The notion of a textbf{$boldsymbol{mathcal{C}}$-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 textbf{$boldsymbol{mathcal{C}}$-Filtration Game of length $boldsymbol{omega_1}$} on a module, paying particular attention to the case where $mathcal{C}$ is the collection of all countably presented, projective modules. We prove that Martins Maximum implies the determinacy of many $mathcal{C}$-Filtration Games of length $omega_1$, which in turn imply the determinacy of certain Ehrenfeucht-Fraiss{e} games of length $omega_1$; this allows a significant strengthening of a theorem of Mekler-Shelah-Vaananen cite{MR1191613}. Also, Martins Maximum implies that if $R$ is a countable hereditary ring, the class of textbf{$boldsymbol{sigma}$-closed potentially projective modules}---i.e., those modules that are projective in some $sigma$-closed forcing extension of the universe---is closed under $<aleph_2$-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 Lowenheim-Skolem number $aleph_1$ in some models in set theory, but fails to be an AEC in other models of set theory.
Download