Skew Kn\"orrer's periodicity Theorem

TL;DR

This paper extends Knörrer's periodicity theorem to a noncommutative setting using twisted matrix algebras and Clifford deformations, revealing new equivalences in the stable categories of Cohen-Macaulay modules.

Contribution

It introduces skew versions of Knörrer's periodicity theorem for noncommutative graded algebras using Clifford deformation techniques.

Findings

Stable category of Cohen-Macaulay modules is equivalent to derived categories involving twisted matrix algebras.

Noncommutative hypersurfaces may not be graded isolated singularities even if related algebras are.

Results generalize classical periodicity to noncommutative algebraic geometry.

Abstract

In this paper, we introduce a class of twisted matrix algebras of $M_2(E)$ and twisted direct products of $E\times E$ for an algebra $E$. Let $A$ be a noetherian Koszul Artin-Schelter regular algebra, $z\in A_2$ be a regular central element of $A$ and $B=A_P[y_1,y_2;\sigma]$ be a graded double Ore extension of $A$. We use the Clifford deformation $C_{A^!}(z)$ of Koszul dual $A^!$ to study the noncommutative quadric hypersurface $B/(z+y_1^2+y_2^2)$. We prove that the stable category of graded maximal Cohen-Macaulay modules over $B/(z+y_1^2+y_2^2)$ is equivalent to certain bounded derived categories, which involve a twisted matrix algebra of $M_2(C_{A^!}(z))$ or a twisted direct product of $C_{A^!}(z)\times C_{A^!}(z)$ depending on the values of $P$. These results are presented as skew versions of Kn\"orrer's periodicity theorem. Moreover, we show $B/(z+y_1^2+y_2^2)$ may not be a noncommutative graded isolated singularity even if $A/(z)$ is.