The spectrum of the Chern subring 

D. J. Green and I. J. Leary

Abstract 
For certain subrings of the mod-p cohomology ring of a compact Lie
group, we give a description of the prime ideal spectrum, analogous to
Quillen's description of the spectrum of the whole ring. Examples of
such subrings include the Chern subring (the subring generated by
Chern classes of all unitary representations), and for finite groups
the subring generated by Chern classes of representations realizable
over any specified field.  

As a corollary, we deduce that the inclusion of the Chern subring in
the cohomology ring is an F-isomorphism for a compact Lie group G if
and only if the following condition holds: For any homomorphism f
between elementary abelian p-subgroups of G such that f(v) is always
conjugate to v, there is an element g of G such that f is equal to
conjugation by g.  

Comm. Math. Helvetici 73 (1998) 406-426.