# Finite Generation of Extensions of Associated Graded Rings Along a Valuation

## TL;DR

This paper investigates conditions under which the associated graded ring along a valuation is finitely generated over a base ring, providing broad results in equicharacteristic zero and for unramified extensions.

## Contribution

It establishes general criteria for finite generation of associated graded rings along valuations in equicharacteristic zero and for unramified extensions of excellent local rings.

## Key findings

- Finite generation of associated graded rings is characterized in equicharacteristic zero.
- Results apply to unramified extensions of excellent local rings.
- The paper provides broad conditions for finiteness in valuation theory.

## Abstract

In this paper we consider the question of when the associated graded ring along a valuation, ${\rm gr}_{\nu^*}(S)$, is a finite ${\rm gr}_{\nu^*}(R)$-module, where $S$ is a normal local ring which lies over a normal local ring $R$ and $\nu^*$ is a valuation of the quotient field of $S$ which dominates $S$. We obtain a very general result in equicharacteristic zero in Theorem 1.5. We also obtain general results for unramified extensions of excellent local rings in Proposition 1.7.