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 $\leq n$.

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