Generators of C G

In this subsection we will introduce some morphisms in C^G and then show that they serve as basic building blocks of generators ofC^G.

Definition 15.9. For g∈G, let us define elements in KK^G(C(G)^H,C(G)^H) by t^H_g :=^h(λ_g)_|C(G)H

i and r^H_g := [χ_g]⊗_C1_C(G)H

In words,t^H_g is given by translation by the group elementgandr_g^H by the outer tensor product with the character χ_g.

Lemma 15.10. Let g∈G and K, H be subgroups of G and x ∈KK^G_∗(C(G)^K,C(G)^H), then t^H_g ·x=x·t^H_g andr^K_g ·x=x·r_g^H.

Proof. The first equality follows since [α_g]·x = x·[β_g] for G-algebras (A, α),(B, β), g∈Gandx∈KK^G(A, B) and the second since the outer tensor product with elements inKK^G(C,C) is commutative.

Definition 15.11. Let K ≤H ≤G, let ι^K_H ∈KK^G(C(G)^H,C(G)^K) be the class of the inclusion C(G)^H ,→ C(G)^K.

Still assuming K ≤ H ≤ G, the right module structure over C(G)^H and an inner product given by

hx, yi(s) = Z

H

x(s+t)y(s+t)dt, x, y∈ C(G)^K (15.12) turn C(G)^K into a Kasparov-C(G)^K-C(G)^H-module, which will be denoted by R^H_K.

Definition 15.13. Let ρ^H_K = [(R^H_K,0, λ)] ∈ KK^G(C(G)^K,C(G)^H) denote the corres-ponding KK-element.

Recall that forH≤G, there are induction and restriction functors Ind^G_H:KK^H →KK^G and Res^H_G: KK^G → KK^H (see [23] Definition 3.1 and Section 3.6 or [27], Section 3.6 for a more functorial approach). Res^H_G is given by restricting the action to the subgroup H. If (B, β) is anH-algebra, then in our case (ofGbeing finite abelian), Ind^G_H(B, β) is given as aC^∗-subalgebra ofC(G, B):

Ind^G_H(B, β) ={f ∈ C(G, B)|f(g+h) =β−h(f(g))},

where G acts by left translation. There is an isomorphism of functors Ind^G_HRes^H_G = (_)⊗_CC(G)^H ([27], Equation (18)). Furthermore we will use the adjointness relations

KK^G(A,Ind^G_HB)∼= KK^H(Res^H_GA, B) and KK^G(Ind^G_HB, A)∼= KK^H(B,Res^H_GA) for aG-algebra A and anH-algebra B ([27], Equations (19) and (20)).

Definition 15.14. LetH and Lbe arbitrary subgroups ofG. Set µ^L_H :=ρ^H+L_H ·ι^L_H+L.

Proposition 15.15. Let L and H be arbitrary subgroups of G. Let (gi)_i∈I be a set of representatives ofG/(H+L) and (s_j)_j∈J a set of representatives of G/(H∩L)^⊥. Then KK^G_∗(C(G)^H,C(G)^L) is a free Z-module, concentrated in degree 0 with basis

t^H_gi ·µ^L_H ·r_s^L_j

(i,j)∈I×J. Proof. We have

ρ^H+L_H ·t^L+H_gi ·ι^L_H+L·r_s^L_j =t^H_gi·ρ^H+L_H ·ι^L_H+L·r^L_sj =t^H_gi ·µ^L_H ·r_s^L_j

by Lemma 15.10. Hence it is sufficient to show that KK^G_∗(C(G)^H,C(G)^L) is a free Z-module, concentrated in degree 0 with basis

ρ^H+L_H ·t^L+H_gi ·ι^L_H+L·r_s^L_j

(i,j)∈I×J.

The unital embeddingu:C,→ C(H)^H∩L= Ind^H_H∩LRes^H∩L_H Cis the unit of the adjoint-ness relation

KK^H∩L(C,Res^H∩L_H (_))∼= KK^H(Ind^H_H∩LC,_).

The counit of the adjointness relation

KK^G(Ind^G_H(_),C(G)^L)∼= KK^H(_,Res^H_GC(G)^L)

will be denoted by c: Ind^G_HRes^H_GC(G)^L → C(G)^L. c is given by [τ]·(ρ^G_H ⊗_C1_C(G)L), where

τ: Ind^G_HRes^H_GC(G)^L∼=C(G)^H ⊗_CC(G)^L, τ(f)(g, g⁰) =f(g)(g⁰−g).

Note that inKK^H, there is an isomorphism to show that there is an isomorphism of Kasparov C(G)^H-C(G)^L-modules

(Ti)^∗R^G_H⊗_C1_C(G)L

∼= (ι^L_H+L◦λgi)∗R^H_H^+L.

In the following, we will identifyC(G)^H⊗_CC(G)^LwithC(G×G)^H×L. Since (T_i)^∗R^G_H⊗_C1_C(G)L)=_vs Ti· C(G)^H ⊗_CC(G)^L, the left-hand-side is given as a vector space by

T_i(1)· C(G)^H ⊗ C(G)^L=_vs{f ∈ C(G×G)^H×L|f(g, g⁰) = 1_H+L−gi(g−g⁰)f(g, g⁰)}.

The left module structure is given byT_i, the right module structure by 1⊗id_C(G)L and the inner product byhf, f⁰i(g⁰) =^R_Gf(g, g⁰)f⁰(g, g⁰) dg.

To describe (ι^L_H+L◦λ_gi)_∗R_H^H+L, let us first set

Λ_i :=ι^L_H+L◦λgi:C(G)^H+L→ C(G)^L, Λ_i(f)(g) =f(g−gi).

Then (ι^L_H+L◦λgi)_∗R^H+L_H =C(G)^H⊗_ΛiC(G)^L. Left and right module structure are given by id_C(G)H ⊗_Λi 1 and 1⊗_Λiid_C(G)L, respectively, and the inner product is given by

hf₀⊗_Λif₀⁰, f1⊗_Λi f₁⁰i(g) =f₀⁰(g) Z

H+L

f0(g−gi+r)f1(g−gi+r)drf₁⁰(g).

