Indecomposables live in all smaller lengths

Bongartz, Klaus

doi:10.1090/S1088-4165-2013-00429-6

Indecomposables live in all smaller lengths
HTML articles powered by AMS MathViewer

by Klaus Bongartz

Represent. Theory 17 (2013), 199-225

DOI: https://doi.org/10.1090/S1088-4165-2013-00429-6

Published electronically: April 5, 2013

PDF | Request permission

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

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
Raymundo Bautista, On algebras of strongly unbounded representation type, Comment. Math. Helv. 60 (1985), no. 3, 392–399. MR 814146, DOI 10.1007/BF02567422
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, DOI 10.1007/BF01389052
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, DOI 10.1112/jlms/s2-26.1.43
Klaus Bongartz, Treue einfach zusammenhängende Algebren. I, Comment. Math. Helv. 57 (1982), no. 2, 282–330 (German). MR 684118, DOI 10.1007/BF02565862
Klaus Bongartz, A criterion for finite representation type, Math. Ann. 269 (1984), no. 1, 1–12. MR 756773, DOI 10.1007/BF01455993
Klaus Bongartz, Critical simply connected algebras, Manuscripta Math. 46 (1984), no. 1-3, 117–136. MR 735517, DOI 10.1007/BF01185198
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, DOI 10.1007/BF01163167
Klaus Bongartz, Indecomposables are standard, Comment. Math. Helv. 60 (1985), no. 3, 400–410. MR 814147, DOI 10.1007/BF02567423
Klaus Bongartz, On degenerations and extensions of finite-dimensional modules, Adv. Math. 121 (1996), no. 2, 245–287. MR 1402728, DOI 10.1006/aima.1996.0053
K. Bongartz, On mild contours in ray categories, to appear in Algebras and Representation Theory, preprint of 26 pages, arXiv:1201. 1434.
K. Bongartz and P. Gabriel, Covering spaces in representation-theory, Invent. Math. 65 (1981/82), no. 3, 331–378. MR 643558, DOI 10.1007/BF01396624
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, DOI 10.24033/bsmf.1975
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, DOI 10.1007/BFb0075257
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
Urs Fischbacher, Zur Kombinatorik der Algebren mit endlich vielen Idealen, J. Reine Angew. Math. 370 (1986), 192–213 (German). MR 852515, DOI 10.1515/crll.1986.370.192
Pierre Gabriel, Des catégories abéliennes, Bull. Soc. Math. France 90 (1962), 323–448 (French). MR 232821, DOI 10.24033/bsmf.1583
Peter Gabriel, Indecomposable representations. II, Symposia Mathematica, Vol. XI (Convegno di Algebra Commutativa, INDAM, Rome, 1971 & Convegno di Geometria, INDAM, Rome, 1972) Academic Press, London, 1973, pp. 81–104. MR 0340377
P. Gabriel, The universal cover of a representation-finite algebra, Representations of algebras (Puebla, 1980) Lecture Notes in Math., vol. 903, Springer, Berlin-New York, 1981, pp. 68–105. MR 654725
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
C. Geiss: Darstellungsendliche Algebren und multiplikative Basen, Diplomarbeit Uni Bayreuth 1990, 126 pages.
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, DOI 10.1007/BF01169585
C. Riedtmann, Algebren, Darstellungsköcher, Überlagerungen und zurück, Comment. Math. Helv. 55 (1980), no. 2, 199–224 (German). MR 576602, DOI 10.1007/BF02566682
Claus Michael Ringel, Tame algebras and integral quadratic forms, Lecture Notes in Mathematics, vol. 1099, Springer-Verlag, Berlin, 1984. MR 774589, DOI 10.1007/BFb0072870
Claus Michael Ringel, The Gabriel-Roiter measure, Bull. Sci. Math. 129 (2005), no. 9, 726–748. MR 2172139, DOI 10.1016/j.bulsci.2005.04.002
Claus Michael Ringel, Indecomposables live in all smaller lengths, Bull. Lond. Math. Soc. 43 (2011), no. 4, 655–660. MR 2820151, DOI 10.1112/blms/bdq128
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
A. V. Roĭter, A generalization of a theorem of Bongartz, Akad. Nauk Ukrain. SSR Inst. Mat. Preprint 17 (1981), 32 (Russian). MR 640749