A topological space X can be seen as a category if we take open sets as its objects and inclusion maps as its morphisms, i.e. HomX(U,V)={{iUV:U↪V},∅,if U⊂V⊂X;otherwise.
Definition 1: A presheaf is a functor F:X∘→AG , where X∘ is the dual category of a topological space X and AG is the category of Abelian groups. In other words, F:X→AG is a contravariant functor. The elements s∈F(U) are called the sections on U . Denote that ρUV:=F(iUV):F(V)→F(U) and ρUV(s)=s∣V for s∈F(U) .
The morphisms φ between two presheaf F and G is just the morphisms between two functors, that is, there exist a family of morphisms φ(U):F(U)→G(U) such that the following diagram is commutative.
Definition 2: If F is a presheaf on X , and if p is a point on X , we define the stalk Fp∈AG of F at p to be the direct limit of the group F(U) for all U⊂X containing p , via the restriction maps ρ .
We can directly construct Fp=limU∋pF(U) that Fp={⟨U,s⟩:p∈U⊂X,s∈F(U)}/∼, where ∼ is defined as follows: Suppose s∈F(U) and t∈F(V) , if there exists a W⊂U∩V=∅ such that s∣W=t∣W , then ⟨U,s⟩=⟨V,t⟩ or s∼t . Fp is indeed a group since the addition can be defined by ⟨U,s⟩+⟨V,t⟩=⟨W,s∣W+t∣W⟩ .
Here F(U)∋s↦⟨U,s⟩=sp defines a family of morphisms ρUp:F(U)→Fp for any U⊂X . We can then vertify the universal property of direct product that
Now, suppose φ:F→G is a functor. The diagram
is commutative, then the universal property of direct product
gives the existence of the morphism φp:Fp→Gp , and the diagram
is commutative.
Definition 3: A sheaf is a presheaf that for any open set U⊂X , the complex 0→F(U)d0i∈I∏F(Ui)d1i,j∈I∏F(Ui∩Uj) is exact for any open cover {Ui} of U , where d0:d1:si∈I∏si↦↦i∈I∏s∣Ui,i,j∈I∏(si∣Ui∩Uj−sj∣Ui∩Uj).
The definition can be rewritten as: For any open cover {Ui} of any open set U⊂X ,
If ∀i∈I , s∣Ui=0 , then s=0 .
If ∀i,j∈I , si∣Ui∩Uj=sj∣Ui∩Uj , then there’s a section s∈F(U) such that s∣Ui=si .
It is not so difficult to vertify that this definition is equivalent to the old one.
The following proposition (which would be false for presheves) illustrates the local nature of a sheaf.
Propostion 1: Suppose φ:F→G is a morphism of sheaves on a topological space X . Then φ is a isomorphism if and only if the induced map on the stalk φp:Fp→Gp is an isomorphism for every p∈X .
Proof.
p.63 on Hartshorne.
Definition 4: Let φ:F→G be a morphism of presheaves. We define the presheaf kernel of φ , presheaf of cokernel of φ , and presheaf image of φ to be the presheaves given by U↦ker(φ(U)) , U→coker(φ(U)) , and U↦Im(φ(U)) respectively.
Then
Proposition 2:
If φ:F→G is a morphism of sheaves, then the presheaf U↦ker(φ(U)) is a sheaf.
Proof.
Let {Ui} be an open cover of U , and si is local section on Ui .
Suppose s∈ker(φ(U)) and s∣Ui=0 , since F is a sheaf, s=0 .
Suppose si∣Ui∩Uj=sj∣Ui∩Uj , we need to show that there exists a global section s∈ker(φ(U)) such that s∣Ui=si . Since F is a sheaf, it’s nature that there’s s∈F(U) such that s∣Ui=si . The last thing to vetify is φ(U)(s)=0 . Restrict φ(U)(s) on Ui , then ρUUi′∘φ(U)(s)=φ(Ui)(ρUUis)=φ(Ui)(si)=0, so φ(U)(s)∈G(U) vanishes locally. Since G is a sheaf, it also vanishes globally, i.e. φ(U)(s)=0 .
Thus U↦ker(φ(U)) is a sheaf.
However, the presheaves coker(φ) and Im(φ) need not to be sheaves. Actually, the key point in the proof above is that ker is compatible with the sheaf property of G . Then we come to an important notion of a sheaf associated to a presheaf, i.e. sheafification.
Roughly speaking, the sheafification of a presheaf F is the “smallest” sheaf with the same stalks as F . Because of the “smallest”, sheafification should have the universal property.
Propostion 3: Given a presheaf F , there is a sheaf F+ and a morphism θ make the diagram
commutative for any sheaf G . F+ is called the sheaf associated to the preshead F or the sheafification of F .
Proof.
We construct the sheaf F+ as follows. For any open set U , let F+(U) be the set of functions s:U→∪p∈UFp , such that
for each p∈U , s(p)∈Fp , and
for each p∈U , there is a neighborhood V⊂U of P , and an element t∈F(V) , such that ∀q∈V , s(q)=tq:=t+(q) .
