ON THE NONEXISTENCE OF SMITH-TODA COMPLEXES

LEE S. NAVE

Abstract. Let p be a prime. The Smith-Toda complex V (k) is a finite spectrum whose BP -homology is isomorphic to BP_{∗}/(p, v_{1}, …, v_{k}). For example, V (−1) is the sphere spectrum and V (0) the mod p Moore spectrum. In this paper we show that if p > 5, then V ((p+3)/2) does not exist and V ((p+1)/2), if it exists, is not a ring spectrum. The proof uses the new homotopy fixed point spectral sequences of Hopkins and Miller.

1. Introduction

Let p be a prime and recall the Brown-Peterson spectrum BP. It is a p-local ring spectrum with coefficient ring BP_{∗} ≅ ℤ_{(p)}[v_{1}, v_{2}, …], where v_{i} is the ith Hazewinkel generator in degree 2(p^{i} − 1). If X is a spectrum, BP_{∗}X is a comodule over the Hopf algebroid BP_{∗}BP. (See [Rav86] for details.) Smith [Smi70] and Toda [Tod71] considered the existence of finite spectra V (k) with BP_{∗}V (k) ≅ BP_{∗}/I_{k+1} (as BP_{∗}- modules, hence as BP_{∗}BP -comodules), where I_{k+1} = (p, v_{1}, …, v_{k}). For example, V (−1) = S^{0} and V (0) = M(p), the mod p Moore spectrum.

The construction of V (1) is originally due to Adams [Ada66]. Smith constructed V (2) for p > 3 and Toda constructed V (3) for p > 5. Furthermore, these results are sharp. The first negative results were obtained by Toda, who showed that V (1) cannot be constructed when p = 2 and likewise for V (2) when p = 3. Later, Ravenel [Rav86, 7.5.1] showed that V (3) does not exist when p = 5.

Recently, Hopkins, Mahowald, and Miller established the nonexistence of V (p−2) for all p > 3. In this paper, we prove

Theorem 1.1. If p > 5, V ((p + 3)/2) does not exist. If V ((p + 1)/2) exists, it is not a ring spectrum.

The proof uses consequences of new work of Hopkins and Miller, not yet published, which we will briefly describe. An account of this work is given in [Rez98]. Throughout this paper, we restrict ourselves to the category of p-local spectra. We call a spectrum p-compact if it is the p-localization of a finite spectrum. (The pcompact spectra comprise the small objects in the p-local stable homotopy category, in the sense of [HPS97].) We write S^{0} for the p-local sphere spectrum.

First we must recall the relevant parts of Morava’s theory ([Mor85], see also [Dev95]). Fix n ≥ 1, set q = p^{n}, and let E_{n∗} ≡ W 𝔽_{q}[[u_{1}, …, u_{n−1}]][u, u^{−1}], with grading |u| = −2 and |u_{i}| = 0. Here W 𝔽_{q} denotes the Witt vectors with coefficients in 𝔽_{q}. E_{n∗} is a graded BP_{∗}-algebra via the map θ : BP_{∗} → E_{n∗} given by

Date: October 30, 1998.

1991 Mathematics Subject Classification. 55N22 (Primary) 55T15, 55P42 (Secondary).

θ(v_{i}) = \begin{cases}u^{1−p^{i}}u_{i}, & i < n \\ u^{1−p^{n}}, & i = n \\ 0, & i > n. \end{cases}

We will write v_{i} for θ(v_{i}) ∈ E_{n∗}. By the Landweber exact functor theorem [Lan76], E_{n∗} ⊗_{BP_{∗}} (−) is a homology theory and we write E_{n} for the representing spectrum. If X is a p-compact spectrum, Morava’s theory provides an action of S_{n} on E_{n∗}X, where S_{n} is the nth Morava stabilizer group. Furthermore, Gal ≡ Gal(𝔽_{q}/𝔽_{p}) acts on E_{n∗}X via its action on W 𝔽_{q}, making E_{n∗}X into a G_{n}-module, where G_{n} ≡ S_{n} ⋊ Gal. In fact, this action is induced by an action (in the homotopy category) of G_{n} on E_{n} ([Dev97], see discussion p. 767).

Morava’s change of rings theorem, in conjunction with [MS95], leads to a spectral sequence

H_{c}^{∗}(G_{n}; E_{n∗}X) ⟹ π_{∗}L_{K(n)}X,

where L_{K(n)} denotes localization with respect to K(n), the nth Morava K-theory spectrum. This spectral sequence resembles a homotopy fixed point spectral sequence, but with two crucial differences. First, the group G_{n} is not discrete. In fact, it is profinite and the E_{2} term involves continuous cohomology. Second, G_{n} acts on E_{n} only up to homotopy. To form homotopy fixed point spectra in the usual way requires that the action be “on the nose”.

