ON TRACE MAPS

John Rognes

October 15th 1998

1. The trace map to Hochschild homology

Let R be a ring, always associative and unital. We can think of R as a ℤ-algebra.

1.1 Algebraic K-theory. The algebraic K-groups of the ring R are the homotopy groups K_{∗}(R) = π_{∗}K(R) of a space K(R), defined as

K(R) = ΩB (∐_{P} B Aut_{R}(P))

where P ranges through the isomorphism classes of finitely generated projective Rmodules [Quillen]. Restricting attention to the finitely generated free R-modules, we have

K^{f}(R) = ΩB (∐_{n≥0} BGL_{n}(R)) .

In either case the coproduct is a topological monoid with respect to direct sum of modules and automorphisms of such. For a topological monoid M, the bar construction BM is the simplicial space with q-simplices

[q] ↦ M × ⋯ × M

(q factors). Denoting a typical q-simplex by [m_{1}| … |m_{q}], the face maps are given by

d_{i}([m_{1}| … |m_{q}]) = \{ \begin{matrix}[m_{2}| … |m_{q}] & for i = 0, \\ [m_{1}| … |m_{i}m_{i+1}| … |m_{q}] & for 0 < i < q and \\ [m_{1}| … |m_{q−1}] & for i = q, \end{matrix}

and the degeneracy maps by

s_{j}([m_{1}| … |m_{q}]) = [m_{1}| … |m_{j}|1|m_{j+1}| … |m_{q}] .

The inclusion of the 1-skeleton BM^{(1)} ≅ ΣM → BM has an adjoint map ι: M → ΩBM, which is a group completion when π = π_{0}(M) is central in the Pontryagin algebra H_{∗}(M). Then there is an isomorphism [Segal–McDuff]

H_{∗}(M)[π^{−1}] ⟶^{≅} H_{∗}(ΩBM) .

Typeset by AMS-TEX

When the monoid π_{0}(M) is in fact a group, we say that M is grouplike. In this case the group completion map ι is a homotopy equivalence, hence does not alter the homotopy type of M.

Returning to algebraic K-theory, there is a homotopy equivalence

K(R) ≃ K_{0}(R) × BGL(R)^{+}

where BGL(R)^{+} denotes Quillen’s plus construction applied to

BGL(R) = colim_{n} BGL_{n}(R) .

The connected components of K(R) and K^{f}(R) are homotopy equivalent, hence their positive-dimensional homotopy groups are isomorphic.

1.2 Hochschild homology. The Hochschild homology groups of the ring R are the homology groups HH_{∗}(R) = H_{∗}(Z_{∗}(R)) of a chain complex Z_{∗}(R) called the Hochschild complex or the cyclic complex of R. This is the natural homology theory for associative and unital algebras, in our case over the ground ring ℤ. Alternatively the Hochschild homology groups can be thought of as the homotopy groups of a space HH(R), which we now define.

Let HH(R)_{•} be the simplicial abelian group

[q] ↦ HH(R)_{q} = R ⊗_{ℤ} R ⊗_{ℤ} ⋯ ⊗_{ℤ} R

(q + 1 factors). The face maps are given by

d_{i}(r_{0} ⊗ r_{1} ⊗ ⋯ ⊗ r_{q}) = \{ \begin{matrix}r_{0}r_{1} ⊗ ⋯ ⊗ r_{q} & for i = 0, \\ r_{0} ⊗ ⋯ ⊗ r_{i}r_{i+1} ⊗ ⋯ ⊗ r_{q} & for 0 < i < q and \\ r_{q}r_{0} ⊗ r_{1} ⊗ ⋯ ⊗ r_{q−1} & for i = q, \end{matrix}

and the degeneracy maps by

s_{j}(r_{0} ⊗ r_{1} ⊗ ⋯ ⊗ r_{q}) = r_{0} ⊗ ⋯ ⊗ r_{j} ⊗ 1 ⊗ r_{j+1} ⊗ ⋯ ⊗ r_{q} .

The cyclic complex Z_{∗}(R) is the associated chain complex, with Z_{q}(R) = HH(R)_{q} and boundary maps

∂_{q} = ∑_{i=0}^{q}(−1)^{i}d_{i} .

