TOPOLOGICAL CYCLIC HOMOLOGY OF 𝕊-ALGEBRAS

John Rognes

April 5th 2001

1. Γ-spaces and 𝕊-algebras

Let Γ^{op} be the category of finite pointed sets k_{+} = {0, 1, …, k} for k ≥ 0, and base-point preserving functions. We often write ∗ = 0_{+} and S^{0} = 1_{+}.

Γ^{op} admits a wedge sum ∨: Γ^{op} ×Γ^{op} → Γ^{op} taking (k_{+}, l_{+}) to the concatenation (k + l)_{+} ≅ k_{+} ∨ l_{+}, and a smash product ∧: Γ^{op} × Γ^{op} → Γ^{op} taking (k_{+}, l_{+}) to the lexicographically ordered product (kl)_{+} ≅ k_{+} ∧ l_{+}.

Let S_{∗} be the category of spaces, i.e., pointed simplicial sets. A Γ-space X is a functor X: Γ^{op} → S_{∗} with X(0_{+}) = ∗, i.e., a pointed functor. Let ΓS_{∗} be the category of Γ-spaces. The morphisms are the natural transformations of functors.

As an example, let A be an abelian group. A functor HA: k_{+} ↦ A ⊕ ⋯ ⊕ A (k summands A) is given on a morphism f: k_{+} → l_{+} by HA(f)(a_{1}, …, a_{k}) = (b_{1}, …, b_{l}), where b_{j} = ∑_{f(i)=j} a_{i}. We call HA the Eilenberg–Mac Lane Γ-space of A. This yields an embedding H: Ab → ΓS_{∗}, where Ab is the category of abelian groups.

We call X(1_{+}) = X(S^{0}) the underlying space of X. A Γ-space X is special if the canonical map X(k_{+} ∨ l_{+}) → X(k_{+}) × X(l_{+}) is a weak equivalence for all k and l. Equivalently the canonical map X(k_{+}) → X(1_{+})^{k} is a weak equivalence for all k, so for X special X(k_{+}) has the homotopy type of the product of k copies of the underlying space of X. In this case π_{0}X(1_{+}) naturally becomes a commutative monoid. If this monoid has inverses, i.e., is a commutative group, then we say that X is very special.

A Γ-space X extends to a functor Ens_{∗} → S_{∗} taking a pointed set T to the colimit colim_{k_{+}→T} X(k_{+}). It extends further to an endofunctor S_{∗} → S_{∗} taking a pointed simplicial set K to the diagonal of the simplicial space [q] ↦ X(K_{q}), i.e., the simplicial set [q] ↦ X(K_{q})_{q}. We also denote these extensions by X.

There is a natural map X(K)∧L → X(K ∧L) for K, L ∈ S_{∗}. Let S^{1} = Δ^{1}/∂Δ^{1}. Taking K = S^{n} = S^{1} ∧ ⋯ ∧ S^{1} (n factors S^{1}) and L = S^{1} we obtain the structure maps of a (pre-)spectrum n ↦ X(S^{n}), briefly denoted X(S). The homotopy groups of the Γ-space X are defined as the homotopy groups of this spectrum, i.e., as

π_{k}(X) = π_{k}(X(S)) = colim_{n} π_{k+n}X(S^{n}) .

A map X → Y is called a stable equivalence if the induced map π_{k}(X) → π_{k}(Y) is an isomorphism for all k ∈ ℤ.

Typeset by AMS-TEX

If X is special then the adjoint structure map X(S^{n}) → ΩX(S^{n+1}) is a weak equivalence for all n ≥ 1 (X is a semi-Ω-spectrum), and if X is very special then this map is a weak equivalence for all n ≥ 0 (X is an Ω-spectrum).

In the Eilenberg–Mac Lane example, HA(S^{n}) is a K(A, n)-space and HA(S) is the Eilenberg–Mac Lane spectrum of A. Its homotopy is π_{k}(HA) = A for k = 0 and zero otherwise.

