Abstract
We develop the fundamentals of hereditary noetherian categories with Serre duality over an arbitrary field k, where the category of coherent sheaves over a smooth projective curve over k serves as the prime example and others are coming from the representation theory of finite dimensional algebras. The proper way to view such a category is to think of coherent sheaves on a possibly non-commutative smooth projective curve. We define for each such category notions like function field and Euler characteristic, determine its Auslander-Reiten components and study stable and semistable bundles for an appropriate notion of degree. We provide a complete classification of hereditary noetherian categories for the case of positive Euler characteristic by relating these to finite dimensional representations of (locally bounded) hereditary k-algebras whose underlying valued quiver admits a positive additive function.
Similar content being viewed by others
References
Auslander, M., Reiten, I., Smalø, SO.: Representation theory of artin algebras. Cambridge Studies in Advanced Mathematics. V. 36, Cambridge University Press, Cambridge 1995
Gabriel, P.: Indecomposable representations II. Symposia Mat. Inst. Naz. Alta Mat. 11, 81–104 (1973)
Geigle, W., Lenzing, H.: A class of weighted projective curves arising in representation theory of finite dimensional algebras. In: Singularities, representations of algebras, and vector bundles, Lecture Notes Math. 1273, Springer pp. 265–297, 1987
Geigle, W., Lenzing, H.: Perpendicular categories with applications to representations and sheaves. J. Algebra 144, 273–343 (1991)
Happel, D., Reiten, I.: A combinatorial characterization of hereditary categories containing simple objects, Representations of algebras (Sao Paulo 1999., Lect. Notes in Pure and Applied Math 224, Dekker, New York 2002, pp. 91–97
Happel, D., Reiten, I., Smalø, S.: Tilting in abelian categories and quasitilted algebras. Mem. Amer. Math. Soc. 575, 1996
Happel, D., Ringel, C.M.: Tilted Algebras. Trans. Amer. Math. Soc. 274, 399–443 (1982)
Hartshorne, R.: Algebraic Geometry. Springer Verlag 1977
Lenzing, H.: Curve singularities arising from the representation theory of tame hereditary algebras. In: Representation theory I, Finite dimensional algebras, Proc. 4th Int. Conf., Ottawa Can. 1984, Lect. Notes Math. pp. 199–231
Lenzing, H.: Hereditary noetherian categories with a tilting complex. Proc. Am. Math. Soc. 125, 1893-1901 (1997)
Lenzing, H., de la Peña, J.A.: Wild canonical algebras. Math. Z. 224, 403–425 (1997)
Nasatyr, E.B., Steer, B.: Orbifold Riemann surfaces and the Yang-Mills-Higgs equations. Ann. Scuola Norm. Sup. Pisa. Cl. Sci. 22, 595–643 (1995)
I. Reiten and M. Van den Bergh. Noetherian hereditary categories satisfying Serre duality, J. Amer. Math. Soc. 15 (2002), 295–366.
Seshadri, C.S.: Fibrés vectoriels sur les courbes algébriques. Astérisque, 96 (1982)
Strauss, H.: On the perpendicular category of a partial tilting module. J. Algebra 144, 43–66
Thurston, W.P.: The geometry and topology of three-manifolds. Electronic Version 1.1 – March 2002, http://www.msri.org/publications/books/gt3m/.
Author information
Authors and Affiliations
Corresponding author
Additional information
Dedicated to Otto Kerner on the occasion of his 60th birthday
Rights and permissions
About this article
Cite this article
Lenzing, H., Reiten, I. Hereditary noetherian categories of positive Euler characteristic. Math. Z. 254, 133–171 (2006). https://doi.org/10.1007/s00209-006-0938-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00209-006-0938-6