TOPOLOGICAL CYCLIC HOMOLOGY OF 𝕊-ALGEBRAS

John Rognes

November 16th–23rd 1998

1. Some categories.

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_{+}.

Let I ⊂ Γ^{op} be the subcategory of injective functions k_{+} → l_{+}. Its admits a wedge sum ∨: I × I → I taking (k_{+}, l_{+}) to their concatenation (k + l)_{+}.

Let C be a small ΓS_{∗}-category. In the case when C has only one object ∗ we let A = \underline{C}(∗, ∗) ∈ ΓS_{∗}. The composition makes A a monoid in (ΓS_{∗}, ∧, 𝕊), which is what we also call an 𝕊-algebra.

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

Λ([p], [q]) = Δ([p], [q]) × C_{p+1} .

As a category, Λ is generated by the morphisms δ_{i}: [q−1] → [q] and σ_{j}: [q+1] → [q] in Δ, together with the cyclic structure maps τ_{q}: [q] → [q] of order q + 1, subject to a list of relations, which we omit.

A cyclic set X is a functor X: Λ^{op} → Ens, and similarly for cyclic objects in other categories. We denote the images of δ_{i}, σ_{j} and τ_{q} by d_{i}, s_{j} and t_{q}, respectively. The omitted relations in Λ translate to the following cyclic identities, in addition to the simplicial identities:

d_{i}t_{q} = t_{q−1}d_{i−1}  0 < i ≤ q
d_{0}t_{q} = d_{q}
s_{j}t_{q} = t_{q+1}s_{j−1}  0 < j ≤ q
s_{0}t_{q} = t_{q+1}^{2}s_{q}
t_{q}^{q+1} = 1

By restriction to Δ^{op} ⊂ Λ^{op}, a cyclic object determines an underlying simplicial object.

2. Topological Hochschild homology.

For x = k_{+} ∈ obI let

S^{x} = ⋀_{x\∗} S^{1} ≅ S^{k} ,

and let Ω^{x}Y = \underline{S_{∗}}(S^{x}, Y) ≅ Ω^{k}X be the simplicial mapping space. For any (q + 1)- tuple x = (x_{0}, x_{1}, …, x_{q}) let ∨x = x_{0} ∨ x_{1} ∨ ⋯ ∨ x_{q}.

Typeset by AMS-TEX

For each q ≥ 0 and x = (x_{0}, x_{1}, …, x_{q}) ∈ I^{q+1} let

V (C, x) = ⋁_{c_{0},...,c_{q}} \underline{C}(c_{0}, c_{q})(S^{x_{0}}) ∧ \underline{C}(c_{1}, c_{0})(S^{x_{1}}) ∧ ⋯ ∧ \underline{C}(c_{q}, c_{q−1})(S^{x_{q}}) .

In the case of an 𝕊-algebra A this simplifies to

V (A, x) = A(S^{x_{0}}) ∧ A(S^{x_{1}}) ∧ ⋯ ∧ A(S^{x_{q}}) .

The association

x ↦ Ω^{∨x}V (C, x)

is a functor I^{q+1} → S_{∗}. Its homotopy colimit defines the simplicial space

T HH(C)_{q} = hocolim_{x∈I^{q+1}} Ω^{∨x}V (C, x) .

This can be extended to a simplicial Γ-space, by defining

k_{+} ↦ T HH(C)_{q}(k_{+}) = hocolim_{x∈I^{q+1}} Ω^{∨x}(V (C, x) ∧ k_{+}) .

In the case of an 𝕊-algebra, its associated spectrum is stably equivalent to that of A ∧ ⋯ ∧ A = A^{∧(q+1)}.

There is a cyclic category [q] ↦ I^{q+1}, with simplicial structure maps d_{i}: I^{q+1} → I^{q} given by

d_{i}(x_{0}, …, x_{q}) = \{ \begin{matrix}(x_{0}, …, x_{i} ∨ x_{i+1}, …, x_{q}) & for 0 ≤ i < q, \\ (x_{q} ∨ x_{0}, x_{1}, …, x_{q−1}) & for i = q, \end{matrix}

together with

s_{j}(x_{0}, …, x_{q}) = (x_{0}, …, x_{j}, ∗, x_{j+1}, …, x_{q}) .

