DETECTING K-THEORY BY CYCLIC HOMOLOGY

WOLFGANG LÜCK AND HOLGER REICH

Abstract. We discuss which part of the rationalized algebraic K-theory of a group ring is detected via trace maps to Hochschild homology, cyclic homology, periodic cyclic or negative cyclic homology. Key words: algebraic K-theory of group rings, Hochschild homology, cyclic homology, trace maps. Mathematics Subject Classification 2000: 19D55.

Dedicated to memory of Michel Matthey.

0. Introduction and statement of results

Fix a commutative ring k, referred to as the ground ring. Let R be a k-algebra, i.e. an associative ring R together with a unital ring homomorphism from k to the center of R. We denote by HH_{∗}^{⊗_{k}}(R) the Hochschild homology of R relative to the ground ring k, and similarly by HC_{∗}^{⊗_{k}}(R), HP_{∗}^{⊗_{k}}(R) and HN_{∗}^{⊗_{k}}(R) the cyclic, the periodic cyclic and the negative cyclic homology of R relative to k. Hochschild homology receives a map from the algebraic K-theory, which is known as the Dennis trace map. There are variants of the Dennis trace taking values in cyclic, periodic cyclic and negative cyclic homology (sometimes called Chern characters), as displayed in the following commutative diagram.

\xymatrix{ & HN_{∗}^{⊗_{k}}(R) \ar[r] \ar[d]^{h} & HP_{∗}^{⊗_{k}}(R) \ar[d] \\ K_{∗}(R) \ar@{-->}[ur]^{ntr} \ar[r]^{dtr} & HH_{∗}^{⊗_{k}}(R) \ar[r] & HC_{∗}^{⊗_{k}}(R).}

(0.1)

For the definition of these maps, see [18, Chapters 8 and 11] and Section 5 below.

In the following we will focus on the case of group rings RG, where G is a group and we refer to the k-algebra R as the coefficient ring. We investigate the following question.

Question 0.2. Which part of K_{∗}(RG) ⊗_{ℤ} ℚ can be detected using linear trace invariants like the Dennis trace to Hochschild homology, or its variants with values in cyclic homology, periodic cyclic homology and negative cyclic homology ?

For any group G, we prove “detection results”, which state that certain parts of K_{∗}(RG) ⊗_{ℤ} ℚ can be detected by the trace maps in diagram 0.1, accompanied by “vanishing results”, which state that a complement of the part which is then known to be detected is mapped to zero. For the detection results, we only make assumptions on the coefficient ring R, whereas for the vanishing results we additionally need the Farrell-Jones Conjecture for RG as an input, compare Example 1.2. Modulo the Farrell-Jones Conjecture, we will give a complete answer to Question 0.2 for instance in the case of Hochschild and cyclic homology, when the coefficient ring R is an algebraic number field F or its ring of integers O_{F}. We will also give partial results for periodic cyclic and negative cyclic homology.

Date: August 31, 2005.

All detection results are obtained by using only the Dennis trace with values in HH_{∗}^{⊗_{k}}(RG), whereas all vanishing results hold even for the trace with values in HN_{∗}^{⊗_{ℤ}}(RG), which, in view of diagram (0.1), can be viewed as the best among the considered trace invariants. (Note that for a k-algebra R every homomorphism k′ → k of commutative rings leads to a homomorphism HN_{∗}^{⊗_{k′}} (R) → HN_{∗}^{⊗_{k}}(R). Similar for Hochschild, cyclic and periodic cyclic homology.) We have no example where the extra effort that goes into the construction of the variants with values in cyclic, periodic cyclic or negative cyclic homology yields more information about K_{∗}(RG) ⊗_{ℤ} ℚ than one can obtain by Hochschild homology, see also Remark 0.16 and 0.17 below.

We will now explain our main results. We introduce some notation.

Notation 0.3. Let G be a group and H a subgroup. We write ⟨g⟩ for the cyclic subgroup generated by g ∈ G. We denote by (g) and by (H) the conjugacy class of g respectively of H in G. Let con G be the set of conjugacy classes of elements of G. The set of conjugacy classes of finite cyclic subgroups of G will be denoted by (FCyc).

Let Z_{G}H and N_{G}H denote the centralizer and the normalizer of H in G, respectively. The Weyl group W_{G}H is defined as the quotient N_{G}H/H · Z_{G}H and coincides for an abelian subgroup H with N_{G}H/Z_{G}H.

Let C be a finite cyclic group. We will define in (1.11) an idempotent θ_{C} ∈ A(C) ⊗_{ℤ} ℚ in the rationalization of the Burnside ring A(C) of C. Since there is a natural action of A(C) on K_{∗}(RC), we obtain a corresponding direct summand

θ_{C}(K_{∗}(RC) ⊗_{ℤ} ℚ) ⊆ K_{∗}(RC) ⊗_{ℤ} ℚ .

In Lemma 7.4, we prove that θ_{C}(K_{∗}(RC) ⊗_{ℤ} ℚ) is isomorphic to the Artin defect

coker ( ⨁_{D≨C} ind_{D}^{C}: ⨁_{D≨C} K_{∗}(RD) ⊗_{ℤ} ℚ → K_{∗}(RC) ⊗_{ℤ} ℚ) ,

which measures the part of K_{∗}(RC) ⊗_{ℤ} ℚ which is not obtained by induction from proper subgroups of C.

The conjugation action of N_{G}C on C induces an action of the Weyl group W_{G}C = N_{G}C/Z_{G}C on K_{∗}(RC) ⊗_{ℤ} ℚ and thus on θ_{C}(K_{∗}(RG) ⊗_{ℤ} ℚ). There is an obvious W_{G}C-action on BZ_{G}C = Z_{G}C\EN_{G}C. These actions are understood in the following statement.

Theorem 0.4 (Main Detection Result). Let G be a group, k a commutative ring and R a k-algebra. Suppose that the underlying ring of R is from the following list :

(i) a finite dimensional semisimple algebra R over a field F of characteristic zero;

(ii) a commutative complete local domain R of characteristic zero; (iii) a commutative Dedekind domain R in which the order of every finite cyclic

subgroup of G is invertible and whose quotient field is an algebraic number field.

Then there exists an injective homomorphism

⨁_{(C)∈(FCyc)} H_{∗}(BZ_{G}C; ℚ) ⊗_{ℚ[W_{G}C]} θ_{C}(K_{0}(RC) ⊗_{ℤ} ℚ) → K_{∗}(RG) ⊗_{ℤ} ℚ (0.5)

whose image is detected by the Dennis trace map

dtr: K_{∗}(RG) ⊗_{ℤ} ℚ → HH_{∗}^{⊗_{k}}(RG) ⊗_{ℤ} ℚ , (0.6) in the sense that the composition of the map (0.5) with dtr is injective. Also the composition with the map to HC_{∗}^{⊗_{k}}(RG) ⊗_{ℤ} ℚ remains injective.

Examples of rings R appearing in the list of Theorem 0.4 are :

• fields of characteristic zero;

• the group ring F H of a finite group H over a field F of characteristic zero;

• the ring ℤ_{p} of p-adic integers;

• for the given G, the localization S^{−1}O_{F} of the ring of integers O_{F} in an algebraic number field F, for instance S^{−1}ℤ, where S is the multiplicative set generated by the orders of all finite cyclic subgroups of G.

Depending on the choice of the coefficient ring R, the description of the source of the map 0.5 can be simplified. We mention two examples. Let ℚ_{∞} be the field obtained from ℚ by adjoining all roots of unity.

Theorem 0.7 (Detection Result for ℚ and ℂ as coefficients). For every group G, there exist injective homomorphisms

⨁_{(C)∈(FCyc)} H_{∗}(BN_{G}C; ℚ) → K_{∗}(ℚG) ⊗_{ℤ} ℚ,

⨁_{(g)∈con G,|g|<∞} H_{∗}(BZ_{G}⟨g⟩; ℚ_{∞}) → K_{∗}(ℂG) ⊗_{ℤ} ℚ_{∞} .

The image of these maps is detected by the Dennis trace map with ℚ and ℂ as ground ring, respectively. The coefficient field ℚ (resp. ℂ) can be replaced by any field of characteristic zero (resp. any field containing ℚ_{∞}).

Theorem 0.7 for ℚ_{∞} and ℂ as coefficient fields is the main result of the paper by Matthey [27]. The techniques there are based on so-called delocalization and the computation of the Hochschild homology and of the cyclic homology of group rings with commutative coefficient rings containing ℚ (see [38, Section 9.7] and [4]). They are quite different from the ones used in the present paper and are exactly suited for the cases studied there and do not seem to be extendable to the situations considered here. Both maps appearing in Theorem 0.7 are optimal in the sense of Theorem 0.10 and of Theorem 0.12 below, provided that the Farrell-Jones Conjecture holds rationally for K_{∗}(ℚG) and K_{∗}(ℂG) respectively.

The Main Detection Theorem 0.4 is obtained by studying the following commutative diagram :

\xymatrix{H_{∗}^{G}(\underline{E}G; KR) \ar[d]_{H_{∗}^{G}(\underline{E}G;dtr)} \ar[r]^{assembly} & K_{∗}(RG) \ar[d]^{dtr} \\ H_{∗}^{G}(\underline{E}G; HH^{⊗_{ℤ}}R) \ar[r]^{assembly} & HH_{∗}^{⊗_{ℤ}}(RG)}

(0.8)

Here, the horizontal arrows are generalized assembly maps for K-theory and Hochschild homology respectively, and the left vertical arrow is a suitable version of the Dennis trace map. The G-space \underline{E}G is a model for the so-called classifying space for proper G-actions. Moreover, H_{∗}^{G}(−; KR) and H_{∗}^{G}(−; HH^{⊗_{ℤ}}R) are certain Ghomology theories. We will explain the diagram in more detail in Section 1. We will prove that the lower horizontal arrow in (0.8) is split injective, see Theorem 1.7. In fact, Theorem 1.7 gives a complete picture of the generalized assembly map for Hochschild and cyclic homology. We will also compute the left-hand vertical arrow after rationalization, compare Theorem 1.13 and Propositions 3.3, 3.4 and 3.5. According to this computation, the left-hand side in (0.5) is a direct summand in H_{∗}(\underline{E}G; KR) ⊗_{ℤ} ℚ on which, for R as in Theorem 0.4, the map

H_{∗}^{G}(\underline{E}G; KR) ⊗_{ℤ} ℚ → H_{∗}^{G}(\underline{E}G; HH^{⊗_{ℤ}}R) ⊗_{ℤ} ℚ (0.9) is injective. This will prove Theorem 0.4. Now, suppose that R is as in case (i) of Theorem 0.4, with F a number field. Then, it turns out that the map (0.9) vanishes on a complementary summand. According to the Farrell-Jones Conjecture for K_{∗}(RG), the upper horizontal arrow in (0.8) should be an isomorphism (this uses that R is a regular ring with ℚ ⊆ R). Combining these facts, we will deduce the following result.

Theorem 0.10 (Vanishing Result for Hochschild and cyclic homology). Let G be a group, F an algebraic number field, and R be finite dimensional semisimple F -algebra. Suppose that for some n ≥ 0, the Farrell-Jones Conjecture holds rationally for K_{n}(RG), see Example 1.2 below.

Then Theorem 0.4 is optimal for the Hochschild homology trace invariant, in the sense that the Dennis trace map

dtr: K_{n}(RG) ⊗_{ℤ} ℚ → HH_{n}^{⊗_{ℤ}}(RG) ⊗_{ℤ} ℚ (0.11)

vanishes on a direct summand that is complementary to the image of the injective map (0.5) in degree n. Consequently, also the trace taking values in rationalized cyclic homology HC_{n}^{⊗_{ℤ}}(RG) ⊗_{ℤ} ℚ vanishes on this complementary summand.

One might still hope that the refinements of the Dennis trace map with values in periodic cyclic or negative cyclic homology detect more of the rationalized algebraic K-theory of RG. But one can show that this is not the case if one additionally assumes a finiteness condition on the classifying space \underline{E}G. Recall that the Gspace \underline{E}G is called cocompact if the orbit space G\\underline{E}G is compact, in other words, if it consists of finitely many G-equivariant cells. Cocompact models for \underline{E}G exist for many interesting groups G such as discrete cocompact subgroups of virtually connected Lie groups, word-hyperbolic groups, arithmetic subgroups of a semisimple connected ℚ-algebraic group, and mapping class groups (see for instance [21]).

Theorem 0.12 (Vanishing Result for periodic and negative cyclic homology). Let F be an algebraic number field, and R a finite dimensional semisimple F - algebra. Suppose that for some n ≥ 0, the Farrell-Jones Conjecture holds rationally for K_{n}(RG). Suppose further that there exists a cocompact model for the classifying space for proper G-actions \underline{E}G.

Then also the refinements of the Dennis trace with values in HP_{n}^{⊗_{ℤ}}(RG) ⊗_{ℤ} ℚ and in HN_{n}^{⊗_{ℤ}}(RG) ⊗_{ℤ} ℚ vanish on a direct summand which is complementary to the image of the injective map (0.5) in degree n.

The next result is well-known. It shows in particular that the rational group homology H_{∗}(BG; ℚ) is contained in K_{∗}(RG) ⊗_{ℤ} ℚ for all commutative rings R of characteristic zero.

Theorem 0.13 (Detection Result for commutative rings of characteristic zero). Let R be a ring such that the canonical ring homomorphism ℤ → R induces an injection HH_{0}^{⊗_{ℤ}}(ℤ) = ℤ ↪ HH_{0}^{⊗_{ℤ}}(R) = R/[R, R], for instance a commutative ring of characteristic zero.

Then, for any group G, there exists an injective homomorphism

H_{∗}(BG; ℚ) → K_{∗}(RG) ⊗_{ℤ} ℚ (0.14)

whose composition with the Dennis trace map (0.6) is injective for every choice of a ground ring k such that R is a k-algebra. The corresponding statement holds with Hochschild homology replaced by cyclic homology.

Special cases of this result are treated for example in [30, Proposition 6.3.24 on page 366].

According to the Farrell-Jones Conjecture, the image of (0.14) should only be a very small part of the rationalized K-theory of RG. The following result illustrates that, for certain coefficient rings, including ℤ, one cannot expect to detect more by linear traces than achieved in Theorem 0.13.

Theorem 0.15 (Vanishing Result for integral coefficients). Let S^{−1}O_{F} be a localization of a ring of integers O_{F} in an algebraic number field F with respect to a (possibly empty) multiplicatively closed subset S. Assume that no prime divisor of the order |H| of a nontrivial finite subgroup H of G is invertible in S^{−1}O_{F}. Suppose that for some n ≥ 0, the Farrell-Jones Conjecture holds rationally for K_{n}(S^{−1}O_{F} [G]).

Then the Dennis trace (0.11) vanishes on a summand in K_{n}(S^{−1}O_{F} [G]) ⊗_{ℤ} ℚ which is complementary to the image of the map (0.14) in degree n. Consequently, the analogous statement holds for the trace with values in HC_{n}^{⊗_{ℤ}}(S^{−1}O_{F} [G]) ⊗_{ℤ} ℚ.

The most interesting case in Theorem 0.15 is R = ℤ. We remark that rationally, the Farrell-Jones Conjecture for K_{∗}(ℤG) is known in many cases, for example for every subgroup G of a discrete cocompact subgroup of a virtually connected Lie group [13]. For a survey of known results about the Farrell-Jones Conjecture, we refer the reader to [22].

Remark 0.16. There are further trace invariants (or Chern characters) given by maps ch_{n,r}: K_{n}(RG) → HC_{n+2r}^{⊗_{k}}(RG), for fixed n, r ≥ 0, see [18, 8.4.6 on page 272 and 11.4.3 on page 371]. This will however produce no new detection results in the spirit of the above statements, since there is a commutative diagram

\xymatrix{ &  & K_{n}(RG) \ar[dll]_{ntr} \ar[dl] \ar[dr]_{_{ch_{n,r}}} \ar[drr] \\ HN_{n}^{⊗_{k}}(RG) \ar[r] & HP_{n}^{⊗_{k}}(RG) \ar[rr]^{\overline{S}} &  & HC_{n+2r}^{⊗_{k}}(RG) \ar[r]^{S^{r}} & HC_{n}^{⊗_{k}}(RG).}

Remark 0.17. In [2], Bökstedt, Hsiang and Madsen define the cyclotomic trace, a map out of K-theory which takes values in topological cyclic homology. The cyclotomic trace map can be thought of as an even more elaborate refinement of the Dennis trace map. In contrast to the Dennis trace, it seems that the cyclotomic trace has the potential to detect almost all of the rationalized K-theory of an integral group ring. This question is investigated in detail in [23].

The paper is organized as follows :

1. Outline of the method
2. Proofs
3. The trace maps for finite cyclic groups
4. Notation and general machinery
5. The trace maps
6. Equivariant homology theories, induction and Mackey structures
7. Evaluating the equivariant Chern character
8. Comparing different models
9. Splitting assembly maps

References

1. Outline of the method

This paper is concerned with comparing generalized assembly maps for K-theory, via the Dennis trace or its refinements, with generalized assembly maps for Hochschild homology, for cyclic, periodic cyclic or negative cyclic homology. Before we explain the general strategy behind our results we briefly explain the concept of a generalized assembly map; for more details the reader is referred to [8] and [22, Section 2 and 6].

