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Rapid tour through sheaves

Let XX be a topological space. We have a category Opens(X)\operatorname{Opens}(X) whose objects are open subsets and Hom(U,V)={UVUVU⊈V\operatorname{Hom}(U,V) = \begin{cases} U \to V & U \subseteq V \\ \emptyset & U \not \subseteq V \end{cases}

Definition. Let C\mathcal C be a category. A pre-sheaf valued in C\mathcal C F\mathcal F on XX is a functor F:Opens(X)opC\mathcal F : \operatorname{Opens}(X)^{op} \to \mathcal C A morphisms of pre-sheaves is a natural transformation of the functors.

We can unpack this as follows:

  • For every open UU we have an object F(U)\mathcal F(U) of C\mathcal C. We can this sections of F\mathcal F over UU and will also write this as Γ(U,F)\Gamma(U,\mathcal F).
  • If UVU \subseteq V, we have a morphism rVU:F(V)F(U)r_{VU} : \mathcal F(V) \to \mathcal F(U) with rUU=1F(U)r_{UU} = 1_{\mathcal F(U)}. We usually call this map restriction from VV to UU and is also often denoted by rVU(s)=sUr_{VU}(s) = s|_U
  • If UVWU \subseteq V \subseteq W, the diagram

commutes.

For a morphism f:FGf : \mathcal F \to \mathcal G we have morphisms f(U):F(U)G(U)f(U) : \mathcal F(U) \to \mathcal G(U) with the commutative diagrams

Generally, when we just say pre-sheaf without specifying the values, we will mean sets or abelian groups.

Examples

  • Take an object CC from C\mathcal C, then we have the constant pre-sheaf C\underline{C} C(U)=CrVU=1C.\underline{C}(U) = C \\ r_{VU} = 1_C.

  • Let CxC_x be a set of objects of C\mathcal C indexed by XX. We have the pre-sheaf UxUCxU \mapsto \prod_{x \in U} C_x where rVU:xVCxxUCxr_{VU} : \prod_{x \in V} C_x \to \prod_{x \in U} C_x is the projection onto the components with xUx \in U.

  • We have the pre-sheaf of continuous functions O(U)={f:XYf continuous}\mathcal O(U) = \lbrace f : X \to Y \mid f \text{ continuous} \rbrace with target YY. If YY is a topological ring, then O\mathcal O is a pre-sheaf of rings.

  • Fix xXx \in X and CCC \in \mathcal C. Assume that C\mathcal C has a terminal object \ast. Then we have ixC(U)={CxUx∉U.i_{x \ast} \underline{C} (U) = \begin{cases} C & x \in U \\ \ast & x \not \in U. \end{cases} This works, in particular, if C\mathcal C is an abelian category.

Definition. A pre-sheaf F\mathcal F is a sheaf if given an open cover UαU_\alpha of VV and sections sαF(Uα)s_\alpha \in \mathcal F(U_\alpha) satisfying sαUαUβ=sβUαUβs_{\alpha}|_{U_\alpha \cap U_\beta} = s_{\beta}|_{U_\alpha \cap U_\beta} for all α,β\alpha, \beta, there exists a unique sF(V)s \in \mathcal F(V) with sUα=sαs|_{U_\alpha} = s_\alpha for all α\alpha.

Morphisms of sheaves are morphisms of the underlying pre-sheaves.

We have a forgetful functor Sh(X,C)PreSh(X,C)\operatorname{Sh}(X,\mathcal C) \to \operatorname{PreSh}(X,\mathcal C) which often has a left adjoint. To give a (somewhat) explicit description, we need an auxiliary notion.

Definition. Let xXx \in X and F\mathcal F a pre-sheaf on XX with values in C\mathcal C. Then the stalk of F\mathcal F at xx is Fx=colimxUF(U)\mathcal F_x = \operatorname{colim}_{x \in U} \mathcal F(U)

In general, this need not exist. For common targets, like Ab\operatorname{Ab}, this is not an issue.

Assume that C\mathcal C has small colimits and limits. Let F~(U)xUFx\widetilde{\mathcal F}(U) \subset \prod_{x \in U} \mathcal F_x where (sx)(s_x) is a section of F~\widetilde{\mathcal F} over UU if there exists a cover VαV_\alpha of UU and sections sαF(Vα)s_\alpha \in \mathcal F(V_\alpha) with (sα)y=sy(s_\alpha)_y = s_y for all yVαy \in V_\alpha. We call F~\widetilde{F} the sheafification of F\mathcal F.

