inhaltsverzeichnis - uni-regensburg.de...etale cohomology prof. dr. uwe jannsen summer term 2015...
TRANSCRIPT
Etale cohomology
Prof. Dr. Uwe JannsenSummer 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 etale site 49
8 The etale site of a field 57
9 Henselian rings 62
10 Examples of etale sheaves 72
11 The decomposition theorem 80
12 Cech 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 regardedobjects. Often such invariants are given by cohomology groups.
This is a long standing approach in topology, where one considers singular cohomologygroups H i(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 lookstopologically like a sphere with g handles:
g=1 g=2
then dimQH1(X,Q) = 2g. These cohomology groups can also be obtained as sheaf cohomo-
logy (of a constant sheaf).
Riemann surfaces can also be regarded as complex algebraic curves, i.e., as algebraic curvesover a field C of complex numbers. For any algebraic varieties X over any field k (or anyscheme) one can consider sheaf cohomology with respect to the Zariski topology. This is usefulfor coherent sheaves, for example for the Grothendieck-Serre duality and the Riemann-Rochtheorem.
However, the Zariski cohomology of an algebraic variety X over C does not give the singularcohomology of a topological space X(C); this is due to the fact that this topology is muchfiner than the Zariski topology. Furthermore one wants to obtain an analogous topology forvarieties over any field k. For fields with positive characteristic, Serre showed that thereexists no cohomology theory H∗(−,Q), such that H1(X,Q) has the dimension 2g for asmooth projective curve of genus g. But Weil had postulated such a theory to show the Weilconjectures 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 etalecohomology. For any prime ` 6= char(k) this provides cohomology groups H i(X,Q`) thathave the properties postulated by Weil. With these, Deligne eventually proved the Weilconjectures.
1
2 Grothendieck topologies/Sites
Grothendieck’s approach for the etale cohomology (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 atthe 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 valuesin 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 categoryX 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. ofrings, resp. ...).
Example 2.2 Let X be a topological space. Then one can assign to X the following categoryX: 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: Becauseof 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.
(a): A Grothendieck topology on X consists of a set T of families (Uiϕi→ U)i∈I of mor-
phisms 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 Vexist, 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 .
(T3) If ϕ : U ′ → U is an isomorphism, then (U ′ϕ→ 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: Thefibre 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., thecategory 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
F (U)α→∏i
F (Ui)α1
⇒α2
∏i,j
F (Ui ×U Uj)
is exact, where the arrow α1 on the right side is induced by the first projections Ui×UUj → Uiand the arrow α2 is induced by the second projection Ui ×U Uj → Uj (This means that αis the difference kernel of α1 and α2, see appendix 4.A below). Morphisms of sheaves aremorphisms of the underlying presheaves.
Remark 2.6 Let C be the category of sets. If, for s ∈ F (U), we denote the component ofα(s) in F (Ui) by s|Ui and for (si) ∈
∏i
F (Ui), we denote the images of si and sj in F (Ui×UUj)
by si|Ui×UUj and sj|Ui×UUj respectively, then we literally obtain the same conditions as forthe usual sheaves on topological spaces, except that we replace Ui ∩ Uj with Ui ×U Uj: Theconditions are:
(i) If s, t ∈ F(U) and s|Ui = t|Ui for all i, then s = t.
(ii) If (si)i∈I ∈∏i
F(Ui) with si|Ui×UUj = sj|Ui×UUj for all i, j ∈ I, then there is an s ∈ F(U)
with s|Ui = si for all i ∈ I.
Definition 2.7 (a) A morphism f : (X ′, T ′) → (X , T ) of sites is a (covariant) functorf 0 : X → X ′ (!) which has the following properties:
(S1) If (Uiϕi−→ U) is in T , then (f 0(Ui)
f0(ϕi)−→ f 0(U)) is in T ′.(S2) If (Ui → U) is in T and V → U is a morphism in T , then the canonical morphism
f 0(Ui ×U V )→ f 0(Ui)×f0(U) f0(V )
is an isomorphism for all i.
Example 2.8 If f : X ′ → X is a continuos map between of topological spaces, we obtain amorphism f : S(X ′)→ S(X) of the associated sites (Example 2.4) by
f−1 : X → X ′
U 7→ f−1(U) .
3
3 Constructions for presheaves and sheaves
For a category X let Pr(X ) be the category of abelian presheaves on X .
Definition 3.1 (Push-forward) Let f : (X ′, T ′) → (X , T ) be a morphism of sites and letP ′ : X ′ → Ab be an abelian presheaf. Then the direct image (or push-forward) fPP
′ of P ′ isdefined as the presheaf
fPP′ = P ′f 0 : X f0→ X ′ P
′→ Ab .
Explicitly we have (fPP′)(U) = P ′(f 0(U)) for U in X and fP (ϕ) = P ′(f 0(ϕ)) : P ′(f 0(U2))→
P ′(f 0(U1)) for ϕ : U1 → U2 in X ). For a morphism ψ : P ′1 → P ′2 of abelian presheaves on Xone obtains a morphism
(3.1.1) fPψ : fPP′1 → fPP
′2
as follows: For U in X define
(fPψ)U : (fPP′1)(U) → (fPP
′2)(U)
q q qψf0(U) : P ′1(f 0(U)) → P ′2(f 0(U)) .
One can see easily that this produces a morphism of presheaves (3.1.1) and that one obtainsa functor
fP : Pr(X ′) → Pr(X )P ′ 7→ fPP
′
ψ 7→ fPψ .
Proposition 3.2 The functor
fP : Pr(X ′)→ Pr(X )
has a left adjointfP : Pr(X )→ Pr(X ′) .
For presheaves P ∈ Pr(X ) and P ′ ∈ Pr(X ′) we thus have isomorphisms
(3.2.1) HomX ′(fPP, P ′) ∼= HomX (P, fPP
′) ,
functorially in P and P ′. For a presheaf P on X , fPP is called the inverse image (or pull-back) of P .
Proof of 3.2: For U ′ in X ′ consider the following category IU ′ : Objects are pairs (U, ψ),where U is an object in X and
ψ : U ′ → f 0(U)
is a morphism in X ′. A morphism (U1, ψ1) → (U2, ψ2) is a morphism ϕ : U1 → U2 in X forwhich the diagram
(3.2.2) f 0(U1)
f0(ϕ)
U ′
ψ1
;;xxxxxxxxx
ψ2 ##FFFFFFFFF
f 0(U2)
4
is commutative. Then we have a functor
(3.2.3)
P : IopU ′ → Ab
(U, ψ) 7→ P (U)ϕ 7→ P (ϕ)
(where IopU ′ denotes the dual category of IU ′) and define
(fPP )(U ′) = lim→(U,ψ)∈Iop
U′
P (U)
as the inductive limit over IopU ′ (The idea is that (fPP )(U ′) is the inductive limit of all setsP (U), where “U ′ is contained in f 0(U)”, see Example 3.4 below).
If ϕ′ : U ′ → V ′ is a morphism in X ′, we obtain a functor
IV ′ → IU ′ ,
by mapping an object (V, V ′ → f(V )) in IV ′ to the object (V, U ′ϕ′→ V ′ → f(V )), and
mapping a morphism ϕ : V1 → V2 to the same morphism.
This gives a morphism
(fPP )(V ′) = lim→IopV ′
P (U)→ lim→IopU′
P (U) = (fPP )(U ′) .
With this fPP becomes a contravariant functor
fPP : X ′ → Ab
i.e., an abelian presheaf on X ′.Now we prove the adjointness. Let P ′ be an abelian presheaf on X ′ and let
(3.2.4) v : fPP → P ′
be a morphism of abelian presheaves. For all U in X one obtains the homomorphism
(3.2.5) vf0(U) : (fPP )(f 0(U))→ P ′(f 0(U)) = (fPP′)(U) .
Furthermore the pair (U, idf0(U)) is an object of If0(U), and we obtain a canonical homomor-phism
(3.2.6) P (U)→ lim→(V,ψ)∈Iop
f0(U)
P (V ) = (fPP )(f 0(U)) ,
and by composition of (3.2.6) and (3.2.5) a homomorphism
(3.2.7) P (U)→ (fPP′)(U) ,
which is obviously functorial in U , so that we get a morphism of abelian presheaves on X
(3.2.8) w : P → fPP′ .
5
Conversely, consider a morphism w as in (3.2.8), and let U ′ ∈ ob(X ′). Then for every object(U, ψ : U ′ → f 0(U)) in IU ′ one has the homomorphism
P (U)wU−→ (fPP
′)(U) = P ′(f 0(U))P ′(ψ)−→ P ′(U ′) .
This homomorphism is functorial in (U, ψ) and gives a homomorphism (universal propertiesof the direct limit)
(fPP )(U ′) = lim→(U,ψ)∈Iop
U′
P (U)→ P ′(U ′) ,
which itself is functorial in U ′ and therefore gives a morphism
v : fPP → P ′
of abelian presheaves on X ′.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 limitsexist in C, e.g., C = Set, Rg, ....
Example 3.4 Let f : X ′ → X be a continuous map of topological spaces and
f : S(X ′)→ S(X), U 7→ f−1(U) ,
the corresponding morphism of sites. Then
fP : Pr(X ′)→ Pr(X), fP : Pr(X)→ Pr(X ′)
are the usual functors. This is obvious for fP : One has (fPP′)(U) = P ′(f−1(U)). For fP one
obtains the usual construction: for U ′ ⊆ X ′, IU ′ is the ordered set (!) of the open sets U ⊆ Xwith f(U ′) ⊆ U , thus U ′ ⊆ f−1(U), and fPP (U ′) = lim→
f(U ′)⊆U
P (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 ) → Pr(X ) .
Theorem 3.5 The embedding i has a left adjoint
a = aT : Pr(X )→ Sh(X , T ) .
Thus for all presheaves P and all sheaves F one has isomorphisms, functorial in P and F ,
HomPr(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 -morphismsfj : Vj → Uε(j).
With the refinements as morphisms and the obvious compositions, we obtain the categoryT (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
H0(U, P ) = ker(∏i
P (Ui)α1
⇒α2
∏i,j
P (Ui ×U Uj))
is called the zeroth Cech cohomology of P with respect to U. Here let α1 and α2 bedefined as in Definition 2.5.
(b) CallH0(U, P ) = lim→
U
H0(U, P )
the zeroth Cech cohomology of P for U , where the direct limit runs over the categoryT (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 allU = (Ui → U) in T (U) the canonical homomorphism
P (U)→ H0(U, P )
is an isomorphism. In this case P (U)→ H0(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) := H0(U, P ) .
This produces a presheaf, since for ϕ : V → U in X we have a canonical homomorphism
(3.5.1) ϕ∗ : H0(U, P )→ H0(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
(3.5.2) H0(U, P )→ H0(UV , P ) ,
and by passing to the limit over the coverings in U we obtain (3.5.1).
A morphism of abelian presheavesψ : 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
(3.5.4) H0(U, P1)→ H0(U, P2) .
This is compatible with refinements and by passing to the limit over T (U)op gives a map
(3.5.5) ψU : H0(U, P1)→ H0(U, P2) .
For every morphism ϕ : V → U , the diagram
ψU : H0(U, P1) //
ϕ∗
H0(U, P2)
ϕ∗
ψV : H0(V, P1) // H0(V, P2)
is commutative. This gives (3.5.3). One can easily see that this defines a functor
Pr(X ) → Pr(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
P (U)→∏i
P (Ui)
is injective. (Equivalently, P (U)→ H0((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):
1) For every element s ∈ H0(U, P ) there exists a covering U = (Ui → U) in T (U) and anelement s ∈ H0(U, P ) which is mapped to s under
H0(U, P )→ H0(U, P )
(In this case we say that s is represented by s).
2) If s is represented by s1 ∈ H0(U1, P ) and s2 ∈ H0(U2, P ) (with U1,U2 ∈ T (U)), then thereare refinements U3 → U1,U3 → U2, such that s1 and s2 have the same image in H0(U3, P ).
Proof of 3.10 (a): Let (Ui → U)i be a covering in T and let s ∈ ker(P (U)→∏i
P (Ui)). We
have to show s = 0. There exists a covering (Vj → U)j and an element s ∈ H0((Vj → U)j, P )which represents s.
8
Let si be the image of s under
H0((Vj → U)j, P )→ H0((Vj ×U Ui → Ui)j, P ) .
This represents s|Ui = 0 ∈ H0(Ui, P ). By the preliminary remark there exists a refinementfor every i ∈ I
fi : (Wik → Ui)k → (Vj ×U Ui → Ui)j
such that f ∗i maps si to 0 in H0((Wik → Ui)k, P ).
By composition of the coverings (Wik → Ui)k and (Ui → U)i (axiom (T2)) we obtain acovering (Wik → U)k and via the fi a refinement
f : (Wik → U)k → (Vj → U)j .
Then, underf ∗ : H0((Vj → U)j, P )→ H0((Wik → U)k, P ) ,
s is mapped to 0 by construction. Thus s = 0.
(b): This is given by the canonical homomorphisms
P (U)∼→ H0((U
id→ U), P )→ H0(U, P ) = P (U) .
(c) Let P be a separated abelian presheaf.
Claim 3.10.1 : For every covering U = (Ui → U) in T , H0(U, P )→ H0(U, P ) is injective.
Proof By the preliminary remark it suffices to show the injectivity of
f ∗ : H0((Ui → U), P )→ H0((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 tworefinements
(Vj ×U Ui → U)pr2−→ (Ui → U)
(Vj ×U Ui → U)pr1−→ (Vj → U)
f→ (Ui → U) .
By the following Lemma 3.11 the two induced homomorphisms
H0((Ui → U), P )pr∗2⇒pr∗1f
∗H0((Vj ×U Ui → U), P )
are equal. It thus suffices to prove the injectivity of pr∗2; then pr∗1f∗ and hence also f ∗ is
injective. But pr∗2 is the restriction of∏i
P (Ui)pr∗2−→
∏i
∏j
P (Vj ×U Ui)
9
to H0((Ui → U), P ), and for every i, P (Ui) →∏j
P (Vj ×U Ui) is injective, because P is
separated.
If we apply Claim 3.10.1 to the covering (U → U), then the injectivity of
P (U) = H0((U → U), P )→ H0(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
(3.10.2) P (U)→∏i
P (Ui)⇒∏i,j
P (Ui ×U Uj)
is exact. By (a) P is separated, thus the first map is injective. Now let
(si) ∈ ker(∏i
P (Ui)⇒∏i,j
P (Ui ×U Uj) .
For each i choose a covering (Vik → Ui) and an element si ∈ H0((Vik → Ui), P ), whichrepresents si ∈ P (Ui). Let s1
ij be the image of si under
H0((Vik → Ui), P )→ H0((Vik ×U Uj → Ui ×U U), P ) ,
and let s2ij be the image of sj under
H0((Vik → Uj), P )→ H0((Ui ×U Vik → Ui ×U Uj), P ) .
The elements represented by s1ij and s2
ij in H0(Ui×U Uj, P ) are equal to the images of si andsj, respectively, and thus are equal. It follows from Claim 3.10.1 that s1
ij and s2ij have the
same image inH0((Vik ×U Vj` → Ui ×U Uj), P ) ⊆
∏k,`
P (Vik ×U Vj`)
This implies that
s′ = (si) ∈ ker(∏i,k
P (Vik)⇒∏i,k,j,`
P (Vjk ×U Vj`)) = H0((Vik → U), P ) .
The element s′ ∈ P (U) represented by s′ is then mapped to (si) under P (U) →∏i
P (Ui).
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 (theassignment P 7→ P , the morphism P → P ), we obtain a commutative diagram
Pψ //
F
oρ1
ρ
Pψ //
F
oρ2
aP =≈P
≈ψ //
≈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
P can //
ψ AAAAAAAA aP
ρ−1aψ =: ψ′
F ,
in which ψ′ is unique: For this it suffices to show that in a commutative diagram
P can //
ψ ???????? P
µ
F
the morphism µ is unique (By applying it twice it follows that ψ′ is unique). Because of theadditivity it suffices to show this for ψ = 0. But if ψ = 0 and (Ui → U) is a covering in T ,then the commutative diagram
P (U) //
ψU
H0((Ui → U), P ) //
µ
wwoooooooooooo
∏i P (Ui)
ψ
pri // P (Ui)
ψUi=0
F (U) ∼ // H0((Ui → U), F )
//∏
i F (Ui)pri // F (Ui)
implies that µ = 0.
Lemma 3.11 Letf, g : (U ′j → U)→ (Ui → U)
be two refinements of coverings in the Grothendieck topology T . Then for every abelianpresheaf P the induced maps
f ∗, g∗ : H0((Ui → U), P )→ H0((U ′j → U), P )
11
are equal.
Proof Let f = (ε, (fj)) and g = (δ, (gj)). We have a diagram
∏i
P (Ui)d0=α1−α2 //
f∗
g∗
∏i1,i2
P (Ui1 ×U Ui2)
f∗
g∗
∆1
uullllllllllllll
∏j
P (U ′j)d0=α1−α2 //
∏j1,j2
P (Uj1 ×U Uj2) ,
where ∆1 is defined by(∆1s)j = P ((fj, gj)U)(sε(j),δ(j)) ,
with the canonical morphism
(fj, gj)U : U ′j → Uε(j) ×U Uδ(j) .
One checks that∆1 d0 = g∗ − f ∗ .
Thus f ∗ and g∗ agree on ker d0 = H0((Ui → U), P ), which proves the claim.
With this result we are able to understand the limit
H0(U, P ) = lim→T (U)0
H0(U, P ) .
better: For two coverings U,U′ in T (U) call U′ finer than U (Notation U′ ≥ U), if there is arefinement f : U′ → U. Define the equivalence relation ∼ on the set ob(T (U)) of the coveringsof U by
U ∼ U′ ⇔ U ≤ U′ and U′ ≤ 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: Fortwo 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 haveU,V ≤W.
By Lemma 3.11, for U′ ≥ U we further obtain, by choice of a refinement f : U′ → U, auniquely determined homomorphism
(3.11.1) H0(U, P )→ H0(U′, P ) .
12
Corollary 3.12 The zeroeth Cech Cohomology
H0(U, P ) = lim→T (U)0
H0(U, P ) ,
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 ′, T ′)→ (X , T )
be a morphism of sites.
Lemma 3.13 If F ′ is an abelian sheaf on (X ′, T ′), then fPF′ is again a sheaf.
Proof : Left to the readers!
Lemma/Definition 3.14 (a) f∗F′ := fPF
′ is called the direct image (or push-forward) ofF ′ (with respect to f).
(b) For an abelian sheaf F on (X , T ), f ∗F := afPF is called the (sheaf-theoretic) inverseimage (or pull-back) of F (with respect to f).
(c) The functorf ∗ = afP : Sh(X , T )→ Sh(X ′, T ′)
is left adjoint to the functor
f∗ : Sh(X ′, T ′)→ Sh(X , T ) .
