Baer and Mittag-Leffler modules over tame hereditary algebras

We develop a structure theory for two classes of infinite dimensional modules over tame hereditary algebras: the Baer modules, and the Mittag-Leffler ones. A right R-module M is called Baer if Ext1R,(M, T) = 0 for all torsion modules T, and M is Mittag-Leffler in case the canonical map M ⊗R ∏i∈I Qi → ∏i∈I (M ⊗RQi) is injective where {Qi}i∈I are arbitrary left R-modules. We show that a module M is Baer iff M is p-filtered where p is the preprojective component of the tame hereditary algebra R. We apply this to prove that the universal localization of a Baer module is projective in case we localize with respect to a complete tube. Using infinite dimensional tilting theory we then obtain a structure result showing that Baer modules are more complex then the (infinite dimensional) preprojective modules. In the final section, we give a complete classification of the Mittag-Leffler modules. © 2009 Springer-Verlag.