is AS-regular of global dimension $n$ for some $n \in {\ensuremath{\mathbb{N}}}$;

-   it is Koszul;

-   it is Auslander regular;

-   it is Cohen-Macaulay.

Furthermore, $A$ and $A^{G,\mu}$ have the same Hilbert series.

### Examples of twists {#subsec: examples}

Having described some of our results regarding properties preserved under cocycle twists, let us now turn to examples. We will be primarily concerned with twisting AS-regular algebras of global dimension 4, whose classification is an ongoing and central project in noncommutative algebraic geometry. This term will refer throughout to the school that was created in the wake of the seminal papers of Artin, Tate and Van den Bergh in the early 1990’s [@artin1990some; @artin1991modules].

The algebras that will form the main focus of this thesis are the *4-dimensional Sklyanin algebras*, whose twists are studied in Chapter \[chap: sklyanin\]. Such algebras were studied in [@smith1992regularity; @smith1993irreducible; @levasseur1993modules], not to mention [@tate1996homological], where they are viewed as a special case of a more general construction.

In the following definition we use the notation $[x,y] := xy-yx$ and $[x,y]_{+} :=
xy+yx$ to denote certain types of commutator in an algebra.

\[def: sklyanin relations\] Let $k$ be an algebraically closed field with $\text{char}(k)\neq 2$ and 