Lemma 21.52.3. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $U$ be an object of $\mathcal{C}$. Assume the functors $\mathcal{F} \mapsto H^ p(U, \mathcal{F})$ commute with direct sums. Then $\mathcal{O}$-module $j_!\mathcal{O}_ U$ is a compact object of $D^+(\mathcal{O})$ in the following sense: if $M = \bigoplus _{i \in I} M_ i$ in $D(\mathcal{O})$ is bounded below, then $\mathop{\mathrm{Hom}}\nolimits (j_{U!}\mathcal{O}_ U, M) = \bigoplus _{i \in I} \mathop{\mathrm{Hom}}\nolimits (j_{U!}\mathcal{O}_ U, M_ i)$.
Proof. Since $\mathop{\mathrm{Hom}}\nolimits (j_{U!}\mathcal{O}_ U, -)$ is the same as the functor $\mathcal{F} \mapsto \mathcal{F}(U)$ by Modules on Sites, Equation (18.19.2.1) it suffices to prove that $H^ p(U, M) = \bigoplus H^ p(U, M_ i)$. Let $\mathcal{I}_ i$, $i \in I$ be a collection of injective $\mathcal{O}$-modules. By assumption we have
\[ H^ p(U, \bigoplus \nolimits _{i \in I} \mathcal{I}_ i) = \bigoplus \nolimits _{i \in I} H^ p(U, \mathcal{I}_ i) = 0 \]
for all $p$. Since $M = \bigoplus M_ i$ is bounded below, we see that there exists an $a \in \mathbf{Z}$ such that $H^ n(M_ i) = 0$ for $n < a$. Thus we can choose complexes of injective $\mathcal{O}$-modues $\mathcal{I}_ i^\bullet $ representing $M_ i$ with $\mathcal{I}_ i^ n = 0$ for $n < a$, see Derived Categories, Lemma 13.18.3. By Injectives, Lemma 19.13.4 we see that the direct sum complex $\bigoplus \mathcal{I}_ i^\bullet $ represents $M$. By Leray acyclicity (Derived Categories, Lemma 13.16.7) we see that
\[ R\Gamma (U, M) = \Gamma (U, \bigoplus \mathcal{I}_ i^\bullet ) = \bigoplus \Gamma (U, \bigoplus \mathcal{I}_ i^\bullet ) = \bigoplus R\Gamma (U, M_ i) \]
as desired. $\square$
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