Étale Cohomology I I HÉt(X,F ) := H (XÉt,F )

Etale´ cohomology

Prof. Dr. Uwe Jannsen Summer Term 2015

Inhaltsverzeichnis

1 Introduction 1

2 Grothendieck topologies/Sites 2

3 Constructions for presheaves and sheaves 4

4 The abelian categories of sheaves and presheaves 14

4.A Representable functors, limits, and colimits 17

4.B Filtered categories 30

5 Cohomology on sites 32

6 Spectral sequences 37

7 The ´etalesite 49

8 The ´etalesite of a ﬁeld 57

9 Henselian rings 62

10 Examples of ´etalesheaves 72

11 The decomposition theorem 80

12 Cechˇ cohomology 86

13 Comparison of sites 94

14 Descent theory and the multiplicative group 98

15 Schemes of dimension 1 105

1 Introduction

In mathematics, one often looks for invariants which characterize or classify the regarded objects. Often such invariants are given by cohomology groups. This is a long standing approach in topology, where one considers singular cohomology groups Hi(X, Q) of a topological space X, which are deﬁned by explicit ‘cycles’ and ‘boun- daries’. These suﬃce to determine the genus g of a (compact) Riemann surface: If X looks topologically like a sphere with g handles:

g=1 g=2

1 then dimQ H (X, Q) = 2g. These cohomology groups can also be obtained as sheaf cohomology (of a constant sheaf). Riemann surfaces can also be regarded as complex algebraic curves, i.e., as algebraic curves over a field C of complex numbers. For any algebraic varieties X over any field k (or any scheme) one can consider sheaf cohomology with respect to the Zariski topology. This is useful for coherent sheaves, for example for the Grothendieck-Serre duality and the Riemann-Roch theorem. However, the Zariski cohomology of an algebraic variety X over C does not give the singular cohomology of a topological space X(C); this is due to the fact that this topology is much finer than the Zariski topology. Furthermore one wants to obtain an analogous topology for varieties over any field k. For fields with positive characteristic, Serre showed that there exists no cohomology theory H∗(−, Q), such that H1(X, Q) has the dimension 2g for a smooth projective curve of genus g. But Weil had postulated such a theory to show the Weil conjectures for varieties over finite fields by a fixpoint formula, as it is known in topology. The solution was found by Grothendieck, together with M. Artin, by creating the étale i cohomology. For any prime ` 6= char(k) this provides cohomology groups H (X, Q`) that have the properties postulated by Weil. With these, Deligne eventually proved the Weil conjectures.

1 2 Grothendieck topologies/Sites

Grothendieck’s approach for the ´etalecohomology (and since then for many other theories) was to leave the setting of topological spaces. He noticed that one only needs the notion of ‘coverings’ with certain properties, to deﬁne sheaves and their cohomologies, by replacing at the same time ‘open set’ by ‘object in a category’.

Deﬁnition 2.1 Let X be a category and C another category. A presheaf on X with values in C is a contravariant functor P : X → C . Morphisms of presheaves are morphisms of functors. (Here we ignore –actually non-trivial – set theory problems by assuming that the category X is small). If C is the category Ab of abelian groups (resp. the category Rg of rings, resp. ...), then one speaks of presheaves of abelian groups [for short: abelian presheaves] (resp. of rings, resp. ...).

Example 2.2 Let X be a topological space. Then one can assign to X the following category X: Objects are open sets U ⊆ X. Morphisms are the inclusions V ⊆ U. Then one can see that a presheaf in Grothendieck’s sense is just a classical presheaf: Because of the contravariant functoriality, one has an arrow P (U) → P (V ) for every inclusion V ⊆ U. The properties of a functor provide the properties of presheaves for these ‘restrictions’ resU,V .

Deﬁnition 2.3 Let X be a category.

ϕi (a): A Grothendieck topology on X consists of a set T of families (Ui → U)i∈I of morphisms in X , called coverings of T , such that the following properties hold:

(T1) If (Ui → U)i∈I in T and V → U is a morphism in X , then all ﬁbre products Ui ×U V exist, and (Ui ×U V → V )i∈I is in T .

(T2) If (Ui → U)i∈I is in T and (Vij → Ui)j∈Ji is in T for all i ∈ I, then the family

(Vij → U)i,j obtained by the compositions Vij → Ui → U is in T . ϕ (T3) If ϕ : U 0 → U is an isomorphism, then (U 0 → U) is in T . (b) A site is a pair S = (X , T ) with a category X and a Grothendieck topology T on X . One denotes the underlying category X also by Cat(S) and the topology also by Cov(S), thus S = (Cat(S), Cov(S)). Sometimes (X , T ) is called a Grothendieck topology as well.

Example 2.4 If one takes the usual coverings (Ui)i∈I of open sets U ⊆ X in example 2.2, then the corresponding families (Ui ,→ U)i∈I form a Grothendieck topology on X. Note: The ﬁbre product of open sets U ⊆ X,V ⊆ X is the intersection U ∩ V .

Deﬁnition 2.5 Let S = (X , T ) be a site, and let C be a category with products (e.g., the category of sets or of abelian groups). A presheaf

F : X → C

2 is called a sheaf (with respect to T ), if for every covering (Ui → U)i∈I in T the diagram

α Q α1 Q F (U) → F (Ui) ⇒ F (Ui ×U Uj) i α2 i,j is exact, where the arrow α1 on the right side is induced by the ﬁrst projections Ui ×U Uj → Ui and the arrow α2 is induced by the second projection Ui ×U Uj → Uj (This means that α is the diﬀerence kernel of α1 and α2, see appendix 4.A below). Morphisms of sheaves are morphisms of the underlying presheaves.

Remark 2.6 Let C be the category of sets. If, for s ∈ F (U), we denote the component of Q α(s) in F (Ui) by s|Ui and for (si) ∈ F (Ui), we denote the images of si and sj in F (Ui ×U Uj) i by si|Ui×U Uj and sj|Ui×U Uj respectively, then we literally obtain the same conditions as for the usual sheaves on topological spaces, except that we replace Ui ∩ Uj with Ui ×U Uj: The conditions are:

Deﬁnition 2.7 (a) A morphism f :(X 0, T 0) → (X , T ) of sites is a (covariant) functor f 0 : X → X 0 (!) which has the following properties: 0 ϕi 0 f (ϕi) 0 0 (S1) If (Ui −→ U) is in T , then (f (Ui) −→ f (U)) is in T .

(S2) If (Ui → U) is in T and V → U is a morphism in T , then the canonical morphism

0 0 0 f (Ui ×U V ) → f (Ui) ×f 0(U) f (V ) is an isomorphism for all i.

Example 2.8 If f : X0 → X is a continuos map between of topological spaces, we obtain a morphism f : S(X0) → S(X) of the associated sites (Example 2.4) by

f −1 : X → X0 U 7→ f −1(U) .

3 3 Constructions for presheaves and sheaves

For a category X let P r(X ) be the category of abelian presheaves on X .

Definition 3.1 (Push-forward) Let f :(X 0, T 0) → (X , T ) be a morphism of sites and let 0 0 0 0 P : X → Ab be an abelian presheaf. Then the direct image (or push-forward) fP P of P is defined as the presheaf 0 0 0 0 0 f 0 P fP P = P f : X → X → Ab . 0 0 0 0 0 0 0 Explicitly we have (fP P )(U) = P (f (U)) for U in X and fP (ϕ) = P (f (ϕ)) : P (f (U2)) → 0 0 0 0 P (f (U1)) for ϕ : U1 → U2 in X ). For a morphism ψ : P1 → P2 of abelian presheaves on X one obtains a morphism 0 0 (3.1.1) fP ψ : fP P1 → fP P2 as follows: For U in X define 0 0 (fP ψ)U :(fP P1)(U) → (fP P2)(U) q q q 0 0 0 0 ψf0(U) : P1(f (U)) → P2(f (U)) . One can see easily that this produces a morphism of presheaves (3.1.1) and that one obtains a functor 0 fP : P r(X ) → P r(X ) 0 0 P 7→ fP P ψ 7→ fP ψ .

Proposition 3.2 The functor 0 fP : P r(X ) → P r(X ) has a left adjoint f P : P r(X ) → P r(X 0) . For presheaves P ∈ P r(X ) and P 0 ∈ P r(X 0) we thus have isomorphisms P 0 ∼ 0 (3.2.1) HomX 0 (f P,P ) = HomX (P, fP P ) , functorially in P and P 0. For a presheaf P on X , f P P is called the inverse image (or pull- back) of P .

0 0 Proof of 3.2: For U in X consider the following category IU 0 : Objects are pairs (U, ψ), where U is an object in X and ψ : U 0 → f 0(U) 0 is a morphism in X . A morphism (U1, ψ1) → (U2, ψ2) is a morphism ϕ : U1 → U2 in X for which the diagram

0 (3.2.2) f (U1) x; ψ1 xx xx xx xx U 0 f 0(ϕ) FF FF FF ψ FF 2 F# 0 f (U2)

4 is commutative. Then we have a functor op P : IU 0 → Ab (3.2.3) (U, ψ) 7→ P (U) ϕ 7→ P (ϕ)

op (where IU 0 denotes the dual category of IU 0 ) and deﬁne (f P P )(U 0) = lim P (U) → (U,ψ)∈Iop U0

op P 0 as the inductive limit over IU 0 (The idea is that (f P )(U ) is the inductive limit of all sets P (U), where “U 0 is contained in f 0(U)”, see Example 3.4 below). If ϕ0 : U 0 → V 0 is a morphism in X 0, we obtain a functor

IV 0 → IU 0 ,

0 0 0 ϕ 0 by mapping an object (V,V → f(V )) in IV 0 to the object (V,U → V → f(V )), and mapping a morphism ϕ : V1 → V2 to the same morphism. This gives a morphism

(f P P )(V 0) = lim P (U) → lim P (U) = (f P P )(U 0) . → → Iop Iop V 0 U0

With this f P P becomes a contravariant functor

f P P : X 0 → Ab i.e., an abelian presheaf on X 0. Now we prove the adjointness. Let P 0 be an abelian presheaf on X 0 and let

(3.2.4) v : f P P → P 0 be a morphism of abelian presheaves. For all U in X one obtains the homomorphism

P 0 0 0 0 (3.2.5) vf 0(U) :(f P )(f (U)) → P (f (U)) = (fP P )(U) .

Furthermore the pair (U, idf 0(U)) is an object of If 0(U), and we obtain a canonical homomorphism

(3.2.6) P (U) → lim P (V ) = (f P P )(f 0(U)) , → (V,ψ)∈Iop f0(U) and by composition of (3.2.6) and (3.2.5) a homomorphism

0 (3.2.7) P (U) → (fP P )(U) , which is obviously functorial in U, so that we get a morphism of abelian presheaves on X

0 (3.2.8) w : P → fP P .

5 Conversely, consider a morphism w as in (3.2.8), and let U 0 ∈ ob(X 0). Then for every object 0 0 (U, ψ : U → f (U)) in IU 0 one has the homomorphism

0 wU 0 0 0 P (ψ) 0 0 P (U) −→ (fP P )(U) = P (f (U)) −→ P (U ) .

This homomorphism is functorial in (U, ψ) and gives a homomorphism (universal properties of the direct limit) (f P P )(U 0) = lim P (U) → P 0(U 0) , → (U,ψ)∈Iop U0 which itself is functorial in U 0 and therefore gives a morphism

v : f P P → P 0 of abelian presheaves on X 0. Finally, one easily shows that the mappings v 7→ w and w 7→ v are inverse to each other.

Remark 3.3 The same holds for presheaves with values in a category C, if all direct limits exist in C, e.g., C = Set, Rg, ....

Example 3.4 Let f : X0 → X be a continuous map of topological spaces and

f : S(X0) → S(X),U 7→ f −1(U) , the corresponding morphism of sites. Then

0 P 0 fP : P r(X ) → P r(X), f : P r(X) → P r(X )

0 0 −1 P are the usual functors. This is obvious for fP : One has (fP P )(U) = P (f (U)). For f one 0 0 obtains the usual construction: for U ⊆ X , IU 0 is the ordered set (!) of the open sets U ⊆ X with f(U 0) ⊆ U, thus U 0 ⊆ f −1(U), and f P P (U 0) = lim P (U). → f(U 0)⊆U

For a site (X , T ) let Sh(X , T ) be the category of abelian sheaves (with respect to T ) on X . We obtain a fully faithful embedding

i = iT : Sh(X , T ) ,→ P r(X ) .

Theorem 3.5 The embedding i has a left adjoint

a = aT : P r(X ) → Sh(X , T ) .

Thus for all presheaves P and all sheaves F one has isomorphisms, functorial in P and F ,

∼ HomP r(P, iF ) → HomSh(aP, F ) .

For a presheaf P , aP is called the associated sheaf (with respect to T ).

For the proof we need some preparations.

6 Deﬁnition 3.6 A reﬁnement

(Vj → U)j∈J → (Ui → U)i∈I of coverings of U is a map ε : J → I of the index sets and a family (fj)j∈J of U-morphisms fj : Vj → Uε(j).

With the reﬁnements as morphisms and the obvious compositions, we obtain the category T (U) of the coverings of U (with respect to the topology T ).

Deﬁnition 3.7 Let U in X and P be an abelian presheaf on X .

(a) For every covering U = (Ui → U) in T

α ˇ 0 Q 1 Q H (U,P ) = ker( P (Ui) ⇒ P (Ui ×U Uj)) i α2 i,j ˇ is called the zeroth Cech cohomology of P with respect to U. Here let α1 and α2 be deﬁned as in Deﬁnition 2.5. (b) Call Hˇ 0(U, P ) = lim Hˇ 0(U,P ) → U the zeroth Cechˇ cohomology of P for U, where the direct limit runs over the category T (U)op.

Remark 3.8 A presheaf P on X is a sheaf for T if and only if for all U in X and all U = (Ui → U) in T (U) the canonical homomorphism P (U) → Hˇ 0(U,P ) is an isomorphism. In this case P (U) → Hˇ 0(U, P ) is an isomorphism as well.

Proof of Theorem 3.5 Let P be an abelian presheaf on X . For U in X deﬁne

P˜(U) := Hˇ 0(U, P ) .

This produces a presheaf, since for ϕ : V → U in X we have a canonical homomorphism

(3.5.1) ϕ∗ : Hˇ 0(U, P ) → Hˇ 0(V,P ) , because for every covering U = (Ui → U) of U we obtain the covering UV := (Ui ×U V → V ) of V , thus an induced homomorphism

ˇ 0 ˇ 0 (3.5.2) H (U,P ) → H (UV ,P ) , and by passing to the limit over the coverings in U we obtain (3.5.1). A morphism of abelian presheaves ψ : P1 → P2 induces a canonical morphism of presheaves ˜ ˜ ˜ (3.5.3) ψ : P1 → P2

7 as follows: For every covering U = (Ui → U), ψ induces a homomorphism

ˇ 0 ˇ 0 (3.5.4) H (U,P1) → H (U,P2) .

This is compatible with reﬁnements and by passing to the limit over T (U)op gives a map

˜ ˇ 0 ˇ 0 (3.5.5) ψU : H (U, P1) → H (U, P2) .

For every morphism ϕ : V → U, the diagram

˜ ˇ 0 ˇ 0 ψU : H (U, P1) / H (U, P2)

ϕ∗ ϕ∗ ˜ ˇ 0 ˇ 0 ψV : H (V,P1) / H (V,P2)

is commutative. This gives (3.5.3). One can easily see that this deﬁnes a functor

P r(X ) → P r(X ) P 7→ P˜ ψ 7→ ψ˜ .

Deﬁnition 3.9 A presheaf P is called separated with respect to T , if for every covering (Ui → U) in T the homomorphism Q P (U) → P (Ui) i

ˇ 0 is injective. (Equivalently, P (U) → H ((Ui → U),P ) is injective).

Lemma 3.10 (a) If P is an abelian presheaf, then P˜ is separated. (b) There exists a canonical morphism P → P˜. (c) If P is a separated abelian presheaf, then P → P˜ is a monomorphism and P˜ is a sheaf. (d) If F is a sheaf, then F → F˜ is an isomorphism.

Preliminary Remark for the Proof: We will see later (see 3.11 and 3.12): ˇ 0 1) For every element s ∈ H (U, P ) there exists a covering U = (Ui → U) in T (U) and an element s ∈ Hˇ 0(U,P ) which is mapped to s under

Hˇ 0(U,P ) → Hˇ 0(U, P )

(In this case we say that s is represented by s). ˇ 0 ˇ 0 2) If s is represented by s1 ∈ H (U1,P ) and s2 ∈ H (U2,P ) (with U1, U2 ∈ T (U)), then there ˇ 0 are reﬁnements U3 → U1, U3 → U2, such that s1 and s2 have the same image in H (U3,P ).

˜ Q ˜ Proof of 3.10 (a): Let (Ui → U)i be a covering in T and let s ∈ ker(P (U) → P (Ui)). We i ˇ 0 have to show s = 0. There exists a covering (Vj → U)j and an element s ∈ H ((Vj → U)j,P ) which represents s.

8 Let si be the image of s under

ˇ 0 ˇ 0 H ((Vj → U)j,P ) → H ((Vj ×U Ui → Ui)j,P ) .

ˇ 0 This represents s|Ui = 0 ∈ H (Ui,P ). By the preliminary remark there exists a reﬁnement for every i ∈ I fi :(Wik → Ui)k → (Vj ×U Ui → Ui)j ∗ ˇ 0 such that fi maps si to 0 in H ((Wik → Ui)k,P ).

By composition of the coverings (Wik → Ui)k and (Ui → U)i (axiom (T2)) we obtain a covering (Wik → U)k and via the fi a reﬁnement

f :(Wik → U)k → (Vj → U)j .

Then, under ∗ ˇ 0 ˇ 0 f : H ((Vj → U)j,P ) → H ((Wik → U)k,P ) , s is mapped to 0 by construction. Thus s = 0. (b): This is given by the canonical homomorphisms

P (U) →∼ Hˇ 0((U →id U),P ) → Hˇ 0(U, P ) = P˜(U) .

ˇ 0 ˇ 0 Claim 3.10.1 : For every covering U = (Ui → U) in T , H (U,P ) → H (U, P ) is injective.

Proof By the preliminary remark it suﬃces to show the injectivity of

∗ ˇ 0 ˇ 0 f : H ((Ui → U),P ) → H ((Vj → U),P ) for every reﬁnement f :(Vj → U) → (Ui → U). For this consider the covering

(Vj ×U Ui → U) which is the composition of the coverings (Vj ×U Ui → Vi) and (Ui → U). It has the two reﬁnements pr2 (Vj ×U Ui → U) −→ (Ui → U)

pr1 f (Vj ×U Ui → U) −→ (Vj → U) → (Ui → U) . By the following Lemma 3.11 the two induced homomorphisms

pr∗ ˇ 0 2 ˇ 0 H ((Ui → U),P ) ⇒ H ((Vj ×U Ui → U),P ) ∗ ∗ pr1 f

∗ ∗ ∗ ∗ are equal. It thus suﬃces to prove the injectivity of pr2; then pr1f and hence also f is ∗ injective. But pr2 is the restriction of

pr∗ Q 2 Q Q P (Ui) −→ P (Vj ×U Ui) i i j

9 ˇ 0 Q to H ((Ui → U),P ), and for every i, P (Ui) → P (Vj ×U Ui) is injective, because P is j separated. If we apply Claim 3.10.1 to the covering (U → U), then the injectivity of

P (U) = Hˇ 0((U → U),P ) → Hˇ 0(U, P ) = P˜(U)

follows and therefore the ﬁrst claim of (c). ˜ Now we prove that P is a sheaf. Let (Ui → U) be a covering. We have to show that ˜ Q ˜ Q ˜ (3.10.2) P (U) → P (Ui) ⇒ P (Ui ×U Uj) i i,j is exact. By (a) P˜ is separated, thus the ﬁrst map is injective. Now let Q ˜ Q ˜ (si) ∈ ker( P (Ui) ⇒ P (Ui ×U Uj) . i i,j

ˇ 0 For each i choose a covering (Vik → Ui) and an element si ∈ H ((Vik → Ui),P ), which ˜ 1 represents si ∈ P (Ui). Let sij be the image of si under

ˇ 0 ˇ 0 H ((Vik → Ui),P ) → H ((Vik ×U Uj → Ui ×U U),P ) ,

2 and let sij be the image of sj under

ˇ 0 ˇ 0 H ((Vik → Uj),P ) → H ((Ui ×U Vik → Ui ×U Uj),P ) .

1 2 ˇ 0 The elements represented by sij and sij in H (Ui ×U Uj,P ) are equal to the images of si and 1 2 sj, respectively, and thus are equal. It follows from Claim 3.10.1 that sij and sij have the same image in ˇ 0 Q H ((Vik ×U Vj` → Ui ×U Uj),P ) ⊆ P (Vik ×U Vj`) k,` This implies that

0 Q Q ˇ 0 s = (si) ∈ ker( P (Vik) ⇒ P (Vjk ×U Vj`)) = H ((Vik → U),P ) . i,k i,k,j,`

0 ˜ 0 ˜ Q ˜ The element s ∈ P (U) represented by s is then mapped to (si) under P (U) → P (Ui). i This proves the second claim of (c). (d) follows immediately from Remark 3.8. This ﬁnishes the proof of 3.10.

Lemma 3.10 implies Theorem 3.5: If P is an abelian presheaf, then we deﬁne

≈ aP = P .

≈ By 3.10 (a) P˜ is separated, by 3.10 (c) P is a sheaf. Furthermore by 3.10 (b) we obtain a canonical morphism of abelian presheaves

≈ can : P → P˜ → P = aP .

10 If now F is an abelian sheaf and

ψ : P → F (= iF )

is a morphism of abelian presheaves, then, by functoriality of the used constructions (the assignment P 7→ P˜, the morphism P → P˜), we obtain a commutative diagram

ψ P / F

ρ1 o ψ˜ P˜ / F˜ ρ

ρ2 o ≈ Ù ≈ ψ ≈ aP = P / F = aF

where we have isomorphisms on the right hand side by 3.10 (d). Now, if we deﬁne

≈ aψ = ψ ,

then we obtain a commutative diagram

can P / aP A AA {{ AA {{ ψ AA {{ −1 0 A }{{ ρ aψ =: ψ F, in which ψ0 is unique: For this it suﬃces to show that in a commutative diagram

can P / P˜ ? ?? ?? ψ ?? µ ? F

the morphism µ is unique (By applying it twice it follows that ψ0 is unique). Because of the additivity it suﬃces to show this for ψ = 0. But if ψ = 0 and (Ui → U) is a covering in T , then the commutative diagram

ˇ 0 Q pri P (U) / H ((Ui → U),P ) / i P (Ui) / P (Ui) o µ ooo ψU oo ψ ψU =0 ooo i wooo ∼ ˇ 0 Q pri F (U) / H ((Ui → U),F ) / i F (Ui) / F (Ui)

implies that µ = 0.

Lemma 3.11 Let 0 f, g :(Uj → U) → (Ui → U) be two reﬁnements of coverings in the Grothendieck topology T . Then for every abelian presheaf P the induced maps

∗ ∗ ˇ 0 ˇ 0 0 f , g : H ((Ui → U),P ) → H ((Uj → U),P )

11 are equal.

Proof Let f = (ε, (fj)) and g = (δ, (gj)). We have a diagram

0 Q d =α1−α2 Q P (Ui) / P (Ui1 ×U Ui2 ) i i1,i2 1 ∗ ∗ ∆ ll ∗ ∗ f g lll f g lll lll ulll0 Q 0 d =α1−α2 Q P (Uj) / P (Uj1 ×U Uj2 ) , j j1,j2 where ∆1 is deﬁned by 1 (∆ s)j = P ((fj, gj)U )(sε(j),δ(j)) , with the canonical morphism

0 (fj, gj)U : Uj → Uε(j) ×U Uδ(j) .

One checks that ∆1 ◦ d0 = g∗ − f ∗ . ∗ ∗ 0 ˇ 0 Thus f and g agree on ker d = H ((Ui → U),P ), which proves the claim.

With this result we are able to understand the limit

Hˇ 0(U, P ) = lim Hˇ 0(U,P ) . → T (U)0 better: For two coverings U, U0 in T (U) call U0 finer than U (Notation U0 ≥ U), if there is a refinement f : U0 → U. Define the equivalence relation ∼ on the set ob(T (U)) of the coverings of U by U ∼ U0 ⇔ U ≤ U0 and U0 ≤ U . Then the set of the equivalent classes

T (U)0 = ob(T (U))/ ∼ becomes an ordered set, with the ordering induced by ≤. This ordering is inductive: For two coverings U = (Ui → U)i and V = (Vj → U)j there is a common reﬁnement W = (Ui ×U Uj → U)i,j with the obvious reﬁnements

U ← W → V , given by the maps i ←p(i, j) 7→ j and the projections Ui ← Ui ×U Uj → Uj; hence we have U, V ≤ W. By Lemma 3.11, for U0 ≥ U we further obtain, by choice of a reﬁnement f : U0 → U, a uniquely determined homomorphism

(3.11.1) Hˇ 0(U,P ) → Hˇ 0(U0,P ) .

12 Corollary 3.12 The zeroeth Cechˇ Cohomology

Hˇ 0(U, P ) = lim Hˇ 0(U,P ) , → T (U)0 is the inductive limit over the inductively ordered set T (U)0.

This implies the claims in the preliminary remark for the proof of Lemma 3.10.

Now we deﬁne the push-forward maps and pull-back maps for sheaves. Let

f :(X 0, T 0) → (X , T ) be a morphism of sites.

0 0 0 0 Lemma 3.13 If F is an abelian sheaf on (X , T ), then fP F is again a sheaf.

Proof: Left to the readers!

0 0 Lemma/Deﬁnition 3.14 (a) f∗F := fP F is called the direct image (or push-forward) of F 0 (with respect to f). (b) For an abelian sheaf F on (X , T ), f ∗F := af P F is called the (sheaf-theoretic) inverse image (or pull-back) of F (with respect to f). (c) The functor f ∗ = af P : Sh(X , T ) → Sh(X 0, T 0) is left adjoint to the functor

0 0 f∗ : Sh(X , T ) → Sh(X , T ) .

