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=APy₁, y₂; be a graded double Ore extension of A. We use the Clifford deformation C₀^! (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 (C₀^! (z) ) or a twisted direct product of C₀^! (z) C₀^! (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.