Lemma 38.7.6 (05FR)—The Stacks project

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Table of contents
Part 2: Schemes
Chapter 38: More on Flatness
Section 38.7: Localization and universally injective maps
Lemma 38.7.6 (cite)

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Lemma 38.7.6. Assumption and notation as in Lemma 38.7.1. Assume moreover that $N$ is projective as an $R$ -module. 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. Pick $s \in \Sigma$ . By Lemma 38.7.5 the map $s : N \to N$ is universally injective. 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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