Lemma 15.76.3. Let $R$ be a ring. Let $I \subset R$ be an ideal. Let $E^\bullet $ be a complex of $R/I$-modules. Let $K$ be an object of $D(R)$. Assume that
$E^\bullet $ is a bounded above complex of projective $R/I$-modules,
$K \otimes _ R^\mathbf {L} R/I$ is represented by $E^\bullet $ in $D(R/I)$, and
$I$ is a nilpotent ideal.
Then there exists a bounded above complex $P^\bullet $ of projective $R$-modules representing $K$ in $D(R)$ such that $P^\bullet \otimes _ R R/I$ is isomorphic to $E^\bullet $.
Proof.
We apply Lemma 15.76.2 using the class $\mathcal{P}$ of all projective $R$-modules. Properties (1) and (2) of the lemma are immediate. Property (3) follows from Nakayama's lemma (Algebra, Lemma 10.20.1). Property (4) follows from the fact that we can lift projective $R/I$-modules to projective $R$-modules, see Algebra, Lemma 10.77.5. To see that (5) holds it suffices to show that $K$ is in $D^{-}(R)$. Since we are given that $K \otimes _ R^\mathbf {L} R/I$ is in $D^{-}(R/I)$ because $E^\bullet $ is bounded above, this follows from Lemma 15.67.20.
$\square$
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