Stable Tameness of Two-Dimensional Polynomial Automorphisms Over a Regular Ring

TL;DR

This paper proves that all two-dimensional polynomial automorphisms over a regular ring are stably tame, with stronger results for Dedekind Q-algebras, and shows that stable tameness is a local property.

Contribution

It establishes the stable tameness of two-dimensional polynomial automorphisms over regular rings and demonstrates local stability properties, extending previous understanding.

Findings

All two-dimensional polynomial automorphisms over a regular ring are stably tame.

Over an Artinian ring, automorphisms with Jacobian determinant one are stably tame.

Stable tameness is a local property, meaning local tameness implies stable tameness.

Abstract

In this paper it is established that all two-dimensional polynomial automorphisms over a regular ring R are stably tame. In the case R is a Dedekind Q-algebra, some stronger results are obtained. A key element in the proof is a theorem which yields the following corollary: Over an Artinian ring A all two-dimensional polynomial automorphisms having Jacobian determinant one are stably tame, and are tame if A is a Q-algebra. Another crucial ingredient, of interest in itself, is that stable tameness is a local property: If an automorphism is locally tame, then it is stably tame.