The extension to a cyclic structure is given by the cyclic permutation

t_{q}(x_{0}, …, x_{q}) = (x_{q}, x_{0}, …, x_{q−1}) .

Likewise there are maps

d_{i}: Ω^{∨x}V (C, x) ⟶ Ω^{∨d_{i}(x)}V (C, d_{i}(x))
s_{j}: Ω^{∨x}V (C, x) ⟶ Ω^{∨s_{j}(x)}V (C, s_{j}(x))
t_{q}: Ω^{∨x}V (C, x) ⟶ Ω^{∨t_{q}(x)}V (C, t_{q}(x))

for 0 ≤ i, j ≤ q. Here d_{i} uses the composition

\underline{C}(c_{i}, c_{i−1})(S^{x_{i}}) ∧ \underline{C}(c_{i+1}, c_{i})(S^{x_{i+1}}) ⟶^{∘} \underline{C}(c_{i+1}, c_{i−1})(S^{x_{i}∨x_{i+1}})

for 0 < i < q. When i = 0 the same formula applies, interpreting c_{−1} as c_{q}. The last face map d_{q} also involves twist maps cyclically permuting the smash factors in V (𝕊, x) = S^{x_{0}} ∧ ⋯ ∧ S^{x_{q}} ≅ S^{∨x} and V (C, x). Next, s_{j} inserts the unit map below into V (C, x):

S^{0} ⟶^{1_{c_{j}}} \underline{C}(c_{j}, c_{j})(S^{0})

taking the non-basepoint to the identity map on c_{j}. The cyclic map t_{q} cyclically permutes the smash factors in V (𝕊, x) ≅ S^{∨x} and in V (C, x).

These structures induce maps

d_{i}: T HH(C)_{q} ⟶ T HH(C)_{q−1}
s_{j}: T HH(C)_{q} ⟶ T HH(C)_{q+1}
t_{q}: T HH(C)_{q} ⟶ T HH(C)_{q}

making [q] ↦ T HH(C)_{q} a cyclic space, denoted T HH(C). This also extends to a cyclic Γ-space ([q], k_{+}) ↦ T HH(C)_{q}(k_{+}). In more detail, an r-simplex in the nerve of I^{q+1} is a chain x^{0} ← ⋯ ← x^{r} = x of composable maps, where each x^{s} is a (q + 1)-tuple (x_{0}^{s}, …, x_{q}^{s}) for 0 ≤ s ≤ r. Each morphism ϕ: [p] → [q] in Λ induces a map ϕ^{∗}: T HH(C)_{q} → T HH(C)_{p} taking

(x^{0} ← ⋯ ← x^{r} = x, f: S^{∨x} → V (C, x))

to

(ϕ^{∗}(x^{0}) ← ⋯ ← ϕ^{∗}(x^{r}) = ϕ^{∗}(x), ϕ^{∗}(f): S^{∨ϕ^{∗}(x)} → V (C, ϕ^{∗}(x)))

(Here δ_{i}^{∗} = d_{i}, σ_{j}^{∗} = s_{j} and τ_{q}^{∗} = t_{q}.)

3. The circle action.

The geometric realization of a cyclic set X: Λ^{op} → Ens has a natural S^{1}-action. Similarly for cyclic spaces, cyclic Γ-spaces, etc.

The orbit of a 0-simplex a ∈ X_{0} is the 1-simplex t_{1}s_{0}(a), which has both ends at a, since d_{0}t_{1}s_{0}(a) = d_{1}s_{0}(a) = a and d_{1}t_{1}s_{0}(a) = d_{0}s_{0}(a) = a. The S^{1}-action moves the 0-simplex a in |X| evenly along the 1-simplex t_{1}s_{0}(a) in |X|.

