$\mu$-abstract elementary classes of modules

We prove several new results in the theory of $\mu$-AECs, focusing mainly on (almost) stability, with the primary objective of undertaking a systematic study of $\mu$-AECs of $R$-modules. Our main results are the following.
1. We show that, under suitable syntactic assumptions, all tame $\mu$-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 $\mu$-AECs.
2. We then turn to applications to concrete $\mu$-AECs of $R$-modules. Our main result in this direction is that $(R$-Mod$, \leq_{pp}^\mu)$ has a stable independence relation and is a stable and tame $\mu$-AEC, where $\leq_{pp}^\mu$ denotes the $\mu$-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 $\mu$-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 $\mu$-AECs of the form $(K, \leq_\oplus)$, 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 $(K, \leq_\oplus)$ 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 $(K, \leq_{pp})$ to be stable when $K$ is closed under pure-injective envelopes. This generalizes, by a substantially different proof, results of Mazari-Armida [45].