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Home , Flat topology, Grothendieck topology, Injective object, Sheaf (mathematics), Sheaf cohomology

on a Grothendieck Site

P. McFaddin Fall 2012

Contents

0 Sheaves on a Topological 2

1 Sites and Sheaves 3 1.1 Sites ...... 3 1.2 Sheaves ...... 5

2 6 2.1 Categorical Notions ...... 7 2.2 Cohomology ...... 8

Introduction

Our first goal in this talk is to motivate the definitions of a Grothendieck site and a sheaf on a site as generlizations of a and a sheaf on a topological space. In particular, we hope to make clear the analogies given in the following chart:

Topological Space X Site C

U ⊂ X open U ∈ Ob C

S U = Ui an open {Ui → U} a cover

f : X → Y continuous , f : C → D , f −1 takes open sets to open sets, f takes covers to covers, commutes with intersection commutes with pullbacks

Sheaf F : Op(X) → A Sheaf F : C → A contravariant contravariant functor satisfies an equalizer diagram satisfies an equalizer diagram

1 We will then develop some tools necessary to define cohomology groups of a sheaf on a site, laying foundations with some categorical technicalities along the way.

0 Sheaves on a Topological Space

Definition 0.1. Let X be a toplogical space. A presheaf of sets F on X is the following data: (a) for every open U ⊂ X, a F (U) called sections of F over U

(b) for every V ⊂ U open sets, a map ρUV : F (U) → F (V ) called restriction which satisfy the following: (0) F (∅) = {∗}, the set with one element

(1) ρUU = idF (u)

(2) if W ⊂ V ⊂ U then ρUW = ρVW ◦ ρUV

If s ∈ F (U) and V ⊂ U, we will often write s|V in place of ρUV (s).

Examples.

• The sheaf of R-valued continuous functions on a topological space X, given by F (U) = {f : U → R|f is continuous} for U ⊂ X open.

• The sheaf of regular functions OX on a X. • The sheaf of holomorphic functions (or holomorphic differential forms) on a complex . • The sheaf of C∞-functions on a smooth manifold.

Definition 0.2. A presheaf F is called a sheaf if the following additional conditions are satisfied: S (3) (uniqueness) if U ⊂ X is open, {Vi} is an open cover of U (so that U = Vi) and s ∈ F (U)

satisfies s|Vi = 0 for all i, then s = 0.

(4) (gluing) if U ⊂ X is open, {Vi} is an open cover of U and if there is a collection of sections

{si} such that si|Vi∩Vj = sj|Vi∩Vj for all i, j then there exists a s ∈ F (U) such that

s|Vi = si for all i. Definition 0.3. A morphism of sheaves (or presheaves) ϕ : F → G is a collection of ϕU : F (U) → G(U), for each U ⊂ X open, which commute with the restriction maps:

ϕU F (U) / G(U)

ρUV ρUV

 ϕV  F (V ) / G(V )

We can rephrase the above definitions more categorically with an aim to generalize. Let X be a topological space and let Op(X) denote the whose objects are open of X and whose morphisms are inclusions of open sets. • A presheaf F on X is a functor F : Op(X)op → Sets.

2 • A morphism of presheaves is a ϕ : F → G.

• A sheaf is a presheaf such that for every open cover {Ui} of U, the sequence Y Y F (U) → F (Ui) ⇒ F (Ui ∩ Uj) i i,j

is an equalizer diagram.

Remark 0.4. In general, the equalizer of two maps f, g : B ⇒ C is an object A and a morphism e : A → B with f ◦ e = g ◦ e such that the pair (A, e) is universal with respect to this property (any other object which equalizes f, g has a unique map to A factoring through e). In the , the equalizer can be realized as

Eq(f, g) = {x ∈ B|f(x) = g(x)}.

Even this improved definition of a sheaf relies on our ability to take intersections of open sets, but this is not possible for objects in an arbitrary category. Thus, we will have to adjust our definition in order to successfully generalize.

Definition 0.5. Let C be a category and let f : X → Z ← Y : g be a diagram in C . The pullback of f along g (or g along f) is the universal object T making

T / Y

g  f  X / Z commute.

