The Stacks project

Lemma 10.82.4. Let

\[ 0 \to M_1 \to M_2 \to M_3 \to 0 \]

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 on

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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  • 2 comment(s) on Section 10.82: Universally injective module maps

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