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 The proofs are applications of an even more general category-theoretic principle going back to an unpublished 1977 preprint of Ulmer.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 Šaroch and Št’ovíček. We also discuss totally acyclic complexes of injectives and Gorenstein-injective modules, obtaining various cardinality estimates for the accessibility rank under various assumptions.