Hence (ι^L_H+L ◦λ_gi)_∗R^H_H^+L as a vector space is a quotient and (T_i)^∗R^G_H ⊗_C1_C(G)L

is a subspace of C(G×G)^H×L. Using the description of the inner products above, a straightforward calculation show that

φ:C(G×G)^H×L→ C(G×G)^H×L, f 7→ (g, g⁰)7→1_H+L−gi(g−g⁰)f(g, g⁰) factors through an injective map Φ :C(G)^H⊗_ΛiC(G)^L→ C(G)^H⊗_CC(G)^L, which inter-twines the inner products. It is also easy to check that im Φ = T_i(1)· C(G)^H ⊗ C(G)^L (i.e., that Φ is surjective) and that it intertwines left and right multiplication. Therefore, Φ is an isomorphism of KasparovC(G)^H-C(G)^L-modules

(ι^L_H+L◦λgi)∗R^H_H^+L−^∼=→(Ti)^∗R^G_H ⊗_C1_C(G)L

. 15.3 Relations in C^G

In this subsection we will derive some properties of the basic building blocks for gener-ators ofC^G.

Lemma 15.16. One has I(r^H_g ) =t^H_g^⊥ and I(t^H_g ) =r^H_g ^⊥. Proof. By definition,I(r^H_g ) = [X^H]⁻¹·Gn [χg]⊗_C1_C(G)H

·[X^H]. Letλ^H denote the translation action restricted to C(G)^H. Using the formulas on Baaj-Skandalis duality (formula (15.4) and (15.5)), it is easy to check that

Gn [χ_g]⊗_C1_C(G)H

= [(^dλ^H)_g]∈KK^G(GnC(G)^H, GnC(G)^H),

where^dλ^H is the dual action ofG on GnC(G)^H. Since [αg]·x =x·[βg] for G-algebras (A, α),(B, β),g∈Gandx∈KK^G(A, B), we obtain the first equality. ApplyingI yields the second equality.

Lemma 15.17. Let h∈H, then t^H_h = 1_C(G)H and r_h^H⊥ = 1_C(G)H⊥.

Proof. t^H_h = 1_C(G)H forh∈H is trivial, the other equality follows by applying I.

Lemma 15.18. One hasI(ι^K_H) =ρ^K_H^⊥⊥ and I(ρ^H_K) =ι^H_K^⊥⊥.

Proof. By definition,I(ι^H_K) = (X^H)⁻¹·(Gnι^K_H)·(X^K). Hence we have to show that the KasparovGnC(G)^H-C(G)^K⊥-modules (Gnι^K_H)^∗X^KandX^H⊗_C(G)H⊥R^K_H⊥^⊥are isomorphic.

Recall that X^K = X^H = C(G) and R^K_H⊥^⊥ = C(G)^H⊥ as vector spaces. A lengthy computation using the description of X^K and X^H given by the formulas (23.2) - (23.5) and (15.12) shows that

(Gnι^K_H)^∗X^K →X^H⊗_C(G)H⊥R^K_H^⊥⊥, f 7→f ⊗_C(G)H⊥

1

|K^⊥|^1/2,

is a well-defined isomorphism of Kasparov GnC(G)^H-C(G)^K⊥-modules. This shows I(ι^K_H) =ρ^K_H^⊥⊥. The other equality follows by applying I.

Lemma 15.19. Let K ≤H ≤L≤G. We haveι^H_L ·ι^K_H =ι^K_L and ρ^H_K·ρ^L_H =ρ^L_K. Proof. The first equality is obvious, the second one follows by applyingI.

In the next lemma, we will show that ι^K_H and ρ^H_K behave nicely under induction.

Recall that Ind^G_H is a functor fromKK^H toKK^G. To avoid confusion, we will denote the morphisms ι^K_H andρ^H_K ofKK^H byι^K_H(H) andρ^H_K(H), respectively, and similarly for the corresponding morphisms in KK^G.

Lemma 15.20. Let K ≤H ≤G. Under the identification Ind^G_HC(H)^K ∼=C(G)^K, f 7→(g7→f(g,0)) we have

Ind^G_H(ι^K_H(H)) =ι^K_H(G) and Ind^G_H(ρ^H_K(H)) =ρ^H_K(G)

Proof. The first equality follows straight from the definition. For the second equality, recall that ρ^H_K(H) = [R_K^H(H)] and that the Kasparov bimodule R^H_K(H) is equal to C(H)^K as a vector space. The isomorphism Ind^G_HC(H)^K ∼= C(G)^K, regarded as an isomorphism of vector spaces, can be applied to the Kasparov bimodule representing Ind^G_Hρ^H_K(H) and yields an isomorphism of KasparovC(G)^K-C(G)^H-bimodulesR^H_K(G)∼= Ind^G_HR^H_K(H).

Proposition 15.21. Let K ≤H≤G, then ι^K_H ·ρ^H_K= ^X

g∈K^⊥/H^⊥

r^H_g ∈KK^G(C(G)^H,C(G)^H).

Proof. Note thatι^{0}_G ·ρ^G_{0} ∈KK^G(C,C) is the class of the representation ofGonL²(G) via left translation. Hence, by the Peter–Weyl Theorem [32], we have

ι^{0}_G ·ρ^G_{0} = ^X

φ∈^Gb [φ].

Pullback via the quotient map π:H H/K yields π^∗: KK^H/K → KK^H. Applying π^∗ to the equality above (for G = H/K) and using the identification C(H/K) ∼= C(H)^K, we obtain

ι^K_H ·ρ^H_K = ^X

φ∈\H/K

[φ◦π]∈KK^H(C(H)^H,C(H)^H).

Applying Ind^G_H and Lemma 15.20, we obtain ι^K_H ·ρ^H_K = ^X

Proof. This follows by applyingI to the statement of the last proposition.