The orbit of a 1-simplex b ∈ X_{1}, is the union of the two 2-simplices t_{2}s_{1}(b) and t_{2}^{2}s_{0}(b) along their common edge d_{2}t_{2}s_{1}(b) = t_{1}(b) = d_{0}t_{2}^{2}s_{0}(b). The resulting square has d_{0}t_{2}s_{1}(b) = b = d_{2}t_{2}^{2}s_{0}(b) at two sides, hence is really a cylinder. Its remaining two sides are d_{1}t_{2}s_{1}(b) = t_{1}s_{0}(d_{0}(b)) and d_{1}t_{2}^{2}s_{0}(b) = t_{1}s_{0}(d_{1}(b)), i.e., the orbits of the two end-points of b. The S^{1}-action moves the 1-simplex b in |X| evenly across this square combining t_{2}s_{1}(b) and t_{2}^{2}s_{0}(b) in |X|.

More generally, a q-simplex c ∈ X_{q} is moved across the union of (q+1) simplices, each of dimension (q + 1), given by t_{q+1}^{q+1−j}s_{j}(c) for 0 ≤ j ≤ q. The power of t_{q+1} is chosen to map the degenerate simplex s_{j}(c) to one that is generally not degenerate. These simplices fit together to a copy of S^{1} × Δ^{q}, mapped into |X|, describing the orbit of c under the S^{1}-action.

What are the S^{1}-fixed points of this action on |X| ? A 0-simplex a is fixed only if the 1-simplex t_{1}s_{0}(a) is degenerate, i.e., if and only if

t_{1}s_{0}(a) = s_{0}(a) .

For a nondegenerate q-simplex c ∈ X_{q}, an interior point ξ of {c} × Δ^{q} is moved across an interior part of the simplex d = t_{q+1}^{q+1}s_{0}(c). If d is nondegenerate, this part of the orbit is not constant, so ξ is not in the fixed point set. We cannot have d = s_{j}(e) for j ≠ 0 because then c = d_{0}s_{j}(e) would be degenerate. If d = s_{0}(e) we have e = d_{0}(d) = c and for q ≥ 1 no interior orbit in S^{1} × Δ^{q} is contained in a line through the 0th vertex of d. Thus ξ is not in the fixed point set in this case either.

Lemma 3.1. Let X: Λ^{op} → Ens be a cyclic set. Its S^{1}-fixed point set is the discrete subset

|X|^{S^{1}} = {a ∈ X_{0} | t_{1}s_{0}(a) = s_{0}(a)} .

4. Circle fixed points.

The cyclic structure on T HH(C) thus determines an S^{1}-action on its geometric realization |T HH(C)|. Its S^{1}-fixed point set is given by the lemma above. Explicitly, a point in T HH(C)_{0} is a tuple

a = (x^{0} ← ⋯ ← x^{r} = x, f: S^{x} → ⋁_{c∈obC} \underline{C}(c, c)(S^{x}))

with x^{s} ∈ I for 0 ≤ s ≤ r. It degenerates to

s_{0}(a) = ((x^{0}, ∗) ← ⋯ ← (x^{r}, ∗) = (x, ∗),

f ∧ 1_{c}: S^{x∨∗} → ⋁_{c∈obC} \underline{C}(c, c)(S^{x}) ∧ \underline{C}(c, c)(S^{0}) ⊆ V (C, (x, ∗)))

whose cyclic twist is

t_{1}s_{0}(a) = ((∗, x^{0}) ← ⋯ ← (∗, x^{r}) = (∗, x),

1_{c} ∧ f: S^{∗∨x} → ⋁_{c∈obC} \underline{C}(c, c)(S^{0}) ∧ \underline{C}(c, c)(S^{x}) ⊆ V (C, (∗, x))) .

These are equal if and only if all x^{r} = ∗, so x = ∗ with S^{x} = S^{0}, and f: S^{0} → ⋁_{c∈obC} \underline{C}(c, c)(S^{0}) takes the non-basepoint to 1_{c} ∈ \underline{C}(c, c)(S^{0}) for some c ∈ obC. (The base point identifies with the identity map when there is a zero object c = ∗ in C. We now assume this.) This identifies to a copy of obC. Taking the Γ-space structure into account leads to T HH(C)(k_{+})^{S^{1}} ≅ obC ∧ k_{+}.

Lemma 4.1. T HH(C)^{S^{1}} = obC as spaces, and T HH(C)^{S^{1}} = Σ^{∞}(obC) as Γ-spaces.

