Lemma 15.41.4 (07QI): Composition of regular maps

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Table of contents
Part 1: Preliminaries
Chapter 15: More on Algebra
Section 15.41: Regular ring maps
Lemma 15.41.4: Composition of regular maps (cite)

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Lemma 15.41.4 (Composition of regular maps). Let $A \to B$ and $B \to C$ be regular ring maps. If the fibre rings of $A \to C$ are Noetherian, then $A \to C$ is regular.

Proof. Let $\mathfrak p \subset A$ be a prime. Let $\kappa (\mathfrak p) \subset k$ be a finite purely inseparable extension. We have to show that $C \otimes _ A k$ is regular. By Lemma 15.41.3 we may assume that $A = k$ and we reduce to proving that $C$ is regular. The assumption is that $B$ is regular and that $B \to C$ is flat with regular fibres. Then $C$ is regular by Algebra, Lemma 10.112.8. Some details omitted. $\square$

Comments (2)

Comment #7860 by Rankeya Datta on October 31, 2022 at 18:44

It's strange to say let $A \to B \to C$ be regular maps, because that makes it seem like the composition is also regular, which is what one wants to show. Perhaps it is better to say let $A \to B$ and $B \to C$ be regular maps.

Comment #8078 by Stacks Project on January 23, 2023 at 14:52

Thanks and fixed here.

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