Resolutions in factorization categories

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.