THE S^{1}-TATE SPECTRUM FOR J

Lars Hesselholt and Ib Madsen

0.Introduction

0.1. For any compact Lie group G and any G-equivariant spectrum X_{G}, Greenlees and May have introduced the Tate spectrum ℍ̂(G; X), the G-fixed set of a Gequivariant spectrum denoted t(X_{G}) in [6]. In this paper we evaluate the homotopy groups of this construction when G is the circle group and X_{G} = J_{G}, the S^{1}- equivariant non-connective image of J-spectrum completed at p. When there is no danger of confusion we will omit the subscribt G and just write J instead of J_{G}.

Our interest in the Tate spectrum for J comes in part from the connection of this construction with the topological cyclic homology theory TC(R, p), demonstrated in [4] when the ring R is equal to the integers. While we at the time of writing do not know of any direct interplay between ℍ̂(S^{1}; J) and the topological cyclic homology theory we do believe that the calculations below will aid the calculation of TC(R, p) for R = ℤ[v, v^{−1}]. In any event the Tate spectrum is an interesting construction in its own right, and ℍ̂(S^{1}; J) is a non-trivial example. From this point of view one would like to evaluate ℍ̂(G; J) for any compact Lie group G. This seems to be a non-trivial task. We remark that ℍ̂(G; J) is a rational spectrum when G is a finite group, e.g. ℍ̂(C_{p^{n}}; J) = H(ℚ_{p}[C_{p^{n}}], 0) ∨ H(ℚ_{p}[C_{p^{n}}], −1).

Let K_{G} be p-completion in the sense of [3, ch.IV] of the G-equivariant nonconnected spectrum which represents equivariant K-theory. We choose an integer g which generates (ℤ/p^{2})^{×} and let J_{G} be the homotopy fiber of ψ^{g} − 1: K_{G} → K_{G} where ψ^{g} is the stable Adams operation. There is a map from the equivariant sphere spectrum S_{G}^{0} into J_{G} which is a kind of an equivariant K-theoretic localization (all fixed sets are K-local). In the rest of this paper p is a fixed odd prime.

We emphasize that K_{G} denotes the p-completion of the non-connected G-spectrum which represents G-equivariant K-theory. Thus K_{G} and J_{G} as well as their underlying non-equivariant spectra are all p-complete.

0.2. The Tate spectrum fits into the norm cofibration, cf. [6], which in our case

Typeset by AMS-TEX

takes the form

ΣJ_{hS^{1}} ⟶^{N} J^{hS^{1}} ⟶^{Ψ} ℍ̂(S^{1}; J).

Since K_{G} and hence also J_{G} are split G-spectra in the sense that the fixed sets contain the non-equivariant J as a direct factor,

J_{hS^{1}} ≃ J ∧ ℂP+^{∞}, J^{hS^{1}} ≃ F (ℂP+^{∞}, J)

so the norm cofibration takes a particular simple form. The homotopy types of these two spectra are given in terms of K-theory, cf. §1, 2 below, so our calculation thus amounts to evaluating N.

Our main result is the following formula

π_{n}ℍ̂(S^{1}; J) = \{ \begin{matrix}ℚ/ℤ_{(p)} ⊕ S(ℚ_{p}) & if n ≡ −1 (mod 2p − 2) and n ≠ −1 \\ S(ℚ_{p}) & otherwise when n is odd \\ ℤ_{p} & if n = 0 \\ 0 & if n ≠ 0 is even. \end{matrix}

Here S(ℚ_{p}) = \underrightarrow{lim}(∏_{s∈S} ℚ_{p} ×∏_{s∉S} ℤ_{p}), the colimit taken over finite subsets S ⊂ ℕ.

0.3. The homotopy groups of the p-completion ℍ̂(S^{1}; J)p^{∧} can be evaluated from the formula of [3], recalled in 1.1 below. Let N (ℤ_{p}) ⊂ ∏_{s=0}^{∞} ℤ_{p} be the sequences which converges to 0 in the p-adic topology, i.e. the profinite completion of ⨁_{s=0}^{∞} ℤ_{p}. Then

π_{n}ℍ̂(S^{1}; J)p^{∧} = \{ \begin{matrix}ℤ_{p} ⊕ ∏ ℤ_{p}/N (ℤ_{p}) & if n ≡ 1 (mod 2p − 2) \\ ∏ ℤ_{p}/N (ℤ_{p}) & otherwise when n is odd \\ ℤ_{p} & if n = 0 (mod 2p − 2) \\ 0 & otherwise if n is even. \end{matrix}

0.4. The term ∏ ℤ_{p}/N (ℤ_{p}) is not surprising in view of the norm cofibration since one has (cf. §1, 2 below)

(J_{hS^{1}})p^{∧} ≃ J ∨ ( ⋁_{s=0}^{∞} K)p^{∧}

J^{hS^{1}} ≃ J ∨ ∏_{s=0}^{∞} ΣK.

The surprise in 0.3 is the lack of torsion. The homotopy groups indicate that one might have

ℍ̂(S^{1}; J)p^{∧} ≃ K^{[1]} ∨ ΣK^{[1]} ∨ (∏_{r=0}^{∞} ΣK)/( ⋁_{r=0}^{∞} ΣK)p^{∧}.

We have not so far been able to prove this so we leave it as a conjecture (see also §5 below). There is a similar conjecture before p-completion which we leave for the reader to formulate.

0.5. Our main tool for proving 0.2 is a spectral sequence

Ê_{s,t}^{2} = \{ \begin{matrix}H^{−s}(BS^{1}; π_{t}J) & when s ≤ 0 \\ H_{s−2}(BS^{1}; π_{t}J) & when s ≥ 2 \end{matrix}

which approximates π_{∗}ℍ̂(S^{1}; J). The spectral sequence has non-zero terms in all quadrants, and in general there are convergence problems in such a situation. In §3 we examine the convergence. The results are based on an unpublished paper by J.M.Boardman, [1]. We show that the spectral sequence above converges, but oddly enough that the corresponding spectral sequence with 𝔽_{p}-coefficients diverges very badly. Nevertheless it is the spectral sequence with 𝔽_{p}-coefficients, along with the norm cofibration which supply the necessary information for determining N_{∗} and for solving the extension problems to obtain the results above. We are much indebted to Peter May for drawing our attension to [1], and for warning us of convergence problems.

1.The homotopy fixed points of J

1.1. We begin with a short review of localization, cf. [2].

A spectrum Z is said to be E-local if F (V, Z) ≃ ∗ whenever V ∧E ≃ ∗. Homotopy limits of E-local spectra are again E-local. In particular any product of E-local spectra is E-local and if two of the spectra in a cofibration Z → V → W are Elocal so is the third. If E is a ring spectrum and Z is an E-module spectrum then Z is E-local.

The E-localization functor L_{E} associates to any spectrum V its E-localization L_{E}V. It comes with a natural transformation V → L_{E}V such that E ∧ V → E ∧ L_{E}V is an equivalence and such that F (L_{E}V, Z) → F (V, Z) is an equivalence whenever Z is E-local. Furthermore if W is an E-local spectrum which satisfies these two conditions then W ≃ L_{E}V. The functor L_{E} preserves cofibrations and L_{E}(F ∧ V) ≃ F ∧ L_{E}(V) for any finite complex F. In particular L_{E} commutes with finite wedge sums.

Consider first the arithmetic localizations L_{S^{0}R} where S^{0}R is a Moore spectrum for the ring R. If R ⊂ ℚ is a subring of the rationals then

L_{S^{0}R}V = V ∧ S^{0}R = V R,

the spectrum with coefficients in R. The homotopy groups are π_{∗}(V R) = π_{∗}(V)⊗R. When R = ℤ/p we prefer to write Z/p = ZR. Then,

L_{S^{0}/p}V = Vp^{∧} = F (S^{−1}ℚ/ℤ_{(p)}, V)

is the p-completion of V in the sense of [3]. The homotopy groups are given by the split short exact sequence

0 → Ext(ℚ/ℤ_{(p)}, π_{n}V) → π_{n}(Vp^{∧}) → Hom(ℚ/ℤ_{(p)}, π_{n−1}V) → 0.

For any spectrum E we have the homotopy cartesian square

L_{Eℤ_{(p)}} V ⟶ L_{E/p}V
↓  ↓
L_{Eℚ}V ⟶ L_{Eℚ}(L_{E/p}V).

When E = S^{0} this is the usual arithmetic square for V. One may use this to prove that if either R is torsion or Eℚ ≄ ∗ then L_{ER}V = L_{S^{0}R}(L_{E}V). In particular L_{E/p}V = (L_{E}V)p^{∧}. Also one may see that L_{Eℤ_{(p)}} V = L_{E_{p}^{∧}} V provided (Ep^{∧})ℚ ≄ ∗.

The K-localization of the sphere spectrum is related to the J-spectrum. More precisely we have ([2], [8])

J = L_{K/p}S^{0} = (L_{K}S^{0})p^{∧}.

It has homotopy groups

π_{n}J = \{ \begin{matrix}ℤ_{p} & if n = 0, −1 \\ ℤ/p^{v_{p}(n+1)+1} & if n ≡ −1 (mod 2p − 2) \\ 0 & otherwise \end{matrix}

with v_{p}(−) denoting the p-adic valuation.

