Noncommutative Knörrer’s periodicity theorem and noncommutative quadric hypersurfaces

Mori, Izuru; Ueyama, Kenta

doi:10.2140/ant.2022.16.467

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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

DOI: 10.2140/ant.2022.16.467

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

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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

Vol. 16, No. 2, 2022