Given two Γ-spaces X and Y, their smash product X ∧ Y is the Γ-space

k_{+} ↦ colim_{m_{+}∧n_{+}→k_{+}} X(m_{+}) ∧ Y (n_{+}) .

This is the left Kan extension of the external smash product X ¯∧Y : Γ^{op}×Γ^{op} → ΓS_{∗} taking (m_{+}, n_{+}) to X(m_{+}) ∧ Y (n_{+}), over the smash product ∧: Γ^{op} × Γ^{op} → Γ^{op}.

There is a stable homotopy equivalence X(S)∧Y (S) ≃ (X ∧Y)(S), so the smash product of Γ-spaces models the smash product of spectra in the stable homotopy category.

Given Γ-spaces X, Y and Z, and a morphism f: X ∧ Y → Z, the composite

X(m_{+}) ∧ Y (n_{+}) → (X ∧ Y)(m_{+} ∧ n_{+}) ⟶^{f} Z(m_{+} ∧ n_{+})

is a natural transformation of functors Γ^{op} × Γ^{op} → S_{∗}, i.e., a morphism of bi-Γ-spaces. This correspondence is a bijection, so each such morphism X ¯∧Y → Z ∘ ∧ of bi-Γ-spaces comes from a unique morphism X ∧ Y → Z of Γ-spaces.

The morphism of bi-Γ-spaces above also extends to a natural transformation

X(K) ∧ Y (L) → Z(K ∧ L)

of functors S_{∗} × S_{∗} → S_{∗}.

The inclusion Γ^{op} → S_{∗} interpreting k_{+} as a constant pointed simplicial set defines a Γ-space 𝕊 called the sphere Γ-space. The extended endo-functor 𝕊: S_{∗} → S_{∗} is the identity, and the associated spectrum is the sphere spectrum n ↦ S^{n}. Thus its homotopy π_{k}(𝕊) = colim_{n} π_{k+n}(S^{n}) equals the stable homotopy groups of spheres.

The category ΓS_{∗} of Γ-spaces equipped with the smash product pairing ∧: ΓS_{∗}× ΓS_{∗} → ΓS_{∗} and the unit object 𝕊 is a symmetric monoidal category (ΓS_{∗}, ∧, 𝕊). This thus has similar formal properties to the category Ab of abelian groups, with the tensor product pairing ⊗: Ab × Ab → Ab and the unit object ℤ.

A monoid (R, μ, η) in (Ab, ⊗, ℤ) is an abelian group R equipped with a product μ: R⊗R → R and a unit map η: ℤ → R satisfying associativity and unit conditions. This is precisely an associative ring with unit, or a ℤ-algebra. It is a commutative ring, or a commutative ℤ-algebra if μ ∘ T = μ, where T: R ⊗ R → R ⊗ R is the twist isomorphism.

Likewise, an 𝕊-algebra A is by definition a monoid (A, μ, η) in (ΓS_{∗}, ∧, 𝕊). It is thus a Γ-space A, equipped with a product μ: A ∧ A → A and a unit η: 𝕊 → A, satisfying associativity and unit conditions. If μ ∘ T = μ, then A is a commutative 𝕊-algebra.

The extended endofunctor A: S_{∗} → S_{∗} is now a functor with smash product (FSP). It comes equipped with a product map

A(K) ∧ A(L) → A(K ∧ L)

and a unit map K → A(K), which are natural in K, L ∈ S_{∗}. These satisfy strict associativity and unit conditions.

The associated spectrum A(S) of an 𝕊-algebra A becomes a ring spectrum, with product A(S) ∧ A(S) ≃ (A ∧ A)(S) → A(S) and unit S = 𝕊(S) → A(S), but an 𝕊-algebra is a stricter structure, defined in the category of Γ-spaces and strict maps, not just in the stable homotopy category.

When R is a (commutative) ring, the Eilenberg–Mac Lane Γ-space HR becomes a (commutative) 𝕊-algebra. The sphere Γ-space 𝕊 is the initial (commutative) 𝕊algebra.

2. Cyclic objects

Let Λ be Connes’ cyclic category, with objects {[q] | q ≥ 0} and morphism sets

Λ([p], [q]) = Δ([p], [q]) × C_{p+1} .

