$a, b \in A$ and $g, h \in G$.

Cocycle twists have been studied in many other places and generalised vastly – they can be described as Zhang twists for example (see Theorem \[thm: cocycleaszhang\]), which were studied in detail by Zhang in [@zhang1998twisted]. One can also recover them from the *twisted $H$-comodule algebra* construction of [@montgomery1993hopf §7.5] by using the Hopf algebra $H=kG$, the group algebra of a finite abelian group.

Let us exhibit some of the results already in the literature in relation to the preservation of properties under cocycle twisting. Although some of the results hold more generally, we state them in the setting of Construction 1.

\[prop: propspreservedalready\] Assume that $A$ and $G$ are subject to the assumptions at the beginning of §\[subsec: theoryoftwist\] and let $\mu$ be a normalised 2-cocycle. Then if $A$ has one of the following properties, so does $A^{G,\mu}$:

-   it is finitely generated as a $k$-algebra [@montgomery2005algebra cf. Proposition 3.1(1)];

-   it satisfies the polynomial identity (PI) property (cf. [@zhang1998twisted Proposition 5.6], [@montgomery2005algebra Proposition 3.1(4)]);

-   it is noetherian (cf. [@zhang1998twisted Proposition 5.1], [@montgomery2005algebra Proposition 3.1(3)]).

One can replicate this construction without using an identification of $G$ and $G^{\vee}$ by instead 