Generalized Kn{o}rrers Periodicity Theorem

Let $A$ be a noetherian Koszul Artin-Schelter regular algebra, and let $fin A_2$ 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 $underline{text{mcm}},A/(f)congunderline{text{mcm}},(Aotimes B)/(f+g)$ of the stable categories of maximal Cohen-Macaulay modules. This result may be viewed as a generalization of Kn{o}rrers periodicity theorem. As an application, we study the double branch cover $(A/(f))^#=A[x]/(f+x^2)$ of a noncommutative conic $A/(f)$.