A family of subgroups of a given group G is a non-empty collection of subgroups which is closed under conjugation and finite intersections. Given a family F of subgroups, there always exists a G-CW -complex E_{F}(G) all of whose isotropy groups lie in F and which has the property that for all H ∈ F, the fixed subspace E_{F}(G)^{H} is a contractible space. A G-CW -complex with these properties is unique up to G-homotopy because it receives a G-map from every G-CW -complex all whose isotropy groups lie in F and this G-map is unique up to G-homotopy. If F = Fin is the family of finite subgroups, then one often writes \underline{E}G for E_{Fin}(G). For a survey on these spaces, see for instance [21].

Let OrG denote the orbit category of G. Objects are the homogenous spaces G/H considered as left G-spaces, and morphisms are G-maps. A functor E, from the orbit category OrG to the category of spectra, is called an OrG-spectrum. Each OrG-spectrum E gives rise to a G-homology theory H_{∗}^{G}(−; E), compare [22, Section 6] and the beginning of Section 6 below. Given E and a family F of subgroups of G, the so-called generalized assembly map

\xymatrix{H_{∗}^{G}(E_{F}(G); E) \ar[r]^{assembly} & H_{∗}^{G}(pt; E)}

(1.1)

is merely the homomorphism induced by the map E_{F}(G) → pt. The group H_{∗}^{G}(pt; E) can be canonically identified with π_{∗}(E(G/G)).

Example 1.2 (The Farrell-Jones Conjecture). Given an arbitrary ring R and an arbitrary group G, there exists a non-connective K-theory OrG-spectrum, denoted by K^{−∞}R(?), such that there is a natural isomorphism

π_{n}(K^{−∞}R(G/H)) ≅ K_{n}(RH)

for all H ≤ G and all n ∈ ℤ, compare [22, Theorem 6.9]. The Farrell-Jones Conjecture for K_{n}(RG), [13, 1.6 on page 257], predicts that the generalized assembly map

\xymatrix{H_{n}^{G}(E_{VCyc}(G); K^{−∞}R) \ar[r]^{assembly} & H_{n}^{G}(pt; K^{−∞}R) ≅ K_{n}(RG)}

is an isomorphism. Here VCyc stands for the family of all virtually cyclic subgroups of G. A group is called virtually cyclic if it contains a cyclic subgroup of finite index.

In Section 5, we will construct the following commutative diagram of connective OrG-spectra and maps (alias natural transformations) between them :

\xymatrix{ & HN^{⊗_{k}}R \ar[r] \ar[d]^{h} & HP^{⊗_{k}}R \ar[d] \\ KR \ar[r]^{dtr} \ar[ur]^{ntr} & HH^{⊗_{k}}R \ar[r] & HC^{⊗_{k}}R.}

(1.3)

Decisive properties of these constructions are that for all n ≥ 0, we have natural isomorphisms

π_{n}(KR(G/H))  ≅ K_{n}(RH)
π_{n}(HH^{⊗_{k}}R(G/H)) ≅ HH_{n}^{⊗_{k}}(RH)
π_{n}(HC^{⊗_{k}}R(G/H)) ≅ HC_{n}^{⊗_{k}}(RH)
π_{n}(HP^{⊗_{k}}R(G/H)) ≅ HP_{n}^{⊗_{k}}(RH)
π_{n}(HN^{⊗_{k}}R(G/H)) ≅ HN_{n}^{⊗_{k}}(RH)

(1.4) and all negative homotopy groups vanish. Note that we need to distinguish between the non-connective version K^{−∞}R and the connective version KR. Under the identifications above, the maps of OrG-spectra in (1.3) evaluated at an orbit G/H induce, on the level of homotopy groups, the maps in (0.1) with R replaced by the corresponding group ring RH.

Remark 1.5. We found it technically convenient to work, at the level of spectra, with the connective versions of periodic cyclic and negative cyclic homology. Since we are mainly interested in the trace maps (whose source will be the connective K-theory spectrum), we do not lose any information.

Since the assembly map (1.1) is natural in the functor E, we obtain, for each family of subgroups F of a group G and for each n ≥ 0, the commutative diagram

\xymatrix{H_{n}^{G}(E_{F}(G); KR) \ar[r]^{assembly} \ar[d]_{H_{n}^{G}(E_{F} (G);ntr)} & H_{n}^{G}(pt; KR) ≅ K_{n}(RG) \ar[d]|{ntr} \\ H_{n}^{G}(E_{F}(G); HN^{⊗_{k}}R) \ar[r]^{assembly} \ar[d]_{H_{n}^{G}(E_{F} (G);h)} & H_{n}^{G}(pt; HN^{⊗_{k}}R) ≅ HN_{n}^{⊗_{k}}(RG) \ar[d]|{h} \\ H_{n}^{G}(E_{F}(G); HH^{⊗_{k}}R) \ar[r]^{assembly} & H_{n}^{G}(pt; HH^{⊗_{k}}R) ≅ HH_{n}^{⊗_{k}}(RG).}

(1.6)

The vertical compositions are the corresponding versions of the Dennis trace map.

Our investigation relies on two main ingredients. The first ingredient are splitting and isomorphism results for the assembly maps of Hochschild and cyclic type.

Theorem 1.7 (The Isomorphism Conjecture for HH and HC). Let k be a commutative ring, R a k-algebra, and G a group. Then the generalized Hochschild homology assembly map

\xymatrix{H_{∗}(E_{F}(G); HH^{⊗_{k}}R) \ar[r]^{assembly} & H_{∗}(pt; HH^{⊗_{k}}R) ≅ HH_{∗}^{⊗_{k}}(RG)}

is split injective for every family F. If F contains the family of all (finite and infinite) cyclic subgroups, then the map is an isomorphism. The analogous statement holds for HC in place of HH.

The fact that the definition of periodic cyclic and of negative cyclic homology involves certain inverse limit processes prevents us from proving the analogous result in these cases without assumptions on the group G. But we still have the following statement.

Addendum 1.8 (Splitting Results for the HP and HN-assembly maps). Suppose that there exists a cocompact model for the classifying space E_{F}(G). Then the statement of Theorem 1.7 also holds for HP and HN in place of HH.

The proofs of Theorem 1.7 and Addendum 1.8 are presented in Section 9.

Remark 1.9. We do not know any non-trivial example where the isomorphism statement in Addendum 1.8 applies, i.e. where F contains all (finite and infinite) cyclic groups and where, at the same time, E_{F}(G) has a cocompact model.

The second main ingredient of our investigation is the rational computation of equivariant homology theories from [20]. For varying G, our G-homology theories like H_{∗}^{G}(−) = H_{∗}^{G}(−; KR) or H_{∗}^{G}(−) = H_{∗}^{G}(−; HH^{⊗_{k}}R) are linked by a so-called induction structure and form an equivariant homology theory in the sense of [20]. Moreover, these homology theories admit a Mackey structure. In Section 6, we review these notions and explain some general principles which allow us to verify that G-homology theories like the ones we are interested in indeed admit induction and Mackey structures. In particular, Theorems 0.1 and 0.2 in [20] apply and yield an explicit computation of H_{∗}^{G}(\underline{E}G)⊗_{ℤ} ℚ. In Section 7, we review this computation and discuss a simplification which occurs in the case of K-theory, Hochschild, cyclic, periodic cyclic and negative cyclic homology, due to the fact that in all these special cases, we have additionally a module structure over the Swan ring.

In order to state the result of this computation, we introduce some more notation. For a finite group G, we denote by A(G) the Burnside ring which is additively generated by isomorphism classes of finite transitive G-sets. Let (sub G) denote the set of conjugacy classes of subgroups of G.

The counting fixpoints ring homomorphism

χ_{G}: A(G) → ∏_{(sub G)} ℤ (1.10)

which is induced by sending a G-set S to (|S^{H}|)_{(H)} becomes an isomorphism after rationalization, compare [35, page 19]. For a finite cyclic group C, we consider the idempotent

(1.11)  θ_{C} = (χ_{C} ⊗_{ℤ} ℚ)^{−1}((δ_{CD})_{D}) ∈ A(C) ⊗_{ℤ} ℚ ,

where (δ_{CD})_{D} ∈ ∏_{sub C} ℚ is given by δ_{CC} = 1 and δ_{CD} = 0 if D ≠ C.

Recall that K_{∗}(RC) and similarly Hochschild, cyclic, periodic cyclic and negative cyclic homology of RC are modules over the Burnside ring A(C). The action of a C-set S is in all cases induced from taking the tensor product over ℤ with the corresponding permutation module ℤS. In Lemma 7.4 below, we prove that θ_{C}(K_{∗}(RC) ⊗_{ℤ} ℚ) is isomorphic to the ℚ-vector space

(1.12) coker ( ⨁_{D≨C} ind_{D}^{C}: ⨁_{D≨C} K_{∗}(RD) ⊗_{ℤ} ℚ → K_{∗}(RC) ⊗_{ℤ} ℚ) ,

which is known as the Artin defect of K_{∗}(RC) ⊗_{ℤ} ℚ.

In Section 7 we establish the following result.

Theorem 1.13. For each n ≥ 0, the following diagram commutes and the arrows labelled ch^{G} are isomorphisms :

\xymatrix{⨁_{\begin{matrix}p,q≥0 \\ p+q=n \end{matrix}} ⨁_{(C)∈(FCyc)}H_{p}(BZ_{G}C; ℚ) ⊗_{ℚ[W_{G}C]} θ_{C}(K_{q}(RC) ⊗_{ℤ} ℚ) \ar@{-->}[dr]^{_{ch_{G}}}_{≅} \ar[dd]_{_{dtr_{∗}}} \\  & H_{n}^{G}(\underline{E}G; KR) ⊗_{ℤ} ℚ \ar[dd]^{H_{n}^{G}(\underline{E}G;dtr)⊗_{ℤ}ℚ} \\ ⨁_{\begin{matrix}p,q≥0 \\ p+q=n \end{matrix}} ⨁_{(C)∈(FCyc)}H_{p}(BZ_{G}C; ℚ) ⊗_{ℚ[W_{G}C]} θ_{C}( HH_{q}^{⊗_{k}}(RC) ⊗_{ℤ} ℚ) \ar@{-->}[dr]^{_{ch_{G}}}_{≅} \\  & H_{n}^{G}(\underline{E}G; HH^{⊗_{k}}R) ⊗_{ℤ} ℚ.}

The left-hand vertical arrow is induced by the Dennis trace maps for finite cyclic groups and respects the double direct sum decompositions. The right-hand vertical arrow is induced by the OrG-spectrum Dennis trace dtr, compare (1.3). There are similar diagrams and isomorphisms corresponding to each of the other maps in diagram (1.3).

Remark 1.14. The (−1)-connected covering map of OrG-spectra KR → K^{−∞}R induces for every orbit G/H an isomorphism

π_{n}(KR(G/H)) → π_{n}(K^{−∞}R(G/H))

if n ≥ 0. The source is trivial for n < 0. This map induces the following commutative diagram.

\xymatrix{⨁_{\begin{matrix}p,q≥0 \\ p+q=n \end{matrix}} ⨁_{(C)∈(FCyc)}H_{p}(BZ_{G}C; ℚ) ⊗_{ℚ[W_{G}C]} θ_{C}(K_{q}(RC) ⊗_{ℤ} ℚ) \ar@{-->}[dr]^{_{ch_{G}}}_{≅} \ar[dd] \\  & H_{n}^{G}(\underline{E}G; KR) ⊗_{ℤ} ℚ \ar[dd] \\ ⨁_{\begin{matrix}p,q∈ℤ \\ p+q=n \end{matrix}} ⨁_{(C)∈(FCyc)}H_{p}(BZ_{G}C; ℚ) ⊗_{ℚ[W_{G}C]} θ_{C}(K_{q}(RC) ⊗_{ℤ} ℚ) \ar@{-->}[dr]^{_{ch_{G}}}_{≅} \\  & H_{n}^{G}(\underline{E}G; K^{−∞}R) ⊗_{ℤ} ℚ.}

Here the arrows labelled ch^{G} are isomorphisms. Note the restriction p, q ≥ 0 for the sum in the upper left hand corner.

1.1. General strategy. We now explain the strategy behind all the results that appeared in the introduction. If we combine the diagram appearing in Theorem 1.13 with diagram (1.6), for each n ≥ 0, we get a commutative diagram

\xymatrix{⨁_{\begin{matrix}p,q≥0 \\ p+q=n \end{matrix}} ⨁_{(C)∈(FCyc)}H_{p}(BZ_{G}C; ℚ) ⊗_{ℚ[W_{G}C]} θ_{C}(K_{q}(RC) ⊗_{ℤ} ℚ) \ar@{-->}[dr] \ar[dd]_{_{dtr_{∗}}} \\  & K_{n}(RG) ⊗_{ℤ} ℚ \ar[dd]^{dtr} \\ ⨁_{\begin{matrix}p,q≥0 \\ p+q=n \end{matrix}} ⨁_{(C)∈(FCyc)}H_{p}(BZ_{G}C; ℚ) ⊗_{ℚ[W_{G}C]} θ_{C}( HH_{q}^{⊗_{ℤ}}(RC) ⊗_{ℤ} ℚ) \ar@{-->}[dr] \\  & HH_{n}^{⊗_{ℤ}}(RG) ⊗_{ℤ} ℚ.}

Because of Theorem 1.7 and the isomorphism statement in Theorem 1.13 the lower horizontal map is injective. There is an analogue of the commutative diagram above, where the upper row is the same and HH is replaced by HC in the bottom row. Also in this case we know that the lower horizontal map is injective because of Theorem 1.7 and 1.13.

Observe that W_{G}C is always a finite group, hence ℚ[W_{G}C] is a semisimple ring, so that every module over ℚW_{G}C is flat and the functor H_{p}(BZ_{G}C; ℚ)⊗_{ℚW_{G}C} (−) preserves injectivity.

For q ≥ 0 given, we see that suitable injectivity results about the maps

θ_{C}(K_{q}(RC) ⊗_{ℤ} ℚ) → θ_{C}( HH_{q}^{⊗_{k}}(RC) ⊗_{ℤ} ℚ) (1.15)

for the finite cyclic subgroups C ≤ G lead to the proof of detection results in degree n. These maps (1.15) will be studied in Section 3.

If R is a regular ring containing ℚ, then the family VCyc of virtually cyclic subgroups can be replaced by the family Fin of finite subgroups and the nonconnective K-theory OrG-spectrum K^{−∞}R(?) by its connective version KR(?) in the statement of the Farrell-Jones Conjecture, i.e., in this case, the Farrell-Jones Conjecture for K_{n}(RG), for some n ∈ ℤ, is equivalent to the statement that the assembly map

\xymatrix{H_{n}^{G}(\underline{E}G; KR) \ar[r]^{assembly} & K_{n}(RG)}

is an isomorphism if n ≥ 0 and to the statement that K_{n}(RG) = 0 if n ≤ −1 (see [22, Proposition 2.14]). As a consequence, the upper horizontal arrow in the diagram above (where n ≥ 0) is bijective if the Farrell-Jones Conjecture is true rationally for K_{n}(RG).

So for q ≥ 0 given, we see that suitable vanishing results about the maps (1.15) (and about their analogues involving cyclic homology) combined with the assumption that the Farrell-Jones conjecture holds rationally for K_{n}(RG) lead to the proof of vanishing results in degree n.

2. Proofs

Based on the strategy explained in the previous paragraphs we now give the proofs of the theorems stated in the introduction, modulo the following results : Theorem 1.7 and Addendum 1.8 (proved in Section 9); Theorems 1.13 (proved in Section 7, using Sections 4–6); and the results of Section 3 (which is self-contained, except for Lemma 7.4 whose proof is independent of the rest of the paper).

2.1. Proof of Theorem 0.4. After the general strategy 1.1, the necessary injectivity result to complete the proof appears in Proposition 3.3 below. □

2.2. Proof of Theorem 0.10. The result follows directly from the general strategy 1.1 and the vanishing result stated as Proposition 3.5 below. □

2.3. Proof of Theorem 0.12. The proof is completely analogous to that of Theorem 0.10. The extra condition that there is a cocompact model for \underline{E}G is only needed to apply Addendum 1.8 in place of Theorem 1.7. □

2.4. Proof of Theorem 0.7. The next lemma explains why Theorem 0.7 for ℚ as coefficients follows from Theorem 0.4. The case of ℂ as coefficients is proven similarly, compare [20, Example 8.11]. □

Lemma 2.1. (i) Let C be a finite cyclic group. Then one has

θ_{C}(K_{0}(ℚC) ⊗_{ℤ} ℚ) ≅ ℚ

and every group automorphism of C induces the identity on ℚ. (ii) For any group G and finite cyclic subgroup C ≤ G, the map

H_{∗}(BZ_{G}C; ℚ) ⊗_{ℚ[W_{G}C]} ℚ ⟶^{≅} H_{∗}(BN_{G}C; ℚ)

induced by the inclusion Z_{G}C ↪ N_{G}C is an isomorphism. Here ℚ carries the trivial W_{G}C-action.

