Lemma 15.73.8. Let $A \to B$ be a ring map. Assume that $B$ is perfect as an $A$-module. Let $K^\bullet $ be a perfect complex of $B$-modules. Then $K^\bullet $ is perfect as a complex of $A$-modules.

**Proof.**
Using Lemma 15.73.2 this translates into the corresponding results for pseudo-coherent modules and modules of finite tor dimension. See Lemma 15.65.12 and Lemma 15.63.11 for those results.
$\square$

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