Lemma 38.7.2 (05DF)—The Stacks project

Loading web-font TeX/Math/Italic

Table of contents
Part 2: Schemes
Chapter 38: More on Flatness
Section 38.7: Localization and universally injective maps
Lemma 38.7.2 (cite)

numberstags

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 $s : N \to N$ . $\square$

Comments (0)

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$ ). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Comments (0)

Post a comment