THE COHOMOLOGY OF THE MORAVA STABILIZER GROUP 𝕊_{2} AT THE PRIME 3

VASSILY GORBOUNOV, STEPHEN F. SIEGEL, AND PETER SYMONDS

Abstract. We compute the cohomology of the Morava stabilizer group 𝕊_{2} at the prime 3 by resolving it by a free product ℤ/3 ∗ ℤ/3 and analyzing the “relation module.”

1. Introduction and Statement of the Main Result

The applications of the main theorem of this paper in homotopy theory are due to the Morava Change of Rings Theorem [7]. Let p be a prime, and denote by 𝕊_{n} the group of units in the maximal order of a cyclic division algebra over ℚ_{p} of index n and Hasse invariant \frac{1}{n}. The Morava Theorem says essentially that the cohomology of 𝕊_{n} with coefficients in a certain representation describes the Bousfield localization functor L_{K(n)}. This is the localization of stable homotopy theory with respect to the spectrum of the n-th Morava K-theory, K(n) [2]. The functors L_{K(n)} play an important role in homotopy theory [10]. At present the case n = 1 is completely understood for all primes p. The next case, n = 2, has been partially investigated for primes p ≥ 5 (see for example [15], [16]). The functor L_{K(2)} for small primes is harder to study because the group 𝕊_{2} is of infinite cohomological dimension.

In this paper we deal with the prime 3 only. As the first step of the analysis of L_{K(2)} one needs to compute the continuous cohomology of a certain canonical subgroup 𝕊_{2}^{0} of 𝕊_{2} with trivial coefficients. We compute these cohomology groups in the course of the proof of the Theorem 1.1 below. This “almost” computes, according to the Morava Change of Ring Theorem, the homotopy groups of the localization of the Toda Smith complex V (1); for more details see [9]. The rest of the calculation of π_{∗}L_{K(2)}S^{0} consists of two Bockstein spectral sequences. We will not pursue this here.

The research of the second author was supported by an NSF postdoctoral fellowship.

Theorem 1.1. H_{c}^{∗}(𝕊_{2}; 𝔽_{3}) is freely generated as a graded-commutative 𝔽_{3}-algebra by elements Z (in degree 1), C,E (in degree 3), and X (in degree 4). Its Poincaré series is \frac{(1+t)(1+t^{3})^{2}}{1−t^{4}} .

We obtain this by first calculating the cohomology of the canonical subgroup, 𝕊l, of 𝕊_{2}.

Theorem 1.2. H_{c}^{∗}(𝕊l; 𝔽_{3}) has generators e_{1}, e_{2} (in degree 1), x_{1}, x_{2}, a (in degree 2), c_{1}, c_{2} (in degree 3). The product of any two generators with different subscripts is 0 and in addition there are relations

a^{2} = ae_{1} = ae_{2} = ac_{1} = ac_{2} = 0, c_{1}e_{1} = ax_{1}, c_{2}e_{2} = ax_{2}.

Its Poincaré series is \frac{1+t+t^{2}+t^{3}}{1−t}.

A computation of H_{c}^{∗}(𝕊l; 𝔽_{3}) was sketched in [9], but the multiplicative structure given there does not agree with the one above. The question of the cohomology of 𝕊l was first reopened by Henn in connection with a deep theorem of his on the cohomology of profinite groups [4], and he also obtained the result stated here. Our calculation proceeds by more classical methods.

The structure of H_{c}^{∗}(𝕊l; 𝔽_{3}) can also be described as follows: it was shown in [3] that if j is the quotient map

𝕊l ⟶^{j} 𝕊l/𝕊l′ ≅ ℤ/3 ⊕ ℤ/3,

(where 𝕊l′ denotes the commutator subgroup [𝕊l, 𝕊l] of 𝕊l), then there exists a homomorphism ℤ/3 ∗ ℤ/3 ⟶^{i} 𝕊l such that ji is onto. The image R = j^{∗}H^{∗}(ℤ/3 ⊕ ℤ/3) ⊂ H_{c}^{∗}(𝕊l) is mapped isomorphically onto H^{∗}(ℤ/3∗ℤ/3) by i^{∗} (R is the subring generated by 1, e_{1}, e_{2}, x_{1}, x_{2}). The kernel R′ = Ker i^{∗} (generated by a, c_{1}, c_{2}, as an R-module) is additively like R but with the degrees increased by 2. The structure of R′ as an R-module is as given above and R′^{2} = 0 : this determines the ring H_{c}^{∗}(𝕊l) ≅ R ⊕ R′.

