The Stacks project

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

  1. 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.

  2. 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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View Definition 46.8.5 as pdf