Proof For sheaves F ′ on (X ′, T ′) and F on (X , T ) we have canonical isomorphisms
HomSh(f∗F, F ′)
∼= HomSh(af
PF, F ′)∼= HomPr(f
PF, iF ′)
∼= HomPr(F, fP iF
′)∼= HomSh(F, f∗F
′) ,
functorial in F and F ′.
13
4 The abelian categories of sheaves and presheaves
Let X be a category.
Theorem 4.1 (a) The category Pr(X ) of abelian presheaves on X is an abelian category.
(b) A sequence of abelian presheaves
0→ P ′ → P → P ′′ → 0
is exact, if and only if for all U ∈ ob(X ) (:= objects of X ) the sequence
0→ P ′(U)→ P (U)→ P ′′(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 ofpresheaves kerP ϕ (i.e., (kerϕ)(U) = ker(ϕU : F1(U)→ F2(U)) for all U in X .
(c) The cokernel of a morphism ϕ : F1 → F2 of abelian sheaves is equal to a cokerPϕ, i.e., thesheaf associated to the presheaf cokernel cokerPϕ ( defined by (cokerPϕ)(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 inX 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 theassociated sheaf. For (d) we note that ϕ is an epimorphism if and only if cokerϕ = 0, i.e.,if a(cokerPϕ) = 0. This means that there is a covering (Ui → U) for every U in X andevery s ∈ (cokerPϕ)(U) with s|Ui = 0 for all i. Since (cokerPϕ)(Ui) = coker(ϕUi : F1(Ui)→F2(Ui)), the proposition follows.
Theorem 4.3 (a) There exist arbitrary limits (inverse limits) and colimits (direct limits) inPr(X ) and Sh(X , T ).
(b) The functor i : Sh(X , T ) → Pr(X ) is left exact.
(c) The functor a : Pr(X )→ Sh(X , T ) is exact.
Proof (a) In Pr(X ) we have
(lim←PPi)(U) = lim←
i
P (Ui) ,
14
and a similar formula for direct limits. If now (Fi)i∈I is a diagram of sheaves, then lim←i
PPi is
again a sheaf, because inverse limits commute with each other. Hence lim←i
Fi = lim←i
PFi. The
direct limit islim→i
Fi = a(lim→i
PFi) ,
since we have the universal property for every sheaf G
HomSh(a lim→PFi, G) ∼= HomPr(lim→
i
Fi, iG) ∼= lim← HomPr(Fi, iG) ∼= lim← HomSh(Fi, G)
(b) It follows from the adjunction of i and a that i is left exact and a is right exact (seeLemma 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→U∈T(U)0
H0(U, P ) ,
the functor P 7→ H0(U, P ) is left exact, and forming an inductive limit of abelian groups isan exact functor (see also Annex 4.B).
Theorem 4.4 Let f : (X ′, T ′)→ (X , T ) be a morphism of sites.
(a) fP : Pr(X ′)→ Pr(X ) is exact and fP : Pr(X )→ Pr(X ′) is right exact.
(b) If finite limits exist in X and X ′, and if f 0 : X → X ′ commutes with these, then fP isexact.
(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 ′, and if f 0 : X → X ′ commutes with these, then f ∗ isexact.
Proof (a) The exactness of fP follows from 4.1, and because of the adjunction, fP is rightexact (see below).
(b) It follows from the assumption that the category IU ′ (see Proof of 3.2) is cofiltered forevery object U ′ in X ′ (see Annex 4.B.2). In fact, in IU ′ finite limits exist by assumption:For this one has to show the existence of finite products and difference kernels. But forobjects U ′ → f 0(U1) and U ′ → f 0(U2) in IU ′ the product is the product morphism U ′ →f 0(U1)× f 0(U2) = f 0(U1 × U2), and for morphisms
f 0(U1)
f0(α)
f0(β)
U ′
;;xxxxxxxxx
##FFFFFFFFF
f 0(U1)
15
(with U1
β
⇒αU2) the difference kernel is
f 0(ker(α, β)) = ker(f 0(α), f 0(β))
U ′
44jjjjjjjjjjjjjjjjjj
**TTTTTTTTTTTTTTTTTTTT
f 0(U1)
(check the universal properties!). If I ′U is cofiltered, IopU ′ is filtered. Therefore forming thedirect limit over IopU ′ is exact (see Appendix 4.B.3).
(c) Since f∗F′ = fP iF
′, the claim follows for f∗, because i is left exact (4.3(b)) and fP isexact by 4.4 (a). Furthermore f ∗ is right exact, because f ∗ is left adjoint to f∗ (see 4.5).
(d) By (b), fP is exact, therefore f ∗ = afP i is left exact, because a is exact and i is leftexact.
Lemma 4.5 Let A and B be abelian categories, and let F : A → B and G : B → A befunctors such that G is right adjoint to F (⇔ F is left adjoint to G). Then G is left exactand F right exact.
Proof By assumption we have bi-functorial isomorphisms
HomA(A,GB) ∼= HomB(FA,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) isisomorphic to the sequence
(4.5.4) 0→ HomB(FA,B1)→ HomB(FA,B2)→ HomB(FA,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 thecontravariant Hom-functor
hX = HomC(−, X) : C → SetsA 7→ HomC(A,X) ,
i.e., if there is a bijection, functorial in A,
F (A) = HomC(A,X) .
(b) A covariant functorG : C → Sets
is called representable, if it is isomorphic to the covariant Hom-functor
hX = HomC(X,−) : C → SetsA 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) .
By definition hX and hX are representable. In the situation 4.A.1(a) (resp. (b)) X is calledrepresenting object for F (resp. G). The object X – if it exists – is in each case uniqueup 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) IfG : C → Sets
is a covariant functor, then for every object Y in C one has a canonical bijection
eY : Hom(hY , G)∼→ G(Y )
ϕ 7→ ϕY (idY ) .
17
Proof (a): The inverse mX of eX assigns the following morphism mX(a) := ϕa of functorsto an object a ∈ F (X)
ϕaA : hX(A) = HomC(A,X) → F (A)f 7→ ϕaA(f) := F (f)(a) .
Note: For f : A→ X we obtain F (f) : F (X)→ F (A), because F is contravariant. Note thatϕa is indeed a morphism of functors: For a morphism g : A→ A′ in C we have a commutativediagram
HomC(A′, X)
ϕaA′ //
g∗
F (A′)
F (g)
HomC(A,X)
ϕaA // F (A)
f ′ //_
F (f ′)(a)_
F (g)(F (f ′)(a))
f ′g // F (f ′g)(a) ,
since F (f ′g) = F (g) F (f ′).
We have eXmX = id: For a ∈ F (X) we have eX(ϕa) = ϕaX(idX) = a, because F (idX) =idF (X). Conversely we have mXeX = id: Let ϕ : hX → F be given and let eX(ϕ) = ϕX(idX) ∈F (X), and let ϕeX(ϕ) : hX → F be constructed as above. For every A ∈ ob(C) the maps
ϕeX(ϕ)A = ϕA HomC(A,X)→ F (A)
are equal, becauseϕeX(ϕ)A (f) = F (f)(ϕX(idX)) = ϕA(f) ,
since the diagram
idX ∈ HomC(X,X)_
f∗
ϕX // F (X)
F (f)
f ∈ HomC(A,X)ϕA // F (A)
commutes (ϕ is a morphism of functors).
The proof of (b) is analogous.
By applying 4.A.2 to F = hY resp. G = hX , we get the following:
Corollary 4.A.3 For objects X, Y in C one has canonical bijections
HomC(X, Y )∼→ Hom(hX , hY )
HomC(X, Y )∼→ Hom(hY , hX)
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 subca-tegory of the representable functors.
Now we define fiber products and fiber sums.
Definition 4.A.5 LetY
β
Xα // 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
X ×S Ypr2 //
pr1
Y
β
Xα // S
(pr1 resp. pr2 are called the first resp. second projection).
(b) If
Wα //
β
Y
β
X α // S
is another commutative diagram, then there is a unique morphism γ : W → X ×S Y withpr1γ = α and pr2γ = β, i.e., such that the diagram
Wα
&&
∃!γ
$$II
II
I
β
X ×S Ypr2 //
pr1
Y
β
Xα // S
commutes.
Remark 4.A.6 (a) Fiber products do not always exist; but if they exist, they are uniqueup to canonical isomorphism (Exercise!).
19
(b) In the category Sets of sets fiber products exist: For maps Mα→ T
β← 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 theproperty 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 )
β∗
g_
HomC(Z,X)
α∗ // HomC(Z, S) βg
f // αf
.
(d) A slightly different interpretation is given as follows: Let C/S be the category of objectsin C over S: objects in C/S are objects X in C together with a morphism α : X → S; onecan regard α itself as objects, because by α, X is already given. A morphism from α : X → Sto α′ : X ′ → S is a morphism f : X → X ′, for which the diagram
Xf //
α@@@@@@@@ X ′
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, becausethe universal properties correspond.
Lemma 4.A.7 The properties of Lemma 1.14 (commutativity, associativity, transitivity andfunctoriality) 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
Xα
AAAAAAAA
f
Yβ
||||||||
g
S
X ′α′
??~~~~~~~~Y ′ ,
β′
``AAAAAAAA
20
i.e., X,X ′, Y and Y ′ are objects over S (by α, α′, β and β′) and f and g are morphisms ofobjects over S (commutativity of triangles). If the fiber products exist, we obtain a diagram
X ×S Ypr2 //
pr1
∃!h
&&NN
NN
N Y
g
X ′ ×S Y ′
pr′2 //
pr′1
Y ′
β′
X
f // X ′α′ // S ,
where the internal and external square are commutative (Note that α′f = α and β′g = β).By the universal property of X ′×SY ′ there exists a unique morphism h : X×SY → X ′×SY ′which makes the entire diagram commutative; we call this f × g, and it fulfills the claim ofLemma 1.14 (d).
Remark 4.A.7 By reversing all arrows one obtains the notion of a fiber sum X∐
S Y for adiagram
Sβ //
α
Y
X
where one has the dual universal properties and functorial properties.
Example 4.A.8 For every diagram of ring homomorphisms
Rβ //
α
B
A
,
the fiber sum in the category of rings exists and is given by the tensor product: One has acommutative diagram
Rβ //
α
B
A // A⊗R B
and for every diagram of rings
Rβ //
α
B
g
A //
f ,,
A⊗R B∃!h
$$HH
HH
H
C
with fα = gβ there exists a uniquely determined ring homomorphism h as indicated abovethat 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 forma set.
Examples 4.A.10 Often, those small categories are “really small” in a sense that one canwrite all objects and morphisms in them.
(a) The discrete category I over a set I has the elements of I as objects and only theidentities as morphisms.
(b) Let•
@@@@@@@
•
•
??~~~~~~~
be the category with three objects (marked by points) and apart from for the identities onlyhas 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 ∈ Gas 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 onemorphism i→ j, if i ≤ j.
Definition 4.A.11 Let I be a small category, and let C be any category. A diagram in Cover 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 Iwith 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 bya diagram
Xg−→ Y
β←− Z
with morphisms α and β in C. Morphisms of such diagrams are commutative diagrams
X
α // Y
Z
βoo
X ′α′ // Y ′ Z ′ .
β′oo
(b) Consider the small category • −→ • (2 objects, apart from the identities only one arrow).Diagrams over this category in C are simply morphisms
Af−→ B
22
in C, where morphisms of these are commutative diagrams
A
f // B
A′
f ′ // B′
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 covariantfunctor
X : (I,≤) → C
is the same as an inductive system over I in C. A contravariant functor
X ′ : (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 → Ci 7→ A
i→ j 7→ idA .
Example 4.A.14 In Example 4.A.12 (a) I = • −→ • ←− • the constant object A is
AidA−→ A
idA←− A .
Definition 4.A.15 (a) One says that the limit (or inverse limit) of an I-object (Ai)i∈I existsin CI , if the contravariant functor
C → SetsX 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 limi∈I
Ai .
(b) One says that the colimit (or direct limit) of (Ai)i∈I exists, if the covariant functor
C → SetsX 7→ HomCI ((Ai)i∈I , X)
is representable. The representing object is called the colimit of (Ai)i∈I , notation
colim (Ai)i∈I = colimi∈I
Ai .
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 obviouslygiven 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
X
ϕi>>
ϕj AAAAAAAA
Aj
commutes, where the vertical morphism belongs to i → j (one has a functor a : I → C, wewrite a(i) = Ai, and then the morphism on the right is a(i→ j)).
(b) If lim (Ai)i∈I exists (other notation limIa), and if we fix isomorphisms functorial in X
(4.A.16.1) α : HomC(X, limI
(Ai))∼→ HomCI (X, (Ai)) ,
then forX = lim (Ai)i∈I , the image of idlim(Ai) gives an element ϕuniv ∈ HomCI (lim(Ai), (Ai)),by (a), therefore morphisms pi : lim(Ai)→ Ai for all i ∈ I and commutative triangles
(4.A.16.2) Ai
lim(Ai)i∈I
pi
::tttttttttt
pj$$IIIIIIIIII
Aj
for every morphism i → j in I. The morphism pi is called i-th projection. If we have anelement α ∈ HomCI (X, (Ai)), i.e. morphisms ϕi : X → Ai for all i ∈ I and commutativediagrams
(4.A.16.3) Ai
X
ϕi>>
ϕj AAAAAAAA
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)).
(c) For colim (Ai) one obtains an analogous conclusion, by reversing all arrows.
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 morphismsbetween i 6= j), and one has
limI
(Ai) =∏i∈IAi ,
the product of the Ai if this exists in C, because the universal properties of limI
(Ai) (4.A.16
(b)) and∏i∈IAi are identical. Similarly,
colimI
(Ai) =∐i∈IAi ,
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), thenfor every I-object in C, hence every inductive system (Ai)i∈I over I in C,
colim (Ai)i∈I = lim→i∈I
Ai
is the inductive limit of the system, if it exists. Dually, for every I-object (Ai)i∈I in C (whereI notes the dual category to I), hence every projective system over I in C, one sees that
lim (Ai)i∈I = lim←i∈I
Ai
is the projective limit of the system, if it exists.
(c) Consider the category • −→ • ←− • of 4.A.10 (b). For a corresponding diagram
Z
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 Cis
Aα
⇒βB .
Definition 4.A.18 (a) If it exists, the limit of the above diagram is called the differencekernel 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. IfX
γ→ A
is another morphism with αγ = βγ, then there is a unique morphism γ′ : X → ker(α, β)which makes the diagram
ker(α, β) i // A
X
γ
??∃γ′
ddII
II
I
commutative.
(b) One has a morphism Bπ→ coker(α, β) with πα = πβ. If
Bρ→ X
is another morphism with ρα = ρβ, then there is a unique morphism ρ′ : coker(α, β) → X,which makes the diagram
B //
ρ ???????? coker(α, β)
ρ′yys
ss
ss
X
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 addi-tive categories. Like there we have:
Lemma 4.A.20 Let α, β : A→ B be morphisms.
26
(a) If ker(α, β) exists, theni : ker(α, β)→ A
is a monomorphism.
(b) If coker(α, β) exists, thenπ : 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 uniquemorphism 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 limitsand colimits.
Theorem 4.A.21 (a) In C, there exist arbitrary (resp. arbitrary finite) limits if and only ifall 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 colimitsturn 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 sourceand t(f) := B the target of f . For a I-diagram a : I → C consider the morphisms
(4.A.21.1)∏
i∈ob(I)
a(i)α
⇒β
∏f∈mor(I)
a(t(f))
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∏
i∈ob(i)
a(i)prt(f)−→ a(t(f)) ,
the f -component of β is the morphism∏i∈ob(i)
a(i)prs(f)−→ a(s(f))
a(f)−→ a(t(f))
27
Claim: ker(α, β) = limi∈I
a(i).
Proof : By the universal property of ker(α, β), a morphism Z → ker(α, β) corresponds to amorphism
ϕ : Z →∏
i∈ob(I)
a(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)
a(f)
Z
ϕi==||||||||
ϕj !!BBBBBBBB
a(j)
commutes. This just gives the universal property of limi∈I
a(i) (see 4.A.16) for Z = ker(α, β),
because the argument above shows that all diagrams (4.A.21.2) factorize through the diagram
a(i)
a(f)
ker(α, β)
99tttttttttt
%%KKKKKKKKK
a(j) .
Theorem 4.A.22 Let C be category with finite products. The following properties areequivalent:
(a) C possesses fiber products.
(b) C possesses difference kernels.
Proof (a) ⇒ (b): For Af
⇒gB consider the fiber product diagram
Kp2 //
p1
A
(id,g)
A(id,f)// A×B
Then Kp2→ 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) Af
@@@@@@@@
C
B
g
??~~~~~~~~
.
In the associated diagram
(2) Af
2''PPPPPPPPPPPPPPP
K
qA
1
77nnnnnnnnnnnnnnn q //
qB
3
''PPPPPPPPPPPPPPP A×BfpA //gpB
//
pA
OO
pB
C
B
g4
77nnnnnnnnnnnnnnn
let K be the difference kernel of fpA and gpB, and let qA = pAq and qB = qBq. Then alltriangles 1 to 4 are commutative. We claim that K forms a fiber product of diagram (1) viaqA and qB.
Since fpAq = gpBq (by assumption!), we have fqA = fpAq = gpBq = gqB. If one hasfurther 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
Af
''PPPPPPPPPPPPPPP
D` //
h22
k ,,
Kq //
qA
;;wwwwwwwwww
qB##GGGGGGGGGG A×B
OO
C
B
g
77nnnnnnnnnnnnnnn
is commutative, and the uniqueness of ` follows, because q is a monomorphism (universalproperty of the difference kernel).
29
4.B Filtered categories
The following definition generalizes the notion of an inductively ordered set and an inductivelimit. 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 formj
i
88rrrrrrrrrrrrr
%%LLLLLLLLLLLLL
j′
can be extended to a commutative diagram
j
&&MMMMMMMMMMMMMM
i
99rrrrrrrrrrrrr
%%LLLLLLLLLLLLL k
j′
88qqqqqqqqqqqqqq.
(2) Every diagram of the form
iβ
⇒αj
can be extended to a commutative diagram
iβ
⇒αj
γ→ k
(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 ofj and j′ 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 filteredif 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, whereC = Set or C = Ab. Then the direct limit
lim→i∈I
f(i)
exists in C, and the formation of the direct limit is exact (exchanges with finite limits andcolimits).
Proof Explicitly, we havelim→i∈I
f(i) = (∐i∈If(i))/ ∼ ,
where for x ∈ f(i) and y ∈ f(j) the following holds.
x ∼ y ⇔ i→ k, j → k, exist, then x and yhave 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 Pr(X ) and Sh(X , T ) have enough injectives, i.e.,every object P in Pr(X ) has a monomorphism P → I into an injective presheaf I, and theanalogous 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 functorA → AbA 7→
∏i∈IHomA(Ei, A)
is faithful, i.e., for all objects A,A′ in A the map
HomA(A,A′)→ Hom(∏i∈I
HomA(Ei, A),∏i∈I
HomA(Ei, A′))
is injective.
