Lemma 15.75.5. 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 finite stably free $R/I$-modules,

$K \otimes _ R^\mathbf {L} R/I$ is represented by $E^\bullet $ in $D(R/I)$,

$K^\bullet $ is pseudo-coherent, and

every element of $1 + I$ is invertible.

Then there exists a bounded above complex $P^\bullet $ of finite stably free $R$-modules representing $K$ in $D(R)$ such that $P^\bullet \otimes _ R R/I$ is isomorphic to $E^\bullet $. Moreover, if $E^ i$ is free, then $P^ i$ is free.

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