The Stacks project

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

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

  2. $R \to S$ is of finite type,

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

  4. $N$ is flat over $R$,

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

  6. 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.

Proof. By Lemma 38.8.4 the map $N \to N^\wedge $ is universally injective. By Lemma 38.8.2 the module $N^\wedge $ is Mittag-Leffler. By Algebra, Lemma 10.89.7 we conclude that $N$ is Mittag-Leffler. Hence $N$ is countably generated, flat and Mittag-Leffler as an $R$-module, whence projective by Algebra, Lemma 10.93.1. $\square$

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