Lemma 10.112.8 (031E)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 10: Commutative Algebra
Section 10.112: Homomorphisms and dimension
Lemma 10.112.8 (cite)

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Lemma 10.112.8. Let $R \to S$ be a local homomorphism of local Noetherian rings. Assume

$R$ is regular,
$S/\mathfrak m_ RS$ is regular, and
$R \to S$ is flat.

Then $S$ is regular.

Proof. By Lemma 10.112.7 we have $\dim (S) = \dim (R) + \dim (S/\mathfrak m_ RS)$ . Pick generators $x_1, \ldots , x_ d \in \mathfrak m_ R$ with $d = \dim (R)$ , and pick $y_1, \ldots , y_ e \in \mathfrak m_ S$ which generate the maximal ideal of $S/\mathfrak m_ RS$ with $e = \dim (S/\mathfrak m_ RS)$ . Then we see that $x_1, \ldots , x_ d, y_1, \ldots , y_ e$ are elements which generate the maximal ideal of $S$ and $e + d = \dim (S)$ . $\square$

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