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.

**1.**A. A. Beilinson,*Coherent sheaves on 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**

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