Nevertheless, Hopkins and Miller have shown that if G ⊂ G_{n} is a finite subgroup one may form the homotopy fixed point spectrum E_{n}^{hG} and if X is a p-compact spectrum there is a spectral sequence

H^{∗}(G; E_{n∗}X) ⟹ π_{∗}(E_{n}^{hG} ∧ X).

Subsequent work of Devinatz and Hopkins, in preparation, has shown that this can be done for any closed subgroup G ⊂ G_{n}, provided one works with continuous cohomology. More precisely, the spectral sequence is the K(n)-local E_{n}^{Gal}-based Adams spectral sequence for E_{n}^{hG}∧X, where E_{n}^{Gal} is the Landweber exact spectrum with coefficient ring ℤ_{p}[[u_{1}, …, u_{n−1}]][u, u^{−1}].

This construction is natural in G and agrees with the usual spectral sequence when G = S_{n}. Also, the spectral sequence is multiplicative when X is a ring spectrum. A suitable geometric boundary theorem holds as well: Suppose

X ⟶^{f} Y ⟶^{g} Z ⟶^{h} ΣX

is a cofiber sequence of p-compact spectra with E_{n∗}(h) = 0. The resulting short exact sequence

0 ⟶ E_{n∗}X ⟶ E_{n∗}Y ⟶ E_{n∗}Z ⟶ 0

gives rise to a connecting homomorphism H_{c}^{∗}(G; E_{n∗}Z) →^{δ} H_{c}^{∗+1}(G; E_{n∗}X). Suppose z ∈ π_{∗}(E_{n}^{hG} ∧ Z) is detected by z̄ ∈ H_{c}^{∗}(G; E_{n∗}Z). Then δ(z̄) is a permanent cycle and h(z) ∈ π_{∗−1}(E_{n}^{hG} ∧ X) is detected by δ(z), or else h(z) is in higher filtration.

For the rest of the paper, we take p odd and n = p−1. In this case, S_{n} has exactly one maximal finite subgroup G, unique up to conjugation, namely G ≅ ℤ/p⋊ℤ/n^{2}, where the action of the right factor on the left is given by

ℤ/n^{2} → ℤ/n ≅ Aut(ℤ/p).

(See [Hew95], [Hew], also [Mil].) If X is a p-compact spectrum, we refer to the spectral sequence

H^{∗}(G; E_{n∗}X) ⟹ π_{∗}(E_{n}^{hG} ∧ X)

as the spectral sequence for X.

2. Preliminaries

We begin with a discussion of the spectral sequence for S^{0}. These results are due to Hopkins and Miller. Because they have yet to appear in print, we include some details for the reader’s convenience. Full proofs will also appear in the author’s thesis. We remind the reader that n = p − 1, q = p^{n}, and G ≅ ℤ/p ⋊ ℤ/n^{2}.

Our interest in this work began when Hopkins showed us how to compute the E_{2} term of this spectral sequence. To wit,

H^{∗}(G; E_{n∗}) ≅ 𝔽_{q}[Δ^{±}]⟨α⟩[β], (2.1)

where |Δ| = (0, 2pn^{2}), |α| = (1, 2n), and |β| = (2, 2pn). (The cohomological degree is given first.) Actually, this presentation ignores much of H^{0}(G; E_{n∗}). More precisely, H^{0}(G; E_{n∗})/p has a splitting with 𝔽_{q}[Δ^{±}] as a direct summand. With this splitting, the projection H^{0}(G; E_{n∗}) → 𝔽_{q}[Δ^{±}] gives 𝔽_{q}[Δ^{±}]⟨α⟩[β] the structure of an H^{0}(G; E_{n∗})-algebra, which is isomorphic to H^{∗}(G; E_{n∗}) in positive cohomological degrees.

The calculation of H^{∗}(G; E_{n∗}) involves choosing coordinates for E_{n∗} on which the action of G is easily described. Let 𝔪 = (p, u_{1}, …, u_{n−1}) be the maximal ideal of E_{n∗}. There are elements w, w_{1}, …, w_{n−1} ∈ E_{n∗} with

w ≡ u mod (p, 𝔪^{2})
w_{i} ≡ u_{i} mod (p, u_{1}, …, u_{i−1}, 𝔪^{2})

and G action given by

σ(w) = w + ww_{n−1},
σ(ww_{i}) = ww_{i} + ww_{i−1}, for 2 ≤ i ≤ n − 1,
τ(w) = ηw,

and the relation (1 + σ + ⋯ + σ^{p−1})w = 0, where σ generates ℤ/p, τ generates ℤ/n^{2}, and η is a primitive n^{2} root of unity.