2. Background Information

We briefly recall some facts about the group of units of a maximal order in a division algebra. A full account can be found in [11], for example. Consider a cyclic algebra 𝔻 over ℚ_{p} of index n and Hasse invariant \frac{1}{n}. It can be constructed as follows. Let 𝕎 be the totally unramified extension of ℚ_{p} of degree n (so 𝕎 ≅ ℚ_{p}(ζ) where ζ is a (p^{n} − 1)-st root of unity). The Galois group of the extension 𝕎/ℚ_{p} is a cyclic group of order n: it is generated by the Frobenius homomorphism σ. We form the crossed product algebra of 𝕎 and Gal(𝕎/ℚ_{p}). This amounts to introducing a variable S which commutes with 𝕎 according to the formula wS = Sw^{σ} and satisfies S^{n} = a ∈ ℚ_{p}^{×}. To define 𝔻 we set S^{n} = p. 𝔻 is a division algebra over ℚ_{p} of rank n^{2}.

Let 𝕆 be the maximal order in 𝔻: it is generated by S and the integers of 𝕎. Its maximal ideal is 𝕆S and 𝕆/𝕆S ≅ 𝔽_{p^{n}}. We are interested in three groups contained in 𝕆. The first is the group of units of 𝕆, which we denote by 𝕊_{n}. The second is the subgroup of strict units in 𝕊_{n}. It consists of the elements a ∈ 𝕊_{n} such that a ≡ 1 mod S. We denote it by 𝕊_{n}^{0}. The third is the kernel of the reduced norm restricted to 𝕊_{n}^{0}. We denote this subgroup by 𝕊l. It is a pro-p group because it is p-filtered and compact ([6], II, 2.1.3).

Let H_{r} denote the subgroup of 𝕊l consisting of elements congruent to 1 modulo S^{r}. By definition H_{1} = 𝕊l. According to [12], there are injective maps ρ_{r} : H_{r}/H_{r+1} → 𝕆/𝕆S ≅ 𝔽_{p^{n}} given by ρ_{r}(1+aS^{r}) ≡ a modS, (a ∈ 𝕆). They are also surjective unless n|r, in which case the image consists of those elements of trace 0 over the prime field.

From now on we shall only consider p = 3 and n = 2. Then |H_{1}/H_{2}| = 9, |H_{2}/H_{3}| = 3, |H_{3}/H_{4}| = 9, and according to [12], [H_{1}, H_{1}] = H_{2}, [H_{1}, H_{2}] = H_{3}, [H_{2}, H_{2}] = H_{4}. Now 𝕎 contains an 8-th root of unity ζ, and so z = \frac{ζS−1}{2} ∈ 𝔻 is a cube root of unity. Thus X = z and Y = z^{S} are two elements of 𝕊l of order 3. Their images x = j(X), y = j(Y) generate 𝕊l/𝕊l′ ≅ ℤ/3 ⊕ ℤ/3, (where 𝕊l′ = [𝕊l, 𝕊l] = H_{2}). We now define i : ℤ/3 ∗ ℤ/3 → 𝕊l to be the map which takes the generators 𝕏 and 𝕐 to X and Y. This map is in fact injective [3], but we do not need to know that here.

There is a group D of automorphisms of 𝕊l which has order 8 and is generated by (i) conjugation by S, which interchanges X and Y, and (ii) conjugation by Sζ, which interchanges X and X^{2} and fixes Y.

Note that the action of D lifts to an action on ℤ/3 ∗ ℤ/3.

The natural cohomology theory for a profinite group is the cohomology on continuous cochains [13], denoted by H_{c}^{∗}, and that is what we use here. It agrees with the usual cohomology on a finite group.

Any maximal finite subgroup of 𝕊l is cyclic of order 3, so the Krull dimension of H_{c}^{∗}(𝕊l; 𝔽_{3}) is one [8], [7].