Finally we recall that K is ‘smashing’, i.e. that L_{K}V = V ∧ L_{K}S^{0}. This is not

the case for K/p, since L_{K/p}V = (L_{K}S^{0} ∧ V)p^{∧} ≠ (L_{K}S^{0})p^{∧} ∧ V.

1.2. The functor F (X, −) preserves cofibrations so the definition of J in 0.1 gives us a cofibration for computing F (ℂP^{∞}, J).

We recall that π_{∗}F (ℂP+^{∞}, K) = ℤ[u, u^{−1}] ⊗ ℤ_{p}[[λ]] where u is the Bott class and λ = H − 1 is the reduced Hopf bundle; deg u = −2 and deg λ = 0. The ideal generated by λ corresponds to π_{∗}F (ℂP^{∞}, K), and ψ^{g} is the ring homomorphism determined by

ψ^{g}(u) = g^{−1}u, ψ^{g}(λ) = (1 + λ)^{g} − 1.

Note that all elements are concentrated in even degrees so that the homotopy groups of F (ℂP^{∞}, J) are given by short exact sequences

0 → π_{−2i}F (ℂP^{∞}, J) → ℤ̃_{p}[[λ]] ⟶^{ψ^{g}−g^{i}} ℤ̃_{p}[[λ]] → π_{−2i−1}F (ℂP^{∞}, J) → 0.

Here we have written ℤ̃_{p}[[λ]] for the ideal in ℤ_{p}[[λ]] generated by λ.

1.3. Modulo p we have (T − 1)^{p^{n}} = T^{p^{n}} − 1 and therefore the power series ring 𝔽_{p}[[λ]] is isomorphic to the inverse limit of group rings

\underleftarrow{lim}_{n}𝔽_{p}[T]/(T^{p^{n}} − 1) = \underleftarrow{lim}_{n}𝔽_{p}[C_{p^{n}}],

with the isomorphism given by 1+λ ↦ T. We thank Steffen Bentzen for reminding us that there is the following p-adic analog of this.

Proposition. There are ring-homomorphisms ℤ_{p}[[λ]] → ℤ_{p}[C_{p^{n}}] which maps 1 + λ to the generator T ∈ C_{p^{n}} and the induced map

ℤ_{p}[[λ]] → \underleftarrow{lim}_{n}ℤ_{p}[C_{p^{n}}]

is an isomorphism of rings.

Proof. By the Atiyah-Segal completion theorem K(BG) = K_{G}(EG) = RG_{IG}^{∧}. The map induced by the inclusion of groups

ℤ[T, T^{−1}] = RS^{1} → RC_{p^{n}} = ℤ[C_{p^{n}}]

maps T to the generator T ∈ C_{p^{n}}. We may use p-complete theories instead to get a commutative diagram of rings

ℤ_{p}[T, T^{−1}] ⟶ ℤ_{p}[C_{p^{n}}]
c↓  ↓_{c′}
ℤ_{p}[[T − 1]] ⟶ ℤ_{p}[C_{p^{n}}]_{IC_{p^{n}}}^{∧}

where the vertical maps are completion in the augmentation ideals. To compute c′ we apply the exact sequence of limit-systems (in m)

0 → (T − 1)^{m} → ℤ_{p}[T]/(T^{p^{n}} − 1) → ℤ_{p}[T]/(T^{p^{n}} − 1, (T − 1)^{m}) → 0.

Since all the groups involved are compact we may take the limit and maintain exactness. Thus c′ is an isomorphism and we obtain maps ℤ_{p}[[λ]] → ℤ_{p}[C_{p^{n}}] and consequently

ℤ_{p}[[λ]] → \underleftarrow{lim}_{n}ℤ_{p}[C_{p^{n}}].

Finally this is an isomorphism since K(\underrightarrow{lim} BC_{p^{n}}) = \underleftarrow{lim} K(BC_{p^{n}}) and because the map

\underrightarrow{lim}_{n}BC_{p^{n}} = Bℚ/ℤ_{(p)} → BS^{1}

induced by group inclusions becomes a homotopy equivalence upon p-completion. □

Of course the proposition may also be proved by purely algebraic means; cf. [9]. The inverse of the map in the proposition can be defined as follows. Consider ℤ_{p}[λ] in the (p, λ)-adic topology. Since λ ∈ (p, λ) and since

\frac{(1 + λ)^{p^{n+1}} − 1}{(1 + λ)^{p^{n}} − 1} = (1 + λ)^{p^{n}(p−1)} + (1 + λ)^{p^{n}(p−2)} + ⋯ + (1 + λ)^{p^{n}} + 1 ∈ (p, λ)

induction gives (1 + λ)^{p^{n}} − 1 ∈ (p, λ)^{n+1}. Thus the map ℤ_{p}[T] → ℤ_{p}[λ] which sends T to 1 + λ is continuous if we give ℤ_{p}[T] the linear topology defined by the descending chain of ideals (T − 1) ⊃ (T^{p} − 1) ⊃ ⋯ ⊃ (T^{p^{n}} − 1) ⊃ · · ·. The induced map on completions is inverse to the map in the proposition,

\underleftarrow{lim}ℤ_{p}[T]/(T^{p^{n}} − 1) → ℤ_{p}[[λ]].

1.4. Let C_{p^{n}} = ⟨T ⟩ denote the cyclic group of order p^{n} and let G_{n} = Aut(C_{p^{n}}) denote its group of automorphisms. We recall that G_{n} = ⟨f⟩ is cyclic of order ϕ(p^{n}) = p^{n−1}(p − 1) with the action of the generator f given by T · f = T^{g}.

We may extend the action of G_{n} to a representation on the group algebra ℤ_{p}[C_{p^{n}}] by linearity. The decomposition of C_{p^{n}} into orbits under G_{n} gives a splitting as G_{n}-modules

ℤ_{p}[C_{p^{n}}] = ∏_{k=0}^{n} ℤ_{p}[T^{p^{k}}G_{n}] = ∏_{k=0}^{n} ℤ_{p}[G_{n−k}].

The G_{n}-module structure on ℤ_{p}[G_{n−k}] is as follows. The projection C_{p^{n}} ↠ C_{p^{n−k}} induces a group homomorphism G_{n} → G_{n−k}, the induced map on group algebras turns ℤ_{p}[G_{n−k}] into a G_{n}-module. In this setup ψ^{g} corresponds to the action of the generator f.

Lemma. Let H = ⟨h⟩ be a finite cyclic group and u ∈ ℤ_{p}^{×} a unit of infinite order. Then there is a short exact sequence

0 → ℤ_{p}[H] ⟶^{h−u} ℤ_{p}[H] → ℤ_{p}/(u^{|H|} − 1)ℤ_{p} → 0

and this sequence is natural with respect to projections H ↠ H̄.

Proof. Suppose h − u had a kernel. Then this would be an H-module on which H acted by the rule x · h = ux. But u has infinite order and H is finite, hence the kernel is zero.

In the cokernel we have the identities h^{s} = uh^{s−1}. Thus the cokernel has only one generator as a ℤ_{p}-module and is subject to the relation u^{|H|} = 1. □

1.5. We return to the calculation of the homotopy groups of F (ℂP^{∞}, J) by the short exact sequences of 1.2.

Proposition. There is an exact sequence

0 → ℤ̃_{p}[[λ]] ⟶^{ψ^{g}−g^{i}} ℤ̃_{p}[[λ]] ⟶^{π} ∏_{k=0}^{∞} ℤ_{p}⟨x_{k}⟩ → 0

Here x_{k} = π((1 + λ)^{p^{k}} − 1) and π((1 + λ)^{s} − 1) = γ_{s}x_{v_{p}(s)}, where γ_{s} ∈ ℤ_{p}^{×} is a unit.

Proof. We only deal with the case i ≠ 0 leaving the simpler case i = 0 to the reader. Then g^{i} is a unit in ℤ_{p} of infinite order and we can use lemma 1.4. In proposition 1.3 we may restrict ourselves to augmentation ideals,

ℤ̃_{p}[[λ]] ≅ \underleftarrow{lim}_{n}ℤ̃_{p}[C_{p^{n}}].

We let G_{n} be as in 1.4 and let π_{n}: G_{n} → G_{n−1} be the map on automorphism groups induced by the reduction map red_{n}: C_{p^{n}} ↠ C_{p^{n−1}}. The reduction map also induces a map ℤ_{p}[red_{n}]: ℤ_{p}[C_{p^{n}}] → ℤ_{p}[C_{p^{n−1}}] and this decomposes as

∏_{k=0}^{n} ℤ_{p}[π_{n−k}]: ∏_{k=0}^{n} ℤ_{p}[G_{n−k}] → ∏_{k=0}^{n−1} ℤ_{p}[G_{n−1−k}].

Since lim^{(1)} vanishes for compact groups lemma 1.4 gives us an exact sequence

0 → ℤ̃_{p}[[λ]] ⟶^{ψ^{g}−g^{i}} ℤ̃_{p}[[λ]] → \underleftarrow{lim}_{n} ∏_{k=0}^{n−1} ℤ_{p}/(g^{iϕ(p^{n−1−k})} − 1)ℤ_{p} → 0.

