CHERN APPROXIMATIONS FOR GENERALISED GROUP COHOMOLOGY

Let G be a nite group, and let E be a generalised cohomology theory, subject to certain technical conditions (\admissibility " in the sense of [4]). Our aim in this paper is to dene and study a certain ring C(E;G) that is in a precise sense the best possible approximation to E0BG that can be built using only knowledge of the complex representation theory of G. There is a natural map C(E;G) −! E0BG, whose image is the subring C(E;G) E0BG generated over E0 by all Chern classes of all complex representations. There is ample precedent for considering this subring in the parallel case of ordinary cohomology; see for example [13, 14, 3]. However, although the generators of C(E;G) 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 Q ⊗ C(E;G). In the few cases that we have been able to analyse completely, either Q ⊗ C(E;G) 6 = Q ⊗ E0BG for easy character-theoretic reasons, or we have C(E;G) = E0BG. Rather than working directly with rings, we will study the formal schemes X(G) = spf(E0BG) and XCh(G) = spf(C(E;G)); note that the map C(E;G) −! E0BG corresponds to a map X(G) −! XCh(G). See [4, 9, 8] for foundational material on formal schemes. Suitably interpreted, our mai