3. Resolutions

The fact that ji is onto implies that Im i is dense in the pro-3 topology. So we have an epimorphism of pro-3 groups

{ \atop ℤ/3 ∗ ℤ/3} ⟶ 𝕊l ( denotes pro-3 completion).

Let K denote the kernel of this map. The kernel of ℤ/3 ∗ ℤ/3 ⟶ ℤ/3 ⊕ ℤ/3 is free on the four generators [𝕏^{i}, 𝕐^{j}], (1 ≤ i, j ≤ 2) [14]. The completion of the corresponding short exact sequence remains exact, since ℤ/3⊕ℤ/3 is finite. This leads to a diagram with exact rows and columns:

\begin{aligned}K & ⟶ & F̂_{4} & ⟶ & 𝕊l′ \\ ∥ & & ↓ & & ↓ \\ K & ⟶ & { \atop ℤ/3 ∗ ℤ/3} & ⟶ & 𝕊l \\ & & ↓ & & ↓ \\ & & ℤ/3 ⊕ ℤ/3 & & ℤ/3 ⊕ ℤ/3. \end{aligned} (3.1)

Now K is a closed subgroup of a free pro-3 group (i.e. the pro-3 completion of a free group), so is itself a free pro-3 group ([13], Cor. 2 to I, Prop. 24).

According to the theory in ([6], V 2.5.7), 𝕊l′ is equi-3-valued (with the usual 3-adic valuation), so H_{c}^{∗}(𝕊l′) ≅ Λ^{∗}H_{c}^{1}(𝕊l′). 𝕊l′/[𝕊l′, 𝕊l′] = H_{2}/H_{4} has order 27 and is easily seen to have exponent 3, so it has rank 3, and thus so does H_{c}^{1}(𝕊l′).

Let N denote 𝕊l′/[𝕊l′, 𝕊l′] as 𝔽_{3}E-module (where E = 𝕊l/𝕊l′ ≅ ℤ/3 ⊕ ℤ/3). Then H_{c}^{2}(𝕊l′) ≅ N by duality.

We want to be able to describe modules such as N explicitly. For this purpose note that if we set X = x − 1 and Y = y − 1, then 𝔽_{3}E ≅ 𝔽_{3}[X, Y |X^{3} = Y^{3} = 0]. The augmentation ideal I is generated by X and Y, and, because E is a p-group, I is the radical.

Since [H_{1}, H_{2}] = H_{3} and |H_{2}/H_{3}| = 3, we must have N/IN ≅ 𝔽_{3}. If we consider the image of annN in I/I^{2} ≅ E, we see that it cannot be 1-dimensional, since then it could not be invariant under the group of automorphisms D. Hence ann N = I^{2} and N ≅ 𝔽_{3}E/I^{2}.

Now consider the spectral sequence

H_{c}^{p}(𝕊l′; H_{c}^{q}(K)) ⇒ H_{c}^{p+q}(F̂_{4}).

There are only two rows, since K is a free pro-3 group, and we deduce that

H_{c}^{r}(𝕊l′; H_{c}^{1}(K)) = 0, r ≥ 2,
H_{c}^{1}(𝕊l′; H_{c}^{1}(K)) ≅ H_{c}^{3}(𝕊l′) ≅ 𝔽_{3},

and there are short exact sequences

0 ⟶ H_{c}^{1}(𝕊l′) ⟶ H_{c}^{1}(F̂_{4}) ⟶ E_{∞}^{0,1} ⟶ 0, (3.2)

0 ⟶ E_{∞}^{0,1} ⟶ H_{c}^{1}(K)^{𝕊l′} ⟶ N ⟶ 0. (3.3)

Sequence (3.2) shows that E_{∞}^{0,1} ≅ 𝔽_{3}. Let M denote H_{c}^{1}(K)^{𝕊l′} as an 𝔽_{3}E-module, so we have

0 ⟶ 𝔽_{3} ⟶ M ⟶ N ⟶ 0.

