Lemma 38.9.1. Let $R$ be a ring. Let $I \subset R$ be an ideal. Let $R \to S$ be a ring map, and $N$ an $S$-module. Assume

$R$ is Noetherian and $I$-adically complete,

$R \to S$ is of finite type,

$N$ is a finite $S$-module,

$N$ is flat over $R$,

$N/IN$ is projective as a $R/I$-module, and

for any prime $\mathfrak q \subset S$ which is an associated prime of $N \otimes _ R \kappa (\mathfrak p)$ where $\mathfrak p = R \cap \mathfrak q$ we have $IS + \mathfrak q \not= S$.

Then $N$ is projective as an $R$-module.

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