Chern approximations for generalised group cohomology 
Neil P. Strickland 

Abstract: Let G be a finite group, and let E be a generalised cohomology 
theory, subject to certain technical conditions. We study a certain ring 
C(E,G) that is the best possible approximation to E^0BG that can be built 
using only knowledge of the complex representations of G. There is a natural 
map C(E,G) -> E^0BG, whose image is the subring of E^0BG generated over E^0 
by all Chern classes of such representations.

There is ample precedent for considering this subring in the parallel case of 
ordinary cohomology. However, although the generators of this subring come 
from representation theory, the same cannot be said for the relations; one 
purpose of our construction is to remedy this. We also also develop a kind of 
generalised character theory which gives good information about the 
rationalisation of C(E,G). In the few cases that we have been able to analyse 
completely, either C(E,G) is rationally different from E^0BG for easy 
character-theoretic reasons, or we have C(E,G)=E^0BG.

Rather than working directly with rings, we will study the formal schemes 
X(G)=spf(E^0BG) and XCh(G)=spf(C(E,G)). Suitably interpreted, our main 
definition is that XCh(G) is the scheme of homomorphisms from the 
Lambda-semiring R^+(G) of complex representations of G to the Lambda-semiring 
scheme of divisors on the formal group associated to E. 

Math Subject Class: 55R40; 55N20