To obtain this representation of E_{n∗} requires the following formulae for the action of G on the generators u, u_{i}, which we also record for later use. Recall that an element g ∈ S_{n} has a unique expression of the form g = ∑_{j=0}^{∞} a_{j}S^{j}, where a_{j} ∈ W 𝔽_{q}, a_{j}^{q} = a_{j}, and a_{0} is a unit. We have [DH95, Proposition 3.3 and Theorem 4.4]

g(u) ≡ a_{0}u + a_{n−1}^{χ}uu_{1} + ⋯ + a_{1}^{χ^{n−1}}uu_{n−1} mod (p, 𝔪^{2})

and

g(uu_{i}) ≡ a_{0}^{χ^{i}}uu_{i} + ⋯ + a_{i−1}^{χ}uu_{1} mod (p, 𝔪^{2}),

where χ denotes the Frobenius automorphism of W 𝔽_{q}. Now it turns out that σ = ∑_{j=0}^{∞} a_{j}S^{j} with a_{0} ≡ 1 mod p and a_{1} ∈ (W 𝔽_{q})^{×}. Also, τ = η. Then for 1 ≤ i ≤ n,

\begin{aligned}σ(uu_{i}) & ≡ uu_{i} + c_{i}uu_{i−1} & mod (p, u_{1}, …, u_{i−2}, 𝔪^{2}) \\ τ(uu_{i}) & ≡ η^{p^{i}}uu_{i} & mod (p, 𝔪^{2}), \end{aligned} (2.2)

where c_{i} ∈ (W 𝔽_{q})^{×} and we make the conventions that u_{0} = p and u_{n} = 1.

To get the desired representation of E_{n∗}, Hopkins and Miller start with s = (1 − σ)(v_{1})/p ∈ E_{n∗}. From the definition of the action of S_{n} on E_{n∗} [Dev95, (5.2)] and the formula η_{R}(v_{1}) = v_{1} + pt_{1} in BP_{∗}BP, we have s = t_{1}(σ^{−1}), where we write t_{i} for the image of t_{i} under the map of Hopf algebroids

(BP_{∗}, BP_{∗}BP) ⟶ (E_{n∗}^{Gal}, Map_{c}(S_{n}, E_{n∗})^{Gal}).

This map is a composition of several Hopf algebroid maps. By unraveling the definitions of these maps in [Dev95], it is clear that if g ∈ S_{n} is given by g = ∑_{j=0}^{∞} a_{j}S^{j}, then t_{i}(g) ≡ a_{0}^{−1}a_{i}u^{1−p^{i}} mod 𝔪.

In particular, s ≡ cu^{1−p} mod 𝔪, where c ∈ (W 𝔽_{q})^{×}. Let

t = s ∏_{j=0}^{p−1} σ^{j}(u)  and  w = \frac{1}{n^{2}} ∑_{j=1}^{n^{2}} η^{−j}τ^{j}(t).

Then w ≡ cu mod (p, 𝔪^{2}), (1 + σ + ⋯ + σ^{p−1})w = 0, and τ(w) = ηw. Finally, let w_{i} = (σ − 1)^{n−i}(w).

The differentials are determined by the Toda differential in the Adams-Novikov spectral sequence (ANSS) and the nilpotence of β_{1} ∈ π_{∗}S^{0} as follows. There is a map from the ANSS to the spectral sequence for S^{0},

Ext_{BP_{∗}BP} (BP_{∗}, BP_{∗}) ⟶ H^{∗}(G; E_{n∗}).

By [Rav78], this map sends α_{1} to α, β_{1} to β, and β_{p/p} to Δβ. (We will omit the phrase “up to multiplication by a unit” from this discussion.) (See [Rav86, Section 1.3] for notation.) Because α_{1} and β_{1} are permanent cycles, so are α and β.

Let K denote the kernel of the projection H^{0}(G; E_{n∗}) → 𝔽_{q}[Δ^{±}]. Because β·K = 0, d_{2p−1} vanishes on K. Also, from the Toda differential d_{2p−1}(β_{p/p}) = α_{1}β_{1}^{p} ([Tod67],[Tod68]) and the evident sparseness in our spectral sequence, we conclude that

d_{2p−1}(Δ) = αβ^{p−1}.

These facts and the multiplicative structure of the spectral sequence completely determine d_{2p−1}.

