Coderived and contraderived categories of locally presentable abelian DG-categories

Abstract

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. We also describe compact generators of the coderived categories of quasi-coherent matrix factorizations over coherent schemes.

LaTeX 2e with xy-pic and one mathb symbol; 76 pages, 1 figure; v.2: a discussion of quasi-coherent matrix factorizations over coherent schemes added in a new Section 9; new Corollary 0.4, Sections 1.10 and 2.7, Examples 3.15, 6.12, 7.8, 8.8, and 8.10 inserted; a paragraph added at the end of Section 2.1, 4th paragraph of the introduction expanded; v.3: several misprints corrected