Lemma 10.82.4. Let

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

is universally exact if and only if it is split.

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.
$\square$

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## Comments (1)

Comment #1398 by Fred Rohrer on

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