manuscripta mathematica

, Volume 60, Issue 3, pp 323–347

Tilting sheaves in representation theory of algebras

  • Dagmar Baer


For a nonsingular weighted projective varietyX we introduce the notion of a tilting sheaf onX. We characterize tilting sheaves as those sheavesM ∈ coh (X) that lead to an equivalence φ: Db(coh (X))→ Db(mod (B)) with φ(M)=B, whereB is the endomorphism algebra ofM. An induced comparison theorem interrelates directly certain subcategories of coh (X) and mod(B), respectively. For a weighted projective spaceX, these subcategories control vector bundles up to twist and, ifM is itself a bundle, also all coherent sheaves.

There are self-equivalences of Db(mod (B)), induced by the twist in the category of sheaves, that can be considered as a generalization of Coxeter functors. In special situations, these functors are related to the Auslander-Reiten translation and serve as “higher” Auslander-Reiten functors.

For the projectiven-space, reduction procedures lead to up to twist descriptions of coherent sheaves by modules over algebrasS[t, n] of global dimensionn−t,0≤tn−1. Db(coh (Pn)) is equivalent to the homotopy category of bounded complexes of suitable modules over each of these algebras.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    M. Auslander, M. I. Platzeck, and I. Reiten. Coxeter functors without diagrams.Trans. Amer. Math. Soc., 250:1–46, 1979Google Scholar
  2. [2]
    M. Auslander and I. Reiten. Representation theory of Artin algebras III.Comm.Algebra, 3:239–294, 1975Google Scholar
  3. [3]
    D. Baer. Zerlegungen von Moduln und Injektive über Ringoiden.Arch. Math., 36:495–501, 1981Google Scholar
  4. [4]
    A. A. Beilinson. Coherent sheaves onP n and problems of linear algebra.Funct. Anal. Appl., 12:214–216, 1979Google Scholar
  5. [5]
    A. A. Beilinson. The derived category of coherent sheaves onP n.Sel. Math. Sov., 3:233–237, 1983/84Google Scholar
  6. [6]
    M. Beltrametti and L. Robbiano.Introduction to the theory of weighted projective spaces. Preprint MPI Bonn, 1985Google Scholar
  7. [7]
    K. Bongartz. Tilted algebras.Representations of algebras, 26–38, 1981. Lecture Notes in Mathematics 903Google Scholar
  8. [8]
    S. Brenner and M. C. R. Butler. Generalization of the Bernstein-Gelfand-Ponomarev reflection functors.Representations of algebras, 103–169, 1980. Lecture Notes in Mathematics 832Google Scholar
  9. [9]
    E. Cline, B. Parshall, and L. Scott. Derived categories and Morita theory.J. Algebra, 104:397–409, 1986Google Scholar
  10. [10]
    C. Delorme. Espaces projectifs anisotropes.Bull. Soc. Math. France, 103:203–223, 1975Google Scholar
  11. [11]
    I. Dolgachev. Weighted projective varieties.Group actions and vector fields, 34–71, 1982. Lecture Notes in Mathematics 956Google Scholar
  12. [12]
    P. Gabriel. Des catégories abéliennes.Bull. Soc. math. France, 90:323–448, 1967Google Scholar
  13. [13]
    W. Geigle and H. Lenzing. A class of weighted projective curves arising in representation theory of finite dimensional algebras. In G.-M. Greuel and G. Trautmann, editors,Singularities, representations of algebras, and vector bundles, pages 265–297, 1987. Lecture Notes in Mathematics 1273Google Scholar
  14. [14]
    A. Grothendieck.Cohomologie l-adique et fonctions L, Séminaire de Géométrie Algébrique du Bois-Marie 1965–66, SGA 5. Springer Lecture Notes in Mathematics 589, 1977Google Scholar
  15. [15]
    D. Happel. On the derived category of a finite dimensional algebra.Comm. Math. Helv., 1986. To appearGoogle Scholar
  16. [16]
    D. Happel and C. M. Ringel. Tilted algebras.Trans. Amer. Math. Soc., 274:399–443, 1982Google Scholar
  17. [17]
    R. Hartshorne.Residues and duality. Springer-Verlag, Berlin-Heidelberg-New York, 1966Google Scholar
  18. [18]
    K. Hulek. On the classification of stable rank-r vector bundles over the projective plane.Progress in Mathematics, 7:113–144, 1980Google Scholar
  19. [19]
    M. M. Kapranov. Derived category of coherent sheaves on Grassmann manifolds.Funct. Anal. Appl., 17:145–146, 1983Google Scholar
  20. [20]
    Y. Miyashita. Tilting modules of finite projective dimension.Math. Z., 193:113–146, 1986Google Scholar
  21. [21]
    R. Mulczinski. Eine neue algebraische Methode zur Untersuchung von Vektorbündeln auf der projektiven Ebene.Thesis, 1978Google Scholar
  22. [22]
    C. Okonek, M. Schneider, and H. Spindler.Vector bundles on complex projective spaces. Birkhäuser, Boston-Basel-Stuttgart, 1980Google Scholar
  23. [23]
    J.-P. Serre. Faisceaux algébriques cohérents.Annals of Math., 61:197–278, 1955Google Scholar
  24. [24]
    J. L. Verdier. Catégories dérivées.Séminaire géométrie algébrique, 4 1/2, 262–311, 1977. Lecture Notes in Mathematics 569Google Scholar

Copyright information

© Springer-Verlag 1988

Authors and Affiliations

  • Dagmar Baer
    • 1
  1. 1.Fachbereich Mathematik-InformatikUniversität-Gesamthochschule PaderbornPaderbornWest Germany

Personalised recommendations