Noncommutative Kn\"{o}rrer periodicity and noncommutative Kleinian singularities

TL;DR

This paper extends Kn"{o}rrer's Periodicity Theorem to noncommutative settings, establishing a bijection between certain Cohen-Macaulay modules over noncommutative algebras and their double branched covers, with applications to noncommutative Kleinian singularities.

Contribution

It introduces a noncommutative version of Kn"{o}rrer's Periodicity Theorem, linking Cohen-Macaulay modules over noncommutative algebras and their double branched covers.

Findings

Established a bijection between indecomposable non-free maximal Cohen-Macaulay modules over noncommutative algebras and their double covers.

Extended twisted matrix factorizations to the noncommutative setting.

Applied the theory to noncommutative Kleinian singularities.

Abstract

We establish a version of Kn\"{o}rrer's Periodicity Theorem in the context of noncommutative invariant theory. Namely, let $A$ be a left noetherian AS-regular algebra, let $f$ be a normal and regular element of $A$ of positive degree, and take $B=A/(f)$. Then there exists a bijection between the set of isomorphism classes of indecomposable non-free maximal Cohen-Macaulay modules over $B$ and those over (a noncommutative analog of) its second double branched cover $(B^\#)^\#$. Our results use and extend the study of twisted matrix factorizations, which was introduced by the first three authors with Cassidy. These results are applied to the noncommutative Kleinian singularities studied by the second and fourth authors with Chan and Zhang.