Hereditary noetherian categories with a tilting complex

Lenzing, Helmut

doi:10.1090/S0002-9939-97-04122-1

Hereditary noetherian categories with a tilting complex
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by Helmut Lenzing

Proc. Amer. Math. Soc. 125 (1997), 1893-1901

DOI: https://doi.org/10.1090/S0002-9939-97-04122-1

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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

I. N. Bernšteĭn, I. M. Gel′fand, and S. I. Gel′fand, Algebraic vector bundles on $\textbf {P}^{n}$ and problems of linear algebra, Funktsional. Anal. i Prilozhen. 12 (1978), no. 3, 66–67 (Russian). MR 509387
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
Werner Geigle and Helmut Lenzing, A class of weighted projective curves arising in representation theory of finite-dimensional algebras, Singularities, representation of algebras, and vector bundles (Lambrecht, 1985) Lecture Notes in Math., vol. 1273, Springer, Berlin, 1987, pp. 265–297. MR 915180, DOI 10.1007/BFb0078849
Werner Geigle and Helmut Lenzing, Perpendicular categories with applications to representations and sheaves, J. Algebra 144 (1991), no. 2, 273–343. MR 1140607, DOI 10.1016/0021-8693(91)90107-J
Dieter Happel, Triangulated categories in the representation theory of finite-dimensional algebras, London Mathematical Society Lecture Note Series, vol. 119, Cambridge University Press, Cambridge, 1988. MR 935124, DOI 10.1017/CBO9780511629228
D. Happel, I. Reiten and S. Smalø, Tilting in abelian categories and quasitilted algebras, Mem. Amer. Math. Soc. 120 (1996), no. 575.
Dieter Happel and Claus Michael Ringel, Tilted algebras, Trans. Amer. Math. Soc. 274 (1982), no. 2, 399–443. MR 675063, DOI 10.1090/S0002-9947-1982-0675063-2
H. Lenzing and H. Meltzer, Tilting sheaves and concealed-canonical algebras, CMS Conf. Proc. 18 (1996), 455–473.
H. Lenzing and J. A. de la Peña, Algebras with a separating tubular family, Math. Z., to appear.
Claus Michael Ringel, The canonical algebras, Topics in algebra, Part 1 (Warsaw, 1988) Banach Center Publ., vol. 26, PWN, Warsaw, 1990, pp. 407–432. With an appendix by William Crawley-Boevey. MR 1171247