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)

 
 

 

Hereditary noetherian categories
with a tilting complex


Author: Helmut Lenzing
Journal: Proc. Amer. Math. Soc. 125 (1997), 1893-1901
MSC (1991): Primary 14G14, 16G20; Secondary 18F20, 18E30
DOI: https://doi.org/10.1090/S0002-9939-97-04122-1
MathSciNet review: 1423314
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We are characterizing the categories of coherent sheaves on a weighted projective line as the small hereditary noetherian categories without projectives and admitting a tilting complex. The paper is related to recent work with de la Peña (Math. Z., to appear) characterizing finite dimensional algebras with a sincere separating tubular family, and further gives a partial answer to a question of Happel, Reiten, Smalø (Mem. Amer. Math. Soc. 120 (1996), no. 575) regarding the characterization of hereditary categories with a tilting object.


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

  • 1. A. A. Beilinson, Coherent sheaves on $P_n$ and problems of linear algebra, Funct. Anal. Appl. 12 (1979), 214-216. MR 80c:14010b
  • 2. P. Gabriel, Indecomposable representations II, Symposia Mat. Inst. Naz. Alta Mat. 11 (1973), 81-104. MR 49:5132
  • 3. W. Geigle and H. Lenzing, A class of weighted projective curves arising in representation theory of finite dimensional algebras. In Singularities, representations of algebras, and vector bundles. Lect. Notes Math. 1273 (1987), 265-297, Springer-Verlag, Berlin-Heidelberg-New York. MR 89b:14049
  • 4. -, Perpendicular categories with applications to representations and sheaves, J. Algebra 144 (1991), 273-343. MR 93b:16011
  • 5. D. Happel, Triangulated categories in the representation theory of finite dimensional algebras, London Math. Soc. Lect. Notes 119, Cambridge University Press, 1988. MR 89e:16035
  • 6. D. Happel, I. Reiten and S. Smalø, Tilting in abelian categories and quasitilted algebras, Mem. Amer. Math. Soc. 120 (1996), no. 575. CMP 96:08
  • 7. D. Happel and C. M. Ringel, Tilted Algebras, Trans. Amer. Math. Soc. 274 (1982), 399-443. MR 84d:16027
  • 8. H. Lenzing and H. Meltzer, Tilting sheaves and concealed-canonical algebras, CMS Conf. Proc. 18 (1996), 455-473. CMP 96:12
  • 9. H. Lenzing and J. A. de la Peña, Algebras with a separating tubular family, Math. Z., to appear.
  • 10. C. M. Ringel, The canonical algebras with an appendix by William Crawley-Boevey. In Topics in Algebra, Banach Center Publ. 26 (1990), 407-432. MR 93e:16022

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 14G14, 16G20, 18F20, 18E30

Retrieve articles in all journals with MSC (1991): 14G14, 16G20, 18F20, 18E30


Additional Information

Helmut Lenzing
Affiliation: Universität-GH Paderborn, Fachbereich Mathematik-Informatik, D-33095 Pader- born, Germany
Email: helmut@uni-paderborn.de

DOI: https://doi.org/10.1090/S0002-9939-97-04122-1
Keywords: Weighted projective line, coherent sheaf, almost-split sequence, Auslander-Reiten theory, finite dimensional algebra, quiver, derived category, exceptional object, tilting complex.
Received by editor(s): February 9, 1995
Dedicated: In memory of Maurice Auslander
Communicated by: Eric M. Friedlander
Article copyright: © Copyright 1997 American Mathematical Society

American Mathematical Society