Lemma 10.70.1. Let $R$ be a ring. Let $M$ be an $R$-module.

There exists an exact complex

\[ \ldots \to F_2 \to F_1 \to F_0 \to M \to 0. \]with $F_ i$ free $R$-modules.

If $R$ is Noetherian and $M$ finite over $R$, then we can choose the complex such that $F_ i$ is finite free. In other words, we can find an exact complex

\[ \ldots \to R^{\oplus n_2} \to R^{\oplus n_1} \to R^{\oplus n_0} \to M \to 0. \]

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