The Stacks project

Lemma 15.13.1. Let $(R, I)$ be a henselian pair. Let $\overline{P}$ be a finite projective $R/I$-module. Then there exists a finite projective $R$-module $P$ such that $P/IP \cong \overline{P}$.

Proof. This follows from the fact that we can lift the finite projective $R/I$-module $\overline{P}$ to a finite projective module $P'$ over some $R'$ ├ętale over $R$ with $R/I = R'/IR'$, see Lemma 15.9.11. Then, since $(R, I)$ is a henselian pair, the ├ętale ring map $R \to R'$ has a section $\tau : R' \to R$ (Lemma 15.11.6). Setting $P = P' \otimes _{R', \tau } R$ finishes the proof. $\square$

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