The Stacks project

Proof. Flatness is preserved under any base change, see Algebra, Lemma 10.38.7. Consider a prime $\mathfrak p' \subset R'$ lying over $\mathfrak p \subset R$. The residue field extension $\kappa (\mathfrak p) \subset \kappa (\mathfrak p')$ 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 10.30.8 and the assumption on the fibre rings of $R \to \Lambda $. Geometric regularity of the fibres is preserved by Algebra, Lemma 10.160.1. $\square$


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