Finally v_{p}(g^{iϕ(p^{s})} − 1) = s + 1 + v_{p}(i) when s > 0, in particular v_{p}(g^{iϕ(p^{s})} − 1) tends to infinity with s. □

It follows that π_{n}F (ℂP^{∞}, J) is an infinite product of ℤ_{p}’s in odd degrees and zero in even degrees. We end this paragraph with a determination of the homtopy type.

1.6. We define a_{s}(x) ∈ ℤ[x] recursively by the formulae a_{0}(x) = x − 1, a_{s+1}(x) = a_{s}(x^{p})−p^{s}a_{s}(x). If ψ^{p}: ℤ[x] → ℤ[x] is the ring homomorphism given by ψ^{p}(x) = x^{p} then a_{s+1}(x) = (ψ^{p}−p^{s})(ψ^{p}−p^{s−1}) ⋯ (ψ^{p}−1)a_{0}(x). This shows that a_{s}(x) belongs to the augmentation ideal in ℤ[x−1]. Therefore we can define maps a_{s}: ℂP^{∞} → K by the formula a_{s} = a_{s}(H) where H is the Hopf bundle. We let b: Σ^{2}K → K represent the Bott isomorphism K^{2}(−) ≅ K^{0}(−) and define a map f_{s}: ΣK → F (ℂP^{∞}, J) as the adjoint of a desuspension of the composition

ℂP^{∞} ∧Σ^{2}K ⟶^{a_{s}∧b} K ∧ K ⟶^{μ} K ⟶^{∂} ΣJ.

We can take the wedge sum of all the maps f_{s} to get

f: ⋁_{s=0}^{∞} ΣK → F (ℂP^{∞}, J).

We would like to extend f over the product. We shall need a duality result.

Since ℚ/ℤ_{(p)} is divisible and hence injective

D^{n}(X) = Hom(π_{n−2}(L_{K}S^{0} ∧ X), ℚ/ℤ_{(p)}) = Hom(π_{n−2}(L_{K}X), ℚ/ℤ_{(p)})

is a cohomology theory. Now Hom takes sums in the first variable to products, so D^{∗} is completely additive and is represented by a spectrum D. The composition

(L_{K}S^{0})^{n}(X) = π_{−n}F (X, L_{K}S^{0}) = π_{−n}(L_{K}X, L_{K}S^{0}) = [Σ^{−n+2}L_{K}X, Σ^{2}L_{K}S^{0}]

⟶^{π_{0}} Hom(π_{n−2}(L_{K}X), ℚ/ℤ_{(p)}) = D^{n}(X)

is induced by a map δ: L_{K}S^{0} → D and comparing homotopy groups we get

Lemma. There is a cofibration of spectra L_{K}S^{0} ⟶^{δ} D → H(ℚ_{p}/ℚ, 0). □

We can now prove our main result in this paragraph.

Theorem. Up to homotopy there exists a unique extension f̂ of the map f to the product

f̂: ∏_{s=0}^{∞} ΣK ⟶^{≃} F (ℂP^{∞}, J),

and this is a homotopy equivalence

Proof. The n’th derivative a_{s}^{(n)}(1) = 0 when s > n. Indeed this is true when s = 0 and we may differentiate the recursion formula to get

a_{s+1}^{(n)}(x) = a_{s}^{(n)}(x^{p})p^{n}x^{n(p−1)} − p^{s}a_{s}^{(n)}(x) + ∑_{k=1}^{n−1} g_{k}(x)a_{s}^{(k)}(x^{p})

which gives the induction step. Thus (x − 1)^{s}|a_{s}(x) such that a_{s}|_{ℂP^{n}} ≃ 0 when s > n and consequently the same holds for the composition

ΣK ⟶^{f_{s}} F (ℂP^{∞}, J) ⟶^{ι^{∗}} F (ℂP^{n}, J).

Now as F (ℂP^{∞}, J) = holim_{⟵} F (ℂP^{n}, J) a choice of null-homotopies gives the desired extension of f. Furthermore the homotopy class of this extension is uniquely determined if and only if

\underleftarrow{lim}_{n}^{(1)}[S^{1} ∧ ∏_{s=0}^{∞} ΣK, F (ℂP^{n}, J)] = 0.

We proceed to show that each group in the limit system is zero. We have by Bott periodicity,

[S^{1} ∧ ∏_{s=0}^{∞} ΣK, F (ℂP^{n}, J)] = [ℂP^{n} ∧ ∏_{s=0}^{∞} K, J] = [∏_{s=0}^{∞}(ℂP^{n} ∧K), J].

where the last equality holds becauce ℂP^{n} is a finite complex (so has an S-dual). To calculate the J-cohomology we first notice that Z = ∏(ℂP^{n} ∧K) is K-local. Indeed ℂP^{n} ∧K is K-local and a product of K-local spectra is again K-local. Thus D^{n}(Z) = Hom(π_{n−2}(Z), ℚ/ℤ_{(p)}). For abelian groups A, B with A torsion free and B divisible, Hom(A, B) is divisible as we have a short exact sequence

0 → Hom(A/nA, B) → Hom(A, B) ⟶^{n} Hom(A, B) → 0.

Now π_{ev}(Z) is torsion free and π_{od}(Z) = 0, so D^{ev}(Z) is divisible and D^{od}(Z) = 0. The groups H^{∗}(Z; ℚ_{p}/ℚ) are ℚ-vectorspaces, and by the lemma (L_{K}S^{0})^{ev}(Z) is divisible and (L_{K}S^{0})^{od}(Z) is a ℚ-vectorspace. We can calculate

[Z, J] = π_{0}F (Z, J) = π_{0}(F (Z, L_{K}S^{0})p^{∧})

from the short exact sequence (cf. 1.1)

0 → Ext(ℚ/ℤ_{(p)}, (L_{K}S^{0})^{0}(Z)) → [Z, J] → Hom(ℚ/ℤ_{(p)}, (L_{K}S^{0})^{1}(Z)) → 0.

The Ext-group is zero as a divisible abelian group is injective. The Hom-group vanishes since there are no maps from a torsion group into a torsion free group. We conclude that [Z, J] = 0.

In view of proposition 1.5 we only have left to show that x^{p^{s}} − 1 is a linear combination of the polynomials a_{t}(x), t ≤ s. Since x − 1 = a_{0}(x) the case s = 0 is trivial. Now suppose x^{p^{s}} − 1 = ∑_{t=0}^{s} n_{t}a_{t}(x) then

x^{p^{s+1}} − 1 = ∑_{t=0}^{s} n_{t}a_{t}(x^{p}) = ∑_{t=0}^{s} n_{t}(a_{t+1}(x) + p^{t}a_{t}(x))

and we are done by induction. □

2.The homotopy orbits of J

2.1. Since J_{S^{1}} is a split S^{1}-spectrum the homotopy orbits are

J_{hS^{1}} = J ∧ BS_{+}^{1} ≃ J ∨ (J ∧ ℂP^{∞}).

The ‘arithmetic’ square of 1.1 gives a cartesian square

L_{K}S^{0} ∧ ℂP^{∞} ⟶ L_{K/p}S^{0} ∧ ℂP^{∞}
↓  ↓
L_{K}S^{0}ℚ ∧ ℂP^{∞} ⟶ (L_{K}S^{0})p^{∧}ℚ ∧ ℂP^{∞} .

which relates J ∧ ℂP^{∞} to the K-localization of the suspension spectrum S^{0} ∧ ℂP^{∞}. We recall form 1.1 that p-completed and p-localized K-theory gives the same localization functors L_{E}. The spectrum L_{K}S^{0} ∧ ℂP^{∞} is calculated in [8], [7], we give a short review.

2.2. The maps a_{s}: ℂP^{∞} → K of 1.5 induce a map a: ℂP^{∞} → ∏_{s=0}^{∞} K to the product. Now as a_{s}|_{ℂP^{n}} ≃ 0 when s > n and ℂP^{∞} = \underrightarrow{lim} ℂP^{n} this map factors through the inclusion of the sum and we obtain

a: ℂP^{∞} → ⋁_{s=0}^{∞} K.

It is shown in [7] that this map induces an isomorphism on K_{∗}(−; 𝔽_{p}) and thus that a: S^{0} ∧ ℂP^{∞} → (⋁ K)p^{∧} is a K/p-localization. Alternatively

(L_{K}S^{0} ∧ ℂP^{∞})p^{∧} ⟶^{≃} ( ⋁_{s=0}^{∞} K)p^{∧}.

However as (L_{K}S^{0}∧ℂP^{∞})ℚ = H(ℚ, 0)∧ℂP^{∞} this is far from a rational equivalence. In order to state the integral result we need a new spectrum. We let bu denote the connected cover of K, i.e. bu = K[2, ∞). Its K-localization is the homotopy fiber of the composition K → Kℚ → Kℚ(−∞, 0], cf. [7], [8]. The homotopy groups π_{∗}L_{K}(bu) are ℤ_{(p)} in even degrees ≥ 2 and ℚ/ℤ_{(p)} in odd degrees ≤ −1.

