On Kn\"orrer periodicity for quadric hypersurfaces in skew projective spaces

TL;DR

This paper extends Kn"orrer's periodicity theorem to certain noncommutative graded algebras, showing the structure of Cohen-Macaulay modules over specific quadric hypersurfaces depends on geometric components for n ≤ 5.

Contribution

It proves that for n ≤ 5, the stable category of Cohen-Macaulay modules over skew polynomial algebras with quadratic forms is determined by the number of P^1 components of the point scheme.

Findings

Structure determined by the number of P^1 components for n ≤ 5

Extension of Kn"orrer's periodicity to noncommutative settings

Connection between algebraic and geometric properties of point schemes

Abstract

We study the structure of the stable category $\mathsf{\underline{CM}}^{\mathbb Z}(S/(f))$ of graded maximal Cohen-Macaulay module over $S/(f)$ where $S$ is a graded ($\pm 1$)-skew polynomial algebra in $n$ variables of degree 1, and $f =x_1^2 + \cdots +x_n^2$. If $S$ is commutative, then the structure of $\mathsf{\underline{CM}}^{\mathbb Z}(S/(f))$ is well-known by Kn\"orrer's periodicity theorem. In this paper, we prove that if $n\leq 5$, then the structure of $\mathsf{\underline{CM}}^{\mathbb Z}(S/(f))$ is determined by the number of irreducible components of the point scheme of $S$ which are isomorphic to ${\mathbb P}^1$.