HOCHSCHILD AND CYCLIC HOMOLOGY VIA FUNCTOR HOMOLOGY

By T. PIRASHVILI and B. RICHTER

1 Preliminaries and the main result

1.1 Introduction

The aim of the present paper is to show that Hochschild homology and cyclic homology of any associative algebra in any characteristic can be described via homological algebra of functor category over the category of non-commutative sets constructed by Fiedorowicz and Loday [2]. Our results should be considered as a different version of the theorem of Connes [1], where cyclic (co)homology is also described via functor (co)homology, but in the category of cyclic modules. Our results and ideas are non-commutative versions of the recent development of commutative algebra homology via functor homology given in [4], [5], [6],[7].

Both authors were supported by the TMR network K-theory and algebraic groups, ERB FMRX CT-97-0107. The first author was partially supported by the grant INTAS-99-00817. He is specially grateful to the MPI at Bonn for hospitality. The second author thanks Jean-Louis Loday for his invitation to Strasbourg.

1.2 Category of non-commutative sets

We introduce the category F(as), which we call the category of non-commu-tative sets. Lemma 1.1 shows that this category is isomorphic to the category ΔS introduced by Fiedorowicz and Loday in [2] (see also [3]). Objects of the category F(as) are finite sets

[n] = {0, …, n}, n ≥ 0.

A morphism [n] → [m] in F(as) is a map f : [n] → [m] together with a total ordering of the preimages f^{−1}(j) for all j ∈ [m]. If f : [n] → [m] and g : [m] → [k] are morphisms in F(as), then the composite of g and f as a map is gf, while the total ordering in (gf)^{−1}(i), i ∈ [n] is given via the ordered union of ordered sets:

(gf)^{−1}(i) = ∐_{j∈g^{−1}(i)} f^{−1}(j).

Clearly there is a forgetful functor F(as) → F. Here F denotes the category of finite sets. Objects of the category F are still the sets [n], n ≥ 0, while morphisms in F are just set maps.

We let Γ(as) (resp. Γ) be the subcategory of F(as) (resp. F) whose morphisms f : [n] → [m] preserve the zero element, that is f(0) = 0. Again, there is a forgetful functor Γ(as) → Γ.

If g : [n] → [m] is an order preserving map (in the usual ordering of [n] and [m]) then the restriction of the total ordering of [n] to g^{−1}(i), i ∈ [m] allows us to consider g as a morphism in F(as). In this way one obtains a functor Δ → F(as), which is the identity on objects and an inclusion on morphisms. Here Δ is the standard category of simplicial topology: Objects of Δ are the sets [n], n ≥ 0, while morphisms are non-decreasing maps. Thus one can identify Δ with a subcategory of F(as).

A morphism of the category F(as) is called injective (resp. bijective, surjective) if it so as a set map. Clearly the forgetful functor F(as) → F is bijective on injective morphisms. In particular any bijection [n] → [n] is a morphism in F(as).

Lemma 1.1 Any morphism f : [n] → [m] in F(as) has a unique decomposition g ∘ h, where h is a bijection and g is a morphism in Δ.

Proof. It is enough to observe that there exist a unique order preserving map g : [n] → [m] such that

Card(g^{−1}(i)) = Card(f^{−1}(i)), i ∈ [m]

and for this g there exists a unique bijection h with f = g ∘ h. Conversely, if f = g ∘ h with bijective h, then for each i ∈ [n] the number of elements in g^{−1}(i) and f^{−1}(i) are the same and the lemma is proved. □

If f : [n] → [m] is a morphism in F(as) and f = g ∘ h as in lemma 1.1 then we write g = μ(f) and h = ω(f).

Corollary 1.2 The symmetric group Σ_{n+1} acts freely on

Hom_{F(as)}([n], [m])

and the set of orbits can be identify with Hom_{Δ}([n], [m]) via the map f ↦ μ(f).

Another consequence of lemma 1.1 is that the category F(as) is isomorphic to the category ΔS considered in [3] and [2].

