Remote Access Representation Theory
Green Open Access

Representation Theory

ISSN 1088-4165

 
 

 

Indecomposables live in all smaller lengths


Author: Klaus Bongartz
Journal: Represent. Theory 17 (2013), 199-225
MSC (2010): Primary 16G10, 16G20, 20C05
DOI: https://doi.org/10.1090/S1088-4165-2013-00429-6
Published electronically: April 5, 2013
MathSciNet review: 3038490
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We show that there are no gaps in the lengths of the indecomposable objects in an abelian $ k$-linear category over a field $ k$ provided all simples are absolutely simple. To derive this natural result we prove that any distributive minimal representation-infinite $ k$-category is isomorphic to the linearization of the associated ray category which is shown to have an interval-finite universal cover with a free fundamental group so that the well-known theory of representation-finite algebras applies.


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

  • [1] 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 (96c:16015)
  • [2] Raymundo Bautista, On algebras of strongly unbounded representation type, Comment. Math. Helv. 60 (1985), no. 3, 392-399. MR 814146 (87b:16029), https://doi.org/10.1007/BF02567422
  • [3] R. Bautista, P. Gabriel, A. V. Roĭter, and L. Salmerón, Representation-finite algebras and multiplicative bases, Invent. Math. 81 (1985), no. 2, 217-285. MR 799266 (87g:16031), https://doi.org/10.1007/BF01389052
  • [4] R. Bautista and F. Larrión, Auslander-Reiten quivers for certain algebras of finite representation type, J. London Math. Soc. (2) 26 (1982), no. 1, 43-52. MR 667243 (83k:16014), https://doi.org/10.1112/jlms/s2-26.1.43
  • [5] Klaus Bongartz, Treue einfach zusammenhängende Algebren. I, Comment. Math. Helv. 57 (1982), no. 2, 282-330 (German). MR 684118 (84a:16051), https://doi.org/10.1007/BF02565862
  • [6] Klaus Bongartz, A criterion for finite representation type, Math. Ann. 269 (1984), no. 1, 1-12. MR 756773 (86k:16023), https://doi.org/10.1007/BF01455993
  • [7] Klaus Bongartz, Critical simply connected algebras, Manuscripta Math. 46 (1984), no. 1-3, 117-136. MR 735517 (85j:16026), https://doi.org/10.1007/BF01185198
  • [8] Klaus Bongartz, Indecomposables over representation-finite algebras are extensions of an indecomposable and a simple, Math. Z. 187 (1984), no. 1, 75-80. MR 753421 (86b:16025), https://doi.org/10.1007/BF01163167
  • [9] Klaus Bongartz, Indecomposables are standard, Comment. Math. Helv. 60 (1985), no. 3, 400-410. MR 814147 (87d:16039), https://doi.org/10.1007/BF02567423
  • [10] Klaus Bongartz, On degenerations and extensions of finite-dimensional modules, Adv. Math. 121 (1996), no. 2, 245-287. MR 1402728 (98e:16012), https://doi.org/10.1006/aima.1996.0053
  • [11] K. Bongartz, On mild contours in ray categories, to appear in Algebras and Representation Theory, preprint of 26 pages, arXiv:1201. 1434.
  • [12] 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
  • [13] O. Bretscher and P. Gabriel, The standard form of a representation-finite algebra, Bull. Soc. Math. France 111 (1983), no. 1, 21-40 (English, with French summary). MR 710374 (85g:16014)
  • [14] Otto Bretscher and Gordana Todorov, On a theorem of Nazarova and Roĭter, Representation Theory, I (Ottawa, Ont., 1984) Lecture Notes in Math., vol. 1177, Springer, Berlin, 1986, pp. 50-54. MR 842458 (87i:16056), https://doi.org/10.1007/BFb0075257
