The Stacks project

Remark 17.13.5. Let $(X, \mathcal{O}_ X)$ be a ringed space. Let $Z \subset X$ be a closed subset. For an $\mathcal{O}_ X$-module $\mathcal{F}$ we can consider the submodule of sections with support in $Z$, denoted $\mathcal{H}_ Z(\mathcal{F})$, defined by the rule

\[ \mathcal{H}_ Z(\mathcal{F})(U) = \{ s \in \mathcal{F}(U) \mid \text{Supp}(s) \subset U \cap Z\} \]

Observe that $\mathcal{H}_ Z(\mathcal{F})(U)$ is a module over $\mathcal{O}_ X(U)$, i.e., $\mathcal{H}_ Z(\mathcal{F})$ is an $\mathcal{O}_ X$-module. By construction $\mathcal{H}_ Z(\mathcal{F})$ is the largest $\mathcal{O}_ X$-submodule of $\mathcal{F}$ whose support is contained in $Z$. Applying Lemma 17.13.4 to the morphism of ringed spaces $(Z, \mathcal{O}_ X|_ Z) \to (X, \mathcal{O}_ X)$ we may (and we do) view $\mathcal{H}_ Z(\mathcal{F})$ as an $\mathcal{O}_ X|_ Z$-module on $Z$. Thus we obtain a functor

\[ \textit{Mod}(\mathcal{O}_ X) \longrightarrow \textit{Mod}(\mathcal{O}_ X|_ Z), \quad \mathcal{F} \longmapsto \mathcal{H}_ Z(\mathcal{F}) \text{ viewed as an }\mathcal{O}_ X|_ Z\text{-module on }Z \]

This functor is left exact, but in general not exact. All of the statements made above follow directly from Lemma 17.5.2. Clearly the construction is compatible with the construction in Remark 17.6.2.

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