ON THE K-THEORY OF FINITE ALGEBRAS

OVER WITT VECTORS OF PERFECT FIELDS

Lars Hesselholt^{1} and Ib Madsen

Contents

Introduction i

1. The topological Hochschild spectrum 1

2. Witt vectors 12

3. Topological cyclic homology 18

4. Topological cyclic homology of perfect fields 22

5. Topological cyclic homology of finite W (k)-algebras 32

6. Pointed monoids 36

7. A formula for TC(L[ϵ]) 40

8. Topological cyclic homology of k[ϵ] 45 Appendix A: Spectra and prespectra 49 Appendix B: Continuity properties for K-theory 50 References 52

Introduction

The purpose of this paper is two fold. Firstly, it gives a thorough introduction to the topological cyclic homology theory, which to a ring R associates a spectrum TC(R). We determine TC(k) and TC(k[ϵ]) where k is a perfect field of positive characteristic and k[ϵ] its dual numbers, and set the stage for further calculations. Secondly, we show, as conjectured in [M], that the cyclotomic trace from Quillen’s K(R) to TC(R) becomes a homotopy equivalence after p-adic completion when R is a finite algebra over the Witt vectors W (k) of a perfect field of characteristic p > 0. This involves a recent relative result of R. McCarthy, stated in theorem A below, the calculation of TC(k) and Quillen’s theorem about K(k), and continuity results for TC(R) and K(R), the latter basically due to Suslin and coworkers. In particular, we obtain a calculation of the tangent space of K(k), i.e. the homotopy fiber of the map from K(k[ϵ]) to K(k) given by ϵ ↦ 0.

The functor TC(R), and more generally TC(L) where L is a ‘functor with smash product’, for short FSP, was initially introduced in [BHM], but its more formal properties were maybe not so well elucidated in that paper. The present account focuses upon the concept of cyclotomic spectra. These are a special class of equivariant S^{1}-spectra for which the associated fixed point spectrum (suitably defined) with respect to finite subgroups of the circle are equivalent to the original spectrum. The defining extra property is analogous to the property shared by free loop spaces LX, namely that the fixed set (LX)^{C} is homeomorphic to LX, for C finite. Indeed, the S^{1}-equivariant suspension spectrum of the free loop space is a cyclotomic spectrum. More generally, Bökstedt’s topological Hochschild homology spectrum THH(L) is always a cyclotomic spectrum, so they are in rich supply. The construction TC(−), given in paragraph 3, can be applied to any cyclotomic spectrum, and applied to THH(R), or more generally to THH(L), gives TC(R) or TC(L). If R (or L) is commutative then TC(R) (or TC(L)) is a homotopy commutative ring spectrum. It is (−2)-connected in the sense that π_{i} TC(R) = 0 for i ≤ −2; in general π_{−1} TC(R) ≠ 0.

Typeset by AMS-TEX

1 Supported in part by the Danish Natural Science Research Council

Theorem A. (McCarthy) Let R → R̄ be a surjection of rings whose kernel is nilpotent. Then the square

K(R) ⟶^{trc} TC(R)
↓  ↓
K(R̄) ⟶^{trc} TC(R̄)

becomes homotopy cartesian after profinite completion.

The proof of this result is unfortunately indirect. It is based upon Goodwillie’s calculus of functors and a reduction of his to the case where R is a split extension of R̄ by a square zero ideal.

Let k be a perfect field of characteristic p > 0 and let F: W (k) → W (k) be the Frobenius homomorphism of its (p-typical) Witt vectors. The kernel of F − 1 is the Witt vectors of 𝔽_{p} = k^{⟨F ⟩}, i.e. ker(F − 1) = ℤ_{p}. If k is finite then coker(F − 1) = ℤ_{p}; it vanishes if k is algebraicly closed, but can be a large group in general. In §4.5 below we calculate TC(k) to be

Theorem B. Topological cyclic homology of a perfect field k of positive characteristic is the generalized Eilenberg-MacLane spectrum

TC(k) = Hℤ_{p} ∨ Σ^{−1}H(coker(F − 1)).

It follows that the connective cover TC(k)[0, ∞) is H(ℤ_{p}, 0); this is also the value of K(k)p^{∧} by [K], [Q], and the cyclotomic trace trc: K(k)p^{∧} → TC(k)[0, ∞) is an equivalence. For a ℤ_{p}-algebra we define continuous versions of K(R) and TC(R) to be

K^{top}(R) = holim_{⟵} K(R/p^{i}), TC^{top}(R) = holim_{⟵} TC(R/p^{i}),

cf. [W].

Theorem C. Suppose that A is a W (k)-algebra which is finitely generated as a W (k)-module. Then

(i) K^{top}(A)p^{∧} ≃ TC^{top}(A)p^{∧}[0, ∞)

(ii) TC^{top}(A)p^{∧} ≃ TC(A)p^{∧},

K^{top}(A)p^{∧} ≃ K(A)p^{∧}. (iii)

The first part of this result follows from the two previous theorems. The second part is proved in §5 below. The final third part is a recast of results from [SuY]. This uses quite different methods from the rest of the paper, and is proved in Appendix B. In conclusion we have

Theorem D. For the rings of theorem C, K(A)p^{∧} ≃ TC(A)p^{∧}[0, ∞).

It is fair to remark that TC(R)p^{∧} is of course not very easy to evaluate. It does however lend itself to analysis by the well-tried methods of algebraic topology more readily than K(R) does. This is demonstrated here for R = k[ϵ] and in [BM], [BM1] when R is the Witt vectors of a finite field. One might hope in the future to get a thorough grasp of TC(A) for the rings of theorem C, and maybe even a closed formula when A is a k-algebra.

We next describe the tangent space of algebraic K-theory,

K(k[ϵ], (ϵ)) = hofiber(K(k[ϵ]) → K(k)), ϵ ↦ 0,

when k is a perfect field of characteristic p > 0. We have K_{∗}(k[ϵ], (ϵ)) ⊗ ℚ ≅ HC_{∗−1}(k[ϵ], (ϵ)) ⊗ ℚ = 0 by a theorem of Goodwillie, [G2], and on the other hand, by theorem A, K(k[ϵ], (ϵ))^{∧} ≃ TC(k[ϵ], (ϵ))^{∧}. Since the latter turns out to be rationally trivial we get in turn

K(k[ϵ], (ϵ)) ≃ TC(k[ϵ], (ϵ)).

We evaluate the right hand side in paragraph 7 below. The result is best stated in terms of the big Witt vectors. Let W(R) denote the multiplicative group of the power series with constant term 1, and let W_{n}(R) be the quotient of big Witt vectors of length n, i.e.

W_{n}(R) = (1 + XR[[X]])^{×}/(1 + X^{n+1}R[[X]])^{×}.

The second Verschibung V_{2}: W_{n−1}(k) → W_{2n−1}(k) is induced from X ↦ X^{2}. If we write TC_{n}(R) = π_{n} TC(R) then we have from §8.2:

Theorem E. For the dual numbers k[ϵ], TC(k[ϵ], (ϵ)) is a generalized Eilenberg-MacLane spectrum with

TC_{2n−1}(k[ϵ], (ϵ)) ≅ W_{2n−1}/V_{2}W_{n−1}(k),

the even dimensional homotopy groups being zero.

We remark that for p = 2 the groups TC_{2n−1}(k[ϵ], (ϵ)) are k-vector spaces but that for p > 2 there is higher torsion in general. We also note that our results are in agreement with the Evens-Friedlander calculation of K_{i}(𝔽_{p}[ϵ]) for i ≤ 3 and p ≥ 5, [EF]. Indeed the above theorem gives TC_{3}(𝔽_{p}[ϵ]) = ℤ/p ⊕ ℤ/p for p ≠ 3 and TC_{3}(𝔽_{3}[ϵ]) = ℤ/9.

Let us finally mention the following general result is proved in §2.3:

Theorem F. For any commutative ring A,

π_{0} THH(A)^{C_{p^{n}}} ≅ W_{n+1}(A),

the p-typical Witt vectors of length n + 1.

The cyclotomic structure of THH(A) induces two maps

R, F: THH(A)^{C_{p^{n}}} → THH(A)^{C_{p^{n−1}}} .

In earlier writings on topological cyclic homology, and in particular in [BHM], R was called Φ and F was called D. The reason for the change of notation is that π_{0}(R) and π_{0}(F) under the identifications of theorem F become the restriction map and Frobenius homomorphism, respectively, from W_{n+1}(A) to W_{n}(A). Thus the new notation is in agreement with the notation used for Witt vectors.

We say that a spectrum T is connective if π_{i}(T) = 0 when i < 0. A space will mean a compactly generated topological space which is weakly Hausdorff, i.e. the diagonal X ⊂ X ×X is closed when the product is given the compactly generated topology. We shall use equivalence to mean a map which induces isomorphisms on homotopy groups, and a G-equivalence to be a G-equivariant map which induces an equivalence on H-fixed sets for all closed subgroups H ⊂ G. Unless otherwise stated, G will denote the circle group S^{1}.

It is a pleasure to acknowledge the help we have received from M. Bökstedt at various points in time during the preparation of this paper.

iv

1. The Topological Hochschild Spectrum

1.1. Throughout this paper G will denote the circle group, C_{r} or just C the cyclic group of order r and J the quotient G/C. We recall briefly some notions from equivariant stable homotopy theory. The standard reference is [LMS].

A G-prespectrum indexed on a ‘complete G-universe’ U is a collection of G-spaces t(V), one for each finite dimensional sub inner product G-space V ⊂ U, together with a transitive system of G-maps

σ: t(V) → Ω^{W −V} t(W).

Here W − V denotes the orthogonal complement of V in W. It is a G-spectrum if the structure maps σ are all homeomorphisms. A map f: t → t′ of G-prespectra consists of G-maps f(V): t(V) → t′(V) which commute strictly with the structure maps. The category of G-prespectra indexed on U is denoted GPU and GSU denotes the full subcategory of G-spectra. The forgetful functor l: GSU → GPU has an idempotent left adjoint L, spectrification. It is given by the colimit over the structure maps

Lt(V) = \underrightarrow{lim}_{W ⊂U}Ω^{W −V} t(W),

provided that each σ is an inclusion, i.e. induces a homeomorphism onto its image. This, for instance, is the case when t is good as discussed in appendix A. We show in lemma A.1 that any G-prespectrum can be replaced by an equivalent one which is good. Thus we shall tacitly assume that our G-spectra are of the form T = Lt for some good G-prespectrum t.

Suppose that C is a closed subgroup in G with quotient J and let t ∈ GPU. There are two possible notions of an associated fixed point prespectrum in JPU^{C}, in [LMS] denoted t^{C} and Φ^{C}t, respectively. For each V ⊂ U^{C} we choose W ⊂ U such that W^{C} = V and such that the union of the W as V runs through the f. d. sub inner product spaces of U^{C} is all of U. Then the V ’th spaces are

t^{C}(V) = t(V)^{C}, (Φ^{C}t)(V) = t(W)^{C},

respectively, and the structure maps are the evident ones. There is a natural map

s_{C}: t^{C} → Φ^{C}t

which on V ’th spaces is given by the composite

t(V)^{C} ⟶^{σ} (Ω^{W −V} t(W))^{C} → t(W)^{C}

where the map on the right is induced from the inclusion of the C-fixed set 0 = (W − V)^{C} ⊂ W − V. If T ∈ GSU is a G-spectrum then T^{C} ∈ JSU, but to get Φ^{C}T ∈ JSU we must spectrify; Φ^{C}T = LΦ^{C}(lT).

Lemma 1.1. Suppose t is a good prespectrum and let T = Lt. Then there is a canonical homeomorphism

(Φ^{C}T)(V) ≅ \underrightarrow{lim}_{W ⊂U}Ω^{W^{C}−V} t(W)^{C},

and the maps in the colimit on the right are closed inclusions.

Proof. We have

T (W) = \underrightarrow{lim}_{Z⊃W} Ω^{Z−W} t(Z),

so

\underrightarrow{lim}_{W ⊃V} Ω^{W^{C}−V} T (W^{C}) = \underrightarrow{lim}_{W ⊃V} Ω^{W^{C}−V} ( \underrightarrow{lim}_{Z⊃W} Ω^{Z−W} t(Z))^{C} ≅ \underrightarrow{lim}_{Z⊃W ⊃V} Ω^{W^{C}−V} (Ω^{Z−W} t(Z))^{C}.

The colimit on the right runs over f.d. sub inner product spaces W, Z ⊂ U such that Z ⊃ W ⊃ V. In this index category, the full subcategory of pairs Z ⊃ W with Z = W is cofinal, so

\underrightarrow{lim}_{W ⊃V} Ω^{W^{C}−V} T (W^{C}) ≅ \underrightarrow{lim}_{Z⊂U}Ω^{Z^{C}−V} t(Z)^{C}.

These spaces form a G-spectrum, which therefore is Φ^{C}T, compare [LMS]. □

We recall that the smash product of a G-space X and a G-prespectrum t is the G-prespectrum whose V ’th space is X ∧ t(V) with the obvious structure maps. For a G-spectrum T we write X ∧ T for the G-spectrum L(X ∧ T). We note that if T = Lt, then X ∧ T ≅ L(X ∧ t).

Let j: U^{G} → U^{C} be the inclusion of the G-trivial universe and let D be a J-spectrum. We call j^{∗}D with its J-action forgotten the underlying non-equivariant spectrum of D.

Proposition 1.1. Suppose C is a cyclic p-group. For G-spectra T there is a cofibration sequence of nonequivariant spectra

T_{hC} ⟶^{N} T^{C} ⟶^{s_{Cp}^{C/Cp}} (Φ^{C_{p}}T)^{C/C_{p}}.

Here T_{hC} = EC_{+} ∧_{C} j^{∗}T is the homotopy orbit spectrum.

Proof. Let ẼG be the mapping cone,

EG_{+} ⟶^{π} S^{0} ⟶^{ι} ẼG,

where π maps EG to the non-basepoint in S^{0}. We can smash with T and obtain a cofibration sequence of G-spectra which in turn induces a cofibration sequence of non-equivariant spectra

[EG_{+} ∧ T]^{C} ⟶^{π_{∗}} T^{C} ⟶^{ι_{∗}} [ẼG ∧ T]^{C}.

The map s_{C_{p}} : T^{C_{p}} → Φ^{C_{p}}T factors as

T^{C_{p}} ⟶^{ι_{∗}} [ẼG ∧ T]^{C_{p}} ⟶^{s̄_{Cp}} Φ^{C_{p}}T,

where s̄_{C_{p}} is the map which on V ’th spaces is the map

\underrightarrow{lim}_{W ⊂U}(Ω^{W −V} (ẼG ∧ t(W)))^{C_{p}} → \underrightarrow{lim}_{W ⊂U}Ω^{W^{Cp} −V} t(W)^{C_{p}}

induced from the inclusion W^{C_{p}} ⊂ W. Here we have used lemma 1.1 to identify the right hand term. We claim that this is an equivalence. The maps in both limit systems are closed inclusions, so it is enough to prove that the map at step W in the limit is an equivalence, for all W. This, on the other hand, is a fibration with fiber the equivariant mapping space

F (S^{W −V} /S^{W^{Cp} −V} , ẼG ∧ t(W))^{C_{p}}.

Regarded as C-spaces, W^{C_{p}} ⊂ W is the singular set, so the S^{W −V} /S^{W^{Cp} −V} is a based free C-CW-complex. An induction over the C-cells shows that it is enough to consider

F (S^{k} ∧ C_{+}, ẼG ∧ t(W))^{C} ≅ F (S^{k}, ẼG ∧ t(W)).

Finally, this is contractible since ẼG is non-equivariantly contractible.

The identification of the first term goes in two steps. Let i: U^{C} → U be the inclusion. The forgetful functor i^{∗}: GSU → GSU^{C} has a left adjoint i_{∗} given by

i_{∗}D = L(W ↦ S^{W −W^{C}} ∧ D(W^{C})).

Since the functors i^{∗} and F (X, −), the pointed mapping space functor, commute the same hold for their left adjoints i_{∗} and X ∧ −. Thus the counit of the adjunction i_{∗} ⊣ i^{∗} induces a map

e: i_{∗}(EG_{+} ∧ i^{∗}T) → EG_{+} ∧ T.

It follows from [LMS], II.2.8 and II.2.12, that e is a G-equivalence. Finally we have the transfer equivalence

τ: EG_{+} ∧_{C} i^{∗}T ≃ [i_{∗}(EG_{+} ∧ i^{∗}T)]^{C}

of [LMS], p.97. Combined with e this identifies the first term. □

Example. It is illuminating to consider the case of a suspension G-spectrum Σ_{G}^{∞}X. We let E_{G}H denote a universal H-free G-space, that is E_{G}H^{K} ≃ ∗ when H ∩ K = 1 and E_{G}H^{K} = ∅ when H ∩ K ≠ 1. Then on the one hand we have the tom Dieck-Segal splitting

(Σ_{G}^{∞}X)^{C} ≃_{J} ⋁_{H≤C} Σ_{J}^{∞}(E_{G/H}(C/H)_{+} ∧_{C/H} X^{H}),

[tD1], and on the other hand, Φ^{C}(Σ_{G}^{∞}X) = Σ_{J}^{∞}X^{C} by lemma 1.1. Moreover, the map s_{C}: (Σ_{G}^{∞}X)^{C} → Φ^{C}(Σ_{G}^{∞}X) is simply the projection onto the summand H = C.

1.2. Suppose C is finite of order r. Then the r’th root ρ_{C}: G → J is an isomorphism of groups, and a J-space X may be viewed as a G-space ρ_{C}^{∗}X through ρ_{C}. We also use ρ_{C} to view J-spectra as G-spectra.

When D is a J-spectrum indexed on U^{C}, then the G-spaces

ρ_{C}^{∗}D((ρ_{C}^{−1})^{∗}(V)),

for V ⊂ ρ_{C}^{∗}U^{C}, form a G-spectrum indexed on ρ_{C}^{∗}U^{C}. From now on we fix our universe. Let ℂ(n) = ℂ with G acting through the n’th power map, g · z = g^{n}z. Then we set

U = ⨁_{n∈ℤ,α∈ℕ} ℂ(n)_{α},

and note that

ρ_{C}^{∗}U^{C} = ⨁_{n∈rℤ,α∈ℕ} ℂ(n/r)_{α}.

Identifying ℤ and rℤ in the usual way we get U = ρ_{C}^{∗}U^{C}. Thus a J-spectrum D indexed on U^{C} determines a G-spectrum indexed on U and we denote this ρ_{C}^{#}D.

Definition 1.2. A cyclotomic spectrum is a G-spectrum T indexed on U together with a G-equivalence

r_{C}: ρ_{C}^{#}Φ^{C}T → T

for every finite C ⊂ G, such that for any pair of finite subgroups the diagram

ρ_{C_{r}}^{#}Φ^{C_{r}}ρ_{C_{s}}^{#}Φ^{C_{s}}T  ρ_{C_{rs}}^{#}Φ^{C_{rs}}T
ρ_{Cr}^{#} Φ^{Cr} r_{Cs} ↓  ^{r_{Crs}} ↓
ρ_{C_{r}}^{#}Φ^{C_{r}}T  ⟶^{r_{Cr}}  T

commutes.

Lemma 1.2. Let t be a good G-prespectrum and let T = Lt. Then T is a cyclotomic spectrum if for each index space V ⊂ U and each finite subgroup C ⊂ G there is a G-map

r_{C}(V): ρ_{C}^{∗}t(V)^{C} → t(ρ_{C}^{∗}V^{C})

subject to the following conditions

i) For each pair V ⊂ W ⊂ U the diagram

S^{ρ_{C}^{∗}(W −V)^{C}} ∧ ρ_{C}^{∗}t(V)^{C} ⟶^{1∧r_{C}(V)} S^{ρ_{C}^{∗}(W −V)^{C}} ∧ t(ρ_{C}^{∗}V^{C})
ρ_{C}^{∗}(σ̃)^{C} ↓  ↓σ̃
ρ_{C}^{∗}t(W)^{C}  ⟶^{r_{C}(W)}  t(ρ_{C}^{∗}W^{C})

commutes. ii) For each pair of finite subgroups the diagram

ρ_{C_{rs}}^{∗}t(V)^{C_{rs}} ⟶^{ρ_{Cs}^{∗} (r_{Cr} (V))^{Cs}} ρ_{C_{s}}^{∗}t(ρ_{C_{r}}^{∗}V^{C_{r}})^{C_{s}}
r_{Crs} (V)↓  ↓^{r_{Cs} (ρ_{Cr}^{∗} V^{Cr})}
t(ρ_{C_{rs}}^{∗}V^{C_{rs}})  t(ρ_{C_{s}}^{∗}(ρ_{C_{r}}^{∗}V^{C_{r}})^{C_{s}})

commutes iii) For any V ⊂ U the induced map on colimits

\underrightarrow{lim}_{W ⊂U}Ω^{ρ_{C}^{∗}W^{C}−V} ρ_{C}^{∗}t(W)^{C} → \underrightarrow{lim}_{W ⊂U}Ω^{ρ_{C}^{∗}W^{C}−V} t(ρ_{C}^{∗}W^{C})

is a G-equivalence. □

Proof. The map in iii) composed with the isomorphism of lemma 1.1 gives a G-equivalence

r_{C}(V): (ρ_{C}^{#}Φ^{C}T)(V) → T (V).

Because of i) the maps r_{C}(V) form a map r_{C}: ρ_{C}^{#}Φ^{C}T → T and this is a G-equivalence. Finally, the diagrams in definition 1.2 commutes by ii). □

We call a G-prespectrum t with the structure above a cyclotomic prespectrum. A map of cyclotomic (pre)spectra is a map of G-(pre)spectra which strictly commutes with the r-maps.

Example. The free loopspace L(X) is the space of unbased maps from S^{1} to X. Rotation of loops defines a G-action on L(X). Suppose C is a subgroup of G of order r. Then there is an equivariant homeomorphism

Δ_{r}: L(X) → ρ_{C}^{∗}L(X)^{C}; Δ_{r}(λ)(z) = λ(z^{r}).

We can use this to give the suspension prespectrum of L(X) the structure of a cyclotomic prespectrum. Indeed we define

r_{C}(V): ρ_{C}^{∗}(S^{V} ∧ L(X)_{+})^{C} = S^{ρ_{C}^{∗}V^{C}} ∧ ρ_{C}^{∗}L(X)_{+}^{C} ⟶^{1∧Δ_{C}^{−1}} S^{ρ_{C}^{∗}V^{C}} ∧ L(X)_{+}

and i), ii) and iii) in the lemma/definition are readily verified.

The map s_{C_{r}} from 1.1 and the cyclotomic structure map r_{C_{r}} give rise to a map of G-spectra

ρ_{C_{rs}}^{#}T^{C_{rs}} = ρ_{C_{s}}^{#}(ρ_{C_{r}}^{#}T^{C_{r}})^{C_{s}} → ρ_{C_{s}}^{#}(ρ_{C_{r}}^{#}Φ^{C_{r}}T)^{C_{s}} → ρ_{C_{s}}^{#}T^{C_{s}}

and hence a map

(1.2.1)  R_{r}: T^{C_{rs}} → T^{C_{s}}

of the underlying non-equivariant spectra, which will play a fundamental role in the following. We call it the r’th restriction map.

Let Z ⊂ U be a representation. Then, slightly more general, we let T_{Z} denote the smash product Gspectrum T ∧ S^{Z}. The cyclotomic structure maps give a G-equivalence

(1.2.2)  r_{C,Z}: ρ_{C}^{#}Φ^{C}T_{Z} → T_{ρ_{C}^{∗}Z^{C}} .

Indeed, by lemma 1.1

ρ_{C}^{#}Φ^{C}(T ∧ S^{Z}) ≅ ρ_{C}^{#}(Φ^{C}T) ∧ ρ_{C}^{∗}S^{Z^{C}} .

We note that T_{Z}(V − Z) ≃_{G} T (V). Again we get a map of non-equivariant spectra

(1.2.3)  R_{r,Z}: T_{Z}^{C_{rs}} → T_{ρ_{Cr}^{∗} Z^{Cr}}^{C_{s}} .

We can restate proposition 1.1 for cyclotomic spectra as

Theorem 1.2. For any cyclotomic spectrum T and any f.d. sub inner product space Z ⊂ U there is a cofibration sequence of non-equivariant spectra

(T_{Z})_{hC_{p^{n}}} ⟶^{N} T_{Z}^{C_{p^{n}}} ⟶^{R_{p,Z}} T_{ρ_{Cp}^{∗} Z^{Cp}}^{C_{p^{n−1}}} ,

where (T_{Z})_{hC_{p^{n}}} is the homotopy orbit spectrum. □

1.3. Suppose T is a cyclotomic spectrum, then so is ρ_{C}^{#}Φ^{C}T but in general ρ_{C}^{#}T^{C} is not. We proceed to explain the situation. First we recall the notion of a family of subgroups.

A collection F of subgroups of G is called a family if it is closed under passage to subgroups. A map f: X → Y of G-spaces (G-spectra) is called an F-equivalence if the induced map f^{H} on H-fixed points is an equivalence for all H ∈ F, or equivalently, if f ∧ EF_{+} is a G-equivalence. Here EF is the join of the free contractible G/H-spaces E(G/H) for H ∈ F. It is the terminal object among G-spaces with orbit types G/H, H ∈ F, and G-homotopy classes of maps; cf. [tD]. We let F_{p} denote the family of finite p-subgroups of G.

Definition 1.3. ([BM]) A p-cyclotomic spectrum is a G-spectrum T indexed on U together with an F_{p}- equivalence r_{p}: ρ_{C_{p}}^{#}Φ^{C_{p}}T → T

Of course a cyclotomic spectrum is p-cyclotomic for every prime p. Also note that for a p-cyclotomic spectrum, theorem 1.2 holds for the prime p.

Proposition 1.3. Let T be a cyclotomic spectrum. Then ρ_{C}^{#}T^{C} is a p-cyclotomic spectrum for every prime p which does not divide the order of C.

Proof. For point set topological reasons we consider instead the spectrum S = ρ_{C}^{#}L((T^{C})^{τ}); compare appendix A. We want to define a G-map

r_{p}(V): ρ_{C_{p}}^{∗}S(V)^{C_{p}} → S(ρ_{C_{p}}^{∗}V^{C_{p}}),

that is, a G-map

ρ_{C_{p}}^{∗}(ρ_{C}^{∗}T^{τ} ((ρ_{C}^{−1})^{∗}V)^{C})^{C_{p}} → ρ_{C}^{∗}T^{τ} ((ρ_{C}^{−1})^{∗}(ρ_{C_{p}}^{∗}V^{C_{p}}))^{C}.

We have a G-map

ρ_{C_{p}}^{∗}(ρ_{C}^{∗}T^{τ} ((ρ_{C}^{−1})^{∗}V)^{C})^{C_{p}} = ρ_{C}^{∗}(ρ_{C_{p}}^{∗}T^{τ} ((ρ_{C}^{−1})^{∗}V)^{C_{p}})^{C}

⟶^{ρ_{C}^{∗}(r_{Cp})^{C}} ρ_{C}^{∗}T^{τ} (ρ_{C_{p}}^{∗}((ρ_{C}^{−1})^{∗}V)^{C_{p}})^{C}.