1.3 Associative algebras as functors on non-commutative sets

Let A be an associative and unital algebra over a commutative ring K with unit and let M be an A-bimodule. We let

L(A, M) : Γ(as) → mod

and

L(A, A) : F(as) → mod

be the functors given on objects by [n] ↦ M⊗A^{⊗n} and [n] ↦ A^{⊗(n+1)} respectively. Here mod denotes the category of K-modules. In order to describe the action of morphisms on L(A, M) and L(A, A), we need some additional notation. Let I be the same set as [k] but the elements may be ordered differently and let a_{i} ∈ A, i = 1, …, k and a_{0} ∈ M. Then we denote by ∏_{i∈I}^{<} a_{i} the product of the elements a_{i} according to the ordering in I. For a morphism f : [n] → [m] in Γ(as), the action of f on L(A, M) is given by

f_{∗}(a_{0} ⊗ ⋯ ⊗ a_{n}) : = b_{0} ⊗ ⋯ ⊗ b_{m},

where b_{j} = ∏_{f(i)=j}^{<} a_{i}, j = 0, …, n. Moreover for M = A the same formula shows that L(A, A) factors through F(as). One observes that if A is commutative and M is a symmetric A-bimodule the functor L(A, M) factors through the category Γ, while the functor L(A, A) factors through the category F.

1.4 Non-commutative circle

Before we define the non-commutative circle, let us recall the construction of the smallest simplicial model of the circle. Consider the finite pointed simplicial set C : Δ^{op} → Γ which assigns [n] to [n]. The face and degeneracy maps in the simplicial pointed set C are given as follows. The map s_{i} : [n] → [n + 1] is the unique monotone injection, whose image does not contain i + 1, while d_{i} : [n] → [n − 1] is given by

d_{i}(j) = \{ \begin{matrix}j & if j < i, \\ i & if i < n, & resp. 0 if i = n \\ j − 1 & if j > i. \end{matrix}

We claim that C considered as a pointed simplicial set is a simplicial model of the circle. Indeed, it is clear that 0 ∈ [0] and 1 ∈ [1] are non-degenerate simplices. The first one is zero-dimensional and the second one is one-dimensional. On the other hand, if j ∈ [n] and n ≥ 2, then j = s_{0}(j − 1) provided j > 1. Similarly 0 = s_{0}(0) and 1 = s_{1}(1), which shows that all other simplices are degenerate and hence the geometric realization of C is a 1-sphere.

The functor C fits in the commutative diagram (see page 221 of [3], with slightly different notation):

\xymatrix{Δ^{op} \ar[r] \ar[d]^{C} & ΔC^{op} \ar[d] \\ Γ \ar[r] & F}

where Γ → F and Δ^{op} → ΔC^{op} are inclusions. Here ΔC is the category constructed by A. Connes, with the property that contravariant functors from ΔC to mod are cyclic objects in mod (see Section 6.1 of [3]).

The following important observation was made by Loday (see Exercise 6.4.1 on page 222 of [3]). The functor C : Δ^{op} → Γ has a canonical lifting Ĉ : Δ^{op} → Γ(as). So the non-commutative circle is the simplicial object in Γ(as), which is [n] in dimension n. Since any injective map has a unique lift in Γ(as), we have only to define the face maps. If i < n then d_{i}^{−1}(j) is a singleton for all j except j = i. We define the total ordering on d_{i}^{−1}(i) = {i, i + 1} by declaring that i < i + 1. Since d_{n}^{−1}(j) is a singleton for all j except j = 0 we need only to define the total ordering on d_{n}^{−1}(0) = {0, n}, which is now given by n < 0. It is tedious, but straightforward to check that in this way we get in fact a simplicial object in Γ(as). Moreover this simplicial object is compatible with the unique lift of t_{n} : [n] → [n], which is given by t_{n}(i) = i + 1 for i < n and t_{n}(n) = 0. Hence Ĉ is indeed a cyclic object in F(as).

1.5 Definition of Hochschild and cyclic homology of functors

Now we define the Hochschild homology H_{∗}(F) of a functor F : Γ(as) → mod as the homotopy of the simplicial module F ∘ Ĉ. Similarly the cyclic homology HC_{∗}(T) of a functor T : F(as) → mod is defined as the cyclic homology of the cyclic module T ∘ Ĉ. Of course, for such T we can also define the Hochschild homology of T as the Hochschild homology of the composite functor

Γ(as) ⊂ F(as) → mod.

