Lemma 15.69.10. Let $A \to B$ be a flat ring map. Let $M$ be a perfect $A$-module. Then $M \otimes _ A B$ is a perfect $B$-module.

**Proof.**
By Lemma 15.69.3 the assumption implies that $M$ has a finite resolution $F_\bullet $ by finite projective $R$-modules. As $A \to B$ is flat the complex $F_\bullet \otimes _ A B$ is a finite length resolution of $M \otimes _ A B$ by finite projective modules over $B$. Hence $M \otimes _ A B$ is perfect.
$\square$

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