Lemma 33.25.3 (056S)—The Stacks project

Processing math: 0%

Table of contents
Part 2: Schemes
Chapter 33: Varieties
Section 33.25: Schemes smooth over fields
Lemma 33.25.3 (cite)

Smooth over a field implies regular

Lemma 33.25.3. Let $X \to \mathop{\mathrm{Spec}}(k)$ be a smooth morphism where $k$ is a field. Then $X$ is a regular scheme.

Proof. (See also Lemma 33.12.6.) By Algebra, Lemma 10.140.3 every local ring $\mathcal{O}_{X, x}$ is regular. And because $X$ is locally of finite type over $k$ it is locally Noetherian. Hence $X$ is regular by Properties, Lemma 28.9.2. $\square$

Comments (1)

Comment #1049 by Lenny Taelman on September 27, 2014 at 17:07

Suggested slogan: Smooth over a field implies regular

Comments (1)

Post a comment