(b) For every object A in A and every subobject B $ A, there exists a morphism Ei → Awhich does not factorize over B.
Proof of the equivalence of (a) and (b):
(a) ⇒ (b): We have the exact sequence
0→ Bi→ A
π→ A′ = A/B → 0 ,
and by assumption A′ 6= 0. Then π 6= 0, and by (a) there is an i ∈ I, for which the inducedmap
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 havea 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 ⊕i∈IAi exist in A (Since cokernels exist, it follows that arbitrary
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 andif it has a family of generators.
The category of R-modules in Example 5.3 is a Grothendieck category, because it has theproperty (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 = ⊕i∈IEi).
For the ring with unit R = HomA(E,E) then consider the faithful functor
A → ModRA p HomA(E,A) .
For the Proof of 5.1 it thus suffices to show:
Theorem 5.7 Pr(X ) and Sh(X , T ) are Grothendieck categories.
Proof : First we consider the generators.
Lemma/Definition 5.8 (a) For an object U in X define the abelian presheaf ZPU by
ZPU(V ) =⊕
Hom(V,U)
Z = ⊕f∈Hom(V,U)
Zf for V in X .
For every abelian presheaf Q on X , we then have isomorphisms
HomPr(X )(ZPU , Q) ∼= HomZ(Z, Q(U)) = Q(U) ,
functorially in Q (i.e., ZPU represents the functor Q 7→ Q(U)).
(b) We define the abelian sheaf ZU as the associated sheaf
ZU = aZPU .
Then for sheaves F on (X , T ), we have
HomSh(X ,T )(ZU , F ) = F (U) ,
functorially in F .
Proof (a): Every morphism f : ZPU → Q is uniquely determined by fU(1idU ) ∈ Q(U).
(b) follows from (a) by adjointness of a and i.
Corollary 5.9 Pr(X ) and Sh(X , T ) have a family of generators.
33
Proof : (a) If P → P ′ is non-zero in Pr(X ), then P (U)→ P ′(U) is non-zero for some objectU in X .
(b)The same applies for Sh(X , T ).
Lemma 5.10 Pr(X ) and Sh(X , T ) satisfy (AB5).
Proof (AB3) and (AB5) are obvious for Pr(X ) by (the proof of) Theorem 4.3 (a), (sin-ce these properties hold for Ab), and it follows from 5.8 (a) that (ZU)U∈X is a family ofgenerators.
(AB3) holds for Sh(X , T ) (arbitrary limits and colimits exist by 4.3(a)), and (AB5) followsfrom the explicit description of the colimits of the proof of 4.3 (a) and the exactness of thefunctor 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
H i(U,−) := H i(U, T ;−) : Sh(X , T ) → AbF p H i(U, F )
is the i-th right derivative of the left exact functor
F p F (U) =: Γ(U, F ) =: H0(U, F ) .
H i(U, F ) is called i-th cohomology of F on U (or i-th cohomology group on U with coefficientsin F ).
By construction, H i(U, F ) = H i(I ·(U)), where F → I · is an injective resolution F inSh(X , T ).
Example 5.12 If X is a topological space and F a sheaf on X, then H i(X,F ) is the usualcohomology of sheaves on X.
Definition 5.13 Let f : (X ′, T ′)→ (X , T ) a morphism of sites. Then Rif∗ is the i-th rightderivative of the left exact functor
f∗ : Sh(X ′, T ′)→ (Sh(X , T ) .
Rif∗F is called the i-th higher direct image of F under f .
Hence Rif∗F = Hi(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 exactsequence
(5.14.1) 0→ F ′ → F → F ′′ → 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 ′) → H0(U, F ) → H0(U, F ′′)δ→ H1(U, F ′)
. . . → Hn(U, F ′) → Hn(U, F ) → Hn(U, F ′′)δ→ Hn+1(U, F ′) → . . .
This is functorial in U and functorial in short exact sequences (5.14.1).
34
(b) Similarly, for every morphism of sites f : (X ′, T ′) → (X , T ) and every short exactsequence of sheaves on (X ′, T ′)
(5.14.2) 0→ G′ → G→ G′′ → 0
we get a long exact sequence
. . .→ Rnf∗G′ → Rnf∗G→ Rnf∗G
′′ δ→ Rn+1f∗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 → Uin X , one has canonical restriction homomorphisms
(5.15.1) α∗ : H i(U, F )→ H i(V, F ) (i ≥ 0),
which coincide with the restriction F (U) → F (V ) for i = 0, are functorial in F , and arecompatible with the exact sequences of 5.14 (a) (i.e., also compatible with the connectingmorphisms).
(b) By this we obtain an abelian presheaf
H i(F ) : U 7→ H i(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 therestrictions of I ·, it follows that one has the relation (αβ)∗ = β∗α∗ for each further morphismβ : W → V . We obtain (b), since id∗U = id. The other functorialities in (a) are again obviousby construction: For a morphism of sheaves F → G, we obtain a morphism
G // J ·
F
OO
// I ·
OO
of injective resolutions that, by definition, provides the functoriality of the cohomology, i.e.the canonical morphisms H i(U, F ) = H i(I ·(U)) → H i(J ·(U)) = H i(U,G). This shows thecompatibility with α∗. The case of exact sequences follows from an exact diagram of injectiveresolutions
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 ′, T ′) → (X , T ) be a morphism of sites. For every sheaf F ∈Sh(X ′, T ′) and every i ≥ 0, Rif∗F is the sheaf associated to the presheaf
U 7→ H i(f 0(U), F )
35
on X ′.
Proof Let F → I · be an injective resolution in Sh(X ′, T ′). Then Rif∗F = Hi(f∗I·) is the
sheaf associated to the presheaf quotient
U 7→ kerP (f∗Ii → f∗I
i+1)(U)/imP (f∗Ii−1 → f∗I)(U)
qker(I i(f 0(U))→ I i+1(f 0(U)))/im(I i−1(f 0(U))/I i(f 0(U)))
qH i(f 0(U), F ) .
36
6 Spectral sequences
Let A be an abelian category.
Definition 6.1 A spectral sequence in A
Ep,q1 ⇒ Ep+q
consists of
(a) objects Ep,q1 in A for all p, q ∈ Z,
(b) subquotients Ep,qr = Zp,q
r /Bp,qr of Ep,q
1 for all r ≥ 2,
(c) morphisms (called the differentials of the spectral sequence)
dp,qr : Ep,qr → Ep+r,q−r+1
r ,
•
•
q
p
r − 1
r
(p, q)
(p+ r, q − r + 1)
such that
Ep,qr+1 =
ker(dp,qr : Ep,qr → Ep+r,q−1
r )
im(dp−r,q+r−1r : Ep−r,q+r−1
r → Ep,qr )
,
(d) subquotients Ep,q∞ = Zp,q
∞ /Bp,q∞ of Ep,q
1 , such that
Bp,qr ⊆ Bp,q
∞ ⊆ Zp,q∞ ⊆ Zp,q
r
for all r ≥ 1 (hence Ep,q∞ “is smaller than Ep,q
r for all r ≥ 1”),
(e) objects (En)n∈Z in A with descending filtrations
En ⊇ . . . ⊇ F pEn ⊇ F p+1En ⊇ . . .
and isomorphismsEp,q∞
∼→ F pEp+q/F p+1Ep+q
for all p, q ∈ Z.
Definition 6.2 The spectral sequence is called finitely convergent, if
Ep,qr = Ep,q
∞ for r >> 0, for all (p, q) ∈ Z2 ,
37
and if for every n ∈ Z the filtration F pEn is finite, i.e.,
F pEn =
0 for p >> 0,En for p << 0.
Some spectral sequences begin with Ep,q2 ; then Ep,q
1 does not exist, and all Ep,qr are subquo-
tients of Ep,q2 .
We now explain how to work with spectral sequences.
1) The layers: For each r one considers the Er-layer of all terms Ep,qr and their differentials
• • • •
• • • •
q
p
E1-layer: d1
• • • •
• •
• • •
q
p
E2-layer:
d2
•
•q
p
Er-layer:dr
dr r − 1
r
One has drdr = 0 and Er+1 = ker dr/imdr.
2) Limit/convergence: Let
Ep,q1 ⇒ Ep+q (or Ep,q
2 ⇒ Ep+q )
38
be a finitely convergent spectral sequence. We obtain the following picture:
••••
••
n
E1,nr
E0,nr
En,0r
n
p+ q = 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 ⊇ . . .
andF pEn/F p+1En = Ep,q
∞ , p+ q = n .
The term on the left is a subquotient of En, the term on the right is a subquotient of Ep,q1 ;
furthermore we have Ep,q∞ = Ep,q
r for r >> 0.
3) Spectral sequences in the first quadrant: Let Ep,q1 ⇒ Ep+q (resp. Ep,q
2 ⇒ Ep+q) be a finitelyconvergent spectral sequence with Ep,q
1 = 0 (resp. Ep,q2 = 0) for p < 0 or q < 0.
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
0
0 0
Lemma/Definition 6.3 (a) One has Ep,q∞ = Ep,q
r for r > max(p, q + 1).
(b) For Ep,q1 ⇒ Ep+q, there are canonical morphisms
En e→ E0,n1
En,01
e→ En .
These are called the edge morphisms.
(c) For Ep,q2 ⇒ Ep+q, there are edge morphisms
En e→ E0,n2
En,02
e→ En
39
Proof (a):
•
p
p− 1
(p, q) q + 1
q
If r > q + 1, the differential dp,qr starting from Ep,qr is zero (since it ends in a zero object). If
r > p, the differential arriving in Ep,qr (dp−r,q+r−1
r ) is zero. If both properties hold, then wehave Ep,q
r+1 = ker dr/imd2 = Ep,qr /0 = Ep,q
r . Since this holds for all higher r (and the spectralsequence converges), we have Ep,q
r = Ep,q∞ .
(b): If Ep,q∞ = 0 for p < 0, then we have En = F 0En (because of the convergence), and we
have
E0,nr+1 = ker(E0,n
rd0,nr→ Er,n−r+1
r ) ⊆ E0,nr
for all r, i.e. E0,n∞ ⊆ E0,n
1 . We obtain
e : En F 0En/F 1En ∼= E0,n∞ → E0,n
1 .
If Ep,q∞ = 0 for q < 0, then we have F n+1En = 0 (because of the convergence), and En,0
r+1 =
coker(En−r,r−1r
dr→ En,0r ) is a quotient of En
r . Then we have En,01 En,0
∞ and a morphism
e : En,01 En,0
∞∼= F nEn/F n+1E → En .
(c) is analogous.
Lemma 6.4 (Exact sequence of the lower terms) Let Ep,q1 ⇒ Ep+q (or Ep,q
2 ⇒ Ep+q) be afinitely convergent spectral sequence in the first quadrant. Then one has an exact sequence
0→ E1,02
e→ E1 e→ E0,12
d0,12→ E2,02
e→ E2 ,
where e always denotes the edge morphism.
40
Proof The picture
•
• • •
0
0
1 2
(or the Proof of 6.5 (a)) shows
E1,0∞ = E0,1
2 , E0,1∞ = ker d0,1
2 , E2,0∞ = coker d0,1
2 .
From this we obtain exact sequences
0→ E0,12
e→ E1 → ker d0,12 → 0
0→ ker d0,12 → E0,1
2
d0,12→ E2,02 → E2,0
∞ ,
and by splicing together these sequences and composing with E2,0∞ ⊆ E2 we obtain the
claimed sequence.
Theorem 6.5 (a) Let A∗ be a complex in A, and let F pA∗ be a descending filtration bysubcomplexes. Then we have a spectral sequence
Ep,q1 = Hp+q(F pA∗/F p+1A∗)⇒ Ep+q = Hn(A∗) .
(b) If the filtration F p is biregular, i.e.,
F pAn =
0 for p >> 0 ,An for p << 0
for every n ∈ Z, then the spectral sequence is finitely convergent.
(c) The E1-differential
dp,q1 : Ep,q1 = Hp+q(F pA∗/F p+1A∗)→ Hp+q+1(F p+1A∗/F p+2A∗) = Ep+1,q
1
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
Zp,qr = im(Hn(F pA∗/F p+rA∗) αp,r // Hn(F pA∗/F p+1A∗) = Ep,q
1 )
Zp,q∞ =?
OO
im(Hn(F pA∗)
OO
αp // Hn(F pA∗/F p+1A∗))
Bp,q∞ =?
OO
im(Hn−1(A∗/F pA∗) δp //
δ
OO
Hn(F pA∗/F p+1A∗))
Bp,qr =?
OO
im(Hn−1(F p−r+1A∗/F pA∗)
OO
δp,r // Hn(F pA∗/F p+1A∗)) .
Here, the morphisms α, αp,r and the commutative diagram above are induced by the com-mutative diagram
F p/F p+r αp,r // F p/F p+1
F p
OO
αp // F p/F p+1
,
where we write Fm for FmA∗.
The connecting homomorphisms δ, δp and δp,r as well as the other commutative diagramsare induced by the following commutative diagrams and the corresponding long exact coho-mology sequences:
δ : 0 // F p //
αp
A∗ //
A∗/F p // 0
δp : 0 // F p/F p+1 // A∗/F p+1 // A∗/F p // 0
δp,r : 0 // F p/F p+1 // F p−r+1/F p+1 //?
OO
F p−r+1/F p //?
OO
0 .
We obtain a commutative diagram with exact rows
(1) Hn(F p/F p+r+1) // Hn(F p/F p+r) δp+r,r+1
//
αp,r
β
))SSSSSSSSSSSSSSSHn+1(F p+r/F p+r+1)
γ
Hn(F p/F p+r+1)α
p,r+1// Hn(F p/F p+1) // Hn+1(F p+1/F p+r+1)
(2) Hn(F p+1/F p+r) // Hn(F p/F p+r)) αp,r //
δp+r,r+1
β
**UUUUUUUUUUUUUUUUHn(F p/F p+1)
Hn(F p+1/F p+r) δp+r,r// Hn+1(F p+r/F p+r+1)
γ // Hn+1(F p+1/F p+r+1)
From (1) we get Zp,qr+1 ⊆ Zp,q
r , and from (2) we get (by renumbering p+ r p) Bp,qr ⊆ Bp,q
r+1.
This gives the following inclusions
0 = Bp,q1 ⊆ . . . ⊆ Bp,q
r ⊆ Bp,qr+1 ⊆ . . . ⊆ Bp,q
∞ ⊆ Zp,q∞ ⊆ . . . ⊆ Zp,q
r+1 ⊆ Zp,qr ⊆ . . . ⊆ Zp,q
1 = Ep,q1 .
42
We now use the following Lemma
Lemma 6.6 IfC
ϕ
ψ
AAAAAAAA
A′
>> ϕ′ // Aη // A′′
is a commutative diagram in A with exact line, then we have im(ϕ′) ⊆ im(ϕ), as well ascanonically
im(ϕ)/im(ϕ′) ∼= im(ψ) .
Proof : The fist claim is obvious, and since ker η = imϕ′ ⊆ imϕ, η induces an isomorphism
imϕ/imϕ′ = imϕ/ ker η∼→ η(imϕ) = im(ηϕ) = imψ .
By applying 6.6 we get an isomorphism
δp,qr : Zp,qr /Zp,q
r+1 = imαp,r/imαp,r+1(1)∼= imβ
(2)∼= im δp+r,r+1/im δp+r,r = Bp+r,q−r+1r+1 /Bp+r,q−r+1
r .
With this, we define the differential
dp,qr : Ep,qr = Zp,q
r /Bp,qr Zp,q
r /Zp,qr+1
δp,qr→∼Bp+r,q−r+1r+1 /Bp+r,q−r+1
r → Zp+r,q−r+1r /Bp+r,q−r+1
r .
Then we have
ker dp,qr = Zp,qr+1/B
p,qr , im dp,qr = Bp+r,q−r+1
r+1 /Bp+r,q−r+1r
and thereforeker dp,qr
im dp−r,q+r−1r
=Zp,qr+1
Bp,qr+1
= Ep,qr+1 .
Finally, the commutative diagrams with exact rows
(3) Hn(F p)
ρ
''OOOOOOOOOOO
Hn(F p+1) //
88qqqqqqqqqqHn(A∗) // Hn(A∗/F p+1)
(4) Hn−1(A∗/F p) δ // Hn(F p)
αp
ρ
((QQQQQQQQQQQQQ// Hn(A∗)
Hn−1(A∗/F p) δp // 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
Ep,q∞ = Zp,q
∞ /Bp,q∞ = imαp/im δp
(4)∼= im ρ(3)∼= F pHn(A∗)/F p+1Hn(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 acomplex C∗ depends only on Cn−1 → Cn → Cn+1.
(c): For r = 1, the diagrams (1) and (2) are
(1) Hn(F p/F p+2) // Hn(F p/F p+1) δp+1,2//
αp,1q
β
))SSSSSSSSSSSSSSHn+1(F p+1/F p+2)
Hn(F p/F p+2) αp,2 // Hn(F p/F p+1) δp+1,2// Hn+1(F p+1/F p+2)
(2) 0 = Hn(F p+1/F p+1) // Hn(F p/F p+1) αp,1
=//
δp+1,2
β
))TTTTTTTTTTTTTTTHn(F p/F p+1)
0 = Hn(F p+1/F p+1) δp+1,1
// Hn+1(F p+1/F p+2) =// Hn+1(F p+1/F p+2)
From the definition of dp,q1 (with p + q = n) we get that dp,q1 = δp+1,2, the connectinghomomorphism for the exact sequence
0→ F p+1/F p+2 → F p/F p+2 → F p/F p+1 → 0 ,
since dp,q1 = β = δp+1,2 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 beleft exact functors between abelian categories, where A and B have enough injectives andF maps injectives to G-acyclic objects. Then for every object X in A we have a finitelyconvergent spectral sequence
Ep,q2 = RpG(RqFX)⇒ Rp+q(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 objectsCp,q ∈ A
. . . // Cp−1,q+1 // Cp,q+1dp,q+1I // Cp+1,q+1 // . . .
. . . // Cp−1,qdp−1,qI //
OO
Cp,qdp,qI //
dp,qII
OO
Cp+1,q //
dp+1,qII
OO
. . .
. . . // Cp−1,q−1
OO
// Cp,q−1 //
dp,q−1II
OO
Cp+1,q−1
OO
// . . .OO OO OO
(b) A double complex is a corresponding diagram, in which all squares are anticommutative,i.e. dp,q+1
I dp,qII + dp+1,qII dp,qI = 0 for all p, q ∈ Z.
44
(c) The double complex associated to a naive double complex as in (a) is the double complexwhere dp,qII is replaced by (−1)pdp,qII .
(d) If there is an N ∈ Z with Cp,q = 0 for p < N or q < N , then the total complex associatedto a double complex is the complex Tot(C∗,∗) with components
Tot(C∗,∗)n = ⊕p+q=n
Cp,q
and differentiald = dI + dII ,
i.e. d|Cp,q = dp,qI + dp,qII .
