Lemma 10.109.12 (065T)—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 10.109.12 (cite)

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Lemma 10.109.12. Let $R$ be a ring. The following are equivalent

$R$ has finite global dimension $\leq n$ ,
every finite $R$ -module has projective dimension $\leq n$ , and
every cyclic $R$ -module $R/I$ has projective dimension $\leq n$ .

Proof. It is clear that (1) $\Rightarrow$ (2) and (2) $\Rightarrow$ (3). Assume (3). Choose a set $E \subset M$ of generators of $M$ . Choose a well ordering on $E$ . For $e \in E$ denote $M_ e$ the submodule of $M$ generated by the elements $e' \in E$ with $e' \leq e$ . Then $M = \bigcup _{e \in E} M_ e$ . Note that for each $e \in E$ the quotient

$M_ e/\bigcup \nolimits _{e' < e} M_{e'}$

is either zero or generated by one element, hence has projective dimension $\leq n$ by (3). By Lemma 10.109.11 this means that $M$ has projective dimension $\leq n$ . $\square$

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