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The Gabriel-Roiter submodules of simple homogeneous modules

Author: Bo Chen
Journal: Proc. Amer. Math. Soc. 138 (2010), 3415-3424
MSC (2010): Primary 16G20, 16G70
Published electronically: June 4, 2010
MathSciNet review: 2661542
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Abstract: Let $ \Lambda$ be a connected tame hereditary algebra over an algebraically closed field. We show that if $ \Lambda=kQ$ is of type $ \widetilde{\mathbb{A}}_n$, $ \widetilde{\mathbb{D}}_n$, $ \widetilde{\mathbb{E}}_6$ or $ \widetilde{\mathbb{E}}_7$, then every Gabriel-Roiter submodule of a quasi-simple module of rank $ 1$ (i.e. a simple homogeneous module) has defect $ -1$. In particular, any Gabriel-Roiter submodule of a simple homogeneous module yields a Kronecker pair, and thus induces a full exact embedding of the category $ \mod k\widetilde{\mathbb{A}}_1$ into $ \mod\Lambda$, where $ \widetilde{\mathbb{A}}_1$ is the Kronecker quiver. Consequently, we obtain that all quasi-simple modules are Gabriel-Roiter factor modules.

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Additional Information

Bo Chen
Affiliation: Hausdorff Center for Mathematics, Universität Bonn, 53115 Bonn, Germany
Address at time of publication: Mathematisches Institut, Universität zu Köln, Weyertal 86–90, 50931 Köln, Germany

Keywords: Tame hereditary algebras, simple homogeneous modules, defect, Gabriel-Roiter measure.
Received by editor(s): October 7, 2008
Received by editor(s) in revised form: August 7, 2009, and September 28, 2009
Published electronically: June 4, 2010
Dedicated: Dedicated to my wife, Qi, and my twin daughters, Yining and Yimeng
Communicated by: Birge Huisgen-Zimmermann
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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