Lemma 15.41.7. Let $A \to B \to C$ be ring maps. If $A \to C$ is regular and $B \to C$ is flat and surjective on spectra, then $A \to B$ is regular.

Proof. By Algebra, Lemma 10.39.10 we see that $A \to B$ is flat. Let $\mathfrak p \subset A$ be a prime. The ring map $B \otimes _ A \kappa (\mathfrak p) \to C \otimes _ A \kappa (\mathfrak p)$ is flat and surjective on spectra. Hence $B \otimes _ A \kappa (\mathfrak p)$ is geometrically regular by Algebra, Lemma 10.166.3. $\square$

## Comments (2)

Comment #8307 by Rankeya on

Why not just say $B \to C$ is faithfully flat?

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