satisfies the following properties:

-   it can be described as a twisted ring over $\mathcal{E}$, where $\mathcal{E}$ is a sheaf of ${\ensuremath{\mathcal{O}}}_{E^G}$-orders;

-   it has a family of fat point modules of multiplicity 2 parameterised by $E^G$, arising in a natural way from certain coherent sheaves of $\mathcal{E}$-modules.

The importance of this result will be stressed in §\[sec: noncommgeom\] in relation to canonical maps to twisted rings. Briefly, it illustrates that twisted rings can capture geometric properties of noncommutative algebras, in this case $A^{G,\mu}$.

Other interesting examples can be found by studying the algebras classified by Rogalski and Zhang in [@rogalski2012regular]. Such algebras are AS-regular of dimension 4 with three degree 1 generators and an additional ${\ensuremath{\mathbb{Z}}}^{\times 2}$ grading. We disregard algebras with these properties that are normal extensions of AS-regular algebras of dimension 3. The classification up to isomorphism of the remaining algebras consists of 8 families $\mathcal{A}-\mathcal{H}$. Theorem \[thm: rogzhang\] shows that one can relate several of these families using cocycle twists.

\[thm: rogzhang\] Consider Rogalski and Zhang’s classification in [@rogalski2012regular]. For the Klein-four group $G$ and a certain normalised 2-cocycle $\mu$, one has the following isomorphisms of $k$-algebras: $$\mathcal{A}(1,-1)^{G,\mu}\cong \mathcal{D}(1,1),\;\; \mathcal{B}(1)^{G,\mu} \cong \mathcal{C}(1), \;\;
\mathcal{E}(1,\gamma)^{G,\mu}\cong \mathcal{E}(1,-\gamma), \;\; 