16 Projective Resolutions of Length 1

From now on letGdenote a finite cyclicgroup Gof order o. The aim of this section is to show that every module overC^G⊗Z[o⁻¹] has projective dimension 1.

Let us first give an outline of the argument: For every finite cyclic group G, there is a decomposition G = ^LG_i, where each G_i is a finite cyclic group of prime power orderp_i. C^G⊗Z[o⁻¹] can be decomposed into the tensor product of categoriesC^Gi[p⁻¹_i ].

We will introduce the notion of a split category over a family of rings {R_i}_i∈I, which basically means that the category of modules over this category is equivalent to modules over^L_IRi. If everyRi is a direct sum of Dedekind domains, modules over^L_IRi have projective dimension 1. The same is true for modules over a category which is split over{R_i}_i∈I. We will then calculate C^Gi ⊗Z[p_i⁻¹] explicitly in terms of hand-selected generators and relations to show that it is indeed split over a family of Dedekind domains.

Together with the fact that tensor products of split categories are again split, this implies the claim.

16.1 Split Categories and Tensor products

Definition 16.1. Let C be a finite preadditive category and {R_i}_i∈I a finite family of unital commutative rings. Let 1_i denote the unit in Ri. Cis called split over {R_i}_i∈I if and only if:

(1) For allA, B ∈∈C, there are subsets I(A, B)⊆I such that

I(A, B) =I(B, A), I(A, B)∩I(B, C)⊂I(A, C) and I = ^[

A∈∈C

I(A, A).

(2) For A, B∈∈C , there is an isomorphism of abelian groups γ_A^B: ^M

i∈I(A,B)

Ri ∼=C(A, B).

(3) If i∈I(A, B), j ∈I(B, C), r∈R_i, s∈R_j, then composition inC is given by γ_A^B(r1i)·γ_B^C(s1j) =δi,jγ_A^C(rs1i).

A trivial example of a split category over{R_i}_i∈I is the categoryCwith objectsI and morphisms

C(i, j) :=

(R_i ifi=j, 0 otherwise..

Note that the morphism groups of a finite preadditive category C, which is split over {R_i}_i∈I carry a module structure overR:=^L_i∈IR_i: For i∈I,A, B∈∈C,j∈I(A, B) and r∈Ri, s∈Rj set

r1_i·γ_A^B(s1_j) :=δ_i,jγ^B_A(rs1_j).

This module structure is compatible with composition in the sense that forA, B, C∈∈C, r, s∈R and x∈C(A, B) andy∈C(B, C),

(r·x)·(s·y) = (rs)·(x·y).

Lemma 16.2. Let C be split over {R_i}_i∈I and set R:=^L_IR_i. Then there are equival-ences of categories.

Mod(C)'Mod(R), Mod(C)_c'Mod(R)_c

Proof. We have to define additive functors

Mod(C)−^F→Mod(R)−^G→Mod(C)

and show that there are isomorphisms F ◦G∼= id_Mod(R) and G◦F ∼= id_Mod(C). Let us first define G: Mod(R) → Mod(C): For A ∈∈ C and M ∈∈Mod(R), set G(M)(A) :=

L

i∈I(A,A)1_iM and for A, B ∈∈C,j∈I(A, B) and r∈R_j, let G(M)γ_A^B(r1_j):G(M)(A)→G(M)(B)

be given by the composition G(M)(A) = ^M

i∈I(A,A)

1_iM 1_jM −−→^r1j 1_jM ^M

i∈I(B,B)

1_iM =G(M)(B).

It is easy to check that this gives a well-defined additive functor. For the definition of F, choose a mapσ, which assigns toi∈I an objectσ(i)∈∈Csuch thati∈I(σ(i), σ(i)) for alli∈I and define an idempotent

p_i :=γ_σ(i)^σ(i)(1_i)∈C(σ(i), σ(i)).

ForN ∈∈Mod(C), let=N(pi) denote the image of N(pi) :N(σ(j))→N(σ(j)) and set F(N) :=^M

i∈I

=N(p_i).

Forj∈I andr ∈R_j, define multiplication byr1_j by the composition M

i∈I

=N(p_i)=N(p_j)−−−−→ =N^N(rpj) (p_j)^M

i∈I

=N(p_i).

It is not hard to check that this gives a well-defined additive functor.

For M ∈∈ Mod(R), compute F ◦G(M) = ^L_i∈I=G(M)(p_i) = ^L_i∈I1_i ·M. Since for every R-module M, there is a natural isomorphism M ∼= ^L_i∈I1_i ·M, we ob-tain a natural isomorphism F ◦G ∼= id_Mod(R). For N ∈∈ Mod(C), we compute G◦ F(N)(A) = ^L_i∈I(A,A)=N(p_i). For i ∈ I(A, A), N(γ_A^σ(i)(1_i)) yields an isomorphism

=N(p_i)∼=Ng^A_A(i)N(A) with inverseN(γ_σ(i)^A (1_i)). This gives a natural isomorphism Φ_N,A:G◦F(N)(A) = ^M

i∈I(A,A)

=N(pi)∼= ^M

i∈I(A,A)

Ng_A^A(i)N(A)∼=N(A).

We obtain a natural isomorphism Φ_N := (Φ_N,A)_A∈∈C: G ◦ F(N) ∼= N and Φ :=

(Φ_N)_{N∈∈Mod(C)} gives the desired natural isomorphism G◦F ∼= id_Mod(C). This shows that F and G implement an equivalence Mod(C) 'Mod(R). It is clear that F and G preserve the countability condition. Therefore, we also obtain an equivalenceMod(C)_c' Mod(R)c.

Corollary 16.3. Let C be split over {R_i}_i∈I such that every Ri is isomorphic to a countable direct sum of countable Dedekind domains. Then every object inMod(C)_c has a projective resolution of length1.

Let J be a finite index set and for j ∈ J, C_j a finite preadditive category. In the following, a tensor product without subscript will always denote the algebraic tensor product overZ(of rings, categories or modules).

