Equivariant Cohomological Chern Characters

We want to use Theorem 2.14 (b) to show that M is injective and analyse itsstructure. The R[W G H]-module T H M introduced in (2.11) is the same as thekernel of∏M(i K ): M(H) →∏M(K),KHKHwhere for each subgroup K H different from H we denote by i K the inclusion.Suppose that R[W G H]-module T H M is injective for every finite subgroup H ⊆G. For every finite subgroup H ⊆ G choose a retraction ρ H : M(H) → T H Mof the inclusion T H M → M(H). Denote by I = Is(Sub(G, F)) the set ofisomorphism classes of objects in Sub(G; F) which is the same as the set ofconjugacy classes (H) of finite subgroups H of G. Letν = ν(M): M → ∏i(K) ! ◦ T K (M) (5.1)(K)∈Ibe the homomorphism of RSub(G, F)-modules uniquely determined by theproperty that for any (K) ∈ I its composition with the projection onto thefactor indexed by (K) is the adjoint of ρ K : M(K) → T K M for the adjoint pair(i(K) ∗ , i(K) ! ).Theorem 5.2 (Injectivity and Mackey functors). Let G be a group andlet R be a commutative ring such that the order of every finite subgroup of Gis invertible in R. Suppose that the R[W G H]-module T H M is injective for eachfinite subgroup H ⊆ G. Then M is injective as RSub(G, F)-module and themap ν of (5.1) is bijective.Proof. The map ν of (5.1) is the map ν(M) appearing in Theorem 2.14 (b).Because of Theorem 2.14 (b) it suffices to show for each finite subgroup H ⊆ Gthat ν(M)(H) is surjective.Fix for any (K) ∈ I a representative K. Then choose for any W G K · f ∈W G K\ mor(K, H) an element f ∈ conhom(K, H) which represents a morphismf : K → H in Sub(G; F) which belongs to W G K ·f ∈ W G K\ mor(K, H). Noticethat W G K is the automorphism group of the object K in Sub(G; F) and W G K,mor(K, H) and W G K\ mor(K, H) are finite. With these choices we get for everyobject H in Sub(G; F) an identification∏i(K) ! T K M(H) = hom RWG K(R mor(K, H), T K M) =T K M W GK fW G K·f∈W G K\ mor(K,H)where W G K f ⊆ W G K is the isotropy group of f under the W G K-action onmor(K, H). Under this identification ν(H) becomes the mapν(H): M(H) →∏ ∏T K M W GK f(K)∈IW G K·f∈W G K\ mor(K,H)for which the component of ν(H)(m), which belongs (K) ∈ I and W G K · f ∈W G K\ mor(K, H), is ρ K ◦ res f (m) for m ∈ M(H). Notice that the image of21