4. The Structure of M

The short exact sequence 0 → 𝔽_{3} → M → N → 0 shows that either M ≅ N ⊕ 𝔽_{3}, or M is generated by one element. In the latter case, M ≅ 𝔽_{3}E/ ann M, and we shall assume that this holds for the rest of this section. The image of ann M in I^{2}/I^{3} is a subspace S of codimension 1, which must be invariant under the action of the group of automorphisms, D. Since I^{2}/I^{3} has basis {X^{2}, XY, Y^{2}}, it is easy to check that the only possibilities for S are S_{1} = ⟨X^{2}, Y^{2}⟩, S_{2} = ⟨XY, X^{2} + Y^{2}⟩, and S_{3} = ⟨XY, X^{2} − Y^{2}⟩. Let M_{i} = 𝔽_{3}E/S_{i}. We claim that, in fact, M ≅ M_{1}, but this will only become apparent later. Notice, however, that if we extend the field to 𝔽_{9}, then all three modules differ only by an automorphism of the group algebra, for if

ϕ_{2} : X ↦ X + Y, Y ↦ X − Y,

then ϕ_{2}(S_{1}) = S_{2}, and so M_{2}^{ϕ_{2}} ≅ M_{1}. Similarly, if

ϕ_{3} : X ↦ X + iY, Y ↦ X − iY, (where i^{2} = 1)

then ϕ_{3}(S_{1}) = S_{3}, and so M_{3}^{ϕ_{3}} ≅ M_{1}.

5. The Cohomology of M_{1}

Let p be an odd prime, k a field of characterisitc p, and E = ⟨x, y⟩ an elementary abelian p-group of order p^{2}. (We will only need the case where p = 3 and k = 𝔽_{3}, but it is just as easy to prove the result of this section more generally.) The group algebra kE is the truncated polynomial algebra k[X, Y | X^{p} = Y^{p} = 0], where X = x − 1 and Y = y − 1. A minimal projective resolution P →^{ϵ} k of the trivial kE-module k may be constructed as follows. First let P_{n} be the free left kE-module on the n + 1 symbols e_{r,s} where r + s = n and r, s ≥ 0. For notational convenience we set e_{r,s} = 0 if r < 0 or s < 0. Then define

∂(e_{r,s}) = X^{1+(p−2)ν(r+1)}e_{r−1,s} + (−1)^{r}Y^{1+(p−2)ν(s+1)}e_{r,s−1},

where ν(n) is defined to be 0 if n is even and 1 if n is odd, and set ϵ(e_{0,0}) = 1. We have

H^{∗}(E; k) = Hom_{kE}(P, k) = k[x_{1}, x_{2}] ⊗_{k} Λ_{k}[e_{1}, e_{2}]

where x_{1} = e_{0,2}^{∗} (by which we mean x_{1}(e_{0,2}) = 1 and x_{1}(e_{r,s}) = 0 for (r, s) ≠ (0, 2)), x_{2} = e_{2,0}^{∗}, e_{1} = e_{0,1}^{∗}, and e_{2} = e_{1,0}^{∗}. The generators e_{1} and e_{2} in H^{1}(E, k) ≅ Hom(E, k) correspond to maps E ⟶ k with kernels ⟨x⟩ and ⟨y⟩ respectively, and x_{1} and x_{2} are their Bocksteins.

Now let M_{1} = kE/(X^{p−1}, Y^{p−1}). (If p = 3 and k = 𝔽_{3}, this is consistent with the definition of M_{1} given above.)

Proposition 5.1. As a right H^{∗}(E; k)-module, H^{∗}(E; M_{1}) is generated by elements α (degree 0), δ_{1}, δ_{2} (degree 1), and β (degree 2), subject to the relations

αe_{1} = αe_{2} = δ_{2}e_{1} = δ_{1}e_{2} = 0,
δ_{1}e_{1} = αx_{1}, δ_{2}e_{2} = αx_{2},
βe_{1} = −δ_{2}x_{1}, βe_{2} = δ_{1}x_{2}.

In particular, H^{∗}(E; M_{1}) is a free k[x_{1}, x_{2}]-module on α, δ_{1}, δ_{2}, β.

