The Stacks project

Lemma 10.109.5. Let $R$ be a local ring. Let $M$ be an $R$-module. Let $d \geq 0$. The equivalent conditions (1) – (4) of Lemma 10.109.4 are also equivalent to

  1. there exists a resolution $0 \to P_ d \to P_{d - 1} \to \ldots \to P_0 \to M \to 0$ with $P_ i$ free.

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