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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