Proof. We compute an explicit basis for H^{n}(E; M_{1}) = Z^{n}(E, M_{1})/B^{n}(E, M_{1}). First note that M_{1} is a commutative ring, and Hom_{kE}(P_{n}, M_{1}) is a free left M_{1}-module on the generators f_{i}^{n} = e_{i,n−i}^{∗} (0 ≤ i ≤ n). Let ā denote the image in M_{1} of an element a in kE. Then X̄^{p−1} = Ȳ^{p−1} = 0, so for f ∈ Hom_{kE}(P_{n}, M_{1}) and r, s ≥ 0 we have

f∂(e_{r,s}) = ν(r)X̄f(e_{r−1,s}) + (−1)^{r}ν(s)Ȳ f(e_{r,s−1}). (5.1)

Suppose first that n = 2m is even. Let f = ∑_{i=0}^{n} α_{i}f_{i}^{n} (α_{i} ∈ M_{1}) and for notational convenience set α_{−1} = α_{n+1} = 0. Then equation (5.1) implies that f ∈ Z^{n}(E, M_{1}) if and only if

ν(j)α_{j−1}X̄ + (−1)^{j}ν(n + j + 1)α_{j}Ȳ = 0 whenever 0 ≤ j ≤ n + 1.

But this occurs if and only if α_{j}X̄ = 0 = α_{j}Ȳ for all even j, or equivalently α_{j} ∈ (X̄^{p−2}Ȳ^{p−2}) for all even j. Hence

Z^{n}(E, M_{1}) = {∑_{j=0}^{n} α_{j}f_{j}^{n} | α_{j} ∈ M_{1} if j is odd, α_{j} ∈ (X̄^{p−2}Ȳ^{p−2}) if j is even}.

A similar calculation yields

B^{n}(E, M_{1}) = {∑_{j=0}^{n} α_{j}f_{j}^{n} | α_{j} ∈ (X̄, Ȳ) if j is odd, α_{j} = 0 if j is even}.

Hence the images of the n + 1 elements α_{k}^{m} = X̄^{p−2}Ȳ^{p−2}e_{2k,2m−2k}^{∗} (0 ≤ k ≤ m) and β_{k}^{m} = e_{2k+1,2m−2k−1}^{∗} (0 ≤ k < m) of Z^{n}(E, M_{1}) form a basis in H^{n}(E; M_{1}).

Now suppose that n = 2m + 1 is odd. Let S = {(ζ, η) ∈ M_{1} ⊕ M_{1} | ζX̄ = ηȲ } and T = {(ζȲ , ζX̄) | ζ ∈ M_{1}}. Working as in the even case, we get

Z^{n}(E, M_{1}) = {∑_{i=0}^{m}(α_{i}f_{2i}^{n} + β_{i}f_{2i+1}^{n}) | (α_{i}, β_{i}) ∈ S},

B^{n}(E, M_{1}) = {∑_{i=0}^{m}(α_{i}f_{2i}^{n} + β_{i}f_{2i+1}^{n}) | (α_{i}, β_{i}) ∈ T }.

Now S/T is 2-dimensional and is spanned by the images of (X̄^{p−2}, 0) and (0, Ȳ^{p−2}). Hence the images of the n + 1 elements γ_{k}^{m} = X̄^{p−2}e_{2k,2m+1−2k}^{∗}, δ_{k}^{m} = Ȳ^{p−2}e_{2k+1,2m−2k}^{∗} (0 ≤ k ≤ m) of Z^{n}(E, M_{1}) form a basis in H^{n}(E; M_{1}).

We now turn to the module structure. Recall that composition with ϵ is a chain map Hom_{kE}(P, P) → Hom_{kE}(P, k) and this map induces an isomorphism in cohomology H^{∗} Hom_{kE}(P, P) →^{≅} H^{∗}(E; k). The action

H^{∗}(E; M_{1}) ⊗_{k} H^{∗}(E; k) → H^{∗}(E; M_{1})

is induced by the map on the cochain level

Hom_{kE}(P, M_{1}) ⊗_{k} Hom_{kE}(P, P) → Hom_{kE}(P, M_{1})

given by composition. So we must first lift x_{i}, e_{i} to maps x̃_{i} ∈ Z^{2} Hom_{kE}(P, P), ẽ_{i} ∈ Z^{1} Hom_{kE}(P, P). This is accomplished by setting x̃_{1}(e_{r,s}) = e_{r,s−2}, x̃_{2}(e_{r,s}) = e_{r−2,s}, ẽ_{1}(e_{r,s}) = (−1)^{r+s+1}Y^{(p−2)ν(s+1)}e_{r,s−1}, ẽ_{2}(e_{r,s}) = (−1)^{r+1}X^{(p−2)ν(r+1)}e_{r−1,s}.

