Lemma 36.35.5. Let $f : X \to S$ be a morphism of schemes which is flat and locally of finite presentation. A perfect object of $D(\mathcal{O}_ X)$ is $S$-perfect. If $K, M \in D(\mathcal{O}_ X)$, then $K \otimes _{\mathcal{O}_ X}^\mathbf {L} M$ is $S$-perfect if $K$ is perfect and $M$ is $S$-perfect.

**Proof.**
First proof: reduce to the affine case using Lemma 36.35.3 and then apply More on Algebra, Lemma 15.83.3.
$\square$

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