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arXiv · 2107.06438

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

Ji-Wei He·Xin-Chao Ma·Yu Ye
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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 Kn\"{o}rrer's periodicity theorem. As an application, we study the double branch cover $(A/(f))^\#=A[x]/(f+x^2)$ of a noncommutative conic $A/(f)$.

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Ji-Wei He, Xin-Chao Ma, Yu Ye. 2021-07-14. Generalized Kn\"{o}rrer's Periodicity Theorem. https://arxiv.org/abs/2107.06438

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