Lemma 21.52.2 (094C)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 21: Cohomology on Sites
Section 21.52: Compact objects
Lemma 21.52.2 (cite)

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Lemma 21.52.2. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Assume every object of $\mathcal{C}$ has a covering by quasi-compact objects. Then every compact object of $D(\mathcal{O})$ is a direct summand in $D(\mathcal{O})$ of a finite complex whose terms are finite direct sums of $\mathcal{O}$ -modules of the form $j_!\mathcal{O}_ U$ where $U$ is a quasi-compact object of $\mathcal{C}$ .

Proof. Apply Lemma 21.52.1 where $S \subset \mathop{\mathrm{Ob}}\nolimits (\textit{Mod}(\mathcal{O}))$ is the set of modules of the form $j_!\mathcal{O}_ U$ with $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ quasi-compact. Assumption (1) holds by Modules on Sites, Lemma 18.28.8 and the assumption that every $U$ can be covered by quasi-compact objects. Assumption (2) follows as

$\mathop{\mathrm{Hom}}\nolimits _\mathcal {O}(j_!\mathcal{O}_ U, \mathcal{F}) = \mathcal{F}(U)$

which commutes with direct sums by Sites, Lemma 7.17.7. $\square$

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