Proof For sheaves F 0 on (X 0, T 0) and F on (X , T ) we have canonical isomorphisms

∗ 0 ∼ P 0 ∼ P 0 HomSh(f F,F ) = HomSh(af F,F ) = HomP r(f F, iF )

∼ 0 ∼ 0 = HomP r(F, fP iF ) = HomSh(F, f∗F ) , functorial in F and F 0.

13 4 The abelian categories of sheaves and presheaves

Let X be a category.

Theorem 4.1 (a) The category P r(X ) of abelian presheaves on X is an abelian category. (b) A sequence of abelian presheaves

0 → P 0 → P → P 00 → 0

is exact, if and only if for all U ∈ ob(X ) (:= objects of X ) the sequence

0 → P 0(U) → P (U) → P 00(U) → 0

is exact in Ab.

Proof: Left to the readers!

Theorem 4.2 Let (X , T ) be a site. (a) The category Sh(X , T ) of abelian sheaves on X with respect to T is an abelian category.

(b) The kernel of a morphism ϕ : F1 → F2 of abelian sheaves is equal to the kernel of P presheaves ker ϕ (i.e., (ker ϕ)(U) = ker(ϕU : F1(U) → F2(U)) for all U in X . P (c) The cokernel of a morphism ϕ : F1 → F2 of abelian sheaves is equal to a coker ϕ, i.e., the P P sheaf associated to the presheaf cokernel coker ϕ ( deﬁned by (coker ϕ)(U) = coker(ϕU : F1(U) → F2(U)) for all U in X ).

(d) In particular, ϕ : F1 → F2 is an epimorphism in Sh(X , T ) if and only if for every U in X and every s ∈ F2(U) there is a covering (Ui → U) in T and there are sections si ∈ F1(Ui),

mapped to s|Ui by ϕ.

Proof The properties (a) - (c) follow easily from 4.1 and the universal property of the associated sheaf. For (d) we note that ϕ is an epimorphism if and only if cokerϕ = 0, i.e., P if a(coker ϕ) = 0. This means that there is a covering (Ui → U) for every U in X and P P every s ∈ (coker ϕ)(U) with s|Ui = 0 for all i. Since (coker ϕ)(Ui) = coker(ϕUi : F1(Ui) → F2(Ui)), the proposition follows.

Theorem 4.3 (a) There exist arbitrary limits (inverse limits) and colimits (direct limits) in P r(X ) and Sh(X , T ). (b) The functor i : Sh(X , T ) ,→ P r(X ) is left exact. (c) The functor a : P r(X ) → Sh(X , T ) is exact.

Proof (a) In P r(X ) we have

(limP P )(U) = lim P (U ) , ← i ← i i

14 and a similar formula for direct limits. If now (F ) is a diagram of sheaves, then limP P is i i∈I ← i i again a sheaf, because inverse limits commute with each other. Hence lim F = limP F . The ← i ← i i i direct limit is lim F = a(limP F ) , → i → i i i since we have the universal property for every sheaf G

Hom (a limP F ,G) ∼= Hom (lim F , iG) ∼= lim Hom (F , iG) ∼= lim Hom (F ,G) Sh → i P r → i ← P r i ← Sh i i

(b) It follows from the adjunction of i and a that i is left exact and a is right exact (see Lemma 4.5 below). ≈ (c) Since aP = P , it suﬃces to show that the functor P 7→ P˜ is left exact. But we have

P˜(U) = lim Hˇ 0(U,P ) , → U∈T(U)0 the functor P 7→ Hˇ 0(U,P ) is left exact, and forming an inductive limit of abelian groups is an exact functor (see also Annex 4.B).

Theorem 4.4 Let f :(X 0, T 0) → (X , T ) be a morphism of sites. 0 P 0 (a) fP : P r(X ) → P r(X ) is exact and f : P r(X ) → P r(X ) is right exact. (b) If ﬁnite limits exist in X and X 0, and if f 0 : X → X 0 commutes with these, then f P is exact. 0 0 ∗ 0 0 (c) f∗ : Sh(X , T ) → Sh(X , T ) is left exact and f : Sh(X , T ) → (X , T ) is right exact. (d) If ﬁnite limits exist in X and X 0, and if f 0 : X → X 0 commutes with these, then f ∗ is exact.

P Proof (a) The exactness of fP follows from 4.1, and because of the adjunction, f is right exact (see below).

(b) It follows from the assumption that the category IU 0 (see Proof of 3.2) is cofiltered for 0 0 every object U in X (see Annex 4.B.2). In fact, in IU 0 finite limits exist by assumption: For this one has to show the existence of finite products and difference kernels. But for 0 0 0 0 0 objects U → f (U1) and U → f (U2) in IU 0 the product is the product morphism U → 0 0 0 f (U1) × f (U2) = f (U1 × U2), and for morphisms

0 f (U1) x; xx xx xx xx U 0 f 0(α) f 0(β) FF FF FF FF F# 0 f (U1)

15 β (with U1 ⇒ U2) the diﬀerence kernel is α

f 0(ker(α, β)) = ker(f 0(α), f 0(β)) jjj4 jjjj jjjj jjjj U 0 jjj TTTT TTTT TTTT TTTT TTT T* 0 f (U1)

0 op (check the universal properties!). If IU is coﬁltered, IU 0 is ﬁltered. Therefore forming the op direct limit over IU 0 is exact (see Appendix 4.B.3). 0 0 (c) Since f∗F = fP iF , the claim follows for f∗, because i is left exact (4.3(b)) and fP is ∗ ∗ exact by 4.4 (a). Furthermore f is right exact, because f is left adjoint to f∗ (see 4.5). (d) By (b), f P is exact, therefore f ∗ = af P i is left exact, because a is exact and i is left exact.

Lemma 4.5 Let A and B be abelian categories, and let F : A → B and G : B → A be functors such that G is right adjoint to F (⇔ F is left adjoint to G). Then G is left exact and F right exact.

Proof By assumption we have bi-functorial isomorphisms ∼ HomA(A, GB) = HomB(F A, B) for A ∈ ob(A) and B ∈ ob(B). (a) Let

(4.5.1) 0 → B1 → B2 → B3 → 0 be exact in B. We have to show that

(4.5.2) 0 → GB1 → GB2 → GB3 is exact. This means that the sequence

(4.5.3) 0 → HomA(A, GB1) → HomA(A, GB2) → HomA(A, GB3) is exact for all A ∈ ob(A)(⇔ GB1 is the kernel of GB2 → GB3). By adjunction, (4.5.3) is isomorphic to the sequence

(4.5.4) 0 → HomB(F A, B1) → HomB(F A, B2) → HomB(F A, B3) .

This sequence is exact by exactness of (4.5.1). (b) The right exactness of F is shown in a similar (dual) way.

16 4.A Representable functors, limits, and colimits

Let C be a category.

Deﬁnition 4.A.1 (a) A contravariant functor

F : C → Sets

is called representable, if there is an object X in C such that F is isomorphic to the contravariant Hom-functor

hX = HomC(−,X): C → Sets A 7→ HomC(A, X) ,

i.e., if there is a bijection, functorial in A,

F (A) = HomC(A, X) .

(b) A covariant functor G : C → Sets is called representable, if it is isomorphic to the covariant Hom-functor

X h = HomC(X, −): C → Sets A 7→ HomC(X,A)

for an object X in C, i.e., if there is a bijection, functorial in A,

G(A) = HomC(X,A) .

X By deﬁnition hX and h are representable. In the situation 4.A.1(a) (resp. (b)) X is called representing object for F (resp. G). The object X – if it exists – is in each case unique up to canonical isomorphism: This follows from the famous

Lemma 4.A.2 (Yoneda-Lemma) (a) If

F : C → Sets

is a contravariant functor, then one has a canonical bijection for every object X in C

∼ eX : Hom(hX ,F ) → F (X) ϕ 7→ ϕX (idX ) .

(b) If G : C → Sets is a covariant functor, then for every object Y in C one has a canonical bijection

Y ∼ eY : Hom(h ,G) → G(Y ) ϕ 7→ ϕY (idY ) .

17 a Proof (a): The inverse mX of eX assigns the following morphism mX (a) := ϕ of functors to an object a ∈ F (X)

a ϕA : hX (A) = HomC(A, X) → F (A) a f 7→ ϕA(f) := F (f)(a) . Note: For f : A → X we obtain F (f): F (X) → F (A), because F is contravariant. Note that ϕa is indeed a morphism of functors: For a morphism g : A → A0 in C we have a commutative diagram ϕa 0 A0 0 HomC(A ,X) / F (A )

g∗ F (g) a ϕA HomC(A, X) / F (A)

f 0 / F (f 0)(a) _ _

F (g)(F (f 0)(a))

f 0g / F (f 0g)(a) , since F (f 0g) = F (g) ◦ F (f 0). a a We have eX mX = id: For a ∈ F (X) we have eX (ϕ ) = ϕX (idX ) = a, because F (idX ) = idF (X). Conversely we have mX eX = id: Let ϕ : hX → F be given and let eX (ϕ) = ϕX (idX ) ∈ eX (ϕ) F (X), and let ϕ : hX → F be constructed as above. For every A ∈ ob(C) the maps

eX (ϕ) ϕA = ϕA HomC(A, X) → F (A) are equal, because eX (ϕ) ϕA (f) = F (f)(ϕX (idX )) = ϕA(f) , since the diagram ϕX idX ∈ HomC(X,X) / F (X) _ f ∗ F (f)

ϕA f ∈ HomC(A, X) / F (A) commutes (ϕ is a morphism of functors). The proof of (b) is analogous.

X By applying 4.A.2 to F = hY resp. G = h , we get the following:

Corollary 4.A.3 For objects X,Y in C one has canonical bijections ∼ HomC(X,Y ) → Hom(hX , hY )

∼ Y X HomC(X,Y ) → Hom(h , h ) Hence the representing objects are unique up to canonical isomorphism: If ∼ ∼ hX = F = hY ,

18 then by 4.A.2 we get a unique isomorphism X ∼= Y ; the same holds for the covariant functors. From 4.A.3 we get

Corollary 4.A.4 (Yoneda-embedding) The functor

C → C∼ := (contravariant functors F : C → sets) X 7→ hX is fully faithful and gives an embedding of C into C∼. The essential image is the full subcategory of the representable functors.

Now we define fiber products and fiber sums.

Deﬁnition 4.A.5 Let Y

β α X / S be morphisms in C. The ﬁber product of α and β – or of X and Y over S, notation X ×S Y , is characterized by the following properties: (a) There is a commutative diagram

pr2 X ×S Y / Y

pr1 β α X / S

(pr1 resp. pr2 are called the ﬁrst resp. second projection). (b) If α˜ W / Y

β˜ β α X / S

is another commutative diagram, then there is a unique morphism γ : W → X ×S Y with ˜ pr1γ =α ˜ and pr2γ = β, i.e., such that the diagram

W I α˜ I ∃!γ I I I $ pr2 & X ×S Y / Y β˜ pr1 β α X / S commutes.

Remark 4.A.6 (a) Fiber products do not always exist; but if they exist, they are unique up to canonical isomorphism (Exercise!).

19 β (b) In the category Sets of sets ﬁber products exist: For maps M →α T ← N of sets one has

M ×T N = {(m, n) ∈ M × N | α(m) = β(n)} .

(c) The universal property of a ﬁber product X ×S Y in a category C is equivalent to the property that for all other objects Z in C the map

HomC(Z,X ×S Y ) → HomC(Z,X) ×HomC(Z,S) HomC(Z,Y ) γ 7→ (pr1γ, pr2γ)

is bijective. Here the ﬁber product on the right is taken for the maps

HomC(Z,Y ) g . _ β∗

α∗ HomC(Z,X) / HomC(Z,S) βg

f / αf

(d) A slightly diﬀerent interpretation is given as follows: Let C/S be the category of objects in C over S: objects in C/S are objects X in C together with a morphism α : X → S; one can regard α itself as objects, because by α, X is already given. A morphism from α : X → S to α0 : X0 → S is a morphism f : X → X0, for which the diagram

f X / X0 @ @@ }} @@ } α @ }} 0 @@ }} α S ~} commutes.

Then a ﬁber product X ×S Y is the same as a product of X → S and Y → S in C/S, because the universal properties correspond.

Lemma 4.A.7 The properties of Lemma 1.14 (commutativity, associativity, transitivity and functoriality) are valid for ﬁber products in any category C (if they exist).

Proof For example we show functoriality. We have a commutative diagram in C

X Y A AA α β || AA || AA || A }|| f S g ? `A ~~ AA ~~ AA ~~ 0 0 AA ~~ α β A X0 Y 0 ,

20 i.e., X,X0,Y and Y 0 are objects over S (by α, α0, β and β0) and f and g are morphisms of objects over S (commutativity of triangles). If the ﬁber products exist, we obtain a diagram

pr2 X ×S Y / Y N N N∃!h g N N pr0 & 0 0 2 0 pr1 X ×S Y / Y

0 0 pr1 β f α0 X / X0 / S, where the internal and external square are commutative (Note that α0f = α and β0g = β). 0 0 0 0 By the universal property of X ×S Y there exists a unique morphism h : X ×S Y → X ×S Y which makes the entire diagram commutative; we call this f × g, and it fulﬁlls the claim of Lemma 1.14 (d).

` Remark 4.A.7 By reversing all arrows one obtains the notion of a ﬁber sum X S Y for a diagram β S / Y

α X where one has the dual universal properties and functorial properties.

Example 4.A.8 For every diagram of ring homomorphisms

β R / B ,

α A the ﬁber sum in the category of rings exists and is given by the tensor product: One has a commutative diagram β R / B

α A / A ⊗R B and for every diagram of rings

β R / B

α g A / A ⊗R B H H ∃!h H H f H $, C with fα = gβ there exists a uniquely determined ring homomorphism h as indicated above that makes the entire diagram commutative.

21 Now we get to the general theory of limits and colimits.

Deﬁnition 4.A.9 A category I is called small (or a diagram category), if the objects form a set.

Examples 4.A.10 Often, those small categories are “really small” in a sense that one can write all objects and morphisms in them. (a) The discrete category I over a set I has the elements of I as objects and only the identities as morphisms. (b) Let • @ @@ @@ @@ • ? ~~ ~~ ~~ • ~ be the category with three objects (marked by points) and apart from for the identities only has the two indicated arrows (then all compositions are obvious!). (c) For every group G one has the small category G with one object ∗ and all elements σ ∈ G as morphisms, where the composition is given by the group law. (d) For every ordered set (I, ≤) one has the category with objects i ∈ I and exactly one morphism i → j, if i ≤ j.

Deﬁnition 4.A.11 Let I be a small category, and let C be any category. A diagram in C over I (or a I-object in C) is a (covariant) functor X : I → C. The I-objects in C form a category CI , where the morphisms are the morphisms of functors. Often, one describes the objects of I with small letters i, j,... and writes Xi for X(i).

Example 4.A.12 Let C be a category. (a) For the category • −→ • ←− • from 4.A.6 (b) a corresponding diagram in C is given by a diagram g β X −→ Y ←− Z with morphisms α and β in C. Morphisms of such diagrams are commutative diagrams

α β X / Y o Z

α0 β0 X0 / Y 0 o Z0 .

(b) Consider the small category • −→ • (2 objects, apart from the identities only one arrow). Diagrams over this category in C are simply morphisms

f A −→ B

22 in C, where morphisms of these are commutative diagrams

f A / B

f 0 A0 / B0

This is called the category of arrows in C, notation Ar(C). (c) If (I, ≤) is an inductively ordered set, regarded as a category by 4.A.6 (d), then a covariant functor X :(I, ≤) → C is the same as an inductive system over I in C. A contravariant functor

X0 :(I, ≤) → C

is the same as a projective system in C over I.

Now let I be a small category and C any category.

Deﬁnition 4.A.13 For a object A ∈ C deﬁne the constant I-object A as the functor

A : I → C i 7→ A i → j 7→ idA .

Example 4.A.14 In Example 4.A.12 (a) I = • −→ • ←− • the constant object A is

A −→idA A ←−idA A.

Deﬁnition 4.A.15 (a) One says that the limit (or inverse limit) of an I-object (Ai)i∈I exists in CI , if the contravariant functor

C → Sets X 7→ HomCI (X, (Ai)i∈I )

is representable. The representing object is called the limit of (Ai)i∈I , notation

lim (Ai)i∈I oder lim Ai . i∈I

(b) One says that the colimit (or direct limit) of (Ai)i∈I exists, if the covariant functor

C → Sets X 7→ HomCI ((Ai)i∈I ,X)

is representable. The representing object is called the colimit of (Ai)i∈I , notation

colim (Ai)i∈I = colim Ai . i∈I

23 Now we make this elegant deﬁnition more explicit.

Remark 4.A.16 (explicit description) (a) An element of HomCI (X, (Ai)i∈I ) is obviously given by the following: (i) For every i ∈ I one has a morphism in C

ϕi : X → Ai .

(ii) For every morphism i → j in I the diagram

Ai ϕ }> i }} }} }} X A AA AA ϕj AA A Aj

commutes, where the vertical morphism belongs to i → j (one has a functor a : I → C, we write a(i) = Ai, and then the morphism on the right is a(i → j)).

(b) If lim (Ai)i∈I exists (other notation lim a), and if we ﬁx isomorphisms functorial in X I

∼ (4.A.16.1) α : HomC(X, lim (Ai)) → HomCI (X, (Ai)) , I

univ I then for X = lim (Ai)i∈I , the image of idlim(Ai) gives an element ϕ ∈ HomC (lim(Ai), (Ai)), by (a), therefore morphisms pi : lim(Ai) → Ai for all i ∈ I and commutative triangles

(4.A.16.2) A : i tt pi tt tt tt tt lim(Ai)i∈I I II II pj II II I$ Aj

for every morphism i → j in I. The morphism pi is called i-th projection. If we have an element α ∈ HomCI (X, (Ai)), i.e. morphisms ϕi : X → Ai for all i ∈ I and commutative diagrams

(4.A.16.3) Ai ϕ }> i }} }} }} X A AA AA ϕj AA A Aj

for all morphisms i → j, then there exists a uniquely determined morphism

ϕ : X → lim (Ai)

24 (the preimage of α under (4.A.16.1)) with

ϕi = piϕ

for all i (this follows from the choice of ϕuniv and the functoriality of (4.A.16.1)).

Example 4.A.17 (a) If I is a set and I the discrete category associated to I (4.A.10 (a)), then an I-object in C is simply given by a family (Ai)i∈I in C (there are no morphisms between i 6= j), and one has Q lim (Ai) = Ai , I i∈I the product of the Ai if this exists in C, because the universal properties of lim (Ai) (4.A.16 I Q (b)) and Ai are identical. Similarly, i∈I ` colim (Ai) = Ai , I i∈I

is the sum or the co-product of the Ai, if this exist in C. (b) Similarly one can show: If (I, ≤) is an ﬁltered ordered set (regarded as category), then for every I-object in C, hence every inductive system (Ai)i∈I over I in C,

colim (A ) = lim A i i∈I → i i∈I

◦ is the inductive limit of the system, if it exists. Dually, for every I -object (Ai)i∈I in C (where I◦ notes the dual category to I), hence every projective system over I in C, one sees that

lim (A ) ◦ = lim A i i∈I ← i i∈I is the projective limit of the system, if it exists. (c) Consider the category • −→ • ←− • of 4.A.10 (b). For a corresponding diagram

X / Y

in C, it follows from the universal properties that the limit is the ﬁber product

X ×Y Z

in C – if it exists.

We now look at a special, but very important example. Consider the following small category

• ⇒ •

25 (two objects, and apart from both identities only the two indicated arrows). A diagram in C is α A ⇒ B. β

Deﬁnition 4.A.18 (a) If it exists, the limit of the above diagram is called the diﬀerence kernel of α and β, notation ker(α, β) .

(b) If it exists, the colimit of this diagram is called the diﬀerence kernel α and β, notation

coker(α, β) .

Now we describe the universal properties:

Lemma 4.A.19 (a) One has a morphism

ker(α, β) →i A with αi = βi. If γ X → A is another morphism with αγ = βγ, then there is a unique morphism γ0 : X → ker(α, β) which makes the diagram i ker(α, β) / A dI ? I I ∃γ0 I γ I X commutative. (b) One has a morphism B →π coker(α, β) with πα = πβ. If

ρ B → X is another morphism with ρα = ρβ, then there is a unique morphism ρ0 : coker(α, β) → X, which makes the diagram B / coker(α, β) ? ?? s ?? s ρ ?? sρ0 ? s X ys commutative.

Proof: This follows immediately from the explicit description in 4.A.16.

Diﬀerential kernels and cokernels are non-additive analogues of kernel and cokernel in additive categories. Like there we have:

Lemma 4.A.20 Let α, β : A → B be morphisms.

26 (a) If ker(α, β) exists, then i : ker(α, β) → A is a monomorphism. (b) If coker(α, β) exists, then π : B → coker(α, β) is an epimorphism.

Proof (a): Let f, g : Z → ker(α, β) be two morphisms with if = ig. Since αi = βi we have

αif = βif as well. By the universal property of the diﬀerence kernel (4.A.19 (a)) there is a unique morphism h : Z → ker(α, β) with ih = if . Since if = ig by assumption, we deduce that h = f = g, and in particular f = g. The proof of (b) is dual.

Because of the following result, diﬀerence kernels and cokernels play a special role for limits and colimits.

Theorem 4.A.21 (a) In C, there exist arbitrary (resp. arbitrary finite) limits if and only if all difference kernels exist and all (resp. all finite) products exist in C. (b) In C there exist all (resp. all finite) colimits if and only if all difference cokernels and all (resp. all finite) sums exist in C. Here one speaks of finite limits or colimits, if the underlying index category I is finite, i.e., if it has finitely many objects and only finite sets of morphisms.

Proof We only show (a), then (b) follows by passing to the dual category, where colimits turn to limits, sums to products and difference cokernels to difference kernels. Let I be a small (resp. finite) category. Let ob(I) be a set of all objects in I and let mor(I) be the set of all morphisms in I. For a morphism f : A → B in C let s(f) := A be the source and t(f) := B the target of f. For a I-diagram a : I → C consider the morphisms Y α Y (4.A.21.1) a(i) ⇒ a(t(f)) β i∈ob(I) f∈mor(I)

deﬁned as follows (by assumption the considered products exist): The “f-component” of α (according to the universal property of the product) is the morphism

Y prt(f) a(i) −→ a(t(f)) , i∈ob(i)

the f-component of β is the morphism

Y prs(f) a(f) a(i) −→ a(s(f)) −→ a(t(f)) i∈ob(i)

27 Claim: ker(α, β) = lim a(i). i∈I

Proof: By the universal property of ker(α, β), a morphism Z → ker(α, β) corresponds to a morphism Y ϕ : Z → a(i) i∈ob(I) with αϕ = βϕ. By the universal property of the product, ϕ is given by giving morphisms

ϕi : Z → a(i) for all i ∈ I, and αϕ = βϕ means that for every f : i → j in I the diagram

(4.A.21.2) a(i) = ϕi || || || || Z a(f) B BB BB ϕj BB B! a(j)

commutes. This just gives the universal property of lim a(i) (see 4.A.16) for Z = ker(α, β), i∈I because the argument above shows that all diagrams (4.A.21.2) factorize through the diagram

a(i) t9 ttt ttt ttt ker(α, β) a(f) KK KK KK KK K% a(j) .

Theorem 4.A.22 Let C be category with finite products. The following properties are equivalent: (a) C possesses fiber products. (b) C possesses difference kernels.

f Proof (a) ⇒ (b): For A ⇒ B consider the ﬁber product diagram g

p2 K / A

p1 (id,g) (id,f) A / A × B

p Then K →2 A is the diﬀerence kernel of f and g. In fact, consider h : C → A with fh = gh. If one has (id, f)h = (h, fh) = (h, gh) = (id, g)h ,

28 then there exists a uniquely determined morphism ˜ ˜ ˜ h : C → K with p1h = h = p2h .

Furthermore, if q1 : A×B → A is the ﬁrst projection, then p1 = q1(id, f)p1 = q1(id, g)p2 = p2. Hence we obtain a uniquely determined morphism K → A with the desired property. (b) ⇒ (a): Consider a diagram (1) A . @ @@ f @@ @@ @ C ? ~~ ~~ ~~g ~~ B In the associated diagram

(2) 7 A P nnn O PPP qA nn PP f nnn pA PPP nnn 1 2 PPP nn q fpA PP nn P'/ K P / A × B 7/ C PP gpB nn PPP 3 4 nnn PPP pB nnn qB PP nn g PPP nnn P' B nn

let K be the diﬀerence kernel of fpA and gpB, and let qA = pAq and qB = qBq. Then all triangles 1 to 4 are commutative. We claim that K forms a ﬁber product of diagram (1) via qA and qB.

Since fpAq = gpBq (by assumption!), we have fqA = fpAq = gpBq = gqB. If one has further morphisms h : D → A and k : D → B with fh = gk, then for the morphism (h, k): D → A × B one has

fpA(h, k) = fh = gk = gpB(h, k) by deﬁnition. Hence, by the deﬁnition of K, there is a unique morphism ` : D → K with q` = (h, k). Then we have

qA` = pAq` = h and qB` = pBq` = k by deﬁnition. Furthermore the diagram

2 A h ; O PP q ww PP f A ww PPP ww PP ww PPP ` ww q PPP D / K / A × B ' C GG nn7 GG nnn GG nnn qB GG nn g k GG nnn ,# B nn is commutative, and the uniqueness of ` follows, because q is a monomorphism (universal property of the diﬀerence kernel).

29 4.B Filtered categories

The following deﬁnition generalizes the notion of an inductively ordered set and an inductive limit. The dual terms are treated accordingly.