These definitions generalize the classical definition of Hochschild and cyclic homology of associative algebras as follows. Let A be a unital associative algebra and let M be an A-bimodule. One observes that the simplicial module L(A, M)∘ Ĉ is exactly the standard Hochschild complex of A with coefficients in M. Hence

H_{∗}(L(A, M)) ≅ H_{∗}(A, M) and HC_{∗}(L(A, A)) ≅ HC_{∗}(A).

1.6 Functor homology

For any small category C we denote by C-mod the category of all covariant functors from C to mod. Similarly mod-C denotes the category of contravariant functors from C to the category of K-modules. The categories C-mod and mod-C are abelian categories with sufficiently many projective and injective objects. Projective generators of the category C-mod (resp. mod-C) are the functors C^{c} (resp. C_{c}), c ∈ C, where c ∈ Ob(C) and

C^{c} : = K[Hom_{C}(c, −)] and C_{c} : = K[Hom_{C}(−, c)], c ∈ C.

Here K[S] denotes the free K-module generated by a set S.

If F ∈ C-mod and T ∈ mod-C one defines a module T ⊗_{C} F as a quotient of ⨁_{c∈C} T (c) ⊗ F (c) modulo the relation

α^{∗}(x) ⊗ y = x ⊗ α_{∗}(y).

Here α : c → c′ is a morphism in C, x ∈ T (c′) and y ∈ F (c). It is well known (see Section 16.7 of [8]) that the bifunctor

− ⊗_{C} − : (mod − C) × (C − mod) → mod

is right exact with respect to both variables and preserves sums. It is also important to note that

T ⊗_{C} C^{c} ≅ T (c) and C_{c} ⊗_{C} F ≅ F (c).

Moreover the derived functors of − ⊗_{C} − with respect to each variable are isomorphic and we will denote the common value by Tor_{∗}^{C}(−, −).

Let us return to the category of non-commutative sets F(as). For simplicity we will write P_{n} and P^{n} instead of F(as)_{m} and F(as)^{n}. Similarly, we write P̄^{n} and P̄_{n} instead of Γ(as)^{n} and Γ(as)_{n}.

1.7 The functor b

The morphism d_{i} : [n] → [n − 1] of F(as) yields a natural transformation P_{n} → P_{n−1}, which is still denoted by d_{i}. We define the contravariant functor b as the cokernel of the morphism d = d_{0} − d_{1} : P_{1} → P_{0}. We claim that the evaluation of b on the set [m] can identified with the free K-module spanned on all total orderings on {1, …, m}. Indeed, the generators of P_{0}([m]) are morphisms of non-commutative sets [m] → [0] and this is the same as total orderings of [m]. Similarly, generators of P_{1}([m]) are morphisms of non-commutative sets [m] → [1] and can therefore be identified with partitions of [m] into two disjoint subsets (ξ_{0}, ξ_{1}) together with a total ordering on each of them. The map d maps (ξ_{0}, ξ_{1}) to ξ_{0} ∐ ξ_{1} − ξ_{1} ∐ ξ_{0}, where ∐ means the ordered union of ordered sets. Therefore the cokernel of d consists of equivalence classes of total orderings of the set [m] and each equivalence class contains exactly one total ordering of [m] with minimal element 0 and hence the claim is proved.

We let b̄ be the restriction of b on Γ(as). Since the morphisms d_{i} : [n] → [n−1] respect 0 they yield a natural transformation: d_{i} : P̄_{n} → P̄_{n−1}. Thus we can form the cokernel of the map d = d_{0} − d_{1} : P̄_{1} → P̄_{0}. We claim that this is isomorphic to the functor b̄. Indeed, we have P̄_{0}([m]) = P_{0}([m]), thus the evaluation of the cokernel on [m] can identified with the free K-module spanned by the equivalence classes of total orderings of [m]. The equivalence relation is similar to the one above. The only difference is the that now partitions (ξ_{0}, ξ_{1}) of [m] satisfy the property 0 ∈ ξ_{0}. But this has no effect on the quotient.

1.8 The main theorem

The main results of the paper are the identifications of cyclic and Hochschild homology of functors as derived functors of the tensor product with b resp. b̄.

