Coherent analogues of matrix factorizations and relative singularity categories

TL;DR

This paper develops a comprehensive theory of matrix factorizations and relative singularity categories, establishing their relationships, dualities, and functorial properties, with applications to derived categories and Hochschild (co)homology.

Contribution

It introduces new definitions and identifications of relative singularity categories and matrix factorizations, extending existing frameworks to infinite rank and large categories, and proves localization and duality theorems.

Findings

Identified exotic derived categories of matrix factorizations with relative singularity categories.

Proved Thomason-Trobaugh-Neeman localization for coherent matrix factorizations.

Established Serre-Grothendieck duality theorems for matrix factorizations.

Abstract

We define the triangulated category of relative singularities of a closed subscheme in a scheme. When the closed subscheme is a Cartier divisor, we consider matrix factorizations of the related section of a line bundle, and their analogues with locally free sheaves replaced by coherent ones. The appropriate exotic derived category of coherent matrix factorizations is then identified with the triangulated category of relative singularities, while the similar exotic derived category of locally free matrix factorizations is its full subcategory. The latter category is identified with the kernel of the direct image functor corresponding to the closed embedding of the zero locus and acting between the conventional (absolute) triangulated categories of singularities. Similar results are obtained for matrix factorizations of infinite rank; and two different "large" versions of the triangulated category of relative singularities, corresponding to the approaches of Orlov and Krause, are identified in the case of a Cartier divisor. A version of the Thomason-Trobaugh-Neeman localization theory is proven for coherent matrix factorizations and disproven for locally free matrix factorizations of finite rank. Contravariant (coherent) and covariant (quasi-coherent) versions of the Serre-Grothendieck duality theorems for matrix factorizations are established, and pull-backs and push-forwards of matrix factorizations are discussed at length. A number of general results about derived categories of the second kind for CDG-modules over quasi-coherent CDG-algebras are proven on the way. Hochschild (co)homology of matrix factorization categories are discussed in an appendix.