Proposition 10.110.1 (00O7)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 10: Commutative Algebra
Section 10.110: Regular rings and global dimension
Proposition 10.110.1 (cite)

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Proposition 10.110.1. Let $R$ be a regular local ring of dimension $d$ . Every finite $R$ -module $M$ of depth $e$ has a finite free resolution

$0 \to F_{d-e} \to \ldots \to F_0 \to M \to 0.$

In particular a regular local ring has global dimension $\leq d$ .

Proof. The first part holds in view of Lemma 10.106.6 and Lemma 10.104.9. The last part follows from this and Lemma 10.109.12. $\square$

Comments (2)

Comment #2235 by David Savitt on September 08, 2016 at 17:10

This needs a reference to [065T] for the last part of the statement.

Comment #2270 by Johan on November 03, 2016 at 17:36

Thanks. Fixed here.

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