In C = Sets, the pullback can be realized as T = X ×Z Y = {(x, y) ∈ X × Y |f(x) = g(y)}. This element-wise definition actually works for any category whose objects are sets with extra structure (so-called concrete categories). Notice that if C = Op(X) and V,W ⊂ U ⊂ X, then the pullback of V and W over U is given by V ∩ W , since V ×U W = {(x, y) ∈ V × W |x = y ∈ U} = V ∩ W. Thus, to generalize the notion of a sheaf, we use pullbacks (as opposed to intersections). We now attempt to extend our definitions to arbitrary categories.

1 Sites and Sheaves 1.1 Sites Definition 1.1. Let C be a category. A family of morphisms with fixed target in C is given by an object U ∈ Ob C , an indexing set I and for each i ∈ I, a morphism Ui → U of C . We denote a family by {Ui → U}i∈I .

Definition 1.2. A Grothendieck on a category C is given by a set Cov(C ) of families of morphisms with fixed target (called coverings) satsifying the following:

1. if V → U an then {V → U} ∈ Cov(C )

3 2. if {Ui → U}i∈I ∈ Cov(C ) and for all i {Vij → Ui}j∈Ji ∈ Cov(C ) then {Vij → U}i∈I,j∈Ji ∈ Cov(C )

3. if {Ui → U}i∈I ∈ Cov(C ) and V → U is a morphism in C and Ui ×U V exists for all i then {Ui ×U V → V }i∈I ∈ Cov(C ) A Grothendieck site is given by a category C and a choice of . Examples.

• Let X be a topological space and let TX denote the site consisting of the category Op(X) and topology given by [ {Ui → U} ∈ Cov(Op(X)) if and only if Ui = U.

Note that by definition, we allow coverings of the empty set by the empty set, as it is an open subset of X. However, we must also allow the empty covering of the empty set (i.e. taking I = ∅ and {Ui → ∅}i∈I ∈ Cov(Op(X))). It turns out that dissallowing such a covering still defines a site, but the sheaves on this site will, in general, be different than those on TX .

• Small Zariski Site. Let X be a scheme. A Zariski covering of X is a family of morphisms {fi : S Xi → X}i∈I of schemes such that each fi is an open immersion and such that X = fi(Xi) (a so-called surjective family of morphisms). Let Xzar be the category of open immersions into X and let Cov(Xzar) be the set of Zariski coverings of objects in Xzar. • Big Zariski Site. Let X be a scheme. As a category, the big Zariski site is the category of schemes over X and coverings are Zariski coverings.

Etale´ Morphisms Definition 1.3. Let X and Y be nonsingular varieties and let f : X → Y be a morphism. Then f is ´etale at x ∈ X if the induced map df : TxX → Tf(x)Y on tangent spaces is an isomorphism. We say f is ´etaleif is so at every point.

If X and Y are varieties over C, we can view them as complex (using the analytic toplogy). In this case, a map f is ´etaleif it is locally an isomorphism. Thus, any covering space of a compact Riemann surface is an example of an ´etalemorphism. The aim of using ´etalemaps is to generalize the notion of local ismorphism so that we may algebraically define “small neighborhood” as we do in the complex analytic setting. More generally, a map f : X → Y of schemes is called ´etaleif it is flat and unramified.

Examples.

• Etale´ Site. Let X be a scheme. The ´etalesite of X (denoted X´et) is given by the category Et´ /X of ´etale X-schemes (schemes S together with an ´etalemorphism S → X) and with coverings being surjective families of morphisms in Et´ /X. ´ • Big Etale Site. For a scheme X, the big ´etalesite of X is denoted by XEt´ and is given by the category Sch/X of schemes over X with coverings being surjective families of ´etale X-morphisms.

• . Xfl is given by Sch/X with coverings being surjective families of flat X- morphims of finte type.

4 Remark 1.4. We have maps of (defined below):

Xfl → XEt´ → X´et → Xzar.

Definition 1.5. A morphism f : C → C 0 of sites is a functor on the underlying categories satisfying

0 1. if {Ui → U} ∈ Cov(C ) then {f(Ui) → f(U)} ∈ Cov(C )

2. if {Ui → U} ∈ Cov(C ) and U → V is a morphism in C , then the canonical morphism

f(Ui ×U V ) → f(Ui) ×f(U) f(V )

(given by the of the pullback on the right) is an isomorphism.