Deﬁnition 4.B.1 (a) A category I is called pseudo-ﬁltered, if the following holds (1) Every diagram of the form j r8 rr rr rr rr rr i rr LLL LLL LLL LLL L% j0 can be extended to a commutative diagram

j r9 MM rr MM rr MM rr MM rr MM rr MMM rr M& i L k LL qq8 LL qqq LLL qq LL qqq LLL qq % j0 qq .

(2) Every diagram of the form β i ⇒ j α can be extended to a commutative diagram

β γ i ⇒ j → k α (i.e., such that γα = γβ). (b) I is called connected, if, for any objects i and j in I, there is a ﬁnite chain of morphisms

i → i1 ← j1 → i2 ← ... ← j .

(c) I is called filtered, if I is pseudo-filtered and connected. (d) I is called (pseudo-)cofiltered, if Iop is (pseudo-)filtered.

Examples 4.B.2 (a) I is filtered, if I has finite colimits: In (1) one can take the fiber sum of j and j0 over i, in (2) the difference cokernel, and in (b) the sum i → i`j ← j. Accordingly I is cofiltered, if I has finite limits. (b) Let M be an ordered set, considered as category with ≤ as morphisms. Then M is filtered if and only if M is ordered inductively (every morphism x ≤ y is unique).

30 Theorem 4.B.3 Let I be a ﬁltered category and f : I → C a covariant functor, where C = Set or C = Ab. Then the direct limit

lim f(i) → i∈I exists in C, and the formation of the direct limit is exact (exchanges with ﬁnite limits and colimits).

Proof Explicitly, we have lim f(i) = (` f(i))/ ∼ , → i∈I i∈I where for x ∈ f(i) and y ∈ f(j) the following holds.

x ∼ y ⇔ i → k, j → k, exist, then x and y have the same image under f(i) → f(k) and f(j) → f(k).

The exactness follows easily from this.

31 5 Cohomology on sites

Let (X , T ) be a site.

Proposition 5.1 The abelian categories P r(X ) and Sh(X , T ) have enough injectives, i.e., every object P in P r(X ) has a monomorphism P,→ I into an injective presheaf I, and the analogous fact holds for Sh(X , T ).

For this we use a method of Grothendieck. Let A be an abelian category.

Lemma/Deﬁnition 5.2 A family (Ei)i∈I of objects of A is called a family of generators, if the following equivalent conditions hold: (a) The functor A → Ab Q A 7→ HomA(Ei,A) i∈I is faithful, i.e., for all objects A, A0 in A the map

0 Y Y 0 HomA(A, A ) → Hom( HomA(Ei,A), HomA(Ei,A )) i∈I i∈I is injective.

(b) For every object A in A and every subobject B $ A, there exists a morphism Ei → A which does not factorize over B.

Proof of the equivalence of (a) and (b): (a) ⇒ (b): We have the exact sequence

0 → B →i A →π A0 = A/B → 0 , and by assumption A0 6= 0. Then π 6= 0, and by (a) there is an i ∈ I, for which the induced map 0 HomA(Ei,A) → HomA(Ei,A )

is not zero. If ϕ : Ei → A is not in the kernel, then ϕ does not factorize over B. (b) ⇒ (a): Left to the readers!

Examples 5.3 If R is a ring with unit, then E = R is a generator for ModR, since we have a canonical isomorphism ∼ HomR(R,M) → M f 7→ f(1) for every R-module M.

Deﬁnition 5.4 We say that the abelian category A has the property

(AB3), if any direct sums ⊕ Ai exist in A (Since cokernels exist, it follows that arbitrary i∈I direct limits (colimits) exist in A, see 4.A.21 above)

32 (AB4), if (AB3) holds, and forming direct sums is an exact functor, (AB5), if (AB3) holds, and forming inductive limits is an exact functor. We deﬁne the properties (AB3∗), (AB4∗) and (AB5∗) dually.

Deﬁnition 5.5 An abelian category is called a Grothendieck category, if (AB5) holds and if it has a family of generators.

The category of R-modules in Example 5.3 is a Grothendieck category, because it has the property (AB5), as one can see easily, and the second property applies by 5.3.

Theorem 5.6 A Grothendieck category has enough injectives.

Idea of Proof: Because of (AB3), A has a generator E (for a family (Ei)i∈I let E = ⊕ Ei). i∈I For the ring with unit R = HomA(E,E) then consider the faithful functor

A → ModR A p HomA(E,A) .

For the Proof of 5.1 it thus suﬃces to show:

Theorem 5.7 P r(X ) and Sh(X , T ) are Grothendieck categories.

Proof: First we consider the generators.

P Lemma/Deﬁnition 5.8 (a) For an object U in X deﬁne the abelian presheaf ZU by P L ZU (V ) = Z = ⊕ Zf for V in X . Hom(V,U) f∈Hom(V,U)

For every abelian presheaf Q on X , we then have isomorphisms

P ∼ HomP r(X )(ZU ,Q) = HomZ(Z,Q(U)) = Q(U) ,

P functorially in Q (i.e., ZU represents the functor Q 7→ Q(U)).

(b) We deﬁne the abelian sheaf ZU as the associated sheaf

P ZU = aZU . Then for sheaves F on (X , T ), we have

HomSh(X ,T )(ZU ,F ) = F (U) , functorially in F .

P Proof (a): Every morphism f : ZU → Q is uniquely determined by fU (1idU ) ∈ Q(U). (b) follows from (a) by adjointness of a and i.

Corollary 5.9 P r(X ) and Sh(X , T ) have a family of generators.

33 Proof: (a) If P → P 0 is non-zero in P r(X ), then P (U) → P 0(U) is non-zero for some object U in X . (b)The same applies for Sh(X , T ).

Lemma 5.10 P r(X ) and Sh(X , T ) satisfy (AB5).

Proof (AB3) and (AB5) are obvious for P r(X ) by (the proof of) Theorem 4.3 (a), (since these properties hold for Ab), and it follows from 5.8 (a) that (ZU )U∈X is a family of generators. (AB3) holds for Sh(X , T ) (arbitrary limits and colimits exist by 4.3(a)), and (AB5) follows from the explicit description of the colimits of the proof of 4.3 (a) and the exactness of the functor P p aP (see 4.3 (c)). This proves Proposition 5.1.

Deﬁnition 5.11 Let (X , T ) be a site and U an object in X . The functor

Hi(U, −) := Hi(U, T ; −): Sh(X , T ) → Ab i F p H (U, F ) is the i-th right derivative of the left exact functor

0 F p F (U) =: Γ(U, F ) =: H (U, F ) . Hi(U, F ) is called i-th cohomology of F on U (or i-th cohomology group on U with coeﬃcients in F ). By construction, Hi(U, F ) = Hi(I·(U)), where F,→ I· is an injective resolution F in Sh(X , T ).

Example 5.12 If X is a topological space and F a sheaf on X, then Hi(X,F ) is the usual cohomology of sheaves on X.

0 0 i Deﬁnition 5.13 Let f :(X , T ) → (X , T ) a morphism of sites. Then R f∗ is the i-th right derivative of the left exact functor

0 0 f∗ : Sh(X , T ) → (Sh(X , T ) .

i R f∗F is called the i-th higher direct image of F under f. i i · · 0 0 Hence R f∗F = H (f∗I ), where F,→ I is an injective resolution in Sh(X , T ).

Remark 5.14 (a) By the general properties of right derived functors, for each short exact sequence

(5.14.1) 0 → F 0 → F → F 00 → 0 of sheaves on (X , T ) and for every object U in X one has a long exact sequence of cohomology

0 → H0(U, F 0) → H0(U, F ) → H0(U, F 00) →δ H1(U, F 0) ... → Hn(U, F 0) → Hn(U, F ) → Hn(U, F 00) →δ Hn+1(U, F 0) → ...

This is functorial in U and functorial in short exact sequences (5.14.1).

34 (b) Similarly, for every morphism of sites f :(X 0, T 0) → (X , T ) and every short exact sequence of sheaves on (X 0, T 0) (5.14.2) 0 → G0 → G → G00 → 0 we get a long exact sequence

n 0 n n 00 δ n+1 0 ... → R f∗G → R f∗G → R f∗G → R f∗G → ..., which is functorial in the short exact sequences (5.14.2).

Theorem 5.15 (a) Let F be an abelian sheaf on (X , T ). For every morphism α : V → U in X , one has canonical restriction homomorphisms (5.15.1) α∗ : Hi(U, F ) → Hi(V,F )(i ≥ 0), which coincide with the restriction F (U) → F (V ) for i = 0, are functorial in F , and are compatible with the exact sequences of 5.14 (a) (i.e., also compatible with the connecting morphisms). (b) By this we obtain an abelian presheaf Hi(F ): U 7→ Hi(U, F ) for all i ≥ 0.

Proof If F,→ I· is an injective resolution, then we have a homomorphism of complexes I·(U) → I·(V ) , and the maps (5.15.1) are obtained by passing to the cohomology. By transivity of the restrictions of I·, it follows that one has the relation (αβ)∗ = β∗α∗ for each further morphism ∗ β : W → V . We obtain (b), since idU = id. The other functorialities in (a) are again obvious by construction: For a morphism of sheaves F → G, we obtain a morphism G / J · O O

F / I· of injective resolutions that, by deﬁnition, provides the functoriality of the cohomology, i.e. the canonical morphisms Hi(U, F ) = Hi(I·(U)) → Hi(J ·(U)) = Hi(U, G). This shows the compatibility with α∗. The case of exact sequences follows from an exact diagram of injective resolutions 0 00 0 / F _ / F _ / F _ / 0

0 / I· / J · / K· / 0 , by passing to the sections over U in the bottom line.

Theorem 5.16 Let f :(X 0, T 0) → (X , T ) be a morphism of sites. For every sheaf F ∈ 0 0 i Sh(X , T ) and every i ≥ 0, R f∗F is the sheaf associated to the presheaf U 7→ Hi(f 0(U),F )

35 on X 0.

· 0 0 i i · Proof Let F,→ I be an injective resolution in Sh(X , T ). Then R f∗F = H (f∗I ) is the sheaf associated to the presheaf quotient

P i i+1 P i−1 U 7→ ker (f∗I → f∗I )(U)/im (f∗I → f∗I)(U) q ker(Ii(f 0(U)) → Ii+1(f 0(U)))/im(Ii−1(f 0(U))/Ii(f 0(U))) q Hi(f 0(U),F ) .

36 6 Spectral sequences

Let A be an abelian category.

Deﬁnition 6.1 A spectral sequence in A

p,q p+q E1 ⇒ E consists of p,q (a) objects E1 in A for all p, q ∈ Z, p,q p,q p,q p,q (b) subquotients Er = Zr /Br of E1 for all r ≥ 2, (c) morphisms (called the diﬀerentials of the spectral sequence)

p,q p,q p+r,q−r+1 dr : Er → Er ,

r • (p, q) r − 1 • (p + r, q − r + 1)

such that p,q p,q p+r,q−1 p,q ker(dr : Er → Er ) Er+1 = p−r,q+r−1 p−r,q+r−1 p,q , im(dr : Er → Er )

p,q p,q p,q p,q (d) subquotients E∞ = Z∞ /B∞ of E1 , such that

p,q p,q p,q p,q Br ⊆ B∞ ⊆ Z∞ ⊆ Zr

p,q p,q for all r ≥ 1 (hence E∞ “is smaller than Er for all r ≥ 1”), n (e) objects (E )n∈Z in A with descending ﬁltrations En ⊇ ... ⊇ F pEn ⊇ F p+1En ⊇ ... and isomorphisms p,q ∼ p p+q p+1 p+q E∞ → F E /F E for all p, q ∈ Z.

Deﬁnition 6.2 The spectral sequence is called ﬁnitely convergent, if

p,q p,q 2 Er = E∞ for r >> 0, for all (p, q) ∈ Z ,

37 and if for every n ∈ Z the ﬁltration F pEn is ﬁnite, i.e., 0 for p >> 0, F pEn = En for p << 0.

p,q p,q p,q Some spectral sequences begin with E2 ; then E1 does not exist, and all Er are subquo- p,q tients of E2 .

We now explain how to work with spectral sequences. p,q 1) The layers: For each r one considers the Er-layer of all terms Er and their diﬀerentials

E1-layer: d1 ••••

••••

•••• E2-layer:

•• d2

•••

q •

dr r Er-layer: •

dr r − 1

One has drdr = 0 and Er+1 = ker dr/im dr. 2) Limit/convergence: Let

p,q p+q p,q p+q E1 ⇒ E (or E2 ⇒ E )

38 be a ﬁnitely convergent spectral sequence. We obtain the following picture:

• 0,n n • Er • 1,n Er • p + q = n

n,0 Er • n •

The terms which contribute to En are on the line p + q = n. We have a (finite) filtration En ⊇ ... ⊇ F pEn ⊇ F p+1En ⊇ ... and p n p+1 n p,q F E /F E = E∞ , p + q = n . n p,q The term on the left is a subquotient of E , the term on the right is a subquotient of E1 ; p,q p,q furthermore we have E∞ = Er for r >> 0. p,q p+q p,q p+q 3) Spectral sequences in the first quadrant: Let E1 ⇒ E (resp. E2 ⇒ E ) be a finitely p,q p,q convergent spectral sequence with E1 = 0 (resp. E2 = 0) for p < 0 or q < 0.

• • • • • 0 • • • • • • • • • • • • • • • • • • • • • 0 0

p,q p,q Lemma/Deﬁnition 6.3 (a) One has E∞ = Er for r > max(p, q + 1). p,q p+q (b) For E1 ⇒ E , there are canonical morphisms

n e 0,n E → E1 n,0 e n E1 → E . These are called the edge morphisms. p,q p+q (c) For E2 ⇒ E , there are edge morphisms

n e 0,n E → E2 n,0 e n E2 → E

39 Proof (a):

p p − 1 (p, q) q + 1 •

p,q p,q If r > q + 1, the diﬀerential dr starting from Er is zero (since it ends in a zero object). If p,q p−r,q+r−1 r > p, the diﬀerential arriving in Er (dr ) is zero. If both properties hold, then we p,q p,q p,q have Er+1 = ker dr/im d2 = Er /0 = Er . Since this holds for all higher r (and the spectral p,q p,q sequence converges), we have Er = E∞ . p,q n 0 n (b): If E∞ = 0 for p < 0, then we have E = F E (because of the convergence), and we have 0,n 0,n 0,n dr r,n−r+1 0,n Er+1 = ker(Er → Er ) ⊆ Er 0,n 0,n for all r, i.e. E∞ ⊆ E1 . We obtain

n 0 n 1 n ∼ 0,n 0,n e : E F E /F E = E∞ ,→ E1 .

p,q n+1 n n,0 If E∞ = 0 for q < 0, then we have F E = 0 (because of the convergence), and Er+1 = n−r,r−1 dr n,0 n n,0 n,0 coker(Er → Er ) is a quotient of Er . Then we have E1 E∞ and a morphism n,0 n,0 ∼ n n n+1 n e : E1 E∞ = F E /F E,→ E .

p,q p+q p,q p+q Lemma 6.4 (Exact sequence of the lower terms) Let E1 ⇒ E (or E2 ⇒ E ) be a ﬁnitely convergent spectral sequence in the ﬁrst quadrant. Then one has an exact sequence

0,1 1,0 e 1 e 0,1 d2 2,0 e 2 0 → E2 → E → E2 → E2 → E , where e always denotes the edge morphism.

40 Proof The picture

0 •

••• 1 2

(or the Proof of 6.5 (a)) shows

1,0 0,1 0,1 0,1 2,0 0,1 E∞ = E2 ,E∞ = ker d2 ,E∞ = coker d2 .

From this we obtain exact sequences

0,1 e 1 0,1 0 → E2 → E → ker d2 → 0

0,1 0,1 0,1 d2 2,0 2,0 0 → ker d2 → E2 → E2 → E∞ , 2,0 2 and by splicing together these sequences and composing with E∞ ⊆ E we obtain the claimed sequence.

Theorem 6.5 (a) Let A∗ be a complex in A, and let F pA∗ be a descending ﬁltration by subcomplexes. Then we have a spectral sequence

p,q p+q p ∗ p+1 ∗ p+q n ∗ E1 = H (F A /F A ) ⇒ E = H (A ) .

(b) If the ﬁltration F p is biregular, i.e.,

0 for p >> 0 , F pAn = An for p << 0 for every n ∈ Z, then the spectral sequence is ﬁnitely convergent.

p,q p,q p+q p ∗ p+1 ∗ p+q+1 p+1 ∗ p+2 ∗ p+1,q d1 : E1 = H (F A /F A ) → H (F A /F A ) = E1 is the connecting homomorphism for the exact sequence of complexes

0 → F p+1A∗/F p+2A∗ → F pA∗/F p+2A∗ → F pA∗/F p+1A∗ → 0 .

Proof (a): For r ≥ 1, p ∈ Z and q := n − p let F pHn(A∗) = im(Hn(F pA∗) → Hn(A∗))

41 as well as

p,q n p ∗ p+r ∗ αp,r n p ∗ p+1 ∗ p,q Zr = im(H (F A /F A ) / H (F A /F A ) = E1 ) O O

p p,q? n p ∗ α n p ∗ p+1 ∗ Z∞ = im(H (F A ) / H (F A /F A )) O O δ

p p,q? n−1 ∗ p ∗ δ n p ∗ p+1 ∗ B∞ = im(H (A /F A ) / H (F A /F A )) O O

p,r p,q? n−1 p−r+1 ∗ p ∗ δ n p ∗ p+1 ∗ Br = im(H (F A /F A ) / H (F A /F A )) . Here, the morphisms α, αp,r and the commutative diagram above are induced by the commutative diagram αp,r F p/F p+r / F p/F p+1 , O

αp F p / F p/F p+1 where we write F m for F mA∗. The connecting homomorphisms δ, δp and δp,r as well as the other commutative diagrams are induced by the following commutative diagrams and the corresponding long exact cohomology sequences:

δ : 0 / F p / A∗ / A∗/F p / 0

αp δp : 0 / F p/F p+1 / A∗/F p+1 / A∗/F p / 0 O O

? ? δp,r : 0 / F p/F p+1 / F p−r+1/F p+1 / F p−r+1/F p / 0 .

We obtain a commutative diagram with exact rows

δp+r,r+1 (1) Hn(F p/F p+r+1) / Hn(F p/F p+r) / Hn+1(F p+r/F p+r+1) S SSS β p,r SSS γ α SSSS SSSS αp,r+1 ) Hn(F p/F p+r+1) / Hn(F p/F p+1) / Hn+1(F p+1/F p+r+1)

αp,r (2) Hn(F p+1/F p+r) / Hn(F p/F p+r)) / Hn(F p/F p+1) U UUUU β p+r,r+1 UUU δ UUUU UUUU δp+r,r γ * Hn(F p+1/F p+r) / Hn+1(F p+r/F p+r+1) / Hn+1(F p+1/F p+r+1)

p,q p,q p,q p,q From (1) we get Zr+1 ⊆ Zr , and from (2) we get (by renumbering p + r p) Br ⊆ Br+1. This gives the following inclusions

p,q p,q p,q p,q p,q p,q p,q p,q p,q 0 = B1 ⊆ ... ⊆ Br ⊆ Br+1 ⊆ ... ⊆ B∞ ⊆ Z∞ ⊆ ... ⊆ Zr+1 ⊆ Zr ⊆ ... ⊆ Z1 = E1 .

42 We now use the following Lemma

Lemma 6.6 If C }> AA } AA ψ }} ϕ AA } AA }}ϕ0 η A0 / A / A00 is a commutative diagram in A with exact line, then we have im(ϕ0) ⊆ im(ϕ), as well as canonically im(ϕ)/im(ϕ0) ∼= im(ψ) .

Proof: The ﬁst claim is obvious, and since ker η = im ϕ0 ⊆ im ϕ, η induces an isomorphism

im ϕ/im ϕ0 = im ϕ/ ker η →∼ η(im ϕ) = im(ηϕ) = im ψ .

By applying 6.6 we get an isomorphism

(1) (2) p,q p,q p,q p,r p,r+1 ∼ ∼ p+r,r+1 p+r,r p+r,q−r+1 p+r,q−r+1 δr : Zr /Zr+1 = im α /im α = im β = im δ /im δ = Br+1 /Br .

With this, we deﬁne the diﬀerential

p,q p,q p,q p,q p,q p,q p,q δr p+r,q−r+1 p+r,q−r+1 p+r,q−r+1 p+r,q−r+1 dr : Er = Zr /Br Zr /Zr+1 → Br+1 /Br ,→ Zr /Br . ∼ Then we have

p,q p,q p,q p,q p+r,q−r+1 p+r,q−r+1 ker dr = Zr+1/Br , im dr = Br+1 /Br and therefore p,q p,q ker dr Zr+1 p,q p−r,q+r−1 = p,q = Er+1 . im dr Br+1 Finally, the commutative diagrams with exact rows

(3) Hn(F p) q8 OO qq OOO ρ qqq OO qq OOO qq O' Hn(F p+1) / Hn(A∗) / Hn(A∗/F p+1)

δ (4) Hn−1(A∗/F p) / Hn(F p) / Hn(A∗) Q QQQ ρ p QQQ α QQQ QQQ δp ( Hn−1(A∗/F p) / Hn(F p/F p+1) / Hn(A/F p+1)

together with Lemma 6.6 produce the relations F p+1Hn(A∗) ⊆ F pHn(A∗), as well as

(4) (3) p,q p,q p,q p p ∼ ∼ p n ∗ p+1 n ∗ E∞ = Z∞ /B∞ = im α /im δ = im ρ = F H (A )/F H (A ) .

43 This shows all the properties of a spectral sequence. (b): The additional claim about convergence is obvious, since the n-th cohomology of a complex C∗ depends only on Cn−1 → Cn → Cn+1. (c): For r = 1, the diagrams (1) and (2) are

δp+1,2 (1) Hn(F p/F p+2) / Hn(F p/F p+1) / Hn+1(F p+1/F p+2) SS SSS β αp,1 SSS q SSS SSS αp,2 δp+1,2 ) Hn(F p/F p+2) / Hn(F p/F p+1) / Hn+1(F p+1/F p+2)

n p+1 p+1 n p p+1 αp,1 n p p+1 (2) 0 = H (F /F ) / H (F /F ) = / H (F /F ) TT TTTT β δp+1,2 TTTT TTTT p+1,1 T) n p+1 p+1 δ n+1 p+1 p+2 n+1 p+1 p+2 0 = H (F /F ) / H (F /F ) = / H (F /F )

p,q p,q p+1,2 From the deﬁnition of d1 (with p + q = n) we get that d1 = δ , the connecting homomorphism for the exact sequence

0 → F p+1/F p+2 → F p/F p+2 → F p/F p+1 → 0 ,

p,q p+1,2 since d1 = β = δ by the equalities in the diagrams (1) and (2).

An important example for spectral sequences is the following.

Theorem 6.7 (Grothendieck-Leray-spectral sequence) Let F : A → B and G : B → C be left exact functors between abelian categories, where A and B have enough injectives and F maps injectives to G-acyclic objects. Then for every object X in A we have a ﬁnitely convergent spectral sequence

p,q p q p+q E2 = R G(R FX) ⇒ R (G ◦ F )X

The proof needs some preliminary considerations

Deﬁnition 6.8 (a) A naive double complex C∗,∗ in A is a commutative diagram of objects Cp,q ∈ A p,q+1 dI ... / Cp−1,q+1 / Cp,q+1 / Cp+1,q+1 / ... O O O dp,q p+1,q II dII p−1,q p,q dI dI ... / Cp−1,q / Cp,q / Cp+1,q / ... O O O p,q−1 dII ... / Cp−1,q−1 / Cp,q−1 / Cp+1,q−1 / ... O O O

(b) A double complex is a corresponding diagram, in which all squares are anticommutative, p,q+1 p,q p+1,q p,q i.e. dI dII + dII dI = 0 for all p, q ∈ Z.

44 (c) The double complex associated to a naive double complex as in (a) is the double complex p,q p p,q where dII is replaced by (−1) dII . (d) If there is an N ∈ Z with Cp,q = 0 for p < N or q < N, then the total complex associated to a double complex is the complex T ot(C∗,∗) with components

T ot(C∗,∗)n = ⊕ Cp,q p+q=n and diﬀerential d = dI + dII , p,q p,q i.e. d|Cp,q = dI + dII .

The following construction is very important for the treatment and deﬁnition of spectral sequences.

Construction 6.9 Let T ot(I∗,∗) be the total complex associated to the double complex, which is associated to the naive double complex I∗,∗. This has two descending ﬁltrations:

p ∗,∗ n r,s (6.9.1) FI T ot(I ) = ⊕ I , and r+s=n r≥p

p ∗,∗ n r,s (6.9.2) FII T ot(I ) = ⊕ I . r+s=n s≥q Since p ∗,∗ p+1 ∗,∗ p,∗ FI T ot(I )/FI T ot(I ) = I [−p] and p ∗,∗ q+1 ∗,∗ ∗,q FII T ot(I )/FII T ot(I ) = I [−q] the corresponding spectral sequences from Theorem 6.5 are

p,q p+q p,∗ q p,∗ p+q p+q ∗,∗ I E1 = H (I [−p]) = H (I ) ⇒ E = H (T ot(I )) , and for the ﬁrst ﬁltration, and

p,q p+q ∗,q p ∗,q p+q p+q ∗,∗ II E1 = H (I [−q]) = H (I ) ⇒ E = H (T ot(I ))

p,q p+q for the second ﬁltration. The d1-diﬀerential of the spectral sequence I E1 ⇒ E is the morphism p,q q p,∗ q p+1,∗ p+1,q I E1 = H (I ) → H (I ) =IE1 , which is induced by the morphism of complexes

p,∗ p,∗ p+1,∗ dI : I → I .

∗ p ∗ By 6.5 (c), for a ﬁltered complex (A ,F A ) as in 6.5, the d1-diﬀerential

p,q p,q p+q p ∗ p+1 ∗ p+q+1 p+1 ∗ p+2 ∗ p+1,q d1 : E1 = H (F A /F A ) → H (F A /F A ) = E1 is the connecting homomorphism for the exact sequence of complexes

0 → F p+1/F p+2 → F p/F p+2 → F p/F p+1 → 0 .

