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 $\underline{\mathrm{CM}}(A)$ 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 ${\underline{\mathrm{CM}}}^{\mathbb{Z}}(A)$ 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 ${\underline{\mathrm{CM}}}^{\mathbb{Z}}(A)$ 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 ${\underline{\mathrm{CM}}}^{\mathbb{Z}}(A)$ over a smooth quadric hypersurface $A$ in a skew ${\mathbb{P}}^{n-1}$, where $n\le 6$, without high rank property using graphical methods.

31 pages

References in corpus (1)

• Clifford deformations of Koszul Frobenius algebras and noncommutative quadrics

Cited by in corpus (7)

• Clifford deformations of Koszul Frobenius algebras and noncommutative quadrics
• Combinatorial study of stable categories of graded Cohen--Macaulay modules over skew quadric hypersurfaces
• Pre-resolutions of noncommutative isolated singularities
• Noncommutative conics in Calabi-Yau quantum projective planes
• Generalized Knörrer's Periodicity Theorem
• Combinatorial classification of $(\pm 1)$-skew projective spaces
• Noncommutative affine pencils of conics