is particularly surprising given that the point scheme of $A$ consists of the associated elliptic curve $E$ and four extra points [@smith1992regularity Propositions 2.4 and 2.5].

The presence of the modules occurring in Theorem \[thm: sklyanin\] (iii) can be related to a factor ring of $A$. There exist two central elements $\Omega_1,\Omega_2 \in A_2$ by [@smith1992regularity Corollary 3.9]. The factor ring $A/(\Omega_1,\Omega_2)$ is isomorphic to the twisted homogeneous coordinate ring $B:=B(E,{\ensuremath{\mathcal{L}}},\sigma)$, with such rings being defined in Definition \[def: thcr\]. One can use the Noncommutative Serre’s theorem (Theorem \[thm: noncomserrethm\]) to see that the point modules over $A$ that are parameterised by $E$ correspond geometrically to skyscraper sheaves over the elliptic curve.

Since $\Omega_1$ and $\Omega_2$ are fixed by the action of $G$ on $A$ used in Theorem \[thm: sklyanin\], the action of $G$ is compatible with factoring out the ideal $(\Omega_1,\Omega_2)$. We prove the following result, which summarises Theorem \[thm: geomdescthcrtwist\] and Proposition \[prop: fatpointsincohE\]. Any undefined terms are defined in §\[subsec: geomdata\] or §\[subsec: geomdesc\].

\[thm: geometricthcrdesc\] Let $A^{G,\mu}$ be as in Theorem \[thm: sklyanin\]. There is a factor ring of $A^{G,\mu}$, labelled $B^{G,\mu}$, which is a cocycle twist of a factor ring of $A$. The ring $B^{G,\mu}$ 