---
author:
- Andrew Phillip Davies
bibliography:
- 'allrefs.bib'
title: Cocycle twists of algebras
---

Let $A$ be a $k$-algebra where $k$ is an algebraically closed field and $G$ be a finite abelian group whose order is not divisible by the characteristic of $k$. If $G$ acts on $A$ by $k$-algebra automorphisms then the action induces a $G$-grading on $A$ which, in conjunction with a normalised 2-cocycle of the group, can be used to twist the multiplication of the algebra. Such twists can be formulated as Zhang twists [@zhang1998twisted] as well as in the language of Hopf algebras, as done in [@montgomery1993hopf §7.5].

We investigate such cocycle twists with an emphasis on the situation where $A$ also possesses a connected graded structure and the action of $G$ respects this grading. The twisting operation uses the induced $G$-grading of $A$ rather than its [$\mathbb{N}$]{}-grading, as is often the case with Zhang twists. As a result, geometric data encoded by special modules in the category of graded right modules, $\text{grmod}(A)$, may not be preserved. Nevertheless, we show using an alternative construction of the twist and faithful flatness arguments that many properties are preserved; the strongly noetherian property, finite global dimension and Artin-Shelter 