The Stacks project

Lemma 10.110.2. Let $R$ be a Noetherian ring. Then $R$ has finite global dimension if and only if there exists an integer $n$ such that for all maximal ideals $\mathfrak m$ of $R$ the ring $R_{\mathfrak m}$ has global dimension $\leq n$.

Proof. We saw, Lemma 10.109.13 that if $R$ has finite global dimension $n$, then all the localizations $R_{\mathfrak m}$ have finite global dimension at most $n$. Conversely, suppose that all the $R_{\mathfrak m}$ have global dimension $\leq n$. Let $M$ be a finite $R$-module. Let $0 \to K_ n \to F_{n-1} \to \ldots \to F_0 \to M \to 0$ be a resolution with $F_ i$ finite free. Then $K_ n$ is a finite $R$-module. According to Lemma 10.109.3 and the assumption all the modules $K_ n \otimes _ R R_{\mathfrak m}$ are projective. Hence by Lemma 10.78.2 the module $K_ n$ is finite projective. $\square$

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