By restriction to Δ^{op} ⊂ Λ^{op}, a cyclic object X determines an underlying simplicial object, whose geometric realization |X| admits a natural circle action (S^{1}-action).

3. Topological Hochschild homology

Let I ⊂ Γ^{op} be the subcategory of injective functions k_{+} → l_{+}. The wedge sum and smash product functors restrict to the subcategory I.

For x = k_{+} in I we write

S^{x} = S^{1} ∧ ⋯ ∧ S^{1}

(k factors S^{1}). Let Map(S^{x}, Y) ≅ Ω^{k}Y be the (based) simplicial mapping space. Its p-simplices is the set of simplicial maps S^{x} ∧ Δ_{+}^{p} → Y.

Let A be an 𝕊-algebra. For any (q + 1)-tuple x = (x_{0}, …, x_{q}) in I^{q+1} we define a Γ-space k_{+} ↦ G(A, x)(k_{+}) by

G(A, x)(k_{+}) = Map(S^{x_{0}} ∧ ⋯ ∧ S^{x_{q}}, A(S^{x_{0}}) ∧ ⋯ ∧ A(S^{x_{q}}) ∧ k_{+}) .

The association x ↦ G(A, x) is a functor I^{q+1} → ΓS_{∗}, using in part the stabilization maps A(S^{n}) ∧ S^{1} → A(S^{n+1}). Its homotopy colimit defines the Γ-space

T HH(A)_{q} = hocolim_{x∈I^{q+1}} G(A, x) .

There is a stable homotopy equivalence T HH(A)_{q} ≃ A ∧ ⋯ ∧ A (q + 1 factors A).

There are cyclic structure maps making [q] ↦ T HH(A)_{q} a cyclic Γ-space, denoted by T HH(A). These are analogous to the cyclic structure maps defining the Hochschild complex. In particular the face maps use the product μ on A, and the degeneracies use the unit map η.

For example, the face map d_{0} : T HH(A)_{1} → T HH(A)_{0} takes a map f: S^{x_{0}} ∧ S^{x_{1}} ∧ Δ_{+}^{p} → A(S^{x_{0}}) ∧ A(S^{x_{1}}) ∧ k_{+} to d_{0}(f): S^{x_{0}∨x_{1}} ∧ Δ_{+}^{p} → A(S^{x_{0}∨x_{1}}) ∧ k_{+}, by means of the isomorphism S^{x_{0}} ∧ S^{x_{1}} ≅ S^{x_{0}∨x_{1}} and the product A(S^{x_{0}}) ∧ A(S^{x_{1}}) → A(S^{x_{0}∨x_{1}}). The face map d_{1} yields d_{1}(f): S^{x_{1}∨x_{0}} ∧ Δ_{+}^{p} → A(S^{x_{1}∨x_{0}}) ∧ k_{+}, and involves the twist isomorphism x_{0} ∨ x_{1} ≅ x_{1} ∨ x_{0} both in the source and the target.

The cyclic structure gives each space T HH(A)(k_{+}) a natural S^{1}-action. The associated spectrum T HH(A)(S) is thus a spectrum with S^{1}-action. If A is a commutative 𝕊-algebra, then there is a product T HH(A) ∧ T HH(A) → T HH(A ∧ A) → T HH(A) and a unit 𝕊 → A → T HH(A), making T HH(A) a commutative 𝕊algebra. The composite A → T HH(A)_{0} → T HH(A) is then a map of commutative 𝕊-algebras, making T HH(A) a commutative A-algebra.

For example, when A = 𝕊 each iterated degeneracy map 𝕊 ≃ T HH(𝕊)_{0} → T HH(𝕊)_{q} is a stable equivalence. It follows that the inclusion of zero-simplices T HH(𝕊)_{0} → T HH(𝕊) is a stable equivalence, so T HH(𝕊) ≃ 𝕊.

