
Lemma 21.48.4. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $U$ be an object of $\mathcal{C}$. The $\mathcal{O}$-module $j_!\mathcal{O}_ U$ is a compact object of $D(\mathcal{O})$ if there exists an integer $d$ such that

1. $H^ p(U, \mathcal{F}) = 0$ for all $p > d$, and

2. the functors $\mathcal{F} \mapsto H^ p(U, \mathcal{F})$ commute with direct sums.

Proof. Assume (1) and (2). The first means that the functor $F = H^0(U, -)$ has finite cohomological dimension. Moreover, any direct sum of injective modules is acyclic for $F$ by (2). Since we may compute $RF$ by applying $F$ to any complex of acyclics (Derived Categories, Lemma 13.30.2). Thus, if $K_ i$ be a family of objects of $D(\mathcal{O})$, then we can choose K-injective representatives $I_ i^\bullet$ and we see that $\bigoplus K_ i$ is represented by $\bigoplus I_ i^\bullet$. Thus $H^0(U, -)$ commutes with direct sums. $\square$

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