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.
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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
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Partially supported by grants from NSF and EPSRC
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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
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DOI: https://doi.org/10.1007/s10114-999-0061-9