Theorem. ([7], [8]) The K-localization of the suspension spectrum Σ^{∞}ℂP^{∞} is

L_{K}S^{0} ∧ ℂP^{∞} = L_{K}(bu) ∨ ⋁_{s=1}^{∞} Σ^{−1}Kℚ/ℤ.

2.3. In the square 2.1 the rational term is L_{K}S^{0}ℚ ∧ ℂP^{∞} = bu ℚ and the left vertical map is projection onto L_{K}(bu) followed by rationalisation. Thus the mutual cofibers of the vertical maps are Kℚ/ℤ ∨ ⋁_{s=1}^{∞} Kℚ/ℤ. The cartesian square gives the following diagram of homotopy groups when n ≥ 1.

0  0
↓  ↓
ℤ_{(p)}  ⟶  π_{2n}J ∧ ℂP^{∞}
↓  ↓
ℚ  ⟶^{c}  ℚ_{p}
π×0↓  ↓^{ρ}
ℚ/ℤ_{(p)} × ⨁_{s=1}^{∞} ℚ/ℤ_{(p)} ⟶^{id} ℚ_{p}/ℤ_{p} × ⨁_{s=1}^{∞} ℚ/ℤ_{(p)}
↓  ↓
⨁_{s=1}^{∞} ℚ/ℤ_{(p)}  ⟶ π_{2n−1}J ∧ ℂP^{∞}
↓  ↓
0  ℚ_{p}

↓

0

We claim that ρ = π × 0 where π: ℚ_{p} → ℚ_{p}/ℤ_{p} is the canonical projection. Indeed if we restrict ρ to ℚ ⊂ ℚ_{p} this is the case and since c is the completion map and ρ is ℤ_{p}-linear the claim follows. Finally since ⨁ ℚ/ℤ_{(p)} is divisible there are no extension problems.

Corollary. The homotopy groups of the smash product J ∧ ℂP^{∞} are

π_{n}(J ∧ ℂP^{∞}) = \{ \begin{matrix}ℤ_{p} & if n ≥ 2 and even \\ ℚ_{p} × ⨁_{s=1}^{∞} ℚ/ℤ_{(p)} & if n ≥ 1 and odd \\ ⨁_{s=0}^{∞} ℚ/ℤ_{(p)} & if n ≤ −1 and odd \\ 0 & else. \end{matrix}

2.4. We have determined the homotopy groups of the first two terms in the norm cofibration, cf. 0.2. Thus in order to calculate π_{∗}ℍ̂(S^{1}; J) we must evaluate N_{∗} and solve extension problems.

For degree reasons only N_{0} and N_{2i−1}, i ≥ 1 may be non-zero. To settle this we use the spectral sequence Ê^{∗}(S^{1}; J) constructed below and we prove the following result in 4.5.

Proposition. The norm map induces a map on homotopy groups

N_{s}: π_{s−1}J_{hS^{1}} → π_{s}J^{hS^{1}}

which is injective if s = 2i − 1, i ≥ 1 and zero otherwise.

The basic extension problem is now the following. Let n ≤ −1 be an odd integer and suppose n ≢ ±1 (mod 2p − 2). Then the homotopy group of the Tate spectrum is an extension

0 → ∏_{k=0}^{∞} ℤ_{p} → π_{n}ℍ̂(S^{1}; J) → ⨁_{k=0}^{∞} ℚ/ℤ_{(p)} → 0.

We show in 4.4, using the spectral sequence with 𝔽_{p}-coefficients, Ê^{∗}(S^{1}; J/p) that π_{n}ℍ̂(S^{1}; J) is torsion free, and in 5.2 that this in turn determines the group completely.

3.The Tate spectral sequence

3.1. Let G be a compact Lie group of dimension d and V_{G} a G-spectrum. The Tate spectral sequence Ê^{∗}(G; V) is a whole plane spectral sequence which one might expect to approximates the homotopy groups of ℍ̂(G; V), cf. [4], [6]. In this paragraph we study convergence properties.

Let us recall how the spectral sequence is constructed. The G-space ẼG is defined by the cofibration

EG_{+} → S^{0} → ẼG.

It is a G-CW-complex whose k-skeleton is Ẽ_{k}G = cof(E_{k−1}G_{+} → S^{0}). The spectral sequence comes from Greenlees ‘filtration’

⋯ ⟶ F_{−2} ⟶ F_{−1} ⟶ F_{0} ⟶ F_{1} ⟶ F_{2} ⟶ ⋯

of the suspension spectrum of ẼG. Here F_{s} is the suspension spectrum of Ẽ_{s−d}G when s ≥ d, the functional dual of Ẽ_{−s}G when s ≤ 0 and S^{0} when 0 ≤ s ≤ d. The exact couple associated with the spectral sequence is

D_{s,t} = π_{s+t}([F_{s} ∧F (EG_{+}, V)]^{G})
E_{s,t} = π_{s+t}([F_{s} / F_{s−1} ∧F (EG_{+}, V)]^{G}).

One expects that the spectral sequence approximate the colimit D_{n} = \underrightarrow{lim} D_{s,n−s} which is equal to π_{n}ℍ̂(G; V). It has an induced filtration

F_{s} D_{n} = im(D_{s,n−s} → D_{n}).

We recall some notions of convergence from [1].

Definition. We say that the spectral sequence (1) converges to D_{n} conditionally if \underleftarrow{lim}_{s}D_{s,n−s} = \underleftarrow{lim}_{s}^{(1)}D_{s,n−s} = 0.

(2) converges to D_{n} weakly if Ê_{s,n−s}^{∞} = F_{s} D_{n}/ F_{s−1} D_{n}.

(3) converges to D_{n} strongly if it converges to D_{n} weakly and in addition

\underleftarrow{lim}_{s}F_{s}D_{n} = \underleftarrow{lim}_{s}^{(1)}F_{s}D_{n} = 0.

A word of explanation is in place. Condition (1) is rather weak but has the effect that the question of strong convergence can be settled by studying the internal structure of the spectral sequence. If for example the lenght of the non-zero differentials is bounded conditional convergence implies stong convergence. To understand strong convergence note that a filtered group can be viewed as a topological group by taking the stages of the filtration as a base for the neighborhoods of zero. The vanishing of the two groups in (3) is then equivalent to the statements that this topology is Hausdorff and complete respectively. Here complete means that Cauchy sequences converge. Let us finally note that if the topology is complete but not Hausdorff, then the spectral sequence converges strongly to

D̄_{n} = D_{n}/\underleftarrow{lim} F_{s} D_{n}.

3.2. Our first convergence result for the Tate spectral sequence imposes no further conditions on the G-spectrum V.

Lemma. The spectral sequence Ê^{∗}(G; V) is conditionally convergent to π_{∗}ℍ̂(G; V).

Proof. We have a diagram of G-spaces in which the rows are cofibrations

(E_{k−1}G × EG)_{+} ⟶ EG_{+} ⟶ Ẽ_{k}G ∧ EG_{+}
Δ↑^{≃_{G}}  ∥  ↑
E_{k−1}G_{+}  ⟶ EG_{+} ⟶ EG/E_{k−1}G.

It shows that D_{s,n−s} = V_{G}^{−n}(EG, E_{−s−1}) when s ≤ 0. Thus the lemma follows from Milnors short exact sequence

0 → \underleftarrow{lim}_{s}^{(1)}D_{s,n+1−s} → V_{G}^{−n}(EG, EG) → \underleftarrow{lim}_{s}D_{s,n−s} → 0.  □

3.3. In the present general setting the norm cofibration has the form

Σ^{Ad(G)}V_{hG} ⟶^{N} V^{hG} ⟶^{Ψ} ℍ̂(G; V).

It induces a long exact sequence in homotopy groups which we compare to the long exact cohomology sequence of the pair (EG, E_{k}G)

\begin{matrix}⋯ ⟶^{∂} π_{n−d}V_{hG} & ⟶^{N_{∗}} π_{n}V^{hG} ⟶^{Ψ_{∗}} π_{n}ℍ̂(G; V) ⟶^{∂} ⋯ \\ ⋯ ⟶^{δ} V_{G}^{−n}(EG, E_{k}G) & ⟶^{j^{∗}} V_{G}^{−n}(EG) ⟶^{i^{∗}} V_{G}^{−n}(E_{k}G) ⟶^{δ} · · · . \end{matrix}

They have the term π_{n}(V^{hG}) = V_{G}^{−n}(EG) in common, and we can define subgroups

O_{k} = im(j^{∗}: V_{G}^{−n}(EG, E_{k}G) → V_{G}^{−n}(EG)) ⊂ π_{n}V^{hG}.

Taking these as a base for the neighborhoods of zero, π_{n}(V^{hG}) becomes a topological group.

Lemma. Suppose V is a p-complete spectrum with π_{n}(V) finitely generated as a ℤ_{p}-module for all n. Then the group π_{n}(V^{hG}) is complete Hausdorff in the Otopology.

