Resolutions as directed colimits

Abstract

A general principle suggests that "anything flat is a directed colimit of countably presentable flats". In this paper, we consider resolutions and coresolutions of modules over a countably coherent ring $R$ (e.g., any coherent ring or any countably Noetherian ring). We show that any $R$-module of flat dimension $n$ is a directed colimit of countably presentable $R$-modules of flat dimension at most $n$, and any flatly coresolved $R$-module is a directed colimit of countably presentable flatly coresolved $R$-modules. If $R$ is a countably coherent ring with a dualizing complex, then any F-totally acyclic complex of flat $R$-modules is a directed colimit of F-totally acyclic complexes of countably presentable flat $R$-modules. The proofs are applications of an even more general category-theoretic principle going back to an unpublished 1977 preprint of Ulmer. Our proof of the assertion that every Gorenstein-flat module over a countably coherent ring is a directed colimit of countably presentable Gorenstein-flat modules uses a different technique, based on results of Saroch and Stovicek. We also discuss totally acyclic complexes of injectives and Gorenstein-injective modules, obtaining various cardinality estimates for the accessibility rank under various assumptions.

LaTeX 2e, 37 pages; v.2: several misprints corrected, references added and updated; v.3: small additions and corrections; v.4: small corrections, references updated; v.5: a misprint corrected

References in corpus (10)

• Deconstructibility and the Hill lemma in Grothendieck categories
• Locally class-presentable and class-accessible categories
• Coherent rings, fp-injective modules, dualizing complexes, and covariant Serre-Grothendieck duality
• Approximations and locally free modules
• The accessibility rank of weak equivalences
• Flat comodules and contramodules as directed colimits, and cotorsion periodicity
• When does ${\mathrm{Ind}}_{\kappa }(C^I)\simeq {\mathrm{Ind}}_{\kappa }(C{)}^I$?
• Direct limits of Gorenstein injective modules
• Notes on limits of accessible categories
• The cotorsion pair generated by the Gorenstein projective modules and $\lambda$-pure-injective modules

Cited by in corpus (1)

• The categories of corings and coalgebras over a ring are locally countably presentable