Cocycle Twists of Algebras

Let $A$ be a $k$-algebra where $k$ is an algebraically closed field and $G$ be a finite abelian group for which the characteristic of $k$ does not divide $|G|$. 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 as well as in the language of Hopf algebras. 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. We show that many properties are preserved under such twists; for example the strongly noetherian property, finite global dimension and Artin-Shelter regularity. The above concepts are then applied to the 4-dimensional Sklyanin algebras, $A:=A(\alpha,\beta,\gamma)$. We define an action of the Klein four-group on $A$ such that the action restricts to a special geometric factor ring $B$ (a twisted homogeneous coordinate ring). The cocycle twists of these algebras, denoted by $A^{G,\mu}$ and $B^{G,\mu}$ respectively, have very different geometric properties to their untwisted counterparts. While $A$ has point modules parameterised by an elliptic curve $E$ and four extra points, $A^{G,\mu}$ has only 20 point modules (when an automorphism associated to $E$ has infinite order). The point modules over $A$ can be used to construct fat point modules of multiplicity 2 over $A^{G,\mu}$, and there are isomorphisms among such objects corresponding to orbits of a natural action of $G$ on $E$. Furthermore, the ring $B^{G,\mu}$ can be described in terms of Artin and Stafford's classification of noncommutative curves.