Cohomology of Noncommutative Hilbert Schemes

Dedicated to Claus Michael Ringel on the occasion of his sixtieth birthday

Abstract

Noncommutative Hilbert schemes, introduced by M. V. Nori, parametrize left ideals of finite codimension in free algebras. More generally, parameter spaces of finite-codimensional submodules of free modules over free algebras are considered. Cell decompositions of these varieties are constructed, whose cells are parametrized by certain types of forests. Asymptotics for the corresponding Poincaré polynomials and properties of their generating functions are discussed.

This is a preview of subscription content, log in to check access.

References

  1. 1.

    Artin, M.: On Azumaya algebras and finite-dimensional representations of rings, J. Algebra 11 (1969), 523–563.

    Article  MathSciNet  Google Scholar 

  2. 2.

    Bialynicki-Birula, A.: Some theorems on actions of algebraic groups, Ann. of Math. (2) 98 (1973), 480–497.

    MATH  MathSciNet  Google Scholar 

  3. 3.

    Bergman, G.: The diamond lemma for ring theory, Adv. Math. 29(2) (1978), 178–218.

    Article  MATH  MathSciNet  Google Scholar 

  4. 4.

    Duchon, P.: q-grammars and wall polyominoes, Ann. Comb. 3(2–4) (1999), 311–321.

    MATH  MathSciNet  Google Scholar 

  5. 5.

    Duchon, P.: On the enumeration and generation of generalized Dyck words, Formal power series and algebraic combinatorics (Toronto, ON, 1998), Discrete Math. 225(1–3) (2000), 121–135.

    MATH  MathSciNet  Google Scholar 

  6. 6.

    Ellingsrud, G. and Strømme, S. A.: On the homology of the Hilbert scheme of points in the plane, Invent. Math. 87(2) (1987), 343–352.

    Article  MathSciNet  Google Scholar 

  7. 7.

    Flajolet, P. and Louchard, G.: Analytic variations on the Airy distribution, Algorithmica 31(3) (2001), 361–377.

    MathSciNet  Google Scholar 

  8. 8.

    Fulton, W.: Intersection Theory, 2nd edn, Ergeb. Math. Grenzgeb., Springer, Berlin, 1998.

    Google Scholar 

  9. 9.

    Göttsche, L.: Hilbert schemes of zero-dimensional subschemes of smooth varieties, Lecture Notes in Math. 1572, Springer, Berlin, 1994.

  10. 10.

    Haiman, M.: t,q-Catalan numbers and the Hilbert scheme, Selected papers in honor of Adriano Garsia (Taormina, 1994), Discrete Math. 193(1–3) (1998), 201–224.

    MATH  MathSciNet  Google Scholar 

  11. 11.

    King, A.: Moduli of representations of finite-dimensional algebras, Quart. J. Math. Oxford Ser. (2) 45 (1994), 180, 515–530.

    MATH  Google Scholar 

  12. 12.

    Kontsevich, M.: Noncommutative smooth spaces, Talk at the Arbeitstagung Bonn, 1999, Preprint MPI 1999-50-h.

  13. 13.

    Le Bruyn, L.: Nilpotent representations, J. Algebra 197 (1997), 153–177.

    Article  MATH  MathSciNet  Google Scholar 

  14. 14.

    Le Bruyn, L. and Procesi, C.: Semisimple representations of quivers, Trans. Amer. Math. Soc. 317 (1990), 585–598.

    MathSciNet  Google Scholar 

  15. 15.

    Le Bruyn, L. and Seelinger, G.: Fibers of generic Brauer–Severi schemes, J. Algebra 214(1) (1999), 222–234.

    Article  MathSciNet  Google Scholar 

  16. 16.

    Nakajima, H.: Varieties associated with quivers, In: Representation Theory of Algebras and Related Topics (Mexico City, 1994), CMS Conf. Proc. 19, Amer. Math. Soc., Providence, RI, 1996, pp. 139–157.

    Google Scholar 

  17. 17.

    Nori, M. V.: Appendix to the paper by C. S. Seshadri: Desingularisation of the moduli varieties of vector bundles over curves, Intl. Symp. on Algebraic Geometry, 1977, pp. 155–184.

  18. 18.

    Procesi, C.: The invariant theory of n×n matrices, Adv. Math. 19 (1976), 306–381.

    Article  MATH  MathSciNet  Google Scholar 

  19. 19.

    Reineke, M.: The Harder–Narasimhan system in quantum groups and cohomology of quiver moduli, Invent. Math. 152 (2003), 349–368.

    Article  MATH  MathSciNet  Google Scholar 

  20. 20.

    Reineke, M.: Framed quiver moduli, cohomology, and quantum groups, Preprint, 2004, math.AG/0411101.

  21. 21.

    Stanley, R. P.: Enumerative Combinatorics, Vol. 2, Cambridge Stud. Adv. Math. 62, Cambridge University Press, Cambridge, 1999.

    Google Scholar 

  22. 22.

    Takacs, L.: A Bernoulli excursion and its various applications, Adv. Appl. Probab. 23(3) (1991), 557–585.

    MATH  MathSciNet  Google Scholar 

  23. 23.

    Van den Bergh, M.: The Brauer–Severi scheme of the trace ring of generic matrices, In: Perspectives in Ring Theory (Antwerp, 1987), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., Vol. 233, Kluwer Academic, Dordrecht, 1988, pp. 333–338.

    Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Markus Reineke.

Additional information

Mathematics Subject Classifications (2000)

Primary: 16G20; secondary: 14D20.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Reineke, M. Cohomology of Noncommutative Hilbert Schemes. Algebr Represent Theor 8, 541–561 (2005). https://doi.org/10.1007/s10468-005-8762-y

Download citation

Keywords

  • representations of free algebras
  • moduli of representations
  • Hilbert schemes
  • Betti numbers