Higher-dimensional module factorizations and complete intersections

TL;DR

This paper introduces higher-dimensional module factorizations, generalizing classical matrix factorizations, to characterize maximal Cohen-Macaulay modules over complete intersections and extends these concepts to noncommutative rings.

Contribution

It develops a new framework of higher-dimensional module factorizations, including matrix factorizations as special cases, and characterizes the stable category of Cohen-Macaulay modules in this context.

Findings

Higher-dimensional matrix factorizations characterize Cohen-Macaulay modules.

The framework generalizes to noncommutative rings, including quantum complete intersections.

Provides a new perspective on module categories over complete intersections.

Abstract

We introduce higher-dimensional module factorizations associated to a regular sequence. They include higher-dimensional matrix factorizations, which are commutative cubes consisting of free modules with edges being classical matrix factorizations. We characterize the stable category of maximal Cohen-Macaulay modules over a complete intersection via higher-dimensional matrix factorizations over the corresponding regular local ring. The result generalizes to noncommutative rings, including quantum complete intersections.