An 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.

We study cocycle twists of a 4-dimensional Sklyanin algebra A and a factor ring B which is a twisted homogeneous coordinate ring. Twisting such algebras by the Klein four-group G, we show that the twists A G,µ and B G,µ have very different geometric properties to their untwisted counterparts. For example, A G,µ has only 20 point modules and infinitely many fat point modules of multiplicity 2. The ring B G,µ falls under the purview of Artin and Stafford's classification of noncommutative curves, and we describe it using a sheaf of orders over an elliptic curve.