Equivariant Cohomological Chern Characters

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The length l(c) ∈ N ∪ {∞} of an object c is the supremum over all naturalf 1 f 2 f 3numbers l for which there exists a sequence of morphisms c 0 −→ c1 −→ c2 −→f l. . . −→ cl such that no f i is an isomorphism and c l = c. The colength col(c) ∈N ∪ {∞} of an object c is the supremum over all natural numbers l for whichf 1 f 2 f 3 f lthere exists a sequence of morphisms c 0 −→ c1 −→ c2 −→ . . . −→ cl such that nof i is an isomorphism and c 0 = c. If each object c has length l(c) < ∞, we saythat C has finite length. If each object c has colength col(c) < ∞, we say thatC has finite colength.Theorem 2.14. (Structure theorem for projective and injective RCmodules).Let C be an EI-category. Then(a) Suppose that C has finite colength. Let M be a contravariant RC-modulesuch that the R aut(c)-module S c M is projective for all objects c in C.Let σ c : S c M → M(c) be an R aut(c)-section of the canonical projectionM(c) → S c M. Consider the map of RC-modulesµ(M):⊕(c)∈Is(C)i(c) ∗ S c M⊕(c)∈Is(C) i(c)∗σc−−−−−−−−−−−→⊕(c)∈Is(C)i(c) ∗ M(c)⊕(c)∈Is(C) α(c)−−−−−−−−−→ M,where α(c): i(c) ∗ M(c) = i(c) ∗ i(c) ∗ M → M is the adjoint of the identityi(c) ∗ M → i(c) ∗ M under the adjunction (2.5). The map µ(M) is alwayssurjective. It is bijective if and only if M is a projective RC-module;(b) Suppose that C has finite length. Let M be a contravariant RC-modulesuch that the R aut(c)-module T c M is injective for all objects c in C.Let ρ c : M(c) → T c M be an R aut(c)-retraction of the canonical injectionT c M → M(c). Consider the map of RC-modulesν(M): M∏(c)∈Is(C) β(c)−−−−−−−−−→∏(c)∈Is(C)i(c) ! M(c)∏(c)∈Is(C) i(c) !ρ c−−−−−−−−−−−→∏(c)∈Is(C)i(c) ∗ T c Mwhere β(c): M → i(c) ! i(c) ∗ M = i(c) ! M(c) is the adjoint of the identityi(c) ∗ M → i(c) ∗ M under the adjunction (2.6). The map ν(M) is alwaysinjective. It is bijective if and only if M is an injective RC-module.Proof. (a) A contravariant RC-module is the same as covariant RC op -module,where C op is the opposite category of C, just invert the direction of every morphisms.The corresponding covariant version of assertion (a) is proved in [8,Theorem 2.11].(b) is the dual statement of assertion (a). We first show that ν(M) is always12
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
Page 1 and 2: Equivariant Cohomological Chern Cha
Page 3 and 4: The paper is organized as follows:1
Page 5 and 6: Example 1.3 (Rationalizing topologi
Page 7 and 8: Example 1.7 (Equivariant K-theory).
Page 9 and 10: 2. Modules over a CategoryIn this s
Page 11: sends M to the kernel of the map∏
Page 15 and 16: Given a contravariant ROr(G, F)-mod
Page 17 and 18: We get a natural transformation ch
Page 19 and 20: y the composite∏p+q=n chp,q(X,A)H
Page 21 and 22: We want to use Theorem 2.14 (b) to
Page 23 and 24: h −1 im(g)h = im(f) and therefore
Page 25 and 26: if we defineT H (K q (BH)) :=⎛⎞
Page 27 and 28: the G-cohomology theory BHG ∗ . C
Page 29 and 30: References[1] P. Baum, A. Connes, a

Equivariant Cohomological Chern Characters

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