Proof. The statement is equivalent to the vanishing og \underleftarrow{lim} O_{k} and \underleftarrow{lim}^{(1)} O_{k}. We let N_{k} be the kernel of j^{∗}: V_{G}^{−n}(EG, E_{k}G) → V_{G}^{−n}(EG) and obtain a six term exact sequence

0 → \underleftarrow{lim}_{k}N_{k} → \underleftarrow{lim}_{k}V_{G}^{−n}(EG, E_{k}G) → \underleftarrow{lim}_{k}O_{k}

→ \underleftarrow{lim}_{k}^{(1)}N_{k} → \underleftarrow{lim}_{k}^{(1)}V_{G}^{−n}(EG, E_{k}G) → \underleftarrow{lim}_{k}^{(1)}O_{k} → 0.

By lemma 3.2 it suffices to show that \underleftarrow{lim}^{(1)} N_{k} = 0. To this end note that N_{k} is also the image of δ: V_{G}^{−n−1}(E_{k}G) → V_{G}^{−n}(EG, E_{k}G) so is compact in its p-adic topology as E_{k}G is a finite free CW -complex. Finally \underleftarrow{lim}^{(1)} vanishes for compact groups. □

Proposition. Let V be a p-complete spectrum with π_{n}(V) finitely generated as a ℤ_{p}-module for all n. Then the spectral sequence Ê^{∗}(G; V) converges strongly to the cokernel of Ψ_{∗} in the exact sequence

0 ⟶ im N_{∗} ⟶ \overline{im N_{∗}} ⟶^{Ψ_{∗}} π_{n}ℍ̂(G; V).

Here \overline{im N_{∗}} denotes the closure of im N_{∗} in the O-topology. In particular the spectral sequence converges strongly to π_{n}(ℍ̂(G; V)) if im N_{∗} is closed.

Proof. Recall from 3.1 that D_{n} = π_{n}(ℍ̂(G; V)) is filtered by

F_{s} D_{n} = im(D_{s,n−s} → D_{n}) = im(O_{−s−1} ⟶^{Ψ_{∗}} D_{n}).

It follows from (the proof of) [1; 10.1] that the spectral sequence converges weakly to D_{n}. Indeed the criterion is that the group RE_{s,t}^{∞} defined as \underleftarrow{lim}^{(1)} of a certain system of ℤ_{p}-submodules of E_{s,t} vanishes. In our case these submodules are compact, so the criterion is satisfied.

We next claim that the filtration topology on D_{n} is complete, i.e. \underleftarrow{lim}^{(1)} F_{s} D_{n} = 0. To see this notice that

F_{−k−1} D_{n} = Ψ_{∗}(O_{k}) ≅ O_{k}/(O_{k} ∩ im N_{∗}) ≅ (O_{k} + im N_{∗})/ im N_{∗}

so that

0 → im N_{∗} → O_{k} + im N_{∗} → F_{−k−1} D_{n} → 0

is exact. Now \underleftarrow{lim}(O_{k} + im N_{∗}) is precisely the closure of im N_{∗} in the O-topology on π_{n}(V^{hG}), and the six term exaxt sequence used in the above lemma reduces to

0 → im N_{∗} → \overline{im N_{∗}} → \underleftarrow{lim}_{s} F_{s} D_{n} → 0

since im N_{∗} does not depend on s. Thus we are done by the discussion following definition 3.1. □

Corollary. The Tate spectral sequence Ê^{∗}(S^{1}; J) converges strongly to the homotopy groups π_{∗}ℍ̂(S^{1}; J).

Proof. Our calculations in §1 and §2 shows that for dimension reasons im N_{∗} is always compact. But then \overline{im N_{∗}} = im N_{∗} as the O-topology is Hausdorff. □

3.4. We conclude this paragraph with a list of the E^{2}-term of the Tate spectral sequence for various groups G. The cases are

(1) G finite, Ê_{s,t}^{2}(G; V) = Ĥ^{−s}(G; π_{t}V) - the Tate cohomology of G with coefficients in the G-module π_{t}V.

(2) G = S^{1} the circle group, Ê^{2}(S^{1}; V) = ℤ[t, t^{−1}] ⊗ π_{∗}(V), and t ∈ V_{G}^{2}(S^{3}, S^{1}) has bidegree (−2, 0).

(3) G of positive dimension and connected

Ê_{s,t}^{2}(G; V) = \{ \begin{matrix}H^{−s}(BG; π_{t}V) & , s ≤ 0 \\ H_{s−d−1}(BG; π_{t}V) & , s ≥ d + 1 \\ 0 & , 1 ≤ s ≤ d. \end{matrix}

If V_{G} is a ring G-spectrum and G is either finite or S^{1} the Tate spectral sequence is multiplicative in the sense that Ê^{r}(G; V) is a bigraded algebra and d̂^{r} a derivation for the multiplication. For such V the formulas (1) and (2) are algebra isomorphism.

4.The Tate spectral sequence for J/p

4.1. We write X/p = X ∧ S^{0}/p for the spectrum X reduced mod p. Since the Moore spectrum S^{0}/p is finite all the functors in the norm cofibration commute with reduction mod p. In particular π_{∗}(ℍ̂(S^{1}; J); 𝔽_{p}) = π_{∗}ℍ̂(S^{1}; J/p).

The E^{2}-term of the mod p Tate spectral sequence is

Ê^{2}(S^{1}; J/p) = P [t, t^{−1}] ⊗ E{a} ⊗ P [v_{1}, v_{1}^{−1}]

where the bidegrees are deg t = (−2, 0), deg a = (0, 2p − 3) and deg v_{1} = (0, 2p − 2). There are also spectral sequences approximating π_{∗}(J^{hS^{1}}; 𝔽_{p}) and π_{∗}(J_{hS^{1}}; 𝔽_{p}). They are left and right half plane spectral sequences respectively and have E^{2}- terms

E_{s,t}^{2}(J^{hS^{1}}; 𝔽_{p}) = Ê_{s,t}^{2} = P [t] ⊗ E{a} ⊗ P [v_{1}, v_{1}^{−1}]  , s ≤ 0
E_{s,t}^{2}(J_{hS^{1}}; 𝔽_{p}) = Ê_{s+2,t}^{2} = σ^{−2}IP [t^{−1}] ⊗ E{a} ⊗ P [v_{1}, v_{1}^{−1}] , s ≥ 0.

Here σ is the shift operator (σM)_{s,t} = M_{s−1,t} and IP [x] ⊂ P [x] is the augmentation ideal. They are both strongly convergent, cf. 3.1.

There is a fruitful interplay between these spectral sequences and the Tate spectral sequence. We sketch the interplay here and refer the reader to [4] where it is treated thoroughly. For any S^{1}-spectrum X there is a map of spectral sequences

Ψ^{r}: E_{s,t}^{r}(X^{hS^{1}}) ↠ Ê_{s,t}^{r}(S^{1}; X)

which is iso for r = 2 and s ≤ 0 and epi for r > 2 and s ≤ 0. This implies that x ∈ E_{s,t}^{2}(X^{hS^{1}}) survives to E^{∞} precisely when one of two things happens. Either Ψ^{2}x survive to Ê^{∞} or Ψ^{r}x = d̂^{r}y for some y in the right half plane, i.e. s + r > 0. Similarly there is a map of degree −2

∂^{r}: Ê_{s+2,t}^{r}(S^{1}; X) ↪ E_{s,t}^{r}(X_{hS^{1}})

which is iso for r = 2 and s ≥ 0 and mono for r > 2 and s ≥ 0. Let x ∈ Ê_{s,t}^{2}(S^{1}; X). Then ∂^{2}x survives to infinity if and only if x does so or d̂^{r}x ≠ 0 and s − r ≤ 0. We note that what we said above fit in well with the exactness of

π_{n}X^{hS^{1}} ⟶^{Ψ_{∗}} π_{n}ℍ̂(S^{1}; X) ⟶^{∂_{∗}} π_{n−2}X_{hS^{1}}.

From this point of view the norm map N_{∗} should correspond to those differentials in the Tate spectral sequence which cross the line s = 0. Proposition 4.3 below is an instance of such a relation. Our calculations in this paragraph for X = J/p are based on the following result.

Theorem. ([4]) The non-trivial differentials in Ê^{∗}(S^{1}; J/p) are multiplicatively induced from

d^{2(p^{k+1}−1)}t^{p^{k}} = γ_{k} t^{p^{k+1}+p^{k}−1}a v_{1}^{p( \frac{p^{k}−1}{p−1} )}.

Here γ_{k} ∈ 𝔽_{p}^{×} is a unit. □

4.2. Before we can write down the E^{∞}-terms we need some notation. We write r ∈ ℤ as

r = r_{1}(p − 1) + r_{2}; 1 ≤ r_{2} ≤ p − 1

Next we define periodicy elements

π = (t^{p−1}v_{1})^{−r_{2}}, Π = (t^{p−1}v_{1})^{p−r_{2}}.

of total degree zero, and elements w_{r}(k) and z_{r}(k) for k ≥ 0 by

\begin{aligned}w_{r}(0) & = t^{−r_{2}}v_{1}^{r_{1}}; w_{r}(k) = w_{r}(k − 1) · π^{p^{k−1}} = w_{r}(0) · π^{\frac{p^{k}−1}{p−1}} \\ z_{r}(0) & = t^{p−1−r_{2}}a v_{1}^{r_{1}}; z_{r}(k) = z_{r}(k − 1) · Π^{p^{k−1}} = z_{r}(0) · Π^{\frac{p^{k}−1}{p−1}} . \end{aligned} .