Definition 16.4. LetD be a preadditive category. A functor F: ^Q_JC_j →D is called

The tensor product is associative up to an isomorphism of categories.

Lemma 16.7. Let J be a finite index set and for j∈J let {R_i,j}_i∈I

Define γ_A^B by commutativity of the following diagram L

This endows^N_JC_j with the structure of a category, which is split over^nN_JR_ij,j^o

(ij)j∈J∈I. 16.2 A Special Case

Let us consider the caseG=C(pⁿ),n∈Nandpprime. Our aim is to show thatC^G[p⁻¹] is split over a family of Dedekind domains. We will be using facts about cyclotomic polynomials and decompositions of certain rings into Dedekind domains. For better readability, we collected these purely algebraic statements in the appendix, Section 22.

To simplify notation, letCdenote the category with objects {0,1, . . . , n}and morph-isms

C(k, l) := KK^C(pn)C(C(pⁿ)^C(pk)),C(C(pⁿ)^C(pl))[p⁻¹].

Cis obviously isomorphic to C^G[p⁻¹], G=C(pⁿ).

Definition 16.8.

I :={(u, v)| ∃k, 0≤k≤nsuch that 0≤u≤n−k, 0≤v≤k}.

For (u, v)∈I, define

R_(u,v):=Z[p⁻¹][t]/hΦ_p^u(t)i⊗Z[p⁻¹][r]/hΦ_p^v(r)i, where Φ_m denotes the mth cyclotomic polynomial

Φ_m(t) := ^Y

ω mth primitive root of unity

(t−ω).

Having made these definitions, we can be more precise about what we want to prove in this subsection:

Theorem 16.9. C is split over ⁿR_(u,v)|(u, v)∈I^o.

Note that this implies that all countable Z/2-graded modules over C have a pro-jective resolution of length 1: Z[p⁻¹][t]/hΦ_p^u(t)i is isomorphic to the Dedekind domain Z[p⁻¹, θp^u] by Lemma 22.2, and Proposition 22.11 yields an isomorphism of rings

R_(u,v) ∼=Z[p⁻¹, θp^u]⊗Z[p⁻¹, θp^v]∼= ^M

0<k<p^min(u,v),gcd(p,k)=1

Z[p⁻¹, θ_pmax(u,v)].

Therefore, all objects inMod(C)^Z_c^/2 have projective dimension 1 by Corollary 16.3.

The verification of conditions (1)–(3) in the definition of a split category or equival-ently, the proof of Theorem 16.9 will occupy the remainder of this section.

Definition 16.10. For 0≤k, l ≤n, set

I(k, l) :={(u, v)|0≤u≤n−k∨l, 0≤v≤k∧l}.

Herek∨l:= max(k, l) andk∧l:= min(k, l).

It is immediate thatI(k, l) =I(l, k),I(k, l)∩I(l, m)⊂I(k, m) andI =^S_k=0,...,nI(k, k).

Hence condition (1) in Definition 16.1 holds.

The next step is to define the isomorphisms of abelian groups, which are part of condition (2) in Definition 16.1:

γ_k^l: ^M

(u,v)∈I(k,l)

R_(u,v)∼=C(k, l).

γ_k^l will be constructed by using the basic building blocks for generators in C (see Definitions 15.9 and 15.14). To further simplify notation, let us make the following definitions

Definition 16.11. Letedenote the class of 1 in C(pⁿ). For 0≤k, l≤n, define

ktk:=t^C(p_e ^k)∈C(k, k), krk :=r^C(p_e ^k)∈C(k, k), kµl:=µ^C(p_C(p^lk⁾)∈C(k, l).

Define polynomialsψ_n,m ∈Z[n⁻¹][t] by ψ_n,m(t) := 1

n ·t· d

dtΦ_m(t)· ^Y

m⁰|n, m⁰6=m

Φ_m⁰(t)∈Z[n⁻¹][t].

The relevance of theψ_n,ms is that they allow for a decomposition ofZ[n⁻¹]/htⁿ−1i into a direct sum of Dedekind domains (Proposition 22.8 in the appendix). Let us abbreviate ψ_pk,p^u by Ψ_k,u.

Lemma 16.12. Let k≤l, u∈Nand x∈C(k, l). For 0≤u≤n−k, we have Ψ_n−k,u(_kt_k)·x=

(x·Ψ_n−l,u(_lt_l) if u≤n−l,

0 otherwise.

For 0≤v≤l, we have

x·Ψ_l,v(_lrl) =

(Ψ_k,v(_kr_k)·x if v≤k

0 otherwise.

Proof. Note first that _kt_k·x = x·_lt_l. Hence Ψ_n−k,u(_kt_k)·x = x·Ψ_n−k,u(_lt_l). Since (_ktk)^pn−k = 1, the first statement follows by Lemma 22.7. The second statement is proven analogously.

Let us set

ck,l :=

(p^k−l ifk≥l 1 otherwise.

Proposition 16.13. Let 0≤k, l≤n, then there is an isomorphism of abelian groups γ_k^l: ^M

(u,v)∈I(k,l)

R_(u,v)∼=C(k, l), which is given by

γ_k^l(p⊗q1_(u,v))7→c_k,l·p(_kt_k)·Ψ_n−k,u(_kt_k)·_kµ_l·q(_lr_l)·Ψ_l,v(_lr_l).

Here,pandqare polynomials representing elements inZ[p⁻¹][t]/hΦ_pu(t)iandZ[p⁻¹][r]/hΦ_pv(r)i.

Proof. Let 0≤k, l≤n, then

n(_kt_k)^a·_kµ_l·(_lr_l)^b |0≤a < p^n−k∨l,0≤b < p^k∧lo

is a basis ofC(k, l) as a freeZ[p⁻¹]-module by Proposition 15.15. In other words, there is an isomorphism of abelian groups

Z[p⁻¹][t]/ht^pn−k∨l−1i ⊗Z[p⁻¹][r]/hr^pk∧l−1i ∼=C(k, l), p⊗q7→p(_kt_k)·_kµ_l·q(_lr_l).

