Deformation cohomology for cyclic groups acting on polynomial rings

TL;DR

This paper studies the Hochschild cohomology related to graded deformations of polynomial rings under cyclic group actions, especially in cases where the field's characteristic divides the group order, providing a classification of first-order deformations.

Contribution

It classifies infinitesimal graded deformations of skew group algebras for cyclic groups acting on polynomial rings, including challenging cases with characteristic dividing the group order.

Findings

Classification of first-order graded deformations

Analysis of Hochschild cohomology in characteristic dividing cases

Explicit description of deformation space

Abstract

We examine the Hochschild cohomology governing graded deformations for finite cyclic groups acting on polynomial rings. We classify the infinitesimal graded deformations of the skew group algebra $S\rtimes G$ for a cyclic group $G$ acting on a polynomial ring $S$. This gives all graded deformations of the first order. We are particularly interested in the case when the characteristic of the underlying field divides the order of the acting group, which complicates the determination of cohomology.