For degree reasons the next possible differential is d_{2n^{2}+1}. Since β_{1}^{pn+1} = 0 in π_{∗}S^{0} ([Tod67]), [Tod68]), β^{pn+1} is hit by some differential in our spectral sequence. The last differential which could do this is d_{2n^{2}+1} and therefore

d_{2n^{2}+1}(Δ^{p−1}α) = β^{n^{2}+1}.

Again, d_{2n^{2}+1} vanishes on K. This determines d_{2n^{2}+1} completely and the spectral sequence collapses after this point for degree reasons, hence

Proposition 2.1. If s is odd, then E_{∞}^{s,t} = 0 unless 1 ≤ s ≤ 2n − 1 and

t ≡ 2n + (s − 1)pn + 2pn^{2}x mod 2p^{2}n^{2}, where x ≢ −1 mod p.

If s > 0 is even, then E_{∞}^{s,t} = 0 unless 2 ≤ s ≤ 2n^{2} and

t ≡ spn mod 2p^{2}n^{2}.

We also need a handle on the zero line. Using the coordinates w, w_{i} mentioned above, it is easy to show that

Proposition 2.2. H^{0}(G; E_{n∗}) is concentrated in degrees t ≡ 0 mod 2n.

Finally, we will need the following partial description of H^{∗}(G; E_{n∗}V (1)), which is a simple consequence of (2.1).

Proposition 2.3. Let A be the graded abelian group given by

A_{m} = H^{1,2pn+2pn^{2}m}(G; E_{n∗}/I_{2}),

made into an 𝔽_{q}[Δ^{±}]-module via the map E_{n∗} → E_{n∗}/I_{2}. Then A ≅ Σ^{2pn}𝔽_{q}[Δ^{±}].

3. Proof of Theorem 1.1

Suppose V (k) exists. It is well known that the condition BP_{∗}V (k) ≅ BP_{∗}/I_{k+1} is equivalent to H_{∗}(V (k); 𝔽_{p}) being isomorphic to the exterior algebra E(τ_{0}, …, τ_{k}) as a comodule over the mod p dual Steenrod algebra. This point of view allows one to produce V (k − 1) as a skeleton of V (k) and exhibit V (k) as the cofiber of a map f : Σ^{|v_{k}|}V (k − 1)′ → V (k − 1) inducing multiplication by v_{k} in BP -homology. (The notation V (k − 1)′ is meant to distinguish this spectrum from V (k − 1)—they need not be isomorphic.)

Recall that in the Hopf algebroid structure of BP_{∗}BP, v_{k} is invariant mod I_{k}, i.e., η_{r}(v_{k}) ≡ v_{k} mod I_{k}. Then v_{k} ∈ E_{n∗}V (k − 1) ≅ E_{n∗}/I_{k} is fixed by all of S_{n}. In particular, v_{k}^{j} ∈ H^{0}(G; E_{n∗}V (k − 1)) for all j ≥ 0.

The map f induces a map (of degree |v_{k}|) from the spectral sequence for V (k−1)′ to the spectral sequence for V (k − 1), which is multiplication by v_{k} on H^{0}. In particular, if v_{k}^{j} is a permanent cycle in the spectral sequence for V (k − 1)′, then v_{k}^{j+1} is a permanent cycle in the spectral sequence for V (k − 1).

Fix p > 5 and let m = (p + 3)/2. We show that if V (m − 1) exists, then in the spectral sequence for V (m−1), v_{m} is a permanent cycle and v_{m}^{2} is not. Then clearly V (m − 1) cannot be a ring spectrum, and by the discussion above, V (m) cannot exist. In order to simplify the notation, we will ignore the distinction between V (k − 1)′ and V (k − 1). Since our arguments depend only on properties of the spectral sequence for S^{0}, which is unique in the p-local category, no generality is lost.

Before proceeding, we need to establish the vanishing of H^{s,t}(G; E_{n∗}/I_{k}) and π_{r}(E_{n}^{hG}∧V (k)) for various values of s, t, r, and k. We are not able to calculate these groups directly for arbitrary values of k. Instead, we use a brute force approach which reduces the computation to the case k = −1.

Lemma 3.1. If d ∈ ℤ satisfies

d ≡ 2pny mod 2pn^{2}

where y ≢ 1 mod n, then

H^{1,d}(G; E_{n∗}/I_{k}) = 0.

If, in addition, k ≠ p − 1, then

H^{2,d−2n}(G; E_{n∗}/I_{k}) = 0.

