Lemma 21.12.1 (03FB)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 21: Cohomology on Sites
Section 21.12: Cohomology of modules
Lemma 21.12.1 (cite)

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Lemma 21.12.1. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. An injective sheaf of modules is also injective as an object in the category $\textit{PMod}(\mathcal{O})$ .

Proof. Apply Homology, Lemma 12.29.1 to the categories $\mathcal{A} = \textit{Mod}(\mathcal{O})$ , $\mathcal{B} = \textit{PMod}(\mathcal{O})$ , the inclusion functor and sheafification. (See Modules on Sites, Section 18.11 to see that all assumptions of the lemma are satisfied.) $\square$

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