The Stacks project

Lemma 75.13.10. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $L, K$ be objects of $D(\mathcal{O}_ X)$. If either

  1. $L$ in $D^+_\mathit{QCoh}(\mathcal{O}_ X)$ and $K$ is pseudo-coherent,

  2. $L$ in $D_\mathit{QCoh}(\mathcal{O}_ X)$ and $K$ is perfect,

then $R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (K, L)$ is in $D_\mathit{QCoh}(\mathcal{O}_ X)$.

Proof. This follows from the analogue for schemes (Derived Categories of Schemes, Lemma 36.10.8) via the criterion of Lemma 75.5.2, the criterion of Lemmas 75.13.2 and 75.13.5, and the result of Lemma 75.13.9. $\square$

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