Now the representations ρ_{C_{p}}^{∗}((ρ_{C}^{−1})^{∗}V)^{C_{p}} and (ρ_{C}^{−1})^{∗}(ρ_{C_{p}}^{∗}V^{C_{p}}) agree when p does not divide the order of C. □

1.4. In this section we define the topological Hochschild spectrum. It is a cyclotomic spectrum whose zero’th space is naturally C-equivalent to Bökstedt’s topological Hochschild space THH(L).

We briefly recall the definition of THH(L) and refer to [B], [BHM] and [PW] for details. Let I be the category whose objects are the finite cardinals n = {1, 2, …, n} (0 = ∅) and whose morphisms are the injective maps, and let L be a functor with smash product. Then THH(L). is the cyclic space with k-simplices equal to the homotopy colimit

holim_{\overrightarrow{I^{k+1}}} F (S^{i_{0}} ∧ … ∧ S^{i_{k}}, L(S^{i_{0}}) ∧ … ∧ L(S^{i_{k}}))

and with Hochschild-type structure maps. The realization THH(L) is a G-space. More generally we let THH(L; X). be the cyclic space with k-simplices

holim_{\overrightarrow{I^{k+1}}} F (S^{i_{0}} ∧ … ∧ S^{i_{k}}, L(S^{i_{0}}) ∧ … ∧ L(S^{i_{k}}) ∧ X),

where X acts as a dummy for the cyclic structure maps. If X has a G-action then THH(L; X) becomes a G × G-space, and hence a G-space via the diagonal Δ: G → G × G.

We define a G-prespectrum t(L) whose 0’th space is THH(L). Let V be a f.d. sub inner product space of some G-universe U, and let S^{V} be the one-point compactification. Then

t(L)(V) = THH(L; S^{V})

and the obvious maps

σ: t(L)(V) → Ω^{W −V} t(L)(W)

are G-equivariant and form a transitive system. Finally we let T (L) be the associated G-spectrum of the thickened G-prespectrum t^{τ} (L), that is

T (L)(V) = \underrightarrow{lim}_{W ⊂U}Ω^{W −V} t^{τ} (L)(W).

In order to define the cyclotomic structure maps we need the edgewise subdivision of [BHM] §1.

The realization of a cyclic space becomes a G-space upon identifying G with ℝ/ℤ, and hence C = C_{r} may be identified with r^{−1}ℤ/ℤ. Edgewise subdivision associates to a cyclic space Z. a simplicial C-space sd_{C} Z. with k-simplices sd_{C} Z_{k} = Z_{r(k+1)−1}; the generator r^{−1}+ℤ of C acts as τ^{k+1}. The diagonal Δ^{k} → Δ^{k}∗⋯∗Δ^{k} (r factors) induces a natural (non-simplicial) homeomorphism,

D: | sd_{C} Z.| → |Z.|,

of the realizations. Finally, there is a natural ℝ/rℤ-action on | sd_{C} Z.| which extends the simplicial C-action, and the map D is G-equivariant when ℝ/rℤ is identified with ℝ/ℤ through division by r.

We consider the case of THH(L; X).. Let us write G_{k}^{X}(i_{0}, …, i_{k}) for the pointed mapping space

F (S^{i_{0}} ∧ … ∧ S^{i_{k}}, L(S^{i_{0}}) ∧ … ∧ L(S^{i_{k}}) ∧ X).

Then the k-simplices of the edgewise subdivision are the homotopy colimit

sd_{C} THH(L; X)_{k} = holim_{\overrightarrow{I^{r(k+1)}}} G_{r(k+1)−1}^{X}.

We are interested in the subspace of C-fixed points. If X_{α} is a diagram of C-spaces, then the homotopy colimit is again a C-space and its C-fixed set is the homotopy colimit of the C-fixed sets X_{α}^{C}. However the C-action on sd_{C} THH(L; X)_{k} does not arise in this way. We consider instead the composite functor G_{r(k+1)−1}^{X} ∘ Δ_{r} where Δ_{r}: I^{k+1} → (I^{k+1})^{r} is the diagonal functor. This is indeed a diagram of C-spaces and the canonical map of homotopy colimits

b_{k}: holim_{\overrightarrow{I^{k+1}}} G_{r(k+1)−1}^{X} ∘ Δ_{r} → holim_{\overrightarrow{I^{r(k+1)}}} G_{r(k+1)−1}^{X}

197zis a C-equivariant inclusion which induces a homeomorphism of C-fixed sets. Let R be the regular representation ℝC and let iR denote the i-fold direct sum. Then we get

(1.4.1)  sd_{C} THH(L; X)_{k}^{C} ≅ holim_{\overrightarrow{I^{k+1}}} F (S^{i_{0}R} ∧ … ∧ S^{i_{k}R}, L(S^{i_{0}})^{∧r} ∧ … ∧ L(S^{i_{k}})^{∧r} ∧ X)^{C}

with C acting by cyclic permutation on L(S^{i})^{∧r} and by conjugation on the mapping space. Indeed, S^{iR} = (S^{i})^{∧r} as a C-space. This ends our discussion of edgewise subdivision.

Lemma 1.4. Let H be a compact Lie group and let Y. be a simplicial H-space such that Y_{k}^{K} is n(K)- connected for all k, n(K) ≥ 0. Suppose X is a based H-CW-complex with finitely many orbit types, and such that dim X^{K} ≤ n(K) for all K ≤ H. If Y.^{K} is proper in the sense of [May] for the occuring orbit types then the natural map

γ: |F (X, Y.)| → F (X, |Y.|)

is an H-equivalence.

Proof. We prove that γ^{H} is an equivalence by induction over the H-cells in X. Let X_{β} be obtained from X_{α} by adjoining an H-cell H/K_{+} ∧ S^{n}. Then we have a simplicial Hurewicz fibration

F (S^{n}, Y.^{K}) → F (X_{β}, Y.)^{H} → F (X_{α}, Y.)^{H},

and the condition that dim X^{K} ≤ n(K) ensures that its realization is quasi-fibration. We consider the diagram

|F (S^{n}, Y.^{K})| ⟶ |F (X_{β}, Y.)^{H}| ⟶ |F (X_{α}, Y.)^{H}|
γ^{n}↓  _{γ^{H}} ↓  _{γ^{H}} ↓
F (S^{n}, |Y.^{K}|) ⟶ F (X_{β}, |Y.|)^{H} ⟶ F (X_{α}, |Y.|)^{H}.

The map γ^{n} is an equivalence by [May, 12.4] and we are done by induction.

Since an H-CW-complex is also a K-CW-complex for K ≤ H, the same argument shows that γ^{K} is an equivalence. This concludes the proof. □

Proposition 1.4. The canonical map t(L)(V) → T (L)(V) is an F-equivalence, where F is the family of finite subgroups of G.

Proof. We must prove that the prespectrum structure map σ: t(V) → Ω^{W −V} t(W) is a C-equivalence for any C ∈ F. We use edgewise subdivision to get a simplicial C-action and factor σ as

| sd_{C} THH(L; S^{V})| → |Ω^{W −V} sd_{C} THH(L; S^{W})| → Ω^{W −V} | sd_{C} THH(L; S^{W})|.

The right hand map is a C-equivalence by the lemma above. It follows from [L] that the simplicial spaces involved are ‘good’ in the sense of [S] or ‘strictly proper’ in the sense of [May]. Therefore it is enough to show that the map on homotopy colimits

σ̂_{k}: holim_{\overrightarrow{I^{k+1}}} F (S^{i_{0}R} ∧ … ∧ S^{i_{k}R}, L(S^{i_{0}})^{∧r} ∧ … ∧ L(S^{i_{k}})^{∧r} ∧ S^{V})

→ holim_{\overrightarrow{I^{k+1}}} F (S^{i_{0}R} ∧ … ∧ S^{i_{k}R} ∧ S^{W −V} , L(S^{i_{0}})^{∧r} ∧ … ∧ L(S^{i_{k}})^{∧r} ∧ S^{W}),

induced by the adjoints of the evaluation maps, is a C-equivalence. Furthermore we may assume that W − V = lR. We consider the map

τ_{k}: holim_{\overrightarrow{I^{k+1}}} F (S^{i_{0}R} ∧ … ∧ S^{i_{k}R} ∧ S^{lR}, L(S^{i_{0}})^{∧r} ∧ … ∧ L(S^{i_{k}})^{∧r} ∧ S^{lR} ∧ S^{V})

→ holim_{\overrightarrow{I^{k+1}}} F (S^{i_{0}R} ∧ … ∧ S^{(i_{k}+l)R}, L(S^{i_{0}})^{∧r} ∧ … ∧ L(S^{i_{k}+l})^{∧r}),

given by the identification S^{lR} ≅ (S^{l})^{∧r} and the stabilization L(S^{i_{k}}) ∧ S^{l} → L(S^{i_{k}+l}). It is a C-equivalence by [BHM], 3.11 and 3.12, and the approximation theorem [B1], 1.6. The composition τ_{k} ∘ σ̂_{k} is a map in the limit system and induces therefore a C-equivalence on homotopy colimits. It follows that σ̂_{k} is a C-equivalence. □

1.5. In this section we define the cyclotomic structure on t(L) and T (L). For any pointed C-spaces X, Y, we have the obvious map

F (X, Y)^{C} → F (X^{C}, Y^{C})

induced from the inclusion X^{C} ⊂ X of the fixed set. In the case at hand, this gives a simplicial map

r′_{C}: sd_{C} THH(L; X).^{C} → THH(L; X^{C}).

and we define

(1.5.1)  r_{C}(V): ρ_{C}^{∗}t(L)(V)^{C} → t(L)(ρ_{C}^{∗}V^{C})

to be the composite

ρ_{C}^{∗}| THH(L; S^{V})|^{C} ⟶^{D^{−1}} | sd_{C} THH(L; S^{V})|^{C} ⟶^{r′_{C}} | THH(L; S^{ρ_{C}^{∗}V^{C}})|.

The maps r_{C}(V) induce similar maps in the thickened prespectrum t^{τ} (L). In order to show that these makes t^{τ} (L) a cyclotomic prespectrum we need Lemma 1.5. Let j be a G-prespectrum and let J be the G-spectrum associated with j^{τ}. If J^{Γ} ≃ ∗ for any finite subgroup Γ ⊂ G and j(V)^{G} ≃ ∗ for any V ⊂ U then J ≃_{G} ∗.

Proof. Let F be the family of finite subgroups of the circle, then J is F-contractible. Since J ∧ EF_{+} → J is an F-equivalence, J ∧ EF_{+} is also F-contractible. However J ∧ EF_{+} is G-equivalent to an F-CW-spectrum and therefore it is in fact G-contractible by the F-Whitehead theorem, [LMS] p.63. Now

(J ∧ EF_{+})(V) ≅ \underrightarrow{lim}_{W}Ω^{W} (j^{τ} (V + W) ∧ EF_{+}),

and j^{τ} (V) ∧ EF_{+} → j^{τ} (V) is an G-equivalence since j(V)^{G} ≃ ∗. Therefore J ≃_{G} J ∧ EF_{+} and we have already seen that the latter is G-contractible. □

Proposition 1.5. t^{τ} (L) is a cyclotomic prespectrum and T (L) is a cyclotomic spectrum.

Proof. By lemma 1.2 it is enough to show that t^{τ} (L) is a cyclotomic prespectrum. The map r_{C}(V) in (1.5.1) is G-equivariant by construction so we have left to check the three conditions in lemma 1.2. We leave i) and ii) to the reader and prove iii).

We first show that the maps r_{C}(W) induce a weak equivalence

\underrightarrow{lim}_{W ⊂U}(Ω^{ρ_{C}^{∗}W^{C}−V} ρ_{C}^{∗}t^{τ} (L)(W)^{C})^{Γ} → \underrightarrow{lim}_{W ⊂U}(Ω^{ρ_{C}^{∗}W^{C}−V} t^{τ} (L)(ρ_{C}^{∗}W^{C}))^{Γ}

when Γ ⊂ G is finite. Since the maps in both limit systems are closed inclusions it is enough to show that the connectivity of

(Ω^{ρ_{C}^{∗}W^{C}−V} ρ_{C}^{∗}t^{τ} (L)(W)^{C})^{Γ} → (Ω^{ρ_{C}^{∗}W^{C}−V} t^{τ} (L)(ρ_{C}^{∗}W^{C}))^{Γ},

or equivalently,

(Ω^{ρ_{C}^{∗}W^{C}−V} |(sd_{C} THH(L; S^{W}).)^{C}|)^{Γ} → (Ω^{ρ_{C}^{∗}W^{C}−V} | THH(L; S^{ρ_{C}^{∗}W^{C}}).|)^{Γ}

tends to infinity as W runs through the f.d. sub inner product spaces of U. Let π_{C}: G → G/C be the projection and let H = π_{C}^{−1}(ρ_{C}(Γ)) such that |H| = |Γ| · |C|. Then it is proved in [BHM] that sd_{H} = sd_{Γ} sd_{C} and that the diagram

ρ_{C}^{∗}| sd_{H} THH(L; S^{W})^{C}| ⟶^{r̂′_{C}} | sd_{Γ} THH(L; S^{ρ_{C}^{∗}W^{C}})|
D↓≅  ^{D}↓≅
ρ_{C}^{∗}| sd_{C} THH(L; S^{W})^{C}| ⟶^{r′_{C}} | THH(L; S^{ρ_{C}^{∗}W^{C}})|

commutes. In the top row the Γ-action is simplicial, and by lemma 1.4 it is enough to prove that the connectivity of the map

(Ω^{ρ_{C}^{∗}W^{C}−V} (sd_{H} THH(L; S^{W})_{k})^{C})^{Γ} → (Ω^{ρ_{C}^{∗}W^{C}−V} sd_{Γ} THH(L; S^{ρ_{C}^{∗}W^{C}})_{k})^{Γ},

induced from r′_{C}(W), tends to infinity with W. We can use (1.4.1) to identify the homotopy fiber with the homotopy colimit

(*)  holim_{\overrightarrow{I^{k+l}}} F (S^{W^{C}} ∧ S^{iR}/S^{iR^{C}} , L(S^{i_{0}})^{∧r} ∧ … ∧ L(S^{i_{k}})^{∧r} ∧ S^{W})^{H},

where we have written i = i_{0} + … + i_{k}. In general, the connectivity of an equivariant mapping space

A^{H} = F (X, Y)^{H},

where X is an H-CW-complex, is given by

conn(A^{H}) ≥ min{conn(Y^{K}) − dim(X^{K}): K ⊂ H}.

Here conn(Z) denotes the greatest integer such that π_{i}(Z) = 0 whenever i ≤ conn(Z), cf. [A]. In the case at hand,

dim((S^{W^{C}} ∧ S^{iR}/S^{iR^{C}})^{K}) = \{ \begin{matrix}dim(W^{K}), & if K ⊃ C \\ dim(W^{CK}) + i dim(R^{K}), & if K ⊅ C \end{matrix}

whereas, assuming that L is connective,

conn((L(S^{i_{0}})^{∧r} ∧ … ∧ L(S^{i_{k}})^{∧r} ∧ S^{W})^{K}) = dim(W^{K}) + i|H: K| − 1 = dim(W^{K}) + i dim(R^{K}) − 1.

In the case, K ⊃ C the differens tends to infinity as (i_{0}, …, i_{k}) runs through I^{k+1} so (*) is (weakly) contractible for all W ⊂ U. When K ⊅ C, the differens tends to infinity as W runs through the f.d. sub inner product spaces of U.

We define an auxiliary functor a^{C}: GPU → GPU as follows. For each Z ⊂ U^{C} choose V (Z) ⊂ U such that V (Z)^{C} = Z and such that the union of all V (Z) is equal to U, then define a^{C} by

a^{C}t(ρ_{C}^{∗}Z) = ρ_{C}^{∗}t(V (Z))^{C}

with the obvious prespectrum structure maps. The maps r_{C}(V) from (1.5.1) defines a map of G-prespectra

r_{C}: a^{C}t → t

the requirement in lemma 1.2 (iii) becomes that the induced map of the associated G-spectra be a Gequivalence. We now use lemma 1.5 with j equal to the homotopy fiber of r_{C}. We have already shown that J^{Γ} is equivalent to a point, so it remains to show that j(V)^{G} ≃ ∗. For any cyclic space Z., the G-fixed set |Z.|^{G} of the realization may be idenfied with the subspace in Z_{0} consisting of those 0-simplices z for which s_{0}z = τ_{1}s_{0}z. In the case of THH(L; S^{V}) this is S^{V^{G}}, and j(V)^{G} is the homotopy fiber of the identity. □

1.6. In [BHM] C-equivariant deloops of THH(L) were defined using the Γ-space machine of Segal and Shimakawa. We show in this section that the equivariant deloops obtained in this fashion are C-equivalent to the deloops t(L)(V) defined in 1.4, but first we give a brief discussion of Γ_{C}-spaces.

Let Γ_{C} be the category of the finite based C-sets S, whose underlying set is of the form n = {0, 1, …, n}, based at 0. A Γ_{C}-space is a functor A from Γ_{C} to C-spaces. It is special if A(0) ≃_{C} ∗ and if the canonical map is a C-equivalence

A(S ∨ T) → A(S) × A(T),

for any S, T ∈ Γ_{C}. A C-spectrum A defines a special Γ_{C}-space, A(S) = S ∧ A.

Suppose X.: Δ^{op} → Γ_{C} is a finite simplicial C-set, then A(X.) is a simplicial C-space, which we want to realize. To get the correct homotopy type, however, we need that A(S) → A(T) be a closed C-cofibration whenever S ↣ T is an inclusion. In [S1] Segal obtains this by replacing A by a thickened version τA given by

τA(S) = holim_{⟶} ((Γ_{C} ↓S) ⟶^{pr_{1}} Γ_{C} ⟶^{A} Top_{C}),

where (Γ_{C} ↓ S) is the category over S. It has id_{S} as terminal object, so τA(S) → A(S) is a C-homotopy equivalence. Furthermore an injection f: S ↣ T induces an inclusion of over-categories and therefore a closed C-cofibration τA(S) → τA(T).

Alternatively one may consider the two sided bar construction B(A, Γ_{C}, X). It is the realization of a simplicial space B.(A, Γ_{C}, X) with k-simplices

∐_{S_{0},... ,S_{k}} A(S_{0}) × F (S_{0}, S_{1}) × ⋯ × F (S_{k−1}, S_{k}) × F (S_{k}, X),

with the coproduct taken over tuples of finite C-sets in Γ_{C}. We have Lemma 1.6. B(A, Γ_{C}, | X.|) ≃_{C} |τA(X.)|, for any X.: Δ^{op} → Γ_{C}.

Proof. A bisimplicial space Y_{.,.} may be realized as |k ↦ |Y_{k,.}|| or as |l ↦ |Y_{.,l}||, the two realizations are homeomorphic. Hence B(A, Γ_{C}, | X.|) ≅_{C} |B(A, Γ_{C}, X.)|. Now by [Wo] lemma 1.3 the ‘evaluation map’

B(A, Γ_{C}, X_{k}) → A(X_{k})

is a C-equivalence for all k ≥ 0. We want the map on realizations to be a C-equivalence. This requires that the simplicial spaces are ‘good’ in the sense of [S]. The space on the left is good, but the one on the right is not necessarily so. Therefore we must replace it by its thickening τA(X.). □

Following [Sh] we define a C-prespectrum BA whose V ’th space is the quotient

B^{V}A = B(A, Γ_{C}, S^{V})/B(A, Γ_{C}, ∞).

Finally, recall that A(1) is a C-homotopy commutative, C-homotopy associative H-space, with product A(1) × A(1) ≃_{C} A(1 ∨ 1) → A(1).

Proposition 1.6.1. ([Sh]) If A(1) has a C-homotopy inverse, then BA is an Ω-C-spectrum, that is the structure maps induce C-equivalences B^{V}A ≃_{C} Ω^{W −V} B^{W}A. □

We have two Ω-C-spectra with zero’th space THH(L). The first is B THH(L), arising from a special Γ_{C}-structure on THH(L), and the other is t(L), defined in 1.4. We know that t(L) is an Ω-C-spectrum by proposition 1.4. To show that they are equivalent we construct a Ω-C-bispectrum, which contains both.

The Γ_{C}-space on THH(L) constructed in [BHM] works equally well for the space t(L)(V) = THH(L, S^{V}). Specifically, in the notation of [BHM] §4:

t(L)(V, S) = ⋁_{\underline{k}:P_{0}n→ℕ_{0}} | sd_{C}(E.(n, \underline{k})_{+} ∧ THH(L\underline{k}, S^{V}).)|.

Here n is the underlying set of the finite C-set S. In view of [BHM] 4.20 these Γ_{C}-spaces are special, and we obtain Ω-C-spectra Bt(L)(V) for each V. Hence the equivalence follows from the

Proposition 1.6.2. B^{W −V}t(L)(V) ≃_{C} t(L)(W).

Proof. It suffices to treat the case where W −V is the regular representation R = ℝC. We choose a simplicial model S.^{1} for the circle, e.g. S.^{1} = Δ[1]/∂Δ[1] or S.^{1} = Λ[0]. Then S.^{1} ∧ … ∧ S.^{1} (r times) with C acting by cyclic permutation is a simplicial model S.^{R} for S^{R}. From lemma 1.4 and the lemma above we get

Ω^{R}B^{R}t(L)(V ⊕ R) ≃_{C} |Ω^{R}τt(L)(V ⊕ R, S.^{R})| ≃_{C} |τt(L)(V, S.^{R})| ≃_{C} B^{R}t(L)(V).

Since Ω^{R}B^{R} is C-homotopic to the identity functor the proposition follows. □

1.7. We conclude this paragraph with a list of some additional properties of topological Hochschild homology. We shall need the following extension of Bökstedt’s notion of a functor with smash product.

Let L be a functor with smash product. The definition of THH(L; X) does not require the full functoriality of L. In effect, we only need a collection of spaces L(S^{n}), n ≥ 0, with a Σ_{n}-action together with unit and multiplication maps

(1.7.1)  1_{n}: S^{n} → L(S^{n}), μ_{m,n}: L(S^{m}) ∧ L(S^{n}) → L(S^{m+n}),

which are Σ_{n}-equivariant and Σ_{m} × Σ_{n}-equivariant, respectively, and satisfies the relations up to canonical homeomorphism

i) μ_{m,n} ∘ (1_{m} ∧ 1_{n}) = 1_{m+n} ii) σ_{m,n} ∘ μ_{m,n} ∘ (1_{m} ∧ id) = μ_{n,m} ∘ (id ∧1_{m}) ∘ tw iii) μ_{l+m,n} ∘ (μ_{l,m} ∧ id) = μ_{l,m+n} ∘ (id ∧μ_{m,n}) iv) μ_{0,n} ∘ (1_{0} ∧ id) = id = μ_{n,0} ∘ (id ∧1_{0}).

In the commutative case we require, in addition, that

v) σ_{m,n} ∘ μ_{m,n} = μ_{n,m} ∘ tw.

where σ_{m,n} ∈ Σ_{n+m} permutes the first m and last n elements and tw permutes the two smash factores. We call such a set of data an FSP defined on spheres. These are the monoids in the symmetric monoidal category of spectra which has recently been constructed by Jeff Smith, [Sm].

We let L be an FSP defined on spheres and consider a version of topological Hochschild homology where we replace the index category I by the n-fold product I^{n}. By the approximation theorem, [B1] theorem 1.6, this will not change the homotopy type:

(1.7.2)  THH(L^{(n)}; X) ≃ THH(L; X), for n > 0.

In more detail, for n > 0 we let THH(L^{(n)}; X). be the cyclic space with k-simplices

holim_{\overrightarrow{(I^{n})^{k+1}}} F ((S^{i_{10}} ∧ … ∧ S^{i_{n0}}) ∧ … ∧ (S^{i_{1k}} ∧ … ∧ S^{i_{nk}}), L(S^{i_{10}} ∧ … ∧ S^{i_{n0}}) ∧ … ∧ L(S^{i_{1k}} ∧ … ∧ S^{i_{nk}}) ∧ X)

and with Hochschild-type cyclic structure maps. For n = 0 we let

THH(L^{(0)}; S^{0}) = |N_{∧}^{cy}(L(S^{0}))|,

the cyclic bar construction of the pointed monoid L(S^{0}), see (6.1.1) below. In both cases the realization THH(L^{(n)}; X) is a G × Σ_{n}-space, where the Σ_{n}-action is induced from the permutation action on I^{n}. When X = S^{n} we get another Σ_{n}-action induced from the Σ_{n}-action on S^{n}. Hence THH(L^{(n)}; S^{n}) becomes a G × Σ_{n} × Σ_{n}-space which we consider a G × Σ_{n}-space via the diagonal in the second factor.

Proposition 1.7.1. Let L be a commutative FSP defined on spheres. Then the spaces THH(L^{(n)}; S^{n}), n ≥ 0, again form a commutative FSP defined on spheres and the multiplication maps

μ_{m,n}: THH(L^{(m)}; S^{m}) ∧ THH(L^{(n)}; S^{n}) → THH(L^{(m+n)}; S^{m+n})

are G-equivariant when the domain is given the diagonal G-action. Moreover, the restriction and Frobenius maps

R_{r}, F_{r}: THH(L^{(n)}; S^{n})^{C_{rs}} → THH(L^{(n)}; S^{n})^{C_{s}}

are Σ_{n}-equivariant, multiplicative and preserve units.

Proof. Let G_{k}^{X} be as in 1.4 and let μ_{n}: I^{n} → I be the iterated multiplication functor, i.e. concatenation of sets and maps. Then we have

THH(L^{(n)}; X)_{k} = holim_{\overrightarrow{(I^{n})^{k+1}}} G_{k}^{X} ∘ μ_{n}^{k+1}.

We first recall that the canonical map

can: holim_{\overrightarrow{(I^{m})^{k+1}}} G_{k}^{X} ∘ μ_{m}^{k+1} ∧ holim_{\overrightarrow{(I^{n})^{k+1}}} G_{k}^{Y} ∘ μ_{n}^{k+1} → holim_{\overrightarrow{(I^{m})^{k+1}×(I^{n})^{k+1}}} G_{k}^{X} ∘ μ_{m}^{k+1} ∧ G_{k}^{Y} ∘ μ_{n}^{k+1}

is a homeomorphism, when the spaces are given the compactly generated topology. Next, we note that there are natural transformations

G_{k}^{X} ∘ μ_{m}^{k+1} ∧ G_{k}^{Y} ∘ μ_{n}^{k+1} ⟶^{λ} G_{2k+1}^{X∧Y} ∘ (μ_{m}^{k+1} × μ_{n}^{k+1}) ⟶^{σ} G_{k}^{X∧Y} ∘ μ_{m+n}^{k+1}.