Proof. (i) There is a commutative diagram

\xymatrix{A(C) ⊗_{ℤ} ℚ \ar[d]_{χ_{C}⊗_{ℤ}ℚ}^{≅} \ar[r]^{≅} & K_{0}(ℚC) ⊗_{ℤ} ℚ \ar[d]^{≅} \\ ∏_{D∈sub C} ℚ \ar[r]^{≅} & map(sub C, ℚ)}

Here, the upper horizontal map sends a C-set to the corresponding permutation module. The product in the lower left corner is taken over the set sub C of all subgroups of C and the left-hand vertical arrow is given by sending the class of a C-set S to (|S^{D}|)_{D} and is an isomorphism, as already mentioned after (1.10).

The right-hand vertical map is given by sending a rational representation V to its character, i.e. if d generates the subgroup ⟨d⟩, then ⟨d⟩ ↦ tr_{ℚ}(d: V → V). This map is also an isomorphism, compare [33, II. § 12]. The lower horizontal map is the isomorphism given by sending (x_{D})_{D∈sub C} to (D ↦ x_{D}). The diagram is natural with respect to automorphisms of C. By definition, θ_{C} ∈ A(C) ⊗_{ℤ} ℚ corresponds to the idempotent (δ_{CD})_{D} in the lower left-hand corner. Now, the result follows.

(ii) This follows from the Lyndon-Hochschild-Serre spectral sequence of the fibration BZ_{G}C → BN_{G}C → BW_{G}C and from the fact that, the group W_{G}C being finite, for any ℚ[W_{G}C]-module M, the ℚ-vector space H_{p}(C_{∗}(EW_{G}C) ⊗_{ℤ[W_{G}C]} M) is isomorphic to M ⊗_{ℚ[W_{G}C]} ℚ for p = 0 and trivial for p ≥ 1. □

2.5. Proof of Theorem 0.13. The proof is analogous to that of Theorem 0.4, with the exception that we do not use Proposition 3.3 but the following consequences of the hypothesis on R made in the statement : the diagram

\xymatrix{K_{0}(ℤ) \ar[r]^{dtr}_{≅} \ar[d] & HH_{0}^{⊗_{ℤ}}(ℤ) = ℤ \ar[d] \\ K_{0}(R) \ar[r]^{dtr} & HH_{0}^{⊗_{ℤ}}(R)}

commutes, the upper horizontal map is an isomorphism and both vertical arrows are injective. The map (0.14) is now defined as the restriction of the upper horizontal arrow of the diagram appearing in 1.1, in degree n, to the summand for q = 0 and C = {e} and then further to the ℚ-submodule

H_{p}(BG; ℚ) ≅ H_{p}(BG; ℚ) ⊗_{ℚ} (K_{0}(ℤ) ⊗_{ℤ} ℚ) ⊆ H_{p}(BG; ℚ) ⊗_{ℚ} (K_{0}(R) ⊗_{ℤ} ℚ)

(here, p = n). Injectivity of (0.14) is now clear from the general strategy 1.1. □

2.6. Proof of Theorem 0.15. For the given n ≥ 0, the diagram

\xymatrix{ & H_{n}^{G}(E_{Fin}(G); K^{−∞}S^{−1}O_{F} ) \ar[r] \ar[d] & H_{n}^{G}(E_{Fin}(G); K^{−∞}F ) \ar[d]^{≅} \\  & H_{n}^{G}(E_{VCyc}(G); K^{−∞}S^{−1}O_{F} ) \ar[r] \ar[dl]_{assembly}^{_{≅_{ℚ}}} & H_{n}^{G}(E_{VCyc}(G); K^{−∞}F ) \ar[dr]^{assembly} \\ K_{n}(S^{−1}O_{F} [G]) ≅ H_{n}^{G}(pt; K^{−∞}S^{−1}O_{F}) \ar[rrr] \ar[dr]|{dtr} &  &  & H_{n}^{G}(pt; K^{−∞}F) ≅ K_{n}(F G) \ar[dl]|{dtr} \\  & HH_{n}^{⊗_{ℤ}} (S^{−1}O_{F} [G]) \ar[r]^{_{≅_{ℚ}}} & HH_{n}^{⊗_{ℤ}}(F G)}

commutes, where the vertical maps are induced by the, up to G-homotopy, unique G-maps or are given by the trace maps; the horizontal arrows are induced by the inclusion of rings S^{−1}O_{F} ⊆ F. Some explanations are in order for the indicated integral respectively rational isomorphisms.

For every ring R, there are isomorphisms HH_{∗}^{⊗_{ℤ}}(R) ⊗_{ℤ} ℚ ≅ HH_{∗}^{⊗_{ℤ}}(R ⊗_{ℤ} ℚ) and HC_{∗}^{⊗_{ℤ}}(R) ⊗_{ℤ} ℚ ≅ HC_{∗}^{⊗_{ℤ}}(R ⊗_{ℤ} ℚ), because CN_{•}^{⊗_{ℤ}}(R ⊗_{ℤ} ℚ) ≅ CN_{•}^{⊗_{ℤ}}(R) ⊗_{ℤ} ℚ and because the functor (−) ⊗_{ℤ} ℚ commutes with homology and with Tot^{⊕}. (For the notation, see Subsections 4.2 and 4.3 below.) Here, we use that for the total complex occurring in the definition of cyclic homology it does not matter whether one takes Tot^{⊕} or Tot∏. Note that a corresponding statement is false for HP and HN. Hence the bottom horizontal arrow in the diagram above is rationally bijective since S^{−1}O_{F} ⊗_{ℤ} ℚ ≅ F.

The middle left vertical arrow is rationally bijective, since we assume that the Farrell-Jones Conjecture holds rationally for K_{n}(S^{−1}O_{F} [G]).

Since F is a regular ring and contains ℚ, the top right vertical arrow is an isomorphism by [22, Proposition 2.14], see also Subsection 1.1.

Bartels [1] has constructed, for every ring R and every m ∈ ℤ, a retraction

r(R)_{m}: H_{m}^{G}(E_{VCyc}(G); K^{−∞}R) → H_{m}^{G}(E_{Fin}(G); K^{−∞}R)

of the canonical map H_{m}^{G}(E_{Fin}(G); K^{−∞}R) → H_{m}^{G}(E_{VCyc}(G); K^{−∞}R), which is natural in R. We obtain a decomposition, natural in R,

H_{m}^{G}(E_{VCyc}(G); K^{−∞}R) ≅ H_{m}^{G}(E_{Fin}(G); K^{−∞}R) ⊕ ker (r(R)_{m}) .

Therefore, we conclude from the commutative diagram above that the composition

H_{n}^{G}(E_{VCyc}(G); K^{−∞}S^{−1}O_{F} ) ⟶^{≅_{ℚ}} H_{n}^{G}(pt; K^{−∞}S^{−1}O_{F}) ⟶^{dtr} HH_{n}^{⊗_{ℤ}} (S^{−1}O_{F} [G]) ,

after tensoring with ℚ, contains ker(r(S^{−1}O_{F})_{n})⊗_{ℤ}ℚ in its kernel, because ker(r(F)_{n}) = 0. So, to study injectivity properties of the Dennis trace map we can focus attention on the composition

H_{n}^{G}(E_{Fin}(G); K^{−∞}S^{−1}O_{F} ) ⊗_{ℤ} ℚ ↪ H_{n}^{G}(E_{VCyc}(G); K^{−∞}S^{−1}O_{F} ) ⊗_{ℤ} ℚ

⟶^{≅} H_{n}^{G}(pt; K^{−∞}S^{−1}O_{F}) ⊗_{ℤ} ℚ ⟶^{dtr} HH_{n}^{⊗_{ℤ}} (S^{−1}O_{F} [G]) ⊗_{ℤ} ℚ .

By naturality of the bottom isomorphism in Remark 1.14, there is a commutative diagram

\xymatrix{⨁_{\begin{matrix}p,q∈ℤ \\ p+q=n \end{matrix}} ⨁_{(C)∈(FCyc)}H_{p}(BZ_{G}C; ℚ) ⊗_{ℚ[W_{G}C]} θ_{C}(K_{q}(S^{−1}O_{F} [C]) ⊗_{ℤ} ℚ) \ar[dr]^{_{ch_{G}}}_{≅} \ar[dd] \\  & H_{n}^{G}(E_{Fin}(G); K^{−∞}S^{−1}O_{F} ) ⊗_{ℤ} ℚ \ar[dd] \\ ⨁_{\begin{matrix}p,q∈ℤ \\ p+q=n \end{matrix}} ⨁_{(C)∈(FCyc)}H_{p}(BZ_{G}C; ℚ) ⊗_{ℚ[W_{G}C]} θ_{C}(K_{q}(F C) ⊗_{ℤ} ℚ) \ar[dr]^{_{ch_{G}}}_{≅} \\  & H_{n}^{G}(E_{Fin}(G); K^{−∞}F ) ⊗_{ℤ} ℚ.}

Now, consider the composition

(2.2) ⨁_{\begin{matrix}p,q∈ℤ \\ p+q=n \end{matrix}} ⨁_{(C)∈(FCyc)} H_{p}(BZ_{G}C; ℚ) ⊗_{ℚ[W_{G}C]} θ_{C}(K_{q}(S^{−1}O_{F} [C]) ⊗_{ℤ} ℚ)

⟶^{ch^{G}}_{≅} H_{n}^{G}(E_{Fin}(G); K^{−∞}S^{−1}O_{F} )⊗_{ℤ} ℚ ↪ H_{n}^{G}(E_{VCyc}(G); K^{−∞}S^{−1}O_{F} )⊗_{ℤ} ℚ

⟶^{≅} H_{n}^{G}(pt; K^{−∞}S^{−1}O_{F}) ⊗_{ℤ} ℚ ⟶^{dtr} HH_{n}^{⊗_{ℤ}} (S^{−1}O_{F} [G]) ⊗_{ℤ} ℚ .

By the previous two diagrams, the composition (2.2) takes each of the direct summands for q ≤ −1 to zero, since K_{q}(F C) = 0 for q ≤ −1 (the ring F C being regular).

Combining the commutativity of the diagrams occurring in Theorems 1.13 and Remark 1.14 (for R = S^{−1}O_{F}), we deduce that the composition (2.2) restricted to a direct summand with p, q ≥ 0 and with C arbitrary factorizes through the ℚ-vector space

H_{p}(BZ_{G}C; ℚ) ⊗_{ℚ[W_{G}C]} θ_{C}( HH_{q}^{⊗_{ℤ}}(S^{−1}O_{F} [C]) ⊗_{ℤ} ℚ) .

Using the isomorphism HH_{∗}^{⊗_{ℤ}}(S^{−1}O_{F} [G]) ⊗_{ℤ} ℚ ≅ HH_{∗}^{⊗_{ℤ}}(F G) ⊗_{ℤ} ℚ, from the vanishing result stated as Proposition 3.5 below, we conclude that the composition (2.2) vanishes on all summands with q ≥ 1.

Finally, Proposition 3.4 below implies that the composition (2.2) vanishes on all summands with q = 0 and C ≠ {e}, and is injective on the summand for q = 0 and C = {e}. But the restriction of the composition (2.2) to the summand with q = 0 and C = {e} is precisely the composition of the injective map (0.14) with the Dennis trace, simply by Remark 1.14 and by construction of the map (0.14) (see the proof of Theorem 0.13 above). This finishes the proof of Theorem 0.15. □

3. The trace maps for finite cyclic groups

In this section, for a finite cyclic group C, a coefficient k-algebra R, and q ≥ 0, we investigate the trace map

θ_{C}(K_{q}(RC) ⊗_{ℤ} ℚ) → θ_{C}( HH_{q}^{⊗_{k}}(RC) ⊗_{ℤ} ℚ) (3.1)

and its variants using cyclic, periodic cyclic and negative cyclic homology. All results concerning the map (3.1) with q > 0 will in fact be vanishing results stating that the map is the zero map.

Remark 3.2. Note that for a commutative ring k and every k-algebra R, the canonical maps

\xymatrix{HH_{0}^{⊗_{ℤ}}(R) \ar[d]_{≅} \ar[r]^{≅} & HH_{0}^{⊗_{k}}(R) \ar[d]^{≅} \\ HC_{0}^{⊗_{ℤ}}(R) \ar[r]^{≅} & HC_{0}^{⊗_{k}}(R)}

are all isomorphisms, because all four groups can be identified with R/[R, R]. The following results about HH_{0}^{⊗_{ℤ}} hence also apply to other ground rings and to cyclic homology.

Proposition 3.3. Let G be a finite group. Suppose that the ring R is from the following list :

(i) a finite dimensional semisimple algebra R over a field F of characteristic zero;

(ii) a commutative complete local domain R of characteristic zero; (iii) a commutative Dedekind domain R whose quotient field F is an algebraic

number field and for which |G| ∈ R is invertible. Then the trace map K_{0}(RG) → HH_{0}^{⊗_{ℤ}}(RG) is injective in cases (i) and (ii) and is rationally injective in case (iii). This implies in all cases that for a finite cyclic group C, the induced map,

θ_{C}(K_{0}(RC) ⊗_{ℤ} ℚ) → θ_{C}( HH_{0}^{⊗_{ℤ}}(RC) ⊗_{ℤ} ℚ)

is injective. Moreover, in all cases, except possibly in case (ii), the ℚ-vector space θ_{C}(K_{0}(RC) ⊗_{ℤ} ℚ) is non-trivial.

Proof. (i) We first prove injectivity of the trace K_{0}(RG) → HH_{0}^{⊗_{ℤ}}(RG). Since R is semisimple and the order of G is invertible in R, the ring RG is semisimple as well, see for example Theorem 6.1 in [17]. Using the Wedderburn-Artin Theorem [17, Theorem 3.5] and the fact that the trace map is compatible with finite products of rings and with Morita isomorphisms [18, Theorem 1.2.4 on page 17 and Theorem 1.2.15 on page 21], it suffices to show that the trace map

dtr: K_{0}(D) → HH_{0}^{⊗_{ℤ}}(D)

is injective in the case where D is a skew-field which is a finite dimensional algebra over a field F of characteristic zero. The following diagram commutes, where the vertical maps are given by restriction to F :

\xymatrix{K_{0}(D) \ar[d]_{res} \ar[r]^{dtr} & HH_{0}^{⊗_{ℤ}}(D) \ar[d]^{res} \\ K_{0}(F) \ar[r]^{dtr} & HH_{0}^{⊗_{ℤ}}(F)}

The left vertical map can be identified with the map dim_{F} (D) · id: ℤ → ℤ and is hence injective. The trace map K_{0}(F) → HH_{0}^{⊗_{ℤ}}(F) can be identified with the inclusion ℤ → F. This proves injectivity of the Dennis trace K_{0}(RG) → HH_{0}^{⊗_{ℤ}}(RG).

Let R be a finite dimensional F -algebra. Then induction and restriction with respect to the inclusion F G → RG induces maps ind: K_{0}(F G) → K_{0}(RG) and res: K_{0}(RG) → K_{0}(F G) such that res ∘ ind = dim_{F} (R) · id. Hence the map ind: K_{0}(F G) ⊗_{ℤ} ℚ → K_{0}(RG) ⊗_{ℤ} ℚ is injective. For G = C a finite cyclic group, this restricts to an injective map

θ_{C}(K_{0}(F C) ⊗_{ℤ} ℚ) → θ_{C}(K_{0}(RC) ⊗_{ℤ} ℚ) .

Since F is a field of characteristic zero there exists a commutative diagram of ring homomorphisms

\xymatrix{K_{0}(ℚC) ⊗_{ℤ} ℚ \ar[d] \ar[r]^{≅} & map(sub C, ℚ) \ar[d] \\ K_{0}(F C) ⊗_{ℤ} F \ar[r]^{≅} & map(Γ_{F,C}\ con C, F).}

Here, the set con C of conjugacy classes of elements of C identifies with C. Set m = |C| and let μ_{m} ≅ ℤ/mℤ be the group of mth roots of 1 in an algebraic closure of F. The action of the Galois group G(F (μ_{m})|F) on μ_{m} determines a subgroup Γ_{F,C} of (ℤ/mℤ)^{×} ≅ Aut(μ_{m}). An element t ∈ Γ_{F,C} operates on con C by sending (the conjugacy class of) the element c to c^{t}. The set of orbits under this action is Γ_{F,C}\ con C. Note that for F = ℚ, the group Γ_{ℚ,C} is the whole group (ℤ/mℤ)^{×} and Γ_{ℚ,C}\ con C can be identified with sub C, the set of subgroups of C. So, the first line in the diagram is a special case of the second. The right-hand vertical map is contravariantly induced from the quotient map Γ_{F,C}\ con C → sub C and is in particular injective. The horizontal maps are given by sending a representation to its character. They are isomorphisms by [33, II. § 12]. Hence θ_{C}(K_{0}(ℚC) ⊗_{ℤ} ℚ) injects in θ_{C}(K_{0}(F C) ⊗_{ℤ} F). We have shown in Lemma 2.1 that θ_{C}(K_{0}(ℚC) ⊗_{ℤ} ℚ) is non-trivial. Hence θ_{C}(K_{0}(RC) ⊗_{ℤ} ℚ) is non-trivial as well.

(ii) According to Theorem 6.1 in [34], the left-hand vertical map in the commutative diagram

\xymatrix{K_{0}(RG) \ar[r]^{dtr} \ar[d] & HH_{0}^{⊗_{ℤ}}(RG) \ar[d] \\ K_{0}(F G) \ar[r]^{dtr} & HH_{0}^{⊗_{ℤ}}(F G)}

is injective. Here F is the quotient field of R. The bottom map is injective by (i).