By Proposition 22.11, for 0≤k, l≤n, there is an isomorphism M

(u,v)∈I(k,l)

R_(u,v) ∼=Z[p⁻¹][t]/ht^pn−k∨l−1i ⊗Z[p⁻¹][r]/hr^pk∧l−1i,

which is given by p⊗q 7→ p·Ψ_n−k∨l,u ⊗q ·Ψ_k∧l,v forp⊗q ∈ R_(u,v). We obtain an isomorphism of abelian groups

γ˜: ^M

(u,v)∈I(k,l)

R_(u,v)∼=C(k, l) given by

γ˜(p⊗q) =p(_kt_k)·Ψ_n−k∨l,u(_kt_k)·_kµ_l·q(_lr_l)·Ψ_k∧l,v(_lr_l) forp⊗q∈R_(u,v)=Z[p⁻¹][t]/hΦ_pu(t)i ⊗Z[p⁻¹][r]/hΦ_pv(r)i. Setγ_k^l :=c_k,l˜γ.

The last step is to show condition (3) in Definition 16.1, i.e., to show the following statement: If (u, v)∈I(k, l), (u⁰, v⁰)∈I(l, m), r∈R_(u,v), s∈R_(u⁰_,v⁰₎, then composition inCis given by

γ_k^l(r1_(u,v))·γ_l^m(s1_(u⁰_,v⁰₎) =δu,u⁰δv,v⁰γ_k^m(rs1_(u,v)).

Forp∈Z[p⁻¹][t] and x∈C(k, l), Lemma 15.10 implies

p(ktk)·x=x·p(ltl) and p(krk)·x=x·p(lrl).

Therefore, it is sufficient to show the following slightly weaker statement:

Proposition 16.14. Let (u, v) ∈ I(k, l), (u⁰, v⁰) ∈ I(l, m), then composition in C is given by

γ_k^l(1_(u,v))·γ_l^m(1_(u⁰_,v⁰₎) =δ_u,u⁰δ_v,v⁰γ_k^m(1_(u,v)).

We will prove this proposition by a series of lemmas.

Lemma 16.15. Let 0≤k≤n and (u, v),(u⁰, v⁰)∈I(k, k), then γ_k^k(1_(u,v))·γ_k^k(1_(u0,v⁰)) =δ_u,u⁰δ_v,v⁰γ_k^k(1_(u,v)).

Proof. In the special casek=l, the isomorphism of abelian groups

Z[p⁻¹][t]/ht^pn−k−1i ⊗Z[p⁻¹][r]/hr^pk −1i ∼=C(k, k), p⊗q 7→p(_kt_k)·q(_kr_k) from Proposition 15.15 as well as the isomorphism

M

(u,v)∈I(k,l)

R_(u,v)∼=Z[p⁻¹][t]/ht^pn−k−1i ⊗Z[p⁻¹][r]/hr^pk−1i

from Proposition 22.11 are isomorphisms of rings. Sinceγ_k^kis defined as the composition of both, γ^k_k is an isomorphism of rings as well. This shows the claim.

Lemma 16.16. Let 0≤k≤l≤m≤n, (u, v)∈I(k, l), (u⁰, v⁰)∈I(l, m), then γ_k^l(1_(u,v))·γ_l^m(1_(u0,v⁰)) =δ_u,u⁰δ_v,v⁰γ_k^m(1_(u,v)).

If (u, v)∈I(m, l), (u⁰, v⁰)∈I(l, k), then

γ_m^l (1_(u,v))·γ_l^k(1_(u0,v⁰)) =δ_u,u⁰δ_v,v⁰γ_m^k(1_(u,v)).

Proof. Lemma 15.18 implies that µ^l_k·µ^m_l = µ^m_k. Note also that c_k,lc_l,m = c_k,m since k≤l≤m. Now compute

γ_k^l(1_(u,v))·γ_l^m(1_(u0,v⁰))

=ck,m·Ψ_n−k,u(_ktk)·µ^l_k·Ψ_l,v(_lrl)·Ψ_n−l,u⁰(_ltl)·µ^m_l ·Ψ_m,v⁰(_mrm)

=ck,m·µ^l_k·Ψ_n−k,u(_ltl)·Ψ_n−l,u⁰(_ltl)·Ψ_l,v(_lrl)·Ψ_m,v⁰(_lrl)·µ^m_l

=δ_(u,u⁰₎·δ_(v,v⁰₎·c_k,m·µ^l_k·Ψ_n−l,u(_lt_l)·Ψ_l,v(_lr_l)µ^m_l ,

where the last equality uses Lemma 22.7 and Lemma 22.6. By Corollary 16.12, the last expression is equal to

δ_(u,u⁰₎·δ_(v,v⁰₎·c_k,m·Ψ_n−k,u(_kt_k)·µ^l_k·µ^m_l ·Ψ_m,v(_mr_m)

=δ_(u,u⁰₎·δ_(v,v⁰₎·ck,m·Ψ_n−k,u(_ktk)·µ^m_kΨ_m,v(_mrm)

=δ_u,u⁰δ_v,v⁰γ_m^k(1_(u,v)) Lemma 16.17. Let 0≤k≤l≤n, then

µ^l_k·µ^k_l =

n−k

Y

i=n−l+1

Φ_pi(_kt_k) and µ^k_l ·µ^k_l =

l

Y

i=k+1

Φ_pi(_lr_l).

Proof. By Corollary 15.22, we have

µ^l_k·µ^k_l =p^−k

p^l−1

X

i=0

(_ktk)^ipn−l.

