Lemma 75.8.3. Let $S$ be a scheme. Let $X$ be a locally Noetherian algebraic space over $S$. If $L$ is in $D^+_{\textit{Coh}}(\mathcal{O}_ X)$ and $K$ in $D^-_{\textit{Coh}}(\mathcal{O}_ X)$, then $R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (K, L)$ is in $D^+_{\textit{Coh}}(\mathcal{O}_ X)$.
Proof. We can check whether an object of $D(\mathcal{O}_ X)$ is in $D_{\textit{Coh}}(\mathcal{O}_ X)$ étale locally on $X$, see Cohomology of Spaces, Lemma 69.12.2. Hence this lemma follows from the case of schemes, see Derived Categories of Schemes, Lemma 36.11.5. $\square$
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