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1. Factors of cocycle twists of Sklyanin algebras


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      Problem 1.1.

      [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.

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