Equivariant Cohomological Chern Characters

sends M to the kernel of the map∏M(f): M(c) →f : d→cf not an isomorphism∏f : d→cf not an isomorphismM(d).From now on suppose that C is an EI-category, i.e. a small category such thatendomorphisms are isomorphisms. Then we can define the inclusion functorI c : MOD -R[c] → MOD -RC (2.12)by I c (M)(?) = M ⊗ R[c] R mor(?, c) if c ∼ = ? in C and by I c (M)(?) = 0 otherwise.Let B be the RC-R[c]-bimodule, covariant over C and a right module over R[c],given byB(c, ?) = R mor C(c, ?) if c ∼ = ?;0 if c ≁ = ?.Let C be the R[c]-RC-bimodule, contravariant over C and a left module overR[c], given byC(?, c) = R mor C(?, c) if c ∼ = ?;0 if c ≁ = ?.One easily checks that there are natural isomorphismsS c M ∼ = M ⊗ RC B;I c N ∼ = hom R[c] (B, N);T c M ∼ = hom RC (C, M);I c N ∼ = N ⊗ R[c] C.Lemma 2.13. Let C be an EI-category and c, d objects in C.(a) We obtain adjoint pairs (i(c) ∗ , i(c) ∗ ), (i(c) ∗ , i(c) ! ), (S c , I c ) and (I c , T c );∼ = ∼ =(b) There are natural equivalences of functors S c ◦i(c) ∗ −→ id and T c ◦i(c) ! −→id of functors MOD -R[c] → MOD -R[c]. If c ≁ = d, then Sc ◦ i(d) ∗ =T c ◦ i(d) ! = 0;(c) The functors S c and i(c) ∗ send projective modules to projective modules.The functors I c and i(c) ! send injective modules to injective modules.Proof. (a) follows from (2.5), (2.6) and (2.1).(b) This follows in the case T c ◦ i(d) ! from the following chain of canonicalisomorphismsT c ◦ i(d) ! (M) =hom RC (C(?, c), hom R[d] (R mor C (d, ?), M))∼ =−→ hom R[d] (C(?, c) ⊗ RC R mor C (d, ?), M) ∼= −→ hom R[c] (C(c, d), M),and analogously for S d ◦ i(c) ∗ .(c) The functors S c and i(c) ∗ are left adjoint to an exact functor and hencerespect projective. The functors T c and i(c) ! are right adjoint to an exactfunctor and hence respect injective.11