The following construction is very important for the treatment and definition of spectralsequences.
Construction 6.9 Let Tot(I∗,∗) be the total complex associated to the double complex,which is associated to the naive double complex I∗,∗. This has two descending filtrations:
(6.9.1) F pI Tot(I
∗,∗)n = ⊕r+s=nr≥p
Ir,s , and
(6.9.2) F pIITot(I
∗,∗)n = ⊕r+s=ns≥q
Ir,s .
SinceF pI Tot(I
∗,∗)/F p+1I Tot(I∗,∗) = Ip,∗[−p]
andF pIITot(I
∗,∗)/F q+1II Tot(I∗,∗) = I∗,q[−q]
the corresponding spectral sequences from Theorem 6.5 are
IEp,q1 = Hp+q(Ip,∗[−p]) = Hq(Ip,∗)⇒ Ep+q = Hp+q(Tot(I∗,∗)) ,
and for the first filtration, and
IIEp,q1 = Hp+q(I∗,q[−q]) = Hp(I∗,q)⇒ Ep+q = Hp+q(Tot(I∗,∗))
for the second filtration. The d1-differential of the spectral sequence IEp,q1 ⇒ Ep+q is the
morphism
IEp,q1 = Hq(Ip,∗)→ Hq(Ip+1,∗) =IE
p+1,q1 ,
which is induced by the morphism of complexes
dp,∗I : Ip,∗ → Ip+1,∗ .
By 6.5 (c), for a filtered complex (A∗, F pA∗) as in 6.5, the d1-differential
dp,q1 : Ep,q1 = Hp+q(F pA∗/F p+1A∗)→ Hp+q+1(F p+1A∗/F p+2A∗) = Ep+1,q
1
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
0 // Ip+1,∗[−p− 1] α // Tot
Ip+1,∗[−p− 1]↑dp,∗II
Ip,∗[−p− 1]
β // Ip,∗[−p] // 0 ,
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
0 // Ip+1,q //
(−1)p+1dII
OO
Ip+1,q ⊕ Ip,q+1 //
d
OO
Ip,q+1 //
(−1)pdII
OO
0 ,
a // (a, 0), (a, b) // b
where the arrow in the middle, expressed in elements, is given by
(a, b) 7→ ((−1)p+1dIIa+ dIb, (−1)pdIIb) .
By the standard description of the connecting morphism (b ∈ Ip,q+1 with dIIb = 0 is liftedto (0, b) ∈ Ip+1,q ⊕ Ip,q+1, mapped on (dIb, 0) below D, which is the image dIb ∈ Ip+1,q+1),we see that this maps the class of b to the class dIb, hence is induced by dI as claimed. Theanalogous 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 enoughinjectives, then, for any given injective resolutions
A → I∗ and C → K∗
there exists an exact sequence of injective resolutions
0 // I∗ // J∗ // K∗ // 0
0 // A //?
OO
B //?
OO
C //?
OO
0 ,
which is split in each degree.
Proof : One starts with a diagram
0 // I0 // I0 ⊕K0 // K0 // 0
0 // A //?
α
OO
B //
βccHHHHHHHHHH
(β,γ)
OO
C
γ
OO
// 0 ,
46
where β is an extension of α to B (this exists, since I0 is injective). Then one can easily seethat 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 boundedbelow, e.g. An = 0 for n < N , then we have a naive double complex (I∗,∗, dI , dII) withIp,q = 0 for p < N and q < 0 and a morphism of complexes
IN,0 dN,0 // IN+1,0 dN+1,0
// AN+2,0 // . . .
ANdN //
OO
AN+1
OO
dN+1// AN+2 //
OO
. . . ,
such that the following holds:
(a) For all p, Ap → Ip,∗ is an injective resolution.
(b) For all p, ZAp → ZIp,∗, BAp → BIp,∗ and Hp(A∗) → HpI (I∗,∗) are injective reso-
lutions. Here let HpI (I∗,∗) = Hp((I∗,∗, dI)) be the cohomology of Ip−1,∗ dI→ Ip,∗
dI→ Ip+1,∗,
ZAp = ker(Ap → Ap+1), BAp = im(Ap−1 → Ap), ZIp,∗ = ker(Ip,∗dI→ Ip+1,∗) and BIp,∗ =
im(Ip−1,∗ dI→ Ip,∗), so that HpI (I∗,∗) = ZIp,∗/BIp,∗.
Proof Without restriction, let An = 0 be for n < 0. Then we have a chain of morphisms
ZA0 → A0 BA1 → ZA1 → A1 BA2 → . . .
If we choose injective resolutions ZA0 → ZI0,∗ and BA1 → BI1,∗, then by 6.9 we obtain anexact sequence of injective resolutions
0 // ZI0,∗ // I0,∗ // // BI1,∗ // 0
0 // ZA0 //?
OO
A0 // //?
OO
BA1?
OO
// 0 .
If, moreover, we choose an injective resolution H1(A∗) → HI1,∗, then, by 6.9, we get anexact sequence of injective resolutions
0 // BI1,∗ // ZI1,∗ // HI1,∗ // 0
0 // BA1 //?
OO
ZA1 //?
OO
H1(A∗)?
OO
// 0 .
Now we choose an injective resolution B2A∗ → BI2,∗ and we obtain an exact sequence ofinjective resolutions
0 // ZI1,∗ // I1,∗ // BI2,∗ // 0
0 // ZA1 //?
OO
A1 //?
OO
BA2?
OO
// 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 dIis given by the composition
dp,∗I : Ip,∗ BIp+1,∗ → ZIp+1,∗ → Ip+1,∗ ,
so that BIp+1,∗ is really the image of dpiI and ZIp,∗ is the kernel of dp+1,∗I , and moreover
HIp,∗ = ZIp,∗/BIp,∗ = Hp(I∗,∗, dI).
Now we come to the Proof of theorem 6.7. Let F : A → B and G : B → C be leftexact functors between abelian categories with enough injectives, where F maps injectivesto G-acyclic objects.
Let X be an object in A and let X → I∗ an injective resolution. For A∗ = FI∗ we haveHn(A∗) = Hn(FI∗) = RnF (A) by definition. Let
J∗,∗
A∗?
OO
be a Cartan-Eilenberg-resolution as in Theorem 6.11.
We consider the first spectral sequence of the double complex GJ∗,∗
IEp,q1 = Hq(GJp,∗)⇒ Ep+q = Hp+q(Tot(J∗,∗)) .
Since Ap = FIp is G-acyclic, Hq(GJp,∗) = 0 for q > 0 and
H0(GJp,∗) = ker(GJp,0dII→ GJp,1) = GAp .
This implies that the edge morphisms
Rn(GF )(X) = Hn(GFI∗) = Hn(GA∗)→ Hn(Tot(GJ∗,∗))
are all isomorphisms.
We consider the second spectral sequence of the double complex GJ∗,∗,
IIEp,q1 = Hp(GJ∗,q)⇒ Ep,q = Hp,q(Tot(GJ∗,∗))
We have
IIEp,q1 = ZIGJ
p,q/BIGJp,q = G(ZIJ
p,q/BIJp,q)
and by assumptionRnFX = Hn(A∗) → HI(J
n,∗)
is an injective resolution. Therefore we have
IIEp,q2 = Hq
II(GHI(Jn,∗)) = RpG(RqFA)
and we obtain the desired Grothendieck-spectral sequence.
48
7 The etale site
Definition 7.1 (a) A class E of morphisms of schemes is called admissible, if the followingholds
(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 isψ ϕ : Z → X),
(M3) E is closed under base change (If ϕ : Y → X is in E and ψ : X ′ → X any morphism,then the base change ϕ′ : Y ′ = Y ×X X ′ → X ′ is in E).
(b) Let E be admissible. An E-covering (Uigi−→ X)i∈I of a scheme X is a family of E-
morphisms (morphisms in E) with ∪igi(Ui) = X.
Example 7.2 Important examples of admissible classes are
(a) the class (Zar) of all open immersions,
(b) the class (et) of all etale morphisms,
(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 “offinite type” by “of finite presentation”, and the same should be done in the following recol-lection.
Recollection 7.4 (a) A morphism f : Y → X of schemes is called unramified, if it hasthe 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 andx and my and mx are their maximal ideals, and for all y ∈ Y , k(y)/k(x) is a finite separablefield extension.
(ii) f is locally of finite type, and Ω1Y/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 etale, if it has the following equivalent conditions:
(i) f is flat and unramified,
(ii) f is locally of finite type and formally etale.
Lemma/Definition 7.5 (a) Let C be a category of schemes and let E be an admissibleclass 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 thestructural morphism Y → X lies in E, equipped with the E-coverings, and is denoted byXE.
Definition 7.6 In particular, this defines the small etale site Xet of a scheme X. In general,one understands an etale sheaf on X as a sheaf F on the small site Xet and the etalecohomology
H iet(X,F ) := H i(Xet, F )
49
is the cohomology on this site, defined by Definition 5.3.
Remark 7.7 All morphisms in Xet are etale: If
Y1
ϕ1 AAAAAAAf // Y2
ϕ2~~
X
is a morphism of etale X-schemes (i.e., a commutative diagram with etale ϕ1 and ϕ2), thenf is etale.
Lemma/Definition 7.8 Let E and E ′ be two admissible classes of morphisms. A morphism
f : X ′ → X
of schemes defines a morphism of sites
f : X ′E′ → XE
viaf 0 : XE → X ′E′
V 7→ V ×X X ′ ,if the base change V1 ×X X ′ → V2 ×X X ′ is in E ′ for any V1 → V2 in XE. In this case, weget functors
fP : Pr(X ′E) → Pr(XE)
fP : Pr(XE) → Pr(X ′E′)
f∗ : Sh(X ′E′) → Sh(XE)
f ∗ : Sh(XE) → Sh(X ′E′) ,
where fP is left adjoint to fP and f ∗ left adjoint to f∗. The functors fP , fP and f ∗ are exact,
the functor f∗ is left exact.
In particular, this holds for E ′ = E, for example for E ′ = E = et. Hence we have adjointpairs
fP : Pr(X ′) → Pr(X) fP : Pr(X) → Pr(X ′) ,
f∗ : Sh(X ′et) → Sh(Xet), f ∗ : Sh(Xet) → Sh(X ′et) ,
where f∗ is left exact and the other functors are exact.
Proof of the claims: f 0 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-tion, Y ×X X ′ → X ′ is in X ′E′ . Furthermore,
(Ui ×X X ′ → Y ×X X ′)
is an E ′-covering: by assumption, the morphisms are E ′-morphisms, and for every surjectivefamily
(Yiπi−→ Y )
of morphisms of schemes and every morphism of schemes X ′ → X, the family
(Yi ×X X ′π′i−→ Y ×X X ′ =: Y ′)
50
is again a surjective family: If y′ ∈ Y ′, with image y in Y , then there exists an i for whichπ−1i (y) = (Yi)y = Yi×Y k(y) is non-empty. Then we have (π′i)
−1(y′) = (Yi×XX ′)×Y×XX′ y′ =Yi ×Y y′ = (Yi ×Y y) ×y k(y′) = (Yi ×Y k(y)) ×k(y) k(y′) 6= ∅. This shows the property (S1)from 2.7 for f 0. Furthermore, for any Z → X in XE and any X-morphism Z → Y (i.e. anymorphism in XE) the canonical morphism
(Ui ×Y Z)×X X ′ → (Ui ×X X ′)×Y×XX′ (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,and therefore finite limits exist in XE and X ′E′ (see 4.A.22).
Corollary 7.9 By assumption of Lemma/Definition 7.8, the functor f∗ : Sh(X ′E′)→ 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 nextlemma.
Lemma 7.10 Let F : A → B be a left exact functor between abelian categories. If F has aleft 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 everymonomorphism B′ → B the morphism
HomB(B,FI)→ HomB(B′, F I)
is surjective. By the functoriality of the adjunction, we obtain a commutative diagram
HomB(B,FI) //
o
HomB(B′, F I)
o
HomA(GB, I) // HomA(GB′, I)
where GB′ → GB is a monomorphism, because G is left exact. Since I is injective, thebottom map is a monomorphism, and hence the top map, too.
Corollary 7.11 In the setting of Lemma 7.8, for every sheaf F ′ onXE, we have a Grothendieck-Leray-spectral sequence
Ep,q2 = Hp(XE, R
qf∗F ′)⇒ Hp+q(XE′ ,F) .
Proof : This follows from Theorem 6.7, since
H0(XE, f∗F ′) = H0(XE′ ,F ′)
and since the left exact functor f∗ maps injectives to injectives, which then are H0(XE′ ,−)-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 wesay that x is a geometric point over x.
(b) For an etale presheaf P on X and a geometric point as above,
Px := (iPx P )(x) ∈ Ab
is called the stalk of P at x.
Remark 7.13 (a) The functorPr(Xet) → Ab
P 7→ Px
is exact. In fact, by 7.8, iPx : Pr(Xet)→ Pr(xet) is exact. Furthermore, the functor
Pr(xet) → AbQ 7→ Q(x)
is exact.
(b) The adjunction morphism P → (ix)P (iPx )P induces a homomorphism of abelian groups
P (X)→ ((ix)P (ix)PP )(X) = ((ix)
PP )(x) = Px .
Definition 7.14 Let F ∈ Sh(Xet), 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 everylift x→ X to U defines a canonical map, and we denote the image of s ∈ F (U) in Fx againby sx (Obviously, there is always a lift if there exists a point u ∈ U , which is mapped to theimage x ∈ X of x→ X).
Definition 7.15 An etale neighborhood of a geometric point x → X is a commutativediagram
x //
??????? U
etaleX
where U → X is an etale map as indicated. A morphism of etale neighborhoods is a com-mutative diagram
U2
x
??//
@@@@@@@@ U1
X
52
where U2 and U1 are etale over X.
Remark 7.16 A commutative diagram of schemes
X ′ //
BBBBBBBB Y
X
corresponds to an X ′-morphism X ′ → Y ×X X ′, i.e., to a commutative diagram
X ′ //
id BBBBBBBB Y ×X X ′
pr2zztttttttttt
X ′
and hence to a section of pr2 : Y ×X X ′ → X ′ (in short: HomX(X ′, Y )∼→ HomX′(X
′, Y ×XX ′)). This shows that an etale neighborhood of x→ X can be identified with a morphism
x→ U ×X x
in xet, where U → X is etale, and therefore with an object in the category Ix for themorphism of sites xet → Xet, U 7→ U ×X x (see 7.8), which is used for the definition of iPx .Furthermore, the morphisms of etale neighborhoods of x→ X correspond to the morphismsin Ix, to wit, the X-morphisms U1 → U2, for which the diagram
U1 ×X x
x
66nnnnnnnnnnnnnn
((PPPPPPPPPPPPPP
U2 ×X x
is commutative. Combining this with the definition of iPx , we see that for an etale presheafP on X we have
(7.16.1) Px = lim→P (U) ,
where the inductive limits runs over all etale neighborhoods of x→ X.
Lemma 7.17 (a) If U → X is etale and s ∈ F (U) non trivial, then there is a geometricpoint 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 etaleneighborhood Vu → U with s|Vu = 0, and by the separateness of F we get s = 0 (the Vuform a covering of U).
(b) is obvious from this.
53
Lemma 7.18 Let f : S ′ → S be a morphism of sites. If fP maps sheaves to sheaves, thencanonically fPa = afP (more precisely: fP ia = iafP for the embeddings i : Sh(S)→ Pr(S)and i : Sh(S ′)→ Pr(S ′)).
Proof : Let P be a presheaf on S and let F be a sheaf on S ′. Then we have isomorphisms
HomPr(S′)(fP iaP, iF ) ∼= HomPr(S)(iaP, fP iF ) ∼= HomPr(S)(iaP, if∗F )
∼= HomSh(S)(aP, f∗F ) ∼= HomPr(S)(P, if∗F ) ∼= HomPr(S)(P, fP iF )
∼= HomPr(S′)(fPP, iF ) ∼= HomSh(S′)(af
PP, F )
This implies the claim: By assumption, fP iaP = iG for a sheaf G, and the first group isisomorphic to HomSh(S′)(G,F ). The Yoneda-Lemma implies G ∼= afPP , hence fP iaP ∼=iafPP .
Corollary 7.19 For an etale presheaf P on X and a geometric point x of X one hasPx = (aP )x.
Proof : For ix : x → X, one has Px = (iPx P )(x) = (aiPx P )(x)7.15= (iPx aP )(x)
Def.= (aP )x,
because iPx maps sheaves to sheaves: For a sheaf F on X we have
(iPx F )(∐i∈Ix) =
∏i∈I
(iPx F )(x) ,
since for a diagram
V =∐i∈Ix
(fi) //
''OOOOOOOOOOOOOU
X
one has the factorization ∐i∈IU
V =
∐i∈Ix //
∐ifi
88pppppppppppppppU .
Hence the morphisms above are cofinite in the category IV , and for (fPF )(V ) we can formthe limit over these; furthermore F (
∐i
Ui) =∏i
F (Ui).
Corollary 7.20 A sequence
(7.20.1) 0→ F ′ → F → F ′′ → 0
of etale sheaves on X is exact if and only if the sequences of the stalks
(7.20.2) 0→ F ′x → Fx → F ′′x → 0
are exact for all geometric points x of X.
54
Proof (a) If (7.20.1) is exact, then
0→ F ′ → F → F ′′
is exact as an sequence of presheaves, hence
0→ F ′x → Fx → F ′′x
is exact by Remark 7.13 (a).
(b) Conversely, let 0→ F ′x → Fx → F ′′x be exact for all geometric points x of X.
(i) Then F ′ → F is a monomorphism: If U → X is etale and s ∈ F ′(U) is in the kernel ofF ′(U) → F (U), then for every geometric point x of U , the germ sx of s is in the kernel ofF ′x → 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 ′′(U) for U → X etale. By assumption, for everygeometric point x of U , the germ sx lies in the stalk F ′x → Fx. Then for every u ∈ U there isan etale morphism Vu → U , such that s|Vu lies in the subgroup F ′(Vu) ⊆ F (Vu). As (Vu)u∈Uis an etale covering of U and F ′ and F are sheaves, s lies in F ′(U) ⊆ F (U). By (i) and (ii),
0→ F ′ → F → F ′′
is exact.
(c) Let P be the presheaf cokernel of F → F ′′, i.e., let
F (U)→ F ′′(U)→ P (U)→ 0
be exact for all etale morphisms U → X. Then
F → F ′′ → aP → 0
is an exact sequence of sheaves, as the functor a (associated sheaf) is exact and aF = F ,aF ′ = F ′. Then we have the equivalences
F → F ′ 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)⇔ Fx → F ′x surjective for all geometric points x,
because forming stalks is exact on the exact sequence of presheaves
F → F ′ → P → 0
(see 7.13 (a)).
Corollary 7.21 A morphism
(7.21.1) ϕ : F1 → F2
of etale sheaves 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→ (F0)x → (F1)x0→ (F2)x
is then exact for all geometric points x. It follows as in (ii) that for every etale morphismU → X every section s ∈ F1(U) already lies in F0(U), hence ϕU : F1(U)→ F2(U) is the zeromap.
