## The Auslander-Reiten translation in submodule categories

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- by Claus Michael Ringel and Markus Schmidmeier PDF
- Trans. Amer. Math. Soc.
**360**(2008), 691-716 Request permission

## 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.## References

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## Additional Information

**Claus Michael Ringel**- Affiliation: Fakultät für Mathematik, Universität Bielefeld, P.O. Box 100 131, D-33 501 Bielefeld, Germany
- MR Author ID: 148450
- Email: ringel@math.uni-bielefeld.de
**Markus Schmidmeier**- Affiliation: Department of Mathematical Sciences, Florida Atlantic University, Boca Raton, Florida 33431-0991
- MR Author ID: 618925
- ORCID: 0000-0003-3365-6666
- Email: markus@math.fau.edu
- Received by editor(s): April 30, 2005
- Received by editor(s) in revised form: September 30, 2005
- Published electronically: September 5, 2007
- © Copyright 2007
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc.
**360**(2008), 691-716 - MSC (2000): Primary 16G70; Secondary 18E30
- DOI: https://doi.org/10.1090/S0002-9947-07-04183-9
- MathSciNet review: 2346468

Dedicated: Dedicated to Idun Reiten on the occasion of her 60$^{th}$ birthday