Indeed, λ is concatenation of maps and if tw: (I^{m})^{k+1} × (I^{n})^{k+1} → (I^{m+n})^{k+1} denotes the isomorphism of categories given by

tw((i_{10}, …, i_{m0}), …, (i_{1k}, …, i_{mk}), (j_{10}, …, j_{n0}), …, (j_{1k}, …, j_{nk})) = (((i_{10}, …, i_{m0}, j_{10}, …, j_{n0}), …, (i_{1k}, …, i_{mk}, j_{1k}, …, j_{nk})), then σ is the obvious shuffle permutation covering tw followed by multiplication in L. We may now compose can with the map on homotopy colimits induced from λ and σ to get a map

THH(L^{(m)}; X)_{k} ∧ THH(L^{(n)}; Y)_{k} → THH(L^{(m+n)}; X ∧ Y)_{k}.

These maps are Σ_{m}×Σ_{n}-equivariant and form, for varying k, a cyclic map. Accordingly, we get a G×Σ_{m}×Σ_{n}- equivariant map

μ_{m,n}: THH(L^{(m)}; X) ∧ THH(L^{(n)}; Y) → THH(L^{(m+n)}; X ∧ Y)

upon realization. If we let X = S^{m} and Y = S^{n} we obtain the required product map. The unit map is given as the composition

1_{n}: S^{n} → F (S^{0} ∧ … ∧ S^{0}, L(S^{0} ∧ … ∧ S^{0}) ∧ S^{n}) → THH(L^{(n)}; S^{n}) → THH(L^{(n)}; S^{n})

where the first map is given by smashing with the unit map in L, the second is the canonical inclusion in the homotopy colimit and the last is the inclusion of the zero skeleton. We leave it to the reader to verify that the maps 1_{n} and μ_{m,n} in fact make the spaces THH(L^{(n)}; S^{n}) a commutative FSP defined on spheres.

The spaces THH(L^{(n)}; S^{n}) again form a commutative FSP defined on spheres and the Frobenius maps F_{r} are multiplicative. Indeed, the multiplication maps μ_{m,n} are G-equivariant and the unit maps 1_{n} factor through the inclusion of the G-fixed set. Finally, we consider the restriction map which, we remember, is defined as the composite

R_{r}: | THH(L^{(n)}; S^{n}).|^{C_{rs}} ⟶^{D_{rs}^{−1}} | sd_{C_{rs}} THH(L^{(n)}; S^{n}).^{C_{rs}}|

⟶^{r′_{Cr}} | sd_{C_{s}} THH(L^{(n)}; S^{n}).^{C_{s}}| ⟶^{D_{s}} | THH(L^{(n)}; S^{n}).|^{C_{s}}.

The subdivision sd_{C} μ_{m,n} defines a product on the second and third term and the naturality of the homeomorphism D makes it multiplicative. Moreover, sd_{C_{rs}} μ_{m,n} restricts to sd_{C_{s}} μ_{m,n} under r′_{C_{r}} and hence r′_{C_{s}} is multiplicative. □

Next, let L be an FSP, then the associated n × n-matrix FSP is defined by

M_{n}(L)(X) = F (n, n ∧ L(X)),

where n = {0, 1, …, n} with 0 as basepoint. In view of proposition 1.6 above we may restate [BHM] 3.9 and 4.24 as

Proposition 1.7.2. (Morita invariance) T (L) ≃_{G} T (M_{n}(L)). □

For an FSP L and V ⊂ U^{G}, we define the underlying spectrum L^{S} of L by

L^{S}(V) = holim_{\overrightarrow{I}} F (S^{i}, L(S^{i}) ∧ S^{V}).

Lemma 1.7. Suppose f: L_{1} → L_{2} is a natural transformation such that f^{S} is an equivalence of spectra. Then T (f): T (L_{1}) → T (L_{2}) is a G-equivalence. □

2. Witt vectors

2.1. Let A be a commutative ring and let p be a fixed prime. The associated ring W (A) of (p-typical) Witt vectors will play an important role in the sequel, and we briefly recall its definition, refering to [Se], [I] and Bergman’s lecture in [Mu] for details. The underlying set W (A) = A^{ℕ_{0}}; the infinite product. The ring structure is specified by the requirement that the ghost map

w: W (A) → A^{ℕ_{0}},

given by the Witt polynomials,

(2.1.1)  \begin{aligned}w_{0} & = a_{0} \\ w_{1} & = a_{0}^{p} + pa_{1} \\ w_{2} & = a_{0}^{p^{2}} + pa_{1}^{p} + p^{2}a_{2} \\ & : \end{aligned}

be a natural transformation of functors from rings to rings. More concretely,

a + b = (s_{0}(a, b), s_{1}(a, b), …)
a · b = (p_{0}(a, b), p_{1}(a, b), …)

for certain integral polynomials s_{i} and p_{i} which depend only on (a_{0}, …, a_{i}). The integrality follows from the Kummer congruences

x^{p^{n}} ≡ x^{p^{n−1}}(mod p^{n}), x ∈ ℤ.

Hence W (A) is well-defined for any ring. The a_{i} are called the Witt coordinates of the Witt vector a = (a_{0}, a_{1}, …) and the w_{i} are called the ghost or phantom coordinates. The element 1 = (1, 0, …) ∈ W (A) is the unit.

There are operators

(2.1.2)

F: W (A) → W (A),  (Frobenius homomorphism)
V: W (A) → W (A),  (Verschiebung map)
ω: A → W (A),  (Teichmüller character)

characterized by the formulas

\begin{aligned}F (w_{0}, w_{1}, …) & = (w_{1}, w_{2}, …) \\ V (a_{0}, a_{1}, …) & = (0, a_{0}, a_{1}, …) \\ ω(x) & = (x, 0, 0, …). \end{aligned}

Any relation which holds true in ghost coordinates also holds in W (A). This is obvious for a ℤ[1/p]- algebra since the ghost map is a bijection. In general it follows from the functoriality of W: every algebra is the quotient of a p-torsion free algebra which embeds in a ℤ[1/p]-algebra. It follows that F is a ring homomorphims, that V is additive, that ω is multiplicative, and that we have the relations

(2.1.3)  x · V (y) = V (F (x) · y), F V = p, V F = mult_{V (1)} .

Moreover, when A is an 𝔽_{p}-algebra, V (1) = p and F = W (φ) where φ is the Frobenius endomorphism of A, F (a_{0}, a_{1}, …) = (a_{0}^{p}, a_{1}^{p}, …). For any a ∈ W (A),

(2.1.4)  a = ∑_{i=0}^{∞} V^{i}(ω(a_{i})),

where the a_{i} are the Witt coordinates of a.

The additive subgroup V^{n}W (A) of W (A) is an ideal by (2.1.3) whose quotient

W_{n}(A) = W (A)/V^{n}W (A)

is the ring of Witt vectors of lenght n in A. The elements in W_{n}(A) are in 1-1 correspondance with tuples (a_{0}, … a_{n−1}) with addition and multiplication given by the polynomials s_{i} and p_{i}. Hence W (A) is the inverse limit of the W_{n}(A) over the restriction maps

R: W_{n}(A) → W_{n−1}(A), R(a_{0}, …, a_{n−1}) = (a_{0}, …, a_{n−2}).

It follows that W (A) is complete and separated in the topology defined by the ideals V^{n}W (A), n ≥ 1.

Theorem 2.1. (Witt) If k is a perfect field of positive charactesitic p then W (k) is a complete discrete valuation ring with residue field k and uniformizing element p. In particular, W (𝔽_{p}) = ℤ_{p}.

Proof. We have already seen that the ideals V^{n}W (k) define a complete and separated topology on W (k) and that W (k)/V W (k) = k. Therefore, it suffices to show that V^{n}W (k) is generated by p^{n}. Now by (2.1.3)

p^{n} · W (k) = V (1)^{n} · W (k) = V^{n}(F^{n}(W (k))),

and since F = W (φ) is invertible the statement follows. □

We shall also need the ring of big Witt vectors W(A). Its underlying set is A^{ℕ} but its n’th ghost coordinate is w_{n} = ∑_{d|n} da_{d}^{n/d} and again one requires that w: W(A) → A^{ℕ} be a natural transformation of rings. As an abelian group, W(A) may be identified with the multiplicative group of power series with constant term 1. The isomorphism is given by

(2.1.5)  ψ: W(A) ⟶^{≅} (1 + XA[[X]])^{×}, ψ(a_{1}, a_{2}, a_{3}, …)(X) = ∏_{i=1}^{∞}(1 − a_{i}X^{i}).

We call the a_{i} (resp. the w_{i}) the Witt coordinates (resp. the ghost coordinates) of a Witt vector. Again there are Frobenius and Verschiebung operators, one for each n ≥ 0, defined by

(2.1.6)

F_{n}(w_{m}) = w_{nm}
V_{n}(a_{m}) = \{ \begin{matrix}a_{m/n}, & if n|m \\ 0, & otherwise. \end{matrix}

We note that under the isomorphism (2.1.5),

F_{n}(P (X)) = ∏_{i=1}^{n} P (ξ_{i}), V_{n}(P (X)) = P (X^{n}),

where ξ_{1}, …, ξ_{n} are the formal n’th roots of X. The formulas

(2.1.7)  \begin{matrix}F_{r}(xy) = F_{r}(x)F_{r}(y) \\ V_{r}(F_{r}(x)y) = xV_{r}(y) \\ F_{r}V_{r} = r, V_{r}F_{r} = V_{r}(1) \\ F_{r}V_{s} = V_{s}F_{r}, if (r, s)=1. \end{matrix}

are easily verified in ghost coordinates.

We call a subset S ⊂ ℕ a truncation set if it is stable under division. Since w_{n} only involves the a_{d} where d|n, we may replace ℕ above by any truncation set S to obtain a ring W_{S}(A) with underlying set A^{S}. If S ⊂ S′ are two truncation sets then the obvious projection W_{S′}(A) → W_{S}(A) is a ring homomorphism. One can use (2.1.6) to define

F_{n}: W_{S}(A) → W_{S/n}(A), V_{n}: W_{S/n}(A) → W_{S}(A).

We note that W (A) = W_{{1,p,p^{2},... }}(A) such that F = F_{p} and V = V_{p}. Moreover, if ⟨n⟩ = {d | d divides n} then W_{s+1}(A) = W_{⟨p^{s}⟩}(A).

2.2. In section 2.3 below we relate Witt vectors to π_{0}T (A)^{C_{n}} but first we recall some notions from abstract induction theory, cf. [D] and [tD]. We shall only need this when G is the circle group, but in this section G may be any compact Lie group.

We let Or(G; F) denote the category of canonical orbits G/H with H finite, and all G-maps. Let M be an abelian group valued bifunctor on Or(G; F), i.e. M = (M^{∗}, M_{∗}) is a pair of functors from Or(G; F) to abelian groups with M^{∗} contravariant and M_{∗} covariant, and M^{∗}(G/H) = M_{∗}(G/H) for all H. M is called a Mackey functor if i_{∗}i^{∗} = id for any isomorphism i: G/H → G/H and if the double coset formula holds: if H, H′ ⊂ K and K = ∐_{i} Hx_{i}H′ then

(2.2.1)  (π_{H}^{K})^{∗} ∘ (π_{H′}^{K})_{∗} = ∑_{i}(π_{H∩(x_{i}H′x_{i}^{−1})}^{H})_{∗} ∘ (π_{H∩x_{i}H′x_{i}^{−1}}^{x_{i}H′x_{i}^{−1}})^{∗} ∘ r_{x_{i}}^{∗},

where π_{H}^{K}: G/H → G/K is the projection and r_{x}: G/xHx^{−1} → G/H is right multiplication by x.

A Green functor is a Mackey functor M for which M(G/H) is a ring, and such that for all f: G/H → G/K, f^{∗} is a ring homomorphism and f_{∗} is a map of M(G/K)-bimodules when M(G/H) is considered an M(G/K)- bimodule via f^{∗}, i.e.

f_{∗}(xf^{∗}(y)) = f_{∗}(x)y, f_{∗}(f^{∗}(y)x) = yf_{∗}(x),

for any x ∈ M(G/H) and y ∈ M(G/K).

Now, suppose T is a G-spectrum indexed on a complete G-universe U. For H ⊂ G the fixed point spectrum is given by

T^{H} ≅ F (G/H_{+}, T)^{G}.

A G-map f: G/H → G/K induces a map f^{∗}: T^{K} → T^{H}. If H and K are finite, one has the equivariant transfer f^{!}: Σ_{G}^{∞}G/K_{+} → Σ_{G}^{∞}G/H_{+}. The map f^{!} depends on choosing a G-embedding of G/H into some V ⊂ U; one gets f^{!}: S^{V} ∧ G/K_{+} → S^{V} ∧ G/H_{+} and defines f_{∗} to be the composite

T^{H} ≅ F (S^{V} ∧ G/K_{+}, Σ^{V} T)^{G} ⟶^{(f^{!})^{∗}} F (S^{V} ∧ G/K_{+}, Σ^{V} T) ≅ T^{K}.

The homotopy class of f_{∗} is independent of the choice of embedding. Now the statements of [LMS], IV 6.3, 5.6. and 5.8. easily translate to the following

Proposition 2.2. Let G be a compact Lie group and let T be a G-spectrum indexed on a complete Guniverse. The functor which to G/H assigns π_{∗}(T^{H}) and to f: G/H → G/K assigns the homomorphisms f^{∗} and f_{∗} is a Mackey functor on Or(G; F). If T is a G-ring spectrum then this becomes a Green functor.If H ⊂ K and π_{H}^{K}: G/H → G/K is the canonical projection then the composite (π_{H}^{K})_{∗} ∘ (π_{H}^{K})^{∗} is multiplication by (π_{H}^{K})_{∗}(1). □

Let H ⊂ G be a finite subgroup. In section 1.1 we used the norm map N: T_{hH} → T^{H}. We can include H in EH as an orbit to get a map

ι_{H}: T = T ∧_{H} H_{+} → T ∧_{H} EH_{+} = T_{hH}.

Later in the paper we need the following

Lemma 2.2. Let T be as above and let π_{H}: G → G/H be the projection. Then the diagram

T ⟶^{π_{H∗}} T^{H}
↓^{ι_{H}}  ∥
T_{hH} ⟶^{N} T^{H}

is homotopy commutative.

Proof. We consider the diagram

T ∧_{H} H_{+} ⟶^{≃} (T ∧ H_{+})^{H} ⟶^{c} T^{H}
↓  ↓  ∥
T ∧_{H} EH_{+} ⟶^{≃} (T ∧ EH_{+})^{H} ⟶^{c} T^{H}

where the equivalences on the left are as in the proof of proposition 1.1 and where the maps c collapses H (resp. EH) to a point. The left hand square homotopy commutes and the right hand square is strictly commutative. We claim that the following diagram homotopy commutes

F (G_{+}, T)^{G} ⟶^{π_{H∗}} F (G/H_{+}, T)^{G}
∥  ∥
F (H_{+}, T)^{H} ⟶^{π′_{H∗}} F (H/H_{+}, T)^{H}
↓^{≃}  ↓^{≃}
(T ∧ H_{+})^{H} ⟶^{c} (T ∧ H/H_{+})^{H}.

For the lower square this follows from [LMS] V.9.7. For the upper square note that we have a pullback

G ⟶^{π_{H}} G/H
↑^{incl}  ↑^{incl}
H ⟶^{π′_{H}} H/H.

Therefore, the corresponding square of transfers, and hence the upper square homotopy commutes. □

2.3. We apply the general theory discussed above to the topological Hochschild spectrum T (A). Let π_{r}^{rs}: G/C_{s} → G/C_{rs} be the projection, s ≥ 1. We have the maps, with V_{r} only well-defined up to homotopy:

(2.3.1)  \begin{aligned}F_{r} & = (π_{r}^{rs})^{∗}: T (A)^{C_{rs}} → T (A)^{C_{s}}, \\ V_{r} & = (π_{r}^{rs})_{∗}: T (A)^{C_{s}} → T (A)^{C_{rs}}. \end{aligned}

They are called the r’th Frobenius and the r’th Verschiebung, respectively. We shall write F (resp. V) instead of F_{p} (resp. V_{p}) when the subgroups considered are p-groups. We note that F_{r} is just the obvious inclusion map T (A)^{C_{rs}} → T (A)^{C_{s}}. Recall from (1.2.1) the restriction maps R_{r}: T (A)^{C_{rs}} → T (A)^{C_{s}}. On homotopy groups we have

Lemma 2.3.1. For any commutative ring A the following relations hold on π_{∗}(T (A)^{C.}):

(1) F_{r}(xy) = F_{r}(x)F_{r}(y)
(2) V_{r}(F_{r}(x)y) = xV_{r}(y)
(3) F_{r}V_{r} = r, V_{r}F_{r} = V_{r}(1)
(4) F_{r}V_{s} = V_{s}F_{r}, if (r, s)=1
(5) R_{r}F_{s} = F_{s}R_{r}, R_{r}V_{s} = V_{s}R_{r}

Proof. Relations (1), (2), (3) and (4) follow from proposition 2.2 since T (A) is a G-ring spectrum when A is commutative. For example, the double coset formula (2.2.1) shows that

F_{r}V_{r} = 1 + t + t^{2} + … + t^{r−1},

where t ∈ C_{rs} is any generator. But the C_{rs}-action extends to the circle G and is therefore trivial on homotopy groups, so F_{r}V_{r} = r. Finally, (5) is an immediate consequence of the fact that R_{r}: ρ_{C_{rs}}^{#}T (A)^{C_{rs}} → ρ_{C_{s}}^{#}T (A)^{C_{s}} is G-equivariant. □

Proposition 2.3. For any commutative ring A the sequence

0 → π_{0}T (A) ⟶^{V^{n}} π_{0}T (A)^{C_{p^{n}}} ⟶^{R} π_{0}T (A)^{C_{p^{n−1}}} → 0

is exact.

Proof. The fundamental cofibration sequence of theorem 1.2 gives a long exact sequence of homotopy groups

⋯ → π_{1}T (A)^{C_{p^{n−1}}} ⟶^{∂} π_{0}T (A)_{hC_{p^{n}}} ⟶^{N} π_{0}T (A)^{C_{p^{n}}} ⟶^{R} π_{0}T (A)^{C_{p^{n−1}}} → 0.

We claim that the map ι_{C_{p^{n}}}: T (A) → T (A)_{hC_{p^{n}}} induces an isomorphism on π_{0}(−). Indeed, the skeleton filtration of EC_{p^{n}} gives rise to a first quadrant spectral sequence

E^{2} = H_{∗}(C_{p^{n}}; π_{∗}T (A)) ⇒ π_{∗}T (A)_{hC_{p^{n}}}

whose edge homomorphism is induced by ι_{C_{p^{n}}}. Since T (A) is a connective spectrum the claim follows. Moreover, lemma 2.2 shows that V^{n} = N ∘ ι_{C_{p^{n}}}.

It remains to show that V^{n}: π_{0}T (A) → π_{0}T (A)^{C_{p^{n}}} is injective. Since F^{n}V^{n} = p^{n} by lemma 2.2 (3), we are done if A has no p-torsion. To treat the general case suppose that A → Ā is a surjection of rings and that A has no p-torsion. We consider the diagram

π_{1}T (A)_{h_{C_{p^{n}}}} ⟶^{N} π_{1}T (A)^{C_{p^{n}}} ⟶^{R} π_{1}T (A)^{C_{p^{n−1}}} ⟶^{∂} 0
↓  ↓  ↓
π_{1}T (Ā)_{h_{C_{p^{n}}}} ⟶^{N} π_{1}T (Ā)^{C_{p^{n}}} ⟶^{R̄} π_{1}T (Ā)^{C_{p^{n−1}}}

in which the rows are exact. We prove by induction on n that the vertical maps are surjective. For any ring,

π_{i}T (A) ≅ HH_{i}(A), i = 0, 1.

Therefore, the spectral sequence of the skeleton filtration gives an exact sequence

(2.3.2)  HH_{1}(A) ⟶^{ι} π_{1}T (A)_{hC_{p^{n}}} → A/p^{n}A → 0.

For a commutative ring, HH_{1}(A) = Ω_{A/ℤ} and taking Kähler differentials preserves surjections. □

The proposition shows that there is a set bijection π_{0}T (A)^{C_{p^{n}}} ≅ A^{n+1}. We proceed to define a preferred bijection. Consider for any finite subgroup C_{r} ⊂ G the diagonal map (notation as in 1.4)

Δ_{r}: THH(A)_{0} ⟶^{∂} (sd_{C_{r}} THH(A)_{0})^{C_{r}} → |(sd_{C_{r}} THH(A).)^{C_{r}}| ⟶^{D} | THH(A).^{C_{r}}|.

The first map is given by f ↦ f ∧ … ∧f (r factors), the second is the inclusion of the zero-skeleton and D is the homeomorphism from 1.4.

Lemma 2.3.2. The composition R_{r} ∘ Δ_{r} and F_{r} ∘ Δ_{r} are equal to the inclusion of the zero-skeleton, i: THH(A)_{0} → THH(A) and the r’th power endomorphism of the topological monoid THH(A)_{0} followed by i, respectively.

Proof. The claim for R_{r} ∘ Δ_{r} is obvious from the definitions, cf. 1.5. To prove the claim for F_{r} ∘ Δ_{r} recall that for any simplicial space Z. the homeomorphism D: | sd_{C_{r}} Z.| → |Z.| is homotopic to the realization of the simplicial map which in degree k is

d_{0}^{(k+1)(r−1)}: Z_{(k+1)r−1} → Z_{k}.

This follows from the proof of [BHM] proposition 2.5. But the composite

THH(A)_{0} ⟶^{∂} sd_{C_{r}} THH(A)_{0} = THH(A)_{r−1} ⟶^{d_{0}^{r−1}} THH(A)_{0}

is precisely the r’th power endomorphism. □

Theorem 2.3. Let A be a commutative ring. Then there is natural isomorphism of rings

I: W_{n+1}(A) → π_{0}T (A)^{C_{p^{n}}}

such that RI = IR, F I = IF and V I = IV.

Proof. The inclusion of the zero-skeleton π_{0} THH(A)_{0} ≅ π_{0} THH(A) is an isomorphism because A is commutative and both groups are copies of A. Hence by the lemma,

(2.3.3)  R_{r} ∘ Δ_{r} = id, F_{r} ∘ Δ_{r} = r.

Now an easy induction argument based on proposition 2.3 shows that the sequence

0 → π_{0}T (A)^{C_{p^{n−1}}} ⟶^{V} π_{0}T (A)^{C_{p^{n}}} ⟶^{R^{n}} π_{0}T (A) → 0

is exact, and since Δ_{p^{n}} gives a natural splitting of R^{n} (as a set map), we may define a bijection

(2.3.4)  I: W_{n+1}(A) → π_{0}T (A)^{C_{p^{n}}} , I(a_{0}, …, a_{n}) = ∑_{i=0}^{n} V^{i}(Δ_{p^{n−i}}(a_{i})).

As an immediate consequence of (2.3.3) we have that RI = IR, F I = IF and V I = IV. In particular, if we define

w̄: π_{0}T (A)^{C_{p^{n}}} → ∏_{i=0}^{n} A

by w̄_{i} = R^{i}F^{n−i}, then w̄ ∘I = w. It remains to be seen that I is a ring homomorphism. If A has no p-torsion this is obvious because w is injective. In the general case, suppose A → Ā is a surjection of rings where A is without p-torsion and consider the diagram

W_{n+1}(A) ⟶^{I} π_{0}T (A)^{C_{p^{n}}}
↓  ↓
W_{n+1}(Ā) ⟶^{I} π_{0}T (Ā)^{C_{p^{n}}} .

The vertical maps are both surjective and the upper horizontal map is a ring homomorphism. Hence so is the lower horizontal map. □

Recall from proposition 1.3 that ρ_{C_{r}}^{#}T (A)^{C_{r}} is p-cyclotomic if p does not divide r. In analogy with proposition 2.3 we have short exact sequences

0 → π_{0}T (A)^{C_{r}} ⟶^{V_{p^{s}}} π_{0}T (A)^{C_{p^{s}r}} ⟶^{R_{p}} π_{0}T (A)^{C_{p^{s−1}r}} → 0

and induction on the prime divisors of n gives us a natural bijection

I: W_{⟨n⟩}(A) → π_{0}T (A)^{C_{n}}, I(a_{d}|d divides n) = ∑_{d|n} V_{d}(Δ_{n/d}(a_{n/d})).

We can argue as above to get

Addendum 2.3. Let A be a commutative ring. Then

I: W_{⟨n⟩}(A) → π_{0}T (A)^{C_{n}},

is a natural isomorphism of rings such that R_{r}I = IR_{r}, F_{r}I = IF_{r} and V_{r}I = IV_{r}, where ⟨n⟩ denotes the truncation set of natural numbers which divides n.

3. Topological Cyclic Homology

3.1. This section is inspired by T. Goodwillie’s paper [G1].

Let 𝕀 be the category where objects are the natural numbers, ob 𝕀 = {1, 2, 3, … }, and with two morphisms R_{r}, F_{r}: n → m, whenever n = rm, subject to the relations

R_{1} = F_{1} = id_{n}
R_{r}R_{s} = R_{rs}, F_{r}F_{s} = F_{rs}
R_{r}F_{s} = F_{s}R_{r}.

For a prime p, we let 𝕀_{p} be the full subcategory with ob 𝕀_{p} = {1, p, p^{2}, … }. A cyclotomic spectrum T defines a functor from 𝕀 to the category of non-equivariant spectra. Indeed when n = rm we have two maps of non-equivariant spectra

R_{r}, F_{r}: T^{C_{n}} → T^{C_{m}}.

The map R_{r} was defined in (1.2.1) and F_{r} is the inclusion of fixed points spectra. The relations above are a consequence of the compatibility condition in definition 1.2.

Topological cyclic homology at p, denoted TC(T; p), was defined in [BHM]. In the present formulation it is the homotopy limit of the restriction of the functor defined above to 𝕀_{p}.

Definition 3.1. If T is a cyclotomic spectrum, then

TC(T; p) = holim_{\overleftarrow{𝕀_{p}}} T^{C_{p^{s}}} , TC(T) = holim_{\overleftarrow{𝕀}} T^{C_{n}}.

For a functor with smash product L, TC(L) = TC(T (L)) and similarly for TC(L; p).

Remarks. i) The homotopy limit which defines TC(T; p) may be formed in two steps. First we can take the homotopy limit over F_{p} (resp. R_{p}). Since R_{p} and F_{p} commute, R_{p} (resp. F_{p}) induces a self-map of this homotopy limit, and we may take the homotopy fixed points. More precisely, let

(3.1.1)  TR(T; p) = holim_{\overleftarrow{R_{p}}} T^{C_{p^{s}}} , TF(T; p) = holim_{\overleftarrow{F_{p}}} T^{C_{p^{s}}} ,

then F_{p} induces an endomorphism of TR(T; p) and R_{p} an endomorphism of TF(T; p) and

TC(T; p) ≅ TR(T; p)^{h⟨F_{p}⟩} ≅ TF(T; p)^{h⟨R_{p}⟩}.

Here ⟨F_{p}⟩ is the free monoid on F_{p} and X^{h⟨F_{p}⟩} denotes the ⟨F_{p}⟩-homotopy fixed points of X. It is naturally equivalent to homotopy fiber the of id −D, which was the definition used for TC(T; p) in [BHM].

