On Knörrer Periodicity for Quadric Hypersurfaces in Skew Projective Spaces

Ueyama, Kenta

Canadian mathematical bulletin, Tome 62 (2019) no. 4, pp. 896-911

Résumé

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 $\mathbb{P}^{1}$.

Détail

DOI : 10.4153/S0008439518000607

Mots-clés : Knörrer periodicity, stable category, noncommutative quadric hypersurface, skew polynomial algebra, point scheme