Invariants of normal local rings by p-cyclic group actions

TL;DR

This paper investigates the properties of invariant rings under cyclic group actions on Noetherian normal local rings, establishing conditions for regularity and monogeneity related to the augmentation ideal.

Contribution

It provides new criteria linking the principal nature of the augmentation ideal to the structure and regularity of invariant rings under cyclic automorphisms.

Findings

B is a monogenous A-algebra iff the augmentation ideal is principal

If B is regular, then A is regular when the augmentation ideal is principal

Characterization of invariant rings under p-cyclic group actions

Abstract

Let $B$ be a Noetherian normal local ring, and $G\subset\Aut(B)$ a cyclic group of local automorphisms of prime order. Let $A$ be the ring of $G$-invariants of $B$, assume that $A$ is Noetherian. We study the invariant morphism; in particular, we prove that $B$ is a monogenous $A$-algebra if and only if the augmentation ideal of $B$ is principal. If in particular $B$ is regular, we prove that $A$ is regular if the augmentation ideal of $B$ is principal.