Coderived and contraderived categories of locally presentable abelian DG-categories
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866916347390197760 |
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| author | Positselski, Leonid Stovicek, Jan |
| author_facet | Positselski, Leonid Stovicek, Jan |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2210_08237 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Coderived and contraderived categories of locally presentable abelian DG-categories Positselski, Leonid Stovicek, Jan Category Theory Algebraic Geometry Rings and Algebras 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. |
| title | Coderived and contraderived categories of locally presentable abelian DG-categories |
| topic | Category Theory Algebraic Geometry Rings and Algebras |
| url | https://arxiv.org/abs/2210.08237 |