Varieties for cohomology with twisted coefficients
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.
KeywordsGroup Cohomology Support varieties Cohomological varieties Annihilators of cohomology
1991MR Subject Classification20C20 20C06
Unable to display preview. Download preview PDF.
- D J Benson. Representations and Cohomology, I. Cambridge University Press, 1991Google Scholar
- L Evens. The Cohomology of Groups. Oxford University Press, 1991Google Scholar
- J F Carlson. Modules and Group Algebras. ETH Lecture Notes, Birkhäuser Verlag, 1996.Google Scholar
- D J Benson. Cohomology of modules in the principal block of a finite group. New York: J Math, 1995, 1: 196–205Google Scholar
- J F Carlson, C Peng, W W Wheeler. Transfer maps and virtual projectivity. J Algebra, (to appear)Google Scholar
- D Happel. Triangulated Categories in the Representation Theory of Finite Dimensional Algebras. London Math Soc Lecture Notes No 119, Cambridge University Press, 1988Google Scholar
- 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