Now let α = [α_{0}^{0}], δ_{1} = −[γ_{0}^{0}], δ_{2} = −[δ_{0}^{0}], and β = [β_{0}^{1}]. We have αx_{1}^{i}x_{2}^{j} = [α_{j}^{i+j}], δ_{1}x_{1}^{i}x_{2}^{j} = −[γ_{j}^{i+j}], δ_{2}x_{1}^{i}x_{2}^{j} = −[δ_{j}^{i+j}], and βx_{1}^{i}x_{2}^{j} = [β_{j}^{i+j+1}] (i, j ≥ 0). These are easily verified; the first, for example, just follows from the fact that

α_{0}^{0}x̃_{1}^{i}x̃_{2}^{j}(e_{r,s}) = α_{0}^{0}(e_{r−2j,s−2i}) = X̄^{p−2}Ȳ^{p−2}((r, s) = (2j, 2i)) = α_{j}^{i+j}(e_{r,s}).

(Here we are using the computer science notation where, for a proposition P, (P) = 1 if P and (P) = 0 otherwise.) This proves that α, δ_{1}, δ_{2}, and β generate H^{∗}(G; M_{1}) as a k[x_{1}, x_{2}]-module and that H^{∗}(G, M_{1}) is a free k[x_{1}, x_{2}]-module on those four generators.

We now turn to the relations. It is routine to check that the generators satisfy these relations; for example, the last of these follows from the fact that

β_{0}^{1}ẽ_{1}(e_{r,s}) = (−1)^{r+s+1}Ȳ^{(p−2)ν(s+1)}β_{0}^{1}(e_{r,s−1}) = Ȳ^{p−2}((r, s) = (1, 2)) = δ_{0}^{0}x̃_{1}(e_{r,s}).

Now let A^{∗} denote the graded H^{∗}(E; k)-ring defined abstractly by these 4 generators and 8 relations. Since H^{∗}(E; M_{1}) satisfies the relations, there is a surjective H^{∗}(E; k)- homomorphism A^{∗} ↠ H^{∗}(E; M_{1}). We wish to show that this homomorphism is an isomorphism, and to do this it suffices to show that dim_{k}(A^{n}) ≤ dim_{k} H^{n}(E; M_{1}) for all n. This will follow if we can show that A^{∗} is generated as a k[x_{1}, x_{2}]-module by the four generators, because we know that H^{∗}(G; M_{1}) is a free k[x_{1}, x_{2}]-module on those generators. Hence for each θ ∈ {α, δ_{1}, δ_{2}, β} and i ∈ {1, 2} we must show that θe_{i} is in the k[x_{1}, x_{2}]-submodule of A^{∗} generated by α, δ_{1}, δ_{2}, β. But this is exactly what the 8 relations tell us.

The following lemma is easy to check, but will be useful.

Lemma 5.2. We can replace β by any other element of H^{2}(E; M_{1}) linearly independent of αx_{1} and αx_{2} and get the same relations.

6. The Final Calculation for 𝕊l

We assume for now that M = M_{1}: this will be justified later. Consider the spectral sequence

H^{p}(E; H_{c}^{q}(𝕊l′; H_{c}^{1}(K))) ⇒ H_{c}^{p+q}(𝕊l; H_{c}^{1}(K))

Again it has only two rows.

H^{∗−2}(E) ⟶^{d_{2}} H^{∗}(E; M).

H^{∗}(E) is a torsion-free 𝔽_{3}[x_{1}, x_{2}] module, hence so is Ker d_{2}. But this spectral sequence shows that H^{∗}(𝕊l; H^{1}(K)) is finitely generated over H^{∗}(E), and hence over H_{c}^{∗}(𝕊l), which has Krull dimension 1. It maps on to Ker d_{2}, forcing a common bound on dim Ker d_{2} in each degree. Thus Ker d_{2} = 0 and we have:

0 → H^{∗−2}(E) → H^{∗}(E; M) → H_{c}^{∗}(𝕊l; H_{c}^{1}(K)) → 0. (6.1)

Now d_{2}(1) is not a linear combination of αx_{1} and αx_{2} otherwise it would be annihilated by e_{1} and e_{2}. By Lemma 5.2 we may assume that d_{2}(1) = β.

