Following Quillen [26, 27] we use the methods of algebraic geometry to study the ring E~*(BG) where E is a suitable complete periodic complex oriented theory and G is a finite group: we describe its variety in terms of the formal group associated to E, and the category of abelian p-subgroups of G. Our results considerably extend those of Hopkins-Kuhn-Ravenel [16], and this enables us to obtain information about the associated homology of BG. For example if E is the complete 2-periodic version of the Johnson-Wilson theory E(n) the irreducible components of the variety of the quotient E~*(BG)/I_k by the invariant prime ideal I_k = (p,v_1,...,v_(k - 1)) correspond to conjugacy classes of abelian p-subgroups of rank <= n - k. Furthermore, if we invert v_k the decomposition of the variety into irreducible pieces corresponding to minimal primes becomes a decomposition into connected components, corresponding to the fact that the ring splits as a product.
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