The Stacks project

Lemma 17.23.3. Let $(X, \mathcal{O}_ X)$ be a ringed space, let $\mathcal{F}$ be an $\mathcal{O}_ X$-module and let $\mathcal{I} \subset \mathcal{O}_ X$ be an ideal sheaf. If $\mathcal{I} \subset \text{Ann}_{\mathcal{O}_ X}(\mathcal{F})$, then $\mathcal{F}$ has a natural $\mathcal{O}_ X/\mathcal{I}$-module structure which agrees with the usual commutative algebra construction on stalks.

Proof. Applying the universal property of the cokernel of the inclusion $\mathcal{I} \to \mathcal{O}_ X$, we obtain a commutative diagram

\[ \xymatrix{ \mathcal{O}_ X \ar[r] \ar[d] & \mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(\mathcal{F}, \mathcal{F}) \\ \mathcal{O}_ X/\mathcal{I} \ar@{-->}[ur] } \]

of $\mathcal{O}_ X$-modules. By Lemma 17.22.1 the resulting map $\mathcal{O}_ X/\mathcal{I} \to \mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(\mathcal{F}, \mathcal{F})$ corresponds to a map of $\mathcal{O}_ X$-modules

\[ \mathcal{O}_ X/\mathcal{I} \otimes _{\mathcal{O}_ X} \mathcal{F} \longrightarrow \mathcal{F} \]

which means we have an $\mathcal{O}_ X/\mathcal{I}$-module structure on $\mathcal{F}$ compatible with the given $\mathcal{O}_ X$-module structure. We omit the verification of the statement on stalks. $\square$

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