56
8 The etale site 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-setsY 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 surjectivefamilies (Mi →M)i.
Remarks 8.2 (a) With the Krull topology (where the subgroups Gal(ks/L), for finite ex-tensions L/k, form a basis of the neighborhoods of 1 in Gk), Gk is a profinite group, i.e., aprojective limit of finite groups
Gk = lim←L/k fin. gal., L⊆ks
Gal(L/k) .
(b) Gk operates from the left on ks, thus from the right on Spec ks, thus from the left onφ(Y ).
(c) A Gk-set M is called discrete, if for every m ∈M the stabilizer
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 etale, then one has Y =∐i∈I
Spec(Li), where Li/k are finite
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
U =r∐i=1
Spec(Li). This shows that every point is open, so that Y carries the discrete topology.
This implies the first claim.
Proof of Theorem 8.1: 1) A quasi-inverse functor to φ is the functor
ψ : M =∐j∈J
Mj 7→∐j∈J
Spec(HomGk(Mj, ks)) .
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 discreteGk-set is of the form Gk/U with U ≤ Gk open, and the assignment is
Gk/U 7→ Spec(kUs ) .
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
ψ(M) = HomGk(Gk/U, ks)∼→ kUs =: 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 obtainan equivalence of categories between etale k-schemes Y and discrete Gk-sets. From thesefacts it follows that φ is also an equivalence of sites:
2) One has φ(Y ′ ×Y Y ′′) = (Y ′ ×Y Y ′′)(ks) = Y ′(Ks)×Y (ks) Y′′(ks)
3) If SpecL′ → SpecL is etale, then L ⊆ L′ is a separable field extension of k in ks and themap 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 onSpec(k)et:
Fx = lim→k⊆L⊆ks
L/k fin. separable
F (Spec(L)) .
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 subgroupsU ≤ 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 thesurjective families (Mi → M) of G-sets form a Grothendieck topology TG on M(G), theso-called canonical topology. The functor
Φ : (M(G), TG)∼ → M(G)F 7→ MF = F (G)
is an equivalence of categories with quasi-inverse
FM = HomG(−,M)Ψ←pM .
(b) Let G be a profinite group (i.e., a projective limit of finite groups, equipped with theprofinite topology) and let C(G) be the category of the continuous G-sets (with respect tothe discrete topology on this sets). Then the surjective families (Mi →M) of discrete G-setsform a Grothendieck topology, the so-called canonical topology, which again is denoted byTG. The functor
Φ : (C(G), TG)∼ → C(G)F 7→ MF = F (G) := lim→
U≤G open
F (G/U)
is an equivalence of categories with quasi inverse
FM = HomG(−,M)Ψ←pM
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 → Gg′ 7→ g′g
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 (Rgg′) = F (Rg′ Rg) = F (Rg) F (Rg′), as F is contravariant). Obviously, herethe 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) .
In fact, forN =∐i∈INi we have F (N) =
∏i∈IF (Ni), andHomG(N,F (G)) =
∏i∈IHomG(Ni, F (G)).
By considering the orbits, we may thus assume that N = G/U for a subgroup U ⊂ G. Nowwe consider the sheaf condition for the covering G G/U . We have a bijection of G-sets∐
u∈UG → G×G/U G
gu 7→ (g, gu) ;
and hence the diagram
F (G/U)→ F (G) ⇒ F (G×G/U G)∼→∏u∈U
F (G)
f ⇒(. . . , f, . . .)(. . . , uf, . . .)u∈U
is exakt. This gives canonical bijections
F (G/U)∼→ F (G)U = 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 haveMF = lim→
UEG opennormal factor
F (G/U) ,
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
MFM = lim→V≤G open
HomG(G/U,M)∼→ lim→
V≤G open
MU ∼→M .
(iv) By the first case, for every open subgroup U < G and every open normal subgroupU ′ E G with U ′ ⊆ U we have
F (G/U)∼→ f ∈ F (G/U ′) | uf = f for alle u ∈ U/U ′ ,
so thatF (G/U) ∼= (MF )U ,
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
FMF(G/U) = HomG(G/U,MF ) ∼= (MF )U ∼= F (G/U)
f 7→ f(1) .
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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)) = MGL 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 ab-solute Galois group of k and x : Spec(ks)→ Spec(k). Then we have functorial isomorphismsfor all etale abelian sheaves F on Spec(k) and all i ≥ 0
H iet(Spec(k), F )
∼→ H i(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)
F (k) := F (Spec(k))∼→ FGk
x .
(ii) By definition, H i(Gk,−) is the i-th right derived functor of
M 7→ H0(Gk,M) = MGk .
Hence etale cohomology of fields is Galois cohomology.
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9 Henselian rings
Henselian rings, and in particular the strictly henselian rings, play the same role in the etaletopology 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.,〈g0, h0〉 = k[X]), then there are monic polynomials g, h ∈ A[X] with f = g · h, g = g0 andh = h0. Here let f = f mod m in k[X]; similarly for g and h). The polynomials g and h arestrictly coprime (i.e., 〈g, h〉 = A[X]).
(a′) If f ∈ A[X] and f = g0 · h0 where g0 is monic and g0 and h0 are coprime, then thereexist 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.
(b′) If B is a finite A-algebra, then every idempotent e0 ∈ B/mB (i.e., e20 = 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 decomposi-tion
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 ringBi.
(d) If f : Y → X is etale 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).
(d′) Let f1, . . . , fn ∈ A[X1, . . . , Xn] and let a = (a1, . . . , an) ∈ kn with f i(a) = 0 for alli = 1, . . . , n and det(∂f i/∂xj(a)) 6= 0. Then there exists an element c ∈ An with c = a andfi(c) = 0 for i = 1, . . . , n.
Definition 9.2 A morphism f : Y → X of schemes is called quasi-finite, if it is finitelypresented (for noetherian schemes: of finite type) and has finite fibers (i.e., f−1(x) is finitefor all x ∈ X).
If f is etale and finitely presented (noetherian schemes: of finite type), then f is quasi-finite.
Proof of the equivalence of the conditions in 9.1:
(a′) ⇒ (a) is trivial, except for the last sentence in (a). But if f is monic, then A[x]/〈f〉 isfinite over A. Since we have f ∈ 〈g, h〉, i.e., 〈f〉 ⊆ 〈g, h〉, D = A[X]/〈g, h〉 is finite over A aswell, and by the Nakayama-Lemma we have D = 0, since D/mD = k[x]/〈g0, h0〉 = 0.
(a) ⇒ (b): Let B be a finite A-algebra. By the going-up-theorem, every maximal ideal of Blies over m; thus B is local if and only if B/mB is local.
Special case: Let B = A[X]/〈f〉 be with a monic polynomial f .
62
If f is a power of an irreducible polynomial, then B/mB = k[X]/〈f〉 is local, hence B islocal. Otherwise, by (a) we obtain that f = g · h with g, h monic of degrees ≥ 1 and strictlycoprime, and with the Chinese remainder theorem we get
B = A[X]/〈f〉 ∼= A/〈g〉 × A/〈h〉 .
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 anon-trivial idempotent in B/mB (B/mB is an artinian k-algebra, hence a product of localrings). Since b is integral over A, there is a monic polynomial f ∈ A[X] with f(b) = 0. Thenwe have a ring homomorphism by evaluating g ∈ A[X] at b
ϕ : C = A[X]/〈f〉 → B , X 7→ b .
Consider the reduction mod m
ϕ : C/mC = k[X]/〈f〉 → B/mB .
If f =∏i
pnii , with irreducible polynomials pi, then
k[X]/〈f〉 ∼=∏i
k[X]/〈pnii 〉 ,
and for the quotient im (ϕ) we have
im(ϕ) = k[X]/〈g0〉 =∏i
k[X]/〈pmii 〉
with g0 =∏i
pmii | f . This shows that the idempotent b ∈ im(ϕ) lifts to an idempotent
e ∈ C/mC (the decomposition is unique). By the first case there is an idempotent e ∈ Cwith 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 theclaim 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 idempotentsR = 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) ⇔ (b′).
(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
f : Yj→ Y ′
f ′→ X ,
where f ′ 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 ringhomomorphismΓ(U,OX)→ Γ(f−1(U),OY ) is finite.
Proof of Theorem 9.3: Without! See the references in Milne’s ‘Etale Cohomology’, page6.
Now consider f : Y → X = Spec(A), quasi-finite and separated, with a local ring A asabove. Let
Yj→ Y ′
f ′→ X
be a Stein factorization as in Theorem 9.3. Since f ′ is affine, Y ′ = Spec(B′) is affine. By (b),we get
Y ′ =r∐i=1
Yi ,
where Yi = Spec(Bi) for a local finite A-algebra Bi. Let
Y∗ =∐i∈IYi
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 openand 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 etale and let y ∈ Y be a point with f(y) = x andk(x)
∼→ k(y). By replacing Y with an affine open neighborhood of y, f is quasi-finite andseparated without restriction. Then, by (c), we may assume that Y = Spec(B), where B islocal and finite over A. Since f is etale, 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 weobtain an exact sequence
0→ a→ A→ B → 0 ,
with an ideal a ⊆ A. Since B is flat over A,
0→ a⊗A B → A⊗A Bβ→ 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 theidentity. 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.
(d) ⇒ (d′): Let B = A[X1, . . . , Xn]/〈f1, . . . , fn〉 and a = (a1, . . . , an) ∈ kn 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, thusthere 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 ]/〈f1, . . . , fn, bT − 1〉
and det(∂fi/∂Xj)b is the corresponding Jacobian determinant, hence a unit. By the Jacobiancriterion, Bb is etale over A. Furthermore, n gives a maximal ideal of Bb over m with residuefield isomorphic to k = A/m. By (d), a section s : Spec(A) → Spec(Bb) exists, i.e., anA-homomorphism Bb → A and this gives an element c ∈ An with fi(c) = 0 for i = 1, . . . , nand c = a (since n lies over m).
(d′) ⇒ (a′): Let f(X) = anXn + . . .+ a1X + a0 ∈ A[X] and let f = g0 · h0 monic with g0 of
degree ≥ 1. Then we have
f(X) = g(X) · h(X) = (Xr + br−1Xr−1 + . . .+ b0)(csX
s + . . .+ c0)
with r + s = n, if and only if (b0, . . . , br−1, c0, . . . , cs) ∈ An+1 solves the following system ofequations in the n+ 1 variables (X0, . . . , Xr−1, Y0, . . . , Ys):
(9.1.1)
X0Y0 = a0
X0Y1 +X1Y0 = a1
X0Y2 +X1Y1 +X2Y0 = a2...
Xr−1Ys + Ys−1 = an−1
Ys = an
(n+ 1 equations). The corresponding Jacobian is
r s︷ ︸︸ ︷ ︷ ︸︸ ︷
J = det
Y0 0 X0 0Y1 Y0 X1 X0
Y2 Y1 Y0 X2 X0...
......
...
Ys...
Ys...
......
... 1...
... 1
This is just res(G,H), the resultant of the two polynomials
G = xr +Xr−1xr−1 + . . .+X0
H = Ysxs + Ys−1x
s−1 + . . .+ Y0
By (d′) there is a solution of (9.1.1), if Res(g0, h0) 6= 0 in k (because then the vector (b, c) ofthe 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 etaleA-algebras
)∼→(
finite etalek-algebras
).
Proof : This follows with the criteria of 9.1(b), (b′) and (d). Details: left to the readers!
Every field is a henselian ring, as well as every artinian ring (because every artinian ring isa 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 etale A-algebra, and let s0 : B/mB → k be asection of k → B/mB. To find a section
s : B → A ∼= lim←r
A/mr
of A→ B, it suffices to find compatible A-linear maps for all r ≥ 1
sr : B → A/mr .
For r = 1 we take s1 : Bcan−→ B/m
s0−→ A/m = k. If sr is found for some r ≥ 1, then theexistence of sr+1 follows from the formal smoothness of B over A: The lift sr+1 of sr exits inthe diagram
Bsr //
sr+1
##GG
GG
G A/mr
A
OO
// A/mr+1 ,
ϕr
OO
because ker(ϕr) = mr/mr+1 is a nilpotent ideal.
Lemma/Definition 9.7 Let A be a local ring. There is a henselian ring Ah with thefollowing universal properties: There is a local homomorphism i : A → Ah, and any localhomomorphism ϕ : A→ B into a henselian local ring B factorizes uniquely over i (we havea unique homomorphism ϕ, which makes the following diagram commutative):
Ai //
ϕ ???????? Ah
∃! ϕ~~
B
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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 etale (resp. essentially etale)neighborhood of (A,m) is a pair (B, n), such that B is an etale A-algebra (resp. a localizationof an etale 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 (B′, n′) are (essentially) etale neighborhoods of (A,m) withSpec(B′) connected, then there is at most one an A-Homomorphism f : B → B′ withf−1(n′) = n.
(b) If (B, n) and (B′, n′) are (essentially) etale neighborhoods of (A,m), then there is an(essentially) etale neighborhood (B′′, n′′) of A and A-homomorphisms
Bf
))SSSSSSSSSSSSSSSSSSS
B′′
B′f ′
55kkkkkkkkkkkkkkkkkk
with f−1(n′′) = n and (f ′)−1(n′′) = n′.
Proof : (a) follows from the following, more general result:
Lemma 9.10 Let f, g : Y ′ → Y be morphisms of X-schemes, where Y ′ is connected and Yis etale and separated over X. If there is a point y′ ∈ Y ′ with f(y′) = g(y′) = y, and suchthat the maps k(y)→ k(y′) induced by f and g are equal, then we have f = g.
Proof Let Γf ,Γg : Y ′ → Y ′ ×X Y be the graphs of f and g, respectively (Γf = (idY ′ , f)similarly for g). These are sections of pr1 : Y ′ ×X Y → Y ′, and pr1 is etale and separatedas 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 andg = pr2 Γg.
For (b) consider B′′ = B ⊗A B′. The homomorphisms B → k(n) = k and B′ → k(n′) = kinduce a homomorphism α : B′′ k. If n′′ = kerα, then (B′′, n′′) has the required property.
The above implies that the connected etale (resp. essential etale) neighborhoods of A forman inductive system, and we define
(Ah,mh) = lim→(B,n) etale
neighb. of (A,m)
(B, n) = lim→(B,n) ess. etale
neighb. of (A,m)
(B, n) .
Note: the etale A-algebras B are of finite presentation and therefore form an index setwithout restriction.
1) Ah is local with maximal ideal mh: It suffices to show that Ah rmh consists of units. Letx ∈ Ah, represented by y ∈ B, (B, n) etale 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 etale neighborhood of (A,m), and the image of the inverse of y in Bb isan inverse of x, i.e., x is a unit.
2) Obviously we have k∼→ lim→B/nB = Ah/mh.
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 etale, c ⊆ C bean ideal over mh with k = k(mh)
∼→ k(c). Since C is of finite type over Ah = lim→B, there
exists an etale neighborhood (B0, n0) of (A,m) with C = B0⊗AAh (Consider a presentationC = Ah[X1, . . . , Xn]/〈f1, . . . , fm〉 and a B0 such that the finitely many coefficients of the filie in the image of B0 → Ah). Then we obtain a section
C = B0 ⊗A lim→B → lim
→B
of Ah → C: Without restriction, we consider the cofinite family of the etale neighborhoods(B, n) with a (uniquely determined!) morphism (B0, n0) → (B, n), and then the homomor-phism 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 lookfor the homomorphism ϕ, which makes the diagram
Ai //
ϕ???????? Ah
∃! 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 //
ϕ @@@@@@@@ B
∃! ϕB~~~~~~~~
C
commutative. Equivalent: In the commutative diagram
A //
ϕ???????? B ⊗A C
Cψ
::vvvvvvvvv
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 aswanted.
Proposition 9.11 (a) Ah is flat over A.
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(b) Let
A = lim←r
A/mr
Ah = lim←r
Ah/(mh)r
be the completions of A and Ah, respectively. Then A∼→ Ah is an isomorphism.
Proof (a): Ah is flat as direct limit of flat A-algebras (the tensor product commutes withdirect limits).
(b) It suffices to show that A/mr ∼→ B/nr for all etale neighborhoods (B, n) of (A,m). Butwe have A/m
∼→ B/n and mB = n by assumption; hence mrB = nr for all r and
(9.11.1) mr/mr+1 ∼= mr/mr+1 ⊗A/m B/n ∼= mrB/mr+1B = nr/nr+1 .
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
(9.11.2) 0 // mr+1 ⊗A B //
o
mr ⊗A B //
o
mr/mr+1 ⊗A B //
0
0 // mr+1B // mrB // mrB/mr+1B // 0 ,
and the indicated vertical isomorphisms, since, by flatness of B over A, the injection mr → Ainduces an injection mr⊗AB → A⊗AB = B with image mrB (similarly for r+1). Thereforethe vertical map on the right is an isomorphism.
From (9.11.1) the isomorphisms A/mr ∼→ B/mr+1 follows inductively, via the commutativediagram with exakt rows
0 // mr/mr+1
o
// A/mr+1 //
A/mr //
0
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 fieldK. LetKs be a separable closure ofK, Asthe integral closure of A in Ks, m ⊂ As a maximal ideal over m and Zm ⊂ Gal(Ks/K) = GK
the decomposing group. Then we have Ah = AZms .
Definition 9.13 Let X be a scheme and let x ∈ X be a point. An etale neighborhood of xis a pair (U, y), with f : U → X etale, f(y) = x and k(x)
∼→ k(y).
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Lemma 9.14 (a) With the obvious morphisms (U, y)→ (U ′, y′), i.e.
U //
???????? U ′
~~~~~~~~~~, y // y′ ,
X
these form a cofiltered category.
(b) We haveOhX,x = lim→
U etaleneighb. of x
Γ(U,OU) = lim→(U,y) etale
neighb. of x
OU,y .
Proof Analogous to the proof of 9.7.
Definition 9.15 A strictly henselian ring is a henselian ring whose residue field is sepa-rably 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: Ifϕ : A → C is a local morphism in a strict henselian ring (C,mC), and if a k = A/m-embedding
ϕ0 : Ash/msh → C/nC
is given, then there exists a unique local morphism ϕ : Ash → C, which makes the diagram
Ai //
ϕ???????? Ash
∃! 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
OshX,x ∼= OX,x := lim→U etale
neighb. of x
Γ(U,OU) = lim→U etale
neighb. of x
OU, image 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. Letx be the closed point, considered as geometric point of X. Then, for sheaves F on X thereexists a functorial isomorphism
Fx ∼= H0(X,F) ,
and we have Hq(X,F) = 0 for q > 0.
Proof : Let X ′ be an etale neighborhood of x. By 9.16 there exists exactly one local sectionof X ′ → X so that X is initial in the set of all etale neighborhoods of x. By 9.14 (b),
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Fx ∼= H0(X,F), functorial in F . Since the stalk functor is exact, we have Hq(X,F) = 0 forq > 0.
