Categorical matrix factorizations

TL;DR

This paper presents a categorical framework for d-fold matrix factorizations of natural transformations, generalizing classical definitions and analyzing functorial properties between associated triangulated categories.

Contribution

It introduces a purely categorical construction for d-fold matrix factorizations applicable to any even integer d, extending Eisenbud's classical theory.

Findings

For d=2, the functors are full and faithful.

In some cases, the functors are equivalences.

The construction generalizes classical matrix factorizations to higher dimensions.

Abstract

In this paper we give a purely categorical construction of d-fold matrix factorizations of a natural transformation, for any even integer d. This recovers the classical definition of those for regular elements in commutative rings due to Eisenbud. We explore some natural functors between associated triangulated categories, and show that when d=2 these are full and faithful, and in some cases equivalences.