fat point modules over a cocycle twist of it. The machinery we construct is used in several of the examples later in the thesis.

Chapter \[chap: sklyanin\] deals with the main examples of cocycle twists studied in the thesis, namely twists of 4-dimensional Sklyanin algebras, which we denote by $A^{G,\mu}$. In §\[subsec: 4sklyanintwistandptscheme\] such examples are shown to satisfy many good properties (in particular Theorem \[thm: 4sklytwistprops\]), while in Theorem \[prop: finitepointscheme\] their point schemes are determined. We also show that the full version of Odesskii’s example from [@odesskii2002elliptic Introduction] (which was simplified in Example \[ex: odesskii\]), can be described as a cocycle twist of a Sklyanin algebra (Proposition \[prop: odesskiiegdone\]). The existence of fat point modules over $A^{G,\mu}$ is proved in §\[subsec: fatpoints\]; such modules arise as direct sums of point modules over Sklyanin algebras by Proposition \[claim: fatpoints\]. This is just one indication of the interplay between 1-critical modules over $A$ and $A^{G,\mu}$, and we prove several more results in this direction.

In Chapter \[chap: thcrtwist\] we study a twist of the factor ring $B=A/(\Omega_1,\Omega_2)$. Under the induced $G$-grading on $A$, $B$ is a $G$-graded factor ring. Thus one can regard the twist $B^{G,\mu}$ as a factor of 