Lemma 9.18 Let f : X → Y be a finite morphism, where Y is a strictly local scheme. Foran etale sheaf F on X we then have
H0(X,F) ∼=∏x∈Xy
Fx
Hq(X,F) = 0 for q > 0 .
Proof Since X → Y is finite and Y is strictly local, we have
X =∏x
Spec(OX,x) ,
where x runs over the finitely many closed points of X, which are the points over the closedpoints y of Y (see Lemma 9.1). Every ring OX,x ist henselian (see Lemma 9.1), and evenstrictly henselian, since k(x), as finite extension of the spectral closed field k(y), is againseparably closed.
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10 Examples of etale sheaves
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 forevery 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 calleda strict epimorphism, if for all objects Z in X the sequence
Y ×X Ypr1⇒pr2
Yf→ X
is exact, i.e., if f is the difference cokernel of pr1 and pr2, i.e., if for all objects Z in X thesequence
Hom(X,Z)f∗→ Hom(Y, Z)
pr∗1⇒pr∗2
Hom(Y ×X Y, Z)
is exact, i.e., if f ∗ is the difference kernel of pr∗1 and pr∗2 (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 flatand 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 conditionsare equivalent
(a) ϕ is faithfully flat, i.e., for an A-module M we have M = 0 if M ⊗A B = 0.
(b) A sequence M ′ →M →M ′′ of A-modules is exact, if B ⊗AM ′ → B ⊗AM → B ⊗AM ′′
is exact.
(c) Spec(B)→ Spec(A) is surjective.
(d) For any maximal ideal m of A, we have mB 6= B.
Proof (a) ⇒ (b): Assume that M ′ ϕ→ Mψ→ M ′′ 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
Spec(ϕ)−1(p) = Spec(B ⊗A k(p)) 6= ∅ .
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(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 toshow B⊗AN 6= 0 (then we also have B⊗AM 6= 0). But N ∼= A/J for an ideal J $ A, so thatB ⊗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 mand 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 finitetype, 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, thesequence 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. Leth ∈ Hom(Y, Z) be given, with h pr1 = h pr2. We have to show that there is a uniqueg ∈ 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. Letx ∈ X, and let U be an affine open neighborhood of g1(x) = g2(x) in Z. Then there is ana ∈ A with g1(D(a)) = g2(D(a)) ⊆ U . Furthermore, Ba is faithfully flat over Aa. From thefirst 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 itsuffices to define g locally. Consider x ∈ X, y ∈ Y with f(y) = x, and let U ⊆ Z be an affineopen 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
Y ×X Y //// Yf //
h
X = Spec(A)
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). Infact, if we have x1 ∈ D(a), hence x1 = f(y1) with h(y1) ∈ U , and if we have y2 ∈ Y withf(y2) = x1, then, since f(y1) = f(y2), there is an element y′ ∈ Y ×X Y with pr1(y′) = y1
and pr2(y′) = y2 (consider a point in y1 ×x1 y2 and its image in Y ×X Y ). We get
h(y2) = h pr2(y′) = h pr1(y′) = h(y1) ∈ U ,
hence y2 ∈ h−1(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 casewe 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, themorphisms g glue on the covering). Since f is quasi-compact, Y is a finite union Y =Y1 ∪ . . . ∪ Yn of affine open subsets. Let
Y ∗ =n∐i=1
Yi .
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 isobviously injective (Hom(−, Z) is a Zariski sheaf on Y ). The exactness of the top row nowfollows by a diagram chase.
Before we use Theorem 10.6 for the construction of sheaves, we introduce a useful criterionfor a presheaf to be a sheaf.
Proposition 10.8 A presheaf P (of sets or abelian groups) on Xet (resp. Xfl) is a sheaf ifand only if the following two conditions hold:
(a) For any U ∈ Xet (resp. Xfl), the restriction of P to U is a Zariski sheaf.
(b) For any etale covering (U ′ → U), where U and U ′ are affine,
P (U)→ P (U ′)⇒ P (U ′ ×U U ′)
is exact.
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Proof Obviously, these properties are necessary. Conversely, if (a) holds, then for any disjointsum V =
∐j
Vj of schemes we have
P (V ) =∏j
P (Vj) .
For a covering (Ui → U) the sequence
(10.8.1) P (U)→∏i
P (Ui)⇒∏i,j
P (Ui ×U Uj)
is isomorphic to the sequence
(10.8.2) P (U)→ P (U ′)⇒ P (U ′ ×U U ′)
for the covering U ′ → U with U ′ =∐i
Ui, since
(∐Ui)×U (
∐Ui) ∼=
∐i,j
Ui ×U Uj .
If (b) holds, the sequence (10.8.1) is exact for every covering (Ui → U)i∈I with finite I andaffine Ui and U , because then
∐Ui is affine: Spec(A)
∐Spec(B) ∼= Spec(A × B). For any
(U ′j → U) write U as the union of affine open sets Ui, U = ∪iUi, and write f : U ′ =
∐j
U ′j → U .
Then we have f−1(Ui) = ∪k∈Ki
U ′ik with affine open subsets U ′ik ⊆ f−1(Ui). Since U ′ → U is flat
and locally of finite type, U ′ik → Ui is of finite type (since both schemes are affine), thereforef(U ′ik) is open in Ui by Lemma 10.7. Since Ui (as an affine scheme) is quasi-compact, thereis a finite index set Ei ⊆ Ki with Ui = ∪
k∈Eif(U ′ik), i.e., (U ′ik → Ui)k∈Ei is a covering. By
adding all morphisms of the form U ′ik → f(U ′ik) for k ∈ Ki − Ei, we can assume that all Ki
are finite, and U ′ = ∪U ′ik. Now we consider the commutative diagram
P (U) //
P (U ′)////
P (U ′ ×U U ′)
∏i
P (Ui)//
∏i
∏k
P (U ′ik)////
∏i
∏k,`
P (U ′ik ×U U ′i`)
∏i,j
P (Ui ×U Uj)//∏i,j
∏k,`
P (U ′ik ∩ U ′j`) .
By 10.8 (a), the first two columns are exact, and by the first remark about (b), the secondrow is exact (k runs in the finite set Ki for U ′ik). Therefore P (U) → P (U ′) is injective,hence the presheaf P is separated (since (Ui → U) was arbitrary). This in turn implies theinjectivity of the bottom arrow. Now an easy diagram chase shows the exactness of the toprow.
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 Xet and Xfl.
Proof Condition 10.8 (a) is well-known (glueing of morphisms). Condition 10.8 (b) followsfrom Theorem 10.6: If f : U ′ → U is a surjective etale (or flat) X-morphism, with U and U ′
affine, then f is faithfully flat and of finite type, hence the diagram
Hom(U,Z)→ Hom(U ′, Z)⇒ Hom(U ′ ×U U ′, Z)
is exact by 10.6. Let πU : U → X, πU ′ : U ′ → X and πZ : Z → X be the structure morphisms.We then have a commutative diagram
HomX(U,Z) // _
HomX(U ′, Z) _
//// HomX(U ′ ×U U ′, Z) _
Hom(U,Z)
//
(πZ)∗
Hom(U ′, Z)////
(πZ)∗
Hom(U ×U U ′, Z)
Hom(U,X) f∗ // Hom(U ′, X)
πU // πU ′
where the middle row is exact for all schemes Z, so that the bottom morphism f ∗ is injectiveas well. Furthermore we have
HomX(U,Z) = g ∈ Hom(U,Z) | (πZ)∗(g) = πZg = πU ,
and the same for U ′ 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 byG is a sheaf of (abelian) groups on Xet 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.
(b) The sheaf Gm,X = X ×Spec(Z) Gm, with Gm = Spec(Z[T, T−1]) has the value
Gm,X(U) = Γ(U,OU)× = Γ(U,O×U ) ,
for every X-scheme U and is called the multiplicative group over X.
(c) For every n ∈ N let µn be the sheaf X ×Spec(Z) Spec(Z[T ]/〈T n − 1〉). It satiesfies
µn(U) = a ∈ Γ(U,OU)×|an = 1 ,
for every X-scheme U and is called the sheaf of the n-th unit roots.
Lemma 10.12 The sequence of sheaves
1→ µn → Gmn→ Gm
a 7→ an
76
is exact.
Proof : For every X-scheme U ,
1 // µn(U) // Gm(U) n // 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 divisibleby char(k(x))). Then the sequence of etale sheaves on X
1→ µn → Gmn→ Gm → 1
is exact. For any n, the sequence
1→ µn → Gmn→ Gm → 1
of flat sheaves is exact. The sequences above are called the Kummer sequence.
Proof One only has to show that, by the assumptions, Gmn→ 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 opencovering, we may assume that U = Spec(A) is affine and s = a ∈ Γ(U,OU)× = A×. ThenB = A[T ]/〈T n − a〉 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 imageof T in B) with bn = a. Therefore, by the exactness criterion of 4.2 (d), Gm
n→ Gm is anepimorphism of flat sheaves. If n is invertible on X, then n ∈ A× and therefore B is an etaleA-algebra, so that Gm
n→ Gm is an epimorphism of etale sheaves.
Lemma 10.14 Let X be a scheme. For every group G, the corresponding constant Zariski,etale or flat sheaf GX (i.e., the sheaf associated the constant presheaf GP with GP (U) = Gfor all U → X) is representable by the group scheme
GX =∐g∈G
X =∐g∈G
X[g]
with the obvious group law.
Proof Consider the category Sch/X of all X-schemes. One can easily see that for everyX-scheme Y we have:
GX(Y ) = HomX(Y,GX) = ϕ : Y → G | ϕ locally constant
(If ϕ : Y → G is locally constant, then Y =∐g∈G
ϕ−1(g)). Furthermore GX is a sheaf for all
three considered topologies, by 10.10. Finally for the Zariski sheaf GZarX associated to G we
haveGZarX (Y ) = ϕ : Y → G | ϕ locally constant
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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 ofY , we have
GX(Y ) = Gπ0(Y )
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
0→ Z/pZX → Ga,XF−1→ Ga,X → 0 ,
(called Artin-Schreier sequence), where Z/pZX is the constant sheaf, associated to theabelian group Z/pZ, which is represented by the group scheme
Z/pZX
= X ×Spec (Fp) Spec(Fp[T ]/〈T P − T 〉) .
Proof : SinceT p − T =
∏i∈Fp
(T − i)
in Fp[T ], we have an isomorphism
Fp[T ]/〈T p − T 〉 ∼=p−1∏i=0
Fp .
Therefore Spec(Fp[T ]/〈T p − T 〉) is canonically isomorphic to the constant group schemeZ/pZSpec(Fp). Furthermore, for every X-scheme Y we have
HomX(Y,Z/pZX
) = Hom(Y,Z/pZFp
)
= Hom(Fp[T ]/〈T p − T 〉,Γ(Y,OY )) = a ∈ Γ(Y,OY ) | ap − a = 0 .
This shows the exactness of
0→ Z/pZX → Ga,XF−1→ Ga,X .
Furthermore, F − 1 is an epimorphism of sheaves for the etale (and flat) topology, since forU = Spec(A) ⊆ Y affine open and a ∈ A
V = Spec(A[T ]/〈T p − T − a〉)→ U
78
is a covering, so that a = (F − 1)b = bp− b if b is the image of T in B = A[T ]/〈T p− T − a〉.
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
αp,X(U) = a ∈ Γ(U,OU) | ap = 0 .
Then αp,X is represented by the group scheme X ×Spec(Fp) Spec(Fp[T ]/〈T p〉). The sequence
0→ αp,X → Ga,XF→ Ga,X
is exact for the Zariski or etale topology. The sequence
0→ αp,X → Ga,XF→ Ga,X → 0
is exact for the flat topology, but not in general for the etale topology.
Proof : The first claims are obviously true. The morphism F is an epimorphism in the flattopology, since for every Fp-algebra A the algebra B = A[T ]/〈T p − a〉 is faithfully flat overA. For a separably closed field L the Frobenius morphism
LF→ 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 letM 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 theflat topology. Here M⊗OX OU stands for π−1
U M⊗π−1OX OU = π∗UM, the quasi-coherentpull-back. In particular, this gives a sheaf on the small sites Xet and Xfl, called Met andMfl, respectively.
Proof We use the criterion of Theorem 10.8. Condition 10.8 (a) is obviously true, since, byconstruction, 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 OUcorresponds to a B-module B ⊗AM , and we have to show the exactness of
M → B ⊗AM ⇒ B ⊗A B ⊗AM .
But by the theory of flat descent this holds, since A → B is faithfully flat (see TheoremLemma 14.6 below).
Remark 10.19 The case Ga,X = OX, et is a special case.
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11 The decomposition theorem
Let X be a scheme, let i : Y → X be a closed immersion and let j : U → X be the openimmersion of the open complement U = X − Y .
If F is an etale sheaf on X, then F1 = i∗F is an etale sheaf on Y and F2 = j∗F is an
etale sheaf on U . Let Fad→ j∗j
∗F be the adjunction map (which, under the isomorphismHomU(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 etale sheaf on Y , F2
an etale sheaf on U and φ : F1 → i∗j∗F2 is a morphism of etale sheaves on Y . Morphisms
(F1, F2, φ)→ (F ′1, F′2, φ′)
are pairs (ψ1, ψ2), where ψ1 : F1 → F ′1 and ψ2 : F2 → F ′2 are morphisms of sheaves on Yresp. U , such that the diagram
(11.0) F1φ //
ψ1
i∗j∗F2
i∗j∗(ψ2)
F ′1φ′ // i∗j∗F
′2
commutes.
Theorem 11.1 Let the functor
t : Sh(Xet)→ T (X, Y )
be defined as follows:
(i) To a sheaf F ∈ Shet(X) assign the triple (i∗F, j∗F, i∗(ad) : i∗F → i∗j∗j∗F ).
(ii) To a morphism ϕ : F → F ′ in Sh(Xet) assign the morphism in T (X, Y )
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 objectin T (X, Y ), and ψ(ϕ) is a morphism in T (X, Y ), since the diagram
(11.1.1) i∗F
i∗(ϕ)
i∗(adF )// i∗j∗j∗F
i∗j∗j∗(ϕ)
i∗F ′i∗(adF ′ )// i∗j∗j
∗F ′
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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(Xet) bethe 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∗F1i∗(φ) // i∗i
∗j∗F2
is cartesian. This assignment is functorial, since forming the fiber product is functorial: Everymorphism
(ψ1, ψ2) : (F1, F2, φ)→ (F ′1, F′2, φ′)
induces a morphisms(F1, F2, φ)→ s(F ′1, F
′2, φ) ,
because of the commutativity (11.0) and the functoriality of adi. This gives the wantedfunctor s : T (X, Y )→ Sh(Xet).
Now we construct an isomorphism of functors id∼→ st on Sh(Xet).
For every sheaf F in Sh(Xet), the diagram
(11.1.3) Fadj //
adi(F )
j∗j∗F
adi(j∗j∗F )
i∗i∗F
i∗i∗(adj)// i∗i∗j∗j
∗F
commutes, by the functoriality of the adjunction morphism adi (in sheaves). By the universalproperty 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 //
id
(j∗j∗F )x
id
Fx // (j∗j∗F )x
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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) //
β1
i∗j∗F2
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 etale sheaf on Y by the followingLemma. Thus β1 is an isomorphism.
Correspondingly we apply j∗ to (11.1.4) and we obtain the cartesian diagram
j∗s(F1, F2, φ)β2 //
j∗j∗F2
0 // 0 ,
hence an isomorphism
β2 : j∗s(F1, F2, φ)β′2→∼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) //
oβ1
i∗j∗j∗s(F1, F2, φ)
o i∗j∗(β2)
F1
φ // i∗j∗F2 ,
and we obtain an isomorphism
ts(F1, F2, φ)∼→ (F1, F2, φ) .
Lemma 11.2 (a) For an etale sheaf F1 on Y , the adjunction map Adi : i∗i∗F1 → F1 is anisomorphism.
(b) For an etale sheaf F2 on U , the adjunction map Adj : j∗j∗F2 → F2 is an isomorphism.
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Proof (a) By Lemma 9.18, for every y ∈ Y and every geometric point y over y we have acanonical isomorphism
(i∗i∗F)x ∼= (i∗F)f(x)∼= Fx .
(b) For an etale sheaf F on U and an etale morphism X ′ → X we have by definition
jP (j∗F)(U ′) = lim→(X′,ψ)∈Iop
U′
(j∗F)(X ′) ,
where IU ′ is the category of all pairs (X ′, ψ) withX ′ inXet and a morphism ψ : U ′ → U×XX ′,where morphisms are commutative diagrams
U ×X X ′′
(id×f)
U ′
::tttttttttt
$$JJJJJJJJJJ
U ×X X ′
with f : X ′′ → X ′ (see the proof of Proposition 3.2), and where IopU ′ is the dual category toIU ′ .
But in our situation, IU ′ has the initial object U ′ → U×X U ′ (with obvious morphism), sincefor an object ψ : U ′ → U ×X X ′ we have a morphism α : U ′ → X ′ and hence a canonicalmorphism
U ×X U ′
(id×α)
U ′
99tttttttttt
ψ
%%JJJJJJJJJJ
U ×X X ′ .
(Note that U ′ → U → X is an etale morphism). Thus follows
jP (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, withquasi-inverse s.
Remark 11.3 In particular, Theorem 11.1 implies that T (X, Y ) ∼= Sh(Xet) is an abeliancategory. One can easily see that a sequence in T (X, Y )
0→ (F ′1, F′2, φ′)→ (F1, F2, φ)→ (F ′′1 , F
′′2 , φ
′′)→ 0
is exact if and only if the sequences
0→ F ′1 → F1 → F ′′1 → 0
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and0→ F ′2 → F2 → F ′′2 → 0
are exact, since a sequence0→ F ′ → F → F ′′ → 0
is exact if and only if0→ i∗F ′ → i∗F → i∗F ′′ → 0
and0→ j∗F ′ → j∗F → j∗F ′′ → 0
are exact (consider the stalks).
Definition 11.4 Identifying S(Xet) with T (X, Y ), we can define the following six functors
i∗← j!←S(Y et)
i∗→ S(X et)j∗→ S(U et)
i!← j∗←
by
F1
i∗
←p (F1, F2, φ) , (0, F2, 0)j!←p F2
F1i∗7→ (F1, 0, 0) , (F1, F2, φ)
j∗7→ F2
kerφi!
←p (F1, F2, φ) (i∗j∗F2, F2, id)j∗←p F2
One calls j!F the “extension by zero” of F and i!F ⊆ F the “subsheaf of the sections withsupport in Y ”.
It is obvious that these assignments are functorial.
Theorem 11.5 (a) Under the identification between S(Xet) and T (X, Y ), the functors i∗,i∗, j
∗ and j∗ correspond to the functors with the same name between S(Xet) and (Yet), andS(Xet) and S(Uet).
