Lemma 18.30.4 (0936)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 18: Modules on Sites
Section 18.30: Towards constructible modules
Lemma 18.30.4 (cite)

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Lemma 18.30.4. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $U$ be a quasi-compact object of $\mathcal{C}$ . Then the functor $\mathop{\mathrm{Hom}}\nolimits _\mathcal {O}(j_!\mathcal{O}_ U, -)$ commutes with direct sums.

Proof. This is true because $\mathop{\mathrm{Hom}}\nolimits _\mathcal {O}(j_!\mathcal{O}_ U, \mathcal{F}) = \mathcal{F}(U)$ by (18.19.2.1) and because the functor $\mathcal{F} \mapsto \mathcal{F}(U)$ commutes with direct sums by Lemma 18.30.3. $\square$

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