45 In this situation, we have the sequence

0 / F p+1/F p+2 / F p/F p+2 / F p/F p+1 / 0

Ip+1,∗[−p − 1] β p+1,∗ α p,∗ p,∗ 0 / I [−p − 1] / T ot ↑dII  / I [−p] / 0 , Ip,∗[−p − 1] where α and β are the obvious morphisms. We obtain an exact sequence of complexes

0 / Ip+1,q+1 / Ip+1,q+1 ⊕ Ip,q+2 / Ip,q+2 / 0 O O O p+1 p (−1) dII d (−1) dII

0 / Ip+1,q / Ip+1,q ⊕ Ip,q+1 / Ip,q+1 / 0 ,

a / (a, 0), (a, b) / b where the arrow in the middle, expressed in elements, is given by

p+1 p (a, b) 7→ ((−1) dII a + dI b, (−1) dII b) .

p,q+1 By the standard description of the connecting morphism (b ∈ I with dII b = 0 is lifted p+1,q p,q+1 p+1,q+1 to (0, b) ∈ I ⊕ I , mapped on (dI b, 0) below D, which is the image dI b ∈ I ), we see that this maps the class of b to the class dI b, hence is induced by dI as claimed. The analogous claims hold for the second spectral sequence.

Now we remember the well-known

Lemma 6.10 Let 0 → A → B → C → 0 be a short exact sequence in A. If A has enough injectives, then, for any given injective resolutions

A,→ I∗ and C,→ K∗ there exists an exact sequence of injective resolutions

0 / I∗ / J ∗ / K∗ / 0 O O O

? ? ? 0 / A / B / C / 0 , which is split in each degree.

Proof: One starts with a diagram

0 / I0 / I0 ⊕ K0 / K0 / 0 O cHH O O HH β α HH (β,γ) γ HH ? HH 0 / A / B / C / 0 ,

46 where β is an extension of α to B (this exists, since I0 is injective). Then one can easily see that the middle arrow (β, γ) is a monomorphism. By the snake lemma, the sequence 0 → A1 → B1 → C1 → 0 of the cokernels of α, (β, γ) and γ is exact, and one proceeds with A1 ,→ I1 and C1 ,→ K1, etc.

By this we obtain inductively:

Theorem 6.11 (Cartan-Eilenberg-resolution) If A∗ is a complex in A which is bounded n ∗,∗ below, e.g. A = 0 for n < N, then we have a naive double complex (I , dI , dII ) with Ip,q = 0 for p < N and q < 0 and a morphism of complexes

dN,0 dN+1,0 IN,0 / IN+1,0 / AN+2,0 / ... O O O

dN dN+1 AN / AN+1 / AN+2 / ..., such that the following holds: (a) For all p, Ap ,→ Ip,∗ is an injective resolution. p p,∗ p p,∗ p ∗ p ∗,∗ (b) For all p, ZA ,→ ZI ,BA ,→ BI and H (A ) → HI (I ) are injective reso- p ∗,∗ p ∗,∗ p−1,∗ dI p,∗ dI p+1,∗ lutions. Here let HI (I ) = H ((I , dI )) be the cohomology of I → I → I , ZAp = ker(Ap → Ap+1),BAp = im(Ap−1 → Ap),ZIp,∗ = ker(Ip,∗ →dI Ip+1,∗) and BIp,∗ = p−1,∗ dI p,∗ p ∗,∗ p,∗ p,∗ im(I → I ), so that HI (I ) = ZI /BI .

Proof Without restriction, let An = 0 be for n < 0. Then we have a chain of morphisms

0 0 1 1 1 2 ZA ,→ A BA ,→ ZA ,→ A BA ,→ ... If we choose injective resolutions ZA0 ,→ ZI0,∗ and BA1 ,→ BI1,∗, then by 6.9 we obtain an exact sequence of injective resolutions 0 / ZI0,∗ / I0,∗ / / BI1,∗ / 0 O O O

? ? ? 0 / ZA0 / A0 / / BA1 / 0 . If, moreover, we choose an injective resolution H1(A∗) ,→ HI1,∗, then, by 6.9, we get an exact sequence of injective resolutions

0 / BI1,∗ / ZI1,∗ / HI1,∗ / 0 O O O

? ? ? 0 / BA1 / ZA1 / H1(A∗) / 0 . Now we choose an injective resolution B2A∗ ,→ BI2,∗ and we obtain an exact sequence of injective resolutions 0 / ZI1,∗ / I1,∗ / BI2,∗ / 0 O O O

? ? ? 0 / ZA1 / A1 / BA2 / 0 .

47 If we continue like this, we obtain a naive double complex I∗,∗ with a co-argumentation

A∗ → I∗,∗

∗ which is a resolution of A in the category of complexes, and where the ﬁrst diﬀerential dI is given by the composition

p,∗ p,∗ p+1,∗ p+1,∗ p+1,∗ dI : I BI ,→ ZI ,→ I ,

p+1,∗ pi p,∗ p+1,∗ so that BI is really the image of dI and ZI is the kernel of dI , and moreover p,∗ p,∗ p,∗ p ∗,∗ HI = ZI /BI = H (I , dI ).

Now we come to the Proof of theorem 6.7. Let F : A → B and G : B → C be left exact functors between abelian categories with enough injectives, where F maps injectives to G-acyclic objects. Let X be an object in A and let X,→ I∗ an injective resolution. For A∗ = FI∗ we have Hn(A∗) = Hn(FI∗) = RnF (A) by deﬁnition. Let

J ∗,∗ O

? A∗ be a Cartan-Eilenberg-resolution as in Theorem 6.11. We consider the ﬁrst spectral sequence of the double complex GJ ∗,∗

p,q q p,∗ p+q p+q ∗,∗ I E1 = H (GJ ) ⇒ E = H (T ot(J )) . Since Ap = FIp is G-acyclic, Hq(GJ p,∗) = 0 for q > 0 and

H0(GJ p,∗) = ker(GJ p,0 d→II GJ p,1) = GAp .

This implies that the edge morphisms

Rn(GF )(X) = Hn(GF I∗) = Hn(GA∗) → Hn(T ot(GJ ∗,∗))

are all isomorphisms.

We consider the second spectral sequence of the double complex GJ ∗,∗,

p,q p ∗,q p,q p,q ∗,∗ II E1 = H (GJ ) ⇒ E = H (T ot(GJ )) We have p,q p,q p,q p,q p,q II E1 = ZI GJ /BI GJ = G(ZI J /BI J ) and by assumption n n ∗ n,∗ R FX = H (A ) ,→ HI (J ) is an injective resolution. Therefore we have

p,q q n,∗ p q II E2 = HII (GHI (J )) = R G(R FA) and we obtain the desired Grothendieck-spectral sequence.

48 7 The ´etalesite

Deﬁnition 7.1 (a) A class E of morphisms of schemes is called admissible, if the following holds (M1) all isomorphisms are in E, (M2) E is closed under compositions (if ϕ : Y → X and ψ : Z → Y are in E, then so is ψ ◦ ϕ : Z → X), (M3) E is closed under base change (If ϕ : Y → X is in E and ψ : X0 → X any morphism, 0 0 0 0 then the base change ϕ : Y = Y ×X X → X is in E).

gi (b) Let E be admissible. An E-covering (Ui −→ X)i∈I of a scheme X is a family of E- morphisms (morphisms in E) with ∪ gi(Ui) = X. i

Example 7.2 Important examples of admissible classes are (a) the class (Zar) of all open immersions, (b) the class (ét)of all étalemorphisms, (c) the class (fl) of all flat morphisms which are locally of finite type.

Remark 7.3 If the considered schemes are not locally noetherian, one should replace “of ﬁnite type” by “of ﬁnite presentation”, and the same should be done in the following recollection.

Recollection 7.4 (a) A morphism f : Y → X of schemes is called unramiﬁed, if it has the following properties for all y ∈ Y and x = f(y) ∈ X.

(i) f is locally of finite type, mxOY,y = my, where OY,y and OX,x are the local rings at y and x and my and mx are their maximal ideals, and for all y ∈ Y , k(y)/k(x) is a finite separable field extension. 1 (ii) f is locally of finite type, and ΩY/X = 0.

(iii) f is locally of finite type, and the diagonal ∆Y/X : Y,→ Y ×X Y is an open immersion. (iv) f is locally of finite type and formally unramified. (b) A morphism f : Y → X is called étale, if it has the following equivalent conditions: (i) f is flat and unramified, (ii) f is locally of finite type and formally étale.

Lemma/Deﬁnition 7.5 (a) Let C be a category of schemes and let E be an admissible class of morphisms. Then, with the E-coverings, C forms a site, which is denoted by CE. (b) The small E-site of a scheme X consists of all X-schemes Y → X for which the structural morphism Y → X lies in E, equipped with the E-coverings, and is denoted by XE.

Definition 7.6 In particular, this defines the small étalesite Xét of a scheme X. In general, one understands an étalesheaf on X as a sheaf F on the small site Xét and the étale cohomology i i Hét(X,F ) := H (Xét,F )

49 is the cohomology on this site, deﬁned by Deﬁnition 5.3.

Remark 7.7 All morphisms in X´et are ´etale:If

f Y1 / Y2 AA } AA }} ϕ1 AA }}ϕ2 A ~}} X

is a morphism of étale X-schemes (i.e., a commutative diagram with étale ϕ1 and ϕ2), then f is étale.

Lemma/Deﬁnition 7.8 Let E and E0 be two admissible classes of morphisms. A morphism f : X0 → X of schemes deﬁnes a morphism of sites

0 f : XE0 → XE via 0 0 f : XE → XE0 0 V 7→ V ×X X , 0 0 0 if the base change V1 ×X X → V2 ×X X is in E for any V1 → V2 in XE. In this case, we get functors 0 fP : P r(XE) → P r(XE) P 0 f : P r(XE) → P r(XE0 ) 0 f∗ : Sh(XE0 ) → Sh(XE) ∗ 0 f : Sh(XE) → Sh(XE0 ) , P ∗ P ∗ where f is left adjoint to fP and f left adjoint to f∗. The functors fP , f and f are exact, the functor f∗ is left exact. In particular, this holds for E0 = E, for example for E0 = E = ét.Hence we have adjoint pairs 0 P 0 fP : P r(X ) → P r(X) f : P r(X) → P r(X ) , 0 ∗ 0 f∗ : Sh(Xét) → Sh(Xét), f : Sh(Xét) → Sh(Xét) ,

where f∗ is left exact and the other functors are exact.

0 Proof of the claims: f deﬁnes a morphism of sites: If Y → X in XE (hence an E-morphism) and (Ui → Y ) is an E-covering (hence a surjective family of E-morphisms), then, by assump- 0 0 0 tion, Y ×X X → X is in XE0 . Furthermore, 0 0 (Ui ×X X → Y ×X X ) is an E0-covering: by assumption, the morphisms are E0-morphisms, and for every surjective family πi (Yi −→ Y ) of morphisms of schemes and every morphism of schemes X0 → X, the family

0 0 πi 0 0 (Yi ×X X −→ Y ×X X =: Y )

50 is again a surjective family: If y0 ∈ Y 0, with image y in Y , then there exists an i for which −1 0 −1 0 0 0 0 πi (y) = (Yi)y = Yi ×Y k(y) is non-empty. Then we have (πi) (y ) = (Yi ×X X )×Y ×X X y = 0 0 0 Yi ×Y y = (Yi ×Y y) ×y k(y ) = (Yi ×Y k(y)) ×k(y) k(y ) 6= ∅. This shows the property (S1) 0 from 2.7 for f . Furthermore, for any Z → X in XE and any X-morphism Z → Y (i.e. any morphism in XE) the canonical morphism

0 0 0 0 (Ui ×Y Z) ×X X → (Ui ×X X ) ×Y ×X X (Z ×X X ) is an isomorphism. This shows 2.7 (S2). The claims on exactness follow from Theorem 4.4, since finite products and fiber products, 0 and therefore finite limits exist in XE and XE0 (see 4.A.22).

0 Corollary 7.9 By assumption of Lemma/Deﬁnition 7.8, the functor f∗ : Sh(XE0 ) → Sh(XE) maps injective sheaves to injective sheaves.

∗ Proof Since f∗ is left exact, and has the exact left adjoint f , this follows from the next lemma.

Lemma 7.10 Let F : A → B be a left exact functor between abelian categories. If F has a left exact left adjoint functor G : B → A, then F maps injective objects to injective objects.

Proof: Let I be an injective object in A. Then FI is injective if and only if for every monomorphism B0 ,→ B the morphism

0 HomB(B,FI) → HomB(B ,FI)

is surjective. By the functoriality of the adjunction, we obtain a commutative diagram

0 HomB(B,FI) / HomB(B ,FI)

o o

0 HomA(GB, I) / HomA(GB ,I) where GB0 ,→ GB is a monomorphism, because G is left exact. Since I is injective, the bottom map is a monomorphism, and hence the top map, too.

0 Corollary 7.11 In the setting of Lemma 7.8, for every sheaf F on XE, we have a Grothendieck- Leray-spectral sequence

p,q p q 0 p+q E2 = H (XE,R f∗F ) ⇒ H (XE0 , F) .

Proof: This follows from Theorem 6.7, since

0 0 0 0 H (XE, f∗F ) = H (XE0 , F )

0 and since the left exact functor f∗ maps injectives to injectives, which then are H (XE0 , −)- acyclic objects.

Deﬁnition 7.12 Let X be a scheme.

51 (a) A geometric point of X is a morphism

ix : x = Spec(Ω) → X,

where Ω is a separably closed (e.g. an algebraically closed) ﬁeld. If x = ix(x) ∈ X, then we say that x is a geometric point over x. (b) For an ´etalepresheaf P on X and a geometric point as above,

P Px := (ix P )(x) ∈ Ab

is called the stalk of P at x.

Remark 7.13 (a) The functor P r(Xét) → Ab P 7→ Px P is exact. In fact, by 7.8, ix : P r(Xét) → P r(xét) is exact. Furthermore, the functor

P r(x´et) → Ab Q 7→ Q(x) is exact. P (b) The adjunction morphism P → (ix)P (ix )P induces a homomorphism of abelian groups

P P P (X) → ((ix)P (ix) P )(X) = ((ix) P )(x) = Px .

Deﬁnition 7.14 Let F ∈ Sh(X´et), s ∈ F (X), and x be a geometric point.

(a) The image of s under F (X) → Fx is denoted by sx and is called the germ of s at x.

(b) If U → X is ´etale,then, in general, there is no canonical map F (U) → Fx, but every lift x → X to U deﬁnes a canonical map, and we denote the image of s ∈ F (U) in Fx again by sx (Obviously, there is always a lift if there exists a point u ∈ U, which is mapped to the image x ∈ X of x → X).

Definition 7.15 An étaleneighborhood of a geometric point x → X is a commutative diagram x / U ?? ?? ?? étale ? X where U → X is an étalemap as indicated. A morphism of étaleneighborhoods is a commutative diagram U2 ? x / U @ 1 @@ @@ @@ @ X

52 where U2 and U1 are ´etaleover X.

Remark 7.16 A commutative diagram of schemes

X0 / Y B BB BB BB B X

0 0 0 corresponds to an X -morphism X → Y ×X X , i.e., to a commutative diagram

0 0 X / Y ×X X BB t BB tt BB tt id B ttpr2 B tt X0 zt

0 0 0 ∼ 0 and hence to a section of pr2 : Y ×X X → X (in short: HomX (X ,Y ) → HomX0 (X ,Y ×X X0)). This shows that an ´etaleneighborhood of x → X can be identiﬁed with a morphism

x → U ×X x

in xét, where U → X is étale,and therefore with an object in the category Ix for the P morphism of sites xét → Xét, U 7→ U ×X x (see 7.8), which is used for the definition of ix . Furthermore, the morphisms of étaleneighborhoods of x → X correspond to the morphisms in Ix, to wit, the X-morphisms U1 → U2, for which the diagram

U1 ×X x 6 nnn nnn nnn nnn x nn PPP PPP PPP PPP PP( U2 ×X x

P is commutative. Combining this with the deﬁnition of ix , we see that for an ´etalepresheaf P on X we have

(7.16.1) Px = lim P (U) , → where the inductive limits runs over all ´etaleneighborhoods of x → X.

Lemma 7.17 (a) If U → X is ´etaleand s ∈ F (U) non trivial, then there is a geometric point x of U with sx 6= 0 in Fx.

(b) In particular we have F = 0 if and only if Fx = 0 holds for all geometric points x von X.

Proof (a): If there are no such geometric points, then every point u ∈ U has an ´etale

neighborhood Vu → U with s|Vu = 0, and by the separateness of F we get s = 0 (the Vu form a covering of U). (b) is obvious from this.

53 Lemma 7.18 Let f : S0 → S be a morphism of sites. If f P maps sheaves to sheaves, then canonically f P a = af P (more precisely: f P ia = iaf P for the embeddings i : Sh(S) → P r(S) and i : Sh(S0) → P r(S0)).

Proof: Let P be a presheaf on S and let F be a sheaf on S0. Then we have isomorphisms

P ∼ ∼ HomP r(S0)(f iaP, iF ) = HomP r(S)(iaP, fP iF ) = HomP r(S)(iaP, if∗F ) ∼ ∼ ∼ = HomSh(S)(aP, f∗F ) = HomP r(S)(P, if∗F ) = HomP r(S)(P, fP iF ) ∼ P ∼ P = HomP r(S0)(f P, iF ) = HomSh(S0)(af P,F )

This implies the claim: By assumption, f P iaP = iG for a sheaf G, and the ﬁrst group is ∼ P P ∼ isomorphic to HomSh(S0)(G, F ). The Yoneda-Lemma implies G = af P , hence f iaP = iaf P P .

Corollary 7.19 For an ´etalepresheaf P on X and a geometric point x of X one has Px = (aP )x.

P P 7.15 P Def. Proof: For ix : x → X, one has Px = (ix P )(x) = (aix P )(x) = (ix aP )(x) = (aP )x, P because ix maps sheaves to sheaves: For a sheaf F on X we have

P ` Q P (ix F )( x) = (ix F )(x) , i∈I i∈I since for a diagram

(fi) V = ` x / U i∈I OOO OOO OOO OOO O' X one has the factorization ` U ` f p8 i∈I i ppp i ppp ppp ppp pp V = ` x / U. i∈I

P Hence the morphisms above are coﬁnite in the category IV , and for (f F )(V ) we can form ` Q the limit over these; furthermore F ( Ui) = F (Ui). i i

Corollary 7.20 A sequence

(7.20.1) 0 → F 0 → F → F 00 → 0 of ´etalesheaves on X is exact if and only if the sequences of the stalks

0 00 (7.20.2) 0 → Fx → Fx → Fx → 0

are exact for all geometric points x of X.

54 Proof (a) If (7.20.1) is exact, then

0 → F 0 → F → F 00

is exact as an sequence of presheaves, hence

0 00 0 → Fx → Fx → Fx is exact by Remark 7.13 (a). 0 00 (b) Conversely, let 0 → Fx → Fx → Fx be exact for all geometric points x of X. (i) Then F 0 → F is a monomorphism: If U → X is étaleand s ∈ F 0(U) is in the kernel of 0 F (U) → F (U), then for every geometric point x of U, the germ sx of s is in the kernel of 0 Fx → Fx, therefore zero, as this map is injective. But this implies s = 0 by 7.17 (a). (ii) Let s be in the kernel of F (U) → F 00(U) for U → X étale.By assumption, for every 0 geometric point x of U, the germ sx lies in the stalk Fx ,→ Fx. Then for every u ∈ U there is 0 an étalemorphism Vu → U, such that s|Vu lies in the subgroup F (Vu) ⊆ F (Vu). As (Vu)u∈U is an étalecovering of U and F 0 and F are sheaves, s lies in F 0(U) ⊆ F (U). By (i) and (ii),

0 → F 0 → F → F 00

is exact. (c) Let P be the presheaf cokernel of F → F 00, i.e., let

F (U) → F 00(U) → P (U) → 0

be exact for all ´etalemorphisms U → X. Then

F → F 00 → aP → 0

is an exact sequence of sheaves, as the functor a (associated sheaf) is exact and aF = F , aF 0 = F 0. Then we have the equivalences F → F 0 epimorphism of sheaves ⇔ aP = 0 ⇔ (aP )x = 0 for all geometric points x (by 7.17 (b)) ⇔ Px = 0 for all geometric points x (by 7.19) 0 ⇔ Fx → Fx surjective for all geometric points x, because forming stalks is exact on the exact sequence of presheaves

F → F 0 → P → 0

(see 7.13 (a)).

Corollary 7.21 A morphism

(7.21.1) ϕ : F1 → F2

of ´etalesheaves on X is zero if and only if the maps of stalks

(7.21.2) ϕx :(F1)x → (F2)x

55 are zero for all geometric points x of X.

Proof: For the non trivial direction, let F0 be the kernel of ϕ. For the exact sequence

0 → F0 → F1 → F2 , the sequence of stalks 0 0 → (F0)x → (F1)x → (F2)x is then exact for all geometric points x. It follows as in (ii) that for every ´etalemorphism U → X every section s ∈ F1(U) already lies in F0(U), hence ϕU : F1(U) → F2(U) is the zero map.

56 8 The ´etalesite of a ﬁeld

The following theorem is Grothendieck’s version of (inﬁnite) Galois theory.

Theorem 8.1 Let k be a ﬁeld, let ks be a separable closure of k and let Gk := Gal(ks/k) be the absolute Galois group of k. Then the functor

φ = Homk(Spec(ks), −) : Spec(k)´et → C(Gk) = category of discrete Gk-sets Y 7→ φ(Y ) := Y (ks) := Homk(Spec(ks),Y )

is an equivalence of sites, where the Grothendieck topology on C(Gk) is given by the surjective families (Mi → M)i.

Remarks 8.2 (a) With the Krull topology (where the subgroups Gal(ks/L), for finite extensions L/k, form a basis of the neighborhoods of 1 in Gk), Gk is a profinite group, i.e., a projective limit of finite groups

G = lim Gal(L/k) . k ← L/k ﬁn. gal., L⊆ks

(b) Gk operates from the left on ks, thus from the right on Spec ks, thus from the left on φ(Y ).

Stab(m) := {g ∈ Gk | gm = m}

is open (and thus of ﬁnite index) in Gk.

We use the following lemma.

` Lemma 8.3 If Y → Spec(k) is étale,then one has Y = Spec(Li), where Li/k are finite i∈I separable field extensions. If Y is of finite type over k, then I is finite.

Proof: The last claim is well-known. In general, every y ∈ Y has an open neighborhood r ` U = Spec(Li). This shows that every point is open, so that Y carries the discrete topology. i=1 This implies the ﬁrst claim.

Proof of Theorem 8.1: 1) A quasi-inverse functor to φ is the functor ` ` ψ : M = Mj 7→ Spec(HomGk (Mj, ks)) . j∈J j∈J

Here, let Mj be connected, i.e., let Gk operate transitively on Mj, and HomGk (Mj, ks) becomes a k-algebra by the k-algebra structure of ks. Obviously, every connected discrete Gk-set is of the form Gk/U with U ≤ Gk open, and the assignment is

U Gk/U 7→ Spec(ks ) .

57 To show that ψ is quasi inverse to φ it suffices to check this for connected Gk-sets or, respectively, for finite separable field extensions. Let M be a connected discrete Gk-set, without restriction M = Gk/U for an open subgroup U ⊆ Gk. Then

∼ U ψ(M) = HomGk (Gk/U, ks) → ks =: L ⊆ ks α 7→ α(1) is an isomorphism of k-algebras. Conversely ∼ ∼ φ(Spec(L)) = Homk(Spec(ks), Spec(L)) = Homk(L, ks) = Gk/U , is an isomorphism of discrete Gk-sets, by mapping the embedding L,→ ks to 1 ∈ Gk.

By choosing a k-embedding L,→ ks for every finite separable field extension L/k, we obtain an equivalence of categories between étale k-schemes Y and discrete Gk-sets. From these facts it follows that φ is also an equivalence of sites: 0 00 0 00 0 00 2) One has φ(Y ×Y Y ) = (Y ×Y Y )(ks) = Y (Ks) ×Y (ks) Y (ks) 0 0 3) If Spec L → Spec L is étale,then L ⊆ L is a separable field extension of k in ks and the 0 map Homk(L , ks) Homk(L, ks) is surjective, as known from classical algebra.

Corollary 8.4 We have an equivalence of categories

∼ Sh(Spec(k)´et) → (discrete Gk-modules) , F 7→ Fx where Fx is the stalk in the geometric point x = Spec(ks) → Spec(k).

Proof: We show a more general fact.

Deﬁnition 8.5 For a site S = (X , T ), let (X , T )∼ be the category of sheaves of sets on S; this is also called the topos to S.

Theorem 8.6 For a ﬁeld k, the functor

∼ ∼ (Spec(k)´et) → (discrete Gk-sets) , F 7→ Fx is an equivalence of categories.

Corollary 8.4 follows, because in 8.6 the abelian sheaves correspond to the discrete Gk- modules (as the abelian group objects in these categories).

Proof of Theorem 8.6: We have functorial isomorphisms for every sheaf F of sets on Spec(k)´et: F = lim F (Spec(L)) . x → k⊆L⊆ks L/k ﬁn. separable Since the equivalence of categories in Theorem 8.1 assigns

Spec(L) 7→ Gk/U ,

58 where U = Gal(ks/L) ≤ Gk, and, as above, the k ⊆ L ⊆ ks correspond to all open subgroups U ≤ Gk, the claim follows from the following theorem.

Theorem 8.7 (a) Let G be a group and M(G) the category of the left G-sets. Then the surjective families (Mi → M) of G-sets form a Grothendieck topology TG on M(G), the so-called canonical topology. The functor

∼ Φ:(M(G), TG) → M(G) F 7→ MF = F (G)

is an equivalence of categories with quasi-inverse

Ψ FM = HomG(−,M) ←p M.