(iii) Since any Dedekind ring is regular, the ring R is a regular domain in which the order of G is invertible. Hence RG and F G are regular, compare [22, Proof of Proposition 2.14]. For any regular ring S, the obvious map K_{0}(S) → G_{0}(S), with G_{0}(S) the Grothendieck group of finitely generated S-modules, is bijective [7, Corollary 38.51 on page 29]. Therefore, the map K_{0}(RG) → K_{0}(F G) can be identified with the map

G_{0}(RG) → G_{0}(F G) .

This map has a finite kernel and is surjective under our assumptions on R and F [7, Theorem 38.42 on page 22 and Theorem 39.14 on page 51]. We infer that K_{0}(RG) → K_{0}(F G) is rationally bijective. Using the corresponding commutative square involving the trace maps we have reduced our claim to the case (i). □

Proposition 3.4. Let S^{−1}O_{F} be a localization of the ring of integers O_{F} in an algebraic number field F. Then the canonical map

K_{0}(ℤ) ⊗_{ℤ} ℚ ⟶^{≅} K_{0}(S^{−1}O_{F}) ⊗_{ℤ} ℚ

is an isomorphism and the trace map

dtr: K_{0}(S^{−1}O_{F}) ⊗_{ℤ} ℚ → HH_{0}^{⊗_{ℤ}}(S^{−1}O_{F}) ⊗_{ℤ} ℚ

is injective. If C is a non-trivial finite cyclic group and no prime divisor of its order |C| is invertible in S^{−1}O_{F}, then

θ_{C}(K_{0}(S^{−1}O_{F} C) ⊗_{ℤ} ℚ) = 0.

Proof. According to a result of Swan [34, Proposition 9.1], the canonical map K_{0}(ℤ) ⊗_{ℤ} ℚ → K_{0}(S^{−1}O_{F} [G]) ⊗_{ℤ} ℚ is an isomorphism for a finite group G if no prime divisor of |G| ∈ O_{F} occurs in S. As a consequence, the Artin defect (1.12) of K_{0}(S^{−1}O_{F} C) ⊗_{ℤ} ℚ (i.e. in degree 0) vanishes. The result now follows from the identification, which will be proved in Lemma 7.4 below, of θ_{C}(K_{0}(S^{−1}O_{F} C)⊗_{ℤ} ℚ) with the Artin defect. □

We next collect the results which state that the trace map is the zero map in higher degrees. Note that all linear trace maps factorize through HN_{∗}^{⊗_{ℤ}}. The following result implies that they all vanish in positive degrees for suitable rings R.

Proposition 3.5. Let F be an algebraic number field and R a finite dimensional semisimple F -algebra. Then, for every finite cyclic group C and for every n ≥ 1, we have

HH_{n}^{⊗_{ℤ}}(RC) ⊗_{ℤ} ℚ = 0 and HN_{n}^{⊗_{ℤ}}(RC) ⊗_{ℤ} ℚ = 0 .

Proof. Analogously to the proof of Proposition 3.3 (i), one reduces the claim to the case where the ring RC is replaced by a skew-field D which is a finite dimensional algebra over an algebraic number field F. Let \overline{F} be a splitting field for D, i.e. a finite field extension \overline{F} of F such that \overline{F} ⊗_{F} D ≅ M_{n}(\overline{F}), for some n ≥ 1, see [6, Corollary 7.22 on page 155]. Induction and restriction for D ⊆ \overline{F} ⊗_{F} D yield maps ind: K_{∗}(D) → K_{∗}(\overline{F} ⊗_{F} D) and res: K_{∗}(\overline{F} ⊗_{F} D) → K_{∗}(D) such that res ∘ ind = dim_{F} (\overline{F}) · id. Hence ind: K_{∗}(D) → K_{∗}(\overline{F} ⊗_{F} D) is rationally injective. The same procedure applies to Hochschild homology, cyclic, periodic cyclic and negative cyclic homology, and all these induction and restriction maps are compatible with the various trace maps. Applying Morita invariance, it thus suffices to prove that

HH_{n}^{⊗_{ℤ}}(\overline{F}) ⊗_{ℤ} ℚ = 0 and HN_{n}^{⊗_{ℤ}}(\overline{F}) ⊗_{ℤ} ℚ = 0 ,

for every n ≥ 1. For every ℚ-algebra A, there is obviously an isomorphism CN_{•}^{⊗_{ℤ}}(A) ≅ CN_{•}^{⊗_{ℚ}}(A) of cyclic nerves, see Subsection 4.2 for the notation; hence an isomorphism HX_{∗}^{⊗_{ℤ}}(A) ≅ HX_{∗}^{⊗_{ℚ}}(A), where HX stands for HH, HC, HP or HN. So, we may consider HX_{∗}^{⊗_{ℚ}} in place of HX_{∗}^{⊗_{ℤ}} in the sequel.

By the Hochschild-Kostant-Rosenberg Theorem, one has HH_{∗}^{⊗_{ℚ}}(\overline{F}) ≅ Λ∗ ; \overline{F} Ω_{\overline{F} |ℚ}^{1},

compare [18, Theorem 3.4.4 on page 103]. But Ω_{\overline{F} |ℚ}^{1} = 0 because \overline{F} is a finite

separable extension of ℚ ([12, Corollary 16.16]); therefore HH_{∗}^{⊗_{ℚ}}(\overline{F}) ≅ \overline{F} and is concentrated in degree 0. From the long exact sequence

… → HH_{n}^{⊗_{ℚ}}(\overline{F}) → HC_{n}^{⊗_{ℚ}}(\overline{F}) ⟶^{S} HC_{n−2}^{⊗_{ℚ}}(\overline{F}) → HH_{n−1}^{⊗_{ℚ}}(\overline{F}) → …

it follows that HC_{∗}^{⊗_{ℚ}}(\overline{F}) is isomorphic to \overline{F} in each even non-negative degree, and is zero otherwise. Since the periodicity map S is an isomorphism as soon as its target is non-trivial, the periodic cyclic homology is the inverse limit HP_{n}^{⊗_{ℚ}}(\overline{F}) = lim_{k} HC_{n+2k}^{⊗_{ℚ}}(\overline{F}) and hence is concentrated in (all) even degrees, with a copy of \overline{F} in each such degree, compare [18, 5.1.10 on page 163] and also Remark 3.6 below. In the long exact sequence

… → HN_{n}^{⊗_{ℚ}}(\overline{F}) → HP_{n}^{⊗_{ℚ}}(\overline{F}) ⟶^{\overline{S}} HC_{n−2}^{⊗_{ℚ}}(\overline{F}) → HN_{n−1}^{⊗_{ℚ}}(\overline{F}) → …

compare [18, Proposition 5.1.5 on page 160], the map \overline{S} is then an isomorphism whenever its target is non-trivial. It follows that HN_{∗}^{⊗_{ℚ}}(\overline{F}) is concentrated in nonpositive even degrees (with a copy of \overline{F} in each such degree). □

Remark 3.6. We could not decide the question whether for an odd n ≥ 1 and a finite cyclic group C, the map θ_{C}(K_{n}(ℤC) ⊗_{ℤ} ℚ) → θ_{C}(HN_{n}^{⊗_{ℤ}}(ℤC) ⊗_{ℤ} ℚ), or the corresponding map to periodic cyclic homology, is non-trivial. The calculations in [16] and [5] show that a finer analysis of the trace map is needed in order to settle the problem. The difficulty is that the lim^{1}-terms in the computation of HP out of HC might contribute to non-torsion elements in odd positive degrees.

4. Notation and generalities

4.1. Categories and k-linear categories. Let k be a commutative ring. A klinear category is a small category which is enriched over k-modules, i.e. each morphism set hom_{A}(c, d), with c, d ∈ obj A, has the structure of a k-module, composition of morphisms is bilinear and satisfies the usual associativity axiom; moreover, there are unit maps k → hom_{A}(c, c), for every object c, satisfying a unit axiom. Compare [26, I.8 on page 27, VII.7 on page 181]. Let R be a k-algebra. For any small category C, we can form the associated k-linear category RC. It has the same objects as C and the morphism k-modules are obtained as the free R-module over the morphism sets of C, i.e.

hom_{RC}(c, d) = R mor_{C}(c, d) .

In fact, this yields a functor R(−) from small categories to k-linear categories. Given a k-linear category A, we denote by A_{⊕} the k-linear category whose objects are finite sequences of objects of A, and whose morphisms are “matrices” of morphisms in A with the obvious “matrix product” as composition. Concatenation of sequences yields a sum denoted by “ ⊕ ” and we hence obtain, functorially, a k-linear category with finite sums, compare [26, VIII.2 Exercise 6 on page 194]. If we consider a k-algebra R as a k-linear category with one object then R_{⊕} is a small model for the category of finitely generated free left R-modules.

4.2. Nerves and cyclic nerves. Let C be a small category and let A be a k-linear category. The cyclic nerve of C and the k-linear cyclic nerve of A are respectively denoted by

CN_{•} C and CN_{•}^{⊗_{k}} A .

Depending on the context, they are considered as a cyclic set or as a simplicial set, respectively as a cyclic k-module or as simplicial k-module. Recall that by definition, we have

CN_{[q]} C = ∐_{c_{0},c_{1},...,c_{q}∈obj C} mor_{C}(c_{1}, c_{0}) × ⋯ × mor_{C}(c_{q}, c_{q−1}) × mor_{C}(c_{0}, c_{q}),
CN_{[q]}^{⊗_{k}} A = ⨁_{c_{0},c_{1},...,c_{q}∈obj A} hom_{A}(c_{1}, c_{0}) ⊗_{k} ⋯ ⊗_{k} hom_{A}(c_{q}, c_{q−1}) ⊗_{k} hom_{A}(c_{0}, c_{q}).

The simplicial and cyclic structure maps are induced by composition, insertion of identities and cyclic permutations of morphisms. For more details, see [36, 2.3], [14] and [10]. The (ordinary) nerve of a small category C will always be considered as a simplicial category and denoted by N_{•}C. We will write obj N_{•}C for the underlying simplicial set of objects.

4.3. Simplicial abelian groups and chain complexes. If we are given a simplicial abelian group M_{•}, we denote by DK_{∗}(M_{•}) the associated normalized chain complex. For a chain complex of abelian groups C_{∗} which is concentrated in nonnegative degrees, we denote by DK_{•}(C_{∗}) the simplicial abelian group that is associated to it under the Dold-Kan correspondence. For details see [38, Section 8.4]. In particular, recall that there are natural isomorphism DK_{•}(DK_{∗}(M_{•})) ≅ M_{•} and DK_{∗}(DK_{•}(C_{∗})) ≅ C_{∗}.

The good truncation τ_{≥0}C_{∗} of a chain complex C_{∗} is defined as the non-negative chain complex which coincides with C_{∗} in strictly positive degrees, has the 0-cycles Z_{0}(C_{∗}) in degree 0, and only trivial modules in negative degrees. Given a bicomplex C_{∗∗}, we denote by

Tot^{⊕} C_{∗∗} and Tot∏ C_{∗∗}

the total complexes formed using, respectively, the direct sum or the direct product, compare [38, 1.2.6 on page 8].

4.4. Spectra, Γ-spaces and Eilenberg-Mc Lane spectra. For us, a spectrum consists of a sequence E of pointed spaces E_{n}, with n ≥ 0, together with pointed maps s_{n}: S^{1} ∧ E_{n} → E_{n+1}. We do not require that the adjoints σ_{n}: E_{n} → ΩE_{n+1} of these maps are homotopy equivalences. A map of spectra f : E → E′ consists of a sequence of maps f_{n}: E_{n} → E′_{n} such that f_{n+1} ∘ s_{n} = s′_{n} ∘ id_{S^{1}} ∧ f_{n}. One defines in the usual way the homotopy groups as π_{n}(E) = colim_{k} π_{n+k}(E_{k}), with n ∈ ℤ. The spectrum E is connective if π_{n}(E) = 0 for all n < 0. A map of spectra is called a stable weak equivalence, or to be short, an equivalence, if it induces an isomorphism on all homotopy groups. A spectrum of simplicial sets is defined similarly, using pointed simplicial sets in place of pointed spaces. Such spectra can be realized and then yield spectra in the sense above. We denote by S the sphere spectrum (as a spectrum of simplicial sets).

Let Γ^{op} denote the small model for the category of finite pointed sets whose objects are k_{+} = {+, 1, …, k}, with k ≥ 0, and whose morphisms are pointed maps. A Γ-space 𝔼 is a functor from the category Γ^{op} to the category of pointed simplicial sets which sends 0_{+} = {+} to the (simplicial) point. Every Γ-space 𝔼 can be extended in an essentially unique way to an endofunctor of the category of pointed simplicial sets which we again denote by 𝔼. By evaluation on the simplicial spheres, a Γ-space 𝔼 gives rise to a spectrum of simplicial sets denoted by 𝔼(S). The realization |𝔼(S)| is then a spectrum in the sense defined above. A Γ-space 𝔼 is called special if the map 𝔼(k_{+}) → 𝔼(1_{+})×⋯×𝔼(1_{+}) induced by the projections p_{i}: k_{+} → 1_{+}, with i = 1, …, k, is a weak equivalence for every k. Here, p_{i}(j) is 1 if j = i, and is + otherwise. For more information on spectra and Γ-spaces, we refer to [3] and [25].

An important example of a Γ-space is the Eilenberg-Mc Lane Γ-space ℍM_{•} associated to a simplicial abelian group M_{•}. Its value on the finite pointed set k_{+} is given by the simplicial abelian group ℍM_{•}(k_{+}) = ℤ̃[k_{+}] ⊗_{ℤ} M_{•}. Here ℤ[S] denotes the free abelian group generated by the set S, and, if the set S is pointed with s_{0} as base-point, ℤ̃[S] = ℤ[S]/ℤ[s_{0}] is the corresponding reduced group. The spectrum HM_{•} = |ℍM_{•}(S)| is a model for the Eilenberg-MacLane spectrum associated to M_{•}. The Γ-space ℍM_{•} is very special in the sense of [3, page 98] and by [3, Theorem 4.2], the homotopy groups of the spectrum HM_{•} coincide with the (unstable) homotopy groups of (the realization of) ℍM_{•}(S^{0}) and hence of M_{•}, and consequently with the homology groups of the associated chain complex DK_{∗}(M_{•}). So we have natural isomorphisms

π_{∗}(HM_{•}) ≅ π_{∗}(|ℍM_{•}(S^{0})|) ≅ π_{∗}(|M_{•}|) ≅ H_{∗}( DK_{∗}(M_{•})) . (4.1)

4.5. Cyclic, periodic cyclic and negative cyclic homology. If Z_{•} is a cyclic object in the category of abelian groups, then we denote by B_{∗∗}(Z_{•}), B_{∗∗}^{per}(Z_{•}) and B_{∗∗}^{−}(Z_{•}) the cyclic, periodic cyclic and negative cyclic bicomplexes, see [18, pages 161–162]. For the good truncations of the associated total complexes, we write

C_{∗}^{HC}(Z_{•}) = Tot∏ B_{∗∗}(Z_{•})
C_{∗}^{HP}(Z_{•}) = τ_{≥0} Tot∏ B_{∗∗}^{per}(Z_{•})
C_{∗}^{HN}(Z_{•}) = τ_{≥0} Tot∏ B_{∗∗}^{−}(Z_{•}) .

In order to have a uniform notation, it is also convenient to write

C_{∗}^{HH}(Z_{•}) = DK_{∗}(Z_{•}).

There is a commutative diagram of chain complexes

\xymatrix{C_{∗}^{HN}(Z_{•}) \ar[r] \ar[d]_{h_{∗}} & C_{∗}^{HP}(Z_{•}) \ar[d] \\ C_{∗}^{HH}(Z_{•}) \ar[r] & C_{∗}^{HC}(Z_{•})}

(4.2)

where the horizontal arrows are induced by inclusions of sub-bicomplexes and the vertical arrows by projections onto quotient bicomplexes. Let k be a commutative ring. If Z_{•} is the k-linear cyclic nerve CN_{•}^{⊗_{k}}(A) of a k-linear category A, we abbreviate

C_{∗}^{HX^{⊗_{k}}} (A) = C_{∗}^{HX}( CN_{•}^{⊗_{k}}(A)) .

Here HX stands for HH, HC, HP or HN. The corresponding simplicial abelian group and the corresponding Eilenberg-Mc Lane spectrum will be denoted

C_{•}^{HX^{⊗_{k}}} (A) = DK_{•} (C_{∗}^{HX^{⊗_{k}}} (A))
HX^{⊗_{k}}(A) = HC_{•}^{HX^{⊗_{k}}} (A) .

In particular we have the map h: HN^{⊗_{k}}(A) → HH^{⊗_{k}}(A) induced from the map h_{∗} in (4.2). If R is a k-algebra we can consider it as a k-linear category with one object. Then the homology groups of C_{∗}^{HX^{⊗_{k}}} (R) as defined above coincide in non-negative degrees with the groups HX_{∗}^{⊗_{k}}(R) that appear in the literature, for instance in [18]. Often negative cyclic homology HN_{∗}^{⊗_{k}}(R) is denoted by HC_{∗}^{−}(R) or HC_{∗}^{−}(R|k) in the literature.

