Lemma 52.6.5. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\mathcal{I} \subset \mathcal{O}$ be a sheaf of ideals. If $K \in D(\mathcal{O})$ and $L \in D_{comp}(\mathcal{O})$, then $R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {O}(K, L) \in D_{comp}(\mathcal{O})$.
Proof. Let $U$ be an object of $\mathcal{C}$ and let $f \in \mathcal{I}(U)$. Recall that
\[ \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O}_ U)}(\mathcal{O}_{U, f}, R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {O}(K, L)|_ U) = \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O}_ U)}( K|_ U \otimes _{\mathcal{O}_ U}^\mathbf {L} \mathcal{O}_{U, f}, L|_ U) \]
by Cohomology on Sites, Lemma 21.35.2. The right hand side is zero by Lemma 52.6.2 and the relationship between internal hom and actual hom, see Cohomology on Sites, Lemma 21.35.1. The same vanishing holds for all $U'/U$. Thus the object $R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ U}(\mathcal{O}_{U, f}, R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {O}(K, L)|_ U)$ of $D(\mathcal{O}_ U)$ has vanishing $0$th cohomology sheaf (by locus citatus). Similarly for the other cohomology sheaves, i.e., $R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ U}(\mathcal{O}_{U, f}, R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {O}(K, L)|_ U)$ is zero in $D(\mathcal{O}_ U)$. By Lemma 52.6.2 we conclude. $\square$
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