Lemma 10.82.4 (058L)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 10: Commutative Algebra
Section 10.82: Universally injective module maps
Lemma 10.82.4 (cite)

numberstags

$0 \to M_1 \to M_2 \to M_3 \to 0$

be an exact sequence of $R$ -modules. Suppose $M_3$ is of finite presentation. Then

$0 \to M_1 \to M_2 \to M_3 \to 0$

is universally exact if and only if it is split.

Proof. A split short exact sequence is always universally exact, see Example 10.82.2. Conversely, if the sequence is universally exact, then by Theorem 10.82.3 (5) applied to $P = M_3$ , the map $M_2 \to M_3$ admits a section. $\square$

Comments (1)

Comment #1398 by Fred Rohrer on April 09, 2015 at 21:23

In the proof, replace "split sequence" by "split short exact sequence". At the end of the first sentence, insert ", see Example 058J (1)".

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