Derived, coderived, and contraderived categories of locally presentable abelian categories

Abstract

For a locally presentable abelian category $\mathsf{B}$ 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 $\mathsf{D}(\mathsf{B})$ is generated, as a triangulated category with coproducts, by the projective generator of $\mathsf{B}$. For a Grothendieck abelian category $\mathsf{A}$, we construct the injective derived and coderived model structures on complexes. Assuming Vopenka's principle, we prove that the derived category $\mathsf{D}(\mathsf{A})$ is generated, as a triangulated category with products, by the injective cogenerator of $\mathsf{A}$. More generally, we 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 $\kappa$-directed colimits of chains of admissible monomorphisms has Hom sets. In particular, the derived category of any locally presentable abelian category has Hom sets.

arXiv commentLaTeX 2e with xy-pic; 50 pages, 5 commutative diagrams; v.2: Remarks 6.4 and 9.2 inserted, Introduction expanded, many references added; v.3: several misprints corrected

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Cited by in corpus (9)

• Fp-projective periodicity
• Differential graded Koszul duality: an introductory survey
• Flat comodules and contramodules as directed colimits, and cotorsion periodicity
• Coderived and contraderived categories of locally presentable abelian DG-categories
• Flat quasi-coherent sheaves as directed colimits, and quasi-coherent cotorsion periodicity
• Local, colocal, and antilocal properties of modules and complexes over commutative rings
• Pseudo-dualizing complexes of bicomodules and pairs of t-structures
• Contraderived categories of CDG-modules
• A contramodule generalization of Neeman's flat and projective module theorem