(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
Sh(Yet)i∗→∼F ∈ Sh(Xet) | F has support in Y
(f) The functors j∗, j∗, i! and i∗ respect injectives.
Proof: (a) is clear from the identification between S(Xet) 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)
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andHomX((F2, 0, 0), (G1, G2, φ)) ∼= HomY (F2, ker(G1
φ→ i∗j∗G2)) .
(c) follows immediately from Remark 11.2, and
(d) follows from the description of the functors
In (e), the fully faithfulness of i∗ again follows from the description. For j∗, a morphism
(i∗j∗F2, F2, id)→ (i∗j∗F′2, F
′2, id)
is again determined only by F2 → F ′2, 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 ifF2 = 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 etale sheaf F on U one hasj∗j∗F = F .
(b) For a closed immersion i : Y → X and an etale sheaf G on Y one has i∗i∗F = F .
Proof: This follows from Theorem 11.4 (a) and the description of the functors in Definition11.3. In fact, we get
j∗j∗F = j∗(i∗j∗F ,F , id) = F ,
andi∗i∗G = i∗(G, 0, 0) = G .
Corollary 11.7 For every etale sheaf 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 exactsequence
0→ (0, j∗F, 0)→ (i∗F, j∗F, φ)→ (i∗F, 0, 0)→ 0 ,
and (b) corresponds to the exact sequence
0→ (kerφ, 0, 0)→ (i∗F, j∗F, φ)→ (i∗j∗j∗F, j∗F, φ) .
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12 Cech cohomology
The following generalizes the zero-th Cech cohomology (see definition 3.7) and the topologicalCech 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
Cn(U, P ) :=∏
(i0,...,in)∈In+1
P (Ui0 ×U . . .×U Uin)
is called the group of the n-cochains for the covering U with values in P . Define thedifferential
dn : Cn(U, P )→ Cn+1(U, P )
by
(dns)i0,...,in+1 =n+1∑ν=0
(−1)νsi0,...,iν ,...,in+1|Ui0×U ...×UUin+1
where the restriction with respect to the morphism
Ui0 ×U . . .×U Uin+1 → Ui0 ×U . . .×U Uiν ×U . . .×U Uin+1
is taken and a denotes the omission of a. Then dn+1dn = 0 for all n and we obtain a complexC ·(U, P ), called the Cech complex for the covering U with values in P .
(b) The n-th cohomologyHn(U, P ) := Hn(C ·(U, P ))
is called the n-th Cech 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-cohomology from Definition3.7 (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
Hn((Ui → U), P )→ Hn((Vj → U), P ) .
Proof We have a map ε : J → I and morphisms fj : Vj → Uε(j). With this we define themap
fn : Cn((Ui → U), P )→ Cn((Vj − U), P )
as follows: If s = (si0,...,in) ∈ Cn((Ui → U), P ), then define
(fns)j0,...,jn = resfj0×...×fjn (sε(j0),...,ε(jn)) ,
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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 H0(−, P ), this map coincides with the map defined in 3.7!
Lemma 12.5 Iff, g : (Vj → U)→ (Ui → U)
are two refinement maps, then we have
f ∗ = g∗ : Hn((Ui → U), P )→ Hn((Vj → U), P )
for all n ≥ 0.
Proof (compare Lemma 3.11) Let f = (ε, fj) and g = (η, gj). Define
kn : Cn((Ui → U), P )→ Cn−1(Vj → U), P )
by
(kns)j0,...,jn−1 =
p−1∑i=0
(−1)rresfj0×...×(fjr ,gjr )×...×gjn−1sε(j0),...,ε(jr),η(jr),...,η(jn−1)
for (fjr , gjr) : Vjr → Uε(jr) × Uη(jr). Then we have
dn−1kn + kn+1dn = gn − fn ,
i.e., (kn) provides a chain homotopy between (fn) and (gn), and the claim follows.
Definition 12.6 The n-th Cech cohomology of U with values in P is defined as
Hn(U, P ) := Hn(U, T ;P ) = lim→Hn(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 allcoverings U of U , where U′ ≥ U, if there is a refinement map f : U′ → U (see the proof of3.11).
Lemma 12.8 Let
(12.8.1) 0→ P1 → P2 → P3 → 0
be an exact sequence of abelian presheaves (!) on X .
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(a) For every covering U in T there is a long exact cohomology sequence
0→ H0(U, P1)→ H0(U, P2)→ H0(U, P3)δ→ H1(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
0→ H0(U, P1)→ H0(U, P2)→ H0(U, P3)δ→ H1(U, P1)→ H1(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 ,
since for U = (Ui → U)i∈I and every (i0, . . . , in) ∈ In+1 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 cohomo-logy 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 Remark12.7), 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 Cech 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 finitetype (resp. locally of finite presentation). Then (V = Spec(B)→ Spec(A) = U) is a coveringin 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
Hn((V → U),Ga) =
A , n = 0,0 , n > 0 .
One can also obtain the Cech cohomology as a derived functor:
Theorem 12.11 (a) For a covering U = (Ui → U), Hn(U,−) is the n-th right derivative ofthe left exact functor
H0(U,−) : Pr(X ) → AbP 7→ H0(U, P ) .
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(b) For U ∈ ob(X ), Hn(U,−) is the n-th right derivative of the left exact functor
H0(U,−) : Pr(X ) → AbP 7→ H0(U, P ) .
Proof It follows from 12.8 that the functors (Hn(U,−))n≥0 resp. (Hn(U,−))n≥0 form exactδ-functors on Pr(X ). Thus it suffices to show that Hn(U,−) resp. Hn(U,−) is effaceablefor 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 Pr(X ) has enough injectives,it suffices to show:
Lemma 12.12 For n > 0 we have Hn(U, I) = 0 = Hn(U, I), if I is an injective presheaf.
Proof The second claim follows from the first. Furthermore, we have to show that for everycovering U = (Ui → U)i∈I in T the sequence∏
i
I(Ui)→∏i0,i1
I(Ui0i1)→∏
i0,i1,i2
I(Ui0i1i2)→ . . .
is exact, where Ui0,...,in = Uij ×U Ui1 ×U . . . ×U Uin . This sequence can be identified with asequence
(12.12.1)∏i
Hom(ZPUi , I)→∏i0,i1
Hom(ZPUi0,i1 , I)→ . . . ,
(see Lemma/Definition 5.8) which comes from an obvious complex of presheaves
(12.12.2)⊕i
ZPUi ←⊕i0,i1
ZPUi0i1 ←⊕
i0,i1,i2
ZUi0i1i2 ← . . . .
by applying Hom(−, I). The last functor is exact since I is injective. Therefore it suffices toshow the exactness of (12.12.2), hence the exactness of
(12.12.3)⊕i
ZPUi(V )←⊕i0,i1
ZUi0i1 (V )← . . .
for every V in X .
Now, we have ZPW (V ) =⊕
Hom(V,W )
Z for W in X , and we have a canonical morphism Ui0...in →
U for every W = Ui0i1...in = Uij ×U Ui1 ×U . . .×U Uin . This implies
Hom(V,W ) =∐
φ∈Hom(V,U)
Homφ(V,W ) ,
where Homφ(V,W ) is the set of the morphisms ϕ : V → W for which
Vϕ //
φ @@@@@@@@ W
~~
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) =∐i∈IHomφ(V, Ui) ,
then the complex (12.12.3) can described as follows :
⊕φ∈Hom(V,U)
( ⊕S(φ)
Z← ⊕S(φ)×S(φ)
Z← ⊕S(φ)3
Z← . . .)
with the obvious differential in the bracket
1j0,...,jp 7→p∑
ν=0
(−1)ν1j0,...,jν ,...jp .
Now the complex in the bracket is exact: A contracting homotopy is (hp, p ≥ 0), with
hp : ⊕S(φ)p+1
Z → ⊕S(φ)p+2
Z
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
Ep,q2 = Hp(U, Hq(F )) ⇒ Hp+q(U, F )
Ep,q2 = Hp(U,Hq(F )) ⇒ Hp+q(U, F ) .
Here, cohomology and Cech-cohomology are taken with respect to T , and Hq(F ) is thepresheaf
V 7→ Hq(V, F ) .
Proof First spectral sequence: We apply Grothendieck’s Theorem (Theorem 6.8). Obviously,we have
H0(U, H0(F )) = H0(U, F ) ,
since F is a sheaf, hence H0(U,−) is the composition of H0(−) and H0(U,−). Furthermore,for an injective sheaf I, the presheaf H0(I) is acyclic for H0(U,−), i.e.,
RnH0(U, H0(I))12.11= Hn(U, H0(I)) = Hn(U, I) = 0 for n > 0 .
In fact, the embedding i : Sh(X , T ) → Pr(X ) respects injectives, since i has the exact leftadjoint a. Therefore, I is also injective as a presheaf, and the last vanishing follows fromLemma 12.12.
The second spectral sequence follows in an analogous way, or by passing to the limit over allcoverings of U .
Corollary 12.14 There is a spectral sequence
Ep,q2 = H
p(Hq(F ))→ Hp+q(F ) .
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Proof Consider the second spectral sequence in 12.13 for all U in X and note that this isfunctorial (contravariant) in U .
Proposition 12.15 We have
H0(U,Hq(F )) = 0 for q > 0 ,
and therefore also H0(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 cohomo-logy 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
sheaf). But H0(Hq(F )) is a sub-presheaf of aHq(F ) = H
0H
0(Hq(F )) (since H
0(P ) is sepa-
rated for every presheaf P , see Lemma 3.10 (c) and note that by definition P = H0(P )).
Therefore, H0(Hq(F )) = 0 for all q > 0, hence H0(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 iso-morphisms
H0(U, F ) ∼= H0(U, F )H1(U, F ) ∼= H1(U, F )
and an exact sequence
0→ H2(U, F )→ H2(U, F )→ H1(U,H1(F ))→ H3(U, F )→ H3(U, F )
Proof The spectral sequence
Hp(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
q
p
The first claim in 12.16 ist obvious, since F is a sheaf, and from the above shape we get
H1(U, F ) ∼= E1,02 = H1(U,H0(F ))
= H1(U, F ) ,
since E0,12 = 0 and all differentials leaving and entering E1,0
2 are zero. Deducing the lastsequence is analogous to the proof of the usual sequence of the low terms (Lemma 6.7).
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Corollary 12.17 Let X be a scheme and F a quasi-coherent OX-module. Let U = (Ui)i∈Ibe 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
Hn(U,F)∼→ Hn(X,F)
for all n ≥ 0.
Proof We use the spectral sequence
Hp(U, Hq(F))⇒ Hp+q(X,F) .
For q > 0, Hq(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 allU ∈ 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), Hn(U, F ) = 0 forn > 0.
(c) Hn(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 spectralsequence of 12.13 therefore provides an isomorphism
Hn(U, F )∼→ Hn(U, F ) = 0 for n > 0 .
(b) ⇒ (c): Follows by passing to the inductive limit over all coverings of U .
(c) ⇒ (a): By assumption, we have Hn(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
Hp(Hq(F ))⇒ Hp+q(F ) .
By assumption, H2(H0(F )) = H2(F ) = 0, furthermore H1(H1(F )) = 0 and H
0(H2(F )) =
0 by 12.15. From the spectral sequence we get H2(F ) = 0. The same argument showsinductively
Hi(Hj(F )) = 0 for i+ j ≤ n
and hence Hn(F ) = 0.
Corollary 12.21 Let f : (X ′, T ′) → (X , T ) be a morphism of sites. If F ′ is a flabby sheafon (X ′, T ′), them f∗F
′ is flabby, too.
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Proof Let f 0 : X → X ′ be the underlying functor. If U = (Ui → U) is a covering in T , thenU′ = (f 0Ui → f 0U) is a covering in T ′ and we have
(f∗F′)(V ) = F (f 0V )
for all V in X . Then we have
Hn(U, f∗F′) = Hn(U′, F ′) = 0 for n > 0 .
Corollary 12.22 (Leray spectral sequence) (a) If f : (X ′′, T ′) → (X , T ) is a morphism ofsites, then for every sheaf F ′ on (X ′, T ′) and every U ∈ X one has a spectral sequence
Hp(U,Rqf∗F′)⇒ Hp+q(f 0U, F ′) .
(b) If (X ′′, T ′′) → (X ′, T ′) → (X , T ) are morphisms of sites, then for every sheaf F ′′ on(X ′′, T ′′) there is a spectral sequence
Rpf∗Rqg∗F
′′ ⇒ R(gf)∗F′′ .
Proof This follows from Theorem 6.8 (Grothendieck’s spectral sequence), since f∗ mapsflabby sheaves to flabby sheaves, therefore acyclic sheaves for H0(U,−), and g∗ as well mapsflabby sheaves to flabby sheaves, and therefore to acyclic sheaves for f∗. In fact, Rnf∗F
′ isthe associated sheaf to the presheaf U 7→ Hn(f 0U, F ′), and the presheaf is already 0, if F ′ isflabby.
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13 Comparison of sites
Proposition 13.1 (Change of the category) Let (X ′, T ′) be a site, let X ⊆ X ′ be a fullsubcategory, and let T be the restriction of T ′ to X .
We assume:
(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 sitesα : (X ′, T ′)→ (X , T ) ,
which is given by the embedding X → X ′, we then have:
(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 ′, T ′) is fully faithful and left exact.
(c) The canonical homomorphisms
Hn(U, T ;α∗F′) → Hn(U, T ′;F ′)
Hn(U, T ;F ) → Hn(U, T ′;α∗F )
are isomorphisms for all U ∈ X , all F ′ ∈ Sh(X ′, T ′), 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
(αPF )(U) = F (U) ,
since the category IU , over which the limit is formed for (αPF )(U), has the initial object(U, idU). Since F restricted to (X , T ) is a sheaf, we have (α∗F )(U) = (aαPF )(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
(13.1.2) Ep,q2 = Hp(U,Rqα∗F )⇒ Hp+q(U, F ) ,
either by Corollary 12.22, or by the Grothendieck-Leray spectral sequence 6.7. This exists,since H0(U, α∗F ) = H0(U, F ), and since α∗ has a the left exact left adjoint α∗, so that by7.10, α∗ maps injectives to injectives (hence acyclic sheaves).
Since α∗ is exact, we have Rqα∗F = 0 for q > 0, and (13.1.2) provides an edge isomorphism
Hn(U, α∗F )∼→ Hn(U, F ) .
The second claim follows from the fact that the composition
Hn(U, F )→ Hn(U, α∗F )∼→ Hn(U, α∗α
∗F )∼→ Hn(U, F )
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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 ofall X-schemes with the E-coverings as topology (the large E-site) and XE is the category ofall X-schemes U , for which the structural morphism U → X is in E, with the E-coveringsas topology (the small E-site). In particular, the large etale site (Sch/X)et and the smalletale site Xet 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 ′ betopologies on X (every covering for T is a covering for T ′). Let
β : (X , T ′)→ (X , T )
be the morphism of sites, given by idX . Assume that for every covering U = (Ui → U) inT ′ there is a covering (Vj → U) in T , which refines U. Then β∗ : Sh(X , T ′) → Sh(X , T ) isexact and thus
Hn(XT , β∗F ′)∼→ Hn(XT ′ , F ′)
for every sheaf F ′ ∈ Sh(X , T ′).
Proof : The functor β∗ is the identity. We only have to show that every epimorphism F → F ′′
for T ′ is an epimorphism for T as well. If we have U ∈ ob(X ) and s ∈ F ′′(U), then there isa 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 , wethen obtain a commutative diagram
F (Vj) // F ′′(Vj)
F (Ui) //
OO
F ′′(Ui) ,
OO
which shows that s|Vj is in the image of the top map. Thus β∗ is exact. The second claimfollows from the Leray-spectral sequence, since Rpβ∗F
′ = 0 for p > 0, by the exactness ofβ∗.
Corollary 13.4 Let X be a scheme, and let E ⊂ E ′ be two admissible classes of morphisms,so that the condition of refinement of 13.3 holds for the corresponding topologies (E) ⊂ (E ′).For the morphism of small sites
γ : XE′ → XE ,
the functor γ∗ : Sh(XE′)→ Sh(XE) is exact, hence for every sheaf F ′ on XE′ we have
Hn(U,E; γ∗F′)∼→ Hn(U,E ′, F ′) .
Proof Let E − Sch/X be the category of the X-schemes U , whose structural morphismU → X is in E. Then γ factorizes as
γ : XE′β→ (E ′ − Sch/X)E
α→ (E − Sch/X)E = XE ,
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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) (etale morphisms of finite type) ⊂ (et)
(b) (et) ⊂ (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, I2.25).
Theorem 13.6 (quasi-coherent OX-modules) Let X be a scheme and M a quasi-coherentOX-module. Let Met and Mfl be the corresponding sheaves in the etale topology and flattopology, respectively (see 10.18). Then we have
Hn(XZar,M)∼→ Hn(Xet,Met)
∼→ Hn(Xfl,Mfl)
for all n.
Proof We give the proof for the flat topology, the case of the etale 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 showthat Rnf∗Mfl = 0 for n > 0 (then the claim follows from the Leray-spectral sequence). SinceRnf∗Mfl is the Zariski-sheaf associated to the presheaf U 7→ Hn(Ufl,Mfl) (see Theorem 5.16),it suffices to show that Hn(UflMfl) = 0, if U = Spec(A) ⊆ X is affine open. Furthermore, byCorollary 13.4, we can consider the small site UE , where E is the class of flat affine morphismsof finite presentation. We want to show that Mfl is flabby, and by 13.20 it suffices to showthat Hn(U,Mfl) = 0 for all n > 0 and all coverings U = (Ui → U)i∈I in E. By 10.6 andthe quasi compactness of U , we can assume that I is finite (cofinite system of E-coverings!).Then V =
∐i
Ui = Spec(B) is affine and A → B is faithfully flat, and the Cech complex is
the obvious complex
B ⊗AM → B ⊗A B ⊗AM → B ⊗A B ⊗A B ⊗AM → . . . ,
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. Thenthe theorem of implicit functions implies that the set X(C) is a complex manifold: It sufficesto show this locally. But locally we have
X = Spec(C[X1, . . . , Xn]/〈f1, . . . , fm〉) ,m ≤ n ,
where the Jacobian matrix (∂fi∂Xj
(P )
)has rank m for all closed points P of X. Then we have
X(C) ∼= a = (a1, . . . , an) ∈ Cn | fj(a) = 0 ∀ j ,
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and the fi define continuous mapsfi : Cn → C ,
and the matrix above is the usual Jacobi matrix at P . The theorem on implicit functiongives local homomorphisms
X(C) ⊇ V∼→ U ⊆ Cn−m ,
and one obtains charts for X(C) as a manifold.
By Artin and Grothendieck there are isomorphisms for all m and n
Hn(X(C),Z/mZ)∼→ Hn(Xet,Z/mZ) ,
for every smooth variety X/C. Here the left hand side is the (topological) cohomology ofsheaves of the constant sheaf, and the right hand side is the etale cohomology.