There is a similar description of TC(T). Let

(3.1.2)  TR(T) = holim_{\overleftarrow{R}} T^{C_{n}}, TF(T) = holim_{\overleftarrow{F}} T^{C_{n}},

then

TC(T) = TR(T)^{hF} = TF(T)^{hR},

where the decoration hF denotes the homotopy fixed set of the multiplicative monoid of natural numbers acting on TR(T) through the maps F_{s}, s ≥ 1. ii) The inclusion 𝕀_{p} ⊂ 𝕀 induces a map TC(T) → TC(T; p) which is a (spacewise) fibration. Similarly the inclusions {1} ⊂ 𝕀_{p} induce fibrations TC(T, p) → T. In section 3.5 below we prove:

Theorem 3.1. The projections TC(T) → TC(T; p) induce an equivalence of TC(T) with the fiber product of the TC(T; p)’s over T. Moreover, the functors agree after p-completion, TC(T)p^{∧} ≃ TC(T; p)p^{∧}.

3.2. We evaluate the realizations of the index categories 𝕀_{p} and 𝕀:

(3.2.1)  | 𝕀_{p}| ≃ S^{1}, | 𝕀 | ≅ ∏′_{p} | 𝕀_{p}| ≃ ∏′_{p} S^{1},

where ∏′ denotes the weak product over the prime numbers. Indeed, the full subcategory 𝕀_{p,1} ⊂ 𝕀_{p} whose objects are {1, p} has realization | 𝕀_{p,1}| ≅ S^{1}, and by theorem A of [Q] the inclusion functor K: 𝕀_{p,1} → 𝕀_{p} is a homotopy equivalence provided that the under-categories (p^{n}) are contractible for all p^{n} ∈ ob 𝕀_{p}. If we write R^{r}F^{s} for the object (p^{ϵ}, R^{r}F^{s}: p^{n} → p^{ϵ}) in (p^{n}), then (p^{n}) is the category

R^{n} ← R^{n−1} → R^{n−1}D ← R^{n−2}D → … ← F^{n−1} → F^{n}.

Its realization is |(p^{n})| ≅ [0, 2n] which is contractible.

Let S = {p_{1}, …, p_{s}} be a finite set of primes and let 𝕀_{S} be the full subcategory of 𝕀 whose objects are the numbers p_{1}^{n_{1}} … p_{s}^{n_{s}}, n_{i} ≥ 0. Then as categories 𝕀 ≅ \underrightarrow{lim} 𝕀_{S} and 𝕀_{S} ≅ 𝕀_{p_{1}} × … × 𝕀_{p_{s}}. Since realization commutes with colimits and finite products we obtain (3.2.1).

3.3. Let Σ^{−1}ℚ/ℤ be a Moore spectrum with integral homology ℚ/ℤ concentrated in degree −1, and let T be any spectrum. Then the profinite completion of T is the function spectrum

T^{∧} = F (Σ^{−1}ℚ/ℤ, T).

We may replace ℚ/ℤ by its p-part ℚ/ℤ_{(p)} to obtain the p-completion Tp^{∧}. Since ℚ/ℤ is the direct sum over the primes p of its p-parts, the profinite completion T^{∧} is the product of the p-completions Tp^{∧}. One proves immediately that the homotopy groups of T^{∧} are given by the exact sequence

0 → Ext(ℚ/ℤ, π_{s}T) → π_{s}(T^{∧}) → Hom(ℚ/ℤ, π_{s−1}T) → 0.

Let T be G-spectrum indexed on a trivial G-universe and consider the homotopy orbit spectrum T_{hC_{p^{n}}} = T ∧_{C_{p^{n}}} EC_{p^{n}+}. There are transfer maps

t_{n}^{m}: T_{hC_{p^{n}}} → T_{hC_{p^{m}}} , n > m

associated to the projections T_{hC_{p^{m}}} → T_{hC_{p^{n}}}, cf. [LMS, p.186], and we have the following key lemma.

Lemma 3.3. Suppose T is a bounded below G-spectrum. Then the homotopy fiber of t_{n}^{m} is a p-complete spectrum; in particular it is profinite complete.

Proof. We can assume that m = 0, and we will write t_{n} = t_{n}^{0}. Let 𝔅_{p} denote the Serre class of abelian p-groups A which are annihilated by some N_{A} > 0. If we can prove that t_{n∗}: π_{∗}T_{hC_{p^{n}}} → π_{∗}T is an isomorphism modulo 𝔅_{p}, then the homotopy fiber of t_{n} will have homotopy groups in 𝔅_{p}, and therefore it will be p-complete by [BK], p.166.

The composition π_{∗}T ⟶^{pr_{∗}} π_{∗}T_{hC_{p^{n}}} ⟶^{t_{n∗}} π_{∗}T is multiplication p^{n} and therefore an isomorphism modulo 𝔅_{p}. Hence we may as well show, that pr_{∗} is an isomorphism modulo 𝔅_{p}. We have the right halfplane homology type spectral sequence (see §4.2 below)

E_{s,t}^{2} = H_{s}(C_{p^{n}}; π_{t}T) ⇒ π_{s+t}T_{hC_{p^{n}}} .

Since the C_{p^{n}}-action on T is the restriction of the G-action, π_{t}T is a trivial C_{p^{n}}-module and therefore

E_{0,t}^{2} = π_{t}T, E_{s,t}^{2} ∈ 𝔅_{p}, s > 0.

Furthermore the edge homomorphism E_{0,t}^{2} → E_{0,t}^{∞} is the surjection by pr_{∗} of π_{t}T onto its image in π_{t}T_{hC_{p^{n}}}. It follows that the edge homomorphism is an isomorphism modulo 𝔅_{p} and that E_{s,t}^{∞} ∈ 𝔅_{p} when s > 0. Since T is bounded below, pr_{∗} is an isomorphism modulo 𝔅_{p}. □

3.4. Let K: I → J be a functor and ℂ a category which have all limits; then the forgetful functor K^{∗}: ℂ^{J} → ℂ^{I} has a leftadjoint R. If T: I → ℂ is a functor, then the right Kan extension of T along K is the functor

RT (j) = \underleftarrow{lim}((j) ⟶^{pr_{1}} I ⟶^{T} ℂ),

cf. [ML]. We apply this to the inclusion K: 𝕀_{1} → 𝕀 of the full subcategory on {1}. The under-category (n) is the discrete category on the set of morphism 𝕀(n, 1) and a functor from 𝕀_{1} to spectra is just a spectrum T. Thus the right Kan extension is simply a product of copies of T,

RT (n) = F (𝕀(n, 1)_{+}, T),

where 𝕀(n, 1) = {R_{n/d}F_{d}|d divides n}.

If n = p_{1}^{n_{1}} … p_{s}^{n_{s}} then #𝕀(n, 1) = (n_{1} + 1) + … + (n_{s} + 1).

Lemma 3.4. holim_{\overleftarrow{𝕀}} RT ≅ F (|(𝕀)|_{+}, T) ≃ T.

Proof. Let 𝕊 denote the categoty of spectra. We recall from [BK] that holim_{\overleftarrow{𝕀}} is right adjoint to the functor

− ∧ |(𝕀)|_{+}: 𝕊 → 𝕊^{𝕀}

which takes a spectrum T to the diagram n ↦ T ∧ |(𝕀)|_{+}. We have the commutative diagram of functors

𝕊 ⟶^{−∧|(𝕀)|_{+}} 𝕊
−∧|(𝕀)|_{+}↓  ∥
𝕊^{𝕀} ⟶^{K^{∗}} 𝕊^{𝕀_{1}}.

All the functors in the square have right adjoints and accordingly these also commute; this proves the first claim. Finally (1, id: 1 → 1) is terminal object in (𝕀), which therefore has contractible realization |(𝕀)|. □

3.5. From now on T will be a cyclotomic spectrum, e.g. T = T (L). The counit of the adjunction above supplies a map of 𝕀-diagrams ϵ: T^{C_{−}} → RT (−) such that

ϵ_{n}: T^{C_{n}} → F (𝕀(n, 1)_{+}, T)

is the adjoint of the ‘evaluation’ map 𝕀(n, 1)_{+} ∧ T^{C_{n}} → T.

Lemma 3.5. The homotopy fiber of ϵ_{n} is a profinitely complete spectrum.

Proof. Suppose first that n = p^{s} is a prime power. By induction it is enough to show that the iterated homotopy fiber, i.e. the homotopy fiber of the induced map on homotopy fibers, of the square

T^{C_{p^{s}}}  ⟶^{R_{p}} T^{C_{p^{s−1}}}
F_{p}↓  ^{F_{p}}↓
T^{C_{p^{s−1}}} ⟶^{R_{p}} T^{C_{p^{s−2}}}

is profinitely complete. We recall from the theorem 1.2 the cofibration sequence

T_{hC_{p^{s}}} → T^{C_{p^{s}}} ⟶^{R_{p}} T^{C_{p^{s−1}}} .

It determines the horizontal homotopy fibers in the square above. Furthermore the map induced by the vertical arrows F_{p} precisely correspond to the transfer map t_{s}^{s−1}, and so lemma 3.3 shows that the iterated homotopy fiber is a p-complete spectrum.

Next we consider the general case and write n = p^{s}k with (k, p) = 1. Then T^{C_{k}} is a p-cyclotomic spectrum by proposition 1.3 and the lemma follows by induction over the prime divisors in n. □

We let Φ(T) denote the fiber of the fibration TC(T) → T; similarly for Φ(T; p).

Corollary 3.5. Φ(T) is profinitely complete, and Φ(T; p) is p-complete.

Proof. Homotopy limits commute with profinite completion, so by the lemma the homotopy fiber of

ϵ_{∗}: holim_{\overleftarrow{𝕀}} T^{C_{n}} → holim_{\overleftarrow{𝕀}} RT

is a profinitely complete spectrum. Finally under the equivalence of lemma 3.4 we can identify ϵ_{∗} with the projection TC(T) → T. □

Proof of theorem 3.1. We first show that TC(T)p^{∧} ≃ TC(T; p)p^{∧} via the projection. To this end we define a new full subcategory 𝕀_{p′} of 𝕀. It has as objects the set {k|(k, p) = 1} of positive integers prime to p. Then 𝕀 ≅ 𝕀_{p} × 𝕀_{p′} so

TC(T) = holim_{\overleftarrow{𝕀}} T^{C_{n}} ≅ holim_{\overleftarrow{𝕀_{p}}}(holim_{\overleftarrow{𝕀_{p′}}} T^{C_{p^{s}k}}).

We may proceed as in lemma 3.5 and show that the fiber spectrum of the projection

holim_{\overleftarrow{𝕀_{p′}}} T^{C_{p^{s}k}} → T^{C_{p^{s}}}

vanishes after p-completion. This proves the last claim in proposition 3.1. We have left to show that the map from TC(T) to the fiber product over T of TC(T; p), indexed by the primes p, is an equivalence. This is the same as to show that Φ(T) → ∏_{p} Φ(T; p) is a homotopy equivalence. Now a profinitely complete spectrum Φ(T) is equivalent to the product of its p-completions, with p varying over the primes. Since Φ(T)p^{∧} ≃ Φ(T; p)p^{∧} by the above and Φ(T; p)p^{∧} ≃ Φ(T; p) by corollary 3.5, we are done. □

3.6. We recall from 1.7 that if L is a commutative FSP defined on spheres, then the Σ_{n}-spaces THH(L^{(n)}; S^{n}) again form a commutative FSP defined on spheres. The same holds for the C-fixed sets THH(L^{(n)}; S^{n})^{C} and the restriction and Frobenius maps are Σ_{n}-equivariant and multiplicative. In particular, the homotopy limit

TC(L^{(n)}; S^{n}) = holim_{\overleftarrow{𝕀}} THH(L^{(n)}; S^{n})^{C_{r}}

carries a Σ_{n}-action.

Proposition 3.6. Let L be a commutative FSP defined on spheres. Then the spaces TC(L^{(n)}; S^{n}) again form a commutative FSP defined on spheres. The associated spectrum is equivalent to TC(L).

Proof. In view of proposition 1.7 it is enough to prove that a homotopy limit of commutative FSP’s defined on spheres is again a commutative FSP defined on spheres. So let L_{i} be a J-diagram of FSP’s defined on spheres. We define the product on the homotopy limit by

μ_{m,n}: holim_{\overleftarrow{J}} L_{j_{1}}(S^{m}) ∧ holim_{\overleftarrow{J}} L_{j_{2}}(S^{n}) ⟶^{can} holim_{\overleftarrow{J×J}} L_{j_{1}}(S^{m}) ∧ L_{j_{2}}(S^{n})

⟶^{Δ^{∗}} holim_{\overleftarrow{J}} L_{j}(S^{m}) ∧ L_{j}(S^{n}) ⟶ holim_{\overleftarrow{J}} L_{j}(S^{m+n}),

where the second map is induced from the diagonal Δ: J → J × J and the last map is induced from the multiplication in L_{j}. The first map is the canonical map, defined as follows: We have the counits

ϵ_{j}: |(J ↓ j)|_{+} ∧ holim_{\overleftarrow{J}} L_{j}(S^{n}) → L_{j}(S^{n})

and since (J × J ↓ (j_{1}, j_{2})) ≅ (J ↓ j_{1}) × (J ↓ j_{2}) we get

ϵ_{j_{1}} ∧ ϵ_{j_{2}}: |(J × J ↓ (j_{1}, j_{2}))|_{+} ∧ holim_{\overleftarrow{J}} L_{j_{1}}(S^{m}) ∧ holim_{\overleftarrow{J}} L_{j_{2}}(S^{n}) → L_{j_{1}}(S^{m}) ∧ L_{j_{2}}(S^{n}).

The canonical map is the adjoint, cf. [BK]. Similarly the unit is the adjoint of the composition

|(J ↓ j)|_{+} ∧ S^{n} ⟶^{pr_{2}} S^{n} ⟶^{1_{n}} L_{j}(S^{n}).

We prove that the product is commutative and leave the remaining verifications to the reader. We have the commutative diagram

(3.6.1)  [figure: holim_{\overleftarrow{J}} L_{j_{1}}(S^{m}) ∧ holim_{\overleftarrow{J}} L_{j_{2}}(S^{n}); can; Tw; holim_{\overleftarrow{J×J}} L_{j_{1}}(S^{m}) ∧ L_{j_{2}}(S^{n}); _{Tw_{∗}}; holim_{\overleftarrow{J×J}} L_{j_{2}}(S^{n}) ∧ L_{j_{1}}(S^{m}); _{tw_{∗}}; holim_{\overleftarrow{J}} L_{j_{1}}(S^{n}) ∧ holim_{\overleftarrow{J}} L_{j_{2}}(S^{m}); can; holim_{\overleftarrow{J×J}} L_{j_{1}}(S^{n}) ∧ L_{j_{2}}(S^{m}),]

where Tw permutes the smash factores and where tw is the functor which permutes the two factores in J × J. Indeed, the adjoints of the two compositions tw^{∗} ∘ Tw_{∗} ∘ can and can ∘ Tw are equal. Now consider the diagram

holim_{\overleftarrow{J}} L_{j_{1}}(S^{m}) ∧ holim_{\overleftarrow{J}} L_{j_{2}}(S^{n}) ⟶^{Δ^{∗}∘can} holim_{\overleftarrow{J}} L_{j}(S^{m}) ∧ L_{j}(S^{n}) ⟶^{μ_{m,n∗}} holim_{\overleftarrow{J}} L_{j}(S^{m+n})
↓^{Tw}  ↓^{Tw}_{∗}  ↓^{σ_{m,n∗}}
holim_{\overleftarrow{J}} L_{j_{1}}(S^{n}) ∧ holim_{\overleftarrow{J}} L_{j_{2}}(S^{m}) ⟶^{Δ^{∗}∘can} holim_{\overleftarrow{J}} L_{j}(S^{n}) ∧ L_{j}(S^{m}) ⟶^{μ_{n,m∗}} holim_{\overleftarrow{J}} L_{j}(S^{n+m})

The commutative of the square on the left follows from (3.6.1) and the fact that Δ = Δ ∘ tw as functors from J to J × J. Finally, the commutativity of L_{j} implies that the right hand square is commutative. This completes the proof. □

Given any commutative FSP L we have from proposition 3.6 a sequence of spectra

TC(L), TC^{2}(L), TC^{3}(L), …

upon iterating the construction. In view of theorem B of the introduction and the calculation of TC(ℤ_{p}) in [BM] it would seem a very interesting question in homotopy theory to determine the iterates TC^{n}(𝔽_{p}). In particular one may wonder about the so-called chromatic filtration of TC^{2}(ℤ_{p}) or TC^{3}(𝔽_{p}).

4. Topological cyclic homology of perfect fields

4.1. To each ring R there is associated a functor with smash product, which we denote R̃. It takes a based space X to the configuration space of particles in X with labels in R, i.e. the space of formal linear combinations ∑ r_{i}x_{i} modulo the relation r · ∗ = 0 · ∗ = ∗. It is a generalized Eilenberg-MacLane space with

π_{∗}R̃(X) ≅ H̃_{∗}(X; R),

the reduced singular homology groups of X with R-coefficients.

In this paragraph we evaluate TC(R̃) in the case where R = k is a perfect field of characteristic p > 0. We note that TC(k̃) ≃ TC(k̃; p) by 3.5. For T (k̃) and its fixed sets are p-complete by theorem 1.2. In the sequel we write T (R) and TC(R) instead of T (R̃) and TC(R̃).

We begin with the basic calculation when k = 𝔽_{p} is the prime field. The general case follows by a descent argument given in section 4.5 below. The strategy for obtaining information about TC(𝔽_{p}) is to compare the fixed sets which defines it with the corresponding homotopy fixed sets.

For any C-spectrum T ∈ CSU, with C finite, there is a norm cofibration sequence of spectra, which we now recall. Following [GM] one defines

T_{hC}  = j^{∗}T ∧_{C} EC_{+}  (homotopy orbit)
T^{hC}  = F (EC_{+}, T)^{C}  (homotopy fixed points)
ℍ̂(C; T) = [ẼC ∧ F (EC_{+}, T)]^{C} (Tate spectrum)

Here j: U^{C} → U, ẼC is the unreduced suspension of EC (as in §1) and the smash product in the definition of ℍ̂ takes place in CSU, i.e.

ℍ̂(C; T)(V) = \underrightarrow{lim}_{W ⊂U}F (S^{W −V} , ẼC ∧ F (EC_{+}, T (W)))^{C}

One has

[F (EC_{+}, T) ∧ EC_{+}]^{C} ≃ [T ∧ EC_{+}]^{C} ≃ T_{hC},

cf. the proof of proposition 1.1. Thus one can smash the cofibration sequence of C-spaces

(4.1.1)  EC_{+} → S^{0} → ẼC

with F (EC_{+}, T) ∈ CSU and take C-fixed points to get the ‘norm cofibration sequence’ of [GM],

T_{hC} ⟶^{N^{h}} T^{hC} ⟶^{R^{h}} ℍ̂(C; T).

We now assume that T ∈ GSU (where, we remember G = S^{1}), and let C be a cyclic p-subgroup. In proposition 1.1 we identified [ẼC ∧ T]^{C} with (Φ^{C_{p}}T)^{C/C_{p}}. Therefore, we may smash the obvious inclusion

γ: T → F (EG_{+}, T)

to obtain a C/C_{p}-equivariant map γ̂: Φ^{C_{p}}T → ℍ̂(C_{p}; T) and taking fixed sets and using that ℍ̂(C_{p}; T)^{C/C_{p}} = ℍ̂(C; T) we obtain from (4.1.1) a cofibration diagram

(4.1.2)  \begin{matrix}T_{hC} & ⟶^{N} & T^{C} & ⟶ & (Φ^{C_{p}}T)^{C/C_{p}} \\ ∥ & & ↓^{Γ_{C}} & & ↓Γ̂_{C} \\ T_{hC} & ⟶^{N^{h}} & T^{hC} & ⟶^{R^{h}} & ℍ̂(C; T). \end{matrix}

For a cyclotomic spectrum ρ_{C_{p}}^{#}Φ^{C_{p}}T ≃_{G} T and (4.1.2) reduces to

Proposition 4.1. ([BM]) For a (p-)cyclotomic spectrum T there is a commutative diagram

T_{hC_{p^{n}}} ⟶^{N} T^{C_{p^{n}}} ⟶^{R} T^{C_{p^{n−1}}}
∥  ↓^{Γ_{n}}  ↓Γ̂_{n}
T_{hC_{p^{n}}} ⟶^{N^{h}} T^{hC_{p^{n}}} ⟶^{R^{h}} ℍ̂(C_{p^{n}}; T)

in which the rows are cofibration sequences of non-equivariant spectra. □

The point of this is that there are spectral sequences

(4.1.3)  \begin{aligned}Ê_{r,s}^{2}(C; T) & = Ĥ^{−r}(C; π_{s}T) & ⇒ π_{r+s}ℍ̂(C; T) \\ E_{r,s}^{2}(T^{hC}) & = H^{−r}(C; π_{s}T) & ⇒ π_{r+s}T^{hC} \\ E_{r,s}^{2}(T_{hC}) & = H_{r}(C; π_{s}T) & ⇒ π_{r+s}T_{hC} \end{aligned}

which in favorable cases can be used to completely calculate the homotopy exact sequence of the norm fibration sequence, cf. [BM], §2. The spectral sequences are associated with the skeleton filtration, and for Ê^{r} a filtration due to Greenlees. One may then attempt a calculation of the actual fixed points, and hence TC(T; p), starting with a calculation of

Γ̂_{1}: π_{∗}T → π_{∗}ℍ̂(C_{p}; T).

This was the strategy used in [BM] for T = T (ℤ̃_{p}) and will below be used for T = T (𝔽̃_{p}).

The spectral sequences in (4.1.3) are strongly interrelated. For any C-spectrum T there is a map of spectral sequences

(4.1.4)  R^{h,r}: E_{s,t}^{r}(T^{hC}) → Ê_{s,t}^{r}(C; T)

which is an isomorphism for r = 2 and s < 0 and an epimorphism for r > 2 and s < 0. Similarly, there is a map of degree −1

(4.1.5)  ∂^{r}: Ê_{s,t}^{r}(C; T) → E_{s−1,t}^{r}(T_{hC})

which is an isomorphism for r = 2 and s ≥ 2 and a monomorphism for r > 2 and s ≥ 2. The situation for r = 2 and s = 0, 1 is described by the exact sequence

0 → Ê_{1,∗}^{2}(C; T) ⟶^{∂^{2}} E_{0,∗}^{2}(T_{hC}) ⟶^{N} E_{0,∗}^{2}(T^{hC}) ⟶^{R^{h,2}} Ê_{0,∗}^{2}(C; T) → 0,

where N is the norm map N: H_{0}(C; π_{∗}T) → H^{0}(C; π_{∗}T). For r > 2 the relationship is explained in [BM], §2.

4.2. We now recall Bökstedt’s and Breen’s basic result on π_{∗}T (𝔽_{p}) and sketch briefly the proof, in Bökstedt’s formulation.

Since T (R) is the realization of a simplicial space it has a skeleton filtration, and there is a first quadrant spectral sequence

E^{2}(R) = HH_{∗}(A_{R}) ⇒ H_{∗}(T (R); 𝔽_{p})

where A_{R} = H_{∗}(HR; 𝔽_{p}) and H_{∗}(−) is spectrum homology. When R = 𝔽_{p}, A_{R} is the dual Steenrod algebra, i.e. A_{𝔽_{p}} = A where

A = \{ \begin{matrix}S_{𝔽_{p}}\{ξ_{1}, ξ_{2}, … \} ⊗ Λ_{𝔽_{p}}\{τ_{0}, τ_{1}, … \}, & p odd \\ S_{𝔽_{p}}\{ξ_{1}, ξ_{2}, … \}, & p = 2. \end{matrix}

Here deg ξ_{i} = 2(p^{i} − 1) (or 2^{i} − 1 if p = 2), deg τ_{i} = 2p^{i} − 1 and S_{𝔽_{p}} resp. Λ_{𝔽_{p}} denotes the symmetric resp. the exterior algebra over 𝔽_{p}. Since A is a connected Hopf algebra one has with A^{e} = A ⊗ A

HH_{∗}(A) = Tor^{A^{e}}(A, A) ≅ A ⊗ Tor^{A}(𝔽_{p}, 𝔽_{p}),

see [CE] p. 194, and

Tor^{A}(𝔽_{p}, 𝔽_{p}) ≅ \{ \begin{matrix}Λ_{𝔽_{p}}\{σξ_{1}, σξ_{2}, … \}, & p = 2 \\ Λ_{𝔽_{p}}\{σξ_{1}, σξ_{2}, … \} ⊗ Γ_{𝔽_{p}}\{στ_{0}, στ_{1}, … \}, & p odd \end{matrix}

where Γ_{𝔽_{p}}{−} is the divided power algebra, e.g.

Γ_{𝔽_{p}}{στ_{i}} ≅ ⨁_{j≥0} S_{𝔽_{p}}{γ_{p^{j}}(στ_{i})}/(γ_{p^{j}}(στ_{i})^{p}).

The (bi-)degrees of the generators are

deg(σξ_{i}) = (1, 2(p^{i} − 1)) (resp. (1, 2^{i} − 1) for p = 2)
deg(γ_{p^{j}}(στ_{i})) = (p^{j}, p^{j}(2p^{i} − 1)).

Let H𝔽_{p} → T (𝔽_{p}) be the inclusion of the 0-skeleton and consider the composition

(4.2.1)  σ: S_{+}^{1} ∧ H𝔽_{p} → S_{+}^{1} ∧ T (𝔽_{p}) ⟶^{μ} T (𝔽_{p}).

Then σξ_{i} and στ_{i} are the images under σ_{∗} of [S^{1}] ⊗ ξ_{i} and [S^{1}] ⊗ τ_{i}. There are homology operations in H_{∗}(T (𝔽_{p})), which commute with σ. The homology operations in H_{∗}(H𝔽_{p}) were examined by Steinberger in [BMMS] chap. III, theorem 2.3, and twenty years before by L. Kristensen (unpublished). The result we need is that

Q^{p^{i}}(τ_{i}) = τ_{i+1}, Q^{p^{i}}(ξ_{i}) = ξ_{i+1}, βτ_{i} = ξ_{i}.

Here τ_{i} and ξ_{i} are not the usual Milnor generators, but the images of these under the canonical anti automorphism (antipode) of A.

For degree reasons there are no differentials in the spectral sequence when p = 2. In the case of odd primes the first possible non-zero differential is d^{p−1}. Bökstedt proves in [B1] that

d^{p−1}(γ_{p^{j}}(στ_{i})) = (γ_{p^{j−1}}(στ_{i}) · . . . · γ_{p}(στ_{i}))^{p−1} · σξ_{i+1}.