Theorem 1.3 For functors F : Γ(as) → mod and T : F(as) → mod one has natural isomorphisms

H_{∗}F ≅ Tor_{∗}^{Γ(as)}(b̄, F) and Tor_{∗}^{F(as)}(b, T) ≅ HC_{∗}(T).

Proof. We will use the well-known axiomatic characterization of Tor functors. Thanks to Section 1.7, one has an exact sequence

P̄_{1} → P̄_{0} → b̄ → 0

in mod-Γ(as). Tensoring with F and using the isomorphism P̄_{i} ⊗_{Γ(as)} F ≅ F ([i]) one obtains that the first isomorphism in question holds in dimension zero. Clearly, the functors F ↦ H_{∗}F form an exact connected sequence of functors and it is enough to show that H_{∗}F vanishes in positive dimensions for any projective F. Thus one only needs to consider functors like F = P̄_{k}. Propostion 2.2 below gives the result. Similarly Propostion 2.5 below shows that the same argument proves the second isomorphism as well, provided we start with the exact sequence

P_{1} → P_{0} → b → 0

constructed in Section 1.7. □

2 Propositions 2.2 and 2.5

In order to prove the statements needed for the proof of theorem 1.3 we introduce three families of auxiliary simplicial sets X(k), Z(k) and Y (k) for k ≥ 0.

2.1 The simplicial set X(k).

Let X(k) be the composite of Ĉ : Δ^{op} → Γ(as) and Hom_{Γ(as)}([k] , −) : Γ(as) → Sets :

X(k) : Δ^{op} → Γ(as) → Sets.

Proposition 2.1 The simplicial set X(k) is isomorphic to the disjoint union of the k! copies of the standard k-dimensional standard simplex Δ^{k}.

Proof. We will use extensively the fact that the symmetric group Σ_{k} on kletters can be identified with Aut_{Γ(as)}([n], [n]). Thus it acts on Hom_{Γ(as)}([n], [m]) via precomposition. Thanks to Corollary 1.2 this action is free and hence Σ_{k} acts freely on X(k).

The zero simplices of X(k) are the maps in Γ(as) from [k] to [0]; hence they correspond to total orderings of [n]. The one-dimensional simplices are the maps from [k] to [1], thus they consist of a partition of [k] say (ξ_{0}, ξ_{1}) with a given ordering in every part ξ_{i} of the partition. The boundary maps take such elements to the fusion of the partition but with different orderings (see Section 1.7). Therefore given a zero simplex i_{0} < … < i_{k} every zero simplex with a cyclic variation of this ordering is in the same component and all other elements are not connected to i_{0} < … < i_{k}. Therefore any connected component contains exactly one total ordering with minimal element 0. Thus we can identify the set of connected components with Σ_{k} or equivalently with the set of total orderings of the set {1, …, k}.

Furthermore, the induced action of Σ_{k} on the set of connected components of X(k) is free and transitive. Thus all connected components are isomorphic to each others. Moreover the set of vertices of each component has exactly (k + 1)!/k! = (k + 1) elements. On the other hand an n-simplex of X(k) is non-degenerate if the corresponding morphism [k] → [n] is surjective. Therefore X(k) is k-dimensional and the highest-dimensional non-degenerate simplices correspond to isomorphisms [k] → [k] in Γ(as)([k], i.e., to elements in Σ_{k}. Any other non-degenerate simplex is a (maybe iterated) face of a such a permutation. Furthermore, the action of Σ_{k} via precomposition on the set Σ_{k} ⊂ X(k)_{k} is free and transitive and we can conclude that in every component there is exactly one highest-dimensional non-degenerate simplex.

We have only to prove that the connected component X corresponding to the standard ordering of [k] is isomorphic to the standard k-simplex. To this end, we let x be the identity map [k] → [k], which is the unique highest-dimensional non-degenerate simplex of X. Then there is a unique morphism of simplicial sets Δ^{k} → X which takes the unique highest-dimensional non-degenerate simplex of Δ^{k} to x. Obviously this map is surjective. It is also injective, because the induced map on vertices is bijective. □

Proposition 2.2 For any n ≥ 0 one has

\begin{aligned}H_{i}(P̄_{n}) & = & 0, i > 0 \\ H_{0}(P̄_{n}) & ≅ & b̄([n]). \end{aligned}

Proof. Since P̄_{n} is the free K-module on Hom_{Γ(as)}([n] , −), we see that H_{∗}(P̄_{n}) is nothing but the homology of the simplicial set X(n) with coefficient in K and the statement follows from Proposition 2.1.

