$\hat{I}¼$-abstract elementary classes of modules

Abstract

We prove several new results in the theory of $\hat{I}¼$-AECs, focusing mainly on (almost) stability, with the primary objective of undertaking a systematic study of $\hat{I}¼$-AECs of $R$-modules. Our main results are the following. 1. We show that, under suitable syntactic assumptions, all tame $\hat{I}¼$-AECs of $R$-modules (where $R$ is a ring) are almost stable, and are stable if they additionally satisfy a strong amalgamation property. This extends the work of the second author and Shelah [49] to the setting of $\hat{I}¼$-AECs. 2. We then turn to applications to concrete $\hat{I}¼$-AECs of $R$-modules. Our main result in this direction is that $(R$-Mod$,{\le }_{pp}^{\hat{I}}¼)$ has a stable independence relation and is a stable and tame $\hat{I}¼$-AEC, where ${\le }_{pp}^{\hat{I}}¼$ denotes the $\hat{I}¼$-pure submodule relation. We also prove similar stability results for various classes of abelian groups, including the ${\aleph }_1$-AEC of torsion-free abelian groups with the balanced subgroup relation. Moreover, we prove the almost stability of all $\hat{I}¼$-AECs of modules of the form $(R$-Mod$,\preccurlyeq )$, where $\preccurlyeq$ refines the direct summand relation and satisfies a strong form of coherence. 3. Finally, we study $\hat{I}¼$-AECs of the form $(K,{\le }_{\oplus })$, where $K$ is a class of pure-injective $R$-modules (note that this is, in general, not an AEC), and use our results to show that, for many natural choices of $K$, the class $(K,{\le }_{\oplus })$ has a stable independence relation and is therefore stable and tame. We use these results to give a sufficient condition for abstract classes of modules of the form $(K,{\le }_{pp})$ to be stable when $K$ is closed under pure-injective envelopes. This generalizes, by a substantially different proof, results of Mazari-Armida [45].

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