For any simplicial abelian group A_{•} with associated chain complex C_{∗} there is a natural isomorphism [Moore]

π_{∗}(|A_{•}|) ≅ H_{∗}(C_{∗}) ,

and the geometric realization |A_{•}| is a product of Eilenberg–Mac Lane spaces. Hence HH_{∗}(R) ≅ π_{∗}(HH(R)) ≅ H_{∗}(Z_{∗}(R)), where we write HH(R) for the geometric realization of the simplicial abelian group HH(R)_{•}.

1.3 Morita invariance. Let M_{n}(R) be the ring of n × n matrices with entries in R. There are homotopy equivalences

K(R) ≃ K(M_{n}(R))
HH(R) ≃ HH(M_{n}(R))

known as Morita invariance for algebraic K-theory and Hochschild homology, respectively. These are symptoms of the fact that both algebraic K-theory and Hochschild homology for a ring R can be defined in terms of the category P(R) of finitely generated projective R-modules, and there is an equivalence of categories P(R) ≃ P(M_{n}(R)). We shall return to this categorical viewpoint later.

In the case of Hochschild homology, the homotopy equivalence is induced in one direction by the ring inclusion R ≅ M_{1}(R) → M_{n}(R) taking r ∈ R to the n × n matrix with r in the upper left corner and 0’s elsewhere. Its homotopy inverse is induced by the multi-trace map

HH(M_{n}(R))_{q} = M_{n}(R) ⊗_{ℤ} M_{n}(R) ⊗_{ℤ} ⋯ ⊗_{ℤ} M_{n}(R)

⟶^{mtr} R ⊗_{ℤ} R ⊗_{ℤ} ⋯ ⊗_{ℤ} R = HH(R)_{q}

taking a tensor product of n × n matrices to the trace of their external tensor product:

(a_{ij}^{0}) ⊗ (a_{ij}^{1}) ⊗ ⋯ ⊗ (a_{ij}^{q}) ↦ ∑_{i_{0},...,i_{q}} a_{i_{0}i_{1}}^{0} ⊗ a_{i_{1}i_{2}}^{1} ⊗ … a_{i_{q}i_{0}}^{q} .

1.4 Dennis trace map. The Dennis trace map is a natural transformation

K^{f}(R) ⟶^{trd} HH(R)

from algebraic K-theory to Hochschild homology. It is obtained by group completing the following composite map:

∐_{n≥0} BGL_{n}(R) ⟶^{SI} ∐_{n≥0} HH(M_{n}(R)) \underrightarrow{∐ mtr} HH(R) .

In simplicial degree q the left hand map is given by

SI: [g_{1}| … |g_{q}] ↦ g_{0} ⊗ g_{1} ⊗ ⋯ ⊗ g_{q}

where g_{0} = (g_{1} … g_{q})^{−1}. On the left each g_{i} lies in GL_{n}(R), while on the right each g_{i} is considered as an element in M_{n}(R) by means of the inclusion S: GL_{n}(R) ⊂ M_{n}(R). The right hand map is the coproduct of the multitrace maps inducing the equivalences HH(M_{n}(R)) ≃ HH(R).

Applying group completion we get a map

K^{f}(R) = ΩB (∐_{n≥0} BGL_{n}(R)) ⟶ ΩB (HH(R)) .

The Hochschild homology space is a simplicial abelian group, hence grouplike, so the group completion map ι: HH(R) → ΩB (HH(R)) is a homotopy equivalence. Composing with a homotopy inverse we get the Dennis trace map

trd: K^{f}(R) ⟶ HH(R) .

Unfortunately, the induced map on homotopy K_{∗}(R) → HH_{∗}(R) is not very interesting. When R = ℤ we have HH_{i}(ℤ) = 0 for i > 0, so the Dennis trace map does not detect anything of K_{i}(ℤ) in positive dimensions.

2. Cyclic structure

Connes noted that the simplicial abelian group HH(R)_{•} has further structure: it is a cyclic object in the category of abelian groups. This determines an action by the circle S^{1} on HH(R), and was used by Connes to define the cyclic homology HC(R), periodic homology HP (R) and negative cyclic homology HN(R) = HC^{−}(R) of the ring R. These are closely related to the S^{1}-homotopy orbits HH(R)_{hS^{1}}, the S^{1}- Tate construction ℍ̂(S^{1}, HH(R)) and the S^{1}-homotopy fixed points HH(R)^{hS^{1}} of the S^{1}-space HH(R), respectively.

