Cocycle twists of algebras

Abstract

Let $A = \bigoplus_{n=0}^{\infty}A_n$ be a connected graded $k$-algebra over an algebraically closed field $k$ (thus $A_0=k$). Assume that a finite abelian group $G$, of order coprime to the characteristic of $k$, acts on $A$ by graded automorphisms. In conjunction with a suitable cocycle this action can be used to twist the multiplication in $A$. We study this new structure and, in particular, we describe when properties like Artin-Schelter regularity are preserved by such a twist. We then apply these results to examples of Rogalski and Zhang.