(b) Let G be a profinite group (i.e., a projective limit of finite groups, equipped with the profinite topology) and let C(G) be the category of the continuous G-sets (with respect to the discrete topology on this sets). Then the surjective families (Mi → M) of discrete G-sets form a Grothendieck topology, the so-called canonical topology, which again is denoted by TG. The functor

∼ Φ:(C(G), TG) → C(G) F 7→ M = F (G) := lim F (G/U) F → U≤G open is an equivalence of categories with quasi inverse

Ψ FM = HomG(−,M) ←p M

Proof (a) Let G be a discrete group. (i) Then F (G) is a left G-set: for g ∈ G, the right translation with g

Rg : G → G g0 7→ g0g

is a morphism of left G-sets and we deﬁne a left G-operation on F (G) by

gx = F (Rg)(x) .

We have (F (Rgg0 ) = F (Rg0 ◦ Rg) = F (Rg) ◦ F (Rg0 ), as F is contravariant). Obviously, here the assignment F p F (G) is functorial. (ii) FM is a sheaf: easy. ∼ (iii) We have a functorial isomorphism MFM → M, since the map

∼ HomG(G, M) → M f 7→ f(1)

is a bijection of G-sets: gf 7→ gf(1) = f(1g) = f(g) = gf(1).

59 ∼ (iv) We have a functorial isomorphism FMF → F , i.e., ∼ HomG(N,F (G)) → F (N) . ` Q Q In fact, for N = Ni we have F (N) = F (Ni), and HomG(N,F (G)) = HomG(Ni,F (G)). i∈I i∈I i∈I By considering the orbits, we may thus assume that N = G/U for a subgroup U ⊂ G. Now we consider the sheaf condition for the covering G G/U. We have a bijection of G-sets ` G → G ×G/U G u∈U gu 7→ (g, gu); and hence the diagram ∼ Q F (G/U) → F (G) ⇒ F (G ×G/U G) → F (G) u∈U (. . . , f, . . .) f ⇒ (. . . , uf, . . .)u∈U is exakt. This gives canonical bijections

∼ U ∼ F (G/U) → F (G) = {f ∈ F (U), uf = f for all u ∈ U} = HomG(G/U, F (G)) ϕ(1) ←p ϕ as wanted. (b) Let G be proﬁnite. (i) We have M = lim F (G/U) , F → UEG open normal factor and MF it becomes a discrete G-Modul, since, by the ﬁrst case, F (G/U) is a G/U-module.

(ii) It follows easily again that FM is a sheaf. (iii) For M in C(G) we have isomorphisms

M = lim Hom (G/U, M) →∼ lim M U →∼ M. FM → G → V ≤G open V ≤G open

(iv) By the ﬁrst case, for every open subgroup U < G and every open normal subgroup 0 0 U E G with U ⊆ U we have F (G/U) →∼ {f ∈ F (G/U 0) | uf = f for alle u ∈ U/U 0} ,

so that ∼ U F (G/U) = (MF ) , and from the above we obtain

∼ FMF (N) = HomG(N,MF ) → F (N) , since, for N = G/U with U open in G, we have

∼ U ∼ FMF (G/U) = HomG(G/U, MF ) = (MF ) = F (G/U) f 7→ f(1) .

60 Remark 8.8 From 8.1 and (the proof of) 8.4 we get an equivalence of categories

Sh(Spec(k)´et) → C(Gk) = (discrete Gk-modules) F 7→ Fx

with quasi-inverse

M 7→ F with F (Spec(L)) = M GL for L/K ﬁnitely separable.

Corollary 8.9 Let k be a ﬁeld with separable closure ks, and let Gk = Gal(ks/k) be the absolute Galois group of k and x : Spec(ks) → Spec(k). Then we have functorial isomorphisms for all ´etaleabelian sheaves F on Spec(k) and all i ≥ 0

i ∼ i H´et(Spec(k),F ) → H (Gk,Fx) , which are compatible with long exact sequences of cohomology.

Proof This follows from the equivalence of categories

Sh(Spec(k)´et) → C(Gk) = (discrete Gk-modules) F 7→ Fx

and the following facts: (i) We have canonical functorial isomorphisms (see 8.8)

∼ Gk F (k) := F (Spec(k)) → Fx .

i (ii) By deﬁnition, H (Gk, −) is the i-th right derived functor of

0 Gk M 7→ H (Gk,M) = M .

Hence ´etalecohomology of ﬁelds is Galois cohomology.

61 9 Henselian rings

Henselian rings, and in particular the strictly henselian rings, play the same role in the ´etale topology as the local rings do in the Zariski topology.

Let A be a local ring with maximal ideal m and factor ﬁeld k = A/m.

Lemma/Deﬁnition 9.1 Let x be the closed point of X = Spec(A). A is called henselian, if the following equivalent conditions hold.

(a) If f ∈ A[X] is monic and f = g0 · h0 with g0.h0 ∈ k[X] monic and coprime (i.e., hg0, h0i = k[X]), then there are monic polynomials g, h ∈ A[X] with f = g · h, g = g0 and h = h0. Here let f = f mod m in k[X]; similarly for g and h). The polynomials g and h are strictly coprime (i.e., hg, hi = A[X]). 0 (a ) If f ∈ A[X] and f = g0 · h0 where g0 is monic and g0 and h0 are coprime, then there exist g, h ∈ A[X] with monic g, f = g · h, g = g0 and h = h0.

(b) Any finite A-algebra is a direct product of local rings Bi. 0 2 (b ) If B is a finite A-algebra, then every idempotent e0 ∈ B/mB (i.e., e0 = e0) can be lifted to an idempotent e ∈ B. (c) If f : Y → X quasi-finite (see below) and separated, then we have a disjoint decomposition Y = Y0 q Y1 q ... q Yr , where x∈ / f(Y0) and where, for i ≥ 1, Yi is finite over X and Yi = Spec(Bi) for a local ring Bi. (d) If f : Y → X is étale and if Y has a point y with f(y) = x and k(x) →∼ k(y), then f has a section s : X → Y (i.e., fs = idX ). 0 n (d ) Let f1, . . . , fn ∈ A[X1,...,Xn] and let a = (a1, . . . , an) ∈ k with f i(a) = 0 for all n i = 1, . . . , n and det(∂f i/∂xj(a)) 6= 0. Then there exists an element c ∈ A with c = a and fi(c) = 0 for i = 1, . . . , n.

Definition 9.2 A morphism f : Y → X of schemes is called quasi-finite, if it is finitely presented (for noetherian schemes: of finite type) and has finite fibers (i.e., f −1(x) is finite for all x ∈ X).

If f is étaleand finitely presented (noetherian schemes: of finite type), then f is quasi-finite.

Proof of the equivalence of the conditions in 9.1: (a0) ⇒ (a) is trivial, except for the last sentence in (a). But if f is monic, then A[x]/hfi is finite over A. Since we have f ∈ hg, hi, i.e., hfi ⊆ hg, hi, D = A[X]/hg, hi is finite over A as well, and by the Nakayama-Lemma we have D = 0, since D/mD = k[x]/hg0, h0i = 0. (a) ⇒ (b): Let B be a finite A-algebra. By the going-up-theorem, every maximal ideal of B lies over m; thus B is local if and only if B/mB is local. Special case: Let B = A[X]/hfi be with a monic polynomial f.

62 If f is a power of an irreducible polynomial, then B/mB = k[X]/hfi is local, hence B is local. Otherwise, by (a) we obtain that f = g · h with g, h monic of degrees ≥ 1 and strictly coprime, and with the Chinese remainder theorem we get

B = A[X]/hfi ∼= A/hgi × A/hhi .

The claim now follows by induction over the number of prime factors of f. General case: Assume B is not local. Then there is an element b ∈ B, such that b is a non-trivial idempotent in B/mB (B/mB is an artinian k-algebra, hence a product of local rings). Since b is integral over A, there is a monic polynomial f ∈ A[X] with f(b) = 0. Then we have a ring homomorphism by evaluating g ∈ A[X] at b

ϕ : C = A[X]/hfi → B,X 7→ b .

Consider the reduction mod m

ϕ : C/mC = k[X]/hfi → B/mB.

Q ni If f = pi , with irreducible polynomials pi, then i

∼ Q ni k[X]/hfi = k[X]/hpi i , i and for the quotient im (ϕ) we have

Q mi im(ϕ) = k[X]/hg0i = k[X]/hpi i i

Q mi with g0 = pi | f. This shows that the idempotent b ∈ im(ϕ) lifts to an idempotent i e ∈ C/mC (the decomposition is unique). By the ﬁrst case there is an idempotent e ∈ C with e mod m = e, hence ϕ(e) = b. Therefore, ϕ(e) is a non trivial idempotent in B. This gives a decomposition of rings B = Be × B(1 − e) in two non trivial rings, and the claim follows by induction over the (ﬁnite) number of components of B/mB. Note: There is a bijective correspondence for commutative rings with unit R:

decomposition into a product idempotents R = R1 × R2 7→ (1, 0) and (0, 1) R = Re × R(1 − e) ←p e and 1 − e (Note also: e idempotent ⇒ 1 − e idempotent). This shows (b) ⇔ (b0). (b) ⇒ (c): We need:

Theorem 9.3 (Stein-factorisation/Zariskis main theorem) Let f : Y → X be a quasi-ﬁnite, separated morphism of schemes, where X is quasi-compact. Then there is a factorization

j f 0 f : Y ,→ Y 0 → X, where f 0 is ﬁnite and j is an open immersion.

63 Remark 9.4 Let f : Y → X be a morphism of schemes. (a) f is called affine, if for any open set U ⊆ X, f −1(U) is affine as well. (b) f is called finite, if f is affine, and for any affine open set U ⊆ X the ringhomomorphism −1 Γ(U, OX ) → Γ(f (U), OY ) is finite.

Proof of Theorem 9.3: Without! See the references in Milne’s ‘Etale´ Cohomology’, page 6.

Now consider f : Y → X = Spec(A), quasi-finite and separated, with a local ring A as above. Let j f 0 Y ,→ Y 0 → X be a Stein factorization as in Theorem 9.3. Since f 0 is affine, Y 0 = Spec(B0) is affine. By (b), we get r 0 ` Y = Yi , i=1

where Yi = Spec(Bi) for a local ﬁnite A-algebra Bi. Let ` Y∗ = Yi i∈I

0 be the product of those Yi, whose closed point yi lies in Y . Then Y∗ is open and closed in Y and lies in Y , because Yi is the smallest open neighborhood of yi ∈ Yi. Therefore Y∗ is open and closed in Y as well, and we have

Y = Y0 q Y∗ ,

where x∈ / f(Y0), hence (c). (c) ⇒ (d): Let f : Y → X = Spec(A) be étaleand let y ∈ Y be a point with f(y) = x and k(x) →∼ k(y). By replacing Y with an affine open neighborhood of y, f is quasi-finite and separated without restriction. Then, by (c), we may assume that Y = Spec(B), where B is local and finite over A. Since f is étale,we have

B/mB = B/mB = k(y) = k(x) = A/m .

Hence, by the Nakayama-Lemma, B is generated by 1 ∈ B as an A-module. By this we obtain an exact sequence 0 → a → A → B → 0 , with an ideal a ⊆ A. Since B is ﬂat over A,

β 0 → a ⊗A B → A ⊗A B → B ⊗A B → 0 is exact. The homomorphism β can be identiﬁed with the map

i2 : B → B ⊗A B , b 7→ 1 ⊗ b .

This map is injective, since the composition with µ : B ⊗A B → B, b1 ⊗ b2 7→ b1b2 is the identity. It follows that a ⊗A B = 0, therefore a = 0, since A → B is faithfully ﬂat, as a ﬂat

64 homomorphism of local rings (see Corollary 10.4 below). Hence A →∼ B is an isomorphism, and this gives the wanted section. 0 n (d) ⇒ (d ): Let B = A[X1,...,Xn]/hf1, . . . , fni and a = (a1, . . . , an) ∈ k with f i(a) = 0 (i = 1, . . . , n) and det(∂fi/∂Xj(a)) 6= 0 in k. Then a corresponds to a maximal ideal of (B/mB hence also of) B; let this be denoted by n. Then det(∂fi/∂Xj) is a unit in Bn, thus there is an element b ∈ B r n such that det(∂fi/∂Xj) is a unit in Bb. But we have ∼ Bb = A[X1,...,Xn,T ]/hf1, . . . , fn, bT − 1i

and det(∂fi/∂Xj)b is the corresponding Jacobian determinant, hence a unit. By the Jacobian criterion, Bb is ´etaleover A. Furthermore, n gives a maximal ideal of Bb over m with residue ﬁeld isomorphic to k = A/m. By (d), a section s : Spec(A) → Spec(Bb) exists, i.e., an n A-homomorphism Bb → A and this gives an element c ∈ A with fi(c) = 0 for i = 1, . . . , n and c = a (since n lies over m). 0 0 n (d ) ⇒ (a ): Let f(X) = anX + ... + a1X + a0 ∈ A[X] and let f = g0 · h0 monic with g0 of degree ≥ 1. Then we have

r r−1 s f(X) = g(X) · h(X) = (X + br−1X + ... + b0)(csX + ... + c0)

n+1 with r + s = n, if and only if (b0, . . . , br−1, c0, . . . , cs) ∈ A solves the following system of equations in the n + 1 variables (X0,...,Xr−1,Y0,...,Ys):

X0Y0 = a0 X0Y1 + X1Y0 = a1 X0Y2 + X1Y1 + X2Y0 = a2 (9.1.1) . . Xr−1Ys + Ys−1 = an−1 Ys = an (n + 1 equations). The corresponding Jacobian is r s z }| { z }| {  Y0 0 X0 0   Y1 Y0 X1 X0     Y2 Y1 Y0 X2 X0   . . . .   . . . .   . . . .   .  J = det  Y .   s   . . .   Ys . . .     . .   . 1 .  . . 1 This is just res(G, H), the resultant of the two polynomials

r r−1 G = x + Xr−1x + ... + X0 s s−1 H = Ysx + Ys−1x + ... + Y0

0 By (d ) there is a solution of (9.1.1), if Res(g0, h0) 6= 0 in k (because then the vector (b, c) of the coeﬃcients of g0 resp. h0 solves the system (9.1.1) modulo m and we have J(b, c) 6= 0).

65 But by classical algebra, Res(g0, h0) is 0 if and only if deg(g0) < r and deg(h0) < s, or if g0 and h0 have a common factor; by assumption, this is not the case.

Corollary 9.4 If A is henselian, then any local ring B ﬁnite over A is henselian. In particular, any factor ring A/J is henselian.

Proof: This follows with criterion 9.1 (b), since a ﬁnite B-algebra is ﬁnite over A.

Corollary 9.5 If A is henselian, then the functor

B p B ⊗A k = B/mB gives an equivalence of categories finite étale ∼ finite étale → . A-algebras k-algebras

Proof: This follows with the criteria of 9.1(b), (b0) and (d). Details: left to the readers!

Every ﬁeld is a henselian ring, as well as every artinian ring (because every artinian ring is a product of local rings). Furthermore we have:

Proposition 9.6 Every complete local ring is henselian.

Proof We use criterion 9.1 (d). Let B be an étale A-algebra, and let s0 : B/mB → k be a section of k → B/mB. To find a section s : B → A ∼= lim A/mr ← r of A → B, it suffices to find compatible A-linear maps for all r ≥ 1 r sr : B → A/m .

can s0 For r = 1 we take s1 : B −→ B/m −→ A/m = k. If sr is found for some r ≥ 1, then the existence of sr+1 follows from the formal smoothness of B over A: The lift sr+1 of sr exits in the diagram sr B / A/mr O G G O Gsr+1 G ϕr G# A / A/mr+1 , r r+1 because ker(ϕr) = m /m is a nilpotent ideal.

Lemma/Deﬁnition 9.7 Let A be a local ring. There is a henselian ring Ah with the following universal properties: There is a local homomorphism i : A → Ah, and any local homomorphism ϕ : A → B into a henselian local ring B factorizes uniquely over i (we have a unique homomorphismϕ ˜, which makes the following diagram commutative):

i A / Ah ? ?? } ?? } ϕ ?? } ∃!ϕ ˜ ? ~} B

66 The ring Ah is called the henselization of A.

To construct Ah we need

Definition 9.8 Let A be local with maximal ideal m. An étale(resp. essentially étale) neighborhood of (A, m) is a pair (B, n), such that B is an étale A-algebra (resp. a localization of an étale A-algebra) and n ⊆ B is an ideal over m, such that the induced map k = A/m → B/n = k(n) is an isomorphism (then n is a maximal ideal as well).

Lemma 9.9 (a) If (B, n) and (B0, n0) are (essentially) étaleneighborhoods of (A, m) with Spec(B0) connected, then there is at most one an A-Homomorphism f : B → B0 with f −1(n0) = n. (b) If (B, n) and (B0, n0) are (essentially) étaleneighborhoods of (A, m), then there is an (essentially) étaleneighborhood (B00, n00) of A and A-homomorphisms B SSS SSS f SSS SSS SSS SSS S) B00 k5 kkk kkk kkk kkk f 0 kkk B0 kk with f −1(n00) = n and (f 0)−1(n00) = n0.

Proof: (a) follows from the following, more general result:

Lemma 9.10 Let f, g : Y 0 → Y be morphisms of X-schemes, where Y 0 is connected and Y is ´etaleand separated over X. If there is a point y0 ∈ Y 0 with f(y0) = g(y0) = y, and such that the maps k(y) → k(y0) induced by f and g are equal, then we have f = g.

0 0 Proof Let Γf , Γg : Y → Y ×X Y be the graphs of f and g, respectively (Γf = (idY 0 , f) 0 0 similarly for g). These are sections of pr1 : Y ×X Y → Y , and pr1 is ´etaleand separated 0 0 as a base change of Y → X. The assumption implies that Γf (y ) = Γg(y ). Then Γf = Γg ´ (see Milne, Etale cohomology, I Cor. 3.12), and the claim follows, since f = pr2 ◦ Γf and g = pr2 ◦ Γg. 00 0 0 0 For (b) consider B = B ⊗A B . The homomorphisms B → k(n) = k and B → k(n ) = k 00 00 00 00 induce a homomorphism α : B k. If n = ker α, then (B , n ) has the required property.

The above implies that the connected étale(resp. essential étale)neighborhoods of A form an inductive system, and we define (Ah, mh) = lim (B, n) = lim (B, n) . → → (B,n) étale (B,n) ess. étale neighb. of (A, m) neighb. of (A, m) Note: the étale A-algebras B are of finite presentation and therefore form an index set without restriction. 1) Ah is local with maximal ideal mh: It suffices to show that Ah r mh consists of units. Let x ∈ Ah, represented by y ∈ B,(B, n) étale neighborhood of (A, m). If x∈ / mh, then y∈ / n,

67 hence y is a unit in Bn. Therefore there is an element b ∈ B − n such that y is a unit in Bb. Then (Bb, nb) is an étaleneighborhood of (A, m), and the image of the inverse of y in Bb is an inverse of x, i.e., x is a unit. ∼ h h 2) Obviously we have k → lim B/nB = A /m . → 3) A → Ah ist a local homomorphism, since m is mapped into mh. 4) Ah is henselian: We use the section criterion 9.1 (d): Let Ah → C be étale, c ⊆ C be an ideal over mh with k = k(mh) →∼ k(c). Since C is of finite type over Ah = lim B, there → h exists an étaleneighborhood (B0, n0) of (A, m) with C = B0 ⊗A A (Consider a presentation h C = A [X1,...,Xn]/hf1, . . . , fmi and a B0 such that the finitely many coefficients of the fi h lie in the image of B0 → A ). Then we obtain a section

C = B0 ⊗A lim B → lim B → →

of Ah → C: Without restriction, we consider the coﬁnite family of the ´etaleneighborhoods (B, n) with a (uniquely determined!) morphism (B0, n0) → (B, n), and then the homomorphism above is induced by the homomorphisms

B0 ⊗A B → B , b0 ⊗ b 7→ b0b .

5) Universal property: Let (C, nC ) be henselian and ϕ : A → C a local morphism. We look for the homomorphismϕ ˜, which makes the diagram

i A / Ah ? ?? } ?? } ϕ ?? } ∃!ϕ ˜ ? ? ~} C

commutative in a unique way. It suﬃces to show: For all ´etale neighborhoods (B, n) of (A, m), there exists a unique homomorphism ϕB, which makes the diagram

A / B @ @@ ~~ @@ ~~ ϕ @@ ~~∃! ϕ @ ~~ B C

commutative. Equivalent: In the commutative diagram

A / B ⊗A C ? : ?? vv ?? vv ϕ ?? vvψ ? vv C v

∼ there exists a unique A-linear section B ⊗A C → C of ψ. But from k = A/m → B/n ∼ we get a surjective homomorphism ψ : B ⊗A C C/nC and an isomorphism C/nC → (B ⊗A C)/ ker(ψ). Since C is henselian, by 9.1 (d) we obtain a section of C → B ⊗A C as wanted.

Proposition 9.11 (a) Ah is ﬂat over A.

68 (b) Let Aˆ = lim A/mr ← r Aˆh = lim Ah/(mh)r ← r be the completions of A and Ah, respectively. Then Aˆ →∼ Aˆh is an isomorphism.

Proof (a): Ah is flat as direct limit of flat A-algebras (the tensor product commutes with direct limits). (b) It suffices to show that A/mr →∼ B/nr for all étaleneighborhoods (B, n) of (A, m). But we have A/m →∼ B/n and mB = n by assumption; hence mrB = nr for all r and

r r+1 ∼ r r+1 ∼ r r+1 r r+1 (9.11.1) m /m = m /m ⊗A/m B/n = m B/m B = n /n .

The isomorphism in the middle of (9.11.1) follows from the ﬂatness of B over A: By this, the exact sequence 0 → mr+1 → mr → mr/mr+1 → 0 induces an exact top row in the commutative diagram

r+1 r r r+1 (9.11.2) 0 / m ⊗A B / m ⊗A B / m /m ⊗A B / 0

o o 0 / mr+1B / mrB / mrB/mr+1B / 0 ,

and the indicated vertical isomorphisms, since, by ﬂatness of B over A, the injection mr ,→ A r r induces an injection m ⊗A B → A⊗A B = B with image m B (similarly for r +1). Therefore the vertical map on the right is an isomorphism. From (9.11.1) the isomorphisms A/mr →∼ B/mr+1 follows inductively, via the commutative diagram with exakt rows

0 / mr/mr+1 / A/mr+1 / A/mr / 0

o 0 / nr/nr+1 / B/nr+1 / B/nr / 0 .

Remark 9.12 One can show: (a) If A is noetherian, then Ah is noetherian, too.

(b) Let A be integral and normal, with fraction ﬁeld K. Let Ks be a separable closure of K, As the integral closure of A in Ks, m ⊂ As a maximal ideal over m and Zm ⊂ Gal(Ks/K) = GK h Zm the decomposing group. Then we have A = As .

Definition 9.13 Let X be a scheme and let x ∈ X be a point. An étaleneighborhood of x is a pair (U, y), with f : U → X étale, f(y) = x and k(x) →∼ k(y).

69 Lemma 9.14 (a) With the obvious morphisms (U, y) → (U 0, y0), i.e.

U / U 0 , y / y0 , ? ?? ~~ ?? ~~ ?? ~~ ? ~ ~ X

these form a cofiltered category. (b) We have Oh = lim Γ(U, O ) = lim O . X,x → U → U,y U étale (U,y) étale neighb. of x neighb. of x

Proof Analogous to the proof of 9.7.

Deﬁnition 9.15 A strictly henselian ring is a henselian ring whose residue ﬁeld is separably closed.

Lemma 9.16 (a) For every local ring (A, m) there exists a strictly henselian ring (Ash, msh) and a local morphism i : A → Ash, which satisﬁes the following universal properties: If ϕ : A → C is a local morphism in a strict henselian ring (C, mC ), and if a k = A/m- embedding sh sh ϕ0 : A /m → C/nC is given, then there exists a unique local morphismϕ ˜ : Ash → C, which makes the diagram

i A / Ash ? ?? | ?? | ϕ ?? | ∃!ϕ ˜ ? C }| commutative and induces the embedding ϕ0 of the residue ﬁelds. (b) For a scheme X, a point x ∈ X and a geometric point x of X over x we have

Osh ∼= O := lim Γ(U, O ) = lim O . X,x X,x → U → U, image of x U ´etale U ´etale neighb. of x neighb. of x

Proof Analogous to the proof of 9.7.

Lemma 9.17 Let X be a strictly local scheme, i.e., the spectrum of a strictly local ring. Let x be the closed point, considered as geometric point of X. Then, for sheaves F on X there exists a functorial isomorphism ∼ 0 Fx = H (X, F) , and we have Hq(X, F) = 0 for q > 0.

Proof: Let X0 be an ´etaleneighborhood of x. By 9.16 there exists exactly one local section of X0 → X so that X is initial in the set of all ´etaleneighborhoods of x. By 9.14 (b),

70 ∼ 0 q Fx = H (X, F), functorial in F. Since the stalk functor is exact, we have H (X, F) = 0 for q > 0.

Lemma 9.18 Let f : X → Y be a ﬁnite morphism, where Y is a strictly local scheme. For an ´etalesheaf F on X we then have

0 ∼ Q H (X, F) = Fx x∈Xy Hq(X, F) = 0 for q > 0 .

Proof Since X → Y is finite and Y is strictly local, we have Y X = Spec(OX,x) , x where x runs over the finitely many closed points of X, which are the points over the closed points y of Y (see Lemma 9.1). Every ring OX,x ist henselian (see Lemma 9.1), and even strictly henselian, since k(x), as finite extension of the spectral closed field k(y), is again separably closed.

71 10 Examples of ´etalesheaves

First we consider sheaves (of sets), represented by schemes.

Remark 10.1 In a category X , a morphism f : Y → X is an epimorphism if and only if for every object Z in X the morphism

f ∗ : Hom(X,Z) → Hom(Y,Z) is injective.

