Filtration Games and Potentially Projective Modules

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 Martin's Maximum implies the determinacy of many $\mathcal{C}$-Filtration Games of length $\omega_1$, which in turn imply the determinacy of certain Ehrenfeucht-Fra\"iss\'{e} games of length $\omega_1$; this allows a significant strengthening of a theorem of Mekler-Shelah-Vaananen \cite{MR1191613}. Also, Martin's 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 L\"owenheim-Skolem number $\aleph_1$ in some models in set theory, but fails to be an AEC in other models of set theory.