Example. Let f : X → Y be a continuous map of topological spaces. Let TX and TY be the −1 −1 corresponding sites of opens sets. We have a functor f : TY → TX via U 7→ f (U). We claim that this is gives a morphism of sites. S 1. {Ui → U} ∈ Cov(TY ) ⇔ Ui = U. Since the inverse image of a continuous map commutes S −1 −1 −1 −1 with unions, we have f (Ui) = f (U) ⇒ {f (Ui) → f (U)} ∈ Cov(TX ). S 2. {Ui → U} ∈ Cov(TY ) and V → U a morphism in TY ⇔ U = Ui and V ⊂ U ⊂ Y . Since the inverse image of a continuous map commutes with intersections, we have (by translating our notation)

−1 −1 −1 −1 −1 −1 f (Ui ×U V ) = f (Ui ∩ V ) = f (U) ∩ f (V ) = f (Ui) ×f −1(U) f (V ).

1.2 Sheaves Definition 1.6. Let C be a Grothendieck site and D a category. A presheaf on C with values in D is simply a contravariant functor F : C → D.A sheaf is a presheaf such that for all coverings {Ui → U} ∈ Cov(C ) the diagram Y Y F (U) → F (Ui) ⇒ F (Ui ×U Uj) is an equalizer. If D is an (resp. an exact category), we may rephrase this and require that the above diagram be an (resp. an admissible exact sequence).

Remark 1.7. The category of sheaves of sets on X´et is sometimes denoted Xe´et and is called the ´etale of X.

Definition 1.8. A topos is any category which is equivalent (i.e. there is a fully faithful and essentially surjective functor) to the category of sheaves of sets on a Grothendieck site.

0 Remark 1.9. Returning to the example TX , let TX denote the site which is the same as TX except that we dissallow the empty covering of the empty set. Consider X = {x}, a . Then 0 0 0 Ob(TX ) = {∅,X} and Cov(TX ) = {{∅ → ∅}, {X → X}}. A sheaf of sets F on TX is given by a set F (X), a set F (∅) and a map F (X) → F (∅). But this is the same data as a sheaf on the space {∅, {x}, {x, y}}.

5 Sheaf Associated to a Presheaf Let C be a Grothendieck site and let Pshv(C ) be the category of abelian presheaves on C (con- travariant with values in Ab, the category of abelian groups). By definition, the category Shv(C ) of abelian sheaves on C is a full subcategory of Pshv(C ) (since maps of sheaves are just maps of presheaves). Let i : Shv(C ) → Pshv(C ) be the inclusion (or forgetful) functor.

Theorem 1.10. The functor i has a left adjoint, i.e., there is a functor #: Pshv(C ) → Shv(C ) such that for all F ∈ Pshv(C ) and G ∈ Shv(C ) we have

# HomShv(C )(F ,G) = HomPshv(C )(F, iG).

For F ∈ Pshv(C ), we call F # the sheaf associated to the presheaf F or the sheafification of F . It satisfies the following universal property: for any morphism ϕ from a presheaf F to a sheaf G there is a unique map ψ : F # → G such that

F # > # ψ

ϕ F / G commutes. This is precisely the adjoint relationship described above.

2 Sheaf Cohomology

Outline • Let C be a site. Then the category Shv(C ) of abelian sheaves has enough injectives so that any sheaf has an injective .

• The global sections functor ΓU : Shv(C ) → Ab is left-exact.

i • The failure of exactness of ΓU allows us to define the derived functors R ΓU : Shv(C ) → Ab.

i i • We define cohomology to be given by H (U, F ) = R ΓU (F ).

To establish a cohomology theory, we need to know that our category of interest has enough injective objects. These objects are acyclic (i.e. have no cohomology) which allows us to “replace” the object we are studying by an injective resolution (a complex of injective objects)

0 / F / 0 / 0 / ···

   0 / I0 / I1 / I2 / ··· which is quasi-isomorphic (induces an isomorphism on homology) to the complex concentrated in one degree (our original object of interest).

Definition 2.1. An object I of C is injective if I 7→ Hom(A, I) is exact for all objects A of C .

Definition 2.2. An abelian category C is said to have enough injectives if for all objects A of C , there is a of A into an injective object.

