Generalized Kn\"{o}rrer's Periodicity Theorem

Let A be a noetherian Koszul Artin-Schelter regular algebra, and let f ∈ A2 be a central regular element of A. The quotient algebra A/(f) is usually called a (noncommutative) quadric hypersurface. In this paper, we use the Clifford deformation to study the quadric hypersurfaces obtained from the tensor products. We introduce a notion of simple graded isolated singularity and proved that, if B/(g) is a simple graded isolated singularity of 0-type, then there is an equivalence of triangulated categories mcmA/(f) ∼= mcm(A⊗B)/(f + g) of the stable categories of maximal Cohen-Macaulay modules. This result may be viewed as a generalization of Knörrer’s periodicity theorem. As an application, we study the double branch cover (A/(f)) = A[x]/(f + x) of a noncommutative conic A/(f).