Definition 46.8.5. Let $A$ be a ring. Let $M$ be an $A$-module.

A

*pure projective resolution*$P_\bullet \to M$ is a universally exact sequence\[ \ldots \to P_1 \to P_0 \to M \to 0 \]with each $P_ i$ pure projective.

A

*pure injective resolution*$M \to I^\bullet $ is a universally exact sequence\[ 0 \to M \to I^0 \to I^1 \to \ldots \]with each $I^ i$ pure injective.

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