sheaf 1

The main reference: Hartshorne.

A topological space XX 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:UV},if UVX;,otherwise. \operatorname{Hom}_{X}(U,V)=\begin{cases} \bigl\{i_{UV}:U\hookrightarrow V\bigr\},& \text{if }U\subset V\subset X\text{;}\\ \varnothing,&\text{otherwise}. \end{cases}

Definition 1: A presheaf is a functor F:XAG\mathcal{F}:X^\circ\to \textit{AG} , where XX^\circ is the dual category of a topological space XX and AG\textit{AG} is the category of Abelian groups. In other words, F:XAG\mathcal{F}:X\to \textit{AG} is a contravariant functor. The elements sF(U)s\in \mathcal{F}(U) are called the sections on UU . Denote that ρUV:=F(iUV):F(V)F(U)\rho_{UV}:=\mathcal{F}(i_{UV}):\mathcal{F}(V)\to\mathcal{F}(U) and ρUV(s)=sV\rho_{UV}(s)=s|_V for sF(U)s \in \mathcal{F}(U) .

The morphisms φ\varphi between two presheaf F\mathcal{F} and G\mathcal{G} is just the morphisms between two functors, that is, there exist a family of morphisms φ(U):F(U)G(U)\varphi(U):\mathcal{F}(U)\to \mathcal{G}(U) such that the following diagram is commutative.

Definition 2: If F\mathcal{F} is a presheaf on XX , and if pp is a point on XX , we define the stalk FpAG\mathcal{F}_p\in \textit{AG} of F\mathcal{F} at pp to be the direct limit of the group F(U)\mathcal{F}(U) for all UXU \subset X containing pp , via the restriction maps ρ\rho . We can directly construct Fp=limundefinedUpF(U)\mathcal{F}_p=\underrightarrow{\lim}_{U\ni p}\mathcal{F}(U) that Fp={U,s:pUX,sF(U)}/, \mathcal{F}_p=\bigl\{\langle U,s\rangle: p\in U\subset X, s\in \mathcal{F}(U)\bigr\}/\sim, where \sim is defined as follows: Suppose sF(U)s\in\mathcal{F}(U) and tF(V)t\in\mathcal{F}(V) , if there exists a WUVW\subset U\cap V\neq \varnothing such that sW=tWs|_W=t|_W , then U,s=V,t\langle U,s\rangle=\langle V,t\rangle or sts\sim t . Fp\mathcal{F}_p is indeed a group since the addition can be defined by U,s+V,t=W,sW+tW\langle U,s\rangle+\langle V,t\rangle=\langle W,s|_W+t|_W\rangle .

Here F(U)sU,s=sp\mathcal{F}(U)\ni s\mapsto \langle U,s\rangle=s_p defines a family of morphisms ρUp:F(U)Fp\rho_{Up}:\mathcal{F}(U)\to \mathcal{F}_p for any UXU\subset X . We can then vertify the universal property of direct product that

Now, suppose φ:FG\varphi:\mathcal{F} \to \mathcal{G} is a functor. The diagram

is commutative, then the universal property of direct product

gives the existence of the morphism φp:FpGp\varphi_p:\mathcal{F}_p\to \mathcal{G}_p , and the diagram

is commutative.

Definition 3: A sheaf is a presheaf that for any open set UXU\subset X , the complex 0F(U)undefinedd0iIF(Ui)undefinedd1i,jIF(UiUj) 0\rightarrow \mathcal{F}(U) \xrightarrow{d_0}\prod_{i\in I}\mathcal{F}(U_i) \xrightarrow{d_1}\prod_{i,j\in I}\mathcal{F}(U_i\cap U_j) is exact for any open cover {Ui}\{U_i\} of UU , where d0:siIsUi,d1:iIsii,jI(siUiUjsjUiUj). \begin{array}{cccl} d_0:&s&\mapsto& \displaystyle{\prod_{i\in I}s|_{U_i}},\\ d_1:&\displaystyle{\prod_{i\in I}s_i}&\mapsto& \displaystyle{\prod_{i,j\in I}\bigl(s_i|_{U_i\cap U_j}-s_j|_{U_i\cap U_j}\bigr)}. \end{array} The definition can be rewritten as: For any open cover {Ui}\{U_i\} of any open set UXU\subset X ,

  • If iI\forall i\in I , sUi=0s|_{U_i}=0 , then s=0s=0 .

