Skew Kn\"orrer's periodicity Theorem

In this paper, we introduce a class of twisted matrix algebras of M₂(E) and twisted direct products of E× E for an algebra E. Let A be a noetherian Koszul Artin-Schelter regular algebra, z∈ A₂ be a regular central element of A and B=AP[y₁,y₂;σ] be a graded double Ore extension of A. We use the Clifford deformation CA^!(z) of Koszul dual A^! to study the noncommutative quadric hypersurface B/(z+y₁²+y₂²). We prove that the stable category of graded maximal Cohen-Macaulay modules over B/(z+y₁²+y₂²) is equivalent to certain bounded derived categories, which involve a twisted matrix algebra of M₂(CA^!(z)) or a twisted direct product of CA^!(z)× CA^!(z) depending on the values of P. These results are presented as skew versions of Kn\"orrer's periodicity theorem. Moreover, we show B/(z+y₁²+y₂²) may not be a noncommutative graded isolated singularity even if $A/(z)$ is.