Regularity ascends along smooth maps of rings.

Lemma 10.162.10. Let $\varphi : R \to S$ be a ring map. Assume

1. $\varphi$ is smooth,

2. $R$ is a regular ring.

Then $S$ is regular.

Proof. This follows from Lemma 10.162.5 applied for all $(R_ k)$ using Lemma 10.140.3 to see that the hypotheses are satisfied. $\square$

Comment #857 by Bhargav Bhatt on

Suggested slogan: Regularity ascends along smooth maps of rings.

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