  • If i,jI\forall i,j\in I , siUiUj=sjUiUjs_i|_{U_i\cap U_j}=s_j|_{U_i\cap U_j} , then there’s a section sF(U)s\in\mathcal{F}(U) such that sUi=sis|_{U_i}=s_i .

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:FG\varphi:\mathcal{F}\to\mathcal{G} is a morphism of sheaves on a topological space XX . Then φ\varphi is a isomorphism if and only if the induced map on the stalk φp:FpGp\varphi_p:\mathcal{F}_p\to \mathcal{G}_p is an isomorphism for every pXp\in X .

Proof. p.63 on Hartshorne.

Definition 4: Let φ:FG\varphi:\mathcal{F}\to\mathcal{G} be a morphism of presheaves. We define the presheaf kernel of φ\varphi , presheaf of cokernel of φ\varphi , and presheaf image of φ\varphi to be the presheaves given by Uker(φ(U))U\mapsto \ker(\varphi(U)) , Ucoker(φ(U))U\to \operatorname{coker}(\varphi(U)) , and UIm(φ(U))U\mapsto \operatorname{Im}(\varphi(U)) respectively. Then Proposition 2: If φ:FG\varphi:\mathcal{F}\to\mathcal{G} is a morphism of sheaves, then the presheaf Uker(φ(U))U\mapsto \ker(\varphi(U)) is a sheaf.

Proof. Let {Ui}\{U_i\} be an open cover of UU , and sis_i is local section on UiU_i .

Suppose sker(φ(U))s\in \ker(\varphi(U)) and sUi=0s|_{U_i}=0 , since F\mathcal{F} is a sheaf, s=0s=0 .

Suppose siUiUj=sjUiUjs_i|_{U_i\cap U_j}=s_j|_{U_i\cap U_j} , we need to show that there exists a global section sker(φ(U))s\in \ker(\varphi(U)) such that sUi=sis|_{U_i}=s_i . Since F\mathcal{F} is a sheaf, it’s nature that there’s sF(U)s\in \mathcal{F}(U) such that sUi=sis|_{U_i}=s_i . The last thing to vetify is φ(U)(s)=0\varphi(U)(s)=0 . Restrict φ(U)(s)\varphi(U)(s) on UiU_i , then ρUUiφ(U)(s)=φ(Ui)(ρUUis)=φ(Ui)(si)=0, \rho'_{UU_i}\circ \varphi(U)(s)=\varphi(U_i)(\rho_{UU_i}s)=\varphi(U_i)(s_i)=0, so φ(U)(s)G(U)\varphi(U)(s)\in \mathcal{G}(U) vanishes locally. Since G\mathcal{G} is a sheaf, it also vanishes globally, i.e. φ(U)(s)=0\varphi(U)(s)=0 .

Thus Uker(φ(U))U\mapsto \ker(\varphi(U)) is a sheaf.

However, the presheaves coker(φ)\operatorname{coker}(\varphi) and Im(φ)\operatorname{Im}(\varphi) need not to be sheaves. Actually, the key point in the proof above is that ker\ker is compatible with the sheaf property of G\mathcal{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\mathcal{F} is the “smallest” sheaf with the same stalks as F\mathcal{F} . Because of the “smallest”, sheafification should have the universal property.

Propostion 3: Given a presheaf F\mathcal{F} , there is a sheaf F+\mathcal{F}^+ and a morphism θ\theta make the diagram

commutative for any sheaf G\mathcal{G} . F+\mathcal{F}^+ is called the sheaf associated to the preshead F\mathcal{F} or the sheafification of F\mathcal{F} .

Proof. We construct the sheaf F+\mathcal{F}^+ as follows. For any open set UU , let F+(U)\mathcal{F}^+(U) be the set of functions s:UpUFps:U\to \cup_{p\in U}\mathcal{F}_p , such that

for each pUp\in U , s(p)Fps(p)\in \mathcal{F}_p , and

for each pUp\in U , there is a neighborhood VUV\subset U of PP , and an element tF(V)t\in\mathcal{F}(V) , such that qV\forall q\in V , s(q)=tq:=t+(q)s(q)=t_q:=t^+(q) .

