Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Standard components of a Krull-Schmidt category


Authors: Shiping Liu and Charles Paquette
Journal: Proc. Amer. Math. Soc. 143 (2015), 45-59
MSC (2010): Primary 16G70, 16G20, 16G10
DOI: https://doi.org/10.1090/S0002-9939-2014-12200-3
Published electronically: August 28, 2014
MathSciNet review: 3272731
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We provide criteria for an Auslander-Reiten component having sections of a Krull-Schmidt category to be standard. Specializing to the category of finitely presented representations of a strongly locally finite quiver and its bounded derived category, we obtain many new types of standard Auslander-Reiten components. An application to the module category of a finite-dimensional algebra yields some interesting new results.


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

  • [1] Ibrahim Assem, Juan Carlos Bustamante, and Patrick Le Meur, Coverings of laura algebras: the standard case, J. Algebra 323 (2010), no. 1, 83-120. MR 2564830 (2011e:16017), https://doi.org/10.1016/j.jalgebra.2009.08.013
  • [2] Raymundo Bautista, Irreducible morphisms and the radical of a category, An. Inst. Mat. Univ. Nac. Autónoma México 22 (1982), 83-135 (1983). MR 736555 (86g:16041)
  • [3] Raymundo Bautista, Shiping Liu, and Charles Paquette, Representation theory of strongly locally finite quivers, Proc. London Math. Soc. 106 (2013), 97-162. MR 3020740
  • [4] K. Bongartz and P. Gabriel, Covering spaces in representation-theory, Invent. Math. 65 (1981/82), no. 3, 331-378. MR 643558 (84i:16030), https://doi.org/10.1007/BF01396624
  • [5] Dieter Happel, On the derived category of a finite-dimensional algebra, Comment. Math. Helv. 62 (1987), no. 3, 339-389. MR 910167 (89c:16029), https://doi.org/10.1007/BF02564452
  • [6] Dieter Happel and Claus Michael Ringel, Tilted algebras, Trans. Amer. Math. Soc. 274 (1982), no. 2, 399-443. MR 675063 (84d:16027), https://doi.org/10.2307/1999116
  • [7] Helmut Lenzing, Hereditary categories, Handbook of tilting theory, London Math. Soc. Lecture Note Ser., vol. 332, Cambridge Univ. Press, Cambridge, 2007, pp. 105-146. MR 2384609 (2009c:18015), https://doi.org/10.1017/CBO9780511735134.006
  • [8] Shiping Liu, Tilted algebras and generalized standard Auslander-Reiten components, Arch. Math. (Basel) 61 (1993), no. 1, 12-19. MR 1222084 (94g:16017), https://doi.org/10.1007/BF01258050
  • [9] Shiping Liu, Infinite radicals in standard Auslander-Reiten components, J. Algebra 166 (1994), no. 2, 245-254. MR 1279256 (95g:16014), https://doi.org/10.1006/jabr.1994.1150
  • [10] Shiping Liu, Shapes of connected components of the Auslander-Reiten quivers of Artin algebras, Representation theory of algebras and related topics (Mexico City, 1994), CMS Conf. Proc., vol. 19, Amer. Math. Soc., Providence, RI, 1996, pp. 109-137. MR 1388561 (97e:16037)
  • [11] Shiping Liu, Auslander-Reiten theory in a Krull-Schmidt category, São Paulo J. Math. Sci. 4 (2010), no. 3, 425-472. MR 2856194
  • [12] Claus Michael Ringel, Finite dimensional hereditary algebras of wild representation type, Math. Z. 161 (1978), no. 3, 235-255. MR 501169 (80c:16017), https://doi.org/10.1007/BF01214506
  • [13] Claus Michael Ringel, Tame algebras and integral quadratic forms, Lecture Notes in Mathematics, vol. 1099, Springer-Verlag, Berlin, 1984. MR 774589 (87f:16027)
  • [14] Andrzej Skowroński, Generalized standard Auslander-Reiten components, J. Math. Soc. Japan 46 (1994), no. 3, 517-543. MR 1276836 (95d:16022), https://doi.org/10.2969/jmsj/04630517
  • [15] Andrzej Skowroński, Generalized standard Auslander-Reiten components without oriented cycles, Osaka J. Math. 30 (1993), no. 3, 515-527. MR 1240010 (94k:16026)
  • [16] Andrzej Skowroński, On semi-regular Auslander-Reiten components, Bull. Polish Acad. Sci. Math. 42 (1994), no. 2, 157-163. MR 1810701

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 16G70, 16G20, 16G10

Retrieve articles in all journals with MSC (2010): 16G70, 16G20, 16G10


Additional Information

Shiping Liu
Affiliation: Département de Mathématiques, Université de Sherbrooke, Sherbrooke, Québec J1K 2R1, Canada
Email: shiping.liu@usherbrooke.ca

Charles Paquette
Affiliation: Department of Mathematics and Statistics, University of New Brunswick, Fredericton, New Brunswick E3B 5A3, Canada
Email: charles.paquette@usherbrooke.ca

DOI: https://doi.org/10.1090/S0002-9939-2014-12200-3
Keywords: Krull-Schmidt categories, almost split sequences, Auslander-Reiten quiver, standard components, representations of quivers, tilted algebras.
Received by editor(s): August 24, 2012
Received by editor(s) in revised form: January 29, 2013, and March 6, 2013
Published electronically: August 28, 2014
Additional Notes: The first-named author was supported in part by the NSERC of Canada
The second-named author was supported in full by NSERC and AARMS
Dedicated: In memory of Michael C. R. Butler
Communicated by: Birge Huisgen-Zimmermann
Article copyright: © Copyright 2014 American Mathematical Society

American Mathematical Society