2.2 The simplicial set Z(k)

We introduce a simplicial set Z(k), which can be described as follows. The set of m-simplices of Z(k) is Hom_{Δ}([k], [m]). So, in some sense Z(k) = Hom_{Δ}([k], −) is the dual of Δ^{k} = Hom_{Δ}(−, [k]). The degeneracy maps s_{i} : Z(k)_{m} → Z(k)_{m+1}, i = 0, …, k are induced from the degeneracy maps s_{i} : [m] → [m + 1] in the simplicial set C. Since the degeneracy maps are non-decreasing, they indeed induce well-defined maps Z(k)_{m} → Z(k)_{m+1}. The same is true for the face maps d_{0}, …, d_{m−1}. Only the last face map d_{m} : [m] → [m − 1] is not monotonic. In order to define the map d_{m} : Z(k)_{m} → Z(k)_{m−1} we need a different description of the set Z(k)_{m}. Let f : [k] → [m] be a non-decreasing map and let a_{i} be the number of elements in f^{−1}(i), i = 0, …, m. Then k+1 = a_{0}+⋯+a_{m} and a_{i} ≥ 0. It is clear that in this way one obtains a one-to-one correspondence between the elements of Z(k)_{m} and (m + 1)-tuples (a_{0}, …, a_{m}) with the properties a_{i} ≥ 0 and a_{0} + ⋯ + a_{m} = k + 1. Having this identification in mind, one sees that for the maps s_{i} and d_{j}, i = 0, …, m and 0 ≤ j < m one has

\begin{aligned}s_{i}(a_{0}, …, a_{m}) & = & (a_{0}, …, a_{i}, 0, …, a_{m}), & 0 ≤ i ≤ m \\ d_{j}(a_{0}, …, a_{m}) & = & (a_{0}, …, a_{j} + a_{j+1}, …, a_{m}), & 0 ≤ j < m. \end{aligned}

We now define

d_{m}(a_{0}, …, a_{m}) := (a_{m} + a_{0}, …, a_{m}).

In this way we get a well-defined simplicial set Z(k). Clearly Z(0) is isomorphic to the circle C. Actually the simplicial set Z(k) is a cyclic set, where the cyclic structure is compatible with the corresponding cyclic structure on Ĉ and it is given by

t(a_{0}, …, a_{m}) = (a_{m}, a_{0}, …, a_{m−1}).

We let | Z(k) |^{cy} be the cyclic geometric realization of Z(k) (see page 235 of [3]), which is nothing but the Borel construction of the geometric realization of Z(k) with respect to the natural S^{1}-action induced by the cyclic structure:

| Z(k) |^{cy}:= ES^{1}×_{S^{1}} | Z(k) | .

Let us also recall (see loc.cit.) that the cyclic homology of the cyclic module K[Z(k)] is isomorphic to the S^{1}-equivariant homology H_{∗}^{S^{1}}(| Z(k) |, K) of | Z(k) |. In other words

HC_{∗}(K[Z(k)]) ≅ H_{∗}(| Z(k) |^{cy}, K).

Lemma 2.3 i) The space |Z(k)| is weakly homotopy equivalent to the circle. ii) The cyclic realization | Z(k) |^{cy} has the same homotopy type as

K(ℤ/(k + 1)ℤ, 1).

Proof. i) Since Z(k)_{0} is a singleton, we see that Z(k) is connected. The 1simplices of Z(k) are just pairs (i, j) of non-negative integers with i + j = k + 1, while the 2-simplices are triples (a, b, c) of non-negative integers with a + b + c = k + 1. Hence the fundamental group is generated by such pairs (i, j) modulo the relations

(a + b, c)(c + a, b) = (a, b + c).

