论文标题
过滤游戏和潜在的投影模块
Filtration Games and Potentially Projective Modules
论文作者
论文摘要
\ textbf {$ \ boldsymbol {\ Mathcal {c}} $ - 过滤}对象的概念，其中$ \ Mathcal {c} $是某种（通常很小）的对象集合，因为在Grothendieck类别中，由于介绍了平坦的封面构想，因此变得无处不在。 \ textbf {$ \ boldsymbol {\ Mathcal {c}} $ - 长度$ \ boldsymbol {ω_1} $}的滤过游戏，特别注意$ \ Mathcal {c} $是所有可计算出现的所有可计算呈现的，Projective Modules的情况。我们证明，马丁的最大值意味着许多$ \ MATHCAL {C} $ - 长度$ω_1$的过滤游戏的确定性，这又意味着某些Ehrenfeuchtfraïssé游戏的确定性$ω_1$;这可以显着加强Mekler-Shelah-Vaananen定理\ cite {MR1191613}。另外，马丁的最大值意味着，如果$ r $是一个可数的遗传环，则\ textbf {$ \boldsymbolσ$ clucted键的潜在投影模块} - 即，这些模块在某些$σ$ cluct的宇宙强迫延伸下的模块是$ <\ aleph_2 $ -dective limitection n Universe novely-close nociale-leptection-eleph_2 $ -ddirection。我们还举了一个（ZFC可定义的）Abelian群体的示例，在某些模型中，在某些模型中，在某些模型中，构成了一个抽象的小学类（AEC），而在某些模型中，但在其他设置理论的其他模型中未能成为AEC。
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 $\boldsymbol{ω_1}$} 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 $ω_1$, which in turn imply the determinacy of certain Ehrenfeucht-Fraïssé games of length $ω_1$; 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{$\boldsymbolσ$-closed potentially projective modules} -- i.e., those modules that are projective in some $σ$-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 Löwenheim-Skolem number $\aleph_1$ in some models in set theory, but fails to be an AEC in other models of set theory.
