A degree bound for rings of arithmetic invariants

TL;DR

This paper establishes a degree bound for generators of rings of invariants under finite group actions over Noetherian domains, especially when these rings are Cohen-Macaulay, extending classical invariant theory results.

Contribution

It proves that Cohen-Macaulay rings of invariants are generated by elements of bounded degree, specifically at most max(|G|, n(|G|-1)), providing a new degree bound in invariant theory.

Findings

Ring of invariants is generated by elements of degree at most max(|G|, n(|G|-1)) when Cohen-Macaulay.

Existence of homogeneous systems of parameters with elements of degree at most |G| in certain local rings.

Extension of classical bounds to more general Noetherian domains and Cohen-Macaulay cases.

Abstract

Consider a Noetherian domain $R$ and a finite group $G \subseteq Gl_n(R)$. We prove that if the ring of invariants $R[x_1, \ldots, x_n]^G$ is a Cohen-Macaulay ring, then it is generated as an $R$-algebra by elements of degree at most $\max(|G|,n(|G|-1))$. As an intermediate result we also show that if $R$ is a Noetherian local ring with infinite residue field then such a ring of invariants of a finite group $G$ over $R$ contains a homogeneous system of parameters consisting of elements of degree at most $|G|$.