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On Knörrer Periodicity for Quadric Hypersurfaces in Skew Projective Spaces
- Part of
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- Journal:
- Canadian Mathematical Bulletin / Volume 62 / Issue 4 / December 2019
- Published online by Cambridge University Press:
- 03 December 2018, pp. 896-911
- Print publication:
- December 2019
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- Article
- Export citation
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We study the structure of the stable category
$\text{}\underline{\mathsf{CM}}^{\mathbb{Z}}(S/(f))$ 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 
$\text{}\underline{\mathsf{CM}}^{\mathbb{Z}}(S/(f))$ is well known by Knörrer’s periodicity theorem. In this paper, we prove that if 
$n\leqslant 5$, then the structure of 
$\text{}\underline{\mathsf{CM}}^{\mathbb{Z}}(S/(f))$ is determined by the number of irreducible components of the point scheme of 
$S$ which are isomorphic to 
$\mathbb{P}^{1}$.
