The Stacks project

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

  1. $E^\bullet $ is a bounded above complex of finite stably free $R/I$-modules,

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

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

  4. 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.

Proof. We apply Lemma 15.75.2 using the class $\mathcal{P}$ of all finite stably free $R$-modules. Property (1) of the lemma is immediate. Property (2) follows from Lemma 15.3.2. Property (3) follows from Nakayama's lemma (Algebra, Lemma 10.20.1). Property (4) follows from the fact that we can lift finite stably free $R/I$-modules to finite stably free $R$-modules, see Lemma 15.3.3. Part (5) holds because a pseudo-coherent complex can be represented by a bounded above complex of finite free $R$-modules. The final assertion of the lemma follows from Lemma 15.3.5. $\square$

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