The Stacks project

Lemma 10.109.7. Let $R$ be a local Noetherian ring. Let $M$ be a finite $R$-module. Let $d \geq 0$. The equivalent conditions (1) – (4) of Lemma 10.109.4, condition (5) of Lemma 10.109.5, and condition (6) of Lemma 10.109.6 are also equivalent to

  1. there exists a resolution $0 \to F_ d \to F_{d - 1} \to \ldots \to F_0 \to M \to 0$ with $F_ i$ finite free.

Proof. This follows from Lemmas 10.109.4, 10.109.5, and 10.109.6 and because a finite projective module over a local ring is finite free, see Lemma 10.78.2. $\square$

Comments (2)

Comment #2529 by Oleksandr Kravets on

typo: should be

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  • 1 comment(s) on Section 10.109: Rings of finite global dimension

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