The Stacks project

Lemma 15.51.11. Fix $n \geq 1$. Properties (A), (B), (C), (D), and (E) hold for $P(k \to R) =$“$R$ has $(S_ n)$”.

Proof. Let $k \to R$ be a ring map where $k$ is a field and $R$ a Noetherian ring. Let $k'/k$ be a finitely generated field extension. Then the fibres of the ring map $R \to R \otimes _ k k'$ are Cohen-Macaulay by Algebra, Lemma 10.167.1. Hence we may apply Algebra, Lemma 10.163.4 to the ring map $R \to R \otimes _ k k'$ to see that if $R$ has $(S_ n)$ so does $R \otimes _ k k'$. This proves (A). Part (B) follows too: a Noetherian rings has $(S_ n)$ if and only if all of its local rings have $(S_ n)$. Part (C). This follows from Algebra, Lemma 10.163.4 as the fibres of a regular homomorphism are regular and in particular Cohen-Macaulay. Part (D). This follows from Algebra, Lemma 10.164.5. Part (E). This is immediate as the condition does not refer to the ground field. $\square$

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