THE TATE-FARRELL COHOMOLOGY OF THE MORAVA STABILIZER GROUP S_{p−1} WITH COEFFICIENTS IN E_{p−1}

PETER SYMONDS

Abstract. We calculate the Tate-Farrell cohomology of the Morava stabilizer group S_{p−1} with coefficients in the moduli space E_{p−1} for odd primes p.

1. Introduction

We present a calculation motivated by homotopy theory, although our methods are algebraic and involve the Tate cohomology of a profinite group with compact coefficients. As a reference to the background in homotopy theory we suggest [4, 5]. For the Tate-Farrell cohomology of profinite groups with coefficients in compact module we refer to [12], although most of the results are analogues of one for discrete groups, for which see [2].

Let p be an odd prime and n ∈ ℕ and let R be the ring of integers of the unramified extension of ℚ̂_{p} of degree n, so k ≅ 𝔽_{p^{n}}. Let χ be the Frobenius automorphism and Gal = ⟨χ⟩ the Galois group. Let S_{n} denote the (full) nth Morava stabilizer group: this is the group of units in the R-algebra M generated by S subject to the relations S^{n} = p and rS = Sχ(r) for r ∈ R. The Galois group Gal acts on S_{n} simply by χ(rS^{i}) = χ(r)S.

Thus S_{n} is virtually a pro-p group of virtual cohomological dimension n^{2} and type FP_{∞}. If Γ_{n} denotes the commutative one-dimensional p-typical formal group law with p-series

x^{p^{n}}, then S_{n} is isomorphic to the group of automorphisms of Γ_{n} over 𝔽_{p}. It therefore acts on the ring of functions on the Lubin-Tate moduli space of ⋆-isomorphism classes of lifts of Γ_{n}, which is E_{n,0} = R[[u_{1}, …, u_{n−1}]], a profinite RS_{n}-module. We denote the category of such modules by C_{R}(S_{n}). There is also an action of S_{n} on a graded version E_{n,∗} = E_{n,0}[u^{±1}]. This is graded by the power of u, normalized so that u has degree −2 (called the internal degree).

This combines with the action of Gal on E_{n,∗} via its action on the coefficients to give an action of the semi-direct product S_{n} ⋊ Gal on E_{n,∗}, and so each E_{n,r} ∈ C_{R}(S_{n} ⋊ Gal).

We would like to calculate the ring H^{∗}(S_{n}, E_{n,∗})^{Gal}, by which we mean ⨁_{r,s} H^{r}(S_{n} ⋊ Gal, E_{n,s}), since this is the initial term of a spectral sequence which converges to π_{∗}L_{K(n)}, the homotopy groups of the localization of the sphere spectrum at the nth Morava K-theory (all at the prime p). What we will actually do is to calculate the Tate-Farrell cohomology in the case n = p − 1: this is equal to the ordinary cohomology in degrees greater than n^{2}.

Theorem 1.1. For odd p and n = p − 1

Ĥ^{∗}(S_{n}, E_{n,∗})^{Gal} = Ĥ^{∗}(G, E_{n,∗})^{Gal} ⊗ Λ(x_{0}, …, x_{n−1})
= Ĥ^{∗}(S_{n}, ℤ̂_{p}) ⊗ Λ(α) ⊗ 𝔽_{p}[Δ^{±1}]
= 𝔽_{p}[Δ^{±1}, β^{±1}] ⊗ Λ(α, x_{0}, …, x_{n−1}).

The generators will be defined in the course of the calculation.

Remark. It would be natural to regard E_{n,∗} as ˆ⨁_{s}E_{n,s}, the sum in C_{R}(S_{n}), but H^{r}(S_{n}, ˆ⨁_{s}E_{n,s}) ≅ Π_{s}H^{r}(S_{n}, E_{n,s}). Since only the homogeneous parts appear in the spectral sequence, the difference is immaterial, but we conform to the conventional usage.

We will need the following corollary of [12] 7.3 and the remark following it. It is what we would expect from the theory for discrete groups in [2]. Similar results for profinite groups but with discrete coefficients also appear in [11] and [10].

