Lemma 15.69.12. Let $R$ be a ring. Let $a, b \in \mathbf{Z}$. Let $K^\bullet $ be a complex of $R$-modules. Let $R \to R'$ be a faithfully flat ring map. If the complex $K^\bullet \otimes _ R R'$ is perfect, then $K^\bullet $ is perfect.

**Proof.**
Using Lemma 15.69.2 this translates into the corresponding results for pseudo-coherent modules and modules of finite tor dimension. See Lemma 15.63.17 and Lemma 15.62.16 for those results.
$\square$

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