On the structure of the graded algebra associated to a valuation

TL;DR

This paper investigates the structure of the graded algebra linked to a valuation, demonstrating conditions under which it is isomorphic to a polynomial ring with twisted multiplication, and providing counterexamples.

Contribution

It establishes an isomorphism between the graded algebra and a twisted polynomial ring under certain conditions and explores when trivial twisting is possible.

Findings

The graded algebra is isomorphic to a twisted polynomial ring in many cases.

Trivial twisting is not always possible, as shown by a counterexample.

Conditions like free value group or algebraically closed residue field facilitate trivial twisting.

Abstract

The main goal of this paper is to study the structure of the graded algebra associated to a valuation. More specifically, we prove that the associated graded algebra ${\rm gr}_v(R)$ of a subring $(R,\mathfrak{m})$ of a valuation ring $\mathcal{O}_v$, for which $Kv:=\mathcal{O}_v / \mathfrak{m}_v=R / \mathfrak{m}$, is isomorphic to $Kv[t^{v(R)}]$, where the multiplication is given by a twisting. We show that this twisted multiplication can be chosen to be the usual one in the cases where the value group is free or the residue field is closed by radicals. We also present an example that shows that the isomorphism (with the trivial twisting) does not have to exist.