Equivariant Cohomological Chern Characters

Consider a functor F : C → D. Given a RD-module M, define its restrictionwith F to be F ∗ M := M ◦F . Given a contravariant RC-module M, its inductionwith F is the contravariant RD-module F ∗ M given by(F ∗ M)(??) := M(?) ⊗ RC R mor D (??, F (?)), (2.3)and coinduction with F is the contravariant RD-module F ! M given by(F ! M)(??) := hom RC (R mor D (F (?), ??), M(?)). (2.4)Restriction with F can be written as F ∗ N(?) = hom RD (R mor D (??, F (?)), N(??)),the natural isomorphisms sends n ∈ N(F (?)) to the mapR mor D (??, F (?)) → N(??),φ: ?? → F (?) ↦→ N(φ)(n).Restriction with F can also be written as F ∗ N(?) = R mor D (F (?), ??) ⊗ RDN(??), the natural isomorphisms sends φ ⊗ RD n to N(φ)(n). We conclude from(2.2) that (F ∗ , F ∗ ) and (F ∗ , F ! ) form adjoint pairs, i.e. for a RC-module M anda RD-module N there are natural isomorphisms of R-moduleshom RD (F ∗ M, N)hom RD (F ∗ N, M)∼ =−→ hom RC (M, F ∗ N); (2.5)∼ =−→ hom RC (N, F ! M). (2.6)Consider an object c in C. Let aut(c) be the group of automorphism of c.We can think of âut(c) as a subcategory of C in the obvious way. Denote byi(c): âut(c) → Cthe inclusion of categories and abbreviate the group ring R[aut(c)] by R[c] inthe sequel. Thus we obtain functorsThe projective splitting functorsends M to the cokernel of the map⊕M(f):f : c→df not an isomorphismThe injective splitting functori(c) ∗ : MOD -RC → MOD -R[c]; (2.7)i(c) ∗ : MOD -R[c] → MOD -RC; (2.8)i(c) ! : MOD -R[c] → MOD -RC. (2.9)S c : MOD -RC → MOD -R[c] (2.10)⊕f : c→df not an isomorphismM(d) → M(c).T c : MOD -RC → MOD -R[c] (2.11)10