The Stacks project

Lemma 10.109.13. Let $R$ be a ring. Let $M$ be an $R$-module. Let $S \subset R$ be a multiplicative subset.

  1. If $M$ has projective dimension $\leq n$, then $S^{-1}M$ has projective dimension $\leq n$ over $S^{-1}R$.

  2. If $R$ has finite global dimension $\leq n$, then $S^{-1}R$ has finite global dimension $\leq n$.

Proof. Let $0 \to P_ n \to P_{n - 1} \to \ldots \to P_0 \to M \to 0$ be a projective resolution. As localization is exact, see Proposition 10.9.12, and as each $S^{-1}P_ i$ is a projective $S^{-1}R$-module, see Lemma 10.94.1, we see that $0 \to S^{-1}P_ n \to \ldots \to S^{-1}P_0 \to S^{-1}M \to 0$ is a projective resolution of $S^{-1}M$. This proves (1). Let $M'$ be an $S^{-1}R$-module. Note that $M' = S^{-1}M'$. Hence we see that (2) follows from (1). $\square$

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