Definition 10.71.2. Let $R$ be a ring. Let $M$ be an $R$-module.

A (left)

*resolution*$F_\bullet \to M$ of $M$ is an exact complex\[ \ldots \to F_2 \to F_1 \to F_0 \to M \to 0 \]of $R$-modules.

A

*resolution of $M$ by free $R$-modules*is a resolution $F_\bullet \to M$ where each $F_ i$ is a free $R$-module.A

*resolution of $M$ by finite free $R$-modules*is a resolution $F_\bullet \to M$ where each $F_ i$ is a finite free $R$-module.

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