The Auslander-Reiten translation in submodule categories
HTML articles powered by AMS MathViewer
- 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
- Maurice Auslander, Idun Reiten, and Sverre O. Smalø, Representation theory of Artin algebras, Cambridge Studies in Advanced Mathematics, vol. 36, Cambridge University Press, Cambridge, 1995. MR 1314422, DOI 10.1017/CBO9780511623608
- M. Auslander and Sverre O. Smalø, Almost split sequences in subcategories, J. Algebra 69 (1981), no. 2, 426–454. MR 617088, DOI 10.1016/0021-8693(81)90214-3
- I. R. Shafarevich (ed.), Algebra. VIII, Encyclopaedia of Mathematical Sciences, vol. 73, Springer-Verlag, Berlin, 1992. Representations of finite-dimensional algebras; A translation of Algebra, VIII (Russian), Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow; Translation edited by A. I. Kostrikin and I. R. Shafarevich. MR 1239446
- Dieter Happel, Triangulated categories in the representation theory of finite-dimensional algebras, London Mathematical Society Lecture Note Series, vol. 119, Cambridge University Press, Cambridge, 1988. MR 935124, DOI 10.1017/CBO9780511629228
- C.M. Ringel, M. Schmidmeier: Invariant subspaces of nilpotent linear operators. I, J. Reine Angew. Mathematik (Crelle) (to appear), 1–55.
- Claus Michael Ringel and Markus Schmidmeier, Submodule categories of wild representation type, J. Pure Appl. Algebra 205 (2006), no. 2, 412–422. MR 2203624, DOI 10.1016/j.jpaa.2005.07.002
- Andrzej Skowroński, Tame triangular matrix algebras over Nakayama algebras, J. London Math. Soc. (2) 34 (1986), no. 2, 245–264. MR 856509, DOI 10.1112/jlms/s2-34.2.245
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