5. Frobenius maps and T F.

Let C_{r} ⊂ S^{1} be the cyclic subgroup of order r. It turns out that the C_{r}-cyclic fixed point sets of T HH(C) are more accessible than the S^{1}-fixed point sets. Let p be a prime. We think of C_{p^{n−1}} as a subgroup of C_{p^{n}} of index p. The C_{p^{n}}-fixed points for the circle action on T HH(C) are contained in the C_{p^{n−1}}-fixed points, by neglect of structure.

The Frobenius map

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

is defined as the inclusion between these fixed point sets, interpreting a point in T HH(C) 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(C)^{C_{p^{n}}} ⟶^{F} T HH(C)^{C_{p^{n−1}}} ⟶^{F} … ⟶^{F} T HH(C)^{C_{p}} ⟶^{F} T HH(C) .

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 (C, p) = holim_{n} T HH(C)^{C_{p^{n}}}

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

There are natural maps

obC ⟶^{trF_{p}} T F (C, p) ⟶ T HH(C)

including the S^{1}-fixed points of T HH(C) into the limit, and thus the homotopy limit, of the C_{p^{n}}-fixed points. (The notation trF_{p} is not standard, and may be confused with a transfer map.)

Alternatively one may consider the diagram r ↦ T HH(C)^{C_{r}} indexed by the multiplicative monoid of natural numbers, i.e., the category with objects natural numbers r ≥ 1, and a morphism from r to s precisely with r is a multiple of s. Let

T F (C) = holim_{r} T HH(C)^{C_{r}}

where the map T HH(C)^{C_{r}} → T HH(C)^{C_{s}} where r is a multiple of s is the inclusion as subspaces of T HH(C). Then T F (C) is the fiber product over T HH(C) of the T F (C, p) for all primes p, and there is a natural map T F (C) → T F (C, p) inducing a homotopy equivalence after p-adic completion. Thus T F (C) does not really carry any more information than the collection of all the T F (C, p)’s.

There are canonical map

γ: T HH(C)^{C_{r}} → T HH(C)^{hC_{r}} = \underline{S_{∗}}(EC_{r}_{+}, T HH(C))^{C_{r}}

from the fixed points to the homotopy fixed points for the C_{r}-action on T HH(C). We obtain maps

T F (C, p) = holim_{n} T HH(C)^{C_{p^{n}}} ⟶^{γ} holim_{n} T HH(C)^{hC_{p^{n}}} .

After p-adic completion, the natural map

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

is a homotopy equivalence. Hence there is a map

T F (C)_{p}^{∧} ≃ T F (C, p)_{p}^{∧} ⟶ T HH(C)^{hS^{1}} _{p}^{∧}

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

6. Edgewise subdivision.

The S^{1}-action on T HH(C) is not simplicial, which seems to make it hard to analyze the structure of the C_{r}-fixed points for C_{r} ⊂ S^{1}. Edgewise subdivision is a method to replace T HH(C) by another simplicial space sd_{r}T HH(C), which admits a simplicial C_{r}-action, such that there is a natural homeomorphism of geometric realizations

D: |sd_{r}T HH(C)| ⟶^{≅} |T HH(C)|

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(C))^{C_{r}}| ⟶^{≅} |T HH(C)|^{C_{r}}

and (sd_{r}T HH(C))^{C_{r}} provides a simplicial model for the C_{r}-fixed points. Let the functor sd_{r}: Δ → Δ be given by

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

(r summands on the right), and

sd_{r}(ϕ) = ϕ ∐ ⋯ ∐ ϕ

for morphisms ϕ: [p] → [q] in Δ.

Then a simplicial set X: Δ^{op} → Ens defines a new simplicial set sd_{r}X = X ∘ sd_{r}: Δ^{op} → Ens, with

(sd_{r}X)_{q} = X_{(q+1)r−1} .

The ith face map is

sd_{r}(d_{i}) = d_{i} ∘ d_{i+q+1} ∘ ⋯ ∘ d_{i+(q+1)(r−1)} ,

and the jth degeneracy map is

sd_{r}(s_{j}) = s_{j} ∘ s_{j+q+1} ∘ ⋯ ∘ s_{j+(q+1)(r−1)}

for 0 ≤ i, j ≤ q. We call sd_{r}X the r-fold edgewise subdivision of X. Similarly for simplicial objects in other categories.

