On exact categories and applications to triangulated adjoints and model structures

Abstract

We show that Quillen's small object argument works for exact categories under very mild conditions. This has immediate applications to cotorsion pairs and their relation to the existence of certain triangulated adjoint functors and model structures. In particular, the interplay of different exact structures on the category of complexes of quasi-coherent sheaves leads to a streamlined and generalized version of recent results obtained by Estrada, Gillespie, Guil Asensio, Hovey, Jørgensen, Neeman, Murfet, Prest, Trlifaj and possibly others.

38 pages; version 2: major revision, more explanation added at several places, reference list updated and extended, misprints corrected

References in corpus (2)

• Infinite-dimensional vector bundles in algebraic geometry (an introduction)
• Model category structures arising from Drinfeld vector bundles

Cited by in corpus (14)

• Fp-projective periodicity
• Flat quasi-coherent sheaves as directed colimits, and quasi-coherent cotorsion periodicity
• A homotopy theory of additive categories with suspensions
• The derived category with respect to a generator
• Right triangulated categories: As extriangulated categories, aisles and co-aisles
• Priles of one-sided triangulated categories and exact model categories
• Dualizations of approximations, ${\aleph }_1$-projectivity, and Vopěnka's Principles
• Flat Mittag-Leffler modules, and their relative and restricted versions
• Generalized periodicity theorems
• Tor-pairs: products and approximations
• The $Q$-shaped derived category of a ring
• Cotorsion pairs in Hopfological algebra
• On a fat small object argument
• Fishing for complements