They have total degrees 2r and 2r − 1 respectively.

Lemma. The E^{∞}-terms of the mod p spectral sequences are

Ê^{∞}(S^{1}; J/p) = E{t^{−1}a} ⊗ P [v_{1}, v_{1}^{−1}]
E^{∞}(J^{hS^{1}}; 𝔽_{p}) = 𝔽_{p}⟨z_{r}(k)|k ∈ ℕ, r ∈ ℤ⟩ ⊕ P [v_{1}, v_{1}^{−1}]
E^{∞}(J_{hS^{1}}; 𝔽_{p}) = σ^{−2}𝔽_{p}⟨w_{r}(k)|k ∈ ℕ, r ∈ ℤ⟩ ⊕ σ^{−2}𝔽_{p}⟨t^{−1}a v_{1}^{j}|j ∈ ℤ⟩,

where σ^{−2} indicates that degrees are shifted down by 2. Furthermore the differential of the Tate spectral sequence takes w_{r}(k) to γ_{r}(k)z_{r}(k) with γ_{r}(k) ∈ 𝔽_{p}^{×} a unit.

Proof. We first derive Ê^{∞}(S^{1}; J/p). If we write n = m p^{k} with (m, p) = 1 the first non-zero differential on t^{n} gives

d^{∗}t^{n} = t^{(m−1)p^{k}}d^{∗}t^{p^{k}} = t^{p^{k+1}+mp^{k}−1}a v_{1}^{p( \frac{p^{k}−1}{p−1} )}.

Here we have neglected to list the units γ ∈ 𝔽_{p}^{×} and we shall do so throughout the proof. Now for l ∈ ℕ to be of the form

l = p^{k+1} + mp^{k} − 1

we must have

k = v_{p}(l + 1), m = \frac{l + 1}{p^{k}} − p.

It follows that all powers of t are hit by exactly one differential save l = −1. This gives the claim for Ê^{∞}(S^{1}; J/p)

To evaluate E^{∞}(J_{hS^{1}}, 𝔽_{p}) we seek x ∈ Ê_{s,t}^{2}(S^{1}; J/p), s ≥ 2 such that d̂^{r}x ≠ 0 for some r ≥ s, cf. 4.1. So let x = t^{n}v_{1}^{i} with n < 0 and suppose d̂^{r}x = t^{l}a v_{1}^{j} with l ≥ 0 for some r. We write n = −mp^{k} with (m, p) = 1 and have from above

l = p^{k+1} − mp^{k} − 1.

Thus l ≥ 0 if and only if m ≤ p − 1 and since also m ≥ 1 (because n < 0) we obtain x = t^{−mp^{k}}v_{1}^{j} with 1 ≤ m ≤ p − 1, k ∈ ℕ and j ∈ ℤ. Ordering these elements after total degree we get the elements w_{r}(k) defined above.

We can similarly evaluate E^{∞}(J^{hS^{1}}; 𝔽_{p}). Indeed the generators are exacly the t^{l}a v_{1}^{j′} = d^{r}t^{n}v_{1}^{j} we have just found. Again ordering after total degree gives us the elements z_{r}(k). □

We shall see in 4.8 that Ê^{∞}(S^{1}; J/p) is very far from π_{∗}(ℍ̂(S^{1}; J); 𝔽_{p}), so the mod p Tate spectral sequence diverges.

4.3. We next evaluate the norm map. We begin with a lemma.

Lemma. Suppose that in the commutive diagram

0 ⟶ A′ ⟶ A ⟶^{π} A′′ ⟶ 0
N′↓≅  ^{N}↓
0 ⟵ B′ ⟵ B ⟵^{ι} B′′ ⟵ 0

the rows are exact and the map N′ is an isomorphism. Then the additive relation

N′′: A′′ ⟵^{π} A ⟶^{N} B ⟵^{ι} B′′

is a homomorphism. □

Proposition. There are elements w̄_{r}(k) and z̄_{r}(k) which reduce to σ^{−2}w_{r}(k) and z_{r}(k) respectively such that

π_{2s−2}(J_{hS^{1}}; 𝔽_{p}) = ⨁_{k=0}^{∞} 𝔽_{p}⟨w̄_{r}(k)⟩, π_{2s−1}(J^{hS^{1}}; 𝔽_{p}) = ∏_{k=0}^{∞} 𝔽_{p}⟨z̄_{r}(k)⟩.

Furthermore these elements may be chosen such that N_{∗}(w̄_{r}(k)) = γ_{r}(k)z̄_{r}(k) where γ_{r}(k) ∈ 𝔽_{p}^{×} as in 4.5 Proof. We write A = π_{2s−2}(J_{hS^{1}}; 𝔽_{p}) and B = π_{2s−1}(J^{hS^{1}}; 𝔽_{p}). For notational convenience we pick subfiltrations of the filtration associated with the spectral sequence such that

F_{k} A/ F_{k−1} A = 𝔽_{p}⟨σ^{−2}w_{r}(k)⟩, F^{k} B/ F^{k+1} B = 𝔽_{p}⟨z_{r}(k)⟩.

We shall prove the proporsition by induction. So assume that we have chosen elements w̄_{r}(k) for all k and z̄_{r}(k) for k < n which represents σ^{−2}w_{r}(k) and z_{r}(k) such that Nw̄_{r}(k) = γ_{r}(k)z̄_{r}(k) when k < n and such that Nw̄_{r}(k) ∈ F^{n} B when k ≥ n. Repeated use of the lemma gives us a commutative diagram

0 ⟶ F_{n} A/ F_{n−1} A ⟶ A/ F_{n−1} A ⟶ A/ F_{n} A ⟶ 0
≅↓  _{N^{(n)}}↓
0 ⟵ F^{n} B/ F^{n+1} B ⟵ F^{n} B ⟵ F^{n+1} B ⟵ 0

We may define z̄_{r}(k) by the equation N^{(n)}σ^{−2}w_{r}(k) = γ_{r}(k)z̄_{r}(k) and can change our choices of w̄_{r}(k) for k > n by adding an appropriate multiple of w̄_{r}(n) so that we obtain Nw̄_{r}(k) ∈ F^{n+1} B when k > n. Now the proposition follows by induction with n = 0 as a trivial case. □

4.4. In the Tate spectral sequence Ê^{∗}(S^{1}; K) all elements are concentrated in even degrees. Thus the spectral sequence collapses and by proposition 3.3 it converges strongly to π_{∗}ℍ̂(S^{1}; K). It follows that the odd degree homotopy groups of K_{hS^{1}}, K^{hS^{1}} and ℍ̂(S^{1}, K) all are zero.

We may apply the snake lemma to the diagram

0 ⟶ π_{2n}K^{hS^{1}} ⟶ π_{2n}ℍ̂(S^{1}; K) ⟶ π_{2n−2}K_{hS^{1}} ⟶ 0
(ψ^{g})^{hS^{1}} −1↓  ^{ℍ̂(S^{1};ψ^{g})−1}↓  ^{(ψ^{g})_{hS^{1}} −1}↓
0 ⟶ π_{2n}K^{hS^{1}} ⟶ π_{2n}ℍ̂(S^{1}; K) ⟶ π_{2n−2}K_{hS^{1}} ⟶ 0

to obtain a six term exact sequence

\begin{aligned}0 & ⟶ π_{2n}J^{hS^{1}} ⟶ π_{2n}ℍ̂(S^{1}; J) ⟶ π_{2n−2}J_{hS^{1}} \\ & ⟶^{N_{∗}} π_{2n−1}J^{hS^{1}} ⟶ π_{2n−1}ℍ̂(S^{1}; J) ⟶ π_{2n−3}J_{hS^{1}} → 0. \end{aligned}

These statements are true with any coefficients; we take π_{∗} = π_{∗}(−; 𝔽_{p}). Since the O-topology on π_{∗}(J^{hS^{1}}; 𝔽_{p}) is Hausdorff so that {0} ⊂ π_{2n}(J^{hS^{1}}; 𝔽_{p}) is a closed subset proposition 3.3 shows that in even degrees the Tate spectral sequence Ê^{∗}(S^{1}; J/p) converges strongly to π_{∗}(ℍ̂(S^{1}; J); 𝔽_{p}). The calculations of the E^{∞}- terms in 4.2 therefore show that Ψ_{∗}: π_{∗}(J^{hS^{1}}; 𝔽_{p}) → π_{∗}(ℍ̂(S^{1}; J); 𝔽_{p}) is an isomorphism in even degrees. For the odd degree homotopy groups we get an exact sequence

0 ⟶ π_{2n−2}(J_{hS^{1}}; 𝔽_{p}) ⟶^{N_{∗}} π_{2n−1}(J^{hS^{1}}; 𝔽_{p}) ⟶^{Ψ_{∗}} π_{2n−1}(ℍ̂(S^{1}; J); 𝔽_{p})

⟶^{∂_{∗}} π_{2n−3}(J_{hS^{1}}; 𝔽_{p}) ⟶ 0.

