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12. Sklyanin algebras and cocycle twists


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      Problem 12.1.

      [Toby Stafford] Let $S$ be a 4-dimensional Sklyanin algebra, not finite over its centre. Odesskii defines a twist by the Klein four-group $K_4$ acts diagonally on $S \otimes_k M_2(k) = M_2(S)$. Let $A = (M_2(S))^{K_4}$. The central elements of $S$, $\Omega_1$ and $\Omega_2$, remain central in $A$. Consider $R_\Omega := A/(\Omega)$, where $\Omega := c_1 \Omega_1 + c_2 \Omega_2$ for some constants $c_1, c_2$. Andrew Davies showed that $R_{\Omega_1}$ is not a domain; is every $R_\Omega$ not a domain?
        1. Remark. [Toby Stafford] If $R_\Omega$ is a domain for some $\Omega$, then $Q_{\text{gr}}(R_\Omega)_0$ has no valuations, and so does not fall under the classification conjectured by Artin.
            • Remark. [Michel van den Bergh] More generally, when are factors of cocycle deformations of domains still domains?


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                  Problem 12.2.

                  [James Zhang] Suppose $A$ is an AS regular algebra which is Koszul, Auslander regular, Cohen-Macaulay etc. and that there exists a homogeneous central element $\Omega$. If $\text{Proj}(A/(\Omega))$ has finite global dimension, is $A/(\Omega)$ a domain?
                    1. Remark. [James Zhang] Let $E_\Omega$ be a deformation of the Koszul dual of $A$. Then $E_\Omega$ is semisimple if and only if $\text{Proj}(A/(\Omega))$ has finite global dimension.


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                          Problem 12.3.

                          [Toby Stafford] If $R_\Omega$ is not a domain, is $Q_{\text{gr}}(R_\Omega)_0$ isomorphic to $M_2(D_{\text{Skl}})$, where $D_{\text{Skl}}$ is the Sklyanin division algebra.

                              Cite this as: AimPL: Noncommutative surfaces and Artin's conjecture, available at http://aimpl.org/ncsurfaceartin.