Filtration games and potentially projective modules
Volume 260 / 2023
Abstract
The notion of a $\mathcal C$-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.
