$G$-valuations and $G$-valuation rings

Abstract

We aim to construct a non-commutative algebraic geometry in the style of Chevalley by using generalised valuations. To this end, we introduce groupoid valuation rings and associate suitable value functions to them. We show that many results from classical valuation theory can be generalised in a natural way to this context and give several examples. In the final section, we give a very concrete example of what a non-commutative curve would look like in this new setting.