The Stacks project

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

  1. $R$ has finite global dimension $\leq n$,

  2. every finite $R$-module has projective dimension $\leq n$, and

  3. 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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