Models for singularity categories

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 Krauseʼs recollement Kac(Inj(R))→K(Inj(R))→D(R)Kac(Inj(R))→K(Inj(R))→D(R). In the special case of curved mixed ZZ-graded complexes, we show that one of our singular models is Quillen equivalent to Positselskiʼs contraderived model for the homotopy category of matrix factorizations.