Lemma 10.110.4. Let R be a Noetherian local ring. Suppose that the residue field \kappa has finite projective dimension n over R. In this case \dim (R) \geq n.
Proof. Let F_{\bullet } be a finite resolution of \kappa by finite free R-modules (Lemma 10.109.7). By Lemma 10.102.2 we may assume all the maps in the complex F_{\bullet } have to property that \mathop{\mathrm{Im}}(F_ i \to F_{i-1}) \subset \mathfrak m F_{i-1}, because removing a trivial summand from the resolution can at worst shorten the resolution. Say F_ n \not= 0 and F_ i = 0 for i > n, so that the projective dimension of \kappa is n. By Proposition 10.102.9 we see that \text{depth}_{I(\varphi _ n)}(R) \geq n since I(\varphi _ n) cannot equal R by our choice of the complex. Thus by Lemma 10.72.3 also \dim (R) \geq n. \square
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