# Towards a Chevalley-style non-commutative algebraic geometry

## TL;DR

This paper proposes a framework for non-commutative algebraic geometry using generalized valuations, introducing groupoid valuation rings and exploring their properties and connections to existing valuation theories.

## Contribution

It introduces groupoid valuation rings and demonstrates their behavior similar to commutative valuation rings, advancing non-commutative algebraic geometry.

## Key findings

- Groupoid valuation rings behave like their commutative counterparts.
- Several examples illustrate the proposed framework.
- Connections with Dubrovin valuation rings are established.

## Abstract

We aim to construct a non-commutative algebraic geometry by using generalised valuations. To this end, we introduce groupoid valuation rings and associate suitable value functions to them. We show that these objects behave rather like their commutative counterparts. Many examples are given and a tentative connections with Dubrovin valuation rings is established.