Since (_kt_k)^pn−k = 1, we obtain

where the last two equalities used the explicit description Φ_pk(t) = ^Pp−1_i=0 t^pk−1i from Lemma 22.2 and the fact that ^Qn_i=1^Pp−1_j=0t^pi−1j = ^Pp_j=0ⁿ⁻¹tⁱ . Similarly, one computes

where we used Lemma 22.7 and Lemma 22.6 in the third equality. Similarly, using p^l−kQl_i=k+1Φ_pi(_lrl)·Ψ_k,v⁰(_lrl) = Ψ_l,v⁰(_lrl), one computes

γ_l^k(1_(u0,v⁰))·γ^l_k(1_(u,v))

=p^l−k·Ψ_n−l,u⁰(_lt_l)·µ^k_l ·Ψ_k,v⁰(_kr_k)·Ψ_n−k,u(_kt_k)·µ^l_k·Ψ_l,v(_lr_l)

=δ_u,u⁰ ·p^l−k·Ψ_n−l,u(_lt_l)µ^k_l ·µ^l_k·Ψ_k,v⁰(_lr_l)·Ψ_l,v(_lr_l)

=δ_u,u⁰ ·p^l−k·Ψ_n−l,u(_lt_l)·

l

Y

i=k+1

Φ_pi(_lr_l)·Ψ_k,v⁰(_lr_l)·Ψ_l,v(_lr_l)

=δ_u,u⁰δ_v⁰_,v·Ψ_n−l,u(_lt_l)·Ψ_l,v(_lr_l)

=δ_u,u⁰δ_v⁰_,vγ^l_l(1_(u,v)).

Now we are able to prove Proposition 16.14. Let us first recall the statement:

Let 0 ≤ k, l, m ≤ n and (u, v) ∈ I(k, l), (u⁰, v⁰) ∈ I(l, m), then composition in C is given by

γ_k^l(1_(u,v))·γ_l^m(1_(u0,v⁰)) =δ_u,u⁰δ_v,v⁰γ_k^m(1_(u,v)) Proof. After orderingk, l and m, there are six cases to consider:

1. k≤l, l≤m;

2. k≤l, k≤m≤l;

3. k≤l, m≤k;

4. l≤k, m≤l;

5. l≤k, l≤m≤k;

6. l≤k, l≤m.

These cases are not mutually exclusive but still exhaustive. We already dealt with cases (1) and (4) in Lemma 16.16. Consider case (2): Since k ≤ m ≤ l, we know that I(k, m)∩I(m, l) =I(k, l). Using Lemma 16.16 and Lemma 16.18, one computes

γ_k^l(1_(u,v))·γ_l^m(1_(u0,v⁰)) =γ_k^m(1_(u,v))·γ_m^l (1_(u,v))·γ_l^m(1_(u0,v⁰))

=δu,u⁰δv,v⁰γ_k^m(1_(u,v))·γ_m^m(1_(u,v))

=δ_u,u⁰δ_v,v⁰γ_k^m(1_(u,v)).

Similarly, we compute in case (3)

γ_k^l(1_(u,v))·γ_l^m(1_(u0,v⁰)) =γ_k^l(1_(u,v))·γ_l^k(1_(u0,v⁰))·γ_k^m(1_(u0,v⁰))

=δu,u⁰δv,v⁰γ_k^k(1_(u,v))·γ_k^m(1_(u⁰_,v⁰₎)

=δ_u,u⁰δ_v,v⁰γ_k^m(1_(u,v)).

Case (5) and case (6) can be dealt with in a completely analogous manner.

16.3 The General Case

In this subsection, we will show that for a finite cyclic groupGof ordero, every graded countable module overC^G[o⁻¹] has projective dimension 1.

We will do so by showing thatC^G[o⁻¹] is the tensor product of split categories over a family of Dedekind domains and thereby itself is split over a family Dedekind domains.

LetGbe a finite cyclic group, then

G=⊕_i=1,...,nGi

withGi cyclic of prime power order p_iⁱ and pi 6=pj fori6=j. Every subgroup H of G decomposes asH =^L_iH_i, with H_i ≤G_i.

The next theorem shows thatC^G is isomorphic to^Nn_i=1C^Gi.

Theorem 16.19. Let G_i, i= 1, . . . , nbe finite abelian groups andG:=^L_iG_i. For each i, let G_i be a class of subgroups of Gi. Define Gto be the class of subgroups of Gof the formH =^L_iH_i for H_i ∈G_i. Let C be the full subcategory ofKK^G with objectsC(G)^H, H∈G.Analogously let C_i be the full subcategory ofKK^Gi with objects C(G_i)^Hi,H_i∈G_i. Then there is an isomorphism

n

O

i=1

C_i ∼=C.

Proof. Let us show the statement for the casen= 2, the general case follows by induc-tion. We have to define

F:C₁⊗C₂ →C.

Note that pullback via the projectionsπ_i:GG_iinduces additive functorsπ_i^∗:KK^Gi → KK^G.Restriction toC_i and taking the tensor product yields

π^∗₁⊗π^∗₂:C₁⊗C₂→KK^G⊗KK^G. On the other hand, the exterior tensor product inKK^G induces

T:KK^G⊗KK^G→KK^G.

We would like to defineF as the composition of the last two functors, but we cannot quite do so since the target objects only match up to a natural isomorphism: ForC(G₁)^H1⊗ C(G₂)^H2 ∈∈C₁⊗C₂ and H:=H1⊕H2, there is a natural isomorphism

Ψ_H:C(G)^H ∼=π₁^∗C(G₁)^H1⊗_Cπ₁^∗C(G₂)^H2,

where on the right-hand-side we have the exterior products inKK^G, which on the object level is just the spatial tensor product ofC^∗-algebras with the diagonal action.

Let us defineF on objects by

FC(G₁)^H1⊗ C(G₂)^H2:=C(G)^H, H_i ∈G_i, H =H₁⊕H₂.

For H_i, K_i∈G_i,H =H₁⊕H₂ and K =K₁⊕K₂, defineF on morphisms by F_H,K:C₁⊗C₂C(G₁)^H1⊗ C(G₂)^H2,C(G₁)^K1⊗ C(G₂)^K2)→CC(G)^H,C(G)^K

FH,K(x) = [ΨH]·(T ◦(π₁^∗⊗π₂^∗)(x))·[Ψ_K]⁻¹.