It is obvious that this group is an infinite cyclic group generated by (k, 1). Let F : C → Z(k) be the unique simplicial map, which sends the unique non-degenerate 1-simplex to (k, 1). By our description of the fundamental group it is clear that f induces an isomorphism on π_{1}. We have to show that f induces an isomorphism on homology with local coefficients. Let M be a module over the ring ℤ[t, t^{−1}]. The homology of Z(k) with coefficients in M is defined as the homology of the reduced chain complex C_{∗}(Z(k), M), where

C_{n}(Z(k), M) = ⨁_{(a_{0},... ,a_{n})} M.

Here (a_{0}, …, a_{n}) corresponds to a non-degenerate simplex of Z(k), that is a_{0} ≥ 0, a_{i} > 0, i ≥ 1 and a_{0} + ⋯ + a_{n} = k + 1. A typical element of C_{n}(Z(k), M) is denoted by (a_{0}, …, a_{n}; x), where x ∈ M. The boundary map d : C_{n}(Z(k), M) → C_{n−1}(Z(k), M) is given by

d(a_{0}, …, a_{n}; x) = (a_{0} + a_{1}, …, a_{n}; x)

+ ∑_{i=1}^{n−1}(−1)^{i}(a_{0}, …, a_{i} + a_{i+1}, …, a_{n}; x)

+(−1)^{n}(a_{n} + a_{0}, …, a_{n−1}; t^{a_{n}}x).

We let F_{i}C_{n} be the subgroup of C_{n}(Z(k), M) corresponding to the summands (a_{0}, …, a_{n}) with a_{0} ≥ k + 1 − i. Then the boundary formula shows that F_{i}C_{∗} is a subcomplex of C_{∗}(Z(k), M). In this way we obtain a filtered complex

F_{0}C_{∗} ⊂ F_{1}C_{∗} ⊂ ⋯ ⊂ F_{k}C_{∗} = C_{∗}(Z(k), M).

Clearly

F_{1}C_{∗} = (A ← A ← 0 ← ⋯)

with only one nontrivial boundary map given by the multiplication on (1 − t). Thus it suffices to prove that homology of F_{i}C_{∗}/F_{i−1}C_{∗} is zero for i ≥ 2. We have

F_{i}C_{n}/F_{i−1}C_{n} = ⨁_{(k+1−i,a_{1},... ,a_{n})} M

where

a_{1} + ⋯ + a_{n} = i.

The boundary map

δ : F_{i}C_{n}/F_{i−1}C_{n} → F_{i}C_{n−1}/F_{i−1}C_{n−1}

is induced by d. One observes that the first and last summand of d lie in F_{i−1}C_{n−1}, hence they disappear in δ. In particular δ does not depend on the ℤ[t, t^{−1}]-module structure on M. We let h : F_{i}C_{n}/F_{i−1}C_{n} → F_{i}C_{n+1}/F_{i−1}C_{n+1} be the map given by

h(a_{0}, a_{1}, …, a_{n}; x) = 0, if a_{1} = 1

and

h(a_{0}, a_{1}, …, a_{n}; x) = −(a_{0}, 1, a_{1} − 1, …, a_{n}; x) if a_{1} > 1.

here a_{0} = k + 1 − i and i ≥ 2. Then hδ + δh = 1 and part i) is proved.

ii) By part i) we know that the integral homology of Z(k) is ℤ in dimensions 0 and 1 and is zero in dimensions > 1. This fact can be seen also by noting that the simplicial abelian group ℤ[Z(k)] is isomorphic to the degree (k + 1)-part of C_{∗}(ℤ[x], ℤ[x]). Here C_{∗}(R, R) denotes the Hochschild complex of a ring R and the grading of C_{∗}(ℤ[x], ℤ[x]) corresponds to the grading of the polynomial ring Z[x] with deg(x) = 1. It is a classical fact that the Hochschild homology of ℤ[x] is zero in dimensions > 1, while

H_{0}(ℤ[x], ℤ[x]) = ℤ[x], H_{1}(ℤ[x], ℤ[x]) ≅ ℤ[x]dx.