5. The trace maps

The aim of this section is to produce the diagram (1.3), i.e. the trace maps as maps of OrG-spectra. We will concentrate on the part of the diagram involving Ktheory, Hochschild homology and negative cyclic homology. The remaining arrows are obtained by straightforward modifications.

5.1. The trace maps for additive categories. We now review the construction of K-theory for additive categories, and of the map h and the trace maps ntr and dtr for k-linear categories with finite sums, following the ideas of [28], [11] and [9].

The following commutative diagram is natural in the k-linear category A:

\xymatrix{ &  & C_{[0]}^{HN^{⊗_{k}}} (A) = DK_{0}(C_{∗}^{HN^{⊗_{k}}} (A)) \ar[dl]_{h_{0}} \ar[r] & DK_{•}(C_{∗}^{HN^{⊗_{k}}} (A)) = C_{•}^{HN^{⊗_{k}}} (A) \ar[d]^{h_{•}=DK_{•}(h_{∗})} \\ obj A \ar[r]^{_{_{dtr}_{0}}} \ar[urr]^{_{_{ntr}_{0}}} & C_{[0]}^{HH^{⊗_{k}}} (A) = CN_{0}^{⊗_{k}} A \ar[rr] &  & DK_{•}(C_{∗}^{HH^{⊗_{k}}} (A)) ≅ CN_{•}^{⊗_{k}}(A).}

(5.1)

Here, the lower horizontal map dtr_{0} is given by sending an object to the corresponding identity morphism. The lift ntr_{0} of this map is explicitly described on page 286 in [28]. The remaining horizontal maps are just the inclusions of the zero simplices. The vertical maps are induced by the map h_{∗} in diagram (4.2). The isomorphism in the bottom right corner is a special case of the natural isomorphism DK_{•}(DK_{∗}(M_{•})) ≅ M_{•}, compare Subsection 4.3. It will be considered as an identification in the following.

The model for the trace maps, for a given k-linear category with finite sums A will be obtained by replacing A in the diagram above by a suitable simplicial k-linear Γ-category. On the K-theory side, we will use the fact that A has finite sums; on the Hochschild side, we will use the k-linear structure.

Let A be a small category with finite sums. We can then apply the Segal construction which yields a Γ-category Seg A, that is, a functor from Γ^{op} to the category of small categories, compare [9, Definition 3.2] and [32, Section 2].

Recall that we consider the nerve of a category as a simplicial category. Let N_{•}^{iso}A be the simplicial subcategory of N_{•}A for which the objects in N_{[q]}^{iso}A are q-tuples of composable isomorphisms in A, whereas there is no restriction on the morphisms. Observe that obj N_{•}^{iso}C = obj N_{•} iso C, where iso C stands for the subcategory of isomorphisms.

The connective K-theory spectrum K(A) can now be defined as the spectrum associated to the Γ-space obj N_{•}^{iso} Seg A, that is,

(5.2)  K(A) = |(obj N_{•}^{iso} Seg A)(S)| .

For a comparison to other definitions of K-theory, see [37, Section 1.8].

We proceed to discuss the trace maps. Let A be a k-linear category with finite sums. Recall that Δ is the category of finite ordered sets [n] = {0 ≤ 1 ≤ … ≤ n}, with n ≥ 0, and monotone maps as morphisms. Observe that N_{•}^{iso} Seg A is a functor from Δ^{op} × Γ^{op} to k-linear categories and it hence makes sense to apply the cyclic nerve constructions. Since the diagram (5.1) is natural in A we obtain maps of simplicial Γ-spaces (alias natural transformations of functors from Δ^{op} × Δ^{op} × Γ^{op} to the category of pointed sets)

\xymatrix{ & DK_{•} C_{∗}^{HN^{⊗k}} N_{•}^{iso} Seg A \ar[d]^{_{h_{••}}} \\ obj N_{•}^{iso} Seg A \ar[r]^{_{dtr_{••}}} \ar[ur]^{_{ntr_{••}}} & CN_{•}^{⊗_{k}} N_{•}^{iso} Seg A.}

(5.3)

Here obj N_{•}^{iso} Seg A is constant in one of the simplicial directions. Taking the diagonal of the two simplicial directions and passing to the associated spectra yields the model for the trace maps that we will use. It remains to identify the objects on the right in (5.3) with our more standard definitions of Hochschild and negative cyclic homology.

Lemma 5.4. Let A be a k-linear category with finite sums. There is a zigzag of stable weak equivalences, natural in A, between

\xymatrix{HN^{⊗_{k}}(A) = H DK_{•} C_{∗}^{HN^{⊗_{k}}} A \ar[dd]^{h} &  & |(DK_{•} C_{∗}^{HN^{⊗_{k}}} N_{•}^{iso} Seg A)(S)| \ar[dd]^{| h_{••}(S)|} \\  & and \\ HH^{⊗_{k}}(A) = H CN_{•}^{⊗_{k}} A &  & |(CN_{•}^{⊗_{k}} N_{•}^{iso} Seg A)(S)|.}

Proof. Consider, for each q, the inclusion of the zero simplices i: A = N_{[0]}^{iso}A → N_{[q]}^{iso}A. There is a left inverse p (forget everything but the zero-th object) and an obvious natural transformation between i ∘ p and the identity which is objectwise an isomorphism. This induces a special homotopy equivalence ([28, Definition 2.3.2]) and hence in particular a homotopy equivalence of cyclic nerves CN_{•}^{⊗_{k}} A ⟶^{≃} CN_{•}^{⊗_{k}} N_{•}^{iso}A, which passes to a chain homotopy equivalence on the negative cyclic construction, compare [28, Proposition 2.4.1]. So we get rid of the N_{•}^{iso}’s in the expressions above. The rest now follows by applying the following lemma to the map

ℍ(h_{•}): ℍ DK_{•} C_{∗}^{HN^{⊗_{k}}} Seg A → ℍ CN_{•}^{⊗_{k}} Seg A

of bi-Γ-spaces, provided we can prove that the source and the target are both special in both variables (see 4.4). Specialness in the Eilenberg-Mc Lane-variable is standard and follows immediately from the definition of the functor ℍ(−). Being special in the Segal-variable means in the case of the first bi-Γ-space that for every l_{+} and k_{+}, the following composition is a weak equivalence :

ℤ̃[l_{+}] ⊗_{ℤ} ( CN_{•}^{⊗_{k}} Seg A(k_{+})) → ℤ̃[l_{+}] ⊗_{ℤ} ( CN_{•}^{⊗_{k}}(A × ⋯ × A))
→ ℤ̃[l_{+}] ⊗_{ℤ} ( CN_{•}^{⊗_{k}} A × ⋯ × CN_{•}^{⊗_{k}} A)
→ ℤ̃[l_{+}] ⊗_{ℤ} CN_{•}^{⊗_{k}} A × ⋯ × ℤ̃[l_{+}] ⊗_{ℤ} CN_{•}^{⊗_{k}} A .

This is clearly true for the last map. The Segal construction is designed in such a way that Seg A(k_{+}) → A × ⋯ × A is an equivalence of categories. By [28, Proposition 2.4.1], this passes to an equivalence on the cyclic constructions and yields that the first map is an equivalence. Proposition 2.4.9 in [28] deals with the second map. The argument for the second bi-Γ-space is analogous. □

Lemma 5.5. Suppose that (k_{+}, l_{+}) ↦ 𝔸(k_{+}, l_{+}) is a bi-Γ-space which is special in both variables, i.e. for every fixed l_{+}, the Γ-space k_{+} ↦ 𝔸(k_{+}, l_{+}) is special, and similarly in the other variable. Then there is a natural zigzag of stable weak equivalences of spectra of simplicial sets between

𝔸(1_{+}, S) and 𝔸(S, 1_{+}) .

Proof. There is a naive definition of a bi-spectrum (of simplicial sets) as a collection 𝔼 of pointed simplicial sets E_{n,m}, with n ≥ 0 and m ≥ 0, together with horizontal and vertical pointed structure maps σ_{h}: E_{n,m} ∧ S^{1} → E_{n+1,m} and σ_{v}: S^{1} ∧ E_{n,m} → E_{n,m+1} satisfying

σ_{h} ∘ (σ_{v} ∧ S^{1}) = σ_{v} ∘ (S^{1} ∧ σ_{h}) .

After some choice of a poset map μ: ℕ_{0} → ℕ_{0} × ℕ_{0} satisfying a suitable cofinality condition, for example μ(2n) = (n, n) and μ(2n + 1) = (n + 1, n), one can form the diagonal spectrum diag_{μ} 𝔼, compare [15, Section 1.3].

A bi-Γ-space 𝔸 is a functor from Γ^{op} to the category of Γ-spaces, denoted by k_{+} ↦ 𝔸(k_{+}, −), and such that 𝔸(0_{+}, l_{+}) is the (simplicial) point for each l_{+}. Every bi-Γ-space 𝔸 gives rise to a simplicial bi-spectrum 𝔸(S, S′) in the naive sense above, with 𝔸(S, S′)_{n,m} = 𝔸(S^{n}, S^{m}). Here S′ is just a copy of the simplicial sphere spectrum S which we want to distinguish in the notation.

There are maps of bi-Γ-spaces

\xymatrix{𝔸(1_{+}, k_{+} ∧ l_{+}) & k_{+} ∧ 𝔸(1_{+}, l_{+}) \ar[l] \ar[r] & 𝔸(k_{+}, l_{+})}

which extend to maps of bi-spectra. We claim that for every pointed simplicial set Y, the corresponding maps of simplicial spectra

\xymatrix{𝔸(1_{+}, Y ∧ S) & Y ∧ 𝔸(1_{+}, S) \ar[l] \ar[r] & 𝔸(Y, S)} (5.6)

are stable weak equivalences. For the first map, this is [3, Lemma 4.1] and no specialness assumption is needed. For the second map, one argues as follows. For a pointed simplicial set X, the simplicial set 𝔸(1_{+}, X) is at least as connected as X [25, Proposition 5.20]. Now, the composition

k_{+} ∧ 𝔸(1_{+}, X) → 𝔸(k_{+}, X) → 𝔸(1_{+}, X) × ⋯ × 𝔸(1_{+}, X)

is the inclusion of a k-fold wedge into the corresponding k-fold product and hence roughly twice as connected as 𝔸(1_{+}, X). Since the second map is a weak equivalence (by the assumption that 𝔸 is special in the second variable), we conclude that the connectivity of the first map grows faster than n for X = S^{n}. The same statement holds for arbitrary pointed simplicial sets Y in place of k_{+} by a careful version of the Realization Lemma for bisimplicial sets (realization preserves connectivity — compare [36, Lemma 2.1.1]). So, the second map in (5.6) indeed is a stable weak equivalence, proving the claim above.

If we now apply the elementary Lemma 1.28 from [15] (this is a Realization Lemma for bi-spectra), we obtain a zigzag of weak equivalences of spectra of simplicial sets between 𝔸(1_{+}, diag_{μ} S ∧ S′) = diag_{μ} 𝔸(1_{+}, S ∧ S′) and diag_{μ} 𝔸(S, S′). The pointed isomorphism between S^{0} and the 0th simplicial set of the spectrum diag_{μ} S ∧ S′ determines uniquely a map of spectra S → diag_{μ} S ∧ S′ which clearly is an isomorphism. In total, we have constructed a zigzag of stable weak equivalences between 𝔸(1_{+}, S) and diag_{μ} 𝔸(S, S′). The result now follows by symmetry (using specialness in the first variable). □

Summarizing we have that for a k-linear category A with finite sums, the model for the trace maps, at the level of spectra, is given by a commutative diagram of the following form

[figure: |(DK_{•} C_{∗}^{HN^{⊗_{k}}} N_{•}^{iso} Seg A)(S)|; | h_{••}(S)|; ≃; …; ≃; ≃; HN^{⊗_{k}}(A); h; K(A) = |(obj N_{•}^{iso} Seg A)(S)|; dtr; ntr; |(CN_{•}^{⊗_{k}} N_{•}^{iso} Seg A)(S)|; ≃; …; ≃; ≃; HH^{⊗_{k}}(A).]

5.2. The trace maps as maps of spectra over the orbit category. We will now define the OrG-spectra representing K-theory, Hochschild homology and other cyclic homology theories, and the trace maps which appear in (1.3).

Given a G-set S, let G^{G}(S) denote the associated transport groupoid, i.e. the category whose objects are the elements of S and where the set of morphisms from s ∈ S to t ∈ S is given by mor(s, t) = {g ∈ G | gs = t}. Given a k-algebra R we can compose the functor G^{G}(?) with the functors R(−) and (−)_{⊕} (compare Subsection 4.1) to obtain a functor

RG^{G}(?)_{⊕}: OrG → k-Cat_{⊕} , G/H ↦ RG^{G}(G/H)_{⊕} ,

where k-Cat_{⊕} denotes the category of small k-linear categories with finite sums, whose morphisms are k-linear functors (and hence respect the sum, compare [26, VIII.2 Prop.4 on page 193]). The idempotent completion Idem A of a category A has as objects the idempotent endomorphisms in A, i.e. morphisms p: c → c with p ∘ p = p; a morphism from p: c → c to q: d → d is given by a morphism f: c → d with q ∘ f = f ∘ p. The idempotent completion of a k-linear category is again k-linear. For a small category C, the idempotent completion of RC_{⊕} is a k-linear category with finite sums. For an arbitrary ring S, the category Idem S_{⊕} is a small model for the category of finitely generated projective left S-modules.

Let R be a k-algebra and H a subgroup of G. Consider the commutative diagram of k-linear categories

\xymatrix{RH \ar[r] \ar[d] & RH_{⊕} \ar[r] \ar[d] & Idem RH_{⊕} \ar[d] \\ RG^{G}(G/H) \ar[r] & RG^{G}(G/H)_{⊕} \ar[r] & Idem RG^{G}(G/H)_{⊕}.}

(5.7)

The vertical functors are all induced from considering H as the full subcategory of G^{G}(G/H) on the object eH ∈ G/H = obj G^{G}(G/H). All vertical functors are k-linear equivalences and the two right hand functors are cofinal inclusions into the corresponding idempotent completions. Hence it follows from [28, Proposition 2.4.1 and 2.4.2] that all functors in the diagram above induce equivalences if one applies Hochschild homology or one of the cyclic homology theories, i.e. HX^{⊗_{k}}(−). Observe that our K-theory functor K(−) can only be applied to the four categories on the right (they have finite sums). The two right-hand vertical maps induce isomorphisms on all higher K-groups, however, K_{0} may differ for a category with finite sums and its idempotent completion.

Finally, define OrG-spectra KR(?) and HX^{⊗_{k}}R(?) by

(5.8)  KR(G/H) = K Idem RG^{G}(G/H)_{⊕}
(5.9)  HX^{⊗_{k}}R(G/H) = HX^{⊗_{k}}RG^{G}(G/H) .

Here again HX stands for HH, HC , HP or HN. Compare (5.2) and the notation introduced in Subsection 4.5. The discussion above and the one in Subsection 4.5 verify all the isomorphisms claimed in (1.4).

Now, apply the construction of (5.3) in the case where the additive category A is Idem RG^{G}(G/H)_{⊕}. Using the equivalences (discussed above) induced by the map Idem RG^{G}(G/H)_{⊕} ← RG^{G}(G/H) and the equivalences appearing in the diagram at the end of Subsection 5.1, we obtain a commutative diagram of connective OrGspectra of the shape

[figure: ′HN^{⊗_{k}}R; ′′HN^{⊗_{k}}R; ≃; ≃; …; ≃; ≃; HN^{⊗_{k}}R; h; KR; dtr; ntr; ′HH^{⊗_{k}}R; ′′HH^{⊗_{k}}R; ≃; ≃; …; ≃; ≃; HH^{⊗_{k}}R,]

where all arrows labelled with a “ ≃ ” (in particular all those pointing left) are objectwise stable weak equivalences.

6. Equivariant homology theories, induction and Mackey structures

A G-homology theory is a collection of functors H_{∗}^{G}(−) = {H_{n}^{G}(−)}_{n∈ℤ} from the category of (pairs of) G-CW -complexes to the category of abelian groups, which satisfies the G-analogues of the usual axioms for a generalized homology theory, compare [22, 2.1.4].

For example, every OrG-spectrum E = E(?) gives rise to a G-homology theory H_{∗}^{G}(−; E) by setting, for a G-CW -complex X,

H_{∗}^{G}(X; E) = π_{∗}(X_{+}^{?} ∧_{Or G} E(?))

and more generally, for a pair of G-CW -complexes (X, A),

H_{∗}^{G}(X, A; E) = π_{∗}((X_{+}/A_{+})^{?} ∧_{Or G} E(?)) .

Here, for a G-space Y, the symbol Y_{+} denotes the space Y with a disjoint base-point added (viewed as a G-fixpoint), and Y^{?} stands for the fixpoint functor map_{G}(−, Y), considered as a contravariant functor from OrG to the category of spaces; and X_{+}^{?} ∧_{Or G} E(?) is the balanced smash product of a contravariant pointed OrG-space and a covariant OrG-spectrum. It is constructed by applying levelwise the balanced smash product

(6.1) Y ∧_{Or G} Z = \xymatrix{coequ ( ⋁_{f∈mor Or G} Y (t(f)) ∧ Z(s(f)) \ar[r] \ar[r] & ⋁_{G/H∈Or G} Y (H) ∧ Z(H))}

of a contravariant pointed OrG-space Y (?) and a covariant pointed OrG-space Z(?); here, s(f) stands for the source and t(f) for the target of the morphism f ∈ mor OrG, coequ is the coequalizer, and the two indicated maps are defined by f^{∗} ∧ id and id ∧ f_{∗} on the wedge-summand corresponding to f. We repeat that H_{∗}^{G}(pt; E) identifies with π_{∗}(E(G/G)). For details, we refer to [8] and [22, Chapter 6].

