Weighted sums of squares in local rings and their completions, I

Let A be an excellent local ring of real dimension ≤2, let T be a finitely generated preordering in A, and let ${\widehat{T}}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 ${{\rm Ext}^{1}_{R}\,(M, T)\,=\,0}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 Ext¹_R (M, T) = 0{{\rm Ext}^{1}_{R}\,(M, T)\,=\,0} for all torsion modules T, and M is Mittag-Leffler in case the canonical map M?_R ?_{i ? I}Q_i? ?_{i ? I}(M?_RQ_i){M\otimes_R \prod _{i\in I}Q_i\to \prod _{i\in I}(M\otimes_RQ_i)} is injective where {Q_i}_{i ? I}{\{Q_i\}_{i\in 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.