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Auslander generators of iterated tilted algebras


Authors: Flávio Ulhoa Coelho, Dieter Happel and Luise Unger
Journal: Proc. Amer. Math. Soc. 138 (2010), 1587-1593
MSC (2000): Primary 16E05, 16E10, 16G10
DOI: https://doi.org/10.1090/S0002-9939-10-10201-9
Published electronically: January 6, 2010
MathSciNet review: 2587443
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Abstract: Let $ \Lambda$ be an iterated tilted algebra. We will construct an Auslander generator $ M$ in order to show that the representation dimension of $ \Lambda$ is three in case $ \Lambda$ is representation infinite.


References [Enhancements On Off] (What's this?)

  • [A] M. Auslander, Representation dimension of artin algebras, Queen Mary College, Mathematics Notes, University of London, 1971.
  • [AH] I. Assem, D. Happel, Generalized tilted algebras of type $ \mathbb{A}_{n}$, Comm. Algebra 9 (1981), no. 20, 2101-2125. MR 640613 (83a:16023a)
  • [APT] I. Assem, M. I. Platzeck, S. Trepode, On the representation dimension of tilted and laura algebras, J. Algebra 296 (2006), no. 2, 426-439. MR 2201050 (2006k:16024)
  • [ARS] M. Auslander, I. Reiten, S. Smalø, Representation theory of Artin algebras, Cambridge University Press, 1995. MR 1314422 (96c:16015)
  • [AS1] M. Auslander, S. Smalø, Almost split sequences in subcategories, J. Algebra 69 (1981), no. 2, 426-454. MR 617088 (82j:16048a)
  • [AS2] M. Auslander, S. Smalø, Preprojective modules over Artin algebras, J. Algebra 66 (1980), no. 1, 61-122. MR 591246 (83a:16039)
  • [CP] F. U. Coelho, M. I. Platzeck, On the representation dimension of some classes of algebras, J. Algebra 275 (2004), 615-628. MR 2052629 (2005c:16019)
  • [EHIS] K. Erdmann, Th. Holm, O. Iyama, J. Schröer, Radical embeddings and representation dimension, Adv. Math. 185 (2004), no. 1, 159-177. MR 2058783 (2005g:16019)
  • [GL1] W. Geigle, H. Lenzing, Perpendicular categories with applications to representations and sheaves, J. Algebra 144 (1991), no. 2, 273-343. MR 1140607 (93b:16011)
  • [GL2] W. Geigle, H. Lenzing, A class of weighted projective curves arising in representation theory of finite-dimensional algebras, Singularities, representation of algebras, and vector bundles (Lambrecht, 1985), 265-297, Lecture Notes in Math., 1273, Springer, Berlin, 1987. MR 915180 (89b:14049)
  • [H1] D. Happel, Triangulated categories in the representation theory of finite-dimensional algebras, London Math. Soc. Lecture Notes Series, 119, Cambridge Univ. Press, Cambridge, 1988. MR 935124 (89e:16035)
  • [H2] D. Happel, A characterization of hereditary categories with tilting object, Invent. Math. 144 (2001), 381-398. MR 1827736 (2002a:18014)
  • [HRS] D. Happel, J. Rickard, A. Schofield, Piecewise hereditary algebras, Bull. London Math. Soc. 20 (1988), no. 1, 23-28. MR 916069 (89c:16039)
  • [HZ] D. Happel, D. Zacharia, Homological properties of piecewise hereditary algebras, J. of Algebra, to appear.
  • [I] O. Iyama, Finiteness of representation dimension, Proc. Amer. Math. Soc. 131 (2003), 1011-1014. MR 1948089 (2003k:16024)
  • [O] S. Oppermann, Representation dimension of quasitilted algebras, preprint.
  • [Ri] J. Rickard, Morita theory for derived categories, J. London Math. Soc. (2) 39 (1989), no. 3, 436-456. MR 1002456 (91b:18012)
  • [R] C. M. Ringel, Tame algebras and integral quadratic forms, Lecture Notes in Mathematics, 1099, Springer-Verlag, Berlin-Heidelberg, 1984. MR 774589 (87f:16027)
  • [S] U. Seidel, On tilting complexes and piecewise hereditary algebras, Dissertation Technische Universität Chemnitz, Chemnitz, 2003.

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Additional Information

Flávio Ulhoa Coelho
Affiliation: Departamento de Matemática - IME, Universidade de São Paulo, CP 66281 São Paulo, Brazil
Email: fucoelho@ime.usp.br

Dieter Happel
Affiliation: Fakultät für Mathematik, Technische Universität Chemnitz, D-09107 Chemnitz, Germany
Email: happel@mathematik.tu-chemnitz.de

Luise Unger
Affiliation: Fakultät für Mathematik und Informatik, Fernuniversität Hagen, D-58084 Hagen, Germany
Email: luise.unger@fernuni-hagen.de

DOI: https://doi.org/10.1090/S0002-9939-10-10201-9
Received by editor(s): April 7, 2009
Received by editor(s) in revised form: July 24, 2009
Published electronically: January 6, 2010
Additional Notes: The results presented here were obtained while the second and third authors were visiting IME-USP. They thank their coauthor for his kind hospitality during their pleasant stay in São Paulo. The project was made possible by a grant from FAPESP, Brazil. The first author also acknowledges a grant from $CNP_{q}$
Communicated by: Birge Huisgen-Zimmermann
Article copyright: © Copyright 2010 American Mathematical Society

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