Coderived and contraderived categories of locally presentable abelian DG-categories

The concept of an abelian DG-category, introduced by the first-named author in Positselski (Exact DG-categories and fully faithful triangulated inclusion functors. arXiv:2110.08237 [math.CT]), 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 (Adv Math 254:187–232, 2014. arXiv:1205.4473 [math.CT]) and by the present authors (Positselski and Št’ovíček in J Pure Appl Algebra 226(#4):106883, 2022. arXiv:2101.10797 [math.CT]), we define the contraderived category of a locally presentable abelian DG-category \(\textbf{B}\) with enough projective objects and the coderived category of a Grothendieck abelian DG-category \(\textbf{A}\). 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 \(\textbf{A}\), and prove that its coderived category is compactly generated by the absolute derived category of finitely presentable objects of \(\textbf{A}\), thus generalizing a result from the second-named author’s preprint (Št’ovíček in On purity and applications to coderived and singularity categories. arXiv:1412.1615 [math.CT]). 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. We also describe compact generators of the coderived categories of quasi-coherent matrix factorizations over coherent schemes.