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Proof. Flatness is preserved under any base change, see Algebra, Lemma 10.39.7. Consider a prime $\mathfrak p' \subset R'$ lying over $\mathfrak p \subset R$. The residue field extension $\kappa (\mathfrak p')/\kappa (\mathfrak p)$ is finitely generated as $R'$ is of finite type over $R$. Hence the fibre ring

\[ (\Lambda \otimes _ R R') \otimes _{R'} \kappa (\mathfrak p') = \Lambda \otimes _ R \kappa (\mathfrak p) \otimes _{\kappa (\mathfrak p)} \kappa (\mathfrak p') \]

is Noetherian by Algebra, Lemma 10.31.8 and the assumption on the fibre rings of $R \to \Lambda $. Geometric regularity of the fibres is preserved by Algebra, Lemma 10.166.1. $\square$

Comments (2)

Comment #8308 by Rankeya on

Can this statement be changed to 'essentially' of finite type ring map instead of finite type? Obviously, any localization of a regular ring map remains regular, but this would make for an easy citation (old Matsumura book also states this for finite type instead of essentially of finite type base change.

Comment #8932 by on

Why not just quote this lemma and Lemma 15.41.2 if you need base change by a ring map which is essentially of finite type.

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