Filtration games and potentially projective modules
Volume 260 / 2023
Abstract
The notion of a -filtered object, where \mathcal C is some (typically small) collection of objects in a Grothendieck category, has become ubiquitous since the solution of the Flat Cover Conjecture around the year 2000. We introduce the \mathcal C-Filtration Game of length \omega _1 on a module, paying particular attention to the case where \mathcal C is the collection of all countably generated projective modules. We prove that Martin’s Maximum implies the determinacy of many \mathcal C-Filtration Games of length \omega _1, which in turn implies the determinacy of certain Ehrenfeucht–Fraïssé games of length \omega _1; this allows a significant strengthening of a theorem of Mekler–Shelah–Vaananen (1993). Also, Martin’s Maximum implies that if R is a countable hereditary ring, the class of \sigma -closed potentially projective modules—i.e., those modules that are projective in some \sigma -closed forcing extension of the universe—is closed under \lt \aleph _2-directed limits. We also give an example of a (ZFC-definable) class of abelian groups that, under the ordinary subgroup relation, constitutes an Abstract Elementary Class (AEC) with Löwenheim–Skolem number \aleph_1 in some models of set theory, but fails to be an AEC in other models of set theory.