Lemma 10.109.13 (00O8)—The Stacks project

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Part 1: Preliminaries
Chapter 10: Commutative Algebra
Section 10.109: Rings of finite global dimension
Lemma 10.109.13 (cite)

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

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