From proposition 4.3 we get

Theorem. The mod p homotopy groups of the S^{1}-Tate spectrum for J are

π_{n}(ℍ̂(S^{1}; J); 𝔽_{p}) = \{ \begin{matrix}𝔽_{p}⟨x_{j}⟩ & , n = 2j(p − 1) \\ 𝔽_{p}⟨y_{j}⟩ × (∏_{k=0}^{∞} 𝔽_{p}/ ⨁_{k=0}^{∞} 𝔽_{p}) & , n = 2j(p − 1) + 1 \\ ∏_{k=0}^{∞} 𝔽_{p}/ ⨁_{k=0}^{∞} 𝔽_{p} & , n odd, n ≢ 1 (2p − 2) \\ 0 & ,else. \end{matrix}

Here x_{j} and y_{j} represents v_{1}^{j} and t^{−1}av_{1}^{j−1} in Ê^{∞}(S^{1}; J/p) respectively.

If we compare this result with lemma 4.2 we see that the spectral sequence misses the term ∏ 𝔽_{p}/ ⨁ 𝔽_{p} in the odd degree homotopy groups. Thus in contrast to the convergent integral spectral sequence, cf. 3.3 the mod p spectral sequence diverges very badly. We note that this is in accord with proposition 3.3. Indeed the O-topology on π_{2r−1}(J^{hS^{1}}; 𝔽_{p}) is given by the descending sequence of submodules

∏_{k=0}^{∞} 𝔽_{p}⟨z̄_{r}(k)⟩ ⊃ ∏_{k=1}^{∞} 𝔽_{p}⟨z̄_{r}(k)⟩ ⊃ ∏_{k=2}^{∞} 𝔽_{p}⟨z̄_{r}(k)⟩ ⊃ ⋯

so the sum ⨁_{k=0}^{∞} 𝔽_{p}⟨z̄_{r}(k)⟩ is a dense subset.

4.5. The rest of this paragraph is devoted to the integral spectral sequence Ê^{∗}(S^{1}; J) and to the proof of proposition 2.4.

Lemma. For every s ≠ 0 there are infinitely many non-zero differentials of the form

d̂^{r}: Ê_{2s,0}^{r}(S^{1}; J) → Ê_{2s−r,r−1}^{r}(S^{1}; J)

and these are the only non-zero differentials in Ê^{∗}(S^{1}; J). Furthermore the only elements of even total degree in Ê^{∞} are Ê_{0,0}^{∞} = ℤ_{p}.

Proof. The spectral sequence has E^{2}-term

Ê^{2}(S^{1}; J) = ℤ[t, t^{−1}] ⊗ π_{∗}(J)

and π_{∗}(J) is listed in 1.1. We notice that the elements in even total degrees are concentrated on the s-axis, so the differentials must either end or start here. Since all elements away from the lines t = 0, −1 are p-torsion the only differentials which may end on the s-axis are

d̂^{2}: Ê_{2s,−1}^{2}(S^{1}; J) → Ê_{2s−2,0}^{2}(S^{1}; J).

The map J → ΣK allows us to compare the Tate spectral sequence for J with that for ΣK. Since there are no differentials in Ê^{∗}(S^{1}; ΣK) as all elements have odd total degree we may conclude that d̂^{2} = 0 in Ê^{∗}(S^{1}; J). Thus all non-zero differentials in Ê^{∗}(S^{1}; J) originate on the s-axis. As a consequence

Ψ^{r}: E_{2s,0}^{r}(J^{hS^{1}}) → Ê_{2s,0}^{r}(S^{1}; J)

is an isomorphism for all r ≥ 2. Now π_{2s}(J^{hS^{1}}) = 0 when s ≠ 0 and since E^{∗}(J^{hS^{1}}) converges strongly to π_{∗}(J^{hS^{1}}) it follows that Ê_{2s,0}^{∞}(S^{1}; J) = 0 when s < 0. Since the Tate spectral sequence is multiplicative we have in particular

0 = d^{∗}(1) = d^{∗}(t^{2s}t^{−2s}) = d^{∗}t^{2s} · t^{−2s} + t^{2s} · d^{∗}t^{−2s}

and may conclude that Ê_{2s,0}^{∞}(S^{1}; J) = Ê_{−2s,0}^{∞}(S^{1}; J) = 0 when s ≠ 0. This can only happen if infinitely many differentials leave Ê_{2s,0}^{2}(S^{1}; J) because all elements in the upper half plane are p-torsion. □

Proposition 2.4 is an immediate consequence of the following corollary.

Corollary. The only non-zero even degree integral homotopy group of the S^{1}-Tate spectrum for J is π_{0}ℍ̂(S^{1}; J) = ℤ_{p}. □

5.The S^{1}-Tate spectrum for J

5.1. In this paragraph we solve the extension problems associated with the calculation of π_{∗}ℍ̂(S^{1}; J), e.g. 2.4. We use some structure theorems from the theory of infinite abelian groups and begin by presenting these; a standard reference is [5]. Throughout a group will mean an abelian group.

The sum of all divisible subgroups of a group A is again divisible and therefore maximal. Since divisible groups are injective A is the direct sum of the maximal divisible subgroup and its quotient called the reduced group. The p-completion of A is defined by

Ap^{∧} = \underleftarrow{lim}_{n}A/p^{n}A

and the canonical map c: A → Ap^{∧} is called the completion map. We say that A is p-complete if c is an isomorphism.

Theorem. [5; p.168] A reduced p-complete group is the p-completion of a direct sum of cyclic ℤ_{p}-modules. The cardinal numbers of the sets of components ℤ_{p} resp. ℤ/p^{n} are invariants of the group. □

Examples. 1. A = ∏_{k=0}^{∞} ℤ_{p}. Since A is torsion free we have A = (⨁_{i∈I} ℤ_{p})p^{∧}, so have only left to determine card(I). To this end we notice that

A/pA = (⨁_{i∈I} ℤ_{p})p^{∧}/p(⨁_{i∈I} ℤ_{p})p^{∧} = (⨁_{i∈I} 𝔽_{p})p^{∧} = ⨁_{i∈I} 𝔽_{p}.

But A/pA is also a countable product of 𝔽_{p}’s so card(I) = 2^{ℵ_{0}}.

2. A = (⨁_{k=0}^{∞} ℤ_{p})p^{∧}. One readily shows that the image of A under the inclusion

A = (⨁_{k=0}^{∞} ℤ_{p})p^{∧} ⊂ (∏_{k=0}^{∞} ℤ_{p})p^{∧} = ∏_{k=0}^{∞} ℤ_{p}

is the group N (ℤ_{p}) of null sequences in ℤ_{p} with respect to the p-adic valuation, i.e. sequences (x_{k}) such that v_{p}(x_{k}) → ∞ with k.

The next theorem gives the structure of divisible groups. Basic examples are the rational group ℚ and the group ℚ/ℤ_{(p)} of all p^{n}th roots of unity, n > 0.

Theorem. [5; p.104] A divisible group is a direct sum of copies of ℚ and ℚ/ℤ_{(p)} where p runs through all primes. The cardinal numbers of the sets of components ℚ resp. ℚ/ℤ_{(p)} are invariants of the group. □

Example. Let ℚ_{p} denote the field of p-adic numbers and consider the group N (ℚ_{p}) of null sequences in ℚ_{p} with respect to the p-adic valuation. It is a divisible torsion free group so a rational vector space. The dimension is 2^{2^{ℵ_{0}}} .

5.2. The basic extension problem we need to solve is the following

0 → ∏_{k=0}^{∞} ℤ_{p} → E → ⨁_{k=0}^{∞} ℚ/ℤ_{(p)} → 0.

As an obvious candidate we have the group of those sequences (x_{k}) in ℚ_{p} where all but a finite number of elements belong to ℤ_{p}. We write S(ℚ_{p}) for this group,

S(ℚ_{p}) = \underrightarrow{lim}_{|S|<∞}(∏_{s∈S} ℚ_{p} × ∏_{s∉S} ℤ_{p})

with the colimit taken over finite subsets of ℕ.

Proposition. If the underlying group E is torsion free it is isomorphic to S(ℚ_{p}).

Proof. We let (E, p) denote the limit system E ⟵^{p} E ⟵^{p} E ⟵^{p} · · ·. Since E is torsion free we have exact sequences

0 → E ⟶^{p^{n}} E → E/p^{n}E → 0

As n varies these give rise to an exact sequence of limit systems and this in turn supplies an exact sequence

0 → \underleftarrow{lim}(E, p) → E ⟶^{c} Ep^{∧} → \underleftarrow{lim}^{(1)}(E, p) → 0.

Now the definition of E as an extension gives us a six term exact sequence

0 → \underleftarrow{lim}(∏_{k=0}^{∞} ℤ_{p}, p) → \underleftarrow{lim}(E, p) → \underleftarrow{lim}(⨁_{k=0}^{∞} ℚ/ℤ_{(p)}, p)

→ \underleftarrow{lim}^{(1)}(∏_{k=0}^{∞} ℤ_{p}, p) → \underleftarrow{lim}^{(1)}(E, p) → \underleftarrow{lim}^{(1)}(⨁_{k=0}^{∞} ℚ/ℤ_{(p)}, p) → 0.

