Lemma 10.166.3 (07NH)—The Stacks project

Processing math: 100%

Table of contents
Part 1: Preliminaries
Chapter 10: Commutative Algebra
Section 10.166: Geometrically regular algebras
Lemma 10.166.3 (cite)

numberstags

Geometric regularity descends through faithfully flat maps of algebras

Lemma 10.166.3. Let $k$ be a field. Let $A \to B$ be a faithfully flat $k$ -algebra map. If $B$ is geometrically regular over $k$ , so is $A$ .

Proof. Assume $B$ is geometrically regular over $k$ . Let $k'/k$ be a finite, purely inseparable extension. Then $A \otimes _ k k' \to B \otimes _ k k'$ is faithfully flat as a base change of $A \to B$ (by Lemmas 10.30.3 and 10.39.7) and $B \otimes _ k k'$ is regular by our assumption on $B$ over $k$ . Then $A \otimes _ k k'$ is regular by Lemma 10.164.4. $\square$

Comments (1)

Comment #2110 by Matthew Emerton on July 13, 2016 at 01:28

Suggested slogan: Geometric regularity descends through faithfully flat maps of algebras

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$ ). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Comments (1)

Post a comment