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

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 ${\mathbb P}^1$.