The Stacks project

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

It's strange to say let 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 and be regular maps.


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