The addition on F+(U) is that (s+t)(p)=s(p)+t(p) , so F+(U) is indeed a group. If V⊂U , there’s a nature map (function restriction) iUV:F+(U)→F+(V) such that iUV(s)=s∣V , so F+ is a presheaf. Let {Ui} be a open cover of U and si∈F+(Ui) be local sections. If si∣Ui∩Uj=sj∣Ui∩Uj , we can define a function s:U→∪p∈UFp by setting s∣Ui=si . If s∣Ui=0 for all i∈I , then s=0 since it is a function. Thus F+ is a sheaf.
For each s∈F(U) , we can associate it a section s+∈F+(U) by s+(p)=sp , then there’s a morphism θ(U):s↦s+ . ∀s∈F(U) , since iUV(θ(U)(s))=iUV(s+)=s+∣V and θ(V)(ρUV(s))=θ(V)(s∣V)=s+∣V.θ is a morphism.
Let sˉ∈F+(U) , because of the construction of F+ , we can find an open cover {Ui} of U such that sˉ∣Ui=sˉi+ , where sˉi∈F(Ui) . Firstly define ψ(Ui):sˉ∣Ui↦φ(Ui)(sˉi) , and we can use the sheaf condition of G to get a global section s′ on U such that s′∣Ui=φ(Ui)(sˉi) . Then, define ψ(U):sˉ↦s′ , and ψ will become the morphism ψ:F+→G . Finally, because of the construction of ψ , for any s+∈F+(U) , ψ(U)(s+)=φ(U)(s) , so that ψ(U)(θ(s))=ψ(U)(s+)=φ(U)(s) makes the diagram commutative.
Proposition 4: Fp≅Fp+ , so if F is a sheaf, then F≅F+ .
Proof.
There's a morphism θp:Fp→Fp+ , such that θp(⟨U,s⟩)=⟨U,s+⟩ .
It is injective. If θp(⟨U,s⟩)=⟨V,0⟩ , then s+∣V=0 , and sp=s+(p)=0 .
It is surjective. ∀⟨U,sˉ⟩∈Fp+ , there exists an open subset V and t∈F(V) such that ⟨U,sˉ⟩=⟨V,t+⟩ , then θp(⟨V,t⟩)=⟨U,sˉ⟩ .
Definiton 5: If φ:F→G is a morphism of sheaves, we define the kernel(repsectively cokernel, image) of φ , denoted kerφ (repsectively cokerφ , Imφ ), to be the sheaf associated to the presheaf of kernel(respectively, coker, image) of φ .
Definiton 6: We say that a morphism of sheaves φ:F→G is injective(respectively, surjective) if kerφ=0 (respectively, Imφ≅G ).
and
Propositon 5: For any morphism of sheaves φ:F→G , (kerφ)p=ker(φp) and (Imφ)p=Im(φp) for each p .
Proof.
We will prove it in the following steps:
ker(φp)⊂(kerφ)p
Suppose sp=⟨U,s⟩∈ker(φp) , then φp⟨U,s⟩=⟨U,φ(U)s⟩=0 , so there exists W⊂U such that (φ(U)s)∣W=φ(W)(s∣W)=0, thus s∣W∈ker(φ(W)) and sp=⟨W,s∣W⟩∈(kerφ)p .
(kerφ)p⊂ker(φp)
Suppose ⟨U,s⟩∈(kerφ)p , then φp⟨U,s⟩=⟨U,φ(U)s⟩=⟨U,0⟩=0∈Gp , thus ⟨U,s⟩∈ker(φp) .
Im(φp)⊂(Imφ)p
Suppose t=φpsp∈Im(φp) and sp=⟨U,s⟩ , then t=φpsp=φp⟨U,s⟩=⟨U,φ(U)s⟩∈(Imφ)p .
(Imφ)p⊂Im(φp)
Suppose t=⟨U,φ(U)s⟩∈(Imφ)p , then t=ρUp′∘φ(U)s=φpsp∈Im(φp) .
That's all.
Corollary 1: For any morphism of sheaves φ:F→G , it is injective(respectively, surjective) if and only if φp is injective(respectively, surjective) for all p .
Proof. According to Proposition 1 and Proposition 5, kerφ=0 if and only if (kerφ)p=ker(φp)=0 . Similarly, Imφ≅G if and only if (Imφ)p=Im(φp)≅Gg .
Corollary 2: A morphism of sheaves is an isomorphism if and only if it is injective and surjective.
Proof. A morphism of sheaves φ is an isomorphism if and only if φp is an isomorphism for all p . As a morphism of groups, φp is an isomorphism for all p if and only if it is injective and surjective for all p , and according to Corollay 1, if and only if φ is injective and surjective.
Definition 7: Let f:X→Y be a continuous map of topological spoaces. For any sheaf F on X , we define the direct image sheaf f∗F on Y by (f∗F)(U)=F(f−1(U)) for any open set U⊂Y . For any sheaf G on Y , we define the inverse image sheaf f−1G on X to be the sheaf associated to the presheaf U↦limV⊃f(U)G(V) , where U is any open set in X , and the limit is taken over all open sets V⊂Y containing f(U) .