Deconstructible abstract elementary classes of modules and categoricity

IF 0.9 3区数学 Q2 MATHEMATICS Bulletin of the London Mathematical Society Pub Date : 2024-10-16 DOI:10.1112/blms.13172

Jan Šaroch, Jan Trlifaj

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Abstract

We prove a version of Shelah's categoricity conjecture for arbitrary deconstructible classes of modules. Moreover, we show that if $A$ is a deconstructible class of modules that fits in an abstract elementary class $(A, ⪯)$ such that (1) $A$ is closed under direct summands and (2) $⪯$ refines direct summands, then $A$ is closed under arbitrary direct limits. In the Appendix, we prove that the assumption (2) is not needed in some models of ZFC.

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可解构抽象基本类的模块和范畴

我们证明了任意可解构模块类的Shelah范畴猜想的一个版本。此外，我们证明了如果A $\mathcal {A}$是一个可解构的模块类，它适合于一个抽象的基本类(A)，⪯)$ (\mathcal {A},\ precq)$使得(1)A $\mathcal {A}$在直接求和下闭，(2)⪯$\ precq $精炼直接求和，则A $\mathcal {A}$在任意直接极限下闭。在附录中，我们证明了在ZFC的一些模型中不需要假设(2)。

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