Current browse context:
math.CT
Change to browse by:
References & Citations
Mathematics > Category Theory
Title: Coderived and contraderived categories of locally presentable abelian DG-categories
(Submitted on 15 Oct 2022 (v1), last revised 22 Apr 2024 (this version, v2))
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.
Submission history
From: Leonid Positselski [view email][v1] Sat, 15 Oct 2022 09:22:00 GMT (64kb)
[v2] Mon, 22 Apr 2024 16:34:27 GMT (71kb)
Link back to: arXiv, form interface, contact.