Proposition. The assignment FF~\mathcal F \mapsto \widetilde{\mathcal F} is a functor PreSh(X,C)Sh(X,C)\operatorname{PreSh}(X,\mathcal C) \to \operatorname{Sh}(X,\mathcal C) which is left adjoint to the forgetful functor Sh(X,C)PreSh(X,C)\operatorname{Sh}(X,\mathcal C) \to \operatorname{PreSh}(X,\mathcal C)

Proof. (Expand to view)

Note that UxUCx U \mapsto \prod_{x \in U} C_x is a sheaf so we only need to check that the condition: "there exists a cover VαV_\alpha of UU and sections sαF(Vα)s_\alpha \in \mathcal F(V_\alpha) with (sα)y=sy (s_\alpha)_y = s_y for all yVαy \in V_\alpha" cuts out a subsheaf. We leave the reader to convince themselves of this. Next, one checks that if G\mathcal G is a sheaf then any map ϕ:FG\phi: \mathcal F \to \mathcal G admits a unique factorization

Assume that A\mathcal A is an abelian category and sheafification exists. Then we have a an abelian category structure on PreSh(X,A)\operatorname{PreSh}(X,\mathcal A) where 0FGH00 \to \mathcal F \to \mathcal G \to \mathcal H \to 0 if exact if each 0F(U)G(U)H(U)00 \to \mathcal F(U) \to \mathcal G(U) \to \mathcal H(U) \to 0 is exact.

Given a map ϕ:FG\phi : \mathcal F \to \mathcal G, then kerϕ\ker \phi is a sheaf in general but cokϕ\operatorname{cok} \phi is not.

We transfer the abelian category structure to Sh(X,A)\operatorname{Sh}(X, \mathcal A) using sheafification. Precisely, in Sh(X,A)\operatorname{Sh}(X, \mathcal A) cokϕ(U):=cokϕ(U)~\operatorname{cok} \phi (U) := \widetilde{\operatorname{cok} \phi(U)}

We lose exactness on open sets in general but still have a simple criteria for exactness in Sh(X,A)\operatorname{Sh}(X, \mathcal A).

Proposition. A sequence of sheaves FGH\mathcal F \to \mathcal G \to \mathcal H is exact if and only if for each xXx \in X the sequence of stalks FxGxHx\mathcal F_x \to \mathcal G_x \to \mathcal H_x is exact.

Proof. (Expand to view)

We note that F~x=Fx \widetilde{\mathcal F}_x = \mathcal F_x for all xXx \in X. One checks that exactness reduces to this. We leave the details to the reader.

If A\mathcal A has enough injectives then so does Sh(X,A)\operatorname{Sh}(X,\mathcal A). For each xXx \in X, we choose some injection FxI(x)\mathcal F_x \to \mathcal I(x) to get F~FxI(x)\widetilde{F} \to \prod \mathcal F_x \to \prod \mathcal I(x) which still has no kernel.

One then checks that

Lemma. The sheaf I(x)\prod \mathcal I(x) is injective in Sh(X,A)\operatorname{Sh}(X,\mathcal A).

As a consequence, we can define right derived functors. One of particular interest is global sections.

The functor Γ(X,):Sh(X,A)AFF(X)\begin{aligned} \Gamma(X,-) : \operatorname{Sh}(X,\mathcal A) & \to \mathcal A \\ \mathcal F & \mapsto \mathcal F(X) \end{aligned} is left-exact but not right exact in general.

Definition. The i-th (sheaf) cohomology of F\mathcal F is the i-th right derived functor of global sections Hi(X,F):=RiΓ(X,F)H^i(X,\mathcal F) := \mathbf{R}^i\Gamma(X,\mathcal F)

One very useful computational tool is the following.

Proposition. Assume we have an exact sequence of sheaves 0FGH00 \to \mathcal F \to \mathcal G \to \mathcal H \to 0 then we get a long exact sequence of cohomologies 0H0(X,F)H0(X,G)H0(X,H)H1(X,F)0 \to H^0(X,\mathcal F) \to H^0(X,\mathcal G) \to H^0(X,\mathcal H) \to H^1(X,\mathcal F) \to \cdots

Proof. (Expand to view)

We have seen that right derived functors are homological and we know that 0FGH0 0 \to \mathcal F \to \mathcal G \to \mathcal H \to 0 provides a triangle in D(Sh(X,A))D(\operatorname{Sh}(X,\mathcal A)).

Global sections is special case of a more general functor between categories of sheaves coming from continuous maps of topological spaces.

Definition. Given a continuous map f:XYf: X \to Y, we have the pushforward f:Sh(X,C)Sh(X,C)f_\ast : \operatorname{Sh}(X,\mathcal C) \to \operatorname{Sh}(X,\mathcal C) where fF(U):=F(f1U)f_\ast \mathcal F(U) := F(f^{-1}U)

Note that for the map f:Xf: X \to \ast to the one-point space we have fF()=Γ(X,F)f_\ast \mathcal F(\ast) = \Gamma(X,\mathcal F) So the derived pushforward Rf\mathbf{R}f_\ast is generalization of cohomology which can be heuristically viewed as “cohomology along the fibers of ff”.

Definition. Given a sheaf G\mathcal G on YY, we also have the pre-sheaf on XX Ucolimf(U)VG(V)U \mapsto \operatorname{colim}_{f(U) \subseteq V} \mathcal G(V) The inverse image sheaf under ff f1Gf^{-1}\mathcal G is the sheafification of this pre-sheaf.

Proposition. We have an adjunction f1ff^{-1} \vdash f_\ast.