For a group homomorphism α: H → G and an H-CW -complex X, let ind_{α} X be the quotient of G × X by the right action of H given by (g, x)h = (gα(h), h^{−1}x). An equivariant homology theory H_{∗}^{?} = H_{∗}^{?}(−) consists of a G-homology theory for each group G together with natural induction isomorphisms

ind_{α}: H_{∗}^{H}(X, A) ⟶^{≅} H_{∗}^{G}(ind_{α} X, ind_{α} A)

for each group homomorphism α: H → G and each H-CW -pair (X, A) such that ker α acts freely on X. The induction isomorphisms need to verify certain natural axioms, compare [22, 6.1]. We refer to the collection of induction isomorphisms as an “induction structure”.

Suppose that an OrG-spectrum D(?) is a composition of functors D = E∘G^{G}(?), where E: Groupoids → Sp is a functor from the category of small groupoids to the category of spectra. If E is a homotopy functor, i.e. sends equivalences of groupoids to stable weak equivalences of spectra, then, according to [22, Proposition 6.10] and [31], there is a ‘naturally’ defined induction structure for the collection of G-homology theories, one for each group G, given by H_{∗}^{G}(−; E ∘ G^{G}). Hence each homotopy functor E: Groupoids → Sp determines an equivariant homology theory H_{∗}^{?}(−; E ∘ G^{G}).

Given an equivariant homology theory H_{∗}^{?}(−), one can, for each n ∈ ℤ, construct a covariant functor from FGINJ, i.e. the category of finite groups and injective group homomorphisms, to Ab, i.e. the category of abelian groups, by setting

M_{∗}: FGINJ→ Ab , G ↦ H_{n}^{G}(pt) ; (6.2)

for a group monomorphism α: H ↪ G, we define M_{∗}(α) as the composition

\xymatrix{M_{∗}(H) = H_{n}^{H}(pt) \ar[r]^{^{ind^{α}}} & H_{n}^{G}(G/α(H)) \ar[r]^{H_{n}^{G}(pr)} & H_{n}^{G}(pt) = M_{∗}(H) ,}

(6.3) where pr is the projection onto the point.

A Mackey functor M is a pair (M_{∗}, M^{∗}) consisting of a co- and a contravariant functor FGINJ→ Ab which agree on objects, i.e. M_{∗}(H) = M^{∗}(H) (merely denoted by M(H)), and satisfy the following axioms.

(i) For an inner automorphism c_{g}: G → G, h ↦ g^{−1}hg with g ∈ G one has M_{∗}(c_{g}) = id: M(G) → M(G).

(ii) If f: G ⟶^{≅} H is an isomorphism, then one has M_{∗}(f) ∘ M^{∗}(f) = id and

M^{∗}(f) ∘ M_{∗}(f) = id. (iii) There is a double coset formula, i.e., for two subgroups H, K ≤ G, one has

\begin{matrix}M^{∗}(i: K → G) ∘ M_{∗}(i: H → G) = \\ ∑_{KgH∈K\G/H}M_{∗}(c_{g}: H ∩ g^{−1}Kg → K) ∘ M^{∗}(i: H ∩ g^{−1}Kg → H) , \end{matrix}

where c_{g}(h) = g^{−1}hg and i in each case denotes the inclusion.

If, for every n ∈ ℤ, the covariant functor M_{∗} we associated in (6.2) and (6.3) to an equivariant homology theory H_{∗}^{?}(−) can be extended to a Mackey functor, then we say that the equivariant homology theory admits a “Mackey structure”.

Let R be a k-algebra. We will consider compositions of functors of the form

\xymatrix{OrG \ar[r]^{G^{G}(−)} & Groupoids \ar[r]^{R(−)_{⊕}} & k-Cat_{⊕} \ar[r]^{F} & Sp .}

Recall that k-Cat_{⊕} denotes the category of small k-linear categories with finite sums.

The OrG-spectra we are mainly interested in, namely KR(?) and HX^{⊗_{k}}R(?), are defined (up to equivalence for the latter) as such a composition with F being the composite functor K ∘ Idem(−) for the former, see (5.2) and (5.8), and being HX^{⊗_{k}}(−) for the latter, see Subsection 4.5 and the discussion following diagram (5.7), and (5.9). It turns out that the non-connective K-theory OrG-spectrum K^{−∞}R(?) of Example 1.2 is also such a composition. In that case F is the Pedersen-Weibel functor (defined on Cat_{⊕}), compare [29]. Up to equivalence a model for the (−1)-connective covering map of OrG-spectra KR(?) → K^{−∞}R(?) mentioned in 1.2 is induced by a specific natural transformation between the corresponding F’s.

So, consider a functor F: k-Cat_{⊕} → Sp. We call F a homotopy functor if it takes k-linear equivalences to stable weak equivalences of spectra. We call F additive if for every k-linear functors f, g: A → B between k-linear categories with finite sums,

π_{∗}(F(f ⊕ g)) = π_{∗}(F(f)) + π_{∗}(F(g)) (6.4)

holds; here, f ⊕ g: A → B is the composition

\xymatrix{A \ar[r]^{diag} & A × A \ar[r]^{f×g} & B × B \ar[r]^{⊕} & B ,}

where diag denotes the diagonal embedding and ⊕ is the sum in B.

Proposition 6.5. Suppose that F: k-Cat_{⊕} → Sp is a homotopy functor. Then, the composite functor F∘R(−)_{⊕} is a homotopy functor; in particular, it determines an equivariant homology theory whose underlying G-homology theory, for a group G, is given by the OrG-spectrum FRG^{G}(?)_{⊕}, that is, by

H_{∗}^{G}(X, A; FRG^{G}(?)_{⊕}) = π_{∗}((X_{+}/A_{+})^{?} ∧_{Or G} FRG^{G}(?)_{⊕}) .

If F is additive then this equivariant homology theory admits a Mackey structure.

Proof. The first part is clear. For the second, we need to define the contravariant half of the Mackey functor and verify the axioms. For a given ring S let F(S) denote the category of finitely generated free left S-modules, which is of course not a small category. If we consider a group H as a groupoid with one object, then RH_{⊕} is a small model for the category of finitely generated free left RH-modules and there is an inclusion functor i_{H}: RH_{⊕} → F(RH) which is an equivalence of categories. We choose a functor p_{H}: F(RH) → RH_{⊕} such that p_{H} ∘ i_{H} ≃ id and i_{H} ∘ p_{H} ≃ id. Here, f ≃ g indicates that there exists a natural transformation through isomorphisms. Given a homomorphism α: H → G between finite groups, there are the usual induction and restriction functors ind_{α}: F(RH) → F(RG) and res_{α}: F(RG) → F(RH). For n ∈ ℤ, we define induction and restriction homomorphisms

ind_{α}: π_{n}(FRH_{⊕}) → π_{n}(FRG_{⊕}) and res_{α}: π_{n}(FRG_{⊕}) → π_{n}(FRH_{⊕})

as ind_{α} = π_{n}(F(p_{G} ∘ ind_{α} ∘i_{H})) and res_{α} = π_{n}(F(p_{H} ∘ res_{α} ∘i_{G})). Since f ≃ g implies π_{n}(F(f)) = π_{n}(F(g)), this does not depend on the choice of p_{H} and p_{G}.

Unravelling the definitions, one checks that under the identifications

M(H) = π_{n}( pt_{+}^{?} ∧_{Or G} FRG^{G}(?)_{⊕}) ≅ π_{n}(FRG^{G}(H/H)_{⊕}) ≅ π_{n}(FRH_{⊕}) ,

the induction homomorphism M_{∗}(α) from (6.3) coincides with the induction homomorphism we have just constructed. Using the same identifications, we consider the map res_{α} constructed above as a map M(G) → M(H) and denote it by M^{∗}(α). The axioms now follow since each of the remaining equalities corresponds to a wellknown natural isomorphism between functors on categories of finitely generated free left modules; for the third axiom, one uses (6.4), i.e. additivity of F. □

The functors F that are responsible for K^{−∞}(?), K(?) and HX^{⊗_{k}}(?) are homotopy invariant and additive. We hence obtain the corresponding equivariant homology theories with Mackey structures given, at a group G, by H_{∗}^{G}(−; KR), H_{∗}^{G}(−; K^{−∞}R) and by H_{∗}^{G}(−; HX^{⊗_{k}}R). The maps between these theories that are induced from the maps of OrG-spectra that we have discussed above are compatible with the induction and Mackey structures.

7. Evaluating the equivariant Chern character

In this section, we prove Theorem 1.13 which is a slight improvement of results in [20].

In the previous section we have verified that the assumptions of Theorem 0.1 and of Theorem 0.2 in [20] are satisfied in the case where the equivariant homology theory H_{∗}^{?}(−) is given, at a group G, by H_{∗}^{G}(−; KR)⊗_{ℤ} ℚ, by H_{∗}^{G}(−; K^{−∞}R)⊗_{ℤ} ℚ, or by H_{∗}^{G}(−; HX^{⊗_{k}}R)⊗_{ℤ}ℚ. Let M be a Mackey functor, for instance H ↦ H_{n}^{H}(pt) for n ∈ ℤ fixed. For a finite group H, recall the notation

(7.1) S_{H}(M(H)) = coker ( ⨁_{K≨H} ind_{K}^{H}: ⨁_{K≨H} M(K) → M(H))

from [20]. Observe for example in the case of K-theory that this specializes to (1.12). We obtain from [20, Theorems 0.1 and 0.2], for every G-CW -complex X which is proper (i.e. with all stabilizers finite) and every n ∈ ℤ, a canonical isomorphism

H_{n}^{G}(X) ≅ ⨁_{p+q=n} ⨁_{(H)∈(Fin)} H_{p}(Z_{G}H\X^{H}; ℚ) ⊗_{ℚ[W_{G}H]} S_{H}(H_{q}^{H}(pt)) ,

where (Fin) denotes the set of conjugacy classes of finite subgroups of G. This isomorphism is natural in X and also in the equivariant homology theory with Mackey structure H_{∗}^{?}(−) (i.e. for natural transformations of equivariant homology theories respecting the induction and Mackey structures). Now, take X = \underline{E}G. As in Lemma 8.1 in [24], one shows that the projections

Z_{G}H\\underline{E}G^{H} ← EZ_{G}H ×_{Z_{G}H} \underline{E}G^{H} → EZ_{G}H/Z_{G}H = BZ_{G}H

induce isomorphisms on rational homology. Theorems 1.13 now follows from the next two lemmas.

Lemma 7.2. Let R be a ring and let H be a finite group. If H is not cyclic, then

S_{H}(K_{n}(RH) ⊗_{ℤ} ℚ) = 0 and S_{H}( HX_{n}^{⊗_{k}}(RH) ⊗_{ℤ} ℚ) = 0 ,

for all n ∈ ℤ.

Proof. For a group H, let Sw(H, ℤ) be its Swan group, i.e. the Grothendieck group of left ℤH-modules which are finitely generated as abelian groups. Let Sw^{f}(H, ℤ) be the Grothendieck group of left ℤH-modules which are finitely generated free as abelian groups. The obvious map Sw^{f}(H, ℤ) → Sw(H, ℤ) is an isomorphism, see [34, Proposition 1.1 on page 553]. If H is a finite group, then Sw^{f}(H, ℤ), and hence also Sw(H, ℤ), has the structure of a commutative associative ring, where multiplication is induced by the tensor product over ℤ equipped with the diagonal Haction. The tensor product over ℤ equipped with the diagonal action also leads to a Sw^{f}(H, ℤ)-module structure, and hence a Sw(H, ℤ)-module structure, on K_{n}(RH) for each n ∈ ℤ and each coefficient ring R. For an injective group homomorphism α: H ↪ K between finite groups, we have the usual induction and restriction homomorphisms ind_{H}^{K}: Sw(H, ℤ) → Sw(K, ℤ) and res_{H}^{K}: Sw(K, ℤ) → Sw(H, ℤ). It is not difficult to check that with these structures, Sw(−, ℤ) is a Green ring functor with values in abelian groups and that, for each n ∈ ℤ, the functor K_{n}(R(−)) is a module over it (compare [20, Sections 7 and 8]). Now, by a result of Swan [34, Corollary 4.2 on page 560], for every finite group H, the cokernel of the map

⨁_{\begin{matrix}C≨H \\ C cyclic \end{matrix}} ind_{C}^{H}: ⨁_{\begin{matrix}C≨H \\ C cyclic \end{matrix}} Sw(C, ℤ) ⊗_{ℤ} ℚ → Sw(H, ℤ) ⊗_{ℤ} ℚ

is annihilated by |H|^{2}. With suitable elements x_{C} ∈ Sw(C, ℤ), we can hence write

|H|^{2} · [ℤ] = ∑_{\begin{matrix}C≨H \\ C cyclic \end{matrix}} ind_{C}^{H}(x_{C}) .

Therefore, up to multiplication by |H|^{2}, every element y ∈ K_{n}(RH) is induced from proper cyclic subgroups, since

|H|^{2} · y = |H|^{2} · [ℤ] · y = ∑_{\begin{matrix}C≨H \\ C cyclic \end{matrix}} ind_{C}^{H}(x_{C}) · y = ∑_{\begin{matrix}C≨H \\ C cyclic \end{matrix}} ind_{C}^{H}(x_{C} · res_{C}^{H} y) .

The argument for Hochschild homology and its cyclic variants is similar. The module structure over the Swan ring is also in that case induced by the tensor product over ℤ. □

Remark 7.3. More generally, the proof of Lemma 7.2 works for every module over the rationalized Swan group Sw(−, ℤ)⊗_{ℤ}ℚ considered as a Green ring functor. Note that such a statement does not hold in general for modules over the rationalized Burnside ring A(−) ⊗_{ℤ} ℚ viewed as a Green ring functor.

Lemma 7.4. Let C be a finite cyclic group and M a Mackey functor with values in ℚ-modules. Keeping notation as in (1.11) and (7.1), there is a natural isomorphism

θ_{C}(M(C)) ≅ S_{C}(M(C)) .

Proof. Let D be a subgroup of C. The ring homomorphism χ_{C} of (1.10) sends [C/D] to (x_{(E)})_{(E)}, where x_{(E)} = |(C/D)^{E}| and hence x_{(E)} = [C: D] if E ≤ D and is 0 otherwise. Therefore the maps i_{D}^{C} and r_{D}^{C} which make the diagrams

\xymatrix{A(D) \ar[d]_{χ_{D}} \ar[r]^{^{ind{C \atop D}}} & A(C) \ar[d]^{χ_{C}} & A(C) \ar[d]_{χ_{C}} \ar[r]^{^{res{C \atop D}}} & A(D) \ar[d]^{χ_{D}} \\ ∏_{sub D} ℚ \ar[r]^{i_{D}^{C}} & ∏_{sub C} ℚ & ∏_{sub C} ℚ \ar[r]^{r_{D}^{C}} & ∏_{sub D} ℚ}

commutative are easily seen to be given as follows. The map i_{D}^{C} is multiplication by the index [C: D] followed by the inclusion of the factors corresponding to subgroups of C contained in D. The map r_{D}^{C} is the projection onto the factors corresponding to subgroups of C contained in D. In particular, χ_{C}(ind_{D}^{C}(θ_{D})), considered as a function, is supported only on (D) and takes there the value [C: D]. As a consequence, in A(C) ⊗_{ℤ} ℚ, we have

1 = [C/C] = ∑_{D≤C} \frac{1}{[C: D]} ind_{D}^{C}(θ_{D}) .

Each element in the image of the map 1 − θ_{C}: M(C) → M(C) lies in the image of I = ⨁_{D≨C} ind_{C}^{D}, because

(1 − θ_{C})x = ( ∑_{D≨C} \frac{1}{[C: D]} ind_{D}^{C} θ_{D}) x = ∑_{D≨C} \frac{1}{[C: D]} ind_{D}^{C}(θ_{D} res_{D}^{C} x) .

Moreover θ_{C}: M(C) → M(C) vanishes on the image of this map I; indeed, for D ≨ C, it follows from the description of r_{D}^{C} that res_{D}^{C}(θ_{C}) = 0, and therefore

θ_{C} ind_{D}^{C} y = ind_{D}^{C}(res_{D}^{C}(θ_{C})y) = ind_{D}^{C}(0 · y) = 0 .

So, the cokernel S_{C}(M(C)) of I is isomorphic to the image θ_{C}(M(C)) of θ_{C}. □

8. Comparing different models

In order to prove splitting results in Section 9, we will work with a chain complex version and occasionally with a simplicial abelian group version of the equivariant homology theory that is associated to Hochschild homology. In the present section, we define these versions and prove that they all agree.

