Lemma 38.7.2. Assumption and notation as in Lemma 38.7.1. Assume moreover that
$S$ is local and $R \to S$ is a local homomorphism,
$S$ is essentially of finite presentation over $R$,
$N$ is finitely presented over $S$, and
$N$ 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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