Skew Knörrer’s periodicity theorem

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.