Chern approximations for generalised group cohomology
Neil P. Strickland
University of Sheffield
N.P.Strickland@sheffield.ac.uk
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.
Cheers,
Neil