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14 Descent theory and the multiplicative group
Lemma 14.1 Let X be a scheme. There is a canonical isomorphism
H1(XZar,O×X) ∼= Pic(X) ,
where Pic(X) is the Picard group of X.
Proof In view of 12.16 it suffices to construct a canonical isomorphism
(14.1.1) Pic(X)∼→ H1(X,O×X) .
Let L be an invertible OX-module. Then there is an open covering U = (Ui)i∈I of X, suchthat there are isomorphisms
ϕi : OUi∼→ L|Ui for all i ∈ I .
On Ui ∩ Uj, these induce isomorphisms
ϕij = ϕ−1j ϕi : OUi|Ui∩Uj
ϕi→ L|Ui∩Ujϕ−1j→ OUj |Ui∩Uj ,
which correspond to elements sij ∈ O×X(Ui ∩ Uj).
Here we have
(14.1.2)
sik|Uijk · sjk|Uijk · sij|Uijk = 1
qϕ−1i ϕkϕ
−1k ϕjϕ
−1j ϕi ,
i.e., we have a Cech-1-cocycle in
H1(U,O×X) = H1(∏i
O×X(Ui)→∏i,j
O×X(Ui ∩ Uj)→∏i,j,k
O×X(Ui ∩ Uj ∩ Uk))
Conversely, such a cocycle gives glueing isomorphisms
ϕi,j : O(Ui ∩ Uj)·sij→∼O(Ui ∩ Uj) ,
which glue together the free modules OUi on Ui to an invertible OX-module L (the cocyclecondition (14.1.2) gives the cocycle condition/transitivity for ϕij). This assignment is addi-tive. Furthermore, L is trivial if and only if the 1-cocycle (sij) is trivial in H1(U,O×X). Thisgives an isomorphism
H1(U,O×X)∼→
isomorphism classes of invertible OX-modules L,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.
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Lemma 14.3 Let A be a ring and let M be an A-module. If M is a flat A-module, then thefollowing holds:
(1) Ifr∑i=1
aimi = 0 with ai ∈ A and mi ∈ M , then there are an s ∈ N and elements bij ∈ A
and yj ∈M (j = 1, . . . , s) with ∑i
aibij = 0
for all j and mi =∑j
bijyj for all i.
Proof Consider the exact sequence
K → Arf→ A
(b1, . . . , br) 7→r∑i=1
biai ,
K = ker(f). Then
K ⊗AM → M r fM→ M
(n1, . . . , nr) 7→r∑i=1
aini
is exact. By assumption, we have fM(m1, . . . ,mr) = 0, therefore there is an element
s∑j=1
βj ⊗ yi ∈ K ⊗AM
(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 followingconditions 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 letm1, . . . ,mn ∈M be in such way that their images m1, . . . ,mn in M/mM form a basis of this A/m-vectorspace. Then, by the Nakayama-Lemma the morphism
An Mbasis element ei 7→ mi
is surjective. It suffices to show that m1, . . . ,mn ∈ M are linearly independent over A, ifm1, . . . ,mn are linearly independent in M/mM . We use induction over n. Let n = 1 andam1 = 0 for a ∈ A. By Lemma 14.3 there are b1, . . . , bs ∈ A and y1, . . . , ys ∈M with abj = 0for all j and m1 =
∑j
bj · yj. Since m1 6= 0 there is a j with bj /∈ m, i.e., bj ∈ A× a unit. From
abj = 0 we get a = 0.
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Now let n > 1 andn∑i=1
aimi = 0. By Lemma 14.3 there exist y1, . . . , ys ∈ M and bij ∈ A
(i = 1, . . . , s) with
mi =s∑j=1
bijyj ,n∑i=1
aibij = 0 .
Since mn 6= 0 there is a j with bnj /∈ m, i.e., bnj a unit. Then we have
an =n−1∑i=1
ciai , with ci = −bij/bin ,
and therefore
0 =n∑i=1
aimi = a1(m1 + c1mn) + . . .+ an−1(mn−1 + cn−1mn)
Since the considered m1 + c1mn, . . . ,mn−1 + cn−1mn are linear independent over A/m, the
induction hypothesis implies a1 = . . . = an−1 = 0 and hence also an =n−1∑i=1
ciai = 0.
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
0 → Mγ→ B ⊗AM
α1
⇒α2
B ⊗A B ⊗AM
m 7→ 1⊗m, b⊗m 7→7→
1⊗ b⊗mb⊗ 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
(14.6.1) 0→Mγ→ B ⊗AM
α1−α2−→ B ⊗B ⊗M
is exact if and only if the sequence
0→ B ⊗M 1⊗γ−→ B ⊗B ⊗M 1⊗(α1−α2)→ B ⊗B ⊗B ⊗M ,
tensored by B, is exact. Letµ : B ⊗B → B
b1 ⊗ b2 7→ b1b2
be the multiplication map. Then
µ⊗ 1 : B ⊗B ⊗M → B ⊗Mb1 ⊗ b2 ⊗m → b1b2 ⊗m
is a left inverse of 1⊗ γ (µ⊗ 1 1⊗ γ = id), hence 1⊗ γ is injective.
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Now let z =∑i
x1 ⊗ yi ⊗mi be in the kernel of 1⊗ (α1 − α2), so that
∑i
xi ⊗ 1⊗ yi ⊗mi =∑i
xi ⊗ yi ⊗ 1⊗mi .
By applying µ to the first two places we obtain∑i
xi ⊗ yi ⊗mi =∑i
xiyi ⊗ 1⊗mi ,
and hence∑i
xiyi ⊗ 1⊗mi is the image of∑i
xiyi ⊗ ui under 1⊗ γ.
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
M ′ = B ⊗AM
one has a canonical isomorphism of B ⊗B-modules (all tensor products are over A)
(14.7.1)φ : M ′ ⊗B ∼→ B ⊗M ′
(b⊗m)⊗ b′ 7→ b⊗ (b′ ⊗m) .
By this one can retrieve M from M ′:
(14.7.2) M = m′ ∈M ′ | φ(m′ ⊗ 1) = 1⊗m′ ,
because this amounts to the exactness of the sequence (14.6.1)
0→M → B ⊗M α1−α2−→ B ⊗B ⊗M
where
(14.7.3)α1(b⊗m) = 1⊗ b⊗mα2(b⊗m) = b⊗ 1⊗m.
In fact for m′ = b⊗m ∈ B ⊗M = M ′ we have
φ(m′ ⊗ 1)− 1⊗m′ = (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 theremaining positions)
φ1 : B ⊗M ′ ⊗B → B ⊗B ⊗M ′
b⊗ (b2 ⊗m)⊗ b3 7→ b⊗ b2 ⊗ (b3 ⊗m)
φ2 : M ′ ⊗B ⊗B → B ⊗B ⊗M ′
(b⊗m)⊗ b2 ⊗ b3 7→ b⊗ (b2 ⊗m)⊗ b3
andφ3 : M ′ ⊗B ⊗B → B ⊗M⊗B
(b⊗m)⊗ b2 ⊗ b3 7→ b⊗ (b2 ⊗m)⊗ b3
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we get the co-called cocycle conditions
(14.7.4) φ2 = φ1φ3 .
(c) Conversely, let M ′ be a B-module, let
(14.7.5) φ : M ′ ⊗B ∼→ B ⊗M ′
be an isomorphism of B ⊗B-modules, and let
φ1 : B ⊗M ′ ⊗B → B ⊗B ⊗M ′
φ2 : M ′ ⊗B ⊗B → B ⊗B ⊗M ′
φ3 : M ′ ⊗B ⊗B → B ⊗M ′ ⊗B
be the induced isomorphisms, where φi keeps the entry on the i-th position and is definedon the other positions by φ. (For φ2 we have explicitly
m′ ⊗ b2 ⊗ b3 7→∑i
bi ⊗ b2 ⊗m′i ,
if φ(m′ ⊗ b3) =∑i
bi ⊗m′i.)
If now the cocycle condition
(14.7.6) φ2 = φ1φ3
holds, then there is a canonical A-module M with B ⊗AM ∼= M ′, namely the A-module
M = m′ ∈M ′ | φ(m′ ⊗ 1) = 1⊗m′ ,
for which the canonical map
(14.7.7)γ : B ⊗AM
∼→ M ′
b⊗m 7→ bm
is an isomorphism. In fact, letτ : M ′ → B ⊗M
be defined by τ(m′) = 1⊗m′ − φ(m′ ⊗ 1). By definition, we then have an exact sequence
0→M →M ′ τ→ B ⊗M ′ .
If we tensorize with B on the right, we obtain the top row in the following diagram
0 //M ⊗B //
γ
M ′ ⊗B //
φo
B ⊗M ′ ⊗Bo 1⊗φ
0 //M ′ // B ⊗M ′ // B ⊗B ⊗M ′ ,
where the lower sequence is the exact sequence from 14.6, applied to the A-module M ′. Themap γ is defined by γ(m ⊗ b) = bm (and therefore corresponds to the map (14.7.7)). We
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show that γ is an isomorphism. Since the rows are exact (the top row is exact because ofthe flatness of B over A) and both vertical maps on the right are isomorphisms, the claimfollows, 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 m′ ∈M ′ let
φ(m′ ⊗ 1) =∑i
bi ⊗m′i
with bi ∈ B and m′i ∈M ′. Then we have
φ(m′ ⊗ b) = (1⊗ b)φ(m′ ⊗ 1) =∑i
bi ⊗ bm′i ,
and thus the image of this in B ⊗B ⊗M ′ is equal to∑i
1⊗ bi ⊗ bm′i −∑i
bi ⊗ 1⊗ bm′i .
On the “upper way”, m′ ⊗ b is mapped to
1⊗m′ ⊗ b − φ(m⊗ 1)⊗ b= 1⊗m′ ⊗ b −
∑i
bi ⊗m′i ⊗ b
and this, by 1⊗ φ, is mapped to
1⊗ φ(m′ ⊗ b) −∑i
bi ⊗ φ(m′i ⊗ b)
=∑i
1⊗ bi ⊗ bm′i −∑i
bi ⊗ φ(m′i ⊗ b) .
Hence we have to show that∑i
bi ⊗ 1⊗ bm′i =∑i
bi ⊗ φ(m′i ⊗ b) .
But this means that we have
φ2(m⊗ 1⊗ b) = φ1(φ3(m⊗ 1⊗ b))‖ ‖∑
i
bi ⊗ 1⊗ bm′i φ1(∑i
bi ⊗ 1⊗m′i)
‖∑i
bi ⊗ φ(1⊗m′i) ,
which holds because of the assumption that φ2 = φ1φ3.
Theorem 14.8 For every scheme X, the canonical morphisms
H1(XZar,O×X)∼→ H1(Xet,Gm)
∼→ H1(Xfl,Gm)
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are isomorphisms.
Proof for the flat topology (the etale case is analogous). We use the Leray-spectral sequencefor
α : Xfl → XZar .
By the sequence of the lower terms
0→ H1(XZar, α∗Gm)→ H1(Xfl,Gm)→ H0(XZar, R1α∗Gm)
where α∗Gm = O×X , it suffices to show that R1α∗Gm = 0. This means that for all for allx ∈ X the stalk (R1α∗Gm)x = 0. But this stalk is
lim→x∈U
H1(Ufl,Gm) ∼= lim→x∈U
H1(Ufl,Gm) ,
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∈Iwith U ⊆ X open the limit
lim→V⊆U open
H1((Ui ×U V → V,Gm) = 0 .
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 considera class in
(14.8.1) H1(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 Mwith B ⊗A M ∼= B. Since the B-module B is finitely generated and flat, this also holds forM . 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 isconstructed 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.
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15 Schemes of dimension 1
Proposition 15.1 Let X be a regular integral noetherian scheme and let j : SpecK → Xbe the inclusion of the generic point. Then there is an exact sequence of etale sheaves
(15.1.1) 0→ Gm/X → j∗(Gm/k)→ ⊕x∈X1
(ix)∗Z→ 0 ,
where X1 is the set of the points of codimension 1 of X and ix: Spec(k(x)) → X is thecanonical 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 dimOU,y = dimOX,x.(b) U is regular if an only if X is regular.
Proof (a): Without restriction, X = SpecA, A is local and U = SpecB is affine. Let m ⊆ A(maximal) and n ⊆ B be the prime ideals which correspond to x and y. Then SpecBn →SpecA is faithfully flat (10.5), therefore surjective, hence dimBn ≥ dimA. Conversely, byZariski’s main theorem 9.3, B/A is finite without restriction. Then ϕ : A → B induces anintegral ring extension A/ kerϕ → B, and, by Cohen-Seidenberg, dimA ≥ dim(A/ kerϕ)) =dimB ≥ dimBn.
(b) Let m ⊂ A and n ⊂ B be as above. Then by (a) d = dimA = dimBn. On the other handwe have mBn = nBn, and hence isomorphisms
m/m2 ⊗k(m) k(n) ∼= m/m2 ⊗A(m) Bn/mBn = m/m2 ⊗A Bn∼= n/n2 ,
i.e., dimk(m) m/m2 = dimk(n) n/n
2.
Proof of Proposition 15.1: For U → X etale we have morphisms
Gm(U)α→ Gm(U ×X SpecK)
β→ ⊕x∈X(1)
Z(U ×X Spec(K)) ,
where α is the restriction and β is defined as follows: We have U ×X SpecK =∐η∈U0
Spec k(η)
by 15.2 (a) and
U ×X Spec k(x) =∐y∈U1f(y)=x
Spec(k(y))
The component βy : Gm(k(η)) = k(η)× → Z of β at y is 0 if y /∈ η (⇔ η /∈ SpecOU,y) andthe discrete valuation associated to y on k(η)×, if η is the generic point of SpecOU,y, hencek(y) = QuotOU,y. It follows immediately that βα = 0.
By forming the associated sheaf to ⊕x∈X(1)
(ix)∗Z we obtain the wanted sequence. For the
exactness it suffices to prove the exactness if U is replaced by a local ring OU,y for y ∈ U1.
But then the sequence is
0→ O×U,y → (Quot(OU,y))×vy→ Z→ 0
105
and hence is exact.
Now we consider the long exact cohomology sequence associated to
0→ Gm → j∗Gm → ⊕x∈X1
(ix)∗Z→ 0 .
We need the following.
Lemma 15.3 Let X be a quasi compact, quasi separated scheme. Then, for every inductivesystem (Fi)i∈I of abelian etale sheaves on X, we have:
lim→i∈I
Hnet(X,Fi)
∼→ Hnet(X, lim→
i∈I
Fi) .
In particular, etale cohomology commutes with direct sums.
Proof : See Tamme II, Introduction to etale Cohomology, 1.5.3.
S(Xet) is equivalent to S(Xet, f.p.) for the noetherian site of all etale X-schemes of finitepresentation. Thus we have
Hnet(X, ⊕
x∈X1(ix)∗Z) = ⊕
x∈X1Hn
et(X, (ix)∗Z)
Lemma 15.4 Let (Xi)i∈I be a projective system of quasi compact and quasi separatedschemes, with affine transition morphisms. Let i0 ∈ J and F be an etale sheaf on Xi0 . Thenthe natural map
lim→i∈I
Hnet(Xi, F |Xi)
∼→ Hnet(lim←
i
Xi, F |lim←i
X)
is an isomorphism, where F |Xi and F |X=lim←i
Xi denote the pull-backs of F , respectively.
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 etale sheaf on Y and let x be a geometric point of X. Then we have
(Rnf∗F )x ∼= Hnet(Y ×X Spec(OhX,x), F |...) .
Proof Let P be the presheaf U Hnet(Y ×X U, F ) on X. Then we have Rnf∗F = aP (the
associated sheaf), hence
(Rnf∗F )x = Px = lim→U etale neighborhood of x in X
Hnet(Y ×X U, F |Y ×X U)
15.4= Hn
et(Y ×X SpecOshX,x, F |...) .
The claim above now follows from the fact that OshX,x ⊗OX,x K = Kx.
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Note that we haveH i
et(SpecKx,Gm) = H i(Kx, (Ksepx )×)
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
(R1j∗Gm,K)x = H1(Kx, K×x ) = 0 ,
where Kx = Quot(OX,x), since we have Spec(K)×X Spec(OX,x) = Spec(Kx). Therefore
R1j∗Gm,K = 0 .
From the Leray-spectral sequence for j∗ we thus get
H1(X, j∗Gm,K) = H1(K, (Ksep)×) = 0
and an exact sequence
0→ H2(X, j∗Gm,K)→ Br(K)→ H0(X,R2j∗Gm,X)→ H3(X, j∗Gm,X) .
On the other hand, from the sequence (15.1.1) we get an exact sequence
0→ Γ(X,O×X)→ K× → ⊕x∈X1
Z→ H1(X,Gm)→ 0 ,
therefore the known isomorphism (for regular X)
H1(X,Gm)∼→ Pic(X) .
Now let dimX = 1 and let k(x) be perfect for all x ∈ X1.
Lemma 15.8 Rij∗Gm,K = 0 for i ≥ 1.
Proof For all geometric points x of X, Kx = Quot(OX,x), and OX,x is a discrete valuationring with algebraic closed residue field, or a separably closed field. In the first case Kx hasthe cohomological dimension 1, in the second case the cohomological dimension 0. Thus
(Rij∗Gm,K)x = H i(Kx, (Ksepx )×) = 0
for i ≥ 1.
In this case we have isomorphisms
H i(X, j∗Gm,K) ∼= H i(K, (Ksep)×)
for all i.
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The sequence (15.1.1) gives an exact sequence
0→ H2(X,Gm)→ Br(K)→ ⊕x∈X1
H2(k(x),Z)→ H3(X,Gm)→ . . .
since (ix)∗ is exact, and with the Leray spectral sequence for (ix)∗ we get
H i(X, (ix)∗Z) ∼= H i(Spec(k(x),Z)
for all i.
Finally, let X be a smooth projective curve over an algebraically closed field k, and as beforelet 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. Hencecd(k(x)) = 0 for x ∈ X1, H i(X,Gm)
∼→ H i(X, j∗,Gm,K) = 0 for i ≥ 2.
Let n be invertible in k. From the Kummer-sequence
0→ µn → Gmn→ Gm → 0 ,
the next lemma follows by passing to cohomology
Lemma 15.10 (i) H0(X,µn) = µn (since H0(X,Gm) = k×)
(ii) H1(X,µn) = Pic(X)[n] ∼= (Z/nZ)2g
(iii)H2(X,µn) ∼= Pic(X)/nPic(X) = Z/nZ, where g is the genus ofX, thus g = dimkH0(X,Ω1
X/k).
Proof The first isomorphisms are obvious from the long exact cohomology sequences. The
further isomorphisms in (ii) and (iii) follow from the fact that for Pic0(X) = ker(Pic(X)deg
Z) we havePic0(X) ∼= Jac(X)(k) ,
where Jac(X) is the Jacobian variety of X. This is an abelian variety of dimension g, whereg is the genus of X.
Corollary 15.11 The cohomology groups H i(X,Z/n) are finite.
Proof Over an algebraically closed field we can identify Z/n with µn.
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