6 2.1 Categorical Notions Definition 2.3. A category C is additive if the following properties hold:

1. for all objects A, B, the set HomC (A, B) is an with bilinear composition 2. finite products and sums exist

3. C has a zero object (both initial and final) Definition 2.4. An additive category is abelian if: Ab1. each morphism has a and cokernel Ab2. for each morphism u, the canonical morphismu ¯ : coIm u → Im u is an isomorphism We also include a few other properties: L Ab3. for each family {Ai}i∈I , I Ai exists (notice the index set may be arbitrary)

Ab5. Ab3 holds and for any increasingly filtered family of of Ai of A and for each system of morphisms ui : Ai → B into a fixed object B such that ui is induced by uj if Ai ⊂ Aj, there S exists a unique u : Ai → B inducing the ui. (Admitting Generators) An abelian category C admits generators if there is a family of objects {Zi}i∈I (the family of generators) such that for all objects A of C and all subobjects B of A with B 6= A, there is an index i and a morphism Zi → A which does not factor through B,→ A.

Proposition 2.5. Let C be an abelian category satisfying Ab3. Let {Ui} be objects of C and L U := Ui. Then the following are equivalent:

1. {Ui} are generators of C 2. U is a generator of C L ∼ 3. for each object A of C , there is an exact sequence Ui → A → 0 (so that A = quotient of L Ui). Examples.

• Let R be a with , and let M(R) be the category of left R-modules. Then R is a generator of M(R).

• Let C be a site and U an object of C . Define the presheaf ZU : C → Ab via M V 7→ ZU (V ) = Z.

HomC (U,V )

# Then the collection {ZU }U∈C is a family of generators of Pshv(C ). The collection {ZU }U∈C is a family of generators of Shv(C ).

Theorem 2.6 (Grothendieck). Let C be an abelian category satisfying Ab5 and which admits gen- erators. Then C has enough injectives. Corollary 2.7. Let C be a site. Then Shv(C ) has enough injectives and thus every abelian sheaf has an injective resolution.

7 2.2 Cohomology Let C be a site and let F be a presheaf on C . For a fixed U ∈ Ob C we define the presheaf sections functor p ΓU : Pshv(C ) → Ab F 7→ F (U). We will use the following fact: Let C be a category, let C 0 be an abelian category and let Hom(C , C 0) be the abelian category of functors from C to C 0. Then a sequence F 0 → F → F 00 in Hom(C , C 0) is exact if and only if F 0(U) → F (U) → F 00(U) is exact in C 0 for all objects U of C (so that exactness of functors can be checked “pointwise”).

Corollary 2.8. A sequence F 0 → F → F 00 in Pshv(C ) = Hom(C op, Ab) is exact if and only if 0 00 p 0 p F (U) → F (U) → F (U) is exact in Ab for all objects U in C if and only if ΓU (F ) → ΓU (F ) → p 00 p ΓU (F ) is exact in Ab. Thus, ΓU is an . We similarly define the sections functor

ΓU : Shv(C ) → Ab.

Proposition 2.9. The functor ΓU is left-exact. Proof. To prove this, we will make use of the fact that if a functor has a right adjoint, it is right exact. If a functor has a left adjoint, it is left exact (recall the Hom-⊗ adjunction). Notice that p ΓU = ΓU ◦ i, where i : Shv(C ) → Pshv(C ) is the inclusion functor. As we saw, ΓU is exact. Since i has a right adjoint (# = sheafification), i is lef-exact. Thus, Γ is left-exact.

Definition 2.10. Let F be an abelian sheaf on a site C . Since Shv(C ) has enough injectives, we can take an injective resolution of F :

0 → F → I0 → I1 → · · ·

Applying ΓU , for U ∈ Ob(C ), we obtain a complex

0 → ΓU (I0) → ΓU (I1) → · · ·

We define ker (Γ (I ) → Γ (I )) Hi(U, F ) = U i U i+1 . Im (ΓU (Ii−1) → ΓU (Ii))

Remark 2.11. Thus, for each abelian sheaf F on X´et and for each ´etale X-scheme S, we can define q q H (S, F ), which are usually denoted H´et(S, F ) to emphasize the choice of Grothendieck topology. As mentioned above, we have a morphism of sites ε : Xzar → X´et. For each abelian sheaf F on X´et there is a (called the ):

p,q p q ∗ p+q E2 = Hzar(S, R ε (F )) ⇒ H´et (S, F ).

References

[1] , Sur quelques points d’alg`ebrehomologique, I. Tohoku Math. J., Vol. 9, No. 2, pp. 119-221, 1957. [2] Robin Hartshorne, Algebraic , Graduate Texts in Mathematics, no. 52, Springer- Verlag, New York, 1977.

8 [3] G¨unter Tamme, Introduction to Etale´ Cohomology. Springer-Verlag, 1994.

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