  • [15] Urs Fischbacher, Une nouvelle preuve d'un théorème de Nazarova et Roiter, C. R. Acad. Sci. Paris Sér. I Math. 300 (1985), no. 9, 259-262 (French, with English summary). MR 785064 (86g:16029)
  • [16] Urs Fischbacher, Zur Kombinatorik der Algebren mit endlich vielen Idealen, J. Reine Angew. Math. 370 (1986), 192-213 (German). MR 852515 (87k:16023), https://doi.org/10.1515/crll.1986.370.192
  • [17] Pierre Gabriel, Des catégories abéliennes, Bull. Soc. Math. France 90 (1962), 323-448 (French). MR 0232821 (38 #1144)
  • [18] Peter Gabriel, Indecomposable representations. II, Symposia Mathematica, Vol. XI (Convegno di Algebra Commutativa, INDAM, Rome, 1971), Academic Press, London, 1973, pp. 81-104. MR 0340377 (49 #5132)
  • [19] P. Gabriel, The universal cover of a representation-finite algebra, Representations of algebras (Puebla, 1980) Lecture Notes in Math., vol. 903, Springer, Berlin, 1981, pp. 68-105. MR 654725 (83f:16036)
  • [20] P. Gabriel and A. V. Roĭter, Representations of finite-dimensional algebras, Algebra, VIII, Encyclopaedia Math. Sci., vol. 73, Springer, Berlin, 1992, pp. 1-177. With a chapter by B. Keller. MR 1239447 (94h:16001b)
  • [21] C. Geiss: Darstellungsendliche Algebren und multiplikative Basen, Diplomarbeit Uni Bayreuth 1990, 126 pages.
  • [22] Dieter Happel and Dieter Vossieck, Minimal algebras of infinite representation type with preprojective component, Manuscripta Math. 42 (1983), no. 2-3, 221-243. MR 701205 (84m:16022), https://doi.org/10.1007/BF01169585
  • [23] C. Riedtmann, Algebren, Darstellungsköcher, Überlagerungen und zurück, Comment. Math. Helv. 55 (1980), no. 2, 199-224 (German). MR 576602 (82k:16039), https://doi.org/10.1007/BF02566682
  • [24] Claus Michael Ringel, Tame algebras and integral quadratic forms, Lecture Notes in Mathematics, vol. 1099, Springer-Verlag, Berlin, 1984. MR 774589 (87f:16027)
  • [25] Claus Michael Ringel, The Gabriel-Roiter measure, Bull. Sci. Math. 129 (2005), no. 9, 726-748. MR 2172139 (2006g:16039), https://doi.org/10.1016/j.bulsci.2005.04.002
  • [26] Claus Michael Ringel, Indecomposables live in all smaller lengths, Bull. Lond. Math. Soc. 43 (2011), no. 4, 655-660. MR 2820151 (2012e:16015), https://doi.org/10.1112/blms/bdq128
  • [27] A. V. Roĭter, Unboundedness of the dimensions of the indecomposable representations of an algebra which has infinitely many indecomposable representations, Izv. Akad. Nauk SSSR Ser. Mat. 32 (1968), 1275-1282 (Russian). MR 0238893 (39 #253)
  • [28] A. V. Roĭter, A generalization of a theorem of Bongartz, Akad. Nauk Ukrain. SSR Inst. Mat. Preprint 17 (1981), 32 (Russian). MR 640749 (83b:16028)

Similar Articles

Retrieve articles in Representation Theory of the American Mathematical Society with MSC (2010): 16G10, 16G20, 20C05

Retrieve articles in all journals with MSC (2010): 16G10, 16G20, 20C05


Additional Information

Klaus Bongartz
Affiliation: Universität Wuppertal, Germany
Email: bongartz@math.uni-wuppertal.de

DOI: https://doi.org/10.1090/S1088-4165-2013-00429-6
Received by editor(s): March 9, 2012
Received by editor(s) in revised form: October 15, 2012
Published electronically: April 5, 2013
Dedicated: Dedicated to A. V. Roiter and P. Gabriel
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society