Graded rings associated to valuations and direct limits

TL;DR

This paper investigates the structure of graded rings linked to valuations using direct limits, providing a framework to understand complex valuation-related algebraic structures through limits of simpler graded rings.

Contribution

It introduces a method to describe graded rings associated to valuations as direct limits, connecting key polynomials and valuation types in a unified framework.

Findings

Describes the direct limit of graded rings for a family of valuations.

Provides an example with valuation-algebraic valuations as limits of residue-transcendental valuations.

Establishes a new approach to analyze valuation-related graded rings.

Abstract

In this paper, we study the structure of the graded ring associated to a limit key polynomial $Q_n$ in terms of the key polynomials that define $Q_n$. In order to do that, we use direct limits. In general, we describe the direct limit of a family of graded rings associated to a totally ordered set of valuations. As an example, we describe the graded ring associated to a valuation-algebraic valuation as a direct limit of graded rings associated to residue-transcendental valuations.