Proof. If k ≥ p, E_{n∗}/I_{k} = 0 so the result is trivial. Thus assume k < p. We prove first that H^{1,d}(G; E_{n∗}/I_{k}) = 0. Using the long exact sequences in cohomology arising from the short exact sequences of E_{n∗}-modules

0 ⟶ E_{n∗}/I_{i} ⟶^{v_{i}} E_{n∗}/I_{i} ⟶ E_{n∗}/I_{i+1} ⟶ 0,

it suffices to show that H^{s,t}(G; E_{n∗}) = 0 for all (s, t) with 1 ≤ s ≤ k + 1 and

t = d − ∑_{j=1}^{s−1} |v_{i_{j}}|, (3.1)

where 0 ≤ i_{1} < i_{2} < ⋯ < i_{s−1} ≤ k − 1.

First, suppose s is even. By (2.1), H^{s,t}(G; E_{n∗}) = 0 unless t ≡ spn mod 2pn^{2}. Substituting this into (3.1) and reducing mod p leads to s = 2 and i_{1} = 0. In this case, working mod 2pn^{2} leads to y ≡ 1 mod n, contrary to our hypothesis on y.

Now suppose s is odd. Again by (2.1), H^{s,t}(G; E_{n∗}) = 0 unless t ≡ 2n+(s−1)pn mod 2pn^{2}. Substituting this into (3.1) and reducing mod p leads to an equation in s which, keeping in mind all the constraints involved, has no solution.

Similarly, we reduce the proof that H^{2,d−2n}(G; E_{n∗}/I_{k}) = 0 to showing that H^{s,t}(G; E_{n∗}) = 0 for all (s, t) with 2 ≤ s ≤ k + 2 and

t = d − 2n − ∑_{j=1}^{s−2} |v_{i_{j}}|, (3.2)

where 0 ≤ i_{1} < i_{2} < ⋯ < i_{s−2} ≤ k − 1.

As before, we proceed by reducing (3.2) mod p. This time there are no solutions for s, assuming as we are that k < p − 1.

Lemma 3.2. Let m < p − 1. If d ∈ ℤ satisfies

d ≡ 2n + 2pn + 2p^{2}ny mod 2p^{2}n^{2},

where y ≢ 0 mod n, and V (m) exists, then

π_{d−1}(E_{n}^{hG} ∧ V (m)) = 0.

Proof. Using the long exact sequences in homotopy arising from the cofiber sequences

Σ^{|v_{i}|}V (i − 1) ⟶^{v_{i}} V (i − 1) ⟶ V (i),

it suffices to show that in the spectral sequence for S^{0}, E_{∞}^{s,t} = 0 for all (s, t) with

t − s = d − 1 − k − ∑_{j=1}^{k} |v_{i_{j}}|, (3.3)

where 0 ≤ k ≤ m + 1 and 0 ≤ i_{1} < i_{2} < ⋯ < i_{k} ≤ m.

First, suppose s = 0. Reducing (3.3) mod 2n, we have t ≡ −(k + 1). By Proposition 2.2, E_{∞}^{0,t} = 0.

Next, suppose s is odd. By Proposition 2.1, we may take 1 ≤ s ≤ 2n − 1 and t ≡ 2n + (s − 1)pn mod 2pn^{2}. Substituting this into (3.3) and reducing mod 2n, we are led to s = k + 1. Substituting this into (3.3) and reducing mod p yields k = 0 or k = 1 (if i_{1} = 0). The second case cannot occur because s is odd, therefore we may take k = 0 and s = 1. In this case, (3.3) becomes t = d. Reducing this mod 2pn^{2}, we are led to y ≡ −1 mod n. It follows that t ≡ 2n − 2pn^{2} mod 2p^{2}n^{2}. By Proposition 2.1, E_{∞}^{1,t} = 0.

Finally, suppose s > 0 is even. In this case, we may take 2 ≤ s ≤ 2n^{2} and t ≡ spn mod 2p^{2}n^{2}. Working mod 2n as above, we get s = 2nl + k + 1, where 0 ≤ l < n. Note that k must be odd. Substituting these equations for s and t into (3.3) and reducing mod p leads to l = k − 1 (if i_{1} > 0) or l = k − 2 (if i_{1} = 0). To finish the proof, we need to reduce (3.3) mod 2p^{2}n, which is a bit unwieldy. There are four cases:

t − s ≡ 2n + 2pn − (k + 1) − \begin{cases}(2pn + 2n)k, & i_{1} > 1 \\ (2pn + 2n)(k − 1) + 2n, & i_{1} = 1 \\ (2pn + 2n)(k − 1), & i_{1} = 0 and i_{2} > 1 \\ (2pn + 2n)(k − 2) + 2n, & i_{1} = 0 and i_{2} = 1. \end{cases}