We have proved:

Proposition 6.2. H_{c}^{∗}(𝕊l; H_{c}^{1}(K)) is an H^{∗}(E)-module on generators α (degree 0), δ_{1}, δ_{2} (degree 1) with relations

αe_{1} = αe_{2} = δ_{1}e_{2} = δ_{2}e_{1} = δ_{1}x_{2} = δ_{2}x_{1} = 0,
δ_{1}e_{1} = αx_{1}, δ_{2}e_{2} = αx_{2}.

Now consider the spectral sequence

H_{c}^{∗}(𝕊l, H^{∗}(K)) ⇒ H_{c}^{∗}({ \atop ℤ/3 ∗ ℤ/3}).

One has H_{c}^{∗}({ \atop ℤ/3 ∗ ℤ/3}) ≅ H^{∗}(ℤ/3 ∗ ℤ/3). (Consider the short exact sequence F_{4} → ℤ/3∗ℤ/3 → ℤ/3⊕ℤ/3 and its pro-3 completion, and use the Comparison Theorem.) The map i^{∗} : H_{c}^{∗}(𝕊l) → H^{∗}(ℤ/3 ∗ ℤ/3) is an isomorpism in degree 1, by construction. As the right hand side is generated by elements of degree 1 and their Bocksteins, i^{∗} is onto in all degrees and the spectral sequence becomes the short exact sequence

0 ⟶ H_{c}^{∗−2}(𝕊l, H_{c}^{1}(K)) ⟶^{d_{2}} H_{c}^{∗}(𝕊l) ⟶ H^{∗}(ℤ/3 ∗ ℤ/3) ⟶ 0, (6.3)

of right H^{∗}(E)-modules. Set a = d_{2}α, c_{i} = d_{2}δ_{i}. Identify e_{i} and x_{i} with their images under j^{∗}. All that remains is to check that Im j^{∗} = R ≅ H^{∗}(ℤ/3 ∗ ℤ/3), i.e. that e_{1}e_{2} = e_{1}x_{2} = e_{2}x_{1} = x_{1}x_{2} = 0.

The spectral sequence

H^{∗}(E; H_{c}^{∗}(𝕊l′)) ⇒ H_{c}^{∗}(𝕊l)

has four rows. We know that d_{2}(E_{2}^{∗,1}) ⊂ E_{2}^{∗,0} ≅ H^{∗}(E) is contained in Ker(ji)^{∗}, which is generated as an H^{∗}(E)-module by e_{1}e_{2}, e_{1}x_{2}, e_{2}x_{1}, and x_{1}x_{2}. Since H_{c}^{1}(𝕊l′) is dual to N as an E-module it is isomorphic to I^{3}, and we can calculate dim E_{2}^{0,1} = 1 and dim E_{2}^{1,1} = 3. (Use dimension shifting: 𝔽_{3}E/I^{3} has invariants of dimension 3.) But E_{2}^{1,0} yields all of H_{c}^{1}(𝕊l) and thus dim d_{2}(E_{2}^{0,1}) = 1. This accounts for e_{1}e_{2}. Also dim E^{2,0} = 2 and dim H_{c}^{2}(𝕊l) = 3, so dim Ker(d_{2} : E_{2}^{1,1} ⟶ E_{2}^{3,0}) ≤ 1. The image of this map must have dimension ≥ 2, which accounts for e_{1}x_{2} and e_{2}x_{1}. Finally x_{1}x_{2} is the Bockstein of e_{1}x_{2}.

All that remains is to justify our assertion that M ≅ M_{1}. We do this by carrying out the above calculation for each of the other possibilities and obtaining a contradiction.

If M ≅ 𝔽_{3} ⊕ N, then the short exact sequence (6.1) shows that dim H_{c}^{n}(𝕊l; M) = dim H^{n}(E; N) + 2. This is impossible because N has complexity 2, yet H_{c}^{∗}(𝕊l) has Krull dimension 1.

If M ≅ M_{2}, then the structure of H^{∗}(E; M_{2}) as an H^{∗}(E)-module is like that of H^{∗}(E; M_{1}), but twisted by ϕ_{2}. Let us denote the new module structures by ∗. Then

u ∗ v = u(ϕ_{2}^{∗}v), u ∈ H^{∗}(E; M), v ∈ H^{∗}(E).

