By S. A. Amitsur, D. J. Saltman, George B. Seligman
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Stud. 138, Princeton Univ. Press, Princeton, NJ, 1995. [Becker and Gottlieb 1975] J. C. Becker and D. H. Gottlieb, “The transfer map and fiber bundles”, Topology 14 (1975), 1–12. [Benson 1991a] D. J. Benson, Representations and cohomology, I: Basic representation theory of finite groups and associative algebras, Cambridge Studies in Advanced Mathematics 30, Cambridge University Press, Cambridge, 1991. Paperback reprint, 1998. [Benson 1991b] D. J. Benson, Representations and cohomology, II: Cohomology of groups and modules, Cambridge Studies in Advanced Mathematics 31, Cambridge University Press, Cambridge, 1991.
1995; 1996] for infinitely generated modules. Let G be a finite group, and let k be an algebraically closed field of characteristic p. We write VG for the maximal ideal spectrum of H ∗ (G, k). This is a closed homogeneous affine variety. For example, if G ∼ = (Z/p)r then VG = Ar (k), affine r-space over k. 1 can be interpreted as saying that for any finite group G, the natural map lim VE → VG −→ E∈AG is bijective at the level of sets of points. However, it is usually not invertible in the category of varieties.
A discrete group G is said to be a duality group of dimension d over k (see [Bieri 1976]) if there is a dualizing module. This is defined to be a kG-module I such that there are isomorphisms H i (G, M ) ∼ = Hd−i (G, I ⊗k M ) for all kG-modules M . It turns out that such isomorphisms may be taken to be functorial in M if they exist at all, and in that case, I ∼ = H d (G, kG). A Poincar´e duality group is a duality group for which the dualizing module I is isomorphic to the field k with some G-action, and it is orientable if the action is trivial.