Models for singularity categories

TL;DR

This paper develops models for singularity categories of modules over differential graded rings, utilizing abelian model structures, cotorsion pairs, and localization techniques, with applications to recollements and matrix factorizations.

Contribution

It introduces new models for singularity categories based on abelian model structures and connects them to existing frameworks like Positselski's contraderived model.

Findings

Constructed models for singularity categories of DG modules.

Established connections between model structures and recollements.

Showed Quillen equivalence with matrix factorization models.

Abstract

In this article we construct various models for singularity categories of modules over differential graded rings. The main technique is the connection between abelian model structures, cotorsion pairs and deconstructible classes, and our constructions are based on more general results about localization and transfer of abelian model structures. We indicate how recollements of triangulated categories can be obtained model categorically, discussing in detail Krause's recollement for the stable derived category. In the special case of curved mixed Z-graded complexes, we show that one of our singular models is Quillen equivalent to Positselski's contraderived model for the homotopy category of matrix factorizations.