Equivariant Cohomological Chern Characters

injective. We show by induction over the length l(x) of an object x ∈ C thatν(M)(x) is injective. Let u be an element in the kernel of ν(M)(x). Considera morphism f : y → x which is not an isomorphism. Then l(y) < l(x) and byinduction hypothesis ν(M)(y) is injective. Since the composite ν(M)(y) ◦ M(f)factorizes through ν(M)(x), we have u ∈ ker(M(f)). This implies u ∈ T x M.Consider the compositeT x M i −→ M(x) ν(M)(x)−−−−−→∏(c)∈Is(C)i(c) ! T c M(x) pr x−−→ i(x) ! T x M(x) −→ j T x M,where i is the inclusion, pr x is the projection onto the factor belonging to theisomorphism class of x and j is the isomorphism hom R[x] (R mor C (x, x), T x M) −→∼=T x M sending φ to φ(id x ). Since this composite is the identity on T x M and ulies in the kernel of ν(M)(x), we conclude u = 0.In particular we see that an injective RC-module M is trivial if and only ifi(d) ! T d M(x) is trivial for all objects d ∈ C.If ν(M) is bijective and each T c M is an injective R[c]-module, then M isan injective RC-module, since i(c) ! sends injective R[c]-modules to injectiveRC-modules by Lemma 2.13 (c) and the product of injective modules is againinjective.Now suppose that M is injective. Let N be the cokernel of ν(M). We havethe exact sequence0 → M ν(M)−−−→ ∏ (c)∈Is(C) i(c) ∗T c M pr−→ N → 0. (2.15)Since M is injective, this is a split exact sequence of injective RC-modules. Fixan object d. The functors i(d) ! and T d send split exact sequences to split exactsequences. Therefore we obtain a split exact sequence if we apply i(d) ! T d to(2.15). Using Lemma 2.13 (b) the resulting exact sequence is isomorphic to theexact sequence0 → i(d) ! T d M id−→ i(d) ! T d M → i(d) ! T d N → 0.Hence i(d) ! T d N vanishes for all objects d. This implies that N is trivial andbecause of (2.15) that ν(M) is bijective.For more details about modules over a category we refer to [7, Section 9A].3. The Associated Bredon Cohomology TheoryGiven a proper equivariant cohomology theory with values in R-modules, wecan associate to it another proper equivariant cohomology theory with values inR-modules satisfying the disjoint union axiom called Bredon cohomology, which13