If X is cyclic, i.e., extends to X: Λ^{op} → Ens, then sd_{r}X gets a simplicial C_{r}- action given in degree q by the cyclic structure map t_{(q+1)r−1}^{q+1}.

T_{q} = t_{(q+1)r−1}^{q+1} : (sd_{r}X)_{q} = X_{(q+1)r−1} ⟶ X_{(q+1)r−1} = (sd_{r}X)_{q} .

A check of the cyclic identities shows that T_{q}: (sd_{r}X)_{q} → (sd_{r}X)_{q} is simplicial, and that T_{q}^{r} = 1.

The homeomorphism D: |sd_{r}X| → |X| is induced by maps

(sd_{r}X)_{q} × Δ^{q} = X_{(q+1)r−1} × Δ^{q} ⟶ X_{(q+1)r−1} × Δ^{(q+1)r−1}

embedding Δ^{q} diagonally into the r-fold join

Δ^{q} ∗ ⋯ ∗ Δ^{q} ≅ Δ^{(q+1)r−1} .

In [BHM] it is shown that these maps induce a homeomorphism, and that the simplicial C_{r}-action on the left agrees with the C_{r}-action on the right restricted from the S^{1}-action. The name ‘edgewise subdivision’ stems from the fact that each edge of X is divided into r edges in sd_{r}X.

7. Restriction maps and T R.

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(C)^{C_{p^{n}}} ⟶ T HH(C)^{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(C)^{C_{p}} ⟶ T HH(C) .

On q-simplices, this is a map

(R_{p})_{q}: (hocolim_{x∈I^{p(q+1)}} Ω^{∨x}V (C, x))^{C_{p}} ⟶ hocolim_{y∈I^{q+1}} Ω^{∨y}V (C, y) .

An r-simplex in the homotopy colimit on the left 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} ≅ S^{x_{0}} ∧ ⋯ ∧ S^{p(q+1)−1} ⟶

⋁_{c_{0},...,c_{p(q+1)−1}} \underline{C}(c_{0}, c_{p(q+1)−1})(S^{x_{0}}) ∧ ⋯ ∧ \underline{C}(c_{p(q+1)−1}, c_{p(q+1)−2})(S^{x_{p(q+1)−1}}) .

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 V (C, x). 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 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 with

X = S^{∨x} ≅ S^{y_{0}} ∧ ⋯ ∧ S^{y_{q}} ∧ ⋯ ∧ S^{y_{0}} ∧ ⋯ ∧ S^{y_{q}} ≅ (S^{y_{0}} ∧ ⋯ ∧ S^{y_{q}})^{∧p}

Here the generator of C_{p} cyclically permutes the p wedge factors in the last expression, so the C_{p}-fixed points are a diagonal copy of S^{∨y}:

X^{C_{p}} ≅ S^{∨y} ≅ S^{y_{0}} ∧ ⋯ ∧ S^{y_{q}} .

We also let

Y = ⋁_{c_{0},...,c_{p(q+1)−1}} \underline{C}(c_{0}, c_{p(q+1)−1})(S^{y_{0}}) ∧ ⋯ ∧ \underline{C}(c_{p(q+1)−1}, c_{p(q+1)−2})(S^{y_{q}}) .

The C_{p}-fixed points are the summands with c_{i} = c_{q+1+i} = ⋯ = c_{(p−1)(q+1)+i} for 0 ≤ i ≤ q, and wedge factors repeating periodically every (q + 1) factors, i.e.,

Y^{C_{p}} ≅ ⋁_{c_{0},...,c_{q}} \underline{C}(c_{0}, c_{q})(S^{y_{0}}) ∧ ⋯ ∧ \underline{C}(c_{q}, c_{q−1})(S^{y_{q}}) ≅ V (C, y) .

Thus R_{p} takes the C_{p}-invariant (r, q)-simplex determined by x^{0} ← ⋯ ← x^{r} = x and f: S^{∨x} → V (C, x) to the (r, q)-simplex in T HH(C) determined by y^{0} ← ⋯ ← y^{r} = y and f^{C_{p}}: (S^{∨x})^{C_{p}} → V (C, x)^{C_{p}}, identified with a map f^{C_{p}}: S^{∨y} → V (C, y).

