Lemma 10.163.10. Let $\varphi : R \to S$ be a ring map. Assume

$\varphi $ is smooth,

$R$ is a regular ring.

Then $S$ is regular.

** Regularity ascends along smooth maps of rings. **

Lemma 10.163.10. Let $\varphi : R \to S$ be a ring map. Assume

$\varphi $ is smooth,

$R$ is a regular ring.

Then $S$ is regular.

**Proof.**
This follows from Lemma 10.163.5 applied for all $(R_ k)$ using Lemma 10.140.3 to see that the hypotheses are satisfied.
$\square$

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## Comments (1)

Comment #857 by Bhargav Bhatt on