Generalized Knörrer's Periodicity Theorem

Abstract

Let $A$ be a noetherian Koszul Artin-Schelter regular algebra, and let $f\in A_2$ 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 $\underline{\text{mcm}}\,A/(f)\cong\underline{\text{mcm}}\,(A\otimes B)/(f+g)$ 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 $(A/(f))^\#=A[x]/(f+x^2)$ of a noncommutative conic $A/(f)$.