Vol. 16, No. 2, 2022

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Noncommutative Knörrer's periodicity theorem and noncommutative quadric hypersurfaces

Izuru Mori and Kenta Ueyama

Vol. 16 (2022), No. 2, 467–504
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 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 CM ¯(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 CM ¯(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 CM ¯(A) over a smooth quadric hypersurface A in a skew n1, where n 6, without high rank property using graphical methods.

Keywords
Knörrer periodicity, noncommutative quadric hypersurfaces, noncommutative matrix factorizations, maximal Cohen–Macaulay modules
Mathematical Subject Classification
Primary: 16E65, 16G50, 16S38
Milestones
Received: 6 December 2020
Revised: 8 April 2021
Accepted: 13 June 2021
Published: 27 April 2022
Authors
Izuru Mori
Department of Mathematics
Shizuoka University
Shizuoka
Japan
Kenta Ueyama
Department of Mathematics
Hirosaki University
Aomori, Hirosaki
Japan