20.7 Locality of cohomology
The following lemma says there is no ambiguity in defining the cohomology of a sheaf \mathcal{F} over an open.
Lemma 20.7.1. Let X be a ringed space. Let U \subset X be an open subspace.
If \mathcal{I} is an injective \mathcal{O}_ X-module then \mathcal{I}|_ U is an injective \mathcal{O}_ U-module.
For any sheaf of \mathcal{O}_ X-modules \mathcal{F} we have H^ p(U, \mathcal{F}) = H^ p(U, \mathcal{F}|_ U).
Proof.
Denote j : U \to X the open immersion. Recall that the functor j^{-1} of restriction to U is a right adjoint to the functor j_! of extension by 0, see Sheaves, Lemma 6.31.8. Moreover, j_! is exact. Hence (1) follows from Homology, Lemma 12.29.1.
By definition H^ p(U, \mathcal{F}) = H^ p(\Gamma (U, \mathcal{I}^\bullet )) where \mathcal{F} \to \mathcal{I}^\bullet is an injective resolution in \textit{Mod}(\mathcal{O}_ X). By the above we see that \mathcal{F}|_ U \to \mathcal{I}^\bullet |_ U is an injective resolution in \textit{Mod}(\mathcal{O}_ U). Hence H^ p(U, \mathcal{F}|_ U) is equal to H^ p(\Gamma (U, \mathcal{I}^\bullet |_ U)). Of course \Gamma (U, \mathcal{F}) = \Gamma (U, \mathcal{F}|_ U) for any sheaf \mathcal{F} on X. Hence the equality in (2).
\square
Let X be a ringed space. Let \mathcal{F} be a sheaf of \mathcal{O}_ X-modules. Let U \subset V \subset X be open subsets. Then there is a canonical restriction mapping
20.7.1.1
\begin{equation} \label{cohomology-equation-restriction-mapping} H^ n(V, \mathcal{F}) \longrightarrow H^ n(U, \mathcal{F}), \quad \xi \longmapsto \xi |_ U \end{equation}
functorial in \mathcal{F}. Namely, choose any injective resolution \mathcal{F} \to \mathcal{I}^\bullet . The restriction mappings of the sheaves \mathcal{I}^ p give a morphism of complexes
\Gamma (V, \mathcal{I}^\bullet ) \longrightarrow \Gamma (U, \mathcal{I}^\bullet )
The LHS is a complex representing R\Gamma (V, \mathcal{F}) and the RHS is a complex representing R\Gamma (U, \mathcal{F}). We get the map on cohomology groups by applying the functor H^ n. As indicated we will use the notation \xi \mapsto \xi |_ U to denote this map. Thus the rule U \mapsto H^ n(U, \mathcal{F}) is a presheaf of \mathcal{O}_ X-modules. This presheaf is customarily denoted \underline{H}^ n(\mathcal{F}). We will give another interpretation of this presheaf in Lemma 20.11.4.
Lemma 20.7.2. Let X be a ringed space. Let \mathcal{F} be a sheaf of \mathcal{O}_ X-modules. Let U \subset X be an open subspace. Let n > 0 and let \xi \in H^ n(U, \mathcal{F}). Then there exists an open covering U = \bigcup _{i\in I} U_ i such that \xi |_{U_ i} = 0 for all i \in I.
Proof.
Let \mathcal{F} \to \mathcal{I}^\bullet be an injective resolution. Then
H^ n(U, \mathcal{F}) = \frac{\mathop{\mathrm{Ker}}(\mathcal{I}^ n(U) \to \mathcal{I}^{n + 1}(U))}{\mathop{\mathrm{Im}}(\mathcal{I}^{n - 1}(U) \to \mathcal{I}^ n(U))}.
Pick an element \tilde\xi \in \mathcal{I}^ n(U) representing the cohomology class in the presentation above. Since \mathcal{I}^\bullet is an injective resolution of \mathcal{F} and n > 0 we see that the complex \mathcal{I}^\bullet is exact in degree n. Hence \mathop{\mathrm{Im}}(\mathcal{I}^{n - 1} \to \mathcal{I}^ n) = \mathop{\mathrm{Ker}}(\mathcal{I}^ n \to \mathcal{I}^{n + 1}) as sheaves. Since \tilde\xi is a section of the kernel sheaf over U we conclude there exists an open covering U = \bigcup _{i \in I} U_ i such that \tilde\xi |_{U_ i} is the image under d of a section \xi _ i \in \mathcal{I}^{n - 1}(U_ i). By our definition of the restriction \xi |_{U_ i} as corresponding to the class of \tilde\xi |_{U_ i} we conclude.
\square
Lemma 20.7.3. Let f : X \to Y be a morphism of ringed spaces. Let \mathcal{F} be a \mathcal{O}_ X-module. The sheaves R^ if_*\mathcal{F} are the sheaves associated to the presheaves
V \longmapsto H^ i(f^{-1}(V), \mathcal{F})
with restriction mappings as in Equation (20.7.1.1). There is a similar statement for R^ if_* applied to a bounded below complex \mathcal{F}^\bullet .
Proof.
Let \mathcal{F} \to \mathcal{I}^\bullet be an injective resolution. Then R^ if_*\mathcal{F} is by definition the ith cohomology sheaf of the complex
f_*\mathcal{I}^0 \to f_*\mathcal{I}^1 \to f_*\mathcal{I}^2 \to \ldots
By definition of the abelian category structure on \mathcal{O}_ Y-modules this cohomology sheaf is the sheaf associated to the presheaf
V \longmapsto \frac{\mathop{\mathrm{Ker}}(f_*\mathcal{I}^ i(V) \to f_*\mathcal{I}^{i + 1}(V))}{\mathop{\mathrm{Im}}(f_*\mathcal{I}^{i - 1}(V) \to f_*\mathcal{I}^ i(V))}
and this is obviously equal to
\frac{\mathop{\mathrm{Ker}}(\mathcal{I}^ i(f^{-1}(V)) \to \mathcal{I}^{i + 1}(f^{-1}(V)))}{\mathop{\mathrm{Im}}(\mathcal{I}^{i - 1}(f^{-1}(V)) \to \mathcal{I}^ i(f^{-1}(V)))}
which is equal to H^ i(f^{-1}(V), \mathcal{F}) and we win.
\square
Lemma 20.7.4. Let f : X \to Y be a morphism of ringed spaces. Let \mathcal{F} be an \mathcal{O}_ X-module. Let V \subset Y be an open subspace. Denote g : f^{-1}(V) \to V the restriction of f. Then we have
R^ pg_*(\mathcal{F}|_{f^{-1}(V)}) = (R^ pf_*\mathcal{F})|_ V
There is a similar statement for the derived image Rf_*\mathcal{F}^\bullet where \mathcal{F}^\bullet is a bounded below complex of \mathcal{O}_ X-modules.
Proof.
First proof. Apply Lemmas 20.7.3 and 20.7.1 to see the displayed equality. Second proof. Choose an injective resolution \mathcal{F} \to \mathcal{I}^\bullet and use that \mathcal{F}|_{f^{-1}(V)} \to \mathcal{I}^\bullet |_{f^{-1}(V)} is an injective resolution also.
\square
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