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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

The Auslander-Reiten translation in submodule categories


Authors: Claus Michael Ringel and Markus Schmidmeier
Journal: Trans. Amer. Math. Soc. 360 (2008), 691-716
MSC (2000): Primary 16G70; Secondary 18E30
Published electronically: September 5, 2007
MathSciNet review: 2346468
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Abstract: Let $ \Lambda $ be an artin algebra or, more generally, a locally bounded associative algebra, and $ \mathcal{S}(\Lambda )$ the category of all embeddings $ (A\subseteq B)$ where $ B$ is a finitely generated $ \Lambda $-module and $ A$ is a submodule of $ B$. Then $ \mathcal{S}(\Lambda )$ is an exact Krull-Schmidt category which has Auslander-Reiten sequences. In this manuscript we show that the Auslander-Reiten translation in $ \mathcal{S}(\Lambda )$ can be computed within $ \operatorname{mod}\,\Lambda $ by using our construction of minimal monomorphisms. If in addition $ \Lambda $ is uniserial, then any indecomposable nonprojective object in $ \mathcal{S}(\Lambda )$ is invariant under the sixth power of the Auslander-Reiten translation.


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Additional Information

Claus Michael Ringel
Affiliation: Fakultät für Mathematik, Universität Bielefeld, P.O. Box 100131, D-33501 Bielefeld, Germany
Email: ringel@math.uni-bielefeld.de

Markus Schmidmeier
Affiliation: Department of Mathematical Sciences, Florida Atlantic University, Boca Raton, Florida 33431-0991
Email: markus@math.fau.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-07-04183-9
PII: S 0002-9947(07)04183-9
Keywords: Auslander-Reiten sequences, approximations, triangulated categories
Received by editor(s): April 30, 2005
Received by editor(s) in revised form: September 30, 2005
Published electronically: September 5, 2007
Dedicated: Dedicated to Idun Reiten on the occasion of her 60$^{th}$ birthday
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.