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

1. 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.

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

3. 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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