Varieties and local cohomology for chromatic group cohomology rings
John Greenlees and Neil Strickland
Following Quillen, 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. This also gives 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.