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 $\underset{¯}{\mathrm{CM}<!--FUNCTION\; APPLICATION-->}(A)$ 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 ${\underset{¯}{\mathrm{CM}<!--FUNCTION\; APPLICATION-->}}^{\mathbb{Z}}(A)$ 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 ${\underset{¯}{\mathrm{CM}<!--FUNCTION\; APPLICATION-->}}^{\mathbb{Z}}(A)$ 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 ${\underset{¯}{\mathrm{CM}<!--FUNCTION\; APPLICATION-->}}^{\mathbb{Z}}(A)$ over a smooth quadric hypersurface $A$ in a skew ${\mathbb{P}}^{n-1}$, where $n\le 6$, without high rank property using graphical methods.

非交换超曲面，特别是非交换二次超曲面是非交换代数几何的主要研究对象。在交换的情况下，Knörrer 的周期性定理是研究 Cohen-Macaulay 表示理论的有力工具，因为它减少了计算稳定类别时的变量数量$\underset{¯}{\mathrm{厘米}<!--FUNCTION\; APPLICATION-->}(\mathrm{一种})$超曲面上的最大 Cohen-Macaulay 模数$\mathrm{一种}$. 在本文中，我们证明了 Knörrer 周期性定理的非交换分级版本。此外，我们证明了另一种在计算稳定类别时减少变量数量的方法${\underset{¯}{\mathrm{厘米}<!--FUNCTION\; APPLICATION-->}}^{\mathbb{Z}}(\mathrm{一种})$分级最大 Cohen-Macaulay 模数如果$\mathrm{一种}$是非交换二次超曲面。在本文定义的高秩属性下，我们还证明了计算${\underset{¯}{\mathrm{厘米}<!--FUNCTION\; APPLICATION-->}}^{\mathbb{Z}}(\mathrm{一种})$在非交换光滑二次超曲面上$\mathrm{一种}$在多达六个变量中可以减少到一两个变量的情况。另外，我们给出了一个完整的分类${\underset{¯}{\mathrm{厘米}<!--FUNCTION\; APPLICATION-->}}^{\mathbb{Z}}(\mathrm{一种})$在光滑的二次超曲面上$\mathrm{一种}$歪斜${\mathbb{P}}^{n-1}$， 在哪里$n\le 6$，没有使用图形方法的高等级属性。