2.1 Cyclic sets. Let Δ be the simplex category, with objects the ordered sets

[q] = {0 < 1 < ⋯ < q}

for all integers q ≥ 0, and morphisms Δ([p], [q]) the set of order preserving maps f: [p] → [q]. We can alternatively think of [q] as the category {0 → 1 → ⋯ → q} and Δ([p], [q]) as the set of functors from [p] to [q]. The face and degeneracy maps δ_{i}: [q − 1] → [q] and σ_{j}: [q + 1] → [q] for 0 ≤ i, j ≤ q generate the morphisms in Δ, subject to the so-called simplicial identities (which we omit to write down).

A simplicial object X_{•} in a category C is a functor X: Δ^{op} → C from the opposite category of Δ to C. With C = Ens the category of sets (French: ensemble), this is called a simplicial set. With C = Ab the category of abelian groups, this is a simplicial abelian group. For each q ≥ 0 the q-simplices X_{q} in X equal the object X([q]) in C. For each n ≥ 0 there is a represented simplicial set Δ_{•}^{n} with q-simplices Δ_{q}^{n} = Δ([q], [n]), i.e., Δ^{n} is the functor [q] ↦ Δ([q], [n]) from Δ^{op} to Ens. Its geometric realization is the standard n-simplex also denoted Δ^{n}.

Connes’ cyclic category Λ is an extension of Δ, adjoining for each q ≥ 0 an automorphism τ_{q}: [q] → [q] of order (q + 1), i.e., τ_{q}^{q+1} = 1. There is an inclusion Δ → Λ which is the identity on objects. The face and degeneracy maps together with the cyclic maps τ_{q} generate the morphisms of Λ, subject to the simplicial identities, the relation τ_{q}^{q+1} = 1, and some additional identities relating τ_{q} to the face and degeneracy maps (which we again omit to write down).

A cyclic object in a category C is a functor X: Λ^{op} → C from the opposite category of Λ to C. With C = Ens the category of sets, this is called a cyclic set. With C = Ab the category of abelian groups, this is a cyclic abelian group (not to be confused with the other meaning of this expression).

The simplicial abelian group HH(R)_{•} is a functor Δ^{op} → Ab. It admits an extension over Λ^{op}, given by taking τ_{q}: [q] → [q] to the homomorphism

t_{q}: HH(R)_{q} → HH(R)_{q}

of order (q + 1) given by cyclically permuting the tensor factors:

t_{q}(r_{0} ⊗ r_{1} ⊗ ⋯ ⊗ r_{q}) = r_{q} ⊗ r_{0} ⊗ ⋯ ⊗ r_{q−1} .

Hence HH(R)_{•} is a cyclic abelian group. We assert that this defines an S^{1}-action on HH(R).

For each n ≥ 0 there is a represented cyclic set Λ_{•}^{n} defined by the functor [q] ↦ Λ([q], [n]) from Λ^{op} to Ens. Restricting to Δ^{op}, we obtain the underlying simplicial set with q-simplices the set of morphisms [q] → [n] in Λ. By inspection, its geometric realization has the form:

Λ^{n} = |Λ_{•}^{n}| ≅ Δ^{n} × S^{1} .

Hence there is natural S^{1}-action on Λ^{n} given by multiplying in the right hand factor of Δ^{n} × S^{1} using the group structure on S^{1}.

Let X: Λ^{op} → Ens be any cyclic set. The geometric realization of its underlying simplicial set is

|X| = ∐_{q≥0} X_{q} × Δ^{q}/ ∼_{Δ}≅ ∐_{q≥0} X_{q} × Λ^{q}/ ∼_{Λ}

where ∼_{Δ} refers to identifications (f^{∗}(x), ξ) ∼ (x, f_{∗}(ξ)) for x ∈ X_{q}, ξ ∈ Δ^{p} and all morphisms f ∈ Δ([p], [q]), while ∼_{Λ} refers to the same identifications for x ∈ X_{q}, ξ ∈ Λ^{p} and f in the larger set of morphisms Λ([p], [q]). The two identification spaces are homeomorphic for categorical reasons. Each space

X_{q} × Λ^{q} ≅ X_{q} × Δ^{q} × S^{1}

has an S^{1}-action by multiplication in the last factor, and these actions are compatible under the identifications ∼_{Λ}.

Hence there is a natural S^{1}-action on the geometric realization of the underlying simplicial set of any cyclic set. Likewise these is an S^{1}-action on the geometric realization of the underlying simplicial abelian group of any cyclic abelian group, and this applies to HH(R).

