Lemma 15.73.12. Let $R$ be a ring. Let $f_1, \ldots , f_ r \in R$ be elements which generate the unit ideal. Let $K^\bullet $ be a complex of $R$-modules. If for each $i$ the complex $K^\bullet \otimes _ R R_{f_ i}$ is perfect, then $K^\bullet $ is perfect.

**Proof.**
Using Lemma 15.73.2 this translates into the corresponding results for pseudo-coherent modules and modules of finite tor dimension. See Lemma 15.65.16 and Lemma 15.63.14 for those results.
$\square$

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