The resulting map R_{p}: T HH(C)^{C_{p}} → T HH(C) 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(C)^{C_{p^{n}}} ⟶^{R} T HH(C)^{C_{p^{n−1}}} ⟶^{R} … ⟶^{R} T HH(C)^{C_{p}} ⟶^{R} T HH(C) .

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

T R(C, p) = holim_{n} T HH(C)^{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. Also the inclusion of obC into T HH(C) as the S^{1}-fixed points, taking c ∈ obC to the identity map on c, commutes with the restriction maps. Hence there are natural maps

obC ⟶^{trR_{p}} T R(C, p) ⟶ T HH(C)

including the S^{1}-fixed points into the limit, and thus the homotopy limit, of the C_{p^{n}}-fixed points. (Again this notation is nonstandard, but less likely to confuse.)

8. The fundamental cofibration sequence.

For each n ≥ 1 there is a cofiber sequence of spectra

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

Hence a ΓS_{∗}-functor F: C → D inducing a homotopy equivalence on T HH will also induce a homotopy equivalence on the C_{p^{n}}-fixed point subspectra of T HH, for each n ≥ 0. Let us return to this when we discuss equivariant homotopy theory in more detail in connection with the calculation of T C for spaces.

9. Topological cyclic homology T C.

Let p be a prime, and let 𝕀_{p} 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(C)^{C_{p^{n}}}

defines a functor, 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 ΓS_{∗}- category C to be

T C(C, p) = holim_{p^{n}∈𝕀_{p}} T HH(C)^{C_{p^{n}}} .

This comes enriched as a Γ-space by taking into account the Γ-space enrichment of T HH(C) and its fixed points.

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

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

and

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

10. The cyclotomic trace map for symmetric monoidal ΓS_{∗}-categories. The inclusion of obC into the various C_{p^{n}}-fixed points of T HH(C) determines a

map into the limit of the functor above, and thus into the homotopy limit:

trc_{p}: obC ⟶ T C(C, p)

refining the maps trF_{p}, trR_{p} and tr to T F (C, p), T R(C, p) and T HH(C), respectively. When C is a symmetric monoidal ΓS_{∗}-category with weak equivalences wUC ⊆ UC, its algebraic K-theory category K(C, w) = hoN^{w}H̄C is a ΓS_{∗}-category with objects obK(C, w) the ordinary Γ-space defining the algebraic K-theory of C with respect to the weak equivalences. There results a cyclotomic trace map

trc_{p}: obK(C, w) ⟶ T C(K(C, w), p) .

When wUC is a subcategory of strong equivalences, the inclusion H̄C → K(C, w) induces a homotopy equivalence on T HH, and thus on p-primary T C by the fundamental cofiber sequence and induction. Also T C(H̄C, p) is a very special Γ-space in the H̄-direction, so there are stable equivalences of biΓ-spaces

Σ^{∞}T C(C, p) ⟶^{≃} T C(H̄C, p) ⟶^{≃} T C(K(C, w), p) .

Combined, these yield a stable map

trc_{p}: obK(C, w) ⟶ T C(C, p)

called the cyclotomic trace map for C.

11. The cyclotomic trace map for 𝕊-algebras.

When A is an 𝕊-algebra, i.e., a monoid in (ΓS_{∗}, ∧, 𝕊), we can associate to it a symmetric monoidal ΓS_{∗}-category Φ_{A} of finitely generated free A-modules, or rather, their fibrant replacements, and the Γ-spaces of A-module maps between them. These admit a notion of strong equivalence w, for which the morphisms in wUΦ_{A} are maps inducing stable equivalences between the objects of Φ_{A}. ((Check ?))

Then K(A) is defined as obK(Φ_{A}, w), and the inclusion T C(A, p) → T C(Φ_{A}, p) is a homotopy equivalence by Morita equivalence for T HH and the fundamental cofiber sequences. Thus there is a stable map

trc_{p}: K(A) ⟶ T C(A, p)

called the cyclotomic trace map for A. Its homotopy class is well defined, and natural in the 𝕊-algebra A.

Department of Mathematics, University of Oslo, Norway

   E-mail address: rognes@math.uio.no