2.2 Edgewise subdivision. The preceding description of the S^{1}-action on cyclic sets can be made much more explicit and precise, if we are willing to restrict the group action to a finite cyclic subgroup C_{r} ⊂ S^{1} of order r.

Let sd_{r}: Λ → Λ be the functor given on objects by

sd_{r}([q]) = [r(q + 1) − 1] ≅ [q] ∐ ⋯ ∐[q]

(r disjoint summands), and on a morphism f: [p] → [q] by

sd_{r}(f) = f ∐ ⋯ ∐ f: [r(p + 1) − 1] → [r(q + 1) − 1] .

Precomposition by sd_{r} defines an action on cyclic objects in any category, taking X: Λ^{op} → C to sd_{r}(X) = X ∘ sd_{r}: Λ^{op} → C. We call sd_{r}(X) the r-fold edgewise subdivision of X.

We note three facts:

(1) There is a (non-simplicial) homeomorphism

|sd_{r}(X)| ⟶^{≅} |X|

taking a q-simplex {x} × Δ^{q} in |sd_{r}(X)| with x ∈ sd_{r}(X)_{q} = X_{r(q+1)−1} into the r(q + 1) − 1-simplex {x} × Δ^{r(q+1)−1} in |X| by a suitable diagonal embedding

Δ^{q} → Δ^{q} ∗ ⋯ ∗ Δ^{q} ≅ Δ^{r(q+1)−1} .

(2) There is a simplicial C_{r}-action on sd_{r}(X)_{•}, given on q-simplices sd_{r}(X)_{q} = X_{r(q+1)−1} by the action induced by τ_{r(q+1)−1}^{q+1}, which has order r.

(3) The induced C_{r}-action on the geometric realization |sd_{r}(X)| is compatible with the restriction of the natural S^{1}-action on |X| to C_{r} ⊂ S^{1}, via the homeomorphism |sd_{r}(X)| ≅ |X|.

This allows us the describe the restriction of the S^{1}-action on HH(R) to each finite subgroup C_{r} ⊂ S^{1} in a simplicial manner, by replacing HH(R) with its r-fold edgewise subdivision sd_{r}(HH(R)), and using the simplicial C_{r}-action given by the map t_{r(q+1)−1}^{q+1} cyclically shifting the r(q + 1) tensor factors in sd_{r}(HH(R))_{q} by (q + 1) positions to the right. A similar description will apply for related objects similar to HH(R).

2.3 The Goodwillie trace map. The negative cyclic homology HC^{−}(R) is defined as the homology of a bicomplex related to the homotopy fixed point space

HH(R)^{hS^{1}} = Map(ES_{+}^{1}, HH(R))^{S^{1}}

for the S^{1}-action on HH(R). There is a natural map HC^{−}(R) → HH(R) related to the forgetful map

Map(ES_{+}^{1}, HH(R))^{S^{1}} ⊂ Map(ES_{+}^{1}, HH(R)) ≃ HH(R)

and Goodwillie constructed a natural factorization

K^{f}(R) ⟶^{trg} HC^{−}(R) ⟶ HH(R)

of the Dennis trace map trd: K^{f}(R) → HH(R) through negative cyclic homology.

This map takes the S^{1}-action on HH(R) into account, and turns out to carry much more precise information about rational algebraic K-theory than the Dennis trace map:

Consider a nilpotent extension ϕ: R → S of algebras over ℚ. This means we have a surjective ring homomorphism ϕ of ℚ-algebras whose kernel I = ker(ϕ) is a nilpotent ideal in S, i.e., I^{n} = 0 for some n. Then the Goodwillie trace map trg: K(R) → HC^{−}(R) induces a rational homotopy equivalence

hofib(ϕ: K(R) → K(S)) ⟶^{trg} hofib(ϕ: HC^{−}(R) → HC^{−}(S)) ,

i.e., the rationalized relative algebraic K-groups K_{∗}(R → S) ⊗ ℚ and the relative negative cyclic homology groups HC_{∗}^{−}(R → S) are isomorphic. Equivalently the commutative square diagram

\xymatrix{K(R) \ar[r]^{ϕ} \ar[d]^{trg} & K(S) \ar[d]^{trg} \\ HC^{−}(R) \ar[r]^{ϕ} & HC^{−}(S)}

becomes homotopy cartesian after rationalization.