Skew Knörrer’s periodicity theorem

Abstract

In this paper, we introduce a class of twisted matrix algebras of $M_2(E)$ and twisted direct products of $E\times E$ for an algebra $E$. Let $A$ be a noetherian Koszul Artin–Schelter regular algebra, $z\in A_2$ be a regular central element of $A$ and $B=A_P[y_1,y_2;\sigma ]$ be a graded double Ore extension of $A$. We use the Clifford deformation $C_{A^{!}}(z)$ of Koszul dual $A^{!}$ to study the noncommutative quadric hypersurface $B/(z+y_1^2+y_2^2)$. We prove that the stable category of graded maximal Cohen–Macaulay modules over $B/(z+y_1^2+y_2^2)$ is equivalent to certain bounded derived categories, which involve a twisted matrix algebra of $M_2(C_{A^{!}}(z))$ or a twisted direct product of $C_{A^{!}}(z)\times C_{A^{!}}(z)$ depending on the values of $P$. These results are presented as skew versions of Knörrer’s periodicity theorem. Moreover, we show $B/(z+y_1^2+y_2^2)$ may not be a noncommutative graded isolated singularity even if $A/(z)$ is.