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

We study the structure of the stable category ${\underset{‾}{\mathsf{C}\mathsf{M}}}^{\mathbb{Z}}(S\mathrm{/}(f))$ of graded maximal Cohen-Macaulay module over $S\mathrm{/}(f)$ 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 ${\underset{‾}{\mathsf{C}\mathsf{M}}}^{\mathbb{Z}}(S\mathrm{/}(f))$ is well-known by Kn"orrer’s periodicity theorem. In this paper, we prove that if $n\le 5$, then the structure of $$\underset{‾}{\mathsf{C}\mathsf{M}}$^{\mathbb Z}(S/(f))$ is determined by the number of irreducible components of the point scheme of $S$ which are isomorphic to ${\mathbb{P}}^1$.