## 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 $i$th 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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