Resolutions in factorization categories

TL;DR

This paper generalizes matrix factorizations to factorization categories, constructs resolutions, and develops tools like spectral sequences to compute morphisms, extending derived category techniques.

Contribution

It introduces factorization categories, constructs resolutions from component resolutions, and develops spectral sequences for morphism computations in derived categories.

Findings

Resolutions of factorizations are constructed from component resolutions.

Spectral sequences are developed to compute morphism spaces.

Fully-faithfulness results are extended to derived categories of factorizations.

Abstract

Generalizing Eisenbud's matrix factorizations, we define factorization categories. Following work of Positselski, we define their associated derived categories. We construct specific resolutions of factorizations built from a choice of resolutions of their components. We use these resolutions to lift fully-faithfulness statements from derived categories of Abelian categories to derived categories of factorizations and to construct a spectral sequence computing the morphism spaces in the derived categories of factorizations from Ext-groups of their components in the underlying Abelian category.