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, Kn\"orrer'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 $\underline{\operatorname{CM}}(A)$ of maximal Cohen-Macaulay modules over a hypersurface $A$. In this paper, we prove a noncommutative graded version of Kn\"orrer's periodicity theorem. Moreover, we prove another way to reduce the number of variables in computing the stable category ${\underline{\operatorname{CM}}}^{\mathbb Z}(A)$ of graded maximal Cohen-Macaulay modules if $A$ is a noncommutative quadric hypersurface. Under high rank property defined in this paper, we also show that computing ${\underline{\operatorname{CM}}}^{\mathbb Z}(A)$ over a noncommutative smooth quadric hypersurface $A$ in up to six variables can be reduced to one or two variables cases. In addition, we give a complete classification of ${\underline{\operatorname{CM}}}^{\mathbb Z}(A)$ over a smooth quadric hypersurface $A$ in a skew $\mathbb P^{n-1}$, where $n \leq 6$, without high rank property using graphical methods.