Subgroups of polynomial automorphisms with diagonalizable fibers

TL;DR

This paper investigates conditions under which subgroups of polynomial automorphisms are diagonalizable, especially focusing on finite abelian groups over PIDs and fields with roots of unity, extending previous results.

Contribution

It generalizes the diagonalizability of finite abelian polynomial automorphism subgroups from affine PIDs over complex numbers to more general fields with roots of unity.

Findings

Finite abelian subgroups of automorphisms are diagonalizable over PIDs with enough roots of unity.

Extension of Kraft-Russell's result to fields beyond the complex numbers.

Conditions for diagonalizability depend on roots of unity in the base field.

Abstract

Let $R$ be an integral domain over a field $k$, and $G$ a subgroup of the automorphism group of the polynomial ring $R[x_1,..., x_n]$ over $R$. In this paper, we discuss when $G$ is diagonalizable under the assumption that $G$ is diagonalizable over the field of fractions of $R$. We are particularly interested in the case where $G$ is a finite abelian group. Kraft-Russell (2014) implies that every finite abelian subgroup of ${\rm Aut}_R(R[x_1,x_2])$ is diagonalizable if $R$ is an affine PID over $k={\bf C}$. One of the main results of this paper says that the same holds for a PID $R$ over any field $k$ containing enough roots of unity.