The Stacks project

Lemma 38.8.4. 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 a Noetherian ring,

  2. $S$ is a Noetherian ring,

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

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

  5. 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 the map $N \to N^\wedge $ of $N$ into the $I$-adic completion of $N$ is universally injective as a map of $R$-modules.

Proof. This follows from Lemma 38.8.3 because Algebra, Lemma 10.65.5 and Remark 10.65.6 guarantee that the set of associated primes of tensor products $N \otimes _ R Q$ are contained in the set of associated primes of the modules $N \otimes _ R \kappa (\mathfrak p)$. $\square$

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