Rank additivity for quasi-tilted algebras of canonical type

Thomas Hübner

Colloquium Mathematicae (1998)

  • Volume: 75, Issue: 2, page 183-193
  • ISSN: 0010-1354

Abstract

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Given the category X of coherent sheaves over a weighted projective line X = X ( λ , p ) (of any representation type), the endomorphism ring Σ = ( 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 X (Example 4.3).

How to cite

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Hübner, Thomas. "Rank additivity for quasi-tilted algebras of canonical type." Colloquium Mathematicae 75.2 (1998): 183-193. <http://eudml.org/doc/210537>.

@article{Hübner1998,
abstract = {Given the category $\{X\}$ of coherent sheaves over a weighted projective line $\{X\}=\{X\}(\{\lambda \},\{p\})$ (of any representation type), the endomorphism ring $\Sigma = (\mathcal \{T\})$ 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 $\{X\}$ (Example 4.3).},
author = {Hübner, Thomas},
journal = {Colloquium Mathematicae},
keywords = {finite-dimensional algebras; derived categories; finite-dimensional modules; categories of coherent sheaves; weighted projective lines; almost concealed-canonical algebras; tilting sheaves; Tits quivers; indecomposable direct summands; Euler forms; rank functions; representation-infinite quasitilted algebras; rank additivity; tilting complexes},
language = {eng},
number = {2},
pages = {183-193},
title = {Rank additivity for quasi-tilted algebras of canonical type},
url = {http://eudml.org/doc/210537},
volume = {75},
year = {1998},
}

TY - JOUR
AU - Hübner, Thomas
TI - Rank additivity for quasi-tilted algebras of canonical type
JO - Colloquium Mathematicae
PY - 1998
VL - 75
IS - 2
SP - 183
EP - 193
AB - Given the category ${X}$ of coherent sheaves over a weighted projective line ${X}={X}({\lambda },{p})$ (of any representation type), the endomorphism ring $\Sigma = (\mathcal {T})$ 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 ${X}$ (Example 4.3).
LA - eng
KW - finite-dimensional algebras; derived categories; finite-dimensional modules; categories of coherent sheaves; weighted projective lines; almost concealed-canonical algebras; tilting sheaves; Tits quivers; indecomposable direct summands; Euler forms; rank functions; representation-infinite quasitilted algebras; rank additivity; tilting complexes
UR - http://eudml.org/doc/210537
ER -

References

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  1. [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. [2] W. Geigle and H. Lenzing, Perpendicular categories with applications to representations and sheaves, J. Algebra 144 (1991), 273-343. 
  3. [3] D. Happel, I. Reiten and S. O. Smalο, Tilting in abelian categories and quasitilted algebras, Mem. Amer. Math. Soc. 575 (1996). Zbl0849.16011
  4. [4] D. Happel and C. M. Ringel, Tilted algebras, Trans. Amer. Math. Soc. 274 (1982), 399-443. Zbl0503.16024
  5. [5] D. Happel and D. Vossieck, Minimal algebras of infinite representation type with preprojective component, Manuscripta Math. 42 (1983), 221-243. Zbl0516.16023
  6. [6] T. Hübner, Exzeptionelle Vektorbündel und Reflektionen an Kippgarben über projektiven gewichteten Kurven, Dissertation, 1996. 
  7. [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. Zbl0809.16012
  8. [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. Zbl0863.16013
  9. [9] H. Lenzing and J. A. de la Pe na, Wild canonical algebras, Math. Z., to appear. 
  10. [10] H. Lenzing and A. Skowroński, Quasi-tilted agebras of canonical type, Colloq. Math. 71 (1996), 161-181. Zbl0870.16007
  11. [11] H. Meltzer, Exceptional sequences for canonical algebras, Arch. Math. (Basel) 64 (1995), 304-312. Zbl0818.16016
  12. [12] C. M. Ringel, Tame Algebras and Integral Quadratic Forms, Lecture Notes in Math. 1099, Springer, 1984. 

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