Deﬁnition 10.2 Let X be a category with ﬁber products. A morphism f : Y → X is called a strict epimorphism, if for all objects Z in X the sequence

pr1 f Y ×X Y ⇒ Y → X pr2 is exact, i.e., if f is the diﬀerence cokernel of pr1 and pr2, i.e., if for all objects Z in X the sequence pr∗ f ∗ 1 Hom(X,Z) → Hom(Y,Z) Hom(Y ×X Y,Z) ⇒∗ pr2 ∗ ∗ ∗ ∗ is exact, i.e., if f is the diﬀerence kernel of pr1 and pr2 (In particular, this implies that f is injective for all Z, i.e., that f is an epimorphism).

Definition 10.3 A morphism of schemes f : Y → X is called faithfully flat, if f is flat and surjective.

The following lemma shows that this corresponds to the usual notions for aﬃne schemes.

Lemma 10.4 Let ϕ : A → B be a ﬂat ring homomorphism. Then the following conditions are equivalent

(a) ϕ is faithfully ﬂat, i.e., for an A-module M we have M = 0 if M ⊗A B = 0. 0 00 0 00 (b) A sequence M → M → M of A-modules is exact, if B ⊗A M → B ⊗A M → B ⊗A M is exact. (c) Spec(B) → Spec(A) is surjective. (d) For any maximal ideal m of A, we have mB 6= B.

ϕ ψ Proof (a) ⇒ (b): Assume that M 0 → M → M 00 becomes exact after tensoring with B. By the ﬂatness of B over A, we have im(ψϕ) ⊗A B = im((ψ ⊗ id) ◦ (ϕ ⊗ id)) = 0, hence, by (a), we have im(ψϕ) = 0, i.e., ψϕ = 0. Furthermore we have ker(ψ)/im(ϕ) ⊗A B = ker(ψ ⊗ id)/im(ϕ ◦ id) = 0, hence ker(ψ)/im(ϕ) = 0. (b) ⇒ (a): 0 → M → 0 is exact if and only if M = 0.

(a) ⇒ (c): For any prime ideal p ⊆ A, we have B ⊗A k(p) 6= 0, hence

−1 Spec(ϕ) (p) = Spec(B ⊗A k(p)) 6= ∅ .

72 (c) ⇒ (d) is obvious, since the prime ideals over m correspond to the prime ideals of B/mB. (d) ⇒ (a): Let x ∈ M, x 6= 0, and N = Ax ⊆ M. By the ﬂatness of B over A it suﬃces to ∼ show B ⊗A N 6= 0 (then we also have B ⊗A M 6= 0). But N = A/J for an ideal J $ A, so that ∼ B ⊗A N = B/JB. If m ⊆ A is a maximal ideal with J ⊆ m, then we have JB ⊆ mB 6= B, therefore B/JB 6= 0.

Corollary 10.5 A ﬂat morphism of local rings ϕ : A → B is faithfully ﬂat.

Proof This follows from 10.4 (d), since, by assumption, ϕ(m) ⊆ n for the maximal ideals m and n of A and B, respectively.

Theorem 10.6 Let f : Y → X be a morphism of schemes. If f is faithfully ﬂat and of ﬁnite type, then f is a strict epimorphism.

Proof For any scheme Z we have to show the exactness of

Hom(X,Z) → Hom(Y,Z) ⇒ Hom(Y ×X Y,Z) .

First case: Let X = Spec(A),Y = Spec(B) and Z = Spec(C) be aﬃne. In this case, the sequence above can be identiﬁed with the sequence

Hom(C,A) → Hom(C,B) ⇒ Hom(C,B ⊗A B) , and the claim follows from the exactness of the sequence

A → B → B ⊗A B, b 7→ b ⊗ 1 − 1 ⊗ b .

Second case: Let X = Spec(A) and Y = Spec(B) be aﬃne and let Z be arbitrary. Let h ∈ Hom(Y,Z) be given, with h pr1 = h pr2. We have to show that there is a unique g ∈ Hom(X,Z) with gf = h.

First we show the uniqueness of g (if it exists). Let g1, g2 : X → Z be given with g1f = g2f. Then g1 and g2 have to coincide as maps of topological spaces, since f is surjective. Let x ∈ X, and let U be an affine open neighborhood of g1(x) = g2(x) in Z. Then there is an a ∈ A with g1(D(a)) = g2(D(a)) ⊆ U. Furthermore, Ba is faithfully flat over Aa. From the first case we get that g1|D(a) = g2|D(a) as scheme morphisms, hence the uniqueness of g.

Now let h : Y → Z be given with h pr1 = h pr2. Because of the proven uniqueness of g it suffices to define g locally. Consider x ∈ X, y ∈ Y with f(y) = x, and let U ⊆ Z be an affine open neighborhood of h(y) in Z. We now use the following lemma.

Lemma 10.7 Let f : Y → X be ﬂat and of ﬁnite type. Then f is an open map.

Proof See Milne, ‘Etale Cohomology’, p. 14, Th. 2.1.

In our setting, we deduce that f(h−1(U)) is open in X. We have a diagram

73 y x

/ f Y ×X Y / Y / X = Spec(A)

h h(y) ∈ U ⊆ Z

By 10.7, there exists an a ∈ A with x ∈ D(a) ⊆ f(h−1(U)). Then f −1(D(a)) ⊆ h−1(U). In fact, if we have x1 ∈ D(a), hence x1 = f(y1) with h(y1) ∈ U, and if we have y2 ∈ Y with 0 0 f(y2) = x1, then, since f(y1) = f(y2), there is an element y ∈ Y ×X Y with pr1(y ) = y1 0 and pr2(y ) = y2 (consider a point in y1 ×x1 y2 and its image in Y ×X Y ). We get

0 0 h(y2) = h pr2(y ) = h pr1(y ) = h(y1) ∈ U,

−1 hence y2 ∈ h (U). If b ∈ B is the image of a, then D(b) = f −1(D(a)), hence h(D(b)) ⊆ U, and by the first case we obtain g|D(a). As stated previously, these local solutions glue together to a global g. Third case: Let X,Y and Z be arbitrary. We easily reduce to the case where X is affine (choose an affine open covering and its inverse image in Y ; because of the uniqueness, the morphisms g glue on the covering). Since f is quasi-compact, Y is a finite union Y = Y1 ∪ ... ∪ Yn of affine open subsets. Let

n ∗ ` Y = Yi . i=1 Then Y ∗ is aﬃne and Y ∗ → Y is faithfully ﬂat. We obtain a commutative diagram

/ Hom(X,Z) / Hom(Y,Z) / Hom(Y ×X Y,Z)

∗ / ∗ ∗ Hom(X,Z) / Hom(Y ,Z) / Hom(Y ×X Y ,Z) ,

where, by the second case, the bottom row is exact. Furthermore, the middle vertical map is obviously injective (Hom(−,Z) is a Zariski sheaf on Y ). The exactness of the top row now follows by a diagram chase.

Before we use Theorem 10.6 for the construction of sheaves, we introduce a useful criterion for a presheaf to be a sheaf.

Proposition 10.8 A presheaf P (of sets or abelian groups) on X´et (resp. Xﬂ) is a sheaf if and only if the following two conditions hold:

(a) For any U ∈ Xét (resp. Xfl), the restriction of P to U is a Zariski sheaf. (b) For any étalecovering (U 0 → U), where U and U 0 are affine,

0 0 0 P (U) → P (U ) ⇒ P (U ×U U ) is exact.

74 Proof Obviously, these properties are necessary. Conversely, if (a) holds, then for any disjoint ` sum V = Vj of schemes we have j Q P (V ) = P (Vj) . j

For a covering (Ui → U) the sequence Q Q (10.8.1) P (U) → P (Ui) ⇒ P (Ui ×U Uj) i i,j

is isomorphic to the sequence

0 0 0 (10.8.2) P (U) → P (U ) ⇒ P (U ×U U )

0 0 ` for the covering U → U with U = Ui, since i ` ` ∼ ` ( Ui) ×U ( Ui) = Ui ×U Uj . i,j

If (b) holds, the sequence (10.8.1) is exact for every covering (Ui → U)i∈I with finite I and ` ` ∼ affine Ui and U, because then Ui is affine: Spec(A) Spec(B) = Spec(A × B). For any 0 0 ` 0 (Uj → U) write U as the union of affine open sets Ui, U = ∪ Ui, and write f : U = Uj → U. i j −1 0 0 −1 0 Then we have f (Ui) = ∪ Uik with affine open subsets Uik ⊆ f (Ui). Since U → U is flat k∈Ki 0 and locally of finite type, Uik → Ui is of finite type (since both schemes are affine), therefore 0 f(Uik) is open in Ui by Lemma 10.7. Since Ui (as an affine scheme) is quasi-compact, there 0 0 is a finite index set Ei ⊆ Ki with Ui = ∪ f(Uik), i.e., (Uik → Ui)k∈Ei is a covering. By k∈Ei 0 0 adding all morphisms of the form Uik → f(Uik) for k ∈ Ki − Ei, we can assume that all Ki 0 0 are finite, and U = ∪Uik. Now we consider the commutative diagram

0 / 0 0 P (U) / P (U ) / P (U ×U U )

/ Q / Q Q 0 Q Q 0 0 P (Ui) P (Uik) / P (Uik ×U Ui`) i i k i k,`

Q / Q Q 0 0 P (Ui ×U Uj) P (Uik ∩ Uj`) . i,j i,j k,`

By 10.8 (a), the first two columns are exact, and by the first remark about (b), the second 0 0 row is exact (k runs in the finite set Ki for Uik). Therefore P (U) ,→ P (U ) is injective, hence the presheaf P is separated (since (Ui → U) was arbitrary). This in turn implies the injectivity of the bottom arrow. Now an easy diagram chase shows the exactness of the top row.

Corollary 10.9 For every X-scheme Z, the functor represented by Z,

HomX (−,Z): U 7→ Z(U) := HomX (U, Z)

75 is a sheaf of sets on X´et and Xﬂ.

Proof Condition 10.8 (a) is well-known (glueing of morphisms). Condition 10.8 (b) follows from Theorem 10.6: If f : U 0 → U is a surjective étale(or flat) X-morphism, with U and U 0 affine, then f is faithfully flat and of finite type, hence the diagram

0 0 0 Hom(U, Z) → Hom(U ,Z) ⇒ Hom(U ×U U ,Z)

0 is exact by 10.6. Let πU : U → X, πU 0 : U → X and πZ : Z → X be the structure morphisms. We then have a commutative diagram

0 / 0 0 HomX (U, Z) / HomX (U ,Z) HomX (U ×U U ,Z) _ _ / _

0 / 0 Hom(U, Z) / Hom(U ,Z) / Hom(U ×U U ,Z)

(πZ )∗ (πZ )∗ f ∗ Hom(U, X) / Hom(U 0,X)

πU / πU 0 where the middle row is exact for all schemes Z, so that the bottom morphism f ∗ is injective as well. Furthermore we have

HomX (U, Z) = {g ∈ Hom(U, Z) | (πZ )∗(g) = πZ g = πU } ,

and the same for U 0 in place of U. This implies the exactness of the top row in the diagram.

Corollary 10.10 For every (abelian) group scheme G over X, the functor represented by G is a sheaf of (abelian) groups on X´et and Xﬂ.

Examples 10.11 (a) The sheaf Ga,X = X ×Spec(Z) Ga,Z = X ×Spec(Z) Spec(Z[T ]) satisﬁes

Ga,X (U) = Γ(U, OU ) , for every X-scheme U and is called the additive group over X. −1 (b) The sheaf Gm,X = X ×Spec(Z) Gm, with Gm = Spec(Z[T,T ]) has the value

× × Gm,X (U) = Γ(U, OU ) = Γ(U, OU ) , for every X-scheme U and is called the multiplicative group over X. n (c) For every n ∈ N let µn be the sheaf X ×Spec(Z) Spec(Z[T ]/hT − 1i). It satiesﬁes

× n µn(U) = {a ∈ Γ(U, OU ) |a = 1} ,

for every X-scheme U and is called the sheaf of the n-th unit roots.

Lemma 10.12 The sequence of sheaves

n 1 → µn → Gm → Gm a 7→ an

76 is exact.

Proof: For every X-scheme U,

n 1 / µn(U) / Gm(U) / Gm(U)

× × 1 / µn(U) / Γ(U, OU ) / Γ(U, OU )

a / an is exact.

Proposition 10.13 Let n be invertible on the scheme X (⇔ for all x ∈ X, n is not divisible by char(k(x))). Then the sequence of ´etalesheaves on X n 1 → µn → Gm → Gm → 1 is exact. For any n, the sequence n 1 → µn → Gm → Gm → 1 of ﬂat sheaves is exact. The sequences above are called the Kummer sequence.

n Proof One only has to show that, by the assumptions, Gm → Gm is an epimorphism. We use the criterion 4.2 (d). Let U be a scheme and s ∈ Gm(U). By passing to an affine open × × covering, we may assume that U = Spec(A) is affine and s = a ∈ Γ(U, OU ) = A . Then B = A[T ]/hT n − ai is a faithfully flat A-algebra of finite type, hence V = Spec(B) → Spec(A) = U is a flat covering, and for a ∈ A× ⊆ B× there is an element b ∈ B× (the image n n of T in B) with b = a. Therefore, by the exactness criterion of 4.2 (d), Gm → Gm is an epimorphism of flat sheaves. If n is invertible on X, then n ∈ A× and therefore B is an étale n A-algebra, so that Gm → Gm is an epimorphism of étalesheaves.

Lemma 10.14 Let X be a scheme. For every group G, the corresponding constant Zariski, P P ´etaleor ﬂat sheaf GX (i.e., the sheaf associated the constant presheaf G with G (U) = G for all U → X) is representable by the group scheme ` ` GX = X = X[g] g∈G g∈G with the obvious group law.

Proof Consider the category Sch/X of all X-schemes. One can easily see that for every X-scheme Y we have:

GX (Y ) = HomX (Y,GX ) = {ϕ : Y → G | ϕ locally constant} ` −1 (If ϕ : Y → G is locally constant, then Y = ϕ (g)). Furthermore GX is a sheaf for all g∈G Zar three considered topologies, by 10.10. Finally for the Zariski sheaf GX associated to G we have Zar GX (Y ) = {ϕ : Y → G | ϕ locally constant}

77 The claim follows, since this gives a sheaf for all three considered topologies.

Remark 10.15 If X is locally noetherian and π0(Y ) is the set of connected components of Y , we have π0(Y ) GX (Y ) = G for all Y → X, which are locally of ﬁnite type (and therefore again locally noetherian).

Proposition 10.16 Let X be a scheme of characteristic p > 0 (⇔ the morphism X → Spec(Z) factorizes over Spec(Fp) ⇔ for every open U ⊆ X we have pΓ(U, OU ) = 0). Let

F : Ga,X → Ga,X be the Frobenius homomorphism: For U → X ﬂat (or ´etale)let

F : Γ(U, OU ) → Γ(U, OU )

be the ring homomorphism (!) a 7→ ap (this is additive, since pa = 0). Then one has an exact sequence of ﬂat (or ´etale)sheaves

F −1 0 → Z/pZX → Ga,X → Ga,X → 0 ,

(called Artin-Schreier sequence), where Z/pZX is the constant sheaf, associated to the abelian group Z/pZ, which is represented by the group scheme

P /p = X ×Spec ( ) Spec( p[T ]/hT − T i) . Z ZX Fp F

Proof: Since T p − T = Q (T − i) i∈Fp

in Fp[T ], we have an isomorphism

p−1 p ∼ Q Fp[T ]/hT − T i = Fp . i=0

p Therefore Spec(Fp[T ]/hT − T i) is canonically isomorphic to the constant group scheme

Z/pZSpec(Fp). Furthermore, for every X-scheme Y we have

HomX (Y, Z/pZ ) = Hom(Y, Z/pZ ) X Fp p p = Hom(Fp[T ]/hT − T i, Γ(Y, OY )) = {a ∈ Γ(Y, OY ) | a − a = 0} . This shows the exactness of

F −1 0 → Z/pZX → Ga,X → Ga,X . Furthermore, F − 1 is an epimorphism of sheaves for the étale(and flat) topology, since for U = Spec(A) ⊆ Y affine open and a ∈ A

V = Spec(A[T ]/hT p − T − ai) → U

78 is a covering, so that a = (F − 1)b = bp − b if b is the image of T in B = A[T ]/hT p − T − ai.

Lemma/Deﬁnition 10.17 Let X again be a scheme of characteristic p > 0 and let αp,X ⊆ Ga,X be the subsheaf deﬁned by

p αp,X (U) = {a ∈ Γ(U, OU ) | a = 0} .

p Then αp,X is represented by the group scheme X ×Spec(Fp) Spec(Fp[T ]/hT i). The sequence

F 0 → αp,X → Ga,X → Ga,X is exact for the Zariski or ´etaletopology. The sequence

F 0 → αp,X → Ga,X → Ga,X → 0 is exact for the ﬂat topology, but not in general for the ´etaletopology.

Proof: The first claims are obviously true. The morphism F is an epimorphism in the flat p topology, since for every Fp-algebra A the algebra B = A[T ]/hT − ai is faithfully flat over A. For a separably closed field L the Frobenius morphism

L →F L a 7→ ap is not surjective in general, except if L is a perfect ﬁeld.

Lemma/Deﬁnition 10.18 Let X be a scheme and let M be a quasi-coherent OX -module. Then

U 7→ Γ(U, M ⊗OX OU )

(for X-schemes πU : U → X) is a sheaf on (Sch/X)fl, the site of all X-schemes with the −1 ∗ −1 flat topology. Here M ⊗OX OU stands for πU M ⊗π OX OU = πU M, the quasi-coherent pull-back. In particular, this gives a sheaf on the small sites Xét and Xfl, called Mét and Mfl, respectively.

Proof We use the criterion of Theorem 10.8. Condition 10.8 (a) is obviously true, since, by 0 construction, M ⊗OX OU is a Zariski sheaf on U. For 10.8 (b) let U = Spec(B) → Spec(A) be aﬃne and faithfully ﬂat. Then M ⊗OX OU corresponds to an A-module M, M ⊗OX OU corresponds to a B-module B ⊗A M, and we have to show the exactness of

M → B ⊗A M ⇒ B ⊗A B ⊗A M. But by the theory of ﬂat descent this holds, since A → B is faithfully ﬂat (see Theorem Lemma 14.6 below).

Remark 10.19 The case Ga,X = OX, ´et is a special case.

79 11 The decomposition theorem

Let X be a scheme, let i : Y,→ X be a closed immersion and let j : U,→ X be the open immersion of the open complement U = X − Y .

∗ ∗ If F is an étalesheaf on X, then F1 = i F is an étalesheaf on Y and F2 = j F is an ad ∗ étalesheaf on U. Let F → j∗j F be the adjunction map (which, under the isomorphism ∗ ∗ ∼ ∗ HomU (j F, j F ) = HomX (F, j∗j F ), corresponds to idj∗F ). Then i∗(ad) gives a morphism

∗ ∗ ∗ ∗ φF : F1 = i F → i j∗j F = i j∗F2 .

Therefore we can associate the triple (F1,F2, φF ) to every sheaf F on X.

Let T (X,Y ) be the the category of triples (F1,F2, φ), where F1 is an étalesheaf on Y , F2 ∗ an étalesheaf on U and φ : F1 → i j∗F2 is a morphism of étalesheaves on Y . Morphisms

0 0 0 (F1,F2, φ) → (F1,F2, φ )

0 0 are pairs (ψ1, ψ2), where ψ1 : F1 → F1 and ψ2 : F2 → F2 are morphisms of sheaves on Y resp. U, such that the diagram

φ ∗ (11.0) F1 / i j∗F2

∗ ψ1 i j∗(ψ2) 0 0 φ ∗ 0 F1 / i j∗F2 commutes.

Theorem 11.1 Let the functor

t : Sh(X´et) → T (X,Y )

be defined as follows: ∗ ∗ ∗ ∗ ∗ ∗ (i) To a sheaf F ∈ Shét(X) assign the triple (i F, j F, i (ad): i F → i j∗j F ). 0 (ii) To a morphism ϕ : F → F in Sh(Xét) assign the morphism in T (X,Y )

∗ ∗ ∗ 0 ∗ ∗ ∗ 0 ψ(ϕ) = (ψ1 = i (ϕ): i F → i F , ψ2 = j ϕ : j F → j F )

Then t is an equivalence of categories.

Proof First of all, t is well-deﬁned: by the preliminary remark, the triple in (i) is an object in T (X,Y ), and ψ(ϕ) is a morphism in T (X,Y ), since the diagram

∗ ∗ i (adF )∗ ∗ (11.1.1) i F / i j∗j F

∗ ∗ ∗ i (ϕ) i j∗j (ϕ) ∗ ∗ 0 i (adF 0 )∗ ∗ 0 i F / i j∗j F

80 commutes (Functoriality of the adjointness and then applying the functor i∗).

Now we define a pseudoinverse s for t. For a triple (F1,F2, φ), let s(F1,F2, φ) ∈ Sh(Xét) be ∗ the fiber product of i∗F1 and j∗F2 over i∗i j∗F2, so that the diagram

(11.1.2) s(F1,F2, φ) / j∗F2

adi

i∗(φ) ∗ i∗F1 / i∗i j∗F2 is cartesian. This assignment is functorial, since forming the ﬁber product is functorial: Every morphism 0 0 0 (ψ1, ψ2):(F1,F2, φ) → (F1,F2, φ ) induces a morphism 0 0 s(F1,F2, φ) → s(F1,F2, φ) , because of the commutativity (11.0) and the functoriality of adi. This gives the wanted functor s : T (X,Y ) → Sh(X´et).

∼ Now we construct an isomorphism of functors id → st on Sh(X´et).

For every sheaf F in Sh(X´et), the diagram

ad j ∗ (11.1.3) F / j∗j F

∗ adi(F ) adi(j∗j F ) i i∗(ad ) ∗ ∗ j ∗ ∗ i∗i F / i∗i j∗j F commutes, by the functoriality of the adjunction morphism adi (in sheaves). By the universal property of the ﬁber product (11.1.2) for the triple

∗ ∗ ∗ (F1,F2, φ) = (i F, j F, i (adj)) = t(F ) , (11.1.3) induces a canonical morphism

∗ ∗ ∗ α : F → s(i F, j F, i (adj)) = st(F ) . We show that (11.1.3) is cartesian, too; then α is an isomorphism. Here it suﬃces to show that the diagram is cartesian if one passes to the (geometric) stalks, since the stalks form a conservative family. But if x is a geometric point over U, we obtain a diagram

Fx / Fx

0 / 0 , which is cartesian. If x is over Y , we obtain the diagram

∗ Fx / (j∗j F )x

id id

∗ Fx / (j∗j F )x

81 which is cartesian as well. Furthermore we have an isomorphism of functors ts →∼ id. If

∗ (F1,F2, φ : F1 → i j∗F2) is an object in T (X,Y ), then s(F1,F2, φ) is deﬁned by the cartesian diagram

(11.1.4) s(F1,F2, φ) / j∗F

adi

∗ i∗F1 / i∗i j∗F2 .

If we apply the functor i∗, we obtain again a cartesian diagram

∗ ∗ i s(F1,F2, φ) / i j∗F2

β1 id

∗ F1 / i j∗F2 ,

∗ ∗ since i is an exact functor (see 7.8) and i (adi) = id by deﬁnition of the adjunction map adi, ∗ ∼ as well as canonically und functorial i i∗F1 → F1 for every ´etalesheaf on Y by the following Lemma. Thus β1 is an isomorphism. Correspondingly we apply j∗ to (11.1.4) and we obtain the cartesian diagram

∗ β2 ∗ j s(F1,F2, φ) / j j∗F2

0 / 0 , hence an isomorphism β0 ∗ 2 ∗ ∼ β2 : j s(F1,F2, φ) → j j∗F2 → F2 , ∼ where the last isomorphism holds by the following lemma. Finally, the following diagram is commutative:

∗ ∗ ∗ ∗ i (adj): i s(F1,F2, φ) / i j∗j s(F1,F2, φ)

∗ β1 o o i j∗(β2)

φ ∗ F1 / i j∗F2 , and we obtain an isomorphism

∼ ts(F1,F2, φ) → (F1,F2, φ) .

∗ Lemma 11.2 (a) For an ´etalesheaf F1 on Y , the adjunction map Adi : i i∗F1 → F1 is an isomorphism. ∗ (b) For an ´etalesheaf F2 on U, the adjunction map Adj : j j∗F2 → F2 is an isomorphism.

82 Proof (a) By Lemma 9.18, for every y ∈ Y and every geometric point y over y we have a canonical isomorphism ∗ ∼ ∼ (i i∗F)x = (i∗F)f(x) = Fx .

(b) For an étalesheaf F on U and an étalemorphism X0 → X we have by definition

jP (j F)(U 0) = lim (j F)(X0) , ∗ → ∗ (X0,ψ)∈Iop U0

0 0 0 0 where IU 0 is the category of all pairs (X , ψ) with X in X´et and a morphism ψ : U → U×X X , where morphisms are commutative diagrams

00 U ×X X : tt tt tt tt tt U 0 (id×f) J JJJ JJJ JJJ $ 0 U ×X X

00 0 op with f : X → X (see the proof of Proposition 3.2), and where IU 0 is the dual category to IU 0 . 0 0 But in our situation, IU 0 has the initial object U → U ×X U (with obvious morphism), since 0 0 0 0 for an object ψ : U → U ×X X we have a morphism α : U → X and hence a canonical morphism 0 U ×X U 9 tt tt tt tt tt U 0 (id×α) JJ JJ ψ JJJ JJJ % 0 U ×X X . (Note that U 0 → U → X is an ´etalemorphism). Thus follows

P 0 0 0 j (j∗F)(U ) = F(U ×X U ) = F(U ) .

∗ ∼ Since F is already a sheaf, we have j j∗F = F. Since s and t induce inverse maps on morphisms (this can again be checked on the stalks, since these form a conservative family), we get that t is an equivalence of categories, with quasi-inverse s.

