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Essential finite generation of extensions of valuation rings

Rankeya Datta

Abstract

Given a generically finite local extension of valuation rings $V \subset W$, the question of whether $W$ is the localization of a finitely generated $V$-algebra is significant for approaches to the problem of local uniformization of valuations using ramification theory. Hagen Knaf proposed a characterization of when $W$ is essentially of finite type over $V$ in terms of classical invariants of the extension of associated valuations. Knaf's conjecture has been verified in important special cases by Cutkosky and Novacoski using local uniformization of Abhyankar valuations and resolution of singularities of excellent surfaces in arbitrary characteristic, and by Cutkosky for valuation rings of function fields of characteristic $0$ using embedded resolution of singularities. In this paper we prove Knaf's conjecture in full generality.

Essential finite generation of extensions of valuation rings

Abstract

Given a generically finite local extension of valuation rings , the question of whether is the localization of a finitely generated -algebra is significant for approaches to the problem of local uniformization of valuations using ramification theory. Hagen Knaf proposed a characterization of when is essentially of finite type over in terms of classical invariants of the extension of associated valuations. Knaf's conjecture has been verified in important special cases by Cutkosky and Novacoski using local uniformization of Abhyankar valuations and resolution of singularities of excellent surfaces in arbitrary characteristic, and by Cutkosky for valuation rings of function fields of characteristic using embedded resolution of singularities. In this paper we prove Knaf's conjecture in full generality.

Paper Structure

This paper contains 5 sections, 14 theorems, 41 equations.

Table of Contents

  1. Introduction
  2. Conventions and basic terminology
  3. Henselization and base change
  4. Henselian valuation rings and some ramification theory
  5. Proof of Theorem \ref{['thm:main-theorem']}

Key Result

Theorem 1.2

Let $L/K$ be a finite field extension, $\omega$ be a valuation of $L$ with valuation ring $(\mathcal{O}_\omega, \mathfrak{m}_\omega, \kappa_\omega)$ and value group $\Gamma_\omega$, and $\nu$ be the restriction of $\omega$ to $K$ with valuation ring $(\mathcal{O}_\nu, \mathfrak{m}_\nu, \kappa_\nu)$

Theorems & Definitions (33)

  • Conjecture 1.1
  • Theorem 1.2
  • Lemma 2.1
  • Lemma 3.1
  • proof : Indication of proof
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • ...and 23 more