Acta Mathematica Sinica

, Volume 15, Issue 1, pp 81–92 | Cite as

Varieties for cohomology with twisted coefficients

  • Jon F. Carlson


Let G be a finite group and k a field of characteristic p > 0. In this paper we consider the support variety for the cohomology module ExtkG*(M, N) where M and N are kG-modules. It is the subvariety of the maximal ideal spectrum of H*(G, k) of the annihilator of the cohomology module. For modules in the principal block we show that that the variety is contained in the intersections of the varieties of M and N and the difference between the that intersection and the support variety of the cohomology module is contained in the group theoretic nucleus. For other blocks a new nucleus is defined and a similar theorem is proven. However in the case of modules in a nonprincipal block several new difficulties are highlighted by some examples.


Group Cohomology Support varieties Cohomological varieties Annihilators of cohomology 

1991MR Subject Classification

20C20 20C06 


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  1. [1]
    D J Benson. Representations and Cohomology, I. Cambridge University Press, 1991Google Scholar
  2. [2]
    L Evens. The Cohomology of Groups. Oxford University Press, 1991Google Scholar
  3. [3]
    J F Carlson. Modules and Group Algebras. ETH Lecture Notes, Birkhäuser Verlag, 1996.Google Scholar
  4. [4]
    J. L Alperin, L Evens. Varieties and elementary abelian subgroups. J Pure Appl Algebra, 1982, 26: 221–227MATHCrossRefMathSciNetGoogle Scholar
  5. [5]
    D J Benson. Cohomology of modules in the principal block of a finite group. New York: J Math, 1995, 1: 196–205Google Scholar
  6. [6]
    D J Benson, J F Carlson, G R Robinson. On the vanishing of group cohomology. J Algebra, 1990, 131: 40–73MATHCrossRefMathSciNetGoogle Scholar
  7. [7]
    J Rickard. Idempotent modules in the stable category. J London Math Soc, 1997, 56: 149–170.MATHCrossRefMathSciNetGoogle Scholar
  8. [8]
    J F Carlson, C Peng, W W Wheeler. Transfer maps and virtual projectivity. J Algebra, (to appear)Google Scholar
  9. [9]
    D Happel. Triangulated Categories in the Representation Theory of Finite Dimensional Algebras. London Math Soc Lecture Notes No 119, Cambridge University Press, 1988Google Scholar
  10. [10]
    D J Benson, J F Carlson, J Rickard. Thick subcategories of the stable module category. Fund Math, 1997, 153: 59–80MATHMathSciNetGoogle Scholar
  11. [11]
    D J Benson, J F Carlson, J Rickard. Complexity and varieties for infinitely generated modules, II. Math Proc Cambridge Phil Soc, 1996, 120: 597–615CrossRefMathSciNetGoogle Scholar
  12. [12]
    D J Benson. The image of the transfer map. Arch Math, 1993, 61: 7–11MATHCrossRefMathSciNetGoogle Scholar
  13. [13]
    M Broué. Equivalences of blocks of group algebras. Finite Dimensional Algebras and Related Topics (Ottawa, ON, 1992) NATO Adv Sci Inst, Ser C Math Phys Sci, 424, Kluwer Acad Pub, Dordrecht, 1994Google Scholar
  14. [14]
    P Fong. Solvable groups and modular representation theory. Trans Amer Math Soc, 1962, 103: 484–494MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 1999

Authors and Affiliations

  • Jon F. Carlson
    • 1
  1. 1.Department of MathematicsUniversity of GeorgiaAthensUSA

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