This can be viewed as a ‘Kudo principle’ since σξ_{i+1} = βQ^{p^{i}}(στ_{i}) by the above. In any case one gets for odd p

E^{p} = A ⊗ S_{𝔽_{p}}{στ_{i}|i ≥ 0}/((στ_{i})^{p}|i ≥ 0)

and for degree reasons E^{p} = E^{∞}. Finally the homology operations solve the extension problems,

(στ_{i})^{p} = Q^{p^{i}}(στ_{i}) = σQ^{p^{i}}(τ_{i}) = στ_{i+1}

so that

H_{∗}(T (𝔽_{p}); 𝔽_{p}) ≅ A ⊗ S_{𝔽_{p}}{στ_{0}}

and hence π_{∗}T (𝔽_{p}) ≅ S_{𝔽_{p}}{στ_{0}}.

Let [S^{1}] ∈ π_{1}^{S}(S_{+}^{1}) be the image of the generator in π_{2}^{S}(S^{2}) under the boundary map ∂: π_{2}^{S}(S^{2}) → π_{1}^{S}(S_{+}^{1}) of the cofibration S_{+}^{1} → S^{0} → S^{2}. Let σ̄ ∈ π_{2}(T (𝔽_{p}); 𝔽_{p}) be the image of [S^{1}] ∧ τ_{0} under the map in (4.2.1), and let σ ∈ π_{2}T (𝔽_{p}) be the preimage of σ̄ under the reduction to 𝔽_{p}-coefficients, which is an isomorphism. We have proved

Theorem 4.2. ([Br], [B]) π_{∗}T (𝔽_{p}) = S_{𝔽_{p}}{σ}. □

The above calculation shows that T (𝔽_{p}) is a wedge of Eilenberg-MacLane spectra. But this is also clear from the beginning because the composition

T (R) ≃ S^{0} ∧ T (R) → HR ∧ T (R) → T (R) ∧ T (R) ⟶^{μ} T (R)

is homotopic to the identity, so that T (R) is a retract of HR ∧ T (R) which is always a wedge of Eilenberg-MacLane spectra.

4.3. We return to the spectral sequences of 4.1 for π_{∗}(ℍ̂(C_{p^{n}}; T (𝔽_{p}); 𝔽_{p});

(4.3.1)  Ê^{2} = Ĥ^{∗}(C_{p^{n}}; π_{∗}(T (𝔽_{p}); 𝔽_{p})) = Λ_{𝔽_{p}}{u_{n}} ⊗ S_{𝔽_{p}}{t, t^{−1}} ⊗ Λ_{𝔽_{p}}{e_{1}} ⊗ S_{𝔽_{p}}{σ̄},

where deg u_{n} = (−1, 0), deg t = (−2, 0), deg e_{1} = (0, 1) and deg σ̄ = (0, 2). Indeed, the Bockstein exact sequence which relates integral and modulo p homotopy groups gives π_{∗}(T (𝔽_{p}); 𝔽_{p}) ≅ Λ_{𝔽_{p}}{e_{1}} ⊗ S_{𝔽_{p}}{σ}. The Bockstein on e_{1} is 1, so that the odd degree homotopy groups map isomorphically onto the even dimensional ones. We also consider the spectral sequence for π_{∗}ℍ̂(C_{p^{n}}; T (𝔽_{p})), (integral homotopy groups)

(4.3.2)  Ê^{2} = Ĥ^{∗}(C_{p^{n}}; π_{∗}T (𝔽_{p})) = Λ_{𝔽_{p}}{u_{n}} ⊗ S_{𝔽_{p}}{t, t^{−1}} ⊗ S_{𝔽_{p}}{σ}.

There is a map of spectral sequences res: Ê^{r} → Ê^{r} which is injective for r = 2. Both spectral sequences are homology type and lie in the second quadrant, Ê^{r} is multiplicative and Ê^{r} is a module over Ê^{r}.

Lemma 4.3. The non-zero differentials in Ê^{r} are generated from d^{2}e_{1} = tσ̄ in the module structure over Ê^{r}. In particular,

π_{∗}(ℍ̂(C_{p^{n}}; T (𝔽_{p})); 𝔽_{p}) ≅ Λ_{𝔽_{p}}{u_{n}} ⊗ S_{𝔽_{p}}{t, t^{−1}}.

Proof. For degree reasons there are no d^{2}-differentials in Ê^{r}. Therefore, if d^{2}e_{1} = tσ̄ we get

Ê^{3} = Λ_{𝔽_{p}}{u_{n}} ⊗ S_{𝔽_{p}}{t, t^{−1}}

and there can be no further differentials. The idea of the proof is to compare with the spectral sequence which calculates π_{∗}(ℍ̂(G; T (𝔽_{p})); 𝔽_{p}). It has E^{2}-term

Ĥ^{∗}(BG; π_{∗}(T (𝔽_{p}); 𝔽_{p})) ≅ S_{𝔽_{p}}{t, t^{−1}} ⊗ Λ_{𝔽_{p}}{e_{1}} ⊗ S_{𝔽_{p}}{σ̄},

and there is a map from this spectral sequence to Ê^{r} which injects the E^{2}-term. The differential d^{2}: E_{0,1}^{2} → E_{2,2}^{2} in this spectral sequence is the composite

π_{1}(T (𝔽_{p}); 𝔽_{p}) ⟶^{[S^{1}]} π_{2}(S_{+}^{1} ∧ T (𝔽_{p}); 𝔽_{p}) ⟶^{μ} π_{2}(T (𝔽_{p}); 𝔽_{p}),

cf. [BM], §5. The first map is exterior multiplication by [S^{1}] ∈ π_{1}^{S}(S_{+}^{1}) and the second map is induced by the S^{1}-action on T (𝔽_{p}). Hence, d^{2}e_{1} = tσ̄ as claimed. □

Corollary 4.3. The integral homotopy groups π_{∗}ℍ̂(C_{p^{n}}; T (𝔽_{p})) are cyclic ℤ_{p}-modules.

Proof. We may compare the spectral sequence (4.3.2) with the spectral sequence for π_{∗}ℍ̂(G; T (𝔽_{p})) to see that t and σ are permanent cycles. Hence there is a differential

(4.3.3)  d^{2r+1}u_{n} = t^{r+1}σ^{r},

for some r ≥ 1, or there are no differentials at all. (We prove in lemma 4.4 that r = n.) On the other hand, the mod p spectral sequence shows that the extensions in the passage from Ê^{∞} to the actual homotopy groups are maximally non-trivial. Hence the claim. □

We use that Γ̂_{1}: T (𝔽_{p}) → ℍ̂(C_{p}; T (𝔽_{p})) is a map of ring spectra to determine the differential (4.3.3). Since Γ̂ preserves the unit, and as π_{0}T (𝔽_{p}) = 𝔽_{p} and π_{0}ℍ̂(C_{p}; T (𝔽_{p})) is cyclic,

Γ̂_{∗}: π_{0}T (𝔽_{p}) → π_{0}ℍ̂(C_{p}; T (𝔽_{p}))

is an isomorphism. This can only happen if r = 1 in (4.3.3), that is, if

d^{3}u_{1} = t^{2}σ

in (4.3.2). It is then easy to solve the spectral sequence to get

(4.3.4)  π_{∗}ℍ̂(C_{p}; T (𝔽_{p})) = S_{𝔽_{p}}{σ̂, σ̂^{−1}}

where σ̂ is a generator of degree 2.

Proposition 4.3. The map Γ̂_{1}: T (𝔽_{p}) → ℍ̂(C_{p}; T (𝔽_{p})) induces an equivalence of connective covers.

Proof. Since Γ̂ is multiplicative theorem 4.2 and (4.3.4) show that it is enough to prove that

Γ̂_{∗}: π_{2}T (𝔽_{p}) → π_{2}ℍ̂(C_{p}; T (𝔽_{p}))

is an isomorphism. We have T (𝔽_{p}) ≃_{G} ρ_{C_{p}}^{#}Φ^{C_{p}}T (𝔽_{p}) and the following commutative square of G/C_{p}-spectra,

Φ^{C_{p}}T (𝔽_{p}) ⟶^{∂} ΣT (𝔽_{p})_{hC_{p}}
↓_{}_{γ̂^{Cp}}  ∥
ℍ̂(C_{p}; T (𝔽_{p})) ⟶^{∂^{h}} ΣT (𝔽_{p})_{hC_{p}},

cf. (4.1.2). Thus we may instead prove that

∂_{∗}: π_{2}Φ^{C_{p}}T (𝔽_{p}) → π_{1}T (𝔽_{p})_{hC_{p}}

is an isomorphism. One has (by the spectral sequence) π_{i}T (𝔽_{p})_{hC_{p}} ≅ 𝔽_{p} for i = 0, 1, see (2.3.1). Theorem 4.2 translate under the equivalence T (𝔽_{p}) ≃_{G} ρ_{C_{p}}^{#}Φ^{C_{p}}T (𝔽_{p}) to the statement that

μ̄_{∗}: π_{1}^{S}(G/C_{p+}) ⊗ π_{1}(Φ^{C_{p}}T (𝔽_{p}); 𝔽_{p}) → π_{2}(Φ^{C_{p}}T (𝔽_{p}); 𝔽_{p})

is surjective, and the generator of the right hand group is the mod p reduction of the generator of the integral group π_{2}Φ^{C_{p}}T (𝔽_{p}). Since ∂ is a G/C_{p}-equivariant map it is therefore enough if we prove that the two maps

(a)  ¯∂_{∗}: π_{1}(Φ^{C_{p}}T (𝔽_{p}); 𝔽_{p}) → π_{0}(T (𝔽_{p})_{hC_{p}}; 𝔽_{p})
(b)  μ̄_{∗}: π_{1}^{S}(G/C_{p+}) ⊗ π_{0}T (𝔽_{p})_{hC_{p}} → π_{1}T (𝔽_{p})_{hC_{p}}

are epimorphisms. Claim (a) follows from the diagram

π_{1}(Φ^{C_{p}}T (𝔽_{p}); 𝔽_{p}) ⟶^{¯∂_{∗}} π_{0}(T (𝔽_{p})_{hC_{p}}; 𝔽_{p}) ⟶^{N̄_{∗}} π_{0}(T (𝔽_{p})^{C_{p}}; 𝔽_{p})
↑  ↑≅  ↑
π_{1}Φ^{C_{p}}T (𝔽_{p}) ⟶^{∂_{∗}} π_{0}T (𝔽_{p})_{hC_{p}}  ⟶^{N_{∗}} π_{0}T (𝔽_{p})^{C_{p}}.

because π_{0}T (𝔽_{p})^{C_{p}} ≅ ℤ/p^{2} and π_{1}(Φ^{C_{p}}T (𝔽_{p}); 𝔽_{p}) = ℤ/p by theorems 2.3 and 4.2. To prove (b) we note that the map

T (𝔽_{p}) ∧_{C_{p}} G_{+} → T (𝔽_{p}) ∧_{C_{p}} EG_{+},

given by the inclusion G ⊂ EG induces an isomorphism on π_{i}(−) for i = 0, 1, and use the G-homeomorphism

T (𝔽_{p}) ∧_{Cp} G_{+} ≅ |T (𝔽_{p})| ∧ G/C_{p+}

where the bars on the right indicate T (𝔽_{p}) with trivial G-action. □

Addendum 4.3. The maps

Γ_{n}: T (𝔽_{p})^{C_{p^{n}}} → T (𝔽_{p})^{hC_{p^{n}}} , Γ̂_{n}: T (𝔽_{p})^{C_{p^{n−1}}} → ℍ̂(C_{p^{n}}; T (𝔽_{p}))

induce equivalences of connective covers.

Proof. Since the spectra are all p-complete it is enough to show that the maps induce isomorphism on π_{∗}(−; 𝔽_{p}) in non-negative degrees. For n = 1, this follows from the lemma and from a 5-lemma argument based on proposition 4.1. In the general case we have ℍ̂(C_{p^{n}}; T (𝔽_{p})) = ρ_{C_{p}}^{#} ℍ̂(C_{p}; T (𝔽_{p}))^{C_{p^{n−1}}} and Γ̂_{n−1} = γ̂^{n−1}, where γ̂ is the G-equivariant map

T (𝔽_{p}) → ρ_{C_{p}}^{#} ℍ̂(C_{p}; T (𝔽_{p})).

We can now compare with the homotopy fixed point situation via the diagram

T (𝔽_{p})^{C_{p^{n−1}}}  ⟶^{Γ_{n−1}}  T (𝔽_{p})^{hC_{p^{n−1}}}
↓_{}_{γ̂^{C_{p^{n}−1}}}  ↓_{}_{γ̂^{hC_{p^{n}−1}}}
ρ_{C_{p}}^{#} ℍ̂(C_{p}; T (𝔽_{p}))^{C_{p^{n−1}}} ⟶^{G} ρ_{C_{p}}^{#} ℍ̂(C_{p}; T (𝔽_{p}))^{hC_{p^{n−1}}} .

Since γ̂ = Γ̂_{1} is a non-equivariant equivalence on connective covers by the lemma, so is γ̂^{hC_{p^{n−1}}} . Inductively, Γ_{n−1} may be assumed to be an equivalence on connective covers, so it remains to show that G is. There is a commutative diagram

ρ_{C_{p}}^{#} ℍ̂(C_{p}; T (𝔽_{p}))_{hC_{p^{n−1}}} ⟶^{N} ρ_{C_{p}}^{#} ℍ̂(C_{p}; T (𝔽_{p}))^{C_{p^{n−1}}}
∥  ↓^{G}
ρ_{C_{p}}^{#} ℍ̂(C_{p}; T (𝔽_{p}))_{hC_{p^{n−1}}} ⟶^{N^{h}} ρ_{C_{p}}^{#} ℍ̂(C_{p}; T (𝔽_{p}))^{hC_{p^{n−1}}}

and we claim

(i)  π_{∗}(ℍ̂(C_{p^{n}}; T (𝔽_{p})); 𝔽_{p}) ≅ π_{∗}(ρ_{C_{p}}^{#} ℍ̂(C_{p}; T (𝔽_{p}))^{hC_{p^{n−1}}} ; 𝔽_{p})
(ii)  ℍ̂(C_{p^{n−1}}; ρ_{C_{p}}^{#} ℍ̂(C_{p}; T (𝔽_{p}))) ≃ 0

Given these claims, (ii) and the norm cofibration sequence for the C_{p^{n−1}}-spectrum ρ_{C_{p}}^{#} ℍ̂(C_{p}; T (𝔽_{p})) show that N^{h} is an equivalence, and hence that π_{∗}(G; 𝔽_{p}) is a surjection of abstractly isomorphic finite groups, thus an isomophism.

It remains to prove (i) and (ii). This uses the spectral sequences of (4.1.3),

\begin{aligned}H^{∗}(C_{p^{n−1}}; π_{∗}(ρ_{C_{p}}^{#} ℍ̂(C_{p}; T (𝔽_{p})); 𝔽_{p})) & ⇒ & π_{∗}(ρ_{C_{p}}^{#} ℍ̂(C_{p}; T (𝔽_{p}))^{hC_{p^{n−1}}} ; 𝔽_{p}) \\ Ĥ^{∗}(C_{p^{n−1}}; π_{∗}(ρ_{C_{p}}^{#} ℍ̂(C_{p}; T (𝔽_{p})); 𝔽_{p})) & ⇒ & π_{∗}(ℍ̂(C_{p^{n−1}}; ρ_{C_{p}}^{#} ℍ̂(C_{p}; T (𝔽_{p}))); 𝔽_{p}) \end{aligned}

We have already proved that

π_{∗}(ℍ̂(C_{p}; T (𝔽_{p})); 𝔽_{p}) ≅ Λ_{𝔽_{p}}{ê_{1}} ⊗ S_{𝔽_{p}}{σ̂, σ̂^{−1}}

with deg ê_{1} = 1, deg σ̂ = 2. The two E^{2}-terms are consequenctly

E^{2} = Λ_{𝔽_{p}}{u_{n−1}} ⊗ S_{𝔽_{p}}{t} ⊗ Λ_{𝔽_{p}}{ê_{1}} ⊗ S_{𝔽_{p}}{σ̂, σ̂^{−1}}
Ê^{2} = Λ_{𝔽_{p}}{u_{n−1}} ⊗ S_{𝔽_{p}}{t, t^{−1}} ⊗ Λ_{𝔽_{p}}{ê_{1}} ⊗ S_{𝔽_{p}}{σ̂, σ̂^{−1}}.

Combining lemma 4.3 and proposition 4.3 one has that d^{2}(ê_{1}) = tσ̂ in both cases. This differential and its multiplicative consequences are the only ones. Hence

E^{3} = Λ_{𝔽_{p}}{u_{n−1}} ⊗ S_{𝔽_{p}}{σ̂, σ̂^{−1}}

and E^{3} = E^{∞}, so π_{∗}(ℍ̂(C_{p}; T (𝔽_{p}))^{hC_{p^{n−1}}} ; 𝔽_{p}) has a copy of 𝔽_{p} in each degree. Now compare with corollary 4.3 to prove (i). For (ii), note that

d^{2}(ê_{1}t^{−1}σ̂^{−1}) = 1

so that Ê^{3} = 0. □

Remark 4.4. S. Tsalides, [T], has given a quite different and more general proof of addendum 4.3, assuming lemma 4.3.

4.4. We can now give a complete description of the fixed point structure of T (𝔽_{p}). We begin by solving the spectral sequences in (4.3.2).

Lemma 4.4. In the spectral sequence Ê^{r} which converges to π_{∗}ℍ̂(C_{p^{n}}; T (𝔽_{p})) the differentials are multiplicatively generated from d^{2n+1}u_{n} = t^{n+1}σ^{n} and the fact that t and σ are permanent cycles. In particular,

π_{∗}ℍ̂(C_{p^{n}}; T (𝔽_{p})) = S_{ℤ/p^{n}}{σ̂, σ̂^{−1}},

where deg σ̂ = 2.

Proof. We may combine addendum 4.3 and theorem 2.3 to get

π_{0}ℍ̂(C_{p^{n}}; T (𝔽_{p})) ≅ π_{0}T (𝔽_{p})^{C_{p^{n−1}}} ≅ ℤ/p^{n}.

Now the claim for the differentials follows from corollary 4.3 and its proof. We get

Ê^{2n+2} = S_{𝔽_{p}}{t, t^{−1}, σ}/(t^{n+1}σ^{n})

and since all elements are in even total degree there are no further differentials. □

Proposition 4.4. The integral homotopy groups of the fixed point spectra T (𝔽_{p})^{C_{p^{n}}} is a copy of ℤ/p^{n+1} in each positive even degree,

π_{∗}T (𝔽_{p})^{C_{p^{n}}} = S_{ℤ/p^{n+1}}{σ_{n}},

where deg σ_{n} = 2. Moreover, F (σ_{n}) = σ_{n−1}, V (σ_{n−1}) = pσ_{n} and R(σ_{n}) = pλ_{n}σ_{n−1} where λ_{n} ∈ ℤ/p^{n+1} is a unit.

Proof. The claim for the homotopy groups is immediate from addendum 4.3 and the lemma. We have the following commutative square

T (𝔽_{p})^{C_{p^{n}}}  ⟶^{F} T (𝔽_{p})^{C_{p^{n−1}}}
↓_{}_{γ̂^{C_{p^{n}}}}  ↓_{}_{γ̂^{C_{p^{n−1}}}}
ℍ̂(C_{p^{n}}; T (𝔽_{p})) ⟶^{F^{h}} ℍ̂(C_{p^{n−1}}; T (𝔽_{p})),

where the vertical maps are the equivalences of addendum 4.3 and F^{h} is the obvious inclusion of fixed sets. It induces the restriction map in Tate cohomology,

res: Ĥ^{∗}(C_{p^{n}}; π_{∗}T (𝔽_{p})) → Ĥ^{∗}(C_{p^{n−1}}; π_{∗}T (𝔽_{p}))

on the E^{2}-term of the spectral sequences Ê^{r}. Since this is an isomorphism in even degrees it follows that we can choose the generators σ_{n} such that F σ_{n} = σ_{n−1}. Next, V (σ_{n−1}) = V F (σ_{n}) = pσ_{n}. Finally, the calculation of R follows from the exact sequence

π_{2}T (𝔽_{p})^{C_{p^{n}}} ⟶^{R} π_{2}T (𝔽_{p})^{C_{p^{n−1}}} ⟶^{∂} π_{1}T (𝔽_{p})_{hC_{p^{n}}} ⟶^{N} π_{1}T (𝔽_{p})^{C_{p^{n}}}

since π_{1}T (𝔽_{p})_{hC_{p^{n}}} ≅ 𝔽_{p} and π_{1}T (𝔽_{p})^{C_{p^{n}}} = 0. □

4.5. In this section we extend proposition 4.4 to any perfect field k of positive characteristic.

Lemma 4.5. If k is a perfect field of positive characteristic then HH_{∗}(k) = k.

Proof. We choose a transcendence basis {X_{i}|i ∈ I} of k over 𝔽_{p}. Since k is perfect it contains as a subfield the field

l = \underrightarrow{lim}_{r} 𝔽_{p}(X_{i}^{p^{−r}}| i ∈ I).

Moreover, k is a separable algebraic extension of l. For l is perfect by construction, and any algebraic extension of a perfect field is separable. We may write k = \underrightarrow{lim} k_{α} where the colimit runs over the finite extensions l ⊂ k_{α} ⊂ k. Each k_{α}/l is a finite separable extensions and hence étale. Therefore, HH_{∗}(k_{α}) ≅ k_{α} ⊗_{l} HH_{∗}(l), [WG], and since Hochschild homology commutes with filtered colimits,

HH_{∗}(k) ≅ k ⊗_{l} HH_{∗}(l).

Now HH_{∗}(l) = l. Indeed, by [HKR]

HH_{∗}(𝔽_{p}[X_{i}| i ∈ I]) ≅ Ω_{𝔽_{p}[X_{i}| i∈I]/𝔽_{p}}^{∗}

and both sides commute with filtered colimits and localization, so HH_{∗}(l) ≅ Ω_{l/𝔽_{p}}^{∗}. Now since l is perfect Ω_{l/𝔽_{p}} = 0, as dx = d(y^{p}) = py^{p−1}dy = 0. □

We thank Chuck Weibel for help with the argument above.

Corollary 4.5. π_{∗}T (k) ≅ k ⊗ π_{∗}T (𝔽_{p}).

Proof. We consider the spectral sequence E^{r}(R) of 4.2 with R = k. The inclusion 𝔽_{p} → k defines A → A_{k}, and since the target is a k-algebra we get a ring homomorphism

k ⊗ A → A_{k}.

This is in fact an isomorphism. For as an abelian group k is just a direct sum of copies of 𝔽_{p} and taking homology commutes with direct sums. We get

HH_{∗}(A_{k}) ≅ HH_{∗}(k ⊗ A) ≅ HH_{∗}(k) ⊗ HH_{∗}(A) ≅ k ⊗ HH_{∗}(A),

where the last equality is the lemma above. Thus E^{2}(k) ≅ k ⊗ E^{2}(𝔽_{p}) and since E^{r}(k) is a spectral sequence of k-modules

E^{∞}(k) ≅ k ⊗ E^{∞}(𝔽_{p}).

The statement follows. □

Suppose that T is any C-ring spectrum and that X is any C-space. Then (T ∧ X)^{C} is a T^{C}-module spectrum. The action map is the composition

(4.5.1)  T^{C} ∧ (T ∧ X)^{C} ⟶ (T ∧ T ∧ X)^{C} ⟶^{μ∧1} (T ∧ X)^{C}.

When T is T (A) and X is any of the C-spaces in (4.1.1) this shows that

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

is a cofibration sequence of T (A)^{C_{p^{n}}} -module spectra. In particular, the associated long exact sequence of homotopy groups

… ⟶^{∂} π_{i}T (A)_{hC_{p^{n}}} ⟶^{N} π_{i}T (A)^{C_{p^{n}}} ⟶^{R} π_{i}T (A)^{C_{p^{n−1}}} ⟶^{∂} …

is a sequence of W_{n+1}(A)-modules. Moreover, (4.1.3) is a spectral sequence of W_{n+1}(A)-modules,

(4.5.2)  E^{2} = H_{∗}(C_{p^{n}}; (F^{n})^{∗}π_{∗}T (A)) ⇒ π_{∗}T (A)_{hC_{p^{n}}} ,

where F^{n}: W_{n+1}(A) → A is the iterated Frobenius. Indeed, the identification of the E^{1}-term uses the transfer equivalence (T ∧ Σ^{r}C_{+})^{C} ≃ Σ^{r}T, and under this equivalence (4.5.1) becomes

T^{C} ∧ Σ^{r}T ⟶^{incl ∧1} T ∧ Σ^{r}T ⟶^{μ} Σ^{r}T,

which gives (4.5.2).

Theorem 4.5. For any perfect field k of positive characteristic p,

π_{∗}T (k)^{C_{p^{n}}} ≅ S_{W_{n+1}(k)}{σ_{n}}, deg σ_{n} = 2,

and F (σ_{n}) = σ_{n−1}, V (σ_{n−1}) = pσ_{n} and R(σ_{n}) = pλ_{n}σ_{n−1} where λ_{n} ∈ W_{n}(𝔽_{p}) = ℤ/p^{n} is a unit.

Proof. We argue by induction on n starting from the case n = 1 was established in corollary 4.5 above. Let W = W_{n+1}(k) and consider the diagram

\xymatrix{… \ar[r] & W ⊗ π_{i}T (𝔽_{p})_{hC_{p^{n}}} \ar[r]^{N} \ar[d] & W ⊗ π_{i}T (𝔽_{p})^{C_{p^{n}}} \ar[r]^{R} \ar[d] & W ⊗ π_{i}T (𝔽_{p})^{C_{p^{n−1}}} \ar[r] \ar[d] & … \\ … \ar[r] & π_{i}T (k)_{hC_{p^{n}}} \ar[r]^{N} & π_{i}T (k)^{C_{p^{n}}} \ar[r]^{R} & π_{i}T (k)^{C_{p^{n−1}}} \ar[r] & …}

By induction the right hand vertical map is an isomorphism. Indeed,

W_{n+1}(k) ⊗ π_{∗}T (𝔽_{p})^{C_{p^{n−1}}} ≅ W_{n}(k) ⊗ π_{∗}T (k)^{C_{p^{n−1}}} ≅ π_{∗}T (k)^{C_{p^{n−1}}} .

Therefore, we are done by induction if we prove that the left hand vertical map is an isomorphism. We let φ_{k} denote the Frobenius automorphism on k and consider the diagram

W_{n+1}(𝔽_{p}) ⟶^{W_{n+1}(φ_{𝔽p}^{n})} W_{n+1}(𝔽_{p}) ⟶^{R^{n}} 𝔽_{p}
↓  ↓  ↓
W_{n+1}(k) ⟶^{W_{n+1}(φ_{k}^{n})} W_{n+1}(k) ⟶^{R^{n}} k.

The left hand square is cocartesian because the horizontal maps are isomorphisms and the right hand square is cocartesian because p generates the maximal ideal of W_{n+1}(k). Moreover, the compositions of the horizontal maps are equal to F^{n} and therefore we have

W_{n+1}(k) ⊗ (F^{n})^{∗}π_{∗}T (𝔽_{p}) ≅ (F^{n})^{∗}k ⊗ π_{∗}T (𝔽_{p}) ≅ (F^{n})^{∗}T (k).

