Mathematische Zeitschrift

, Volume 265, Issue 1, pp 1–19 | Cite as

Baer and Mittag-Leffler modules over tame hereditary algebras

  • Lidia Angeleri Hügel
  • Dolors Herbera
  • Jan Trlifaj
Article

Abstract

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 \({{\rm Ext}^{1}_{R}\,(M, T)\,=\,0}\) for all torsion modules T, and M is Mittag-Leffler in case the canonical map \({M\otimes_R \prod _{i\in I}Q_i\to \prod _{i\in I}(M\otimes_RQ_i)}\) is injective where \({\{Q_i\}_{i\in I}}\) 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.

Mathematics Subject Classification (2000)

Primary 16G10 16D40 16E60 Secondary 13D07 16E30 18E35 

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Copyright information

© Springer-Verlag 2009

Authors and Affiliations

  • Lidia Angeleri Hügel
    • 1
    • 2
  • Dolors Herbera
    • 3
  • Jan Trlifaj
    • 4
  1. 1.Dipartimento di Informatica e ComunicazioneUniversità degli Studi dell’InsubriaVareseItaly
  2. 2.Dipartimento di Informatica, Settore di MatematicaUniversità di VeronaVeronaItaly
  3. 3.Departament de MatemàtiquesUniversitat Autònoma de BarcelonaBellaterra (Barcelona)Spain
  4. 4.Department of Algebra, Faculty of Mathematics and PhysicsCharles UniversityPrague 8Czech Republic

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