Fis a functor and clearly a bijection on the set of objects. Hence, to verify thatF is an isomorphism, we only have to check that each FH,K is an isomorphism. By Proposition 15.15, both domain and target ofF_H,K are free abelian groups, hence it suffices to check thatF_H,K maps generators to generators. By definition, the domain of F_H,K is equal to

KK^G1C(G₁)^H1,C(G₁)^K1⊗KK^G2C(G₂)^H2,C(G₂)^K2 and the target is given by

KK^GC(G)^H,C(G)^K.

The following three statements are easily verified using the definitions oft,r andµ: Let g_i ∈G_i, then

1. FH,H(t^H_g1¹⊗t^H_g2²) = [Ψ_H]·π₁^∗t^H_g1¹ ⊗_Cπ₂^∗t^H_g2²·[Ψ_H]⁻¹=t^H_(g^1⊕H2

1,g2)

2. F_H,H(r^H_g1¹⊗r^H_g2²) = [Ψ_H]·π₁^∗r_g^H₁¹ ⊗_Cπ^∗₂r^H_g2²·[Ψ_H]⁻¹ =r^H_(g^1⊕H2

1,g2)

3. FH,K(µ^K_H¹

1 ⊗µ^K_H²

2) = [Ψ_H]·π₁^∗µ^K_H¹

1 ⊗_Cπ^∗₂µ^K_H²

2

·[Ψ_K]⁻¹ =µ^K_H^1⊕K2

1⊕H₂.

This, together with the description of the generators of domain and target ofF_H,K given by Proposition 15.15, tells us thatF_H,K maps generators to generators and finishes the proof.

Finally, we will show that every countable graded C^G[o⁻¹]-module has a projective resolution of length 1. First we prove an auxiliary lemma:

Lemma 16.20. Let C and D be equivalent finite preadditive graded categories. If all objects in Mod(C)^Z/2_c have projective dimension1, then the same holds for all objects in Mod(D)^Z_c^/2.

Proof. Letα:C→D and β:D→C be grading preserving additive functors such that there are isomorphismsT:α◦β ∼= id_D andS:β◦α∼= id_D. LetM be a D-module, then α^∗(M) :=M◦α is aC-module. We obtain preadditive functors

β^∗:Mod(C)^Z_c^/2→Mod(D)^Z_c^/2 and α^∗:Mod(C)^Z_c^/2→Mod(C)^Z_c^/2.

T and S induce isomorphisms α^∗◦β^∗ ∼= id and β^∗◦α^∗ ∼= id. α^∗ and β^∗ are exact and preserve the property of being projective.

Theorem 16.21. Let Gbe finite cyclic group of ordero⁻¹. Then every countable graded C^G[o⁻¹]-module has a projective resolution of length 1.

Proof. Note that we can ignore the grading: C^G[o⁻¹] is ungraded, therefore, the category of (countable) graded C^G[o⁻¹]-modules is the direct sum of two copies of the category of (countable) C^G[o⁻¹]-modules. Let G = ⊕_i=1,...,nG_i with G_i cyclic of prime power order p_iⁱ and p_i 6= p_j for i 6= j. In Theorem 16.19, we showed that C^G is isomorphic to ^Nn_i=1C^Gi. Since o = ^Q|G_i| = ^Qp_iⁱ, we have Z[o⁻¹] = ^Nn_i=1Z[p⁻¹_i ], this implies that C^G,Φ[o⁻¹] is isomorphic to ^Nn_i=1C^Gi[p⁻¹_i ]. Hence it suffices to show that count-able^Nn_i=1C^Gi[p⁻¹_i ]-modules have a projective resolution of length 1. By Theorem 16.9, C^Gi[p⁻¹_i ] is split over

n Z

hp⁻¹_i , θp^u_i

i⊗Z

hp⁻¹_i , θp^v_i

io

{(u,v)∈N0×N0|∃k∈N0:k≤_i,u≤_i−k,v≤k}. Therefore, Lemma 16.7 shows that^Nn_i=1C^Gi[p⁻¹_i ] is split over

( _n O

i=1

Z

hp⁻¹_i , θ_pui i

i⊗Z

hp⁻¹_i , θ_pvi i

i )

{(u_i,vi)i=1,...,n∈(N0×N0)ⁿ|∀i∃k_i∈N0:ki≤_i,ui≤_i−k_i,vi≤k_i}

.

In the appendix, we prove the following two facts:

1. Lemma 22.9: Letm, n∈Zbe coprime. Then there is an isomorphism of rings Z[θ_n]⊗Z[θ_m]∼=Z[θ_mn].

2. Proposition 22.11: Letm, n ∈N, n≤m and p a prime number. Then there is an isomorphism of rings

Z[p⁻¹, θpⁿ]⊗Z[p⁻¹, θp^m]∼= ^M

0<k<pⁿ,gct(p,k)=1

Z[p⁻¹, θp^m].

Both statements together imply that ^Nn_i=1Z[p⁻¹_i , θp^ui]⊗Z[p⁻¹_i , θp^vi] is isomorphic to a finite direct sum of countable Dedekind domains. Hence Corollary 16.3 shows that every object inMod(^Nn_i=1C^Gi[p⁻¹_i ])_c has a projective resolution of length 1.

C ^∗ -algebras over Topological Spaces

The aim of this chapter is to describe explicitly the class of finite topological spaces for which there is a UCT short exact sequence, which computes KK(X;A, B) in terms of the filtrated K-theory ofA and B.

We will first review the notion of aC^∗-algebra over a (possibly non-Hausdorff) finite topological space X and give an alternative description of X by a finite directed graph in Section 17. In Section 18 we will explain how KK-theory for C^∗-algebras overX can be interpreted as a triangulated category with countable coproducts, introduce filtrated K-theory and show that this invariant arises as a special case of our general construction.