When A = HR with R a ring, we usually write T HH(R) for T HH(HR). Rationally, we have H_{∗}(T HH(A)_{q}; ℚ) ≅ H_{∗}(A; ℚ) ⊗ ⋯ ⊗ H_{∗}(A; ℚ) ((q + 1) factors H_{∗}(A; ℚ)), and there is an isomorphism H_{∗}(T HH(A); ℚ) ≅ HH_{∗}(H_{∗}(A; ℚ)). When A = HR, H_{∗}(HR; ℚ) ≅ R ⊗ ℚ, so rationally the topological Hochschild homology of HR agrees with the Hochschild homology of R. The difference consists of torsion groups.

4. Frobenius maps and T F

Let C_{r} ⊂ S^{1} be the cyclic subgroup of order r. Let p be a prime. The S^{1}-action on T HH(A) arising from the cyclic structure restricts to a C_{p^{n}}-action for each n ≥ 0. The fixed points for this action is the Γ-space T HH(A)^{C_{p^{n}}}, taking k_{+} to the fixed point space T HH(A)(k_{+})^{C_{p^{n}}}.

We think of C_{p^{n−1}} as a subgroup of C_{p^{n}} of index p. Then the C_{p^{n}}-fixed points for the circle action on T HH(A) are contained in the C_{p^{n−1}}-fixed points, by neglecting part of the invariance.

The Frobenius map

F = F_{p}: T HH(A)^{C_{p^{n}}} ⟶ T HH(A)^{C_{p^{n−1}}}

is defined as the inclusion between these fixed point sets, interpreting a point in T HH(A)(k_{+}) that is fixed by C_{p^{n}} as in particular being fixed by C_{p^{n−1}}.

These assemble to a sequential limit diagram

… ⟶^{F} T HH(A)^{C_{p^{n}}} ⟶^{F} T HH(A)^{C_{p^{n−1}}} ⟶^{F} … ⟶^{F} T HH(A)^{C_{p}} ⟶^{F} T HH(A) .

Replacing each map by a fibration and taking the limit, or more precisely taking the homotopy limit of this diagram, defines the functor T F:

T F (A; p) = holim_{n,F} T HH(A)^{C_{p^{n}}}

where the maps in the limit are the Frobenius maps F = F_{p}.

There are canonical map

Γ_{n}: T HH(A)^{C_{p^{n}}} → T HH(A)^{hC_{p^{n}}} = Map(EC_{p^{n}+}, T HH(A))^{C_{p^{n}}}

from the fixed points to the homotopy fixed points for the C_{p^{n}}-action on T HH(A). We obtain maps

T F (A; p) = holim_{n} T HH(A)^{C_{p^{n}}} ⟶ holim_{n} T HH(A)^{hC_{p^{n}}} .

After p-adic completion, the natural map

T HH(A)^{hS^{1}} ⟶ holim_{n} T HH(A)^{hC_{p^{n}}}

is a homotopy equivalence. Hence there is a canonical map

Γ: T F (A; p)_{p}^{∧} ⟶ T HH(A)^{hS^{1}} _{p}^{∧}

which in some cases is a homotopy equivalence, or a homotopy equivalence on suitable connective covers. Hence T F may be thought of as close to the S^{1}-homotopy fixed points of T HH. In this sense, T F is close to a topological negative cyclic homology.

Even if T HH(A) is of finite type, i.e., each homotopy group is finitely generated, it is usually not the case that T F (A; p) and T HH(A)^{hS^{1}} are of finite type. Hence we shall seek to reduce the size of T F (A; p) further, by taking into account more structure available in T HH(A).

6. Edgewise subdivision

The S^{1}-action on T HH(A) is not simplicial. Edgewise subdivision is a method to replace T HH(A) by another simplicial space sd_{r}T HH(A), which admits a simplicial C_{r}-action, such that there is a natural homeomorphism of geometric realizations

D: |sd_{r}T HH(A)| ⟶^{≅} |T HH(A)|

identifying the simplicial C_{r}-action on the left with the C_{r}-action on the right that comes from restricting the S^{1}-action to the subgroup C_{r} ⊂ S^{1}. Hence there is a homeomorphism

D^{C_{r}}: |(sd_{r}T HH(A))^{C_{r}}| ⟶^{≅} |T HH(A)|^{C_{r}}

and (sd_{r}T HH(A))^{C_{r}} provides a simplicial model for the C_{r}-fixed points.

