F on X´et and any i P Zě0, there is an isomorphism i i H pX´et, F q – H pK,AF q, where the right hand side is the Galois cohomology of K.

Proof. Under the correspondence F ÞÑ AF , taking global sections corresponds to taking ΓK -invariants. The result follows since Galois cohomology is deﬁned as the right derived functors of

A ÞÑ AΓK .

133 References

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