  The concept of an abelian DG-category, introduced by the first-named author
in arXiv:2110.08237, unites the notions of abelian categories and (curved)
DG-modules in a common framework. In this paper we consider coderived and
contraderived categories in the sense of Becker. Generalizing some
constructions and results from the preceding papers by Becker arXiv:1205.4473
and by the present authors arXiv:2101.10797, we define the contraderived
category of a locally presentable abelian DG-category $\mathbf B$ with enough
projective objects and the coderived category of a Grothendieck abelian
DG-category $\mathbf A$. We construct the related abelian model category
structures and show that the resulting exotic derived categories are
well-generated. Then we specialize to the case of a locally coherent
Grothendieck abelian DG-category $\mathbf A$, and prove that its coderived
category is compactly generated by the absolute derived category of finitely
presentable objects of $\mathbf A$, thus generalizing a result from the
second-named author's preprint arXiv:1412.1615. In particular, the homotopy
category of graded-injective left DG-modules over a DG-ring with a left
coherent underlying graded ring is compactly generated by the absolute derived
category of DG-modules with finitely presentable underlying graded modules.

Category Theory;Rings and Algebras