7. Restriction maps and T R

Now consider C_{p} as a subgroup of C_{p^{n}}, with quotient group C_{p^{n−1}}. The restriction map

R = R_{p}: T HH(A)^{C_{p^{n}}} ⟶ T HH(A)^{C_{p^{n−1}}}

is defined by applying C_{p^{n−1}}-fixed points to the geometric realization of a simplicial S^{1}-equivariant map

R_{p}: sd_{p}T HH(A)^{C_{p}} ⟶ T HH(A) .

On q-simplices, this is a map of Γ-spaces

(R_{p})_{q}: (sd_{p}T HH(A)_{q})^{C_{p}} = (T HH(A)_{p(q+1)−1})^{C_{p}} ⟶ T HH(A)_{q} .

An r-simplex in the homotopy colimit defining T HH(A)_{p(q+1)−1}(k_{+}) is a chain of maps x^{0} ← ⋯ ← x^{r} = x = (x_{0}, …, x_{p(q+1)−1}) in I^{p(q+1)}, together with a map

f: S^{x_{0}} ∧ ⋯ ∧ S^{x_{p(q+1)−1}} ∧ Δ_{+}^{r} ⟶ A(S^{x_{0}}) ∧ ⋯ ∧ A(S^{x_{p(q+1)−1}}) ∧ k_{+} .

The generator of the C_{p}-action permutes the factors in I^{p(q+1)} by cyclically shifting them (q + 1) positions to the right, and similarly for the p(q + 1) smash product factors in the source and target of the map f. (The final factors Δ_{+}^{r} and k_{+} are fixed.)

The source of (R_{p})_{q} consists of the C_{p}-invariant chains x^{0} ← ⋯ ← x^{r} = x, together with the C_{p}-equivariant maps f as above.

A p(q + 1)-tuple x ∈ I^{p(q+1)} is C_{p}-invariant precisely when it has the form Δ_{p}(y) = (y, …, y) for y ∈ I^{q+1}. Here Δ_{p}: I^{q+1} → I^{p(q+1)} is the p-fold diagonal embedding. Thus we may assume that the C_{p}-invariant chain x^{0} ← ⋯ ← x^{r} = x arises by applying Δ_{p} to a chain y^{0} ← ⋯ ← y^{r} = y = (y_{0}, …, y_{q}) in I^{q+1}. So

x = Δ(y) = (y_{0}, …, y_{q}, …, y_{0}, …, y_{q})

is y repeated p times.

A C_{p}-equivariant map f: X → Y induces a map f^{C_{p}}: X^{C_{p}} → Y^{C_{p}} by restriction to the C_{p}-fixed point spaces. This is the core of the construction of the restriction maps. We apply this to the source of f:

X = (S^{y_{0}} ∧ ⋯ ∧ S^{y_{q}})^{∧p} ∧ Δ_{+}^{r} .

Here the generator of C_{p} cyclically permutes the p wedge factors, so the C_{p}-fixed points are the diagonal copy

X^{C_{p}} ≅ S^{y_{0}} ∧ ⋯ ∧ S^{y_{q}} ∧ Δ_{+}^{r} .

We also apply this to the target of f:

Y = (A(S^{y_{0}}) ∧ ⋯ ∧ A(S^{y_{q}}))^{∧p} ∧ k_{+} .

The C_{p}-fixed points are the diagonal copy

Y^{C_{p}} ≅ A(S^{y_{0}}) ∧ ⋯ ∧ A(S^{y_{q}}) ∧ k_{+} .