Hence the degree (k+1) part of it is zero in dimensions > 1 and is ℤ in dimensions 0 and 1, spanned respectively by x^{k+1} and x^{k}dx. Therefore the same is true for the integral homology of Z(k). Moreover this shows also that Connes’ homomorphism B : H_{0}(Z(k), ℤ) → H_{1}(Z(k), ℤ) corresponding to the cyclic space Z(k) is the multiplication by (k + 1). Therefore it follows from Connes’ exact sequence that the integral homology of | Z(k) |^{cy} is ℤ in dimension 0 and is ℤ/(k + 1)ℤ in odd dimensions. All other homology groups vanish. Furthermore the fibration

S^{1} →| Z(k) |→| Z(k) |^{cy}

corresponding to the Borel construction shows that

π_{i}(| Z(k) |^{cy}) = 0, if i ≠ 1, 2

and one has an exact sequence

0 → π_{2}(| Z(k) |^{cy}) → ℤ → ℤ → π_{1}(| Z(k) |^{cy}) → 0.

As a consequence we see that π_{1}(| Z(k) |^{cy}) is abelian and therefore it is the same as the first homology of | Z(k) |^{cy}, i.e. ℤ/(k + 1)ℤ. Hence the map ℤ → ℤ is injective and we obtain π_{2}(| Z(k) |^{cy}) = 0.

□

2.3 A simplicial set Y (k)

Let Y (k) be the composite of Ĉ and Hom_{F(as)}([k] , −).

ΔC^{op} → F(as) → Sets.

Clearly the cyclic structure on Ĉ yields a cyclic structure on Y (k).

Lemma 2.4 The underlying simplicial set of the cyclic set Y (k) is weakly homotopy equivalent to the disjoint union of the k! copies of the circle.

Proof. A similar argument as in Proposition 2.1 shows that the connected components of the simplicial set Y (k) are in one-to-one correspondence with Σ_{k}. Thanks to Corollary 1.2 Σ_{n+1} acts freely on

Hom_{F(as)}([n], [m])

and the set of orbits can be identified with Hom_{Δ}([n], [m]). Thus the action of the group Σ_{k+1} ⊂ hom_{F(as)}([k] , [k]) on Y (k) is free and the orbits form a simplicial set, which is isomorphic to Z(k). Then the first part of Lemma 2.3 implies the result.

Proposition 2.5 The space | Y (k) |^{cy} is homotopy equivalent to the discrete space with k! points. Thus

HC_{i}(P_{k}) = 0, i > 0

and HC_{0}(P_{k}) ≅ b([k]).

Proof. The fibration

S^{1} →| Y (k) |→| Y (k) |^{cy}

together with Lemma 2.4 shows that π_{0}(| Y (k) |^{cy}) is a set with k! elements. Since the group Σ_{k+1} acts freely on Y (k) and orbits are Z(k) it follows that Σ_{k+1} acts also freely on | Y (k) |^{cy} and orbits are | Z(k) |^{cy}. Thus the result follows from the second part of Lemma 2.3.

□

References

[1] A. Connes. Cohomologie cyclique et foncteurs Ext^{n}. C.R. Acad. Sci. Paris, 296(1983), 953-958.

[2] Z.Fiedorowicz and J.-L. Loday. Crossed simplicial groups and their associated homology. Trans. Amer. Math. Soc. 326 (1991), no. 1, 57–87.

[3] J. - L. Loday. Cyclic Homology, Grund. Math. Wiss. vol. 301, 2nd edition. Springer (1998).

[4] T. Pirashvili. Hodge decomposition for higher order Hochschild homology. Ann. Sci. École Norm. Sup. (4) 33 (2000), no. 2, 151–179.

[5] T Pirashvili and B. Richter. Robinson-Whitehouse complex and stable homotopy. Topology 39 (2000), no. 3, 525–530.

[6] B. Richter and A. Robinson. Γ-homology of polynomial algebras and of group algebras, preprint (2000)

[7] A. Robinson. On a theorem of Pirashvili and Richter, preprint (1999).

[8] H. Schubert. Kategorien. I, II. Heidelberger Taschenbücher, Bände 65, 66. Springer-Verlag, Berlin-New York (1970).

Adresses Teimuraz Pirashvili A.M. Razmadze Math. Inst., Aleksidze str. 1, Tbilisi, 380093. Republic of Georgia

pira@mathematik.uni-bielefeld.de

Birgit Richter Université Louis Pasteur, 7, rue Réne Descartes, 67084 Strasbourg, France

richter@math.u-strasbg.fr