Lemma 15.72.4. Let $R$ be a ring. Let $I \subset R$ be an ideal. Let $K$ be an object of $D(R)$. Assume that

$K \otimes _ R^\mathbf {L} R/I$ is perfect in $D(R/I)$, and

$I$ is a nilpotent ideal.

Then $K$ is perfect in $D(R)$.

Lemma 15.72.4. Let $R$ be a ring. Let $I \subset R$ be an ideal. Let $K$ be an object of $D(R)$. Assume that

$K \otimes _ R^\mathbf {L} R/I$ is perfect in $D(R/I)$, and

$I$ is a nilpotent ideal.

Then $K$ is perfect in $D(R)$.

**Proof.**
Choose a finite complex $\overline{P}^\bullet $ of finite projective $R/I$-modules representing $K \otimes _ R^\mathbf {L} R/I$, see Definition 15.69.1. By Lemma 15.70.3 there exists a complex $P^\bullet $ of projective $R$-modules representing $K$ such that $\overline{P}^\bullet = P^\bullet /IP^\bullet $. It follows from Nakayama's lemma (Algebra, Lemma 10.19.1) that $P^\bullet $ is a finite complex of finite projective $R$-modules.
$\square$

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