Rank additivity for quasi-tilted algebras of canonical type

• Autorzy: Thomas Hübner
• Języki publikacji: EN

Abstrakty

EN

Given the category $\coh\sym{X}$ of coherent sheaves over a weighted projective line $\sym{X}=\sym{X}(\und{\lambda},\und{p})$ (of any representation type), the endomorphism ring $\mit\Sigma = \End(\cal{T})$ of an arbitrary tilting sheaf - which is by definition an almost concealed canonical algebra - is shown to satisfy a rank additivity property (Theorem 3.2). Moreover, this property extends to the representationinfinite quasi-tilted algebras of canonical type (Theorem 4.2). Finally, it is demonstrated that rank additivity does not generalize to the case of tilting complexes over $\coh\sym{X}$ (Example 4.3).

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Colloquium Mathematicum
• Rocznik: 1998
• Tom: 75
• Numer: 2
• Strony: 183-193

Daty

wydano

1998

otrzymano

1997-04-15

Twórcy

autor

Thomas Hübner

• Fachbereich Mathematik-Informatik, Universität-GH Paderborn 33095, Paderborn, Germany

Bibliografia

• [1] W. Geigle and H. Lenzing, A class of weighted projective curves arising in representation theory of finite dimensional algebras, in: Singularities, Representation of Algebras and Vector Bundles (Lambrecht, 1985), Springer, 1987, 265-297.
• [2] W. Geigle and H. Lenzing, Perpendicular categories with applications to representations and sheaves, J. Algebra 144 (1991), 273-343.
• [3] D. Happel, I. Reiten and S. O. Smalο, Tilting in abelian categories and quasitilted algebras, Mem. Amer. Math. Soc. 575 (1996).
• [4] D. Happel and C. M. Ringel, Tilted algebras, Trans. Amer. Math. Soc. 274 (1982), 399-443.
• [5] D. Happel and D. Vossieck, Minimal algebras of infinite representation type with preprojective component, Manuscripta Math. 42 (1983), 221-243.
• [6] T. Hübner, Exzeptionelle Vektorbündel und Reflektionen an Kippgarben über projektiven gewichteten Kurven, Dissertation, 1996.
• [7] H. Lenzing and H. Meltzer, Sheaves on a weighted projective line of genus one, and representations of a tubular algebra, in: Proc. ICRA VI, 1992, 313-337.
• [8] H. Lenzing and H. Meltzer, Tilting sheaves and concealed-canonical algebras, in: Representation Theory of Algebras, ICRA VII, Cocoyoc 1994, CMS Conf. Proc. 18, 1996, 455-473.
• [9] H. Lenzing and J. A. de la Pe na, Wild canonical algebras, Math. Z., to appear.
• [10] H. Lenzing and A. Skowroński, Quasi-tilted agebras of canonical type, Colloq. Math. 71 (1996), 161-181.
• [11] H. Meltzer, Exceptional sequences for canonical algebras, Arch. Math. (Basel) 64 (1995), 304-312.
• [12] C. M. Ringel, Tame Algebras and Integral Quadratic Forms, Lecture Notes in Math. 1099, Springer, 1984.