Skip to main content
SpringerLink
Log in

Rank varieties and projectivity for a class of local algebras

  • Published:
Mathematische Zeitschrift Aims and scope Submit manuscript

Abstract.

We consider algebras where K is an algebraically closed fields. To any finite dimensional module for this algebra we associate a rank variety. When char(K) = 2 we recover Carlson’s rank variety. The main result states that a module is projective if and only if its rank variety vanishes. This has applications to other algebras, including tensor products of certain Brauer tree algebras and certain parabolic Hecke algebras. In addition, the result has implications for the graph structure of the stable Auslander-Reiten quiver. Mathematics Subject Classification (2000): 16G10, 18G05, 20C20

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Adams, J.F.: Lectures on Exceptional Lie Groups. Chicago Lectures in Mathematics, 1996

  2. Alperin, J.L., Evens, L.: Varieties and elementary abelian groups. J. Pure Appl. Algebra 26, 221–227 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  3. Atiyah, M.F., Bott, R., Shapiro, A.: Clifford Modules. Topology 3, 3–38 (1964)

    Article  Google Scholar 

  4. Benson, D.J.: Representations and cohomology Vol I. Cambridge studies in advanced mathematics 30, (1991)

  5. Benson, D.J.: Representations and cohomology Vol II. Cambridge studies in advanced mathematics 31, (1991)

  6. Benson, D.J., Green, E.L.: Quantum complete intersections, and nonprincipal blocks of finite groups. Preprint 2002

  7. Carlson, J.F.: The varieties and the cohomology ring of a module. J. Algebra 85, 104–143 (1983)

    MathSciNet  MATH  Google Scholar 

  8. Carlson, J.F.: Varieties for modules. The Arcata Conference on Representations of Finite Groups (Arcata, Calif., 1986), Proc. Sympos. Pure Math., 47, Part 1, Amer. Math. Soc., Providence, RI, 1987, pp. 37–44

  9. Chouinard, L.G.: Projectivity and relative projectivity over group rings. J. Pure Appl. Algebra 7, 287–302 (1976)

    Article  MATH  Google Scholar 

  10. Dade, E.C.: Endo-Permutation Modules II. Ann. Math. 108, 317–346 (1978)

    MathSciNet  MATH  Google Scholar 

  11. Dipper, R., Du, J.: Trivial and alternating source modules for Hecke algebras of type A. Proc. London Math. Soc. 66, 479–506 (1993)

    MathSciNet  MATH  Google Scholar 

  12. Du, J.: The Green correspondence for the representations of Hecke algebras of type A r-1. Trans. Amer. Math. Soc. 329, 273–287 (1992)

    MathSciNet  MATH  Google Scholar 

  13. Erdmann, K.: On Auslander-Reiten components for group algebras. J. Pure Appl. Algebra 104, 149–160 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  14. Erdmann, K.: Type A Hecke algebras and related algebras. Preprint 2002. To appear in Arch. Math.

  15. Friedlander, E., Parshall, B.: Geometry of p-nilpotent Lie algebras. J. Algebra 109, 25–45 (1987)

    MathSciNet  MATH  Google Scholar 

  16. Happel, D., Preiser, U., Ringel, C.M.: Vinberg’s characterization of Dynkin diagrams using subadditive functions with application to D Tr-periodic modules. Representation theory, II (Proc. Second Internat. Conf., Carleton Univ., Ottawa, Ont., 1979), Lecture Notes in Math., Vol. 832. Springer, Berlin, 1980, pp. 280–294

  17. Jantzen, J.C.: Kohomologie von p-Lie algebren und nilpotente Elemente. Abh. Math. Sem. U. Hamburg 76, (1986)

  18. Reiten, I., Riedtmann, C.: Skew Group Algebras in the Representation Theory of Artin Algebras. J. Algebra 92, 225–282 (1985)

    Google Scholar 

  19. Webb, P.: The Auslander-Reiten quiver of a finite group. Math. Z. 179, 97–121 (1982)

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. University of Oxford, Mathematics Institute, 24-29 St.Giles, Oxford, England

    Karin Erdmann & Miles Holloway

Authors
  1. Karin Erdmann
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Miles Holloway
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Karin Erdmann.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Erdmann, K., Holloway, M. Rank varieties and projectivity for a class of local algebras. Math. Z. 247, 441–460 (2004). https://doi.org/10.1007/s00209-003-0536-9

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00209-003-0536-9

Keywords

Search

Navigation