We will then review alternative characterizations of the bootstrap class and state the UCT criterion in Section 19. Then, in Section 20, we will define spaces of type A and review some spaces, for which there is no UCT in terms of filtrated K-theory available.

Finally, in Section 21, we will show that spaces of type A are indeed the most general type of finite topological spaces, for which there is a short exact UCT sequence, which computes KK(X,_,_) in terms of filtrated K-theory.

17 C

^∗

-algebras over Finite Topological Spaces

17.1 Basic Notions

Throughout this chapter, let X be a finite topologicalT₀ space. A C^∗-algebra overX is a pair (A, ψ) consisting of aC^∗-algebra A and a continuous map ψ: Prim(A)→X.

Let O(X) denote the set of open subsets of X, partially ordered by ⊆ and let I(A) be the set of closed ^∗-ideals in A, partially ordered by ⊆. (O(X),⊆) and (I(A),⊆) are complete lattices, that is, any subset S has both an infimum^VS and a supremum^WS.

ψinduces a map ψ^∗:O(X)→I(A), which commutes with infima and suprema. By [30], Lemma 2.25,ψ can be recovered fromψ^∗. Hence we obtain an equivalent description of a C^∗-algebra over X as a pair (A, ψ^∗) where

ψ^∗:O(X)→I(A), U 7→A(U), commutes with infima and suprema.

A ^∗-homomorphism f:A → B between two C^∗-algebras over X is X-equivariant if f A(U) ⊆ B(U) for all U ∈ O(X). A subset Y ⊆ X is locally closed if and only if Y =U \V for open subsets V, U ∈O(X) with V ⊆U. We defineA(Y) :=A(U)/A(V) for aC^∗-algebraAoverX; this does not depend on the choice ofU andV by [30] Lemma 2.16.

We adopt the following notations from [30].

O(X) set of open subsets of X, partially ordered by ⊆;

LC(X) set of locally closed subsets of X;

LC(X)_c set of connected, non-empty locally closed subsets ofX;

(A, ψ) C^∗-algebra over X;

A(Y) the subquotient of (A, ψ) associated withY ∈LC(X);

Prim(A) primitive ideal space ofA with hull-kernel topology;

I(A) set of closed^∗-ideals inA, partially ordered by ⊆;

C^∗alg(X) category of C^∗-algebras overX withX-equivariant ^∗-homomorphisms C^∗sep(X) full subcategory of separable C^∗-algebras overX.

17.2 Functoriality

A continuous mapf:X →Y induces a functor

f∗:C^∗alg(X)→C^∗alg(Y).

f∗ is given by (A, ψ) 7→ (A, f◦ψ). We have g∗f∗ = (gf)_∗ whenever it makes sense. If f:X → Y is the embedding of a subset with the subspace topology, we also write i^Y_X instead off∗ and call it induction. Y ∈LC(X) induces a restriction functor

r^Y_X:C^∗alg(X)→C^∗alg(Y),

which is given byr_X^YB(Z) :=B(Z) for all Z∈LC(Y)⊆LC(X). We haver^Z_Y ◦r_X^Y =r_X^Z ifZ⊆Y ⊆X and r_X^X = id.

Induction and restriction are related byr_X^Y ◦i^X_Y = id and various adjointness properties (see [30] Definition 2.19 and Lemma 2.20 for a discussion of induction and restriction).

17.3 Specialization Order

There is the specialization preorder onX, defined byxy ⇐⇒ {x} ⊆ {y}. A subset Y ⊆X is locally closed if and only if it isconvex w.r.t. , i.e., if and only if

xyz, x, z ∈Y ⇒y∈Y holds.

LetX be a space and Y ⊆X, there is a locally closed hull, defined as LC(Y) :={x∈X| ∃y₁, y₂ ∈Y :y₁ xy₂}.

Lemma 17.1. LC(LC(Y)) =LC(Y). LC(Y)is the smallest locally closed set contain-ing Y.

Proof. Obviously Y ⊆LC(Y), lety∈LC(LC(Y)), then there arey1, y2 ∈LC(Y) such that y₁ y y₂. By definition, there are z₁, z₂, z₃, z₄ ∈ Y such that z₁ y₁ z₂, z₃ y₂ z₄, hencez₁y₁yy₂ z₄, therefore,y∈Y. Using the characterization of locally closed sets as convex sets, the second statement is obvious.

A mapf:X₁ →X₂ between two finite topological spaces is continuous if and only if it is monotone with respect to, i.e. if

xy⇒f(x)f(y)

holds. Note that is a partial order if and only if X is T₀. By [30] 2.33, this yields a bijection of partial orders and T₀-topologies on a given finite set. Let us denote the topology associated with ≺by τ≺.

17.4 Representation of Finite Topological Spaces as Directed Graphs A useful way to represent finite partially ordered sets and hence finite T₀ spaces is via finite directed acyclic graphs.

To establish notation, we have collected a few elementary notions of graph theory: A directed graph is a pair Γ = (V, E), where V is a set and E⊆V ×V \∆(V). Elements of V are called vertices and elements of E are called edges. We will also write E(Γ) and V(Γ) to denote the edges and vertices associated with Γ. Hence we are neither allowing loops nor multiple edges to exist. A graph (V⁰, E⁰) is a subgraph of (V, E) if and only if V⁰ ⊆V and E⁰ ={(a, b)∈E |a, b∈V⁰}. A directed path ρ is a a sequence

To establish notation, we have collected a few elementary notions of graph theory: A directed graph is a pair Γ = (V, E), where V is a set and E⊆V ×V \∆(V). Elements of V are called vertices and elements of E are called edges. We will also write E(Γ) and V(Γ) to denote the edges and vertices associated with Γ. Hence we are neither allowing loops nor multiple edges to exist. A graph (V⁰, E⁰) is a subgraph of (V, E) if and only if V⁰ ⊆V and E⁰ ={(a, b)∈E |a, b∈V⁰}. A directed path ρ is a a sequence

Im Dokument Universal Coefficient Theorems in Equivariant KK-theory (Seite 65-0)