Resolutions as directed colimits

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 The proofs are applications of an even more general category-theoretic principle going back to an unpublished 1977 preprint of Ulmer.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.