The Stacks project

Lemma 76.52.5. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$ which is flat and locally of finite presentation. A perfect object of $D(\mathcal{O}_ X)$ is $Y$-perfect. If $K, M \in D(\mathcal{O}_ X)$, then $K \otimes _{\mathcal{O}_ X}^\mathbf {L} M$ is $Y$-perfect if $K$ is perfect and $M$ is $Y$-perfect.

Proof. Reduce to the case of schemes using Lemma 76.52.3 and then apply Derived Categories of Schemes, Lemma 36.35.5. $\square$

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