The Stacks project

Lemma 46.8.2. Let $A$ be a ring.

  1. A module is pure projective if and only if it is a direct summand of a direct sum of finitely presented $A$-modules.

  2. For any module $M$ there exists a universally exact sequence $0 \to N \to P \to M \to 0$ with $P$ pure projective.

Proof. First note that a finitely presented $A$-module is pure projective by Algebra, Theorem 10.82.3. Hence a direct summand of a direct sum of finitely presented $A$-modules is indeed pure projective. Let $M$ be any $A$-module. Write $M = \mathop{\mathrm{colim}}\nolimits _{i \in I} P_ i$ as a filtered colimit of finitely presented $A$-modules. Consider the sequence

\[ 0 \to N \to \bigoplus P_ i \to M \to 0. \]

For any finitely presented $A$-module $P$ the map $\mathop{\mathrm{Hom}}\nolimits _ A(P, \bigoplus P_ i) \to \mathop{\mathrm{Hom}}\nolimits _ A(P, M)$ is surjective, as any map $P \to M$ factors through some $P_ i$. Hence by Algebra, Theorem 10.82.3 this sequence is universally exact. This proves (2). If now $M$ is pure projective, then the sequence is split and we see that $M$ is a direct summand of $\bigoplus P_ i$. $\square$

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