Relative singularity categories, Gorenstein objects and silting theory

We study singularity categories through Gorenstein objects in triangulated categories and silting theory. Let ω be a presilting subcategory of a triangulated category T. We introduce the notion of ω-Gorenstein objects. It is a far extended version in triangulated categories of Gorenstein projective modules and Gorenstein injective modules. We prove that the stable category Gω_:=Gω/addω of Gω modulo addω, where Gω is the subcategory of all ω-Gorenstein objects, is a triangulated category. Moreover, we prove that, under some conditions, the triangulated category Gω_ is triangle equivalent to the relative singularity category of T with respect to the thick subcategory generated by ω. As applications, we obtain the following characterizations of singularity categories which partially extend classical results (usually in the context of Gorenstein rings) to some more general settings. (1) For a ring R of finite Gorenstein global dimension, there are triangle equivalences between GP_ (the stable category of Gorenstein projective modules), GI_ (the stable category of Gorenstein injective modules), and GAddM_ (where M is any big silting complex in Db(ModR)) and the big singularity category DSg(R). (2) For a left coherent ring R of finite Gorenstein global dimension, there are triangle equivalences between Gp_ (the stable category of finitely generated Gorenstein projective left modules), and GaddM_ (where M is any silting complex in Db(R)) and the singularity category Dsg(R).