Again, fix a group G. A construction analogous to the balanced smash product (6.1), but with smash products “ ∧ ” replaced by tensor products over ℤ, and with wedge sums “ ∨ ” replaced by direct sums, yields the notion of balanced tensor product M(?) ⊗_{ℤOr G} N(?) of a co- and a contravariant ℤOrG-module M(?) and N(?). Here by definition a co- or contravariant ℤOrG-module is a co- respectively contravariant functor from OrG to abelian groups. Let C_{∗} = C_{∗}(?) be a covariant ℤOrG-chain complex, i.e. a functor from the orbit category to the category of chain complexes of abelian groups. We define the G-equivariant Bredon hyperhomology of a pair of G-CW -complexes (X, A) with coefficients in C_{∗} as

H_{∗}^{G}(X, A; C_{∗}) = H_{∗}( Tot^{⊕} (C̃_{∗}^{sing}((X_{+}/A_{+})^{?}) ⊗_{ℤOr G} C_{∗}(?))) .

Here, for a pointed G-space Y = (Y, y_{0}) (where y_{0} is a G-fixpoint), the functor which sends G/H to the reduced singular chain complex of Y^{H} is denoted by C̃_{∗}^{sing}(Y^{?}).

In this construction, up to canonical isomorphism, we can replace C̃_{∗}^{sing}(−) by the reduced cellular chain complex C̃_{∗}^{cell}(−) (this will be needed in Subsection 9.2).

For a simplicial ℤOrG-module M_{•} = M_{•}(?), i.e. a covariant functor from OrG to the category of simplicial abelian groups, we define similarly

H_{∗}^{G}(X, A; M_{•}) = π_{∗}(|ℤ̃[S_{•}((X_{+}/A_{+})^{?})] ⊗_{ℤOr G} M_{•}(?)|) .

Here, S_{•} stands for the singular simplicial set associated to a topological space. For a pointed simplicial set Y_{•} = (Y_{•}, y_{0}), we set ℤ̃[Y_{•}] = ℤ[Y_{•}]/ℤ[y_{0}] and the tensor products of simplicial abelian groups are taken degreewise.

For an OrG-spectrum E = E(?), recall that we use the notation

H_{∗}^{G}(X, A; E) = π_{∗}((X_{+}/A_{+})^{?} ∧_{Or G} E(?)) .

A simplicial ℤOrG-module M_{•} = M_{•}(?) gives rise to a ℤOrG-chain complex DK_{∗}M_{•} via the Dold-Kan correspondence and determines an OrG-spectrum HM_{•} via the Eilenberg-Mc Lane functor (see Subsections 4.3 and 4.4). The following proposition specializes to a well-known fact in the case where G is the trivial group.

Proposition 8.1. Let M_{•} be a functor from OrG to simplicial abelian groups. Then, there are natural isomorphisms of G-homology theories defined on pairs of G-CW -complexes

H_{∗}^{G}(−; HM_{•}) ⟶^{≅} H_{∗}^{G}(−; M_{•}) ⟶^{≅} H_{∗}^{G}(−; DK_{∗}M_{•}) .

In particular we have natural isomorphisms

H_{∗}^{G}(−; HX^{⊗_{k}}R) ≅ H_{∗}^{G}(−; C_{•}^{HX^{⊗_{k}} R}) ≅ H_{∗}^{G}(−; C_{∗}^{HX^{⊗_{k}} R}) . (8.2)

Here we have used the notation

(8.3)  C_{•}^{HX^{⊗_{k}} R}(?) = C_{•}^{HX^{⊗_{k}}} RG^{G}(?)
(8.4)  C_{∗}^{HX^{⊗_{k}} R}(?) = C_{∗}^{HX^{⊗_{k}}} RG^{G}(?) ,

for the indicated simplicial ℤOrG-module respectively ℤOrG-chain complex, compare Subsection 4.5, and (5.9).

Proof of Proposition 8.1. We discuss the first natural transformation in the absolute case, i.e. for A = ∅ (the general case is similar). For a spectrum E in simplicial sets we denote by |E| the associated (topological) spectrum. For every G-CW -complex X and every OrG-spectrum E = E(?) in simplicial sets, there is a natural equivalence and, since realization commutes with taking coequalizers, a natural homeomorphism

X_{+} ∧_{Or G} |E| ⟵^{≃} |S_{•}X_{+}| ∧_{Or G} |E| ⟶^{≅} |S_{•}X_{+} ∧_{Or G} E| .

Note that there is an obvious natural isomorphism of spectra

S_{•}X_{+} ∧_{Or G} ℍM_{•}(S) ≅ (S_{•}X_{+} ∧_{Or G} ℍM_{•})(S) .

Observe also that for an (unpointed) simplicial set Y_{•}, there is an isomorphism of Γ-spaces

ℤ[Y_{•}] ⊗_{ℤOr G} ℍM_{•} ≅ ℍ(ℤ[Y_{•}] ⊗_{ℤOr G} M_{•}) . (8.5)

By (4.1), the homotopy groups of the spectrum associated to the right hand side are given by the (unstable) homotopy groups of (the realization of) ℤ[Y_{•}]⊗_{Or G} M_{•}.

So, observing that ℤ̃(S_{•}X_{+}) ≅ ℤ[S_{•}X], to produce the first natural transformation of the statement, it will suffice to define a natural transformation of Γ-spaces

S_{•}X_{+} ∧_{Or G} ℍM_{•} → ℤ̃[S_{•}X_{+}] ⊗_{ℤOr G} ℍM_{•} . (8.6)

More generally, for every contravariant functor Z_{•} = Z_{•}(?) from OrG to pointed simplicial sets and every covariant functor N_{•} = N_{•}(?) from OrG to simplicial abelian groups, we will construct a natural transformation

Z_{•} ∧_{Or G} N_{•} → ℤ̃[Z_{•}] ⊗_{ℤOr G} N_{•} .

To produce the map we use the following facts. The left-hand side is defined as a coequalizer in the category of pointed simplicial sets completely analogous to (6.1), and the right-hand side similarly as a coequalizer in the category of simplicial abelian groups. Let U denote the forgetful functor from simplicial abelian groups to pointed simplicial sets. For a pointed simplicial set X, a simplicial abelian group A and a family A_{i}, i ∈ I of simplicial abelian groups there are obvious natural maps X ∧ UA → U(ℤ̃[X] ⊗ A) and ⋁_{i∈I} UA_{i} → U(⨁_{i∈I} A_{i}). Given two maps f, g: A → B of simplicial abelian groups there is an obvious natural map coequ(Uf, Ug) → U coequ(f, g). Combining these facts one easily constructs the required natural transformation above.

Now, we show that the first natural transformation of the statement is an isomorphism. If a natural transformation between G-homology theories induces an isomorphism when evaluated on all orbits G/H, then it induces an isomorphism for all pairs of G-CW -complexes by a well-known argument. Unravelling the construction of the first natural transformation, it suffices to check that for every orbit G/H, the map

(S_{•}(G/H)_{+} ∧_{Or G} ℍM_{•})(S) → ℍ(ℤ̃[S_{•}(G/H)_{+}] ⊗_{ℤOr G} M_{•})(S)

induced by (8.5) and (8.6) is a stable weak equivalence. Before evaluation at S, both sides are canonically isomorphic to the Γ-space ℍM_{•}(G/H) by suitable analogues of Lemma 9.15. We leave it to the reader to verify that we indeed have G-homology theories here, compare [8, Lemma 4.2].

We now construct the second natural transformation of the statement and prove at the same time that it is an isomorphism. For a bisimplicial abelian group A_{••}, let C_{∗∗}(A_{••}) denote the associated bicomplex, compare [38, page 275]. Note that given two simplicial abelian groups C_{•} and D_{•}, there is a natural isomorphism of bicomplexes DK_{∗}(C_{•}) ⊗_{ℤ} DK_{∗}(D_{•}) ≅ C_{∗∗}(C_{•} ⊗_{ℤ} D_{•}), where C_{•} ⊗_{ℤ} D_{•} is viewed as a bisimplicial abelian group. Note also that DK_{∗} (ℤ̃[S_{•}(X)]) = C̃_{∗}^{sing}(X), for every space X. The degreewise tensor products of simplicial abelian groups appearing in the source of the second natural transformation can be thought of as diagonals of the corresponding bisimplicial sets. Applying all these observations and using again the definition of the balanced tensor product in terms of coequalizers, it suffices to observe that for every pair of maps f_{••}, g_{••}: A_{••} → B_{••} of bisimplicial abelian groups, we have the following chain of isomorphisms

π_{∗}(| coequ(diag f_{••}, diag g_{••})|) ≅ π_{∗}(| diag coequ(f_{••}, g_{••})|)
≅ H_{∗}( Tot^{⊕} C_{∗∗}(coequ(f_{••}, g_{••})))
≅ H_{∗}( coequ(Tot^{⊕} C_{∗∗}(f_{••}), Tot^{⊕} C_{∗∗}(g_{••}))) ,

where the second isomorphism is the Eilenberg-Zilber Theorem as formulated in [38, Theorem 8.5.1 on page 276]. □

9. Splitting assembly maps

In this section, we prove Theorem 1.7 and Addendum 1.8, i.e. the splitting and isomorphism results for the assembly maps in Hochschild, cyclic, periodic cyclic and negative cyclic homology. We begin with the case of Hochschild homology.

9.1. Splitting the Hochschild homology assembly map. Fix a group G and let S be a G-set. Recall that con G denotes the set of conjugacy classes of G. Sending a q-simplex (g_{0}, …, g_{q}) in CN_{•} G^{G}(S) to the conjugacyclass (g_{0} ⋯ g_{q}) yields a map of cyclic sets

CN_{•} G^{G}(S) → con G .

Here con G is considered as a constant cyclic set. The cyclic nerve decomposes, as a cyclic set, into the disjoint union of the corresponding pre-images, namely

CN_{•} G^{G}(S) = ∐_{(c)∈con G} CN_{•(c)} G^{G}(S) . (9.1)

Observe that CN_{•(c)} G^{G}(G/H) ≠ ∅ implies that ⟨c⟩ is subconjugate to H. For every small category C, we have a natural isomorphism

k CN_{•} C ≅ CN_{•}^{⊗_{k}} kC

and, because of the isomorphism RC ≅ R ⊗_{k} kC, also

CN_{•}^{⊗_{k}} RC ≅ (CN_{•}^{⊗_{k}} kC) ⊗_{k} (CN_{•}^{⊗_{k}} R) . (9.2)

We therefore obtain an induced decomposition for the k-linear cyclic nerve of RG^{G}(S), that we denote by

(9.3)  CN_{•}^{⊗_{k}} RG^{G}(S) = ⨁_{(c)∈con G} CN_{•(c)}^{⊗_{k}} RG^{G}(S)
(9.4)  = ⨁_{(c)∈con G} CN_{•(c)}^{⊗_{k}} kG^{G}(S) ⊗_{k} CN_{•}^{⊗} R .

For typographical reasons we introduce the following abbreviation for the corresponding decomposition of simplicial ℤOrG-modules :

(9.5)  C_{•}^{HH^{⊗_{k}} R}(?) = ⨁_{(c)∈con G} C_{•(c)}^{HH^{⊗_{k}} R}(?) ;

see (8.3) for the notation. Using the identifications (8.2) and the decomposition (9.5), the Hochschild homology generalized assembly map

\xymatrix{H_{n}^{G}(E_{F}(G); HH^{⊗_{k}}R) \ar[r]^{assembly} & H_{n}^{G}(pt; HH^{⊗_{k}}R) ≅ HH_{n}^{⊗_{k}}(RG)}

appearing in diagram (1.6) can be identified with the upper horizontal map in the following commutative diagram

\xymatrix{H_{n}^{G}(E_{F}(G); ⨁_{(c)∈con G} C_{•(c)}^{HH^{⊗_{k}} R}) \ar[d]_{id ⊗pr_{F}} \ar[r]^{assembly} & H_{n}^{G}( pt; ⨁_{(c)∈con G} C_{•(c)}^{HH^{⊗_{k}} R}) \ar[d]^{id ⊗pr_{F}} \\ H_{n}^{G}(E_{F}(G);_{⨁ ⟨c⟩∈F^{(c)∈con G}}C_{•(c)}^{HH^{⊗_{k}} R}) \ar[r] & H_{n}^{G}( pt;_{⨁ ⟨c⟩∈F^{(c)∈con G}}C_{•(c)}^{HH^{⊗_{k}} R}).}

(9.6)

Here, the vertical maps are induced by the projection pr_{F} onto the summands for which the cyclic subgroup ⟨c⟩ belongs to the family F. Note that pr_{F} is the identity map if F contains all cyclic subgroups of G. The statement about Hochschild homology in Theorem 1.7 now follows directly from the following two lemmas.

Lemma 9.7. For every family F, the left vertical map in (9.6) is an isomorphism.

Lemma 9.8. For every family F, the bottom map in (9.6) is an isomorphism.

The proofs of Lemmas 9.7 and 9.8 will occupy the rest of this subsection. They rely on the following computation of the cyclic nerve of a transport groupoid.

Let E_{•}G be the simplicial set given by N_{•}G^{G}(G/1). In words : consider G as a category with G as set of objects and precisely one morphism between any two objects, and then take the nerve of this category. This is a simplicial model for the universal free G-space which is usually denoted by EG. For c ∈ G let ⟨c⟩ be the cyclic subgroup generated by c. For h ∈ N_{G}⟨c⟩, let R_{h} ∈ map_{G}(G/⟨c⟩, G/⟨c⟩) be the map given by R_{h}(g⟨c⟩) = gh⟨c⟩. For every G-set S, precomposing with R_{h} yields a left action of Z_{G}⟨c⟩ ≤ N_{G}⟨c⟩ on map_{G}(G/⟨c⟩, S).

Proposition 9.9. For a group G, choose a representative c ∈ (c) for each conjugacy class (c) ∈ con G. Let ⟨c⟩ denote the cyclic subgroup it generates. There is a map of OrG-simplicial sets (depending on the choice)

∐_{(c)∈con G} E_{•}Z_{G}⟨c⟩ ×_{Z_{G}⟨c⟩} map_{G}(G/⟨c⟩,?) → CN_{•} G^{G}(?) .

This map is objectwise a simplicial homotopy equivalence, and is compatible with the decomposition (9.1) of the target.

Remark 9.10. There seems to be no obvious cyclic structure on the source of the map above.

Proof of Proposition 9.9. We first introduce some more notation. Given a groupoid G, we denote by aut G its category of automorphisms, i.e. the category whose objects are automorphisms h: s → s in G and where a morphism from h: s → s to h′: t → t is given by a morphism g: s → t satisfying h′ ∘ g = g ∘ h. In the case where G = G^{G}(S), the conjugacy class (h) ∈ con G associated to an object h: s → s in aut G^{G}(S) does only depend on the isomorphism class of this object. This yields a well-defined map of simplicial sets

N_{•} aut G^{G}(S) ⟶ con G , \xymatrix{s_{0} \ar[d]_{h_{0}} & s_{1} \ar[d]_{h_{1}} \ar[l]_{g_{0}} & … \ar[l]_{g_{1}} & s_{q} \ar[d]_{h_{q}} \ar[l]_{g_{q−1}} \\ s_{0} & s_{1} \ar[l]^{g_{0}} & … \ar[l]^{g_{1}} & s_{q}. \ar[l]^{g_{q−1}}} ⟼ (h_{0}) ,

where con G is considered as a constant simplicial set. Let aut_{(c)} G^{G}(S) denote the full subcategory of aut G^{G}(S) on the objects h: s → s with h ∈ (c). The decomposition of the nerve into pre-images under the map to con G above is given by

N_{•} aut G^{G}(S) = ∐_{(c)∈con G} N_{•} aut_{(c)} G^{G}(S) .

The components of the map in Proposition 9.9 are obtained as the composition of the three maps

\xymatrix{E_{•}Z_{G}⟨c⟩ ×_{Z_{G}⟨c⟩} map_{G}(G/⟨c⟩,?) \ar[r] & N_{•}G^{Z_{G}⟨c⟩}( map_{G}(G/⟨c⟩,?)) \ar@{-->}[dl] \\ N_{•} aut_{(c)} G^{G}(?) \ar[r] & CN_{•(c)} G^{G}(?)}

which are constructed in the following lemma. Proposition 9.9 is an immediate consequence of that lemma. □

Lemma 9.11. Let G be a group and S a G-set.

(i) There is a simplicial isomorphism E_{•}G ×_{G} S → N_{•}G^{G}(S).

(ii) For (c) ∈ con G, choose a representative c ∈ (c). There is an equivalence of categories

G^{Z_{G}⟨c⟩}( map_{G}(G/⟨c⟩, S)) → aut_{(c)} G^{G}(S) ,

which depends on the choice. (iii) For every groupoid G, there is a simplicial isomorphism

N_{•} aut G → CN_{•} G .

If G = G^{G}(S) then the isomorphism commutes with the maps to con G.

All three constructions are natural with respect to S.

