0545361 - MÚ 2023 RIV NL eng J - Journal Article
Positselski, Leonid - Šťovíček, J.
Derived, coderived, and contraderived categories of locally presentable abelian categories.
Journal of Pure and Applied Algebra. Roč. 226, č. 4 (2022), č. článku 106883. ISSN 0022-4049. E-ISSN 1873-1376
R&D Projects: GA ČR(CZ) GA20-13778S
Institutional support: RVO:67985840
Keywords : conventional and exotic derived categories * complete cotorsion pairs * abelian model structures
OECD category: Pure mathematics
Impact factor: 0.8, year: 2022 ; AIS: 0.764, rok: 2022
Method of publishing: Limited access
Result website:
https://doi.org/10.1016/j.jpaa.2021.106883DOI: https://doi.org/10.1016/j.jpaa.2021.106883

For a locally presentable abelian category B with a projective generator, we construct the projective derived and contraderived model structures on the category of complexes, proving in particular the existence of enough homotopy projective complexes of projective objects. We also show that the derived category D(B) is generated, as a triangulated category with coproducts, by the projective generator of B. For a Grothendieck abelian category A, we construct the injective derived and coderived model structures on complexes. Assuming Vopěnka’s principle, we prove that the derived category D(A) is generated, as a triangulated category with products, by the injective cogenerator of A. We also define the notion of an exact category with an object size function and prove that the derived category of any such exact category with exact κ-directed colimits of chains of admissible monomorphisms has Hom sets. Hence the derived category of any locally presentable abelian category has Hom sets.
Permanent Link: http://hdl.handle.net/11104/0322070