Noncommutative Knörrer’s periodicity theoremand noncommutative quadric hypersurfaces

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 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 CMZ(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 CMZ(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 CMZ(A) over a smooth quadric hypersurface A in a skew Pⁿ⁻¹, where n ≤ 6, without high rank property using graphical methods.