The addition on F+(U)\mathcal{F}^+(U) is that (s+t)(p)=s(p)+t(p)(s+t)(p)=s(p)+t(p) , so F+(U)\mathcal{F}^+(U) is indeed a group. If VUV\subset U , there’s a nature map (function restriction) iUV:F+(U)F+(V)i_{UV}:\mathcal{F}^+(U)\to \mathcal{F}^+(V) such that iUV(s)=sVi_{UV}(s)=s|_V , so F+\mathcal{F}^+ is a presheaf. Let {Ui}\{U_i\} be a open cover of UU and siF+(Ui)s_i\in \mathcal{F}^+(U_i) be local sections. If siUiUj=sjUiUjs_i|_{U_i\cap U_j}=s_j|_{U_i\cap U_j} , we can define a function s:UpUFps:U\to \cup_{p\in U}\mathcal{F}_p by setting sUi=sis|_{U_i}=s_i . If sUi=0s|_{U_i}=0 for all iIi\in I , then s=0s=0 since it is a function. Thus F+\mathcal{F}^+ is a sheaf.

For each sF(U)s\in \mathcal{F}(U) , we can associate it a section s+F+(U)s^+\in \mathcal{F}^+(U) by s+(p)=sps^+(p)=s_p , then there’s a morphism θ(U):ss+\theta(U):s\mapsto s^+ . sF(U)\forall s\in \mathcal{F}(U) , since iUV(θ(U)(s))=iUV(s+)=s+V i_{UV}\bigl(\theta(U)(s)\bigr)=i_{UV}(s^+)=s^+|_{V} and θ(V)(ρUV(s))=θ(V)(sV)=s+V. \theta(V)\bigl(\rho_{UV}(s)\bigr)=\theta(V)(s|_V)=s^+|_{V}. θ\theta is a morphism.

Let sˉF+(U)\bar{s}\in \mathcal{F}^+(U) , because of the construction of F+\mathcal{F}^+ , we can find an open cover {Ui}\{U_i\} of UU such that sˉUi=sˉi+\bar{s}|_{U_i}=\bar{s}^+_i , where sˉiF(Ui)\bar{s}_i\in \mathcal{F}(U_i) . Firstly define ψ(Ui):sˉUiφ(Ui)(sˉi)\psi(U_i):\bar{s}|_{U_i}\mapsto \varphi(U_i)(\bar{s}_i) , and we can use the sheaf condition of G\mathcal{G} to get a global section ss' on UU such that sUi=φ(Ui)(sˉi)s'|_{U_i}=\varphi(U_i)(\bar{s}_i) . Then, define ψ(U):sˉs\psi(U):\bar{s}\mapsto s' , and ψ\psi will become the morphism ψ:F+G\psi:\mathcal{F}^+\to \mathcal{G} . Finally, because of the construction of ψ\psi , for any s+F+(U)s^+\in \mathcal{F}^+(U) , ψ(U)(s+)=φ(U)(s)\psi(U)(s^+)=\varphi(U)(s) , so that ψ(U)(θ(s))=ψ(U)(s+)=φ(U)(s) \psi(U)(\theta(s))=\psi(U)(s^+)=\varphi(U)(s) makes the diagram commutative.

Proposition 4: FpFp+\mathcal{F}_p\cong \mathcal{F}^+_p , so if F\mathcal{F} is a sheaf, then FF+\mathcal{F}\cong \mathcal{F}^+ .

Proof. There's a morphism θp:FpFp+\theta_p:\mathcal{F}_p\to \mathcal{F}^+_p , such that θp(U,s)=U,s+\theta_p(\langle U,s\rangle)=\langle U,s^+\rangle .

It is injective. If θp(U,s)=V,0\theta_p(\langle U,s\rangle)=\langle V,0\rangle , then s+V=0s^+|_V=0 , and sp=s+(p)=0s_p=s^+(p)=0 .

It is surjective. U,sˉFp+\forall \langle U,\bar{s}\rangle\in \mathcal{F}^+_p , there exists an open subset VV and tF(V)t\in \mathcal{F}(V) such that U,sˉ=V,t+\langle U,\bar{s}\rangle=\langle V,t^+\rangle , then θp(V,t)=U,sˉ\theta_p(\langle V,t\rangle)=\langle U,\bar{s}\rangle .

Definiton 5: If φ:FG\varphi:\mathcal{F}\to\mathcal{G} is a morphism of sheaves, we define the kernel(repsectively cokernel, image) of φ\varphi , denoted kerφ\ker \varphi (repsectively cokerφ\operatorname{coker} \varphi , Imφ\operatorname{Im} \varphi ), to be the sheaf associated to the presheaf of kernel(respectively, coker, image) of φ\varphi .

Definiton 6: We say that a morphism of sheaves φ:FG\varphi:\mathcal{F}\to\mathcal{G} is injective(respectively, surjective) if kerφ=0\ker\varphi=0 (respectively, ImφG\operatorname{Im}\varphi\cong\mathcal{G} ).