The first, fourth and sixth term vanish. The first because no elements in ∏ ℤ_{p} may be divided by p infinitely often, the fourth term because ∏ ℤ_{p} is compact and the sixth term because (⨁ ℚ/ℤ_{(p)}, p) is Mittag-Leffler, cf. [3; p.256]. Thus c is epi. To evaluate the kernel we use

0 → N (ℤ_{p}) → N (ℚ_{p}) → ⨁ ℚ/ℤ_{(p)} → 0

which gives an exact sequence

0 → N (ℚ_{p}) → \underleftarrow{lim}(⨁_{k=0}^{∞} ℚ/ℤ_{(p)}, p) → \underleftarrow{lim}^{(1)}(N (ℤ_{p}), p).

But \underleftarrow{lim}^{(1)}(N (ℤ_{p}), p) = 0 as N (ℤ_{p}) is p-complete, so ker(c) = N (ℚ_{p}).

We claim that N (ℚ_{p}) is the maximal divisible subgroup of E. Indeed any divisible subgroup of E is contained in ker(c) = N (ℚ_{p}) which is itself divible, hence the claim. The reduced group Ep^{∧} is torsion free and p-complete, so by 5.1 we have only left to determine a cardinal number. We have

Ep^{∧} ≅ E/N (ℚ_{p}) ≅ (∏_{k=0}^{∞} ℤ_{p})/N (ℤ_{p}) = (⨁_{2^{ℵ_{0}}} ℤ_{p})p^{∧}/(⨁_{ℵ_{0}} ℤ_{p})p^{∧} = (⨁_{2^{ℵ_{0}}} ℤ_{p}/ ⨁_{ℵ_{0}} ℤ_{p})p^{∧}.

so the cardinal number we were looking for is 2^{ℵ_{0}}. We have shown that if E is torsion free it is the direct sum of N (ℚ_{p}) and ∏_{k=0}^{∞} ℤ_{p}. Now S(ℚ_{p}) is such a group. □

5.3. We can now state and prove our main theorem.

Theorem. Abstractly the integral homotopy groups of the S^{1}-Tate for J are

π_{n}ℍ̂(S^{1}; J) = \{ \begin{matrix}ℚ/ℤ_{(p)} × S(ℚ_{p}) & if n ≡ −1 (mod 2p − 2) and n ≠ −1 \\ S(ℚ_{p}) & otherwise when n is odd \\ ℤ_{p} & if n = 0 \\ 0 & if n ≠ 0 is even. \end{matrix}

Proof. We have already shown that the groups in even degrees are zero, cf. lemma 4.5. If we compare this with theorem 4.4 we also obtain that π_{n}ℍ̂(S^{1}; J) is torsion free when n is odd and not congruent to −1 (mod 2p−2) and when n = −1. For all these groups the result stated above follows from 5.2 or a slight modification thereof. As an example suppose n ≤ −1 and n ≢ ±1 (mod 2p − 2). Then π_{n}ℍ̂(S^{1}; J) is an extension

0 → ∏_{k=0}^{∞} ℤ_{p} → π_{n}ℍ̂(S^{1}; J) → ⨁_{k=0}^{∞} ℚ/ℤ_{(p)} → 0

and we may apply 5.2.

When n ≡ −1 (mod 2p − 2) we must work a little harder. If n ≤ −2p + 1 the homotopy group is an extension

0 → ℤ/p^{v_{p}(n+1)+1} × ∏_{k=0}^{∞} ℤ_{p} → π_{n}ℍ̂(S^{1}; J) → ⨁_{k=0}^{∞} ℚ/ℤ_{(p)} → 0

but this time not torsion free. In fact the torsion product Tor(π_{n}ℍ̂(S^{1}; J), 𝔽_{p}) = 𝔽_{p} since π_{n+1}(ℍ̂(S^{1}; J); 𝔽_{p}) = 𝔽_{p} by 4.4 and since π_{n+1}ℍ̂(S^{1}; J) = 0. However if we can show that the torsion subgroup in π_{n}ℍ̂(S^{1}; J) is divisible it follows by 5.1 that it is isomorphic to ℚ/ℤ_{(p)}. We may instead consider the p-completed Tate spectrum ℍ̂(S^{1}; J)p^{∧} and ask whether π_{n}(ℍ̂(S^{1}; J)p^{∧}) is torsion free. This group fits into an exact sequence (by applying π_{∗} to the completed norm cofibration)

0 → π_{n+1}(ℍ̂(S^{1}; J)p^{∧}) ⟶^{∂} (⨁_{k=0}^{∞} ℤ_{p})p^{∧} ⟶^{N_{∗}} ℤ/p^{α} × ∏_{k=0}^{∞} ℤ_{p} ⟶^{t_{∗}} π_{n}(ℍ̂(S^{1}; J)p^{∧}) → 0

where α = v_{p}(n + 1) + 1, cf. 2.2 and 2.4. The component ℤ/p^{α} in π_{n}((J^{hS^{1}})p^{∧}) is from the first factor of the decomposition J^{hS^{1}} ≃ J × F (ℂP^{∞}, J). In the mod p spectral sequence such elements are located on the t-axis (s=0), and therefore some generator in ℤ/p^{α} represents the class z_{r}(0) with 2r−1 = n, cf. 4.5. Since z_{r}(0) is in the image of N_{∗} (mod p) by 4.3, ℤ/p^{α} ⊂ im N_{∗} and consequently p^{α}ℤ_{p} is a quotient of π_{n+1}(ℍ̂(S^{1}; J)p^{∧}). We conclude that 𝔽_{p} is a quotient of π_{n+1}(ℍ̂(S^{1}; J)p^{∧}) ⊗ 𝔽_{p}. Now recall from 4.4 that π_{n+1}(ℍ̂(S^{1}; J); 𝔽_{p}) = 𝔽_{p} and from [2] the formula

π_{n+1}(ℍ̂(S^{1}; J); 𝔽_{p}) ≅ π_{n+1}(ℍ̂(S^{1}; J)p^{∧}) ⊗ 𝔽_{p} ⊕ Tor(π_{n}(ℍ̂(S^{1}; J)p^{∧}), 𝔽_{p}).

It follows that the second summand must be zero and hence π_{n}(ℍ̂(S^{1}; J)p^{∧}) is a torsion free group. For the p-completed spectra the case n ≥ 2p−3 is the same. □

Corollary. The homotopy groups of the p-completed Tate spectrum are

π_{n}ℍ̂(S^{1}; J)p^{∧} = \{ \begin{matrix}ℤ_{p} ⊕ ∏ ℤ_{p}/N (ℤ_{p}) & if n ≡ 1 (mod 2p − 2) \\ ∏ ℤ_{p}/N (ℤ_{p}) & otherwise when n is odd \\ ℤ_{p} & if n = 0 (mod 2p − 2) \\ 0 & otherwise if n is even. \end{matrix}

Proof. This follows immediately from the formula on page 3 when we notice that

Hom(ℚ/ℤ_{(p)}, ⨁_{k=0}^{∞} ℚ/ℤ_{(p)}) = Hom(\underrightarrow{lim}_{n}ℤ/p^{n}, ⨁_{k=0}^{∞} ℚ/ℤ_{(p)})

= \underleftarrow{lim}_{n} Hom(ℤ/p^{n}, ⨁_{k=0}^{∞} ℚ/ℤ_{(p)}) = \underleftarrow{lim}_{n} ⨁_{k=0}^{∞} ℤ/p^{n} = N (ℤ_{p}). □

We return to the conjecture we stated in 0.2,

ℍ̂(S^{1}; J)p^{∧} ≃ K^{[1]} ∨ ΣK^{[1]} ∨ (∏_{r=0}^{∞} ΣK)/( ⋁_{r=0}^{∞} ΣK)p^{∧}.

The two sides have isomorphic homotopy groups. We note that there does exist a cofibration

ΣJ ∨ ( ⋁_{s=0}^{∞} ΣK)p^{∧} → J ∨ ∏_{s=0}^{∞} ΣK → K^{[1]} ∨ ΣK^{[1]} ∨ (∏_{r=0}^{∞} ΣK)/( ⋁_{r=0}^{∞} ΣK)p^{∧}.

which could be a candidate for the norm cofibration 0.2. The Adams splitting of p-complete K-theory is a decomposition K ≃ K^{[1]} ∨ K^{[2]} ∨ … ∨ K^{[p−1]}, and ψ^{g} − 1 induces a homotopy equivalence on all components save K^{[1]}. Thus J may equally well be defined by the cofibration

J ⟶ K^{[1]} ⟶^{ψ^{g}−1} K^{[1]}.

There are equivalences K^{[i]} = Σ^{2i−2}K^{[1]} and π_{∗}K^{[1]} = ℤ_{p}[v_{1}, v_{1}^{−1}], so by an infinite exchange of Adams components we get equivalences

( ⋁_{s=0}^{∞} ΣK)p^{∧} ≃ Σ^{−1}K^{[1]} ∨ ( ⋁_{r=0}^{∞} ΣK)p^{∧},  ∏_{s=0}^{∞} ΣK ≃ ΣK^{[[H1]} ∨ ∏_{r=0}^{∞} ΣK.

Now it is obvious how to construct the proposed cofibration.

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