Now the spectral sequence discussed above supplies the conclusion. □

Proof of theorem B. Theorem 4.5 shows that TR(k) = HW (k), with the notation of (3.1.1). Moreover, F: TR(k) → TR(k) corresponds to the Frobenius on Witt vectors, and hence we obtain an exact sequence

0 → TC_{0}(k) → W (k) ⟶^{1−F} W (k) → TC_{−1}(k) → 0.

When k = 𝔽_{p} we have 1 − F = 0, proving TC(𝔽_{p}) ≅ Hℤ_{p} ∨ Σ^{−1}Hℤ_{p}. In particular, TC(𝔽_{p}) is an Eilenberg-MacLane spectrum. For general k, TC(k) is a module spectrum over TC(𝔽_{p}) and hence an Eilenberg-MacLane spectrum. □

Remark 4.5. We may also extend addendum 4.3 and lemma 4.4 to general perfect fields. The map Γ̂_{n} in proposition 4.1 shows that π_{∗}ℍ̂(C_{p^{n}}; T (k)) is a W_{n}(k)-module, and we claim that

(4.5.2)  π_{∗}ℍ̂(C_{p^{n}}; T (k)) ≅ W_{n}(k) ⊗ π_{∗}ℍ̂(C_{p^{n}}; T (𝔽_{p})).

Indeed, the spectral sequence of (4.1.3) is a spectral sequence of W_{n+1}(k)-modules,

Ê^{2} = Ĥ^{∗}(C_{p^{n}}; (F^{n})^{∗}π_{∗}T (k)) ⇒ π_{∗}ℍ̂(C_{p^{n}}; T (k)),

where F^{n}: W_{n+1}(k) → k is the iterated Frobenius. This follows from the discussion preseeding theorem 4.5. Therefore, we can repeat the proof of theorem 4.5 and get that

W_{n+1}(k) ⊗ π_{∗}ℍ̂(C_{p^{n}}; T (𝔽_{p})) ≅ π_{∗}ℍ̂(C_{p^{n}}; T (k)).

Since the W_{n+1}(k)-module structure on π_{∗}ℍ̂(C_{p^{n}}; T (k)) comes from the W_{n}(k)-module structure via the restriction map R: W_{n+1}(k) → W_{n}(k) we get the claimed isomorphism.

Quite similarly, the proof of addendum 4.3 generalizes to show that

(4.5.3)  Γ_{n}: T (k)^{C_{p^{n}}} → T (k)^{hC_{p^{n}}} , Γ̂_{n−1}: T (k)^{C_{p^{n−1}}} → ℍ̂(C_{p^{n}}; T (k))

induce equivalences of connective covers.

5. Topological cyclic homology of finite W (k)-algebras

5.1. If the ring R is given as an inverse limit of rings R_{n}, R = \underleftarrow{lim} R_{n}, then one can define continuous versions of K(R) and TC(R) by setting

K^{top}(R) = holim_{⟵} K(R_{n}), TC^{top}(R) = holim_{⟵} TC(R_{n}),

cf. [W]. One may then ask when the natural maps from K(R) to K^{top}(R) and TC(R) to TC^{top}(R) are equivalences.

The cyclotomic trace from K(R) to TC(R) defines by naturality a corresponding map between the continuous versions, so we have a diagram

K(R) ⟶^{trc} TC(R)
↓  ↓
K^{top}(R) ⟶^{trc} TC^{top}(R).

Let k be a perfect field of positive characteristic p and let W (k) be its ring of Witt vectors. We have the following result about the above diagram:

Theorem 5.1. Let A be a W (k)-algebra which is finitely generated as a W (k)-module.

(i) The cyclotomic trace induces an equivalence K^{top}(A)p^{∧} ≃ TC^{top}(A)p^{∧}[0, ∞). (ii) The natural map TC(A)p^{∧} → TC^{top}(A)p^{∧} is an equivalence. In both statements the superscript top refers to the p-adic topology on A.

We note that since W (k) is a P.I.D. the structure theorem for finitely generated modules shows that A is p-adically complete: A = \underleftarrow{lim} A/A_{n}, where A_{n} = A/p^{n}A.

Proof of 5.1(i). By McCarthy’s theorem A of the introduction it suffices to prove that

trc: K(A_{1})p^{∧} → TC(A_{1})p^{∧}[0, ∞)

is an equivalence. As a finite dimensional k-algebra, A_{1} is artinian, and hence its radical J = rad(A_{1}) is nilpotent. Therefore, by one more application of theorem A we are reduced to prove that K(A_{1}/J)p^{∧} ≃ TC(A_{1}/J)p^{∧}[0, ∞). Now A_{1}/J is semi-simple, and since both functors preserve product it suffices to prove that

trc: K(Ā)p^{∧} → TC(Ā)p^{∧}[0, ∞)

is an equivalence for a central simple k-algebra. If the class of Ā in the Brauer group Br(k) is trivial, i.e. if Ā ≅ M_{n}(k), then we are done by theorem B since both K(−) and TC(−) are Morita invariant, cf. 1.7.

However, Br(k) might not vanish for perfect fields in general; one knows only that the p-primary part of Br(k) vanishes, [Se], chap. X, §4. Let K be a Galois splitting field for Ā and G the Galois group of K/k, and let L = K^{G_{p}} where G_{p} is a Sylow p-subgroup of G. Then we have a commutative diagram

0 ⟶ H^{2}(G; K^{×}) ⟶ Br(k) ⟶ Br(K)
↓^{res}  ↓  ∥
0 ⟶ H^{2}(G_{p}; K^{×}) ⟶ Br(L) ⟶ Br(K).

Since _{p} Br(k) = 0, [Ā] is p′-torsion in H^{2}(G; K^{×}) and since H^{2}(G_{p}; K^{×}) has vanishing p′-torsion one must have [Ā ⊗_{k} L] = 0 in Br(L). Thus

B̄ = Ā ⊗_{k} L ≅ M_{r}(L).

On the other hand, L is perfect (being an algebraic extension of k), so by the previous remarks the middle vertical map in the diagram below is an isomorphims:

(5.1.1)  \begin{matrix}K(Ā)p^{∧} & ⟶^{i_{∗}} & K(B̄)p^{∧} & ⟶^{i^{∗}} & K(Ā)p^{∧} \\ ↓ & & ↓ & & ↓ \\ TC(Ā)p^{∧} & ⟶^{i_{∗}} & TC(B̄)p^{∧} & ⟶^{i^{∗}} & TC(Ā)p^{∧}. \end{matrix}

Both the horizontal compositions are isomorphisms since B̄ is a free Ā-algebra of rank |L : k|, prime to p. This is well-known for K-theory and for TC we may argue as follows. First, the composition

HH_{∗}(Ā) ⟶^{i_{∗}} HH_{∗}(B̄) ⟶^{i^{∗}} HH_{∗}(Ā)

is an isomorphism. The spectral sequence of 4.2 then implies that the composition

T (Ā) ⟶^{i_{∗}} T (B̄) ⟶^{i^{∗}} T (Ā)

is an equivalence. The obvious inductive argument, using the cofibration sequence of theorem 1.2 shows that

T (Ā)^{C_{p^{n}}} ⟶^{i_{∗}} T (B̄)^{C_{p^{n}}} ⟶^{i^{∗}} T (Ā)^{C_{p^{n}}}

is an equivalence. The same will then be the case for the lower horizontal composition in (5.1.1). It follows now from (5.1.1) that K(Ā)p^{∧} ≃ TC(Ā)p^{∧}[0, ∞). □

The proof of theorem 5.1 (ii) occupies the rest of this paragraph. It is based on the corresponding statement for Eilenberg-MacLane spectra,

HA ≃ holim_{⟵} HA_{n}.

Indeed, π_{∗}holim_{⟵} HA_{n} = \underleftarrow{lim} π_{∗}HA_{n} by [BK], XI.7, and Eilenberg-MacLane spectra are characterized by their homotopy groups. Let us write HA^{(r)} for the r-fold smash product of HA.

Lemma 5.1. Let A be as in theorem 5.1. Then the natural map

HA^{(r)} → holim_{⟵} HA_{n}^{(r)},

becomes an equivalence upon p-completion.

Proof. We begin with the special case where A = W (k) and A_{n} = W_{n}(k). Completion of a spectrum at a prime p is the same as localization with respect to the Moore spectrum S^{0}/p, and hence the thing to show is that

π_{∗}(HA^{(r)}; 𝔽_{p}) ⟶^{≅} \underleftarrow{lim} π_{∗}((HA_{n})^{(r)}; 𝔽_{p}),

see [Bo]. We have

HA ∧ S^{0}/p ≃ HA_{1}, HA_{n} ∧ S^{0}/p ≃ HA_{1} ∨ ΣHA_{1},

and moreover the map HA_{1}^{n+1} → HA_{n} induced from the reduction when smashed with S^{0}/p becomes the self-map of HA_{1} ∨ ΣHA_{1} which is the identity on the first factor and trivial on the suspension factor. These remarks follows easily from the cofibration diagram

HA_{n} ⟶^{p} HA_{n} ⟶ HA_{n} ∧ S^{0}/p
↑  ↑  ↑
HA ⟶^{p} HA ⟶  HA_{1}
↑^{p^{n}}  ↑^{p^{n}}  ↑^{0}
HA ⟶^{p} HA ⟶  HA_{1}.

Thus we have

π_{∗}((HA_{n})^{(r)}; 𝔽_{p}) ≅ H_{∗}((HA_{n})^{(r−1)}; k) ⊕ H_{∗−1}((HA_{n})^{(r−1)}; k)

and the maps in the inverse limit system are trivial on the second summand. This gives

\underleftarrow{lim} π_{∗}((HA_{n})^{(r)}; 𝔽_{p}) ≅ \underleftarrow{lim} H_{∗}((HA_{n})^{(r−1)}; k).

Let A = H_{∗}(HA; k). Then

H_{∗}(HA_{n}; k) = A ⊗_{k} Λ_{k}{ϵ_{n}}, deg ϵ_{n} = 1

and the map induced from the reduction map ℤ/p^{n+1} → ℤ/p^{n} sends ϵ_{n+1} to zero. Indeed, the cofiber of HA_{n} ∧ Hk of p^{n} ∧ id: HA ∧ Hk → HA ∧ Hk is HA ∧ C_{n}, where

C_{n} = cofiber(p^{n}: Hk → Hk) = Hk ∨ ΣHk,

and C_{n} → C_{n−1} maps the first wedge summand by the identity and the second trivially. If follows that

\underleftarrow{lim} π_{∗}((HA_{n})^{(r)}; 𝔽_{p}) ≅ A^{⊗(r−1)} ≅ π_{∗}(HA^{(r)}; 𝔽_{p})

where the tensor product if over k.

If A is a free W (k)-module of finite rank we can use that

HA ≃ HW (k) ∨ … ∨ HW (k), HA_{n} ≃ HW_{n}(k) ∨ … ∨ HW_{n}(k)

to get the conclusion. Finally, for general A, let T (A) be the submodule of torsion elements, and let F (A) be the free quotient. Since W (k) is a local P.I.D. and since T (A) is finitely generated p^{e}T (A) = 0 for a suitable exponent e. Hence

HA_{n} = HT (A) ∧ HF (A)_{n}

and the map HA_{n} → HA_{n−1} is the identity on HT (A) for n > e. Since

holim_{⟵} HF (A)_{n}^{(r)} ≃ HF (A)^{(r)}

for all r by the above, the same follows for HA_{n} upon decomposing HA_{n}^{(r)}. □

5.2. We next consider the continuity of THH(R). This is the realization of the simplicial space with ksimplices

THH_{k}(R) = holim_{\overrightarrow{I^{k+1}}} F (S^{i_{0}} ∧ … ∧ S^{i_{k}}, R̃(S^{i_{0}}) ∧ … ∧ R̃(S^{i_{k}}) ∧ X).

The k-simplices is a spectrum with n’th space THH_{k}(R, S^{n}), cf. 1.4, and in fact it is one way to make sense of the smash product HR^{(k+1)}. Thus we can restate lemma 5.1 as

THH_{k}(A)p^{∧} ≃ holim_{⟵} THH_{k}(A_{n}).

We want to prove the similar statement for the geometric realization THH(A) of the simplicial spectrum THH.(A).

In general it is a sticky point to commute realizations with inverse limits. For example realization does not in general commute with infinite products. A counter example is provided by ∏ S.^{1}, where S.^{1} is the simplicial circle with one non-degenerate 1-simplex. However for Kan complexes there are no problems, and we can take advantage of the fact that THH_{k}(R) is equivalent to Ω THH_{k}(R; S^{1}).

More precisely, we consider the tri-simplicial set

X_{k,l}(R) = G_{k} THH_{l}(R; S^{1}),

where G.Y denotes the Kan loop group of the singular set Sin. Y, and write X(R) for the realization of the diagonal complex, X(R) = |∂X(R).| ≃ THH(R).

Lemma 5.2. Suppose that HR^{(k)} = holim_{⟵} HR_{n}^{(k)} for all k ≥ 1. Then

THH(R)p^{∧} ≃ holim_{⟵} THH(R_{n})p^{∧}.

Proof. We may rephrase the assumption to give that

|X_{.,l}(R)|p^{∧} ≃ holim_{⟵} |X_{.,l}(R_{n})|p^{∧}.

Since simplicial groups are Kan we have

holim_{⟵} |X_{.,l}(R_{n})| ≃ | holim_{⟵} X_{.,l}(R_{n})|
holim_{⟵} |δX_{.,.}(R_{n})| ≃ | holim_{⟵} δX_{.,.}(R_{n})|.

Indeed, the homotopy groups of the realization of a Kan complex can be combinatorially defined, the homotopy limit of Kan complexes is again Kan, and one has a spectral sequence

E_{s,r}^{2} = \underleftarrow{lim}^{(−s)}π_{r}δX_{.,.}(R_{n}) ⇒ π_{r+s} holim_{⟵} δX_{.,.}(R_{n}),

see [BK], p. 309. There is also a spectral sequence

E_{s,r}^{2} = \underleftarrow{lim}^{(−s)}π_{r}|δX_{.,.}(R_{n})| ⇒ π_{r+s} holim_{⟵} |δX_{.,.}(R_{n})|,

and it maps to the former by a map which is an isomorphism on E^{2}; the claim follows. Thus we have

THH(R) ≃ |[l] ↦ | holim_{⟵} X_{.,l}(R_{n})|| ≅ |δ holim_{⟵} δX_{.,.}(R_{n})|

≅ | holim_{⟵} δX_{.,.}(R_{n})| ≃ holim_{⟵} |δX_{.,.}(R_{n})|

≃ holim_{⟵} THH(R_{n}).

□

The above lemma works equally well for THH(R; S^{n}), so with the notation of §1, the underlying nonequivariant spectrum of T (R)p^{∧} is equivalent to that of holim_{⟵} T (R_{n})p^{∧}.

Proof of theorem 5.1. We first note that after p-completion

T (A)^{C_{p^{m}}} ≃ holim_{⟵} T (A_{n})^{C_{p^{m}}}

for each m. This follows inductively from theorem 1.2 since for bounded below spectra taking homotopy orbits commutes with homotopy inverse limits,

holim_{⟵}(T (A_{n})_{hC_{p^{m}}}) ≃ (holim_{⟵} T (A_{n}))_{hC_{p^{m}}} .

Second, we have a cofibration sequence of spectra

TC(A)p^{∧} → holim_{\overleftarrow{m}}[T (A)^{C_{p^{m}}}]p^{∧} ⟶^{R−F} holim_{\overleftarrow{m}}[T (A)^{C_{p^{m−1}}} ]p^{∧}

since TC(A)p^{∧} ≃ TC(A; p)p^{∧} by proposition 2.1, and we have a similar cofibration sequence for each A_{n}. Finally, homotopy inverse limits commute. □

Addendum 5.2. Suppose R is a ring which is finitely generated as a ℤ-module and let R_{p} = R ⊗ ℤ_{p}. Then the natural map from TC(R)p^{∧} to TC(R_{p})p^{∧} is a homotopy equivalence.

We leave the argument which is very similar to the proof of theorem 5.1 to the reader, and note that this property clearly distinguishes K(R)^{∧} from TC(R)^{∧} for non-complete rings. For in the commutative square

K(R)p^{∧} ⟶^{trc} TC(R)p^{∧}
↓  ↓
K(R_{p})p^{∧} ⟶^{trc} TC(R_{p})p^{∧}

the left hand vertical map is not in general an equivalence. For example a result of C. Soulé, [So] shows that for R = ℤ and p = 691, K_{22}(ℤ) does not map injectively into K_{22}(ℤ_{p}). In general the Lichtenbaum-Quillen conjecture asserts that the numerators of the Bernoulli numbers enter into the torsion subgroup of K_{i}(ℤ) but they do not enter into the structure of TC(ℤ_{p})p^{∧} ≃ K(ℤ_{p})p^{∧}.

Remark 5.2. Suppose A is a complete discrete valuation rings with perfect residue fields of positive characteristic. One may ask if TC(A)p^{∧} ≃ TC^{top}(A)p^{∧} when the topology is given by powers of the maximal ideal, i.e. A_{n} = A/𝔪^{n}. In the unequal characteristic case this follows from theorem 5.1 since the 𝔪-adic topology agrees with the p-adic topology. However, in the equal characteristic case, where A = k[[X]] lemma 5.1 fails, and it seems unlikely that the theorem should hold. The problem is that

π_{∗}(HA^{(r)}; 𝔽_{p}) = π_{∗}(Hk; 𝔽_{p})[[x]]^{(r)} ≅ π̸_{∗}(Hk; 𝔽_{p})[[x_{1}, …, x_{r}]] ≅ \underleftarrow{lim}_{n} π_{∗}(HA_{n}^{(r)} 𝔽_{p}).

6. Pointed Monoids

6.1. By a pointed monoid we mean a monoid in the monoidal category of based spaces and smash product, that is, a based space Π equipped with a multiplication and unit

μ^{Π}: Π ∧ Π → Π, 1^{Π}: S^{0} → Π

satisfying associativity and unit laws up to coherent isomorphism. The cyclic bar construction of Π is the cyclic space N_{∧}^{cy}(Π) whose k-simplices are the (k + 1)-fold smash product

N_{∧,k}^{cy}(Π) = Π^{∧(k+1)}

with the Hochschild-like structure maps

(6.1.1)  \begin{aligned}d_{i}(π_{0}∧ … ∧π_{k}) & = π_{0}∧ … ∧π_{i}π_{i+1}∧ … ∧π_{k} & , 0 ≤ i < k \\ & = π_{k}π_{0}∧π_{1}∧ … ∧π_{k−1} & , i = k \\ s_{i}(π_{0}∧ … ∧π_{k}) & = π_{0}∧ … ∧π_{i}∧1∧π_{i+1}∧ … ∧π_{k} & , 0 ≤ i ≤ k \\ τ_{k}(π_{0}∧ … ∧π_{k}) & = π_{k}∧π_{0}∧ … ∧π_{k−1}. \end{aligned}

Since it is a cyclic space the C’th edgewise subdivision sd_{C} N_{∧}^{cy}(Π) has a simplicial action by the cyclic group C and completely analogous to [BHM; §2] we have an isomorphism of cyclic spaces

Δ_{C}: N_{∧}^{cy}(Π) → (sd_{C} N_{∧}^{cy}(Π))^{C}.

If Γ is an ordinary monoid then Γ_{+} is a pointed monoid and N_{∧}^{cy}(Γ_{+}) = N^{cy}(Γ)_{+}. Conversely a pointed monoid, for which μ^{M}(x∧y) = ∗ implies that x∧y = ∗, is of this form. We define for each n ≥ 1 a pointed monoid

Π_{n} = {0, 1, v, v^{2}, …, v^{n−1}}

with 0 as basepoint and the multiplication determined by the rule v^{n} = 0. These are not of the form Γ_{+}. In the pointed situation we have no analog of the (usual) bar construction since in general we lack the projections pr_{i}: Π ∧ Π → Π.

Suppose A is a ring and Π is a discrete pointed monoid. Then we can give the quotient A[Π] = A⟨Π⟩/A⟨∗⟩ the structure of a ring with multiplication and unit

μ: A[Π] ⊗ A[Π] → A[Π ∧ Π] ⟶^{A[μ^{Π}]} A[Π], η: ℤ → A[S^{0}] ⟶^{A[1^{Π}]} A[Π].

If Π = Γ_{+} for a discrete group Γ and A is commutative, then A[Π] is the usual group algebra A[Γ]. Note also that A[Π_{n}] is the truncated poloynomial algebra A[v]/(v^{n}). Moreover, A[N_{∧}^{cy}(Π)] ≅ HH(A[Π]), provided that the multiplication map A ⊗ A → A is an isomorphism, so in this case

(6.1.2)  H̃_{∗}(|N_{∧}^{cy}(Π)|; A) ≅ HH_{∗}(A[Π]).

We want to replace the coefficient ring by an FSP.

Definition 6.1. Let L be an FSP and Π a pointed monoid. We define a new FSP denoted L[Π] by

L[Π](X) = L(X) ∧ Π

with the structure maps μ_{X,Y}^{L[Π]} = (μ_{X,Y}^{L} ∧μ^{Π}) ∘ (id ∧ tw ∧ id) and 1_{X}^{L[Π]} = 1_{X}^{L} ∧1^{Π}.

Let us write Ã for the FSP associated with the ring A, cf. section 4.1.

Proposition 6.1. Let A be a ring and Π a discrete pointed monoid. There is a natural transformation b: Ã[Π] → A˜[Π] which induces an equivalence of cyclotomic spectra

T (Ã[Π]) ≃_{G} T (A˜[Π]).

Proof. Let R be any ring. The multiplicative monoid (R, ·) acts on the functor R̃. Indeed R = R̃(S^{0}) and the action is given by the composition

R ∧ R̃(X) = R̃(S^{0}) ∧ R̃(X) ⟶^{μR̃} R̃(S^{0} ∧ X) = R̃(X).

Hence Π ⊂ A[Π] acts on A˜[Π]. Now b(X) is the adjoint of the map

Π → F (Ã(X), A˜[Π]); π ↦ π · η(X).

Note that b(X) is the inclusion of a wedge of copies of Ã(X) indexed by Π − ∗ in the corresponding weak product. The proof that T (b) is a G-equivalence, is completely analogous to the proof of the theorem below. □

If t is a cyclotomic prespectrum, the smash product G-prespectrum t∧|N_{∧}^{cy}(Π)| may be given the structure of a cyclotomic prespectrum. Indeed, the composition

ρ_{C}^{∗}t(V)^{C}∧ρ_{C}^{∗}|N_{∧}^{cy}(Π)|^{C} ⟶^{1∧D^{−1}} ρ_{C}^{∗}t(V)^{C} ∧ ρ_{C}^{∗}| sd_{C} N_{∧}^{cy}(Π)|^{C}

⟶^{φ(V)∧Δ_{C}^{−1}} t(ρ_{C}^{∗}V^{C}) ∧ |N_{∧}^{cy}(Π)|

is G-equivariant, and conditions i), ii) and iii) in lemma 1.2 are trivially satisfied. The spectrification T ∧ |N_{∧}^{cy}(Π)| is a cyclotomic spectrum by the remark following theorem 1.2.

Theorem 6.1. Let L be an FSP and Π a pointed monoid. Then there is a natural equivalence of cyclotomic spectra

T (L[Π]) ≃_{G} T (L) ∧ |N_{∧}^{cy}(Π)|.

Here the smash product on the right has \underrightarrow{lim}_{V ⊂U}Ω^{V} (t^{τ} (L)(V) ∧ |N_{∧}^{cy}(Π)|) as its 0’th space.

Proof. We define a map f(\underline{i}, k, V) as the composition

F (S^{i_{0}} ∧ … ∧S^{i_{k}}, S^{V} ∧L(S^{i_{0}})∧ … ∧L(S^{i_{k}}))∧N_{∧,k}^{cy}(Π)
→ F (S^{i_{0}} ∧ … ∧S^{i_{k}}, S^{V} ∧L(S^{i_{0}})∧ … ∧L(S^{i_{k}})∧N_{∧,k}^{cy}(Π))
→ F (S^{i_{0}} ∧ … ∧S^{i_{k}}, S^{V} ∧L[Π](S^{i_{0}})∧ … ∧L[Π](S^{i_{k}})).

The first map is the adjoint of ev ∧ id while the second map is a ‘twist’ map.

The maps f(\underline{i}, k, V) for different \underline{i}’s in the indexing category I^{k+1} are compatible so we obtain maps f(k, V) on the homotopy colimits. It is straight forward to check that these commute with the face and degeneracy maps such that we get maps of the geometric realizations. The maps f(V) which result form a map of cyclotomic prespectra and we obtain a map of the associated cyclotomic spectra

f: T (L) ∧ |N_{∧}^{cy}(Π)| → T (L[Π]).

In order to prove that f is a G-equivalence, we apply lemma 1.5 with j(V) the homotopy fiber of f(V). We claim that f induces an equivalence on C-fixed points for any finite subgroup C ⊂ S^{1}. Indeed let R = ℝC be the regular representation of C. It follows from (the proof of) [BHM] 6.10 that sd_{C} f(\underline{i}, k, lR)^{C} is 2l − 1 connected. By the approximation lemma, [B] 1.5, the same holds for sd_{C} f(k, lR). Now since the C-action is simplicial the C-fixed points of the realization is the realization of the C-fixed points. Therefore the spectral sequence of [S] shows that f(lR)^{C} is homology 2l − 1 connected. But when l ≥ 1 the domain and codomain for f(lR)^{C} are both simply connected and consequently f(lR)^{C} is 2l − 1 connected. Hence J^{C} ≃_{G} ∗ To see that j(V)^{G} ≃_{G} ∗ note that the G-fixed set of t(L)(V) is S^{V^{G}}. Indeed it is those 0-simplices x ∈ t(L)(V)_{0} for which s_{0}x = t_{1}s_{0}x. □

Remarks. (i) We can write the theorem as a statement for RO(G)-graded homology theories,

T (L[Π])_{∗}(X) ≅ T (L)_{∗}(X ∧ |N_{∧}^{cy}(Π)|)

for any G-space X (ii) The theorem shows in particular that the underlying non-equivariant spectra are equivalent. Combined with Bökstedt’s calculation of T (𝔽_{p}) and T (ℤ), cf. 4.2 and [Br], we obtain (ℤ-graded)

T (𝔽_{p}[Π])_{∗} = ⨁_{i≥0} HH_{∗−2i}(𝔽_{p}[Π]),
T (ℤ[Π])_{∗} = HH_{∗}(ℤ[Π]) ⊕ ⨁_{i≥1} HH_{∗−2i+1}(ℤ/i[Π]).

Results of this form has also been proved by T.Pirashvili and A.Lindenstrauss by different methods.

6.2. We evaluate the cyclic bar construction of the pointed monoid Π_{2}, which in view of the above corresponds to dual numbers. First we need a description of the cyclic n-simplex Λ^{n}.

