Lemma 10.109.5 (0CXD)—The Stacks project

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Part 1: Preliminaries
Chapter 10: Commutative Algebra
Section 10.109: Rings of finite global dimension
Lemma 10.109.5 (cite)

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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

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.

Proof. Follows from Lemma 10.109.4 and Theorem 10.85.4. $\square$

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