Thus a∗x_{1}x_{2} = a(x_{1} +x_{2})(x_{1} −x_{2}) = ax_{1}^{2} −ax_{2}^{2} ≠ 0. But the argument that x_{1}x_{2} = 0 is still valid since it only depends on the additive structure of H_{c}^{∗}(𝕊l; M). The case M ≅ M_{3} is similar.

7. The Cohomology of 𝕊_{2}

Remark 7.1. [9] If p > 3 and n = 2 then 𝕊l is torsion-free so H^{∗}(𝕊l) has finite cohomological dimension by [8]. It contains an open subgroup H_{2}, which is a Poincaré duality group of dimension 3. So by ( [13], V4.7) 𝕊l is also a Poincaré duality group of dimension 3. Since dim H_{c}^{1}(𝕊l) = 2, there is only one possible multiplicative structure, namely that with generators e_{1}, e_{2} (degree 1), f_{1}, f_{2} (degree 2), and relations e_{1}e_{2} = 0, e_{1}f_{1} = e_{2}f_{2}, f_{1}^{2} = f_{2}^{2} = f_{1}f_{2} = 0.

Remark 7.2. The map

𝕊_{2}^{0} ↠^{nrd} (1 + 3ℤ_{3})^{×} ⟶_{≅}^{log} 3ℤ_{3}^{+}

is split by x ↦ exp(\frac{1}{2}x), the image being central in 𝕊_{2}^{0}. Thus 𝕊_{2}^{0} ≅ 𝕊l × ℤ_{3}^{+} and H_{c}^{∗}(𝕊_{2}^{0}) ≅ H_{c}^{∗}(𝕊l) ⊗ Λ(z), degz = 1.

The group 𝕊_{2} is the semi-direct product of 𝕊_{2}^{0} and ℤ/8 (the splitting is obtained by lifting the elements of 𝔽_{9}^{×} to roots of unity in 𝕎). Therefore H_{c}^{∗}(𝕊_{2}; 𝔽_{3}) is isomorphic to the subring of H_{c}^{∗}(𝕊l; 𝔽_{3}) invariant under conjugation by the eighth root of unity ζ ∈ 𝕎.

We must track the action of ζ through our calculation. Notice that conjugation by ζ has the effect X ↦ Y ↦ X^{2} ↦ Y^{2} ↦ X. Thus e_{1}^{ζ} = −e_{2}, e_{2}^{ζ} = e_{1}, and the same for their Bocksteins, x_{1}^{ζ} = −x_{2}, x_{2}^{ζ} = x_{1}.

Regarding all groups as ζ-modules now, (3.3) shows that H^{0}(E; M) ≅ E_{∞}^{0,1} ≅ 𝔽_{3}. From short exact sequence (3.2) we see that ζ acts on H^{0}(E; M) as multiplication by det H_{c}^{1}(F̂_{4})(ζ)/ det N(ζ). But ζ permutes the explicit generators of F̂_{4} transitively, hence det H_{c}^{1}(F̂_{4}) = −1. To calculate detN(ζ), note that under the map ρ_{2} of Section 2, conjugation by ζ acts as the identity, whilst under ρ_{3} it corresponds to multiplication by −ζ on 𝔽_{9}, which has determinant 1 over 𝔽_{3}. This proves that det N(ζ) = −1, and consequently from (6.1), α^{ζ} = −α. Sequence (6.3) then shows that a^{ζ} = −a.

The relations involving the elements c_{i} now force c_{1}^{ζ} = −c_{2}, c_{2}^{ζ} = −c_{1}. Clearly z^{ζ} = z, and this completes the calculation of the action of ζ.

It is now fairly straightforward to calculate the invariants of this action, especially if one notes that, if X = x_{1}^{2}+x_{2}^{2}, then X is invariant and H_{c}^{∗}(𝕊l; 𝔽_{3}) is free over 𝔽_{3}[X]. We obtain Theorem 1.1 by setting Z = z, C = c_{1} − c_{2}, E = e_{1}x_{1} + e_{2}x_{2}, X = x_{1}^{2} + x_{2}^{2}.

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Department of Mathematics, Northwestern University, Evanston IL 60208-2730 E-mail address: vgorb@ms.uky.edu, siegel@math.umass.edu, symonds@ms.uky.edu