For each case, we substitute t = spn, s = 2nl + k + 1, and either l = k − 1 (first two cases) or l = k − 2 (last two cases) and solve for k. The first and third cases yield k ≡ −1 mod 2p, which is not possible. The second and fourth cases yield k ≡ 1 mod 2p, i.e., k = 1. In the fourth case, this is not possible, because we are assuming that i_{2} = 1 in that case, so k > 1. This leaves the second case, with k = 1. Substituting t = spn, s = 2nl + k + 1, l = k − 1, and k = 1 into (3.3) and reducing mod 2p^{2}n^{2} leads to y ≡ 0 mod n, contrary to our hypothesis on y.

Proposition 3.3. Let p > 5 and k ≤ p − 1. If V (k − 1) exists, then v_{k} is a permanent cycle in the spectral sequence for V (k − 1).

Proof. The cases k = 1, 2 and 3 are immediate, because v_{1}, v_{2}, and v_{3} are permanent cycles in the corresponding Adams-Novikov spectral sequences. So let k ≥ 4. There are many steps to the proof, so we summarize the argument first.

Consider the connecting homomorphisms

H^{∗}(G; E_{n∗}V (i)) ⟶^{δ_{i+1}} H^{∗+1}(G; E_{n∗}V (i − 1))

arising from the cofiber sequences

Σ^{|v_{i}|}V (i − 1) ⟶^{v_{i}} V (i − 1) ⟶ V (i) ⟶^{h_{i+1}} Σ^{|v_{i}|+1}V (i − 1).

Let

S^{0} ⟶^{f} V (1) ⟶^{g} V (k − 2)

factor the inclusion of the bottom cell into V (k − 2). We will show that

δ_{k}(v_{k}) = g_{∗}(f_{∗}(Δ^{m})δ_{3}(v_{3})),

where m ≡ 0 mod p, and that this element is nontrivial. Since Δ^{m} and v_{3} are permanent cycles in the spectral sequences for S^{0} and V (2), respectively, and V (1) is a ring spectrum [Smi70], it follows that δ_{k}(v_{k}) is a permanent cycle.

So δ_{k}(v_{k}) detects an element x ∈ π_{|v_{k}|−|v_{k−1}|−1}(E_{n}^{hG} ∧V (k −2)). By Lemma 3.2, π_{|v_{k}|−1}(E_{n}^{hG} ∧ V (k − 2)) = 0, so x is in the image of the map

π_{|v_{k}|}(E_{n}^{hG} ∧ V (k − 1)) ⟶^{h_{k}} π_{|v_{k}|−|v_{k−1}|−1}(E_{n}^{hG} ∧ V (k − 2)),

say x = h_{k}(y). Note that y must have filtration zero, i.e., y ∈ H^{0}(G; E_{n∗}V (k − 1)).

Now δ_{k}(y − v_{k}) = 0, so y − v_{k} pulls back to H^{0}(G; E_{n∗}V (k − 2)). We then show that the map (g ∘f)_{∗} is onto in this bidegree and that everything in H^{0,|v_{k}|}(G; E_{n∗}) is a permanent cycle. Therefore y − v_{k} is a permanent cycle, so v_{k} is a permanent cycle.

Step 1: δ_{k}(v_{k}) ≠ 0. Otherwise, v_{k} is in the image of the map

H^{0}(G; E_{n∗}V (k − 2)) ⟶ H^{0}(G; E_{n∗}V (k − 1)),

i.e., v_{k} lifts to a fixed point γ ∈ u^{s}𝔽_{q}[[u_{k−1}, …, u_{n−1}]], where s = 1 − p^{k}. If μ ∈ u^{s}𝔽_{q}[[u_{k−1}, …, u_{n−1}]] is a monomial, we write μ ∈ γ if μ appears as a term (up to multiplication by a unit) when we express γ as a sum of monomials.

We claim that u^{s}u_{k} ∉ γ, which would then contradict δ_{k}(v_{k}) = 0. From (2.2) we have the formula

(σ − 1)(u^{s}u_{i}) ≡ c_{i}u^{s}u_{i−1} mod (p, u_{1}, …, u_{i−2}, 𝔪^{2}).

Suppose u^{s} ∈ γ. Since (σ − 1)γ = 0 and u^{s}u_{n−1} ∈ (σ − 1)u^{s}, there must be another monomial μ ∈ γ with u^{s}u_{n−1} ∈ (σ − 1)μ. But by the formula above, this isn’t possible, hence u^{s} ∉ γ. Iterating this procedure yields u^{s}u_{i} ∉ γ for k ≤ i ≤ n.

