regularity for example.

The above concepts are then illustrated by applying cocycle twists to the 4-dimensional Sklyanin algebras, $A:=A(\alpha,\beta,\gamma)$, first studied ring-theoretically in [@smith1992regularity]. Such algebras are examples of noncommutative projective surfaces and display many interesting geometric properties. Much of this geometry is controlled by a factor ring $B:=B(E,{\ensuremath{\mathcal{L}}},\sigma)$, which is a twisted homogeneous coordinate ring.

We define an action of the Klein four-group $G=(C_2)^2$ on $A$ such that the action restricts to the factor ring $B$. The cocycle twists under the induced $G$-grading, namely $A^{G,\mu}$ 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 the automorphism $\sigma$ 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 $E$. This is just one example of the interplay between the 1-critical modules over $A$ and $A^{G,\mu}$.

The ring $B^{G,\mu}$ falls under the purview of Artin and Stafford’s classification of noncommutative curves [@artin2000semiprime], and we describe it in geometric 