Lemma 38.7.2. Assumption and notation as in Lemma 38.7.1. Assume moreover that

1. $S$ is local and $R \to S$ is a local homomorphism,

2. $S$ is essentially of finite presentation over $R$,

3. $N$ is finitely presented over $S$, and

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

Then each $s \in \Sigma$ defines a universally injective $R$-module map $s : N \to N$, and the map $N \to \Sigma ^{-1}N$ is $R$-universally injective.

Proof. By Algebra, Lemma 10.128.4 the sequence $0 \to N \to N \to N/sN \to 0$ is exact and $N/sN$ is flat over $R$. This implies that $s : N \to N$ is universally injective, see Algebra, Lemma 10.39.12. The map $N \to \Sigma ^{-1}N$ is universally injective as the directed colimit of the maps $s : N \to N$. $\square$

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