Recall from [G], [J] the isomorphism Λ. ≅ S^{1} × Δ. of cocyclic spaces. It is chosen such that on the right the cosimplicial structure is simply the product of that on Δ. and the identity map on S^{1}. As a consequence the action of τ_{n} is complicated; let ℝ/ℤ be our model of S^{1} and identify Δ^{n} with the convex hull of the standard basis in ℝ^{n+1}, then

τ_{n}(x; u_{0}, …, u_{n}) = (x − u_{0}; u_{1}, …, u_{n}, u_{0}).

We want, however, to choose the isomorphism Λ. ≅ S^{1} × Δ. differently so that the action by τ_{n} becomes diagonal

τ_{n}(x; u_{0}, …, u_{n}) = (x − 1/(n + 1); u_{1}, …, u_{n}, u_{0}).

Let us write Λ^{n} for S^{1} × Δ^{n} with the C_{n+1}-action which is given by Jones’ isomorphism and let S^{1} × Δ^{n} have the diagonal C_{n+1}-action. Then we want a G × C_{n+1}-equivariant homeomorphism F_{n}: Λ^{n} → S^{1} × Δ^{n}, which covers the identity on Δ^{n}. We introduce an auxiliary function f_{n}: Δ^{n} → ℝ and write

F_{n}(x; u_{0}, …, u_{n}) = (x − f_{n}(u_{0}, …, u_{n}); u_{0}, …, u_{n}).

We obtain the following equation for f_{n},

f_{n}(u_{1}, …, u_{n}, u_{0}) − f_{n}(u_{0}, …, u_{n}) = 1/(n + 1) − u_{0}.

For each choice of f_{n}(1, 0, …, 0), the equation has a unique affine solution f_{n}; we choose the affine function f_{n} whose value on (1, 0, …, 0) is 0. This gives us the desired isomorphism Λ. ≅ S^{1} × Δ.. Of course in this description the cosimplicial structure on the right is no longer a product.

We say that a k-simplex v^{i_{0}} ∧ … ∧v^{i_{k}} in N_{∧}^{cy}(Π_{n}) has degree i_{0} + ⋯ + i_{k} and that the basepoint 0 has all degrees. The cyclic structure maps preserve degree, so the simplices of degree s form a cyclic subset N_{∧}^{cy}(Π_{n}; s) and we get a splitting

(6.2.1)  N_{∧}^{cy}(Π_{n}) = ⋁_{s≥0} N_{∧}^{cy}(Π_{n}; s)

as cyclic sets.

Lemma 6.2. As G-spaces |N_{∧}^{cy}(Π_{2}; s)| ≅ S_{+}^{1} ∧_{C_{s}} S(ℝC_{s}), with G acting by multiplication in the first variable.

Proof. Let us write Π_{2} = {0, 1, ϵ}, with ϵ^{2} = 0, and N(s) instead N_{∧}^{cy}(Π_{2}; s). The degree counts the number of ϵ’s in a simplex, so the k-simplices in N(s) different from 0 has exactly s ϵ’s. Thus when k ≤ s − 2 there is only one k-simplex 0 whereas for k = s − 1 there is also the simplex ϵ∧ … ∧ϵ (s times) and this generates N(s) as a cyclic set. It follows that the realization of N(s) is a quotient of Λ^{s−1} and in fact that

|N(s)| ≅ (Λ^{s−1}/∂Λ^{s−1})/C_{s} ≅ (Λ^{s−1}/C_{s})/(∂Λ^{s−1}/C_{s}).

In view of the above description of Λ^{s−1} the claimed homeomorphism is evident. □

For s = 2r even we define an equivariant version of ℝP^{2} defined as the mapping cone

(6.2.2)  S^{1}/C_{r+} ⟶^{π_{r}^{s}} S^{1}/C_{s+} → ℝP^{2}(s).

The regular representation ℝC_{s} splits as ℝ ⊕ W_{s} if s is odd and ℝ ⊕ ℝ_{−} ⊕ W_{s} if s is even, where W_{s} is the maximal complex subrepresentation. We have the

Corollary 6.2. There are G-equivariant homeomorphisms

|N_{∧}^{cy}(Π_{2}; s)| ≅ S^{1}/C_{s+} ∧ S^{W_{s}}, if s is odd
≅ ℝP^{2}(s) ∧ S^{W_{s}}, if s is even,

with G acting diagonally on the spaces on the right.

Proof. When s is odd, S(ℝC_{s}) = S(ℝ ⊕ W_{s}) = S^{W_{s}} and since W_{s} is a complex representation we have the usual G-homeomorphism

S_{+}^{1} ∧_{C_{s}} S^{W_{s}} ⟶^{ζ} (S^{1}/C_{s})_{+} ∧ S^{W_{s}}, ζ(u, w) = (u, uw),

where G acts diagonally on the target. When s = 2r is even, we get similarly

S_{+}^{1} ∧_{C_{s}} S(ℝC_{s}) ≅_{G} (S_{+}^{1} ∧_{C_{s}} S^{ℝ_{−}}) ∧ S^{W_{s}},

and C_{s} acts on ℝ_{−} through the quotient C_{s} → C_{s}/C_{r}, so

S_{+}^{1} ∧_{C_{s}} S^{ℝ_{−}} ≅ (S^{1}/C_{r})_{+} ∧_{C_{s}/C_{r}} S^{R_{−}}.

Finally, the right hand side is the Thom space

Th(S^{1}/C_{r} ×_{C_{s}/C_{r}} ℝ_{−} → S^{1}/C_{s}) = (S^{1}/C_{s}) ∪_{π_{r}^{s}} CS^{1}/C_{r} = ℝP^{2}(s).  □

6.3. We end this paragraph with a partial description of the (realization of the) cyclic sets N_{∧}^{cy}(Π_{n}; s) for n > 2. In particular we calculate their singular homology.

Let R be a commutative ring and suppose A = R[v]/(f(v)), where f(x) is monic. We write x = v ⊗ 1, y = 1⊗v and Δ = (f(x)−f(y))/(x−y). Then there is the following free resolution of A as an A-A-bimodule

0 ← A ⟵^{μ} A ⊗ A ⟵^{x−y} A ⊗ A ⟵^{Δ} A ⊗ A ⟵^{x−y} A ⊗ A ⟵^{Δ} …,

see e.g. [LL]. The Hochschild homology of A is now immediately calculated from the complex

0 ← A ⟵^{0} A ⟵^{f′(x)} A ⟵^{0} A ⟵^{f′(x)} ….

Combined with (6.1.2) we get

H̃_{i}(|N_{∧}^{cy}(Π_{n})|; R) ≅ \{ \begin{matrix}R[v]/(v^{n}), & if i = 0 \\ _{n}R⟨1⟩ ⊕ R⟨v, …, v^{n−1}⟩, & if i > 0 is even \\ R⟨1, v, …, v^{n−2}⟩ ⊕ R/nR⟨v^{n−1}⟩, & if i is odd \end{matrix}

Recall from 6.2 the splitting of N_{∧}^{cy}(Π_{n}) as a cyclic set. It induces a splitting of the realization and we want to calculate the homology of the individual wedge summands |N_{∧}^{cy}(Π_{n}; s)|. We compare the resolution above for A = R[v]/(v^{n}) with the bar-resolution and choose a chain equivalence f_{∗},

… ⟶ A ⊗ A ⟶^{Δ} A ⊗ A ⟶^{x−y} A ⊗ A ⟶^{μ} A ⟶ 0
↓^{f_{2}}  ↓^{f_{1}}  ↓^{f_{0}}  ∥
… ⟶^{b′} A^{⊗4} ⟶^{b′} A^{⊗3} ⟶^{b′} A^{⊗2} ⟶^{μ} A ⟶ 0.

We will not need explicit formulas for f_{i}. The degree defined on N_{∧}^{cy}(Π_{n}) extends such that A, and therefore also A^{⊗s}, become graded rings. Moreover

deg(x − y) = 1, deg Δ = n − 1, deg b′ = 0,

and we immediately get

deg f_{2j} = jn, deg f_{2j+1} = jn + 1.

Next we form the tensor product with the A-A-bimodule A. Since the multiplication μ: A ⊗ A → A has degree 0 the induced chain map f̄_{∗} has deg f̄_{i} = deg f_{i}. We compare with the homology calculation above and get

Lemma 6.3. (i) If (j − 1)n < s < jn then

H̃_{2j−1}(|N_{∧}^{cy}(Π_{n}; s)|; R) ≅ H̃_{2j−2}(|N_{∧}^{cy}(Π_{n}; s)|; R) ≅ R,

(ii) if s = jn then there is an exact sequence

0 → H̃_{2j}(|N_{∧}^{cy}(Π_{n}; s)|; R) → R ⟶^{n} R → H̃_{2j−1}(|N_{∧}^{cy}(Π_{n}; s)|; R) → 0

and these are the only non-zero reduced homology groups.

7. A formula for TC(L[ϵ])

7.1. In §6 we evaluated T (L[ϵ]), the topological Hochschild spectrum. We now determine its fix point structure and give a formula for TC(L[ϵ]). In the first section we recall some equivariant duality theory, and here G may be any compact Lie group.

For any finite subgroup H ⊂ G and any G-spectrum T indexed on a complete G-universe U we have the following duality, natural in T,

(7.1.1)  Σ^{Ad(G)}F (G/H_{+}, T) ≃_{G} T ∧ G/H_{+}.

Here Ad(G) denotes the adjoint representation of G on its Lie algebra and the smash product on the right takes place in GSU.

To define the duality map we choose an embedding of G/H in an orthogonal G-representation V and consider the normal bundle ν. As an H-representation V = L ⊕ L^{⊥}, where L = T_{H}(G/H) is the tangent space. Indeed, H acts by left translation on G/H and hence on L and the embedding identifies L as a sub H-representation of V. Therefore the normal bundle is G ×_{H} L^{⊥} → G/H. In general this is non-trivial.

When H is finite we may identify L with Ad(G). Indeed, left translation by h on G/H coincides with conjugation by h and the projection G → G/H is a local diffeomorphism. Now G/H embeds in V ⊕ L with normal bundle G ×_{H} (L^{⊥} ⊕ L) ≅ G ×_{H} V. The action by G on V gives a trivialization of the normal bundle. Thus the Thom-Pontryagin construction yields a G-map

(7.1.2)  (π_{H}^{G})^{!}: S^{L⊕V} → G/H_{+} ∧ S^{V} ,

and the duality map in (7.1.1) is then given by the composite

F (G/H_{+}, T) ∧ S^{L⊕V} ⟶^{1∧t} F (G/H_{+}, T) ∧ G/H_{+} ∧ S^{V} ⟶^{(ev,1)} T ∧ G/H_{+} ∧ S^{V} .

We refer to [LMS] p. 89 for the proof that this is a G-equivalence. We shall need the

Lemma 7.1. Let H ⊂ K be finite subgroups of G, let π_{H}^{K}: G/H → G/K be the projection and let (π_{H}^{K})^{!}: Σ_{G}^{∞}G/K_{+} → Σ_{G}^{∞}G/H_{+} be the associated equivariant transfer. Then the diagram

Σ^{Ad(G)}F (G/H_{+}, T) ⟶ T ∧ G/H_{+}
↓^{(π_{H}^{K})^{!∗}}  ↓^{1∧π_{H}^{K}}
Σ^{Ad(G)}F (G/K_{+}, T) ⟶ T ∧ G/K_{+}

is G-homotopy commutative.

Proof. We may write (7.1.1) as the composite

Σ^{Ad(G)}Σ_{G}^{∞}G/G_{+} ∧ F (Σ_{G}^{∞}G/H_{+}, T) ⟶^{(π_{H}^{G})^{!}∧1} Σ_{G}^{∞}G/H_{+} ∧ F (Σ_{G}^{∞}G/H_{+}, T) ⟶^{(1,ev)} Σ_{G}^{∞}G/H_{+} ∧ T

where (π_{H}^{G})^{!} is the map of equivariant suspension spectra induced from (7.1.2). The transitivity triangle

\xymatrix{ & Σ_{G}^{∞}G/K_{+} \ar[dd]^{(π_{H}^{K})^{!}} \\ Σ^{Ad(G)}Σ_{G}^{∞}G/G_{+} \ar[ur]^{(π_{K}^{G})^{!}} \ar[dr]^{(π_{H}^{G})^{!}} \\  & Σ_{G}^{∞}G/H_{+}}

is G-homotopy commutative and reduces us to prove the following kind of Frobenius reciprocity: The diagram

\xymatrix{Σ_{G}^{∞}G/H_{+} ∧ F (Σ_{G}^{∞}G/H_{+}, T) \ar[r]^{(ev,1)} & Σ_{G}^{∞}G/H_{+} ∧ T \ar[dd]^{π_{H}^{K}∧1} \\ Σ_{G}^{∞}G/K_{+} ∧ F (Σ_{G}^{∞}G/H_{+}, T) \ar[u]_{(π_{H}^{K})^{!}∧1} \ar[d]^{1∧((π_{H}^{K})^{!})^{∗}} \\ Σ_{G}^{∞}G/K_{+} ∧ F (Σ_{G}^{∞}G/K_{+}, T) \ar[r]^{(ev,1)} & Σ_{G}^{∞}G/K_{+} ∧ T.}

is G-homotopy commutative. This in turn is a straight forward consequence of the standard fact that the square

Σ_{G}^{∞}G/H_{+} ⟶^{(π_{H}^{K},1)} Σ_{G}^{∞}(G/K_{+} ∧ G/H_{+})
↑^{(π_{H}^{K})^{!}}  ↑^{1∧(π_{H}^{K})^{!}}
Σ_{G}^{∞}G/K_{+} ⟶^{Δ} Σ_{G}^{∞}(G/K_{+} ∧ G/K_{+})

is G-homotopy commutative. □

7.2. We return to the calculation of TC(L[ϵ]). Again G will be the circle group. Let T̃(L[ϵ]) be the reduced topological Hochschild homology of L[ϵ], i.e. the homotopy fiber

T̃(L[ϵ]) = hofiber(T (L[ϵ]) → T (L)), ϵ ↦ 0.

Recall that for any representation W ⊂ U we write T_{W} for the smash product G-spectrum T ∧ S^{W}. Then from §6 we have the cofibration sequence of G-spectra

⋁_{r≥1} T (L)_{W_{2r}} ∧ S^{1}/C_{r+} → ⋁_{s≥1} T (L)_{W_{s}} ∧ S^{1}/C_{s+} → T̃(L[ϵ]),

where the first maps takes the summand r to the summand s = 2r by the map induced from the projection π_{r}^{s}: S^{1}/C_{r} → S^{1}/C_{s}. If we take C_{n}-fixed points we still get a cofibration sequence. Moreover, we may replace the wedge sums by the corresponding products and get

(7.2.1)  ∏_{r≥1} (T (L)_{W_{2r}} ∧ S^{1}/C_{r+})^{C_{n}} → ∏_{s≥1} (T (L)_{W_{s}} ∧ S^{1}/C_{s+})^{C_{n}} → T̃(L[ϵ])^{C_{n}}.

This is because T (L)_{W_{s}} ∧ S^{1}/C_{s+} is (s − 2)-connected and hence by theorem 1.2 so is its C_{n}-fixed sets.

Lemma 7.2. For any G-spectrum T indexed on U the inclusion of the G-fixed set induces a natural map

(T ∧ S^{1}/C_{s+})^{G} → holim_{\overleftarrow{F}}(T ∧ S^{1}/C_{s+})^{C_{n}}

which becomes an equivalence after profinite completion. Here the limit on the right runs over the inclusion maps and the smash products are taken in GSU.

Proof. The adjoint representation of G is trivial so the duality of (7.1.1) becomes

(T ∧ S^{1}/C_{s+})^{C_{n}} ≃ ΣF (S^{1}/C_{s+}, T)^{C_{n}}

For C_{n} ⊃ C_{s} we have a cofibration sequence of C_{n}-spaces

C_{n}/C_{s+} → S^{1}/C_{s+} → |S^{1}/C_{n}| ∧ C_{n}/C_{s+},

where the bars on the right indicate S^{1}/C_{n} with trivial G-action. This implies a cofibration sequence of function spectra

F (|S^{1}/C_{n}| ∧ C_{n}/C_{s+}, T)^{C_{n}} → F (S^{1}/C_{s+}, T)^{C_{n}} → F (C_{n}/C_{s+}, T)^{C_{n}},

or equivalently the cofibration sequence

F (S^{1}/C_{n}, T^{C_{s}}) ⟶ F (S^{1}/C_{s+}, T)^{C_{n}} ⟶^{ev_{1}} T^{C_{s}}

and one readily verifies commutativity in the diagram

(7.2.2)  \begin{matrix}F (S^{1}/C_{nr}, T^{C_{s}}) & ⟶ & F (S^{1}/C_{s+}, T)^{C_{nr}} & ⟶^{ev_{1}} & T^{C_{s}} \\ ↓^{(π_{n}^{nr})^{∗}} & & ↓^{F_{r}} & & ∥ \\ F (S^{1}/C_{n}, T^{C_{s}}) & ⟶ & F (S^{1}/C_{s+}, T)^{C_{n}} & ⟶^{ev_{1}} & T^{C_{s}}. \end{matrix}

The homotopy limit of the left hand term is

holim_{\overleftarrow{n}} F (S^{1}/C_{n}, T^{C_{s}}) = F (holim_{\overrightarrow{n}} S^{1}/C_{n}, T^{C_{s}}) = F (S^{1}ℚ, T^{C_{s}}),

where S^{1}ℚ is a Moore space with integral homology ℚ, concentrated in degree one. It vanishes after profinite completion:

F (S^{1}ℚ, T)^{∧} = F (S^{−1}ℚ/ℤ, F (S^{1}ℚ, T)) = F (S^{−1}ℚ/ℤ ∧ S^{1}ℚ, T) ≃ ∗.

Finally, the evaluation maps in (7.2.2) are split by the inclusion of the G-fixed set,

T^{C_{s}} = F (S^{1}/C_{s+}, T)^{G} → F (S^{1}/C_{s+}, T)^{C_{s}}

and the lemma follows by one more application of (7.1.1). □

Proposition 7.2. After profinite completion there is a cofibration sequence of spectra

Σ holim_{\overleftarrow{R}} T (L)_{W_{s}}^{C_{s/2}} ⟶^{V_{2}} Σ holim_{\overleftarrow{R}} T (L)_{W_{s}}^{C_{s}} → T̃C(L[ϵ]),

where the homotopy limits runs over the natural numbers ordered by division and where T (L)_{W_{s}}^{C_{s/2}} is a point when s is odd.

Proof. The lemma gives us a cofibration sequence for T̃F(L[ϵ]) = holim_{\overleftarrow{F}} T̃(L[ϵ])^{C_{n}}. Indeed, from lemma 7.1 we have the commutative square

ΣT^{C_{r}} ⟶^{≃} (T ∧ S^{1}/C_{r+})^{G}
↓^{V_{2}}  ↓^{(π_{r}^{2r})_{∗}}
ΣT^{C_{2r}} ⟶^{≃} (T ∧ S^{1}/C_{2r+})^{G},

where, we remember, V_{2} = ((π_{r}^{2r})^{!})^{∗}. Therefore, upon taking homotopy limits over the inclusion maps in (7.2.1), we get the cofibration sequence

(7.2.3)  ∏_{r≥1} ΣT (L)_{W_{2r}}^{C_{r}} ⟶^{V_{2}} ∏_{s≥1} ΣT (L)_{W_{s}}^{C_{s}} → T̃F(L[ϵ]),

where the first map takes the factor r to the factor s = 2r by the map V_{2}.

The restriction maps

R_{n}: T̃(L[ϵ])^{C_{s}} → T̃(L[ϵ])^{C_{s/n}}

induce self maps of T̃F(L[ϵ]), again denoted R_{n}, and

T̃C(L[ϵ]) = T̃F(L[ϵ])^{hR},

the homotopy fixed points of the multiplicative monoid of natural numbers acting through the maps R_{n}, n ≥ 1. When n divides s,

ρ_{C_{n}}: ρ_{C_{n}}^{∗}S^{1}/C_{s} ⟶^{≅} S^{1}/C_{s/n}, ρ_{C_{n}}^{∗}W_{s}^{C_{n}} = W_{s/n}

and R_{n} maps a factor s (resp. r) in (7.2.1) to the factor s/n (resp. r/n). The factors with s not divisible by n are annihilated by R_{n}. In fact, we have

R_{n} = ΣR_{n,W_{s}}: ΣT (L)_{W_{s}}^{C_{s}} → ΣT (L)_{W_{s/n}}^{C_{s/n}},

where R_{n,W_{s}} are the restriction maps of (1.2.3) associated with T (L). This is direct from the discussion of N_{∧}^{cy}(Π_{2}) in 6.1. Hence the claim. □

Addendum 7.2. After p-completion there are equivalences of spectra

(i) For p odd

T̃C(L[ϵ]) ≃ ∏_{(d,2p)=1} Σ holim_{\overleftarrow{R}} T (L)_{W_{p^{n}d}}^{C_{p^{n}}}

(ii) For p = 2

T̃C(L[ϵ]) ≃ ∏_{(d,2)=1} Σ cofiber(V_{2}: holim_{\overleftarrow{R}} T (L)_{W_{2^{n}d}}^{C_{2^{n−1}}} → holim_{\overleftarrow{R}} T (L)_{W_{2^{n}d}}^{C_{2^{n}}}).

Here W_{s} ⊂ ℝC_{s} is the maximal complex subrepresentation. Moreover, the projection map

holim_{\overleftarrow{R}} T (L)_{W_{p^{n}d}}^{C_{p^{n}}} → T (L)_{W_{p^{m}d}}^{C_{p^{m}}}

is (p^{m+1}d − 1)-connected for p odd and (2^{m+1}d − 2)-connected for p = 2.

Proof. For every k prime to p the map

(7.2.4)  ∏_{d|k} R_{k/d}F_{d}: T (L)_{W_{p^{n}k}}^{C_{p^{n}k}} → ∏_{d|k} T (L)_{W_{p^{n}d}}^{C_{p^{n}}}

becomes an equivalence after p-completion. This follows from the proof of lemma 3.3. Note that (7.2.4) induces an equivalence after p-completion

holim_{\overleftarrow{R}} T (L)_{W_{s}}^{C_{s}} ⟶^{≃} ∏_{(d,p)=1} holim_{\overleftarrow{R}} T (L)_{W_{p^{n}d}}^{C_{p^{n}}}.

We evaluate the cofiber of the map

V_{2}: T (L)_{W_{s}}^{C_{s/2}} → T (L)_{W_{s}}^{C_{s}}

under the equivalence of (7.2.4).

First, suppose that p is an odd prime. The composition

T (L)_{W_{s}}^{C_{s/2}} ⟶^{V_{s}} T (L)_{W_{s}}^{C_{s}} ⟶^{F_{2}} T (L)_{W_{s}}^{C_{s/2}}

induces multiplication by 2 on homotopy groups. Hence the map from the cofiber of V_{2} to the homotopy fiber of F_{2} becomes an equivalence after p-completion. We write s = p^{n}2k with (k, p) = 1 and consider the commutative square

T (L)_{W_{p^{n}2k}}^{C_{p^{n}2k}}  ⟶^{F_{2}}  T (L)_{W_{p^{n}k}}^{C_{p^{n}2k}}
↓^{∏ R_{2k/d}F_{d}}  ↓^{∏ R_{2k/d}F_{d}}
∏_{d|2k} T (L)_{W_{p^{n}d}}^{C^{p^{n}}} ⟶^{pr_{even}} ∏_{d|2k, d even} T (L)_{W_{p^{n}d}}^{C^{p^{n}}}.

It shows that after p-completion

cofiber(T (L)_{W_{s}}^{C_{s/2}} → T (L)_{W_{s}}^{C_{s}}) ≃ ∏_{d|2k, d odd} T (L)_{W_{p^{n}d}}^{C^{p^{n}}}.

Taking homotopy limits over the restriction maps as s runs through the natural numbers we get (i). For p = 2 we have a commutative square

T (L)_{W_{2^{n}k}}^{C_{2^{n−1}k}}  ⟶^{V_{2}} T (L)_{W_{2^{n}k}}^{C_{2^{n}k}}
↓^{∏ R_{k/d}F_{d}}  ↓^{∏ R_{k/d}F_{d}}
∏_{d|k} T (L)_{W_{2^{n}d}}^{C_{2^{n−1}}} ⟶^{V_{2}} ∏_{d|k} T (L)_{W_{2^{n}d}}^{C_{2^{n}}}

from which (ii) follows by taking homotopy limits over the restriction maps. Finally, the claimed connectivity of the projection map follows from theorem 1.2 since taking homotopy orbits preserves connectivity. □

8. Topological cyclic homology of k[ϵ]

8.1. We use the scheme set up in §4 to evalulate the fixed point spectra T (k)_{W}^{C_{p^{n}}} for any complex representation W ⊂ U. We first consider the case k = 𝔽_{p} where we use that (4.1.2) gives a diagram of cofibration sequences

(8.1.1)  \begin{matrix}(T (𝔽_{p})_{W})_{hC_{p^{n}}} & ⟶^{N} & T (𝔽_{p})_{W}^{C_{p^{n}}} & ⟶^{R} & T (𝔽_{p})_{ρ_{Cp}^{∗} W^{Cp}}^{C_{p^{n−1}}} \\ ∥ & & ↓^{Γ_{n,W}} & & ↓Γ̂_{n,W} \\ (T (𝔽_{p})_{W})_{hC_{p^{n}}} & ⟶^{N} & T (𝔽_{p})_{W}^{hC_{p^{n}}} & ⟶^{R} & ℍ̂(C_{p^{n}}; T (𝔽_{p})_{W}). \end{matrix}

Indeed, lemma 1.1 and (1.2.2) gives us the G-equivalences

ρ_{C_{p}}^{#}Φ_{C_{p}}T (𝔽_{p})_{W} ≃_{G} ρ_{C_{p}}^{#}Φ^{C_{p}}T (𝔽_{p})_{W^{Cp}} ≃_{G} T (𝔽_{p})_{ρ_{Cp}^{∗} W^{Cp}} .

We start we the following

Lemma 8.1. Let T be a C-spectrum and X be a finite C-CW-complex. Then the inclusion of the C-singular set X^{sing} ⊂ X induces an equivalence

ℍ̂(C; T ∧ X) ≃ ℍ̂(C; T ∧ X^{sing}).

Proof. Recall from 4.1 that ℍ̂(C; T) is the C-fixed point spectrum of the C-equivariant spectrum

κ_{C}(T) = ẼC_{+} ∧ F (EC_{+}, T).

We prove by induction over the C-cells that κ_{C}(T ∧ (X/X^{sing})) is C-contractible. Since X/X^{sing} is a free C-CW-complex in the based sense, it is enough to show that κ_{C}(T ∧ C_{+}) is C-contractible. Now by (7.1.1)

F (EC_{+}, T ∧ C_{+}) ≃_{C} F (EC_{+}, F (C_{+}, T)) ≅ F (EC_{+} ∧ C_{+}, T),

and EC_{+} ∧ C_{+} is C-contractible. Hence κ_{C}(T ∧ C_{+}) is C-contractible. □

Corollary 8.1. The map Γ̂_{1,W} induces isomorphisms on homotopy groups in dimensions greater than or equal to dim W^{C_{p}}.

