Essential finite generation of extensions of valuation rings

TL;DR

This paper proves Knaf's conjecture that certain valuation ring extensions are finitely generated, advancing understanding of local uniformization and ramification theory in valuation extensions.

Contribution

It establishes the conjecture in full generality, confirming that extensions of valuation rings are essentially finitely generated under specified conditions.

Findings

Knaf's conjecture is proven in full generality.

Extension of valuation rings are essentially finitely generated.

Advances the theory of local uniformization and ramification.

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.