Via these identifications, R_{p} takes a C_{p}-invariant (r, q)-simplex in T HH(A)(k_{+}) determined by Δ_{p}(y^{0}) ← ⋯ ← Δ_{p}(y^{r}) = Δ_{p}(y) and a C_{p}-equivariant map f: X → Y, to the (r, q)-simplex in T HH(A)(k_{+}) determined by y^{0} ← ⋯ ← y^{r} = y and the restricted map f^{C_{p}}: X^{C_{p}} → Y^{C_{p}}, identified as a map f^{C_{p}}: S^{y_{0}} ∧ ⋯ ∧ S^{y_{q}} ∧ Δ_{+}^{r} → A(S^{y_{0}}) ∧ ⋯ ∧ A(S^{y_{q}}) ∧ k_{+}.

The resulting map R_{p}: T HH(A)^{C_{p}} → T HH(A) is a cyclic map, hence S^{1}- equivariant. Taking C_{p^{n−1}}-fixed points for n ≥ 1 defines the various restriction maps, as displayed above. They assemble to a sequential limit diagram

… ⟶^{R} T HH(A)^{C_{p^{n}}} ⟶^{R} T HH(A)^{C_{p^{n−1}}} ⟶^{R} … ⟶^{R} T HH(A)^{C_{p}} ⟶^{R} T HH(A) .

Taking the homotopy limit of this diagram defines the functor T R:

T R(A; p) = holim_{n,R} T HH(A)^{C_{p^{n}}}

where the maps in the limit are the restriction maps R = R_{p}.

Since the restriction maps arise by taking fixed points of an S^{1}-equivariant map, they commute with the forgetful Frobenius maps.

8. Topological cyclic homology T C

Let p be a prime, and let RF be the category with objects 1, p, …, p^{n}, … for n ≥ 0, and commuting morphisms r, f: p^{n} → p^{n−1} for all n ≥ 1. Thus there are (k + 1) distinct morphisms p^{n+k} → p^{n}, given as the various composites r^{i}f^{j} for i + j = k. Then

p^{n} ↦ T HH(A)^{C_{p^{n}}}

defines a functor RF → ΓS_{∗}, taking r to the restriction map R = R_{p} and f to the Frobenius map F = F_{p}. We define the p-primary topological cyclic homology of the 𝕊-algebra A to be

T C(A; p) = holim_{p^{n}∈RF} T HH(A)^{C_{p^{n}}} .

Alternatively, T C may be described as a homotopy equalizer for maps between sequential homotopy limits, thus avoiding the details of how this more complicated homotopy limit is defined.

Since the R- and F -maps commute, the R-maps induce a self-map R of T F (A; p), and the F -maps induce a self map F of T R(A; p). There are homotopy equalizer diagrams

\xymatrix{T C(A; p) \ar[r]^{π} & T F (A; p) \ar[r]^{R} \ar[r]_{1} & T F (A; p)}

and

\xymatrix{T C(A; p) \ar[r]^{π} & T R(A; p) \ar[r]^{F} \ar[r]_{1} & T R(A; p) .}

Thus T C is homotopy equivalent to the homotopy fiber of R − 1 acting on T F, or of F − 1 acting on T R.

9. The norm–restriction sequence

For each n ≥ 1 there is a (homotopy) cofiber sequence of Γ-spaces

T HH(A)_{hC_{p^{n}}} ⟶^{N} T HH(A)^{C_{p^{n}}} ⟶^{R} T HH(A)^{C_{p^{n−1}}} .

Here the homotopy orbit construction T HH(A)_{hC_{p^{n}}} is the Γ-space taking k_{+} to EC_{p^{n}+} ∧_{C_{p^{n}}} T HH(A)(k_{+}), where EC_{p^{n}} is a free contractible C_{p^{n}}-space. Its underlying spectrum is m ↦ EC_{p^{n}+} ∧_{C_{p^{n}}} T HH(A)(S^{m}).

A map F: A → B of 𝕊-algebras inducing a stable equivalence on T HH(−), e.g. a stable equivalence A → B, will induce a stable equivalence on all homotopy orbit spectra T HH(−)_{hC_{p^{n}}}. By the norm–restriction sequence and induction, it also induces a stable equivalence on the C_{p^{n}}-fixed point subspectra of T HH, for each n ≥ 0. Hence it also induces a stable equivalence on T F (−; p), T R(−; p) and T C(−; p).