∼ Remark 11.3 In particular, Theorem 11.1 implies that T (X,Y ) = Sh(X´et) is an abelian category. One can easily see that a sequence in T (X,Y )

0 0 0 00 00 00 0 → (F1,F2, φ ) → (F1,F2, φ) → (F1 ,F2 , φ ) → 0

is exact if and only if the sequences

0 00 0 → F1 → F1 → F1 → 0

83 and 0 00 0 → F2 → F2 → F2 → 0 are exact, since a sequence 0 → F 0 → F → F 00 → 0 is exact if and only if 0 → i∗F 0 → i∗F → i∗F 00 → 0 and 0 → j∗F 0 → j∗F → j∗F 00 → 0 are exact (consider the stalks).

Definition 11.4 Identifying S(Xét) with T (X,Y ), we can define the following six functors

∗ j ←i ←! ∗ i∗ j S(Y ét) → S(Xét) → S(Uét) ! j ←i ←∗ by ∗ i j! F1 ←p (F1,F2, φ) , (0,F2, 0) ←p F2 ∗ i∗ j F1 7→ (F1, 0, 0) , (F1,F2, φ) 7→ F2 ! i j∗ ∗ ker φ ←p (F1,F2, φ)(i j∗F2,F2, id) ←p F2 ! One calls j!F the “extension by zero” of F and i F ⊆ F the “subsheaf of the sections with support in Y ”. It is obvious that these assignments are functorial.

∗ Theorem 11.5 (a) Under the identification between S(Xét) and T (X,Y ), the functors i , ∗ i∗, j and j∗ correspond to the functors with the same name between S(Xét) and (Yét), and S(Xét) and S(Uét). (b) Every functor in 11.3 is left adjoint to the functors below it. ∗ ∗ ! (c) The functors i , i∗, j and j! are exact; j∗ and i are left exact. ∗ ! ∗ (d) The compositions i j!, i j! and j i∗ are zero.

(e) The functors i∗ and j∗ are fully faithful, and i∗ induces an equivalence of categories

i∗ Sh(Y´et) →{F ∈ Sh(X´et) | F has support in Y } ∼

∗ ! (f) The functors j∗, j , i and i∗ respect injectives.

Proof: (a) is clear from the identiﬁcation between S(X´et) and T (X,Y ). ∗ ∗ (b) for (i , i∗) and (j , j∗) follows from (a), i.e., from the fact that these form adjoint pairs. For the two remaining cases we have canonical isomorphisms ∼ HomX ((0,F2, 0), (G1,G2, φ)) = HomU (F2,G2)

84 and ∼ φ ∗ HomX ((F2, 0, 0), (G1,G2, φ)) = HomY (F2, ker(G1 → i j∗G2)) .

In (e), the fully faithfulness of i∗ again follows from the description. For j∗, a morphism

∗ ∗ 0 0 (i j∗F2,F2, id) → (i j∗F2,F2, id)

0 is again determined only by F2 → F2, by the commutative diagram (11.0).

With this, the second claim in (d) follows, since (F1,F2, φ) has support in Y if and only if F2 = 0 and hence φ = 0. (f) follows from the fact that these functors have exact left adjoints.

Corollary 11.6 (a) For an open immersion j : U,→ X and an ´etalesheaf F on U one has ∗ j j∗F = F. ∗ (b) For a closed immersion i : Y,→ X and an ´etalesheaf G on Y one has i i∗F = F.

Proof: This follows from Theorem 11.4 (a) and the description of the functors in Deﬁnition 11.3. In fact, we get ∗ ∗ ∗ j j∗F = j (i j∗F, F, id) = F , and ∗ ∗ i i∗G = i (G, 0, 0) = G .

Corollary 11.7 For every ´etalesheaf F on X, one has exact sequences

∗ ∗ (a) 0 → j!j F → F → i∗i F → 0

! ∗ (b) 0 → i∗i F → F → j∗j F

Proof Again this follows from the description in triples. In fact, (a) corresponds to the exact sequence 0 → (0, j∗F, 0) → (i∗F, j∗F, φ) → (i∗F, 0, 0) → 0 , and (b) corresponds to the exact sequence

∗ ∗ ∗ ∗ ∗ 0 → (ker φ, 0, 0) → (i F, j F, φ) → (i j∗j F, j F, φ) .

85 12 Cechˇ cohomology

The following generalizes the zero-th Cechˇ cohomology (see deﬁnition 3.7) and the topological Cechˇ cohomology. We consider presheaves with values in abelian groups.

Lemma/Deﬁnition 12.1 Let S = (X , T ) be a site.

(a) For a presheaf P on X and a covering U = (Ui → U)i∈I in T and n ≥ 0, the group

n Q C (U,P ) := P (Ui0 ×U ... ×U Uin ) n+1 (i0,...,in)∈I is called the group of the n-cochains for the covering U with values in P . Deﬁne the diﬀerential dn : Cn(U,P ) → Cn+1(U,P ) by n+1 n X ν (d s) = (−1) s ˆ | i0,...,in+1 i0,...,iν ,...,in+1 Ui0 ×U ...×U Uin+1 ν=0 where the restriction with respect to the morphism ˆ Ui0 ×U ... ×U Uin+1 → Ui0 ×U ... ×U Uiν ×U ... ×U Uin+1 is taken anda ˆ denotes the omission of a. Then dn+1dn = 0 for all n and we obtain a complex C·(U,P ), called the Cechˇ complex for the covering U with values in P . (b) The n-th cohomology Hˇ n(U,P ) := Hn(C·(U,P )) is called the n-th Cechˇ cohomology of P for covering U.

Proof that dn+1dn = 0: left to the readers! (standard).

Remark 12.2 For n = 0, we obviously obtain the zero-th Cech-cohomologyˇ from Deﬁnition 3.7 (a).

Lemma 12.3 Let (Vj → U)j∈J and (Ui → U)i∈I be coverings and let

f = (ε, fj):(Vj → U) → (Ui → U)

be a reﬁnement map. This induces maps for all n

ˇ n ˇ n H ((Ui → U),P ) → H ((Vj → U),P ) .

Proof We have a map ε : J → I and morphisms fj : Vj → Uε(j). With this we deﬁne the map n n n f : C ((Ui → U),P ) → C ((Vj − U),P ) n as follows: If s = (si0,...,in ) ∈ C ((Ui → U),P ), then deﬁne (f ns) = res (s ) , j0,...,jn fj0 ×...×fjn ε(j0),...,ε(jn)

86 where the restriction is taken with respect to

fj0 × ... × fjn : Vj0 ×U ... ×U Vjn → Uε(j0) ×U ... ×U Uε(jn) .

These maps commute with the diﬀerentials dn, hence give a morphism of complexes

∗ · · f : C ((Ui → U),P ) → C ((Vj → U),P ) , which induces the desired map in the cohomology.

Remark 12.4 On Hˇ 0(−,P ), this map coincides with the map deﬁned in 3.7!

Lemma 12.5 If f, g :(Vj → U) → (Ui → U) are two reﬁnement maps, then we have

∗ ∗ ˇ n ˇ n f = g : H ((Ui → U),P ) → H ((Vj → U),P ) for all n ≥ 0.

Proof (compare Lemma 3.11) Let f = (ε, fj) and g = (η, gj). Deﬁne

n n n−1 k : C ((Ui → U),P ) → C (Vj → U),P ) by p−1 X (kns) = (−1)rres s j0,...,jn−1 fj0 ×...×(fjr ,gjr )×...×gjn−1 ε(j0),...,ε(jr),η(jr),...,η(jn−1) i=0 for (fjr , gjr ): Vjr → Uε(jr) × Uη(jr). Then we have

dn−1kn + kn+1dn = gn − f n , i.e., (kn) provides a chain homotopy between (f n) and (gn), and the claim follows.

Deﬁnition 12.6 The n-th Cechˇ cohomology of U with values in P is deﬁned as

Hˇ n(U, P ) := Hˇ n(U, T ; P ) = lim Hˇ n(U,P ) → where the limit runs over all coverings of U (in T ).

Remark 12.7 Because of 12.5, this is a limit over the inductively ordered set T (U)0 of all coverings U of U, where U0 ≥ U, if there is a reﬁnement map f : U0 → U (see the proof of 3.11).

Lemma 12.8 Let

(12.8.1) 0 → P1 → P2 → P3 → 0 be an exact sequence of abelian presheaves (!) on X .

87 (a) For every covering U in T there is a long exact cohomology sequence

ˇ 0 ˇ 0 ˇ 0 δ 1 0 → H (U,P1) → H (U,P2) → H (U,P3) → H (U,P1) → ..., which is functorial with respect to reﬁnement maps and morphisms of exact sequences (12.8.1). (b) For every U in X there is a long exact cohomology sequence

ˇ 0 ˇ 0 ˇ 0 δ ˇ 1 ˇ 1 0 → H (U, P1) → H (U, P2) → H (U, P3) → H (U, P1) → H (U, P2) → ..., which is functorial for restriction maps and for morphisms of exact sequences (12.8.1).

Proof (a): We have an exact sequence of complexes

· · · (12.8.2) 0 → C (U,P1) → C (U,P2) → C (U,P3) → 0 ,

n+1 since for U = (Ui → U)i∈I and every (i0, . . . , in) ∈ I the sequence

(12.8.2) 0 → P (Ui0,...,in ) → P2(Ui0,...,in ) → P3(Ui0,...,in ) → 0

is exact, where UI0,...,in := Ui0 ×U ... ×U Uin . The sequence in (a) is the long exact cohomology sequence for (12.8.2). The functorialities follow, since (12.8.2) is functorial in U and in (12.8.1). (b) follows from (a) by passing to the inductive limit over all coverings of U (see Remark 12.7), since forming an inductive limit is an exact functor.

Remark 12.9 For an exact sequence of T -sheaves

(12.9.1) 0 → F1 → F2 → F3 → 0 , one does not obtain a long exact sequence of Cechˇ cohomology groups in general, since (12.9.1) is generally not exact as a sequence of presheaves.

Example 12.10 Let A → B a faithfully flat ring homomorphism, which is locally of finite type (resp. locally of finite presentation). Then (V = Spec(B) → Spec(A) = U) is a covering ˇ in the flat topology. The associated Cech complex for the presheaf Ga,X is the complex

0 → B → B ⊗A B → B ⊗A B ⊗A B → ...

By descent theory (see [Mi] I. 2.17, and see also 14.6 below) we have

A , n = 0, Hˇ n((V → U), ) = Ga 0 , n > 0 .

One can also obtain the Cechˇ cohomology as a derived functor:

ˇ n Theorem 12.11 (a) For a covering U = (Ui → U), H (U, −) is the n-th right derivative of the left exact functor Hˇ 0(U, −): P r(X ) → Ab P 7→ Hˇ 0(U,P ) .

88 (b) For U ∈ ob(X ), Hˇ n(U, −) is the n-th right derivative of the left exact functor

Hˇ 0(U, −): P r(X ) → Ab P 7→ Hˇ 0(U, P ) .

ˇ n ˇ n Proof It follows from 12.8 that the functors (H (U, −))n≥0 resp. (H (U, −))n≥0 form exact δ-functors on P r(X ). Thus it suffices to show that Hˇ n(U, −) resp. Hˇ n(U, −) is effaceable for n > 0: then these give universal δ-functors, this is also known for the right derivatives, and two universal δ-functors are obviously isomorphic. Since P r(X ) has enough injectives, it suffices to show:

Lemma 12.12 For n > 0 we have Hˇ n(U,I) = 0 = Hˇ n(U, I), if I is an injective presheaf.

Proof The second claim follows from the ﬁrst. Furthermore, we have to show that for every covering U = (Ui → U)i∈I in T the sequence Q Q Q I(Ui) → I(Ui0i1 ) → I(Ui0i1i2 ) → ... i i0,i1 i0,i1,i2

is exact, where Ui0,...,in = Uij ×U Ui1 ×U ... ×U Uin . This sequence can be identiﬁed with a sequence

(12.12.1) Q Hom( P ,I) → Q Hom( P ,I) → ..., ZUi ZUi0,i1 i i0,i1 (see Lemma/Deﬁnition 5.8) which comes from an obvious complex of presheaves

L P L P L (12.12.2) ← ← Ui i i ← .... ZUi ZUi0i1 Z 0 1 2 i i0,i1 i0,i1,i2 by applying Hom(−,I). The last functor is exact since I is injective. Therefore it suﬃces to show the exactness of (12.12.2), hence the exactness of

(12.12.3) L P (V ) ← L (V ) ← ... ZUi ZUi0i1 i i0,i1 for every V in X . P L Now, we have ZW (V ) = Z for W in X , and we have a canonical morphism Ui0...in → Hom(V,W )

U for every W = Ui0i1...in = Uij ×U Ui1 ×U ... ×U Uin . This implies ` Hom(V,W ) = Homφ(V,W ) , φ∈Hom(V,U) where Homφ(V,W ) is the set of the morphisms ϕ : V → W for which

ϕ V / W @ @@ }} @@ }} φ @@ }} @ ~}} U is commutative. Furthermore, by the universal property of the ﬁber product, we have

Homφ(V,Ui0 ×U ... ×U Uin ) = Homφ(V,Ui0 ) × ... × Homφ(V,Uin ) .

89 As a consequence, if we deﬁne ` S(φ) = Homφ(V,Ui) , i∈I then the complex (12.12.3) can described as follows :

⊕ ( ⊕ Z ← ⊕ Z ← ⊕ Z ← ...) φ∈Hom(V,U) S(φ) S(φ)×S(φ) S(φ)3 with the obvious diﬀerential in the bracket

p X 1 7→ (−1)ν1 . j0,...,jp j0,...,ˆjν ,...jp ν=0 Now the complex in the bracket is exact: A contracting homotopy is (hp, p ≥ 0), with

hp : ⊕ Z → ⊕ Z S(φ)p+1 S(φ)p+2

1i0,...,ip 7→ 1e,i0,...,ip , where e ∈ S(φ) is a ﬁxed element (check!).

Theorem 12.13 Let U ∈ ob(X ) and U = (Ui → U) be a covering in T and let F be a sheaf (with respect to T ). There are spectral sequences

p,q ˇ p q p+q E2 = H (U,H (F )) ⇒ H (U, F ) p,q ˇ p q p+q E2 = H (U, H (F )) ⇒ H (U, F ) .

Here, cohomology and Cech-cohomologyˇ are taken with respect to T , and Hq(F ) is the presheaf V 7→ Hq(V,F ) .

Proof First spectral sequence: We apply Grothendieck’s Theorem (Theorem 6.8). Obviously, we have Hˇ 0(U,H0(F )) = H0(U, F ) , since F is a sheaf, hence H0(U, −) is the composition of H0(−) and Hˇ 0(U, −). Furthermore, for an injective sheaf I, the presheaf H0(I) is acyclic for Hˇ 0(U, −), i.e.,

RnHˇ 0(U,H0(I)) 12.11= Hˇ n(U,H0(I)) = Hˇ n(U,I) = 0 for n > 0 .

In fact, the embedding i : Sh(X , T ) → P r(X ) respects injectives, since i has the exact left adjoint a. Therefore, I is also injective as a presheaf, and the last vanishing follows from Lemma 12.12. The second spectral sequence follows in an analogous way, or by passing to the limit over all coverings of U.

Corollary 12.14 There is a spectral sequence

p,q ˇ p q p+q E2 = H (H (F )) → H (F ) .

90 Proof Consider the second spectral sequence in 12.13 for all U in X and note that this is functorial (contravariant) in U.

Proposition 12.15 We have

Hˇ 0(U, Hq(F )) = 0 for q > 0 ,

0 and therefore also Hˇ (Hq(F )) = 0 for q > 0.

Proof Let F,→ I· be an injective resolution in Sh(X , T ). Then Hq(F ) is the q-th cohomology presheaf of the complex i(I·). Since a is exact, a commutes with taking the cohomology, hence aHq(F ) = Hq(aiI·) = Hq(I·) = 0 for q > 0 (Here, Hq denotes the q-th cohomology 0 0 0 0 sheaf). But Hˇ (Hq(F )) is a sub-presheaf of aHq(F ) = Hˇ Hˇ (Hq(F )) (since Hˇ (P ) is sepa- 0 rated for every presheaf P , see Lemma 3.10 (c) and note that by deﬁnition P˜ = Hˇ (P )). 0 Therefore, Hˇ (Hq(F )) = 0 for all q > 0, hence Hˇ 0(U, Hq(F )) = 0 for all U.

Corollary 12.16 For every sheaf F on (X , T ) and every U in X there are canonical isomorphisms Hˇ 0(U, F ) ∼= H0(U, F ) Hˇ 1(U, F ) ∼= H1(U, F ) and an exact sequence

0 → Hˇ 2(U, F ) → H2(U, F ) → Hˇ 1(U, H1(F )) → Hˇ 3(U, F ) → H3(U, F )

Proof The spectral sequence

Hˇ p(UHq(F )) ⇒ Hp+q(U, F )

has the following shape (on the q-axis the initial terms are zero, except in the origin)

0 0 • 0 •• •••• p

The ﬁrst claim in 12.16 ist obvious, since F is a sheaf, and from the above shape we get

1 ∼ 1,0 ˇ 1 0 H (U, F ) = E2 = H (U, H (F )) = Hˇ 1(U, F ) ,

0,1 1,0 since E2 = 0 and all diﬀerentials leaving and entering E2 are zero. Deducing the last sequence is analogous to the proof of the usual sequence of the low terms (Lemma 6.7).

91 Corollary 12.17 Let X be a scheme and F a quasi-coherent OX -module. Let U = (Ui)i∈I be an open covering of X such that

Ui0 ∩ Ui1 ∩ ... ∩ Uin is aﬃne for all n and all i0, . . . , in ∈ I. Then one has a canonical isomorphism

Hˇ n(U, F) →∼ Hn(X, F) for all n ≥ 0.

Proof We use the spectral sequence

Hˇ p(U,Hq(F)) ⇒ Hp+q(X, F) .

q For q > 0, H (Ui0 ∩ ... ∩ Uin , F) = 0 by Serre’s vanishing theorem. This implies the claim.

Deﬁnition 12.18 A sheaf F on a site (X , T ) is called ﬂabby, if Hn(U, F ) = 0 for all U ∈ ob(X ) and all n > 0.

Example 12.19 Every injective sheaf is ﬂabby.

Proposition 12.20 For a sheaf F the following conditions are equivalent: (a) F is ﬂabby. (b) For every U in X and every covering U of U (in a coﬁnal family), Hˇ n(U,F ) = 0 for n > 0. (c) Hˇ n(U, F ) = 0 for all U in X and all n > 0.

Proof (a) ⇒ (b): If F is flabby, then Hq(F ) = 0 is flabby for q > 0. The first spectral sequence of 12.13 therefore provides an isomorphism

Hˇ n(U,F ) →∼ Hn(U, F ) = 0 for n > 0 .

(b) ⇒ (c): Follows by passing to the inductive limit over all coverings of U. n (c) ⇒ (a): By assumption, we have Hˇ (F ) = 0 for n > 0. By Corollary 12.16, we have H1(F ) = 0. Now we use induction over n, via the spectral sequence of 12.14

p Hˇ (Hq(F )) ⇒ Hp+q(F ) .

2 0 By assumption, Hˇ 2(H0(F )) = Hˇ (F ) = 0, furthermore Hˇ1(H1(F )) = 0 and Hˇ (H2(F )) = 0 by 12.15. From the spectral sequence we get H2(F ) = 0. The same argument shows inductively i Hˇ (Hj(F )) = 0 for i + j ≤ n and hence Hn(F ) = 0.

Corollary 12.21 Let f :(X 0, T 0) → (X , T ) be a morphism of sites. If F 0 is a ﬂabby sheaf 0 0 0 on (X , T ), them f∗F is ﬂabby, too.

92 0 0 Proof Let f : X → X be the underlying functor. If U = (Ui → U) is a covering in T , then 0 0 0 0 U = (f Ui → f U) is a covering in T and we have

0 0 (f∗F )(V ) = F (f V )

for all V in X . Then we have

ˇ n 0 ˇ n 0 0 H (U, f∗F ) = H (U ,F ) = 0 for n > 0 .

Corollary 12.22 (Leray spectral sequence) (a) If f :(X 00, T 0) → (X , T ) is a morphism of sites, then for every sheaf F 0 on (X 0, T 0) and every U ∈ X one has a spectral sequence

p q 0 p+q 0 0 H (U, R f∗F ) ⇒ H (f U, F ) .

(b) If (X 00, T 00) → (X 0, T 0) → (X , T ) are morphisms of sites, then for every sheaf F 00 on (X 00, T 00) there is a spectral sequence

p q 00 00 R f∗R g∗F ⇒ R(gf)∗F .

Proof This follows from Theorem 6.8 (Grothendieck’s spectral sequence), since f∗ maps 0 flabby sheaves to flabby sheaves, therefore acyclic sheaves for H (U, −), and g∗ as well maps n 0 flabby sheaves to flabby sheaves, and therefore to acyclic sheaves for f∗. In fact, R f∗F is the associated sheaf to the presheaf U 7→ Hn(f 0U, F 0), and the presheaf is already 0, if F 0 is flabby.

93 13 Comparison of sites

Proposition 13.1 (Change of the category) Let (X 0, T 0) be a site, let X ⊆ X 0 be a full subcategory, and let T be the restriction of T 0 to X . We assume: 0 (13.1.1) For every object U in X and every covering (Ui → U) in T , all Ui are already in X . For the morphism of sites α :(X 0, T 0) → (X , T ) , which is given by the embedding X ,→ X 0, we then have: 0 0 ∗ (a) The functor α∗ : Sh(X , T ) → Sh(X , T ) is exact and the adjunction map F → α∗α F is an isomorphism for all F ∈ Sh(X , T ). (b) The functor α∗ : Sh(X , T ) → Sh(X 0, T 0) is fully faithful and left exact. (c) The canonical homomorphisms

n 0 n 0 0 H (U, T ; α∗F ) → H (U, T ; F ) Hn(U, T ; F ) → Hn(U, T 0; α∗F ) are isomorphisms for all U ∈ X , all F 0 ∈ Sh(X 0, T 0), all F ∈ Sh(X , T ) and all n ≥ 0.

Proof (a) α∗ is simply the restriction; so that the exactness is obvious by (13.1.1). Further- more, for every U ∈ X and F ∈ Sh(X , T ) we have

(αP F )(U) = F (U) ,

P since the category IU , over which the limit is formed for (α F )(U), has the initial object ∗ P (U, idU ). Since F restricted to (X , T ) is a sheaf, we have (α F )(U) = (aα F )(U) = F (U). Moreover, ∗ ∗ (α∗α F )(U) = (α F )(U) = F (U) , hence the second claim follows. (b) This follows from the proof above. (c) We have a spectral sequence

p,q p q p+q (13.1.2) E2 = H (U, R α∗F ) ⇒ H (U, F ) , either by Corollary 12.22, or by the Grothendieck-Leray spectral sequence 6.7. This exists, 0 0 ∗ since H (U, α∗F ) = H (U, F ), and since α∗ has a the left exact left adjoint α , so that by 7.10, α∗ maps injectives to injectives (hence acyclic sheaves). q Since α∗ is exact, we have R α∗F = 0 for q > 0, and (13.1.2) provides an edge isomorphism

n ∼ n H (U, α∗F ) → H (U, F ) .

The second claim follows from the fact that the composition

n n ∗ ∼ n ∗ ∼ n H (U, F ) → H (U, α F ) → H (U, α∗α F ) → H (U, F )

94 is the identity (consider injective resolutions).

Example 13.2 One can apply this to the morphism of sites

α :(Sch/X)E → XE ,

where X is a scheme, E is a admissible category of morphisms, (Sch/X)E is the category of all X-schemes with the E-coverings as topology (the large E-site) and XE is the category of all X-schemes U, for which the structural morphism U → X is in E, with the E-coverings as topology (the small E-site). In particular, the large étalesite (Sch/X)ét and the small ∗ étalesite Xét give the “same cohomology” (via α∗ and α , respectively).

Proposition 13.3 (Change of the topology) Let X be a category and let T ⊂ T 0 be topologies on X (every covering for T is a covering for T 0). Let β :(X , T 0) → (X , T )

be the morphism of sites, given by idX . Assume that for every covering U = (Ui → U) in 0 0 T there is a covering (Vj → U) in T , which reﬁnes U. Then β∗ : Sh(X , T ) → Sh(X , T ) is exact and thus n 0 ∼ n 0 H (XT , β∗F ) → H (XT 0 ,F ) for every sheaf F 0 ∈ Sh(X , T 0).

00 Proof: The functor β∗ is the identity. We only have to show that every epimorphism F → F for T 0 is an epimorphism for T as well. If we have U ∈ ob(X ) and s ∈ F 00(U), then there is 0 00 a covering (Ui → U) in T , so that, for all i, s|Ui is in the image of F (Ui) → F (Ui). If now (Vj → U) ∈ T is a reﬁnement of (Ui → U), and if Vj → U factorizes as Vj → Ui → U, we then obtain a commutative diagram

00 F (Vj) / F (Vj) O O

00 F (Ui) / F (Ui) ,

which shows that s|Vj is in the image of the top map. Thus β∗ is exact. The second claim p 0 follows from the Leray-spectral sequence, since R β∗F = 0 for p > 0, by the exactness of β∗.

Corollary 13.4 Let X be a scheme, and let E ⊂ E0 be two admissible classes of morphisms, so that the condition of reﬁnement of 13.3 holds for the corresponding topologies (E) ⊂ (E0). For the morphism of small sites γ : XE0 → XE , 0 the functor γ∗ : Sh(XE0 ) → Sh(XE) is exact, hence for every sheaf F on XE0 we have n 0 ∼ n 0 0 H (U, E; γ∗F ) → H (U, E ,F ) .

Proof Let E − Sch/X be the category of the X-schemes U, whose structural morphism U → X is in E. Then γ factorizes as

β 0 α γ : XE0 → (E − Sch/X)E → (E − Sch/X)E = XE ,

95 and the claim follows from 13.1 (exactness of α∗) and 13.3 (exactness of β∗).

