Proposition 10.110.1. Let $R$ be a regular local ring of dimension $d$. Every finite $R$-module $M$ of depth $e$ has a finite free resolution

In particular a regular local ring has global dimension $\leq d$.

Proposition 10.110.1. Let $R$ be a regular local ring of dimension $d$. Every finite $R$-module $M$ of depth $e$ has a finite free resolution

\[ 0 \to F_{d-e} \to \ldots \to F_0 \to M \to 0. \]

In particular a regular local ring has global dimension $\leq d$.

**Proof.**
The first part holds in view of Lemma 10.106.6 and Lemma 10.104.9. The last part follows from this and Lemma 10.109.12.
$\square$

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## Comments (2)

Comment #2235 by David Savitt on

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