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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