Propositon 5: For any morphism of sheaves φ:FG\varphi:\mathcal{F}\to\mathcal{G} , (kerφ)p=ker(φp)(\ker \varphi)_p=\ker(\varphi_p) and (Imφ)p=Im(φp)(\operatorname{Im} \varphi)_p=\operatorname{Im}(\varphi_p) for each pp .

Proof. We will prove it in the following steps:
  • ker(φp)(kerφ)p\ker(\varphi_p)\subset (\ker \varphi)_p Suppose sp=U,sker(φp)s_p=\langle U,s\rangle\in \ker(\varphi_p) , then φpU,s=U,φ(U)s=0\varphi_p\langle U,s\rangle=\langle U,\varphi(U)s\rangle=0 , so there exists WUW\subset U such that (φ(U)s)W=φ(W)(sW)=0,(\varphi(U)s)|_W=\varphi(W)(s|_W)=0, thus sWker(φ(W))s|_W\in \ker(\varphi(W)) and sp=W,sW(kerφ)ps_p=\langle W,s|_W\rangle\in (\ker \varphi)_p .

  • (kerφ)pker(φp)(\ker \varphi)_p\subset\ker(\varphi_p) Suppose U,s(kerφ)p\langle U,s\rangle\in (\ker \varphi)_p , then φpU,s=U,φ(U)s=U,0=0Gp\varphi_p\langle U,s\rangle=\langle U,\varphi(U)s\rangle=\langle U,0\rangle=0\in \mathcal{G}_p , thus U,sker(φp)\langle U,s\rangle\in \ker(\varphi_p) .

  • Im(φp)(Imφ)p\operatorname{Im}(\varphi_p)\subset (\operatorname{Im} \varphi)_p Suppose t=φpspIm(φp)t=\varphi_ps_p\in\operatorname{Im} (\varphi_p) and sp=U,ss_p=\langle U,s\rangle , then t=φpsp=φpU,s=U,φ(U)s(Imφ)pt=\varphi_ps_p=\varphi_p\langle U,s\rangle=\langle U,\varphi(U)s\rangle\in (\operatorname{Im} \varphi)_p .

  • (Imφ)pIm(φp)(\operatorname{Im} \varphi)_p\subset \operatorname{Im}(\varphi_p) Suppose t=U,φ(U)s(Imφ)pt=\langle U,\varphi(U)s\rangle\in (\operatorname{Im} \varphi)_p , then t=ρUpφ(U)s=φpspIm(φp)t=\rho'_{Up}\circ\varphi(U)s=\varphi_p s_p \in \operatorname{Im}(\varphi_p) .

That's all.

Corollary 1: For any morphism of sheaves φ:FG\varphi:\mathcal{F}\to\mathcal{G} , it is injective(respectively, surjective) if and only if φp\varphi_p is injective(respectively, surjective) for all pp .

Proof. According to Proposition 1 and Proposition 5, kerφ=0\ker \varphi=0 if and only if (kerφ)p=ker(φp)=0(\ker \varphi)_p=\ker(\varphi_p)=0 . Similarly, ImφG\operatorname{Im} \varphi\cong \mathcal{G} if and only if (Imφ)p=Im(φp)Gg(\operatorname{Im} \varphi)_p=\operatorname{Im}(\varphi_p)\cong \mathcal{G}_g .

Corollary 2: A morphism of sheaves is an isomorphism if and only if it is injective and surjective.

Proof. A morphism of sheaves φ\varphi is an isomorphism if and only if φp\varphi_p is an isomorphism for all pp . As a morphism of groups, φp\varphi_p is an isomorphism for all pp if and only if it is injective and surjective for all pp , and according to Corollay 1, if and only if φ\varphi is injective and surjective.

Definition 7: Let f:XYf:X\to Y be a continuous map of topological spoaces. For any sheaf F\mathcal{F} on XX , we define the direct image sheaf fFf_*\mathcal{F} on YY by (fF)(U)=F(f1(U))(f_*\mathcal{F})(U)=\mathcal{F}(f^{-1}(U)) for any open set UYU\subset Y . For any sheaf G\mathcal{G} on YY , we define the inverse image sheaf f1Gf^{-1}\mathcal{G} on XX to be the sheaf associated to the presheaf UlimundefinedVf(U)G(V)U\mapsto \underrightarrow{\lim}_{V \supset f(U)}\mathcal{G}(V) , where UU is any open set in XX , and the limit is taken over all open sets VYV\subset Y containing f(U)f(U) .

Buwai Lee

Buwai Lee