Lemma 10.109.9. Let $R \to S$ be a local homomorphism of local Noetherian rings. Assume that $R \to S$ is flat and that $S$ is regular. Then $R$ is regular.

Proof. Let $\mathfrak m \subset R$ be the maximal ideal and let $\kappa = R/\mathfrak m$ be the residue field. Let $d = \dim S$. Choose any resolution $F_\bullet \to \kappa$ with each $F_ i$ a finite free $R$-module. Set $K_ d = \mathop{\mathrm{Ker}}(F_{d - 1} \to F_{d - 2})$. By flatness of $R \to S$ the complex $0 \to K_ d \otimes _ R S \to F_{d - 1} \otimes _ R S \to \ldots \to F_0 \otimes _ R S \to \kappa \otimes _ R S \to 0$ is still exact. Because the global dimension of $S$ is $d$, see Proposition 10.109.1, we see that $K_ d \otimes _ R S$ is a finite free $S$-module (see also Lemma 10.108.3). By Lemma 10.77.6 we see that $K_ d$ is a finite free $R$-module. Hence $\kappa$ has finite projective dimension and $R$ is regular by Proposition 10.109.5. $\square$

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