# Essentially finite generation of valuation rings in terms of classical invariants

## TL;DR

This paper investigates conditions under which valuation rings in finite extensions are essentially finitely generated, using classical invariants, and explores related graded algebra extensions.

## Contribution

It introduces a necessary condition based on classical invariants for finite generation of valuation rings and identifies cases where this condition is sufficient.

## Key findings

- Necessary condition for finite generation in terms of classical invariants
- Identification of particular cases where the condition is sufficient
- Equivalent, weaker condition for finite generation of graded algebra extensions

## Abstract

The main goal of this paper is to study some properties of an extension of valuations from classical invariants. More specifically, we consider a valued field $(K,\nu)$ and an extension $\omega$ of $\nu$ to a finite extension $L$ of $K$. Then we study when the valuation ring of $\omega$ is essentially finitely generated over the valuation ring of $\nu$. We present a necessary condition in terms of classic invariants of the extension by Hagen Knaf and show that in some particular cases, this condition is also sufficient. We also study when the corresponding extension of graded algebras is finitely generated. For this problem we present an equivalent condition (which is weaker than the one for the finite generation of the valuation rings).