Step 2: The map

H^{1,|v_{k}|−|v_{k−1}|}(G; E_{n∗}V (1)) ⟶^{g_{∗}} H^{1,|v_{k}|−|v_{k−1}|}(G; E_{n∗}V (k − 2))

is onto. It suffices to show that

H^{2,|v_{k}|−|v_{k−1}|−|v_{m}|}(G; E_{n∗}V (m − 1)) = 0

for m = 2, …, k − 2, which follows from Lemma 3.1.

Step 3: By Step 2, there is an element y ∈ H^{1}(G; E_{n∗}V (1)) which maps to δ_{k}(v_{k}) ∈ H^{1}(G; E_{n∗}V (k − 2)). Then y = f_{∗}(Δ^{m})δ_{3}(v_{3}), where m ≡ 0 mod p. This follows from Proposition 2.3 after checking that the degrees work out, i.e.,

|δ_{k}(v_{k})| ≡ |δ_{3}(v_{3})| ≡ 2pn + 2pn^{2} mod 2p^{2}n^{2},

and showing that δ_{3}(v_{3}) ≠ 0, which follows exactly as in Step 1.

Step 4: The map

(g ∘ f)_{∗} : H^{0,|v_{k}|}(G; E_{n∗}) ⟶ H^{0,|v_{k}|}(G; E_{n∗}V (k − 2))

is surjective. For this, it suffices to show that

H^{1,|v_{k}|−|v_{m}|}(G; E_{n∗}V (m − 1)) = 0

for m = 0, 1, …, k − 2, which follows from Lemma 3.1.

Step 5: Everything in H^{0,|v_{k}|}(G; E_{n∗}) is a permanent cycle. Indeed, for degree reasons these elements are in the kernel K of the projection H^{0}(G; E_{n∗}) → 𝔽_{q}[Δ^{±}]. As we discussed in the previous section, everything in K is a permanent cycle.

Proposition 3.4. Let m = (p + 3)/2. If V (m − 1) exists, then v_{m}^{2} supports a differential in the spectral sequence for V (m − 1).

Proof. By Lemma 3.2,

π_{2|v_{m}|−|v_{k}|−1}(E_{n}^{hG} ∧ V (k − 1)) = 0

for k = 3, 4, …, m − 1. It follows that if v_{m}^{2} is a permanent cycle, then v_{m}^{2} is in the image of the map

H^{0,2|v_{m}|}(G; E_{n∗}V (2)) ⟶ H^{0,2|v_{m}|}(G; E_{n∗}V (m − 1)).

We complete the proof by showing that this is not possible.

Set s = 2(1 − p^{m}), i.e., v_{m}^{2} = u^{s}u_{m}^{2}. Suppose γ ∈ H^{0,−2s}(G; E_{n∗}V (2)) and let

A_{n} = { u^{s}u_{3}, u^{s}u_{4}, …, u^{s}u_{n−1}, u^{s} }

and

A_{j} = { u^{s}u_{i}u_{j} : 2m − j ≤ i ≤ j }

for j = m, m + 1, …, n − 1. In the language of Step 1 of the previous proof, we claim that each of these sets is disjoint from γ. This will complete the proof, as A_{m} = {u^{s}u_{m}^{2}}. This is precisely where the hypothesis m = (p + 3)/2 appears.

As in the previous proof, we have u^{s}u_{i} ∉ γ for 4 ≤ i ≤ n. For i = 3, note that τ(u^{s}u_{3}) ≡ η^{s−1+p^{3}}u^{s}u_{3} mod (p, 𝔪^{2}), by (2.2). It is easily verified that s − 1 + p^{3} ≡ (1 − p) mod (p − 1)^{2}, which shows that u^{s}u_{3} ∉ γ, again by (2.2). (Recall that η is an n^{2} root of unity.) Thus the claim is established for j = n.

Now let j < n. For each i with 2m − j − 1 ≤ i ≤ j, set

B_{j,i} = { u^{s}u_{k}u_{j} ∈ A_{j} : k > i }.

From (2.2) we have u^{s}u_{i−1}u_{j} ∈ (σ − 1)u^{s}u_{i}u_{j}. Then if u^{s}u_{i}u_{j} ∈ γ, there must be another monomial μ ∈ γ with u^{s}u_{i−1}u_{j} ∈ (σ − 1)μ. We claim that then

μ ∈ B_{j,i} ∪ A_{j+1} ∪ ⋯ ∪ A_{n}.

