Lemma 10.109.7 (0CXF)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 10: Commutative Algebra
Section 10.109: Rings of finite global dimension
Lemma 10.109.7 (cite)

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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

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 May 02, 2017 at 03:30

typo: $P_i$ should be $F_i$

Comment #2565 by Johan on May 25, 2017 at 18:38

Thanks Oleksandr. Fixed here.

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