Proof. We consider the following commutative diagram

(Φ^{C_{p}}T (𝔽_{p}))_{W^{Cp}} ⟶ Φ^{C_{p}}(T (𝔽_{p})_{W^{Cp}}) ⟶ Φ^{C_{p}}(T (𝔽_{p})_{W})
↓^{γ̂∧1}  ↓γ̂  ↓γ̂
ℍ̂(C_{p}; T (𝔽_{p}))_{W^{Cp}} ⟶ ℍ̂(C_{p}; T (𝔽_{p})_{W^{Cp}}) ⟶ ℍ̂(C_{p}; T (𝔽_{p})_{W}).

The right hand horizontal maps are equivalences by lemma 1.1 and lemma 8.1, respectively, and the left hand horizontal maps are equivalences because S^{W^{Cp}} is a C_{p}-trivial finite C_{p}-CW-complex. Now proposition 4.3 shows that the left hand vertical map induces an isormorphism on π_{i}(−) when i ≥ dim W^{C_{p}}, and the corollary follows. □

We next consider the spectral sequence of 4.1 for π_{∗}(ℍ̂(C_{p^{n}}; T (𝔽_{p})_{W}); 𝔽_{p}). It has E^{2}-term

Ê_{W}^{2} = (Λ_{𝔽_{p}}{u_{n}} ⊗ S_{𝔽_{p}}{t, t^{−1}} ⊗ Λ_{𝔽_{p}}{e_{1}} ⊗ S_{𝔽_{p}}{σ̄})[W],

where the decoration [W] indicates that the bidegrees are shifted (0, dim W). The spectral sequence is a module over the spectral sequence Ê^{r} of (4.3.2), and one may repeat the proof of lemma 4.3 and show that the differentials are generated from d^{2}e_{1}[W] = tσ̄[W] in the module structure over Ê^{r}. It follows that

(8.1.2)  π_{∗}(ℍ̂(C_{p^{n}}; T (𝔽_{p})_{W}); 𝔽_{p}) ≅ (Λ_{𝔽_{p}}{u_{n}} ⊗ S_{𝔽_{p}}{t, t^{−1}})[W],

where again [W] indicates that the degrees are shifted up by dim W. Note also that the proof of corollary 4.3 shows that the integral homotopy groups of ℍ̂(C_{p^{n}}; T (𝔽_{p})_{W}) are cyclic ℤ_{p}-modules.

Addendum 8.1. The maps Γ_{n,W} and Γ̂_{n,W} of (8.1.1) induces isomorphisms on homotopy groups in dimensions greater than or equal to dim W^{C_{p}}.

Proof. We prove the claim by induction over n starting from the case n = 1, which was proved in corollary 8.1. The induction step uses the diagram

T (𝔽_{p})_{ρ_{Cp}^{∗} W^{Cp}}^{C_{p^{n−1}}} ⟶^{Γ̂_{n,W}} ρ_{C_{p}}^{∗} ℍ̂(C_{p}; T (𝔽_{p})_{W})^{C_{p^{n−1}}}
↓^{Γ_{n−1,ρ{∗ \atop Cp} W^{Cp}}}  ↓^{G_{n,W}}
T (𝔽_{p})_{ρ_{Cp}^{∗} W^{Cp}}^{hC_{p^{n−1}}} ⟶^{Γ̂_{n,W}} ρ_{C_{p}}^{∗} ℍ̂(C_{p}; T (𝔽_{p})_{W})^{hC_{p^{n−1}}}

By induction the left hand vertical map induces isomorphism on π_{i}(−) for i ≥ dim W^{C_{p^{2}}} . Moreover, since taking homotopy fixed sets preserves connectivity, it follows from corollary 8.1 that the lower horizontal map induces isomorphism on π_{i}(−) for i ≥ dim W^{C_{p}}. Finally, G_{n,W} is an equivalence. Indeed, when W = 0 this was proved in 4.3, and given (8.1.2), the argument of 4.3 extends verbatim to the case of a general W. This proves the induction step, and hence the addendum. □

We can now repeat the proof of lemma 4.4 and solve the spectral sequence

Ê_{W}^{2} = (Λ_{𝔽_{p}}{u_{n}} ⊗ S_{𝔽_{p}}{t, t^{−1}} ⊗ S_{𝔽_{p}}{σ})[W] ⇒ π_{∗}ℍ̂(C_{p^{n}}; T (𝔽_{p})).

It is a module over the spectral sequence Ê^{r} of (4.3.2) and the differentials are generated from d^{2n−1}u_{n}[W] = t^{n+1}σ^{n}[W]. The extensions in the passage from Ê_{W}^{∞} to the actual homotopy groups are maximally non-trivial so we obtain

(8.1.3)  π_{∗}ℍ̂(C_{p^{n}}; T (𝔽_{p})_{W}) ≅ S_{ℤ/p^{n}}{σ̂, σ̂^{−1}}[W].

We can now evaluate the promised homotopy groups.

Proposition 8.1. Let k be a perfect field of positive characteristic and let W ⊂ U be a complex representation. The non-zero integral homotopy groups of T (k)_{W}^{C_{p^{n}}} are concentrated in even degrees greater than or equal to dim W^{C_{p^{n}}}. They are given by

π_{2i}T (k)_{W}^{C_{p^{n}}} = \{ \begin{matrix}W_{s}(k), & dim W^{C_{p^{n−(s−1)}}} ≤ 2i < dim W^{C_{p^{n−s}}} , s = 1, …, n \\ W_{n+1}(k), & 2i ≥ dim W. \end{matrix}

Moreover, the maps

F: π_{2i}T (k)_{W}^{C_{p^{n}}} → π_{2i}T (k)_{W}^{C_{p^{n−1}}} , V: π_{2i}T (k)_{W}^{C_{p^{n−1}}} → π_{2i}T (k)_{W}^{C_{p^{n}}}

are the Frobenius F: W_{s}(k) → W_{s−1}(k) and the Verschiebung V: W_{s−1}(k) → W_{s}(k), respectively.

Proof. First, suppose k = 𝔽_{p}. We let W̃ denote the representation of C_{p^{n+1}} on W through the reduction map C_{p^{n+1}} → C_{p^{n}}. Then W = ρ_{C^{p}}^{#} W̃^{C_{p}} and addendum 8.1 and (8.1.3) shows that

π_{i}T (𝔽_{p})_{W}^{C_{p^{n}}} = π_{i}ℍ̂(C_{p^{n+1}}; T (𝔽_{p})W̃) = ℤ/p^{n+1},

when i ≥ dim W and even. By theorem 1.2 the restriction map

R_{n,W}: T (𝔽_{p})_{W}^{C_{p^{n}}} → T (𝔽_{p})_{ρ_{Cp}^{∗} W^{Cp}}^{C_{p^{n−1}}}

is (dim W − 1)-connected, and hence a downward induction on n gives the claimed homotopy groups. One may repeat the proof of proposition 4.4 to see that F and V are as claimed.

Next, let k be any perfect field with char k = p. The proof of theorem 4.5 shows that

π_{∗}T (k)_{W}^{C_{p^{n}}} ≅ W_{n+1}(k) ⊗ π_{∗}T (𝔽_{p})_{W}^{C_{p^{n}}} .

Indeed, T (k)_{W}^{C_{p^{n}}} is a T (k)^{C_{p^{n}}} -module spectrum, so in particular, the homotopy groups are W_{n+1}(k)-modules. Since W_{n+1}(k) ⊗ W_{s}(𝔽_{p}) ≅ W_{s}(k) we see that the homotopy groups of T (k)_{W}^{C_{p^{n}}} are as stated. Finally, the diagram

W_{n+1}(k) ⊗ π_{∗}T (𝔽_{p})_{W}^{C_{p^{n}}} ⟶^{≅} π_{∗}T (k)_{W}^{C_{p^{n}}}
↓^{F ⊗F}  ↓^{F}
W_{n}(k) ⊗ π_{∗}T (𝔽_{p})_{W}^{C_{p^{n−1}}} ⟶^{≅} π_{∗}T (k)_{W}^{C_{p^{n−1}}}

commutes, and the proposition follows. □.

8.2. In this section k is a perfect field of characteristic p > 0. Let n = n(i, d) be the unique positive integer with p^{n−1}d ≤ i < p^{n}d.

Theorem 8.2. The homotopy groups of T̃C(k[ϵ]) are concentrated in odd positive degrees. If char k is odd, then

T̃C_{i}(k[ϵ]) ≅ ⨁_{{(d,2p)=1 \atop 1≤d≤i}} W_{n(i,d)}(k), i odd

and if char k = 2, then

T̃C_{i}(k[ϵ]) ≅ k^{⊕(i+1)/2}, i odd.

Proof. Since k is an 𝔽_{p}-algebra TC(k[ϵ]) ≃ TC(k[ϵ])p^{∧} ≃ TC(k[ϵ]; p)p^{∧} and we use addendum 7.2 with L = k̃: for p odd,

(8.2.1)  T̃C(k[ϵ]) ≃ ∏_{(d,2p)=1} Σ holim_{\overleftarrow{R}} T (k)_{W_{p^{m}d}}^{C_{p^{m}}},

and by theorem 1.2,

π_{i}Σ holim_{\overleftarrow{R}} T (k)_{W_{p^{m}d}}^{C_{p^{m}}} ≅ π_{i−1}T (k)_{W_{p^{n}d}}^{C_{p^{n}}}, for i < dim W_{p^{n+1}d} + 1.

On the other hand, if further i − 1 ≥ dim W_{p^{n+1}d}^{C_{p}} = p^{n}d − 1 then by proposition 8.1

π_{i−1}T (k)_{W_{p^{n}d}}^{C_{p^{n}}} ≅ W_{n+1}(k)

when i is odd, and the groups vanish when i is even. Thus for p^{n}d ≤ i < p^{n+1}d and i odd, the d’th factor in (8.2.1) contribute one copy of W_{n}(k). For i < d − 1 the d’th factor does not contribute. This finishes the proof when char k is odd.

Assume now that char k = 2, where by addendum 7.2 (ii),

(8.2.2)  T̃C(k[ϵ]) ≃ ∏_{(d,2)=1} Σ cofiber (V_{2}: holim_{\overleftarrow{R}} T (k)_{W_{2^{m}d}}^{C_{2^{m−1}}} → holim_{\overleftarrow{R}} T (k)_{W_{2^{m}d}}^{C_{2^{m}}}).

This time dim W_{2^{n}d} = 2^{n}d − 2, and the projections

π_{i−1} holim_{\overleftarrow{R}} T (k)_{W_{2^{m}d}}^{C_{2^{m}}} → π_{i−1}T (k)_{W_{2^{n}d}}^{C_{2^{n}}},
π_{i−1} holim_{\overleftarrow{R}} T (k)_{W_{2^{m}d}}^{C_{2^{m−1}}} → π_{i−1}T (k)_{W_{2^{n}d}}^{C_{2^{n−1}}} ,

are isomorphisms when i < dim W_{2^{n+1}d} − 1. We have left to evaluate the Verschiebung map

V_{2}: π_{i−1}T (k)_{W_{2^{n}d}}^{C_{2^{n−1}}} → π_{i−1}T (k)_{W_{2^{n}d}}^{C_{2^{n}}}.

By proposition 8.1

π_{i−1}T (k)_{W_{2^{n}d}}^{C_{2^{n−1}}} ≅ W_{n}(k),
π_{i−1}T (k)_{W_{2^{n}d}}^{C_{2^{n}}} ≅ W_{n+1}(k),

for i ≥ dim W_{2^{n}d} + 1 and i odd, and they vanish for i even. Moreover, the Verschiebung map on the left corresponds to the Verschiebung map on Witt vectors, cf. 2.1. This in an injection with cokernel k ≅ W_{n+1}(k)/W_{n}(k). Hence for n ≥ 1 and an odd i with 2^{n}d − 1 ≤ i < 2^{n+1}d − 1, the d’th factor in (8.2.2) contributes one copy of k to T̃C(k[ϵ]). For d ≤ i < 2d − 1

π_{i}Σ holim_{\overleftarrow{R}} T (k)_{W_{2^{m}d}}^{C_{2^{m}}} ≅ π_{i−1}T (k)_{W_{d}} ≅ k,
π_{i}Σ holim_{\overleftarrow{R}} T (k)_{W_{2^{m}d}}^{C_{2^{m−1}}} = 0

which gives one copy of k in the d’th factor of (8.2.2) when i is odd. Finally, for i < d there is no contribution from the d’th factor. This proves the case char k = 2. □

We are now ready to prove theorem E of the introduction.

Proof of theorem E. In view of theorem 8.2 above it suffices to show for char k = p, an odd prime, that

(8.2.3)  W_{2j−1}(k)/V_{2}W_{j−1}(k) ≅ ⨁_{{(d,2p)=1 \atop 1≤d≤2j−1}} W_{n(2j−1,d)}(k).

For any ℤ_{(p)}-algebra R, and in particular for R = k, we have the Artin-Hasse exponential

E(X) = exp(∑_{s=0}^{∞} X^{p^{s}}/p^{s}) = ∏_{(d,p)=1}(1 − X^{d})^{−μ(d)/d} ∈ W(R),

where μ is the Möbius function given by μ(d) = 0 if d is divisible by a prime square, μ(p_{1} ⋯ p_{r}) = (−1)^{r} if p_{1}, …, p_{r} are distinct primes, and μ(1) = 1. It gives rise to an injective map of sets

Ê: ∏_{n=0}^{∞} R → W(R); Ê(a_{0}, a_{1}, …)(X) = ∏_{s=0}^{∞} E(a_{s}X^{p^{s}}),

whose image is a (non-unital) subring of W(R), isomorphic to the ring of p-typical Witt vectors W (R) (in the induced ring structure).

For any d ≥ 1 with (d, p) = 1 we consider the following slight modification of Ê,

Ê_{d}(a_{0}, a_{1}, …)(X) = ∏_{s=0}^{∞} E(a_{s}X^{p^{s}d})^{1/d},

which again is a (non-unital) ring homomorphism Ê_{d}: W (R) → Ŵ (R). Is is not hard to see that any p(X) ∈ W(R) can be written uniquely as

p(X) = ∏_{n=1}^{∞} E(a_{n}X^{n}),

so using all Ê_{d} we get a decomposition of the ring W(R) as a product of rings

(8.2.4)  W(R) ≅ ∏_{(d,p)=1} W (R).

The i’th Verschiebung map V_{i}: W(R) → W(R) is the map given by V_{i}(f(X)) = f(X^{i}). The quotient W_{i}(R) = W(R)/V_{i}W(R) is again a ring, the ring of big Witt vectors of length i. The V -filtration of W (R) can be compared to the V -filtration on W(R), through Ê_{d}. One finds that

Ê_{d}(V^{n}W (R)) ⊂ V_{i}W(R) ⇔ i < p^{n}d.

When p is odd the image of V_{2} can be compared with the splitting (8.2.4). Indeed, one finds that V_{2}W(R) corresponds to the factors W (R) with d even. When R = k this gives us (8.2.3) and hence theorem E. □

We owe to M. Bökstedt the formula (8.2.3).

Appendix A: Spectra and prespectra

A.1. This appendix concerns the passage from G-prespectra to G-spectra. We introduce a class of good G-prespectra and a functor which replaces a G-prespectrum by one which is good.

The forgetful functor l: GSU → GPU has a left adjoint L: GPU → GSU, which to a G-prespectrum t associates a G-spectrum Lt, see [LMS]. The need for such a functor comes from the fact that many spacewise constructions leave the subcategory of G-spectra. As an example let T be a G-spectrum and X a G-space, then the obvious map

X ∧ T (V) → Ω^{W −V} (X ∧ T (W))

is not in general a homeomorphism. Similarly, a spacewise (homotopy) colimit of G-spectra is not in general a G-spectrum. However, for general G-prespectra the functor L is rather badly behaved; for example one might very well have

π_{n}Lt(V) ≠ \underrightarrow{lim}_{W ⊂U}π_{n}Ω^{W −V} t(W).

We call a G-prespectrum t good if the structure maps

σ̃: Σ^{W −V} t(V) → t(W)

are all closed inclusions. Goodness is preserved by smash products and homotopy colimits, and since the adjoints σ: t(V) → Ω^{W −V} t(W) are inclusions, the spectrification functor takes the simple form,

Lt(V) = \underrightarrow{lim}_{W ⊂U}Ω^{W −V} t(W).

In particular the homotopy groups are what one expects.

Now let t be any G-prespectrum indexed on U and let V ⊂ U be a f.d. sub inner product space. The sub inner product spaces Z ⊂ V form a poset and hence a category, and for Z_{1} ⊂ Z_{2} ⊂ V we have a map of G-spaces

Σ^{V −Z_{2}} σ̃: Σ^{V −Z_{1}}t(Z_{1}) → Σ^{V −Z_{2}}t(Z_{2}).

These data specify a functor and we define

Definition A.1. The thickening t^{τ} of a G-prespectrum t, is the G-prespectrum with V ’th space the homotopy colimit

t^{τ} (V) = holim_{\overrightarrow{Z⊂V}} Σ^{V −Z}t(Z)

and structure maps the compositions

Σ^{W −V} holim_{\overrightarrow{Z⊂V}} Σ^{V −Z}t(Z) ≅ holim_{\overrightarrow{Z⊂V}} Σ^{W −Z}t(Z) → holim_{\overrightarrow{Z⊂W}} Σ^{V −Z}t(Z),

where the last map is induced by the inclusion of the category of sub inner product spaces of V in that of W.

Lemma A.1. t^{τ} is good and comes with a map π: t^{τ} → t of G-prespectra, which is a spacewise G-equivalence.

Proof. The map on homotopy colimits induced by the inclusion of a subcategory is always a closed Gcofibration, hence σ^{τ}: Σ^{W −V} t^{τ} t(V) → t(W) is a cofibration. Since the category of sub inner product spaces of V has V as terminal object, there is a natural G-map π(V): t^{τ} (V) → t(V), with ι(V): t(V) → t^{τ} (V) as G-homotopy inverse. Finally the maps π(V) form a map of G-prespectra. □

Lemma A.2. If T is a cyclotomic spectrum, then T^{τ} is a cyclotomic prespectrum. □

Note that the functor (−)^{τ} produces extremely large spaces, because we use all sub inner product spaces of V. A smaller version is considered in [LMS] p. 37. Alternatively one could topologize the index category.

We call a G-spectrum good if it is the spectrification of a good G-prespectrum, e.g.

T (V) = \underrightarrow{lim}_{W ⊂U}Ω^{W −V} t^{τ} (W).

Let us note that a good G-spectrum is not good regarded as a G-prespectrum. We claim that smashing with a G-space X and taking homotopy colimits preserve good G-spectra. To see this we recall that if a: GPU → GPU is a functor, then the associated functor A: GSU → GSU is the composite Lal. If a has a right adjoint b, then B is the right adjoint of A, and if moreover b preserves G-spectra, i.e. b(lT) ≅ lB(T) for any T ∈ GSU, then

A(Lt) ≅ La(t).

Smash products and homotopy colimits are examples of such functors a. Moreover they both preserve good G-prespectra, and the claim follows.

Appendix B: Continuity properties of K-theory

In this appendix we prove theorem C (iii) of the introduction. The proof amounts to a recollection of facts due primarily to Suslin and coworkers, [Su], [SuY].

Let k be a perfect field of positive characteristic p, and W (k) its Witt-vectors. We consider finite W (k)- algebras, i.e. W (k)-algebras whose underlying W (k)-module is finitely generated.

Theorem B.1. For a finite W (k)-algebra A,

K(A)p^{∧} ≃ TC(A)p^{∧},

where p = char(k).

In view of theorem C (i), (ii) the statement is equivalent to the continuity statement that

(B.2)  K(A)p^{∧} ≃ K^{top}(A)p^{∧},

where the right-hand side is the homotopy limit of K(A/p^{s}A)p^{∧}. We begin by reducing to a special case. Let F denote the fraction field of the local ring W (k), and let E = A ⊗_{W (k)} F.

Lemma B.3. If theorem B.1 is true when E is semisimple then it it true in general.

Proof. Let J(E) be the radical of E. It is nilpotent since E is finite dimensional over F, hence artinian. Then J = J(E) ∩ A is a nilpotent ideal of A, so by theorem A of the introduction the diagram

K(A) ⟶^{trc} TC(A)
↓  ↓
K(A/J) ⟶^{trc} TC(A/J)

is homotopy cartesian after p-completion. But A/J is finite over W (k) and

A/J ⊗_{W (k)} F = E/J(E)

is semisimple. □

So from now on we assume that E = A ⊗_{W (k)} F is semisimple, and hence

(B.4)  E = ∏_{i=1}^{t} M_{l_{i}}(D_{i})

for certain division algebras whose centers F_{i} are finite extensions of F. If Δ_{i} ⊂ D_{i} is the maximal order of D_{i}, cf. [R], ch. 5, then

B = ∏_{i=1}^{t} M_{l_{i}}(Δ_{i})

is the maximal order in E, and A ⊂ B. As F comes from W (k) by inverting p and A ⊗_{W (k)} F = B ⊗_{W (k)} F, p^{s}B ⊂ A for some integers. We give E the topology whose neighborhoods of 0 has {p^{i}A} or equivalently {p^{i}B} as a basis. Let GL_{n}(A, p^{i}A) be the kernel of the reduction map

GL_{n}(A) → GL_{n}(A/p^{i}A).

Then {GL_{n}(A, p^{i}A)} is a basis of the neighborhoods of 1 in GL_{n}(E).

Suppose now first that A is commutative, and consider the variety

X_{i,j}(E) = GL_{n}(E) × ⋯ × GL_{n}(E), i factors.

Let O_{n,i}(E) and O_{n,i}^{c}(E) denote the germs at 1 of rational and continuous E-valued functions on X_{n,i}(E), and let M_{n,i}(E) and M_{n,i}^{c}(E) be the maximal ideals of functions which vanish at 1. To prove (B.2) it suffices to show that the natural map

(B.5)  H_{k}(GL(A); 𝔽_{p}) → \underleftarrow{lim} H_{k}(GL(A/p^{i}A); 𝔽_{p})

is an isomorphism (Here GL(A) is considered as a discrete group). Indeed if this is true with 𝔽_{p} coefficients then it is true for p-adic coefficients, and the pro-Hurewicz theorem of [P] supplies the corresponding theorem for p-completed K-theory.

In section 3 of [Su], (B.5) is derived from the following two statements (B.6)

(i) H̃_{k}(GL(O_{n,i}^{c}(E), M_{n,i}^{c}(E)); 𝔽_{p}) = 0 (ii) H_{k}(GL_{n}(A/p^{σ}); 𝔽_{p}) → H_{k}(GL(A/p^{σ}); 𝔽_{p}) are isomorphisms for n ≫ k and 1 ≤ σ ≤ ∞ (A/p^{∞}A =

A).

A few words of explanation are in order. Write G = GL(O_{n,i}^{c}(E), M_{n,i}^{c}(E)). An element g ∈ G lies in GL_{r}(O_{n,i}^{c}(E), M_{n,i}^{c}(E)) for some r ≥ n, say, and g amounts to a continuous germ from (GL_{n}(E)^{i}, 1) to (GL_{r}(E), 1). Thus for each σ > 0 there exists a τ ≥ σ so that the germ g induces a map

g_{#}: GL_{n}(A, p^{τ} A) → GL_{r}(A, p^{σ}A).

A (finite) chain c ∈ C_{i+1}(G; 𝔽_{p}) = 𝔽_{p}[G^{i+1}] in the bar construction then induces a homomorphism

c_{#}: C_{i}(GL_{n}(A, p^{τ} A); 𝔽_{p}) → C_{i+1}(GL_{r}(A, p^{σ}A); 𝔽_{p}).

Using (B.6) (i), [Su], proposition 2.2 exhibits chains c_{n,i} ∈ C_{i+1}(G; 𝔽_{p}) such that (c_{n,i})_{#} becomes a contracting chain homotopy of the natural inclusion of C_{i}(GL_{n}(A, p^{τ} A); 𝔽_{p}) in C_{i}(GL_{r}(A, p^{σ}A); 𝔽_{p}). Hence for given n, σ, i, there exists r ≥ n, τ ≥ σ such that the natural inclusion induces the zero homomorphism

(B.7)  H_{i}(GL_{n}(A, p^{τ} A); 𝔽_{p}) → H_{i}(GL_{r}(A, p^{τ} A); 𝔽_{p}).

Finally, in theorem 3.6 and corollary 3.7 of [Su] it is shown, via a study of the Hochschild-Serre spectral sequence of

BGL_{n}(A, p^{σ}A) → BGL_{n}(A) → BGL_{n}(A/p^{σ}A),

that (B.6) (ii) and (B.7) implies (B.5).

It remains to discuss (B.6). The first part of the statement follows from [Gab]. Indeed, as A was assumed commutative, E is a product of fields F_{j}, and O_{n,i}^{c}(E) is a product of O_{n,i}^{c}(F_{j}), the germs of F_{j}-valued functions on X_{n,i}(E). Then

GL(O_{n,i}^{c}(E), M_{n,i}^{c}(E)) = ∏_{j=1}^{t} GL(O_{n,i}^{c}(F_{j}), M_{n,i}^{c}(F_{j})).

Since (O_{n,i}^{c}(E), M_{n,i}^{c}(E)) is a henselian pair, [Gab], theorem 1 implies that the reduced homology of each of the t factors above is trivial. Then use the Kunneth theorem.

The second part of (B.6) follows from van der Kallens work on stability, and does not use the fact that A is commutative, cf. [vdK], (2.2) and theorem 4.11.

The general case where A is not commutative is quite similar, only the argument for producing the contracting homotopy (c_{n,i})_{#} is different.

Let O_{n,i}^{h}(F_{j}) denote the henselization of O_{n,i}(F_{j}). It is proved in [SuY], that

GL(O_{n,i}^{h}(F_{j}) ⊗_{F_{j}} D_{j}, M_{n,i}^{h}(F_{j}) ⊗_{F_{j}} D_{j}) = GL(O_{n,i}^{h}(F_{j}) ⊗_{F_{j}} M_{l_{j}}(D_{j}), M_{n,i}^{h}(F_{j}) ⊗_{F_{j}} M_{l_{j}}(D_{j}))

has vanishing homology, and universal chains c_{n,i}^{h} are exhibited. But (O_{n,i}^{h}(F_{j}), M_{n,i}^{h}(F_{j})) maps into (O_{n,i}^{c}(F_{j}), M_{n,i}^{c}(𝔽_{i})) by the universal properties of henselizations, and the images of the chains c_{n,i}^{h} give the required chains c_{n,i}, hence the contracting chain homotopy.

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Dept. of Mathematics  Matematisk Institut
Massachusetts Institute of Technology  Aarhus Universitet
Cambridge, MA 02139  8000 Aarhus C
USA  Denmark
email: larsh@math.mit.edu  email: imadsen@mi.aau.dk