Noncommutative Knörrer’s periodicity theorem and noncommutative quadric hypersurfaces

Abstract

Noncommutative hypersurfaces, in particular, noncommutative quadric hypersurfaces are major objects of study in noncommutative algebraic geometry. In the commutative case, Knörrer’s periodicity theorem is a powerful tool to study Cohen–Macaulay representation theory since it reduces the number of variables in computing the stable category $\underset{¯}{\mathrm{CM}}(A)$ of maximal Cohen–Macaulay modules over a hypersurface $A$. In this paper, we prove a noncommutative graded version of Knörrer’s periodicity theorem. Moreover, we prove another way to reduce the number of variables in computing the stable category ${\underset{¯}{\mathrm{CM}}}^{\mathrm{\mathbb{Z}}}\left(A\right)$ of graded maximal Cohen–Macaulay modules if $A$ is a noncommutative quadric hypersurface. Under the high rank property defined in this paper, we also show that computing ${\underset{¯}{\mathrm{CM}}}^{\mathrm{\mathbb{Z}}}\left(A\right)$ over a noncommutative smooth quadric hypersurface $A$ in up to six variables can be reduced to one or two variable cases. In addition, we give a complete classification of ${\underset{¯}{\mathrm{CM}}}^{\mathrm{\mathbb{Z}}}\left(A\right)$ over a smooth quadric hypersurface $A$ in a skew ${\mathbb{P}}^{n-1}$, where $n\le 6$, without high rank property using graphical methods.