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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