Example 13.5 We have the following examples for Corollary 13.4 (a) (étalemorphisms of finite type) ⊂ (ét) (b) (ét) ⊂ (fl), if X is quasi-compact (Milne, Etale´ Cohomology’, I 3.26). (c) (fpqc) ⊂ (fl), where (fpqc) is the class of the flat “quasi-compact” morphism. (Milne, I 2.25).

Theorem 13.6 (quasi-coherent OX -modules) Let X be a scheme and M a quasi-coherent OX -module. Let Mét and Mfl be the corresponding sheaves in the étale topology and flat topology, respectively (see 10.18). Then we have

n ∼ n ∼ n H (XZar, M) → H (Xét, Mét) → H (Xfl, Mfl)

for all n.

Proof We give the proof for the flat topology, the case of the étale topology is analogous. Let f : Xfl → XZar

be the morphism of sites (see 13.4). It is obvious that f∗Mfl = M, therefore it suffices to show n that R f∗Mfl = 0 for n > 0 (then the claim follows from the Leray-spectral sequence). Since n n R f∗Mfl is the Zariski-sheaf associated to the presheaf U 7→ H (Ufl, Mfl) (see Theorem 5.16), n it suffices to show that H (UflMfl) = 0, if U = Spec(A) ⊆ X is affine open. Furthermore, by Corollary 13.4, we can consider the small site UE , where E is the class of flat affine morphisms of finite presentation. We want to show that Mfl is flabby, and by 13.20 it suffices to show ˇ n that H (U, Mfl) = 0 for all n > 0 and all coverings U = (Ui → U)i∈I in E. By 10.6 and the quasi compactness of U, we can assume that I is finite (cofinite system of E-coverings!). ` ˇ Then V = Ui = Spec(B) is affine and A → B is faithfully flat, and the Cech complex is i the obvious complex

B ⊗A M → B ⊗A B ⊗A M → B ⊗A B ⊗A B ⊗A M → ..., which is exact in degrees ≥ 1, as was noted in 12.10.

Remark 13.7 (comparison isomorphism over C). Let X be a smooth variety over C. Then the theorem of implicit functions implies that the set X(C) is a complex manifold: It suﬃces to show this locally. But locally we have

X = Spec(C[X1,...,Xn]/hf1, . . . , fmi) , m ≤ n , where the Jacobian matrix ∂f i (P ) ∂Xj has rank m for all closed points P of X. Then we have

∼ n X(C) = {a = (a1, . . . , an) ∈ C | fj(a) = 0 ∀ j} ,

96 and the fi deﬁne continuous maps n fi : C → C , and the matrix above is the usual Jacobi matrix at P . The theorem on implicit function gives local homomorphisms ∼ n−m X(C) ⊇ V → U ⊆ C , and one obtains charts for X(C) as a manifold. By Artin and Grothendieck there are isomorphisms for all m and n

n ∼ n H (X(C), Z/mZ) → H (X´et, Z/mZ) ,

for every smooth variety X/C. Here the left hand side is the (topological) cohomology of sheaves of the constant sheaf, and the right hand side is the ´etalecohomology.

97 14 Descent theory and the multiplicative group

Lemma 14.1 Let X be a scheme. There is a canonical isomorphism

1 × ∼ H (XZar, OX ) = P ic(X) , where P ic(X) is the Picard group of X.

Proof In view of 12.16 it suﬃces to construct a canonical isomorphism

∼ ˇ 1 × (14.1.1) P ic(X) → H (X, OX ) .

Let L be an invertible OX -module. Then there is an open covering U = (Ui)i∈I of X, such that there are isomorphisms

∼ ϕi : OUi → L|Ui for all i ∈ I.

On Ui ∩ Uj, these induce isomorphisms

−1 ϕ ϕ ϕ = ϕ−1ϕ : O →i L| →j O , ij j i Ui|Ui∩Uj Ui∩Uj Uj |Ui∩Uj

× which correspond to elements sij ∈ OX (Ui ∩ Uj). Here we have

sik| · sjk| · sij| = 1 Uijk Uijk Uijk (14.1.2) q −1 −1 −1 ϕi ϕkϕk ϕjϕj ϕi , i.e., we have a Cech-1-cocycleˇ in

ˇ 1 × 1 Q × Q × Q × H (U, OX ) = H ( OX (Ui) → OX (Ui ∩ Uj) → OX (Ui ∩ Uj ∩ Uk)) i i,j i,j,k

Conversely, such a cocycle gives glueing isomorphisms

·sij ϕi,j : O(Ui ∩ Uj) → O(Ui ∩ Uj) , ∼

which glue together the free modules OUi on Ui to an invertible OX -module L (the cocycle condition (14.1.2) gives the cocycle condition/transitivity for ϕij). This assignment is addi- ˇ 1 × tive. Furthermore, L is trivial if and only if the 1-cocycle (sij) is trivial in H (U, OX ). This gives an isomorphism ˇ 1 × ∼ isomorphism classes of invertible OX -modules L, H (U, OX ) → which are trivialized on (Ui)

We obtain (14.1.1) by taking the inductive limit over all coverings.

Remark 14.2 The same argument holds again for all (locally) ringed spaces.

98 Lemma 14.3 Let A be a ring and let M be an A-module. If M is a ﬂat A-module, then the following holds: r P (1) If aimi = 0 with ai ∈ A and mi ∈ M, then there are an s ∈ N and elements bij ∈ A i=1 and yj ∈ M (j = 1, . . . , s) with P aibij = 0 i P for all j and mi = bijyj for all i. j

Proof Consider the exact sequence

f K → Ar → A r P (b1, . . . , br) 7→ biai , i=1 K = ker(f). Then r fM K ⊗A M → M → M r P (n1, . . . , nr) 7→ aini i=1 is exact. By assumption, we have fM (m1, . . . , mr) = 0, therefore there is an element

s P βj ⊗ yi ∈ K ⊗A M j=1

(βj ∈ K, yi ∈ M), which is mapped to (m1, . . . , mr).

If we write βj = (bij, . . . , brj) with bij ∈ A, the claim follows.

Remark 14.4 The converse holds as well: If (1) holds, then M is ﬂat.

Lemma 14.5 Let M be a ﬁnite generated module over a local ring A. Then the following conditions are equivalent: (a) M is ﬂat. (b) M is free.

Proof We only have to show (a) ⇒ (b). Let m be the maximal ideal of A and let m1, . . . , mn ∈ M be in such way that their images m1,..., mn in M/mM form a basis of this A/m-vector space. Then, by the Nakayama-Lemma the morphism

n A M basis element ei 7→ mi is surjective. It suﬃces to show that m1, . . . , mn ∈ M are linearly independent over A, if m1,..., mn are linearly independent in M/mM. We use induction over n. Let n = 1 and am1 = 0 for a ∈ A. By Lemma 14.3 there are b1, . . . , bs ∈ A and y1, . . . , ys ∈ M with abj = 0 P × for all j and m1 = bj · yj. Since m1 6= 0 there is a j with bj ∈/ m, i.e., bj ∈ A a unit. From j abj = 0 we get a = 0.

99 n P Now let n > 1 and aimi = 0. By Lemma 14.3 there exist y1, . . . , ys ∈ M and bij ∈ A i=1 (i = 1, . . . , s) with s n P P mi = bijyj , aibij = 0 . j=1 i=1

Since mn 6= 0 there is a j with bnj ∈/ m, i.e., bnj a unit. Then we have

n−1 P an = ciai , with ci = −bij/bin , i=1 and therefore

n P 0 = aimi = a1(m1 + c1mn) + ... + an−1(mn−1 + cn−1mn) i=1

Since the considered m1 + c1mn,..., mn−1 + cn−1mn are linear independent over A/m, the n−1 P induction hypothesis implies a1 = ... = an−1 = 0 and hence also an = ciai = 0. i=1

Now we consider the descent theory for faithfully ﬂat ring homomorphisms.

Theorem 14.6 (Descent theory I) Let A → B be a faithfully ﬂat ringhomomorphism. Then, for every A-module M, the sequence

γ α1 0 → M → B ⊗A M ⇒ B ⊗A B ⊗A M α2 7→ 1 ⊗ b ⊗ m m 7→ 1 ⊗ m, b ⊗ m 7→ b ⊗ 1 ⊗ m is exact. This means that M is the diﬀerence kernel of α1 and α2, i.e., that

M = ker(α1 − α2) .

Proof: By Lemma 10.4, the sequence

γ α1−α2 (14.6.1) 0 → M → B ⊗A M −→ B ⊗ B ⊗ M is exact if and only if the sequence

1⊗γ 1⊗(α −α ) 0 → B ⊗ M −→ B ⊗ B ⊗ M →1 2 B ⊗ B ⊗ B ⊗ M, tensored by B, is exact. Let µ : B ⊗ B → B b1 ⊗ b2 7→ b1b2 be the multiplication map. Then

µ ⊗ 1 : B ⊗ B ⊗ M → B ⊗ M b1 ⊗ b2 ⊗ m → b1b2 ⊗ m is a left inverse of 1 ⊗ γ (µ ⊗ 1 ◦ 1 ⊗ γ = id), hence 1 ⊗ γ is injective.

100 P Now let z = x1 ⊗ yi ⊗ mi be in the kernel of 1 ⊗ (α1 − α2), so that i X X xi ⊗ 1 ⊗ yi ⊗ mi = xi ⊗ yi ⊗ 1 ⊗ mi . i i By applying µ to the ﬁrst two places we obtain X X xi ⊗ yi ⊗ mi = xiyi ⊗ 1 ⊗ mi , i i P P and hence xiyi ⊗ 1 ⊗ mi is the image of xiyi ⊗ ui under 1 ⊗ γ. i i

Theorem 14.7 (Descent theory II) Let A → B be a faithfully ﬂat ringhomomorphism. (a) Let M be a A-module. For a B-module

0 M = B ⊗A M

one has a canonical isomorphism of B ⊗ B-modules (all tensor products are over A)

φ : M 0 ⊗ B →∼ B ⊗ M 0 (14.7.1) (b ⊗ m) ⊗ b0 7→ b ⊗ (b0 ⊗ m) .

By this one can retrieve M from M 0:

(14.7.2) M = {m0 ∈ M 0 | φ(m0 ⊗ 1) = 1 ⊗ m0} , because this amounts to the exactness of the sequence (14.6.1)

0 → M → B ⊗ M α−→1−α2 B ⊗ B ⊗ M where α (b ⊗ m) = 1 ⊗ b ⊗ m (14.7.3) 1 α2(b ⊗ m) = b ⊗ 1 ⊗ m . In fact for m0 = b ⊗ m ∈ B ⊗ M = M 0 we have φ(m0 ⊗ 1) − 1 ⊗ m0 = (b ⊗ 1) ⊗ m − 1 ⊗ b ⊗ m = α2(b ⊗ m) − α1(b ⊗ m) .

(b) For the induced morphisms (φi keeps the entry on the i-th position and is φ on the remaining positions)

0 0 φ1 : B ⊗ M ⊗ B → B ⊗ B ⊗ M b ⊗ (b2 ⊗ m) ⊗ b3 7→ b ⊗ b2 ⊗ (b3 ⊗ m)

0 0 φ2 : M ⊗ B ⊗ B → B ⊗ B ⊗ M (b ⊗ m) ⊗ b2 ⊗ b3 7→ b ⊗ (b2 ⊗ m) ⊗ b3 and 0 φ3 : M ⊗ B ⊗ B → B ⊗ M⊗B (b ⊗ m) ⊗ b2 ⊗ b3 7→ b ⊗ (b2 ⊗ m) ⊗ b3

101 we get the co-called cocycle conditions

(14.7.4) φ2 = φ1φ3 .

(14.7.5) φ : M 0 ⊗ B →∼ B ⊗ M 0 be an isomorphism of B ⊗ B-modules, and let

0 0 φ1 : B ⊗ M ⊗ B → B ⊗ B ⊗ M

0 0 φ2 : M ⊗ B ⊗ B → B ⊗ B ⊗ M

0 0 φ3 : M ⊗ B ⊗ B → B ⊗ M ⊗ B

be the induced isomorphisms, where φi keeps the entry on the i-th position and is deﬁned on the other positions by φ. (For φ2 we have explicitly

0 X 0 m ⊗ b2 ⊗ b3 7→ bi ⊗ b2 ⊗ mi , i

0 P 0 if φ(m ⊗ b3) = bi ⊗ mi.) i If now the cocycle condition

(14.7.6) φ2 = φ1φ3

∼ 0 holds, then there is a canonical A-module M with B ⊗A M = M , namely the A-module

M = {m0 ∈ M 0 | φ(m0 ⊗ 1) = 1 ⊗ m0} , for which the canonical map

γ : B ⊗ M →∼ M 0 (14.7.7) A b ⊗ m 7→ bm is an isomorphism. In fact, let τ : M 0 → B ⊗ M be deﬁned by τ(m0) = 1 ⊗ m0 − φ(m0 ⊗ 1). By deﬁnition, we then have an exact sequence

0 → M → M 0 →τ B ⊗ M 0 .

If we tensorize with B on the right, we obtain the top row in the following diagram

0 / M ⊗ B / M 0 ⊗ B / B ⊗ M 0 ⊗ B

γ o φ o 1⊗φ 0 / M 0 / B ⊗ M 0 / B ⊗ B ⊗ M 0 ,

where the lower sequence is the exact sequence from 14.6, applied to the A-module M 0. The map γ is deﬁned by γ(m ⊗ b) = bm (and therefore corresponds to the map (14.7.7)). We

102 show that γ is an isomorphism. Since the rows are exact (the top row is exact because of the ﬂatness of B over A) and both vertical maps on the right are isomorphisms, the claim follows, if we show that the diagram is commutative. The left hand square commutes, since, by deﬁnition of M, we have

φ(m ⊗ b) = (1 ⊗ b)φ(m ⊗ 1) = (1 ⊗ b)(1 ⊗ m) = 1 ⊗ bm

for m ∈ M and b ∈ B. For the right hand square we have the following for the “lower way”: For m0 ∈ M 0 let

0 X 0 φ(m ⊗ 1) = bi ⊗ mi i

0 0 with bi ∈ B and mi ∈ M . Then we have

0 0 X 0 φ(m ⊗ b) = (1 ⊗ b)φ(m ⊗ 1) = bi ⊗ bmi , i and thus the image of this in B ⊗ B ⊗ M 0 is equal to

X 0 X 0 1 ⊗ bi ⊗ bmi − bi ⊗ 1 ⊗ bmi . i i On the “upper way”, m0 ⊗ b is mapped to

1 ⊗ m0 ⊗ b − φ(m ⊗ 1) ⊗ b 0 P 0 = 1 ⊗ m ⊗ b − bi ⊗ mi ⊗ b i and this, by 1 ⊗ φ, is mapped to

0 P 0 1 ⊗ φ(m ⊗ b) − bi ⊗ φ(mi ⊗ b) i P 0 P 0 = 1 ⊗ bi ⊗ bmi − bi ⊗ φ(mi ⊗ b) . i i Hence we have to show that

X 0 X 0 bi ⊗ 1 ⊗ bmi = bi ⊗ φ(mi ⊗ b) . i i But this means that we have

φ2(m ⊗ 1 ⊗ b) = φ1(φ3(m ⊗ 1 ⊗ b)) k k P 0 P 0 bi ⊗ 1 ⊗ bmi φ1( bi ⊗ 1 ⊗ mi) i i k P 0 bi ⊗ φ(1 ⊗ mi) , i which holds because of the assumption that φ2 = φ1φ3.

Theorem 14.8 For every scheme X, the canonical morphisms

1 × ∼ 1 ∼ 1 H (XZar, OX ) → H (X´et, Gm) → H (Xﬂ, Gm)

103 are isomorphisms.

Proof for the flat topology (the étalecase is analogous). We use the Leray-spectral sequence for α : Xfl → XZar . By the sequence of the lower terms

1 1 0 1 0 → H (XZar, α∗Gm) → H (Xﬂ, Gm) → H (XZar,R α∗Gm)

× 1 where α∗Gm = OX , it suﬃces to show that R α∗Gm = 0. This means that for all for all 1 x ∈ X the stalk (R α∗Gm)x = 0. But this stalk is

lim H1(U , ) ∼= lim Hˇ 1(U , ) , → ﬂ Gm → ﬂ Gm x∈U x∈U where U goes through all open neighborhoods of x.

Since inductive limits commute, it suﬃces to show that for every ﬂat covering (Ui → U)i∈I with U ⊆ X open the limit

lim Hˇ 1((U × V → V, ) = 0 . → i U Gm V ⊆U open

We can assume that U = Spec(A) is affine and further that I is finite and every Ui is affine, and hence, that we have a faithfully flat morphism

Spec(B) → Spec(A) .

Furthermore we can pass to the limit and assume that A = OX,x is a local ring. We consider a class in

1 × × × (14.8.1) H (B → (B ⊗A B) → (B ⊗A B ⊗A B) ) ,

× represented by the 1-cocycle α ∈ (B ⊗A B) . We obtain an isomorphism

∼ (14.8.2) φ : B ⊗A B → B ⊗A B, which fulfills the cocycle condition the property (14.7.6). Therefore there is an A-module M ∼ with B ⊗A M = B. Since the B-module B is finitely generated and flat, this also holds for M. Since A is local, M is a free A-module of degree 1, i.e., M ∼= A. It follows from the descent theory that the isomorphism φ in (14.8.2) is the one which is constructed by (14.7.1) from M. Since M ∼= A, we get that the associated 1-cocycle is trivial, i.e., comes from B×. q.e.d.

104 15 Schemes of dimension 1

Proposition 15.1 Let X be a regular integral noetherian scheme and let j : Spec K,→ X be the inclusion of the generic point. Then there is an exact sequence of ´etalesheaves

(15.1.1) 0 → Gm/X → j∗(Gm/k ) → ⊕ (ix)∗Z → 0 , x∈X1

1 where X is the set of the points of codimension 1 of X and ix: Spec(k(x)) ,→ X is the canonical morphism. We need:

Lemma 15.2 Let f : U → X be ´etale.Then the following holds.

(a) For y ∈ U and x = f(y), we have dim OU,y = dim OX,x. (b) U is regular if an only if X is regular.

Proof (a): Without restriction, X = Spec A, A is local and U = Spec B is affine. Let m ⊆ A (maximal) and n ⊆ B be the prime ideals which correspond to x and y. Then Spec Bn → Spec A is faithfully flat (10.5), therefore surjective, hence dim Bn ≥ dim A. Conversely, by Zariski’s main theorem 9.3, B/A is finite without restriction. Then ϕ : A → B induces an integral ring extension A/ ker ϕ ,→ B, and, by Cohen-Seidenberg, dim A ≥ dim(A/ ker ϕ)) = dim B ≥ dim Bn.

(b) Let m ⊂ A and n ⊂ B be as above. Then by (a) d = dim A = dim Bn. On the other hand we have mBn = nBn, and hence isomorphisms 2 ∼ 2 2 ∼ 2 m/m ⊗k(m) k(n) = m/m ⊗A(m) Bn/mBn = m/m ⊗A Bn = n/n ,

2 2 i.e., dimk(m) m/m = dimk(n) n/n .

Proof of Proposition 15.1: For U → X ´etalewe have morphisms

α β Gm(U) → Gm(U ×X Spec K) → ⊕ Z(U ×X Spec(K)) , x∈X(1) ` where α is the restriction and β is deﬁned as follows: We have U ×X Spec K = Spec k(η) η∈U 0 by 15.2 (a) and a U ×X Spec k(x) = Spec(k(y))

y∈U1 f(y)=x

× The component βy : Gm(k(η)) = k(η) → Z of β at y is 0 if y∈ / {η} (⇔ η∈ / Spec OU,y) and × the discrete valuation associated to y on k(η) , if η is the generic point of Spec OU,y, hence k(y) = Quot OU,y. It follows immediately that βα = 0.

By forming the associated sheaf to ⊕ (ix)∗Z we obtain the wanted sequence. For the x∈X(1) 1 exactness it suﬃces to prove the exactness if U is replaced by a local ring OU,y for y ∈ U . But then the sequence is

× × vy 0 → OU,y → (Quot(OU,y)) → Z → 0

105 and hence is exact.

Now we consider the long exact cohomology sequence associated to

0 → Gm → j∗Gm → ⊕ (ix)∗Z → 0 . x∈X1

We need the following.

Lemma 15.3 Let X be a quasi compact, quasi separated scheme. Then, for every inductive system (Fi)i∈I of abelian ´etalesheaves on X, we have:

lim Hn (X,F ) →∼ Hn (X, lim F ) . → ét i ét → i i∈I i∈I In particular, étalecohomology commutes with direct sums.

Proof: See Tamme II, Introduction to ´etaleCohomology, 1.5.3.

S(Xét) is equivalent to S(Xét,f.p.) for the noetherian site of all étale X-schemes of finite presentation. Thus we have

n n H´et(X, ⊕ (ix)∗Z) = ⊕ H´et(X, (ix)∗Z) x∈X1 x∈X1

Lemma 15.4 Let (Xi)i∈I be a projective system of quasi compact and quasi separated

schemes, with affine transition morphisms. Let i0 ∈ J and F be an étalesheaf on Xi0 . Then the natural map lim Hn (X ,F | ) →∼ Hn (lim X ,F | ) → ét i Xi ét ← i ←lim X i∈I i i is an isomorphism, where F | and F | denote the pull-backs of F , respectively. Xi X=lim← Xi i

Proof: See Milne, Etale´ Cohomology, Lemma 1.16.

Corollary 15.5 Let f : Y → X be a quasi compact, quasi separated morphism of schemes, let F be an ´etalesheaf on Y and let x be a geometric point of X. Then we have

n ∼ n h (R f∗F )x = H´et(Y ×X Spec(OX,x),F |...) .

n n Proof Let P be the presheaf U H´et(Y ×X U, F ) on X. Then we have R f∗F = aP (the associated sheaf), hence

(Rnf F ) = P = lim Hn (Y × U, F |Y × U) ∗ x x → ét X X U étaleneighborhood of x in X 15.4 n sh = Hét(Y ×X Spec OX,x,F |...) .

sh The claim above now follows from the fact that OX,x ⊗OX,x K = Kx.

106 Note that we have i i sep × H´et(SpecKx, Gm) = H (Kx, (Kx ) )

We need some facts from the Galois cohomology.

Lemma 15.6 (Hilbert 90) H1(K, (Ksep)×) = 0 for every ﬁeld K.

Lemma 15.7 For every ﬁeld L, we have H2(L, (Ksep)∗) = Br(L), the Brauer group of L. By Corollary 15.5, for every geometric point x of X we have

1 1 × (R j∗Gm,K )x = H (Kx,Kx ) = 0 ,

where Kx = Quot(OX,x), since we have Spec(K) ×X Spec(OX,x) = Spec(Kx). Therefore

1 R j∗Gm,K = 0 .

From the Leray-spectral sequence for j∗ we thus get

1 1 sep × H (X, j∗Gm,K ) = H (K, (K ) ) = 0 and an exact sequence

2 0 2 3 0 → H (X, j∗Gm,K ) → Br(K) → H (X,R j∗Gm,X ) → H (X, j∗Gm,X ) . On the other hand, from the sequence (15.1.1) we get an exact sequence

× × 1 0 → Γ(X,OX ) → K → ⊕ Z → H (X, Gm) → 0 , x∈X1

therefore the known isomorphism (for regular X)

1 ∼ H (X, Gm) → P ic(X) .

Now let dim X = 1 and let k(x) be perfect for all x ∈ X1.

i Lemma 15.8 R j∗Gm,K = 0 for i ≥ 1.

Proof For all geometric points x of X, Kx = Quot(OX,x), and OX,x is a discrete valuation ring with algebraic closed residue field, or a separably closed field. In the first case Kx has the cohomological dimension 1, in the second case the cohomological dimension 0. Thus

i i sep × (R j∗Gm,K )x = H (Kx, (Kx ) ) = 0 for i ≥ 1. In this case we have isomorphisms

i ∼ i sep × H (X, j∗Gm,K ) = H (K, (K ) ) for all i.

107 The sequence (15.1.1) gives an exact sequence

2 2 3 0 → H (X, Gm) → Br(K) → ⊕ H (k(x), Z) → H (X, Gm) → ... x∈X1 since (ix)∗ is exact, and with the Leray spectral sequence for (ix)∗ we get

i ∼ i H (X, (ix)∗Z) = H (Spec(k(x), Z) for all i.

Finally, let X be a smooth projective curve over an algebraically closed ﬁeld k, and as before let K = K(X) be the function ﬁeld of X.

Theorem 15.9 (Theorem of Tsen) K has the cohomological dimension 1.

More precisely, Tsen showed that K is a so-called C1-ﬁeld, and hence cd(K) ≤ 1. Hence 1 i ∼ i cd(k(x)) = 0 for x ∈ X , H (X, Gm) → H (X, j∗, Gm,K ) = 0 for i ≥ 2.

Let n be invertible in k. From the Kummer-sequence

n 0 → µn → Gm → Gm → 0 , the next lemma follows by passing to cohomology

0 0 × Lemma 15.10 (i) H (X, µn) = µn (since H (X, Gm) = k ) 1 ∼ 2g (ii) H (X, µn) = P ic(X)[n] = (Z/nZ) 2 ∼ 0 1 (iii) H (X, µn) = P ic(X)/n P ic(X) = Z/nZ, where g is the genus of X, thus g = dimk H (X, ΩX/k).

Proof The ﬁrst isomorphisms are obvious from the long exact cohomology sequences. The deg 0 further isomorphisms in (ii) and (iii) follow from the fact that for P ic (X) = ker(P ic(X) Z) we have P ic0(X) ∼= Jac(X)(k) , where Jac(X) is the Jacobian variety of X. This is an abelian variety of dimension g, where g is the genus of X.

Corollary 15.11 The cohomology groups Hi(X, Z/n) are ﬁnite.

Proof Over an algebraically closed ﬁeld we can identify Z/n with µn.

108