To see this, write μ = u^{s}u_{a}u_{b} with a ≤ b. It follows easily from (2.2) that b ≥ j and a ≥ i − 1. If b > j, then

2m − b ≤ 2m − j − 1 ≤ i − 1 ≤ a,

hence μ ∈ A_{b}. If b = j, then in fact from (2.2) we have a ≥ i. The case a = i is ruled out by the assumption that μ ≠ u^{s}u_{i}u_{j}, so μ ∈ B_{j,i}, as claimed.

Suppose inductively that A_{k} is disjoint from γ for k = j + 1, …, n. Suppose also that B_{j,i} ∩ γ = ∅ for some i with 2m − j ≤ i ≤ j. (This is trivially true when i = j, as B_{j,j} = ∅.) Now by the previous paragraph, if u^{s}u_{i}u_{j} ∈ γ, there exists μ ∈ γ ∩ (B_{j,i} ∪ A_{j+1} ∪ ⋯ ∪ A_{n}). This is a contradiction, so u^{s}u_{i}u_{j} ∉ γ, i.e., B_{j,i−1} ∩ γ = ∅. By downward induction on i, it follows that A_{j} is disjoint from γ. By downward induction on j, we obtain A_{m} ∩ γ = ∅, as desired.

References

[Ada66] J. F. Adams, On the groups J(X). IV, Topology 5 (1966), 21–71.

[Dev95] Ethan S. Devinatz, Morava’s change of rings theorem, The Čech centennial (Boston, MA, 1993) (Providence, RI), Contemp. Math., vol. 181, Amer. Math. Soc., Providence, RI, 1995, pp. 83–118.

[Dev97] Ethan S. Devinatz, Morava modules and Brown-Comenetz duality, Amer. J. Math. 119 (1997), no. 4, 741–770.

[DH95] Ethan S. Devinatz and Michael J. Hopkins, The action of the Morava stabilizer group on the Lubin-Tate moduli space of lifts, Amer. J. Math. 117 (1995), no. 3, 669–710.

[Hew] Thomas Hewett, Normalizers of finite subgroups of division algebras over local fields, unpublished, available on the web at the author’s home page: http://www.math.princeton.

        edu/~hewett/.

[Hew95] Thomas Hewett, Finite subgroups of division algebras over local fields, J. Algebra 173 (1995), no. 3, 518–548.

[HPS97] Mark Hovey, John H. Palmieri, and Neil P. Strickland, Axiomatic stable homotopy theory, Mem. Amer. Math. Soc. 128 (1997), no. 610, x+114.

[Lan76] Peter S. Landweber, Homological properties of comodules over MU_{∗}MU and BP_{∗}BP, Amer. J. Math. 98 (1976), no. 3, 591–610.

[Mil] Haynes Miller, Notes on algebra, unpublished notes.

[Mor85] Jack Morava, Noetherian localisations of categories of cobordism comodules, Ann. of Math. (2) 121 (1985), no. 1, 1–39.

[MS95] Mark Mahowald and Hal Sadofsky, v_{n} telescopes and the Adams spectral sequence, Duke Math. J. 78 (1995), no. 1, 101–129.

[Rav78] Douglas C. Ravenel, The non-existence of odd primary Arf invariant elements in stable homotopy, Math. Proc. Cambridge Philos. Soc. 83 (1978), no. 3, 429–443.

[Rav86] Douglas C. Ravenel, Complex cobordism and stable homotopy groups of spheres, Pure and Applied Mathematics, vol. 121, Academic Press Inc., Orlando, Fla., 1986.

[Rez98] Charles Rezk, Notes on the Hopkins-Miller theorem, Homotopy theory via algebraic geometry and group representations (Providence, RI), Contemp. Math., vol. 220, Amer. Math. Soc., Providence, RI, 1998, pp. 313–366.

[Smi70] Larry Smith, On realizing complex bordism modules. Applications to the stable homotopy of spheres, Amer. J. Math. 92 (1970), 793–856.

[Tod67] Hirosi Toda, An important relation in homotopy groups of spheres, Proc. Japan Acad. 43 (1967), 839–842.

[Tod68] Hirosi Toda, Extended p-th powers of complexes and applications to homotopy theory, Proc. Japan Acad. 44 (1968), 198–203.

[Tod71] Hirosi Toda, On spectra realizing exterior parts of the Steenrod algebra, Topology 10 (1971), 53–65.

University of Washington, Department of Mathematics, Box 354350, Seattle, WA 98195–4350

   E-mail address: nave@math.washington.edu