The Stacks project

Lemma 18.11.4. Let $\mathcal{C}$ be a site. Let $\mathcal{O} \to \mathcal{O}'$ be an epimorphism of sheaves of rings. Let $\mathcal{G}_1, \mathcal{G}_2$ be $\mathcal{O}'$-modules. Then

\[ \mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}'}(\mathcal{G}_1, \mathcal{G}_2) = \mathop{\mathrm{Hom}}\nolimits _\mathcal {O}(\mathcal{G}_1, \mathcal{G}_2). \]

In other words, the restriction functor $\textit{Mod}(\mathcal{O}') \to \textit{Mod}(\mathcal{O})$ is fully faithful.

Proof. This is the sheaf version of Algebra, Lemma 10.107.14 and is proved in exactly the same way. $\square$

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