Proof. (i) The isomorphism E_{•}G ×_{G} S → N_{•}G^{G}(S) is given, on level q, by

\xymatrix{[ g_{0} & g_{1} \ar[l]_{g_{0}g_{1}^{−1}} & … \ar[l]_{g_{1}g_{2}^{−1}} & g_{q} , s ] ⟼ (g_{0}s \ar[l]_{g_{q−1}g_{q}^{−1}} & g_{1}s \ar[l]_{g_{0}g_{1}^{−1}} & … \ar[l]_{g_{1}g_{2}^{−1}} & g_{q}s) . \ar[l]_{g_{q−1}g_{q}^{−1}}}

(ii) The functor sends an object ϕ ∈ map_{G}(G/⟨c⟩, S) to the automorphism c: ϕ(e⟨c⟩) → ϕ(e⟨c⟩). Here e is the trivial element in G and c the chosen representative in (c). A morphism z: ϕ → zϕ, with z ∈ Z_{G}⟨c⟩, is taken to the (iso)morphism z^{−1} : ϕ(e⟨c⟩) → z^{−1}ϕ(e⟨c⟩). The functor is full and faithful and every object in the target category is isomorphic to an image object.

(iii) The isomorphism N_{•} aut G → CN_{•} G is given, on level q, by

[figure: s_{0}; h_{0}; s_{1}; h_{1}; g_{0}; …; g_{1}; s_{q}; h_{q}; g_{q−1}; ⟼; s_{0}; s_{1}; g_{0}; …; g_{1}; s_{q}; g_{q−1}; s_{0}; h_{0}(g_{0}···g_{q−1})^{−1}; s_{1}; g_{0}; …; g_{1}; s_{q} .; g_{q−1}]

The compatibility with the maps to con G is clear. □

The following is the linear analogue of Proposition 9.9.

Corollary 9.12. For every conjugacy class (c) ∈ con G there is natural transformation of functors from the orbit category OrG to the category of simplicial k-modules,

k[E_{•}Z_{G}⟨c⟩] ⊗_{kZ_{G}⟨c⟩} k map(G/⟨c⟩,?) ⊗_{k} CN_{•}^{⊗_{k}} R → CN_{•(c)}^{⊗_{k}} RG^{G}(?)

which is objectwise a homotopy equivalence.

Proof. Apply the functor free k-module k(−) to the map in Proposition 9.9 and recall the identification (9.2). □

Observe that we have a decomposition of G-homology theories

H_{∗}^{G}(−; C_{•}^{HH^{⊗_{k}} R}) ≅ ⨁_{(c)∈con G} H_{∗}^{G}(−; C_{•(c)}^{HH^{⊗_{k}} R}) , (9.13)

because the tensor product over the orbit category and homology both commute with direct sums. For each of the summands, we have the following computation.

Proposition 9.14. For every G-CW -complex X and every (c) ∈ con G, there is a natural isomorphism

H_{∗}^{G}(X; C_{•(c)}^{HH^{⊗_{k}} R}) ≅ H_{∗}(X^{⟨c⟩} ×_{Z_{G}⟨c⟩} EZ_{G}⟨c⟩; CN_{•}^{⊗_{k}} R) .

Proof. On the level of simplicial abelian groups, Corollary 9.12, in combination with Lemma 9.15, yields

ℤ̃[S_{•}X_{+}^{?}] ⊗_{ℤOr G} C_{•(c)}^{HH^{⊗_{k}} R}(?) ≃

≃ ℤ̃[S_{•}X_{+}^{?}] ⊗_{ℤOr G} k[E_{•}Z_{G}⟨c⟩] ⊗_{kZ_{G}⟨c⟩} k map(G/⟨c⟩,?) ⊗_{k} CN_{•}^{⊗_{k}} R

≅ ℤ̃[S_{•}X_{+}^{⟨c⟩}] ⊗_{ℤZ_{G}⟨c⟩} ℤ[E_{•}Z_{G}⟨c⟩] ⊗_{ℤ} CN_{•}^{⊗_{k}} R ,

hence the result. □

Lemma 9.15. Let F be a contravariant functor from OrG to simplicial k-modules. Then, for every subgroup H ≤ G, there is a natural isomorphism

F (?) ⊗_{kOr G} k map_{G}(G/H,?) ≅ F (G/H) .

We can now finish the proof of the part of Theorem 1.7 concerned with Hochschild homology.

Proof of Lemmas 9.7 and 9.8. Compute the relevant maps in diagram (9.6) using (9.13) and Proposition 9.14. Observe that by the very definition of E_{F}(G), we have E_{F}(G)^{⟨c⟩} = ∅ if and only if ⟨c⟩ ∈ F. So the projection id ⊗pr_{F} is the zero map exactly on those summand which are anyway trivial. This proves Lemma 9.7. For ⟨c⟩ ∈ F, the map E_{F}(G)^{⟨c⟩} → pt is a homotopy equivalence. Therefore, E_{F}(G)^{⟨c⟩} × EZ_{G}⟨c⟩ → pt ×EZ_{G}⟨c⟩ is an equivalence of free Z_{G}⟨c⟩-spaces and hence remains an equivalence if we quotient out the Z_{G}⟨c⟩-action. This establishes Lemma 9.8. □

The following example gives a further illustration of the computation achieved above.

Example 9.16. Combining (8.2), the isomorphism (9.13) and Proposition 9.14, we get, for every G-CW -complex X, a decomposition

H_{∗}^{G}(X; HH^{⊗_{k}}R) ≅ ⨁_{(c)∈con G} H_{∗}(X^{⟨c⟩} ×_{Z_{G}⟨c⟩} EZ_{G}⟨c⟩; C_{∗}^{HH^{⊗_{k}}} (R)) ,

where each direct summand on the right-hand side is the (non-equivariant) hyperhomology of the space X^{⟨c⟩} ×_{Z_{G}⟨c⟩} EZ_{G}⟨c⟩ with coefficients in the Hochschild complex, i.e. in the k-chain complex C_{∗}^{HH^{⊗_{k}}} (R) = DK_{∗}(CN_{•}^{⊗_{k}} R). In the case R = k, the degree zero inclusion k → C^{HH^{⊗_{k}}} (k) is a homology equivalence and hence a chain homotopy equivalence, because both complexes are bounded below and consist of projective k-modules. Thus, we infer

H_{∗}(X^{⟨c⟩} ×_{Z_{G}⟨c⟩} EZ_{G}⟨c⟩; C_{∗}^{HH^{⊗_{k}}} (k)) ≅ H_{∗}(X^{⟨c⟩} ×_{Z_{G}⟨c⟩} EZ_{G}⟨c⟩; k)

for each conjugacy class (c), and therefore

H_{∗}^{G}(X; HH^{⊗_{k}}R) ≅ ⨁_{(c)∈con G} H_{∗}(X^{⟨c⟩} ×_{Z_{G}⟨c⟩} EZ_{G}⟨c⟩; k) .

In the special case where X = pt and R = k, we rediscover the well-known decomposition of k-modules

(9.17)  HH^{⊗_{k}}(kG) ≅ ⨁_{(c)∈con G} H_{∗}(BZ_{G}⟨c⟩; k) .

If we insert X = E_{F}(G) for an arbitrary family of subgroups F, we obtain

H_{∗}^{G}(E_{F}(G); HH^{⊗_{k}}R) ≅ ⨁_{\begin{matrix}(c)∈con G \\ ⟨c⟩∈F \end{matrix}} H_{∗}(BZ_{G}⟨c⟩; k) ,

because E_{F}(G)^{⟨c⟩}×EZ_{G}⟨c⟩ is a model for EZ_{G}⟨c⟩ if ⟨c⟩ ∈ F and is empty otherwise. The map E_{F}(G) → pt induces the obvious inclusion.

Remark 9.18. Of course one does not need the elaborate setup using spectra, nor Theorem 1.7, in order to prove the well-known decomposition (9.17). But our aim was to compare the Hochschild assembly map with the one for K-theory. There is no chain complex version of the assembly map on the level of K-theory. Furthermore, our effort has the pay off that it can be generalized to topological Hochschild homology and its refinements as explained in [23].

9.2. Splitting cyclic, periodic cyclic and negative cyclic assembly maps. Observe that the sum decomposition (9.3) is compatible with the cyclic structure. Keeping notation as in (8.4), we hence obtain a decomposition

C_{∗}^{HX^{⊗_{k}} R}(?) = ⨁_{(c)∈con G} C_{∗(c)}^{HX^{⊗_{k}} R}(?)

of ℤOrG-chain complexes. Compare with the splitting (9.5). There is consequently a version of diagram (9.6) with C_{•(c)}^{HH^{⊗_{k}} R} replaced everywhere by C_{∗(c)}^{HX^{⊗_{k}} R}, and where the upper horizontal map corresponds to the generalized assembly map for HXhomology. In order to prove the cyclic homology part of Theorem 1.7 and to establish Addendum 1.8, it suffices to obtain the analogues of Lemmas 9.7 and 9.8 with C_{•(c)}^{HH^{⊗_{k}} R} replaced everywhere by C_{∗(c)}^{HX^{⊗_{k}} R}. However, this follows immediately from the following proposition.

Proposition 9.19. Let X → X′ be a map of G-CW -complexes and Z_{•}(?) → Z′_{•}(?) a map of cyclic ℤOrG-modules, i.e. a natural transformation between functors from the orbit category OrG to the category of cyclic abelian groups. Keep notation as at the beginning of Subsection 4.5.

If the induced map

H_{∗}^{G}(X; C_{∗}^{HH}(Z_{•})) → H_{∗}^{G}(X′; C_{∗}^{HH}(Z′_{•}))

is an isomorphism, then the map

H_{∗}^{G}(X; C_{∗}^{HC}(Z_{•})) → H_{∗}^{G}(X′; C_{∗}^{HC}(Z′_{•}))

is an isomorphism. If moreover X and X′ are finite G-CW -complexes, then also the maps

H_{∗}^{G}(X; C_{∗}^{HP}(Z_{•})) ⟶^{≅} H_{∗}^{G}(X′; C_{∗}^{HP}(Z′_{•})),

H_{∗}^{G}(X; C_{∗}^{HN}(Z_{•})) ⟶^{≅} H_{∗}^{G}(X′; C_{∗}^{HN}(Z′_{•}))

are isomorphisms.

Proof. There is a short exact sequence of chain complexes

0 → C_{∗}^{HH}(Z_{•}) → C_{∗}^{HC}(Z_{•}) → C_{∗}^{HC}(Z_{•})[−2] → 0 ,

which is natural in Z_{•}, see [18, 2.5.10 on pages 78–79]. We use here the notation C_{∗}[r] for the chain complex which is shifted down r steps, i.e. (C_{∗}[r])_{n} = C_{n+r}. Since Tot^{⊕} and C̃_{∗}^{cell}(X_{+}^{?})⊗_{ℤOr G}(−) are exact functors (we use here that C̃_{∗}^{cell}(X_{+}^{?}) is a free ℤOrG-module), the maps induced by X → X′ and Z_{•} → Z′_{•} lead to a short exact ladder diagram of chain complexes. The corresponding long exact ladder in homology, the fact that H_{∗}^{G}(X; C_{∗}^{HH}(Z_{•})) and H_{∗}^{G}(X; C_{∗}^{HC}(Z_{•})) are concentrated in non-negative degrees and an easy inductive argument based on the Five-Lemma finish the proof for cyclic homology.

In order to prove the statement for periodic cyclic homology, one uses that the periodic cyclic complex can be considered as the inverse limit of the tower of cyclic complexes

… → C_{∗}^{HC}(Z_{•})[4] → C_{∗}^{HC}(Z_{•})[2] → C_{∗}^{HC}(Z_{•})[0] . (9.20)

For n ≥ 0, we have the following natural maps :

H_{n}^{G}(X; C_{∗}^{HP}(Z_{•})) ≅ H_{n}( Tot^{⊕} (C̃_{∗}^{cell}(X_{+}^{?}) ⊗_{ℤOr G} lim_{r} C_{∗}^{HC}(Z_{•})[2r]))
→ H_{n}( lim_{r} Tot^{⊕} (C̃_{∗}^{cell}(X_{+}^{?}) ⊗_{ℤOr G} C_{∗}^{HC}(Z_{•})[2r]))
→ lim_{r} H_{n}( Tot^{⊕} (C̃_{∗}^{cell}(X_{+}^{?}) ⊗_{ℤOr G} C_{∗}^{HC}(Z_{•})[2r]))
≅ lim_{r} H_{n+2r}^{G}(X; C_{∗}^{HP}(Z_{•})) .

In Lemma 9.21 below, we show that the first map is an isomorphism if X is a finite G-CW -complex. The second map sits in a short exact lim^{1}-lim-sequence, because the maps in the tower (9.20) above are all surjective and the functors C̃_{∗}^{cell}(X_{+}^{?})⊗_{ℤOr G}(−) and Tot^{⊕} preserve surjectivity, compare [38, Theorem 3.5.8 on page 83]. Since we already know the comparison result for the lim- and lim^{1}-terms involving cyclic homology, a Five-Lemma argument yields the result for periodic cyclic homology.

It remains to prove the statement about negative cyclic homology. There is a natural exact sequence of chain complexes [18, 5.1.4 on page 160]

0 → C_{∗}^{HN}(Z_{•}) → C_{∗}^{HP}(Z_{•}) → C_{∗}^{HC}(Z_{•})[−2] → 0 .

Again, one uses that C̃_{∗}^{cell}(X_{+}^{?}) ⊗_{ℤOr G} (−) and Tot^{⊕} are exact functors to produce a long exact ladder in homology and uses the Five-Lemma. □

In the previous proof we used the following statement.

Lemma 9.21. Suppose that X is a finite G-CW -complex. Then the natural map

Tot^{⊕} (C̃_{∗}^{cell}(X_{+}^{?}) ⊗_{ℤOr G} lim_{r} C_{∗}^{HC}(Z_{•})[2r]) ⟶^{≅} lim_{r} Tot^{⊕} (C̃_{∗}^{cell}(X_{+}^{?}) ⊗_{ℤOr G} C_{∗}^{HC}(Z_{•})[2r])

is an isomorphism.

Proof. There exists an exact sequence (by explicit construction of an inverse limit)

0 → lim_{r} C_{∗}^{HC}(Z_{•})[2r] → ∏_{r=0}^{∞} C_{∗}^{HC}(Z_{•})[2r] → ∏_{r=0}^{∞} C_{∗}^{HC}(Z_{•})[2r] .

As C̃_{∗}^{cell}(X_{+}^{?}) ⊗_{ℤOr G} (−) and Tot^{⊕} are exact functors, we see that it suffices to study the natural map

Tot^{⊕} (C̃_{∗}^{cell}(X_{+}^{?}) ⊗_{ℤOr G} ∏_{r=0}^{∞} C_{∗}^{HC}(Z_{•})[2r])

∏_{r=0}^{∞} Tot^{⊕} (C̃_{∗}^{cell}(X_{+}^{?}) ⊗_{ℤOr G} C_{∗}^{HC}(Z_{•})[2r])

Let C∗^{≤p} ⊆ C̃_{∗}^{cell}(X_{+}^{?}) be the OrG-sub-complex which agrees with C̃_{∗}^{cell}(X_{+}^{?}) up to dimension p and is trivial in dimension > p. This yields a finite filtration by our assumption on X. There is an induced map of filtered chain complexes

F_{∗}^{p} = Tot^{⊕} (C∗^{≤p} ⊗_{ℤOr G} ∏_{r=0}^{∞} C_{∗}^{HC}(Z_{•})[2r])

′F_{∗}^{p} = ∏_{r=0}^{∞} Tot^{⊕} (C∗^{≤p} ⊗_{ℤOr G} C_{∗}^{HC}(Z_{•})[2r])

and the induced chain map of filtration quotients F_{∗}^{p}/F_{∗}^{p−1} → ′F_{∗}^{p}/′F_{∗}^{p−1} can be identified with the composition

Tot^{⊕} (C̃_{p}^{cell}(X_{+}^{?}) ⊗_{ℤOr G} ∏_{r=0}^{∞} C_{∗}^{HC}(Z_{•})[2r])

Tot^{⊕} ( ∏_{r=0}^{∞} C̃_{p}^{cell}(X_{+}^{?}) ⊗_{ℤOr G} C_{∗}^{HC}(Z_{•})[2r])

∏_{r=0}^{∞} Tot^{⊕} (C̃_{p}^{cell}(X_{+}^{?}) ⊗_{ℤOr G} C_{∗}^{HC}(Z_{•})[2r])

because the tensor product with a fixed module over the orbit category, Tot^{⊕} and ∏_{r=0}^{∞} all “behave well” (in an obvious sense) with respect to taking quotients. The second map in the composition above is clearly an isomorphism. The first map is an isomorphism because the assumption on X implies that each C̃_{p}^{cell}(X_{+}^{?}) is a finitely generated free ℤOrG-module, compare [19, page 167]. Since the filtrations are finite, this concludes the proof. □

Remark 9.22. If we would only assume that X is a G-CW -complex of finite type instead of being finite, then one would have the same conclusion that the induced map of filtration quotients is an isomorphism for each p, as in the proof of Lemma 9.21, but the second filtration would not necessarily be exhaustive. The 0th module of the complex

∏_{r=0}^{∞} Tot^{⊕} (C̃_{∗}^{cell}(X_{+}^{?}) ⊗_{ℤOr G} C_{∗}^{HC}(Z_{•})[2r])

would for instance contain the infinite product ∏_{r=0}^{∞} C̃_{2r}^{cell}(X_{+}^{?}) ⊗_{ℤOr G} C_{0}^{HC}(Z_{•}), whereas an element that is contained in ′F_{∗}^{p} for some p has to be contained in the corresponding infinite direct sum.

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