The Stacks project

Lemma 15.83.3. Let $R \to A$ be a flat ring map of finite presentation. A perfect object of $D(A)$ is $R$-perfect. If $K, M \in D(A)$ then $K \otimes _ A^\mathbf {L} M$ is $R$-perfect if $K$ is perfect and $M$ is $R$-perfect.

Proof. The first statement follows from the second by taking $M = A$. The second statement follows from Lemmas 15.74.2, 15.66.10, and 15.64.16. $\square$

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