The Stacks project

Lemma 15.116.7. Let $A \to B$ be an extension of discrete valuation rings with fraction fields $K \subset L$. Assume $B$ is essentially of finite type over $A$. Let $K'/K$ be an algebraic extension of fields such that the integral closure $A'$ of $A$ in $K'$ is Noetherian. Then the integral closure $B'$ of $B$ in $L' = (L \otimes _ K K')_{red}$ is Noetherian as well. Moreover, the map $\mathop{\mathrm{Spec}}(B') \to \mathop{\mathrm{Spec}}(A')$ is surjective and the corresponding residue field extensions are finitely generated field extensions.

Proof. Let $A \to C$ be a finite type ring map such that $B$ is a localization of $C$ at a prime $\mathfrak p$. Then $C' = C \otimes _ A A'$ is a finite type $A'$-algebra, in particular Noetherian. Since $A \to A'$ is integral, so is $C \to C'$. Thus $B = C_\mathfrak p \subset C'_\mathfrak p$ is integral too. It follows that the dimension of $C'_\mathfrak p$ is $1$ (Algebra, Lemma 10.112.4). Of course $C'_\mathfrak p$ is Noetherian. Let $\mathfrak q_1, \ldots , \mathfrak q_ n$ be the minimal primes of $C'_\mathfrak p$. Let $B'_ i$ be the integral closure of $B = C_\mathfrak p$, or equivalently by the above of $C'_\mathfrak p$ in the field of fractions of $C'_{\mathfrak p'}/\mathfrak q_ i$. It follows from Krull-Akizuki (Algebra, Lemma 10.119.12 applied to the finitely many localizations of $C'_\mathfrak p$ at its maximal ideals) that each $B'_ i$ is Noetherian. Moreover the residue field extensions in $C'_\mathfrak p \to B'_ i$ are finite by Algebra, Lemma 10.119.10. Finally, we observe that $B' = \prod B'_ i$ is the integral closure of $B$ in $L' = (L \otimes _ K K')_{red}$. $\square$

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