Arithmetic invariants of pseudoreflection groups and regular graded algebras

TL;DR

This paper generalizes the Chevalley-Shephard-Todd theorem to pseudoreflection groups over Dedekind domains, showing invariant rings are polynomial rings when the group order is invertible, and characterizes regular graded algebras as tensor products of blowup algebras.

Contribution

It extends classical invariant theory results to Dedekind domains and characterizes regular graded algebras in this broader setting.

Findings

Invariant rings are polynomial rings over Dedekind domains when group order is invertible.

Regular graded algebras are tensor products of blowup algebras.

Generalization of Chevalley-Shephard-Todd theorem to new algebraic contexts.

Abstract

The main result of this paper is a generalization of the theorem of Chevalley-Shephard-Todd to the rings of invariants of pseudoreflection groups over Dedekind domains. In the special case of a principal ideal domain in which the group order is invertible it is proved that this ring of invariants is isomorphic to a polynomial ring. An intermediate result is that every finitely generated regular graded algebra over a Dedekind domain is isomorphic to a tensor product of blowup algebras.