VARIETIES AND LOCAL COHOMOLOGY FOR CHROMATIC GROUP COHOMOLOGY RINGS

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)/Ik by the invariant prime ideal Ik=(p, v1, …, vk-1) correspond to conjugacy classes of abelian p-subgroups of rank ⩽n−k. Furthermore, if we invert vk 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.