Abstract
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 Ext *kG (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.
Similar content being viewed by others
References
D J Benson. Representations and Cohomology, I. Cambridge University Press, 1991
L Evens. The Cohomology of Groups. Oxford University Press, 1991
J F Carlson. Modules and Group Algebras. ETH Lecture Notes, Birkhäuser Verlag, 1996.
J. L Alperin, L Evens. Varieties and elementary abelian subgroups. J Pure Appl Algebra, 1982, 26: 221–227
D J Benson. Cohomology of modules in the principal block of a finite group. New York: J Math, 1995, 1: 196–205
D J Benson, J F Carlson, G R Robinson. On the vanishing of group cohomology. J Algebra, 1990, 131: 40–73
J Rickard. Idempotent modules in the stable category. J London Math Soc, 1997, 56: 149–170.
J F Carlson, C Peng, W W Wheeler. Transfer maps and virtual projectivity. J Algebra, (to appear)
D Happel. Triangulated Categories in the Representation Theory of Finite Dimensional Algebras. London Math Soc Lecture Notes No 119, Cambridge University Press, 1988
D J Benson, J F Carlson, J Rickard. Thick subcategories of the stable module category. Fund Math, 1997, 153: 59–80
D J Benson, J F Carlson, J Rickard. Complexity and varieties for infinitely generated modules, II. Math Proc Cambridge Phil Soc, 1996, 120: 597–615
D J Benson. The image of the transfer map. Arch Math, 1993, 61: 7–11
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, 1994
P Fong. Solvable groups and modular representation theory. Trans Amer Math Soc, 1962, 103: 484–494
Author information
Authors and Affiliations
Additional information
Partially supported by grants from NSF and EPSRC
Rights and permissions
About this article
Cite this article
Carlson, J.F. Varieties for cohomology with twisted coefficients. Acta Math Sinica 15, 81–92 (1999). https://doi.org/10.1007/s10114-999-0061-9
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/s10114-999-0061-9