Theorem 1.2. Let G be a profinite group of finite virtual cohomological dimension over R. Suppose that G has no subgroup isomorphic to ℤ/p × ℤ/p and only a finite number of conjugacy classes of subgroups isomorphic to ℤ/p, which we denote by S(p). Let M be a module in C_{R}(G). Then the Tate-Farrell cohomology satisfies

Ĥ^{∗}(G, M) ≅ ⨁_{P ∈S(p)} Ĥ^{∗}(N_{G}(P), M).

2. Trivial Coefficients

From now on n = p − 1. The group S_{n} contains an element a of order p and ⟨a⟩ is a maximal finite p-subgroup, unique up to conjugacy. The centralizer C = C_{S_{n}}(a) corresponds to the units in the ring of cyclotomic integers ℤ̂_{p}[a] ⊂ M so, written additively, has the form ℤ̂_{p}^{n} × ℤ/p × ℤ/n. There is an element b of order n^{2} which normalizes ⟨a⟩, with b^{n} generating the ℤ/n in the centralizer: let e be an integer such that b^{−1}ab = b^{e}. Then G = ⟨a, b⟩ is the maximal finite subgroup of order divisible by p.

The subgroup N′ = N_{S_{n}⋊Gal}(⟨a⟩) fits into a short exact sequence ℤ̂_{p}^{n} → N′ → T, where |T | = n^{3} and T fits into a short exact sequence ⟨b⟩ → T → Gal. The second generator c of T can be chosen to centralize a.

The action of T on ℤ̂_{p}^{n} is via a cyclic quotient of order n generated by the image of b, and as a module for this it is free of rank 1 or, equivalently, a sum of rank 1 R-lattices, one for each possible eigenvalue.

First we calculate the Tate-Farrell cohomology with trivial coefficients. We use Λ to denote an exterior algebra over 𝔽_{p}.

Proposition 2.1. For p odd and n = p − 1:

Ĥ^{∗}(S_{n}, ℤ̂_{p}) = Ĥ^{∗}(G, ℤ̂_{p}) ⊗ Λ(x_{0}, …, x_{n−1}) = 𝔽_{p}[β^{±1}] ⊗ Λ(x_{0}, …, x_{n−1}),
Ĥ^{∗}(S_{n}, 𝔽_{p}) = Ĥ^{∗}(G, 𝔽_{p}) ⊗ Λ(x_{0}, …, x_{n−1}) = 𝔽_{p}[β^{±1}] ⊗ Λ(α, x_{0}, …, x_{n−1}),

where |β| = 2n, |x_{i}| = 1 − 2i and |α| = −1.

Proof. By Theorem 1.2 we find that Ĥ^{∗}(S_{n}, ℤ̂_{p}) ≅ Ĥ^{∗}(N, ℤ̂_{p}), where N = N_{S_{n}}(⟨a⟩) ≅ (ℤ̂_{p}^{n} × ⟨a⟩) ⋊ ⟨b⟩. Notice that Ĥ^{∗}(N, ℤ̂_{p}) ≅ Ĥ^{∗}(ℤ̂_{p}^{n} × ⟨a⟩, ℤ̂_{p})^{⟨b⟩}, since b has order coprime to p.

By the Künneth Theorem in Tate-Farrell cohomology ([2] X 3 ex. 4), Ĥ^{∗}(C, ℤ̂_{p}) ≅ Ĥ^{∗}(⟨a⟩, ℤ̂_{p}) ⊗ H^{∗}(ℤ̂_{p}^{n}, 𝔽_{p}).

It is well known that Ĥ^{∗}(⟨a⟩, ℤ̂_{p}) = 𝔽_{p}[ζ^{±1}], where |ζ| = 2. To find the action of b on ζ use dimension shifting to see that H^{2}(⟨a⟩, R) ≅ H^{1}(⟨a⟩, k) ≅ Hom(⟨a⟩, k). Then b acts on the latter by sending f to (x ↦ bf(b^{−1}xb)), so b(ζ) = eζ.

Now a basis y_{0}, …, y_{n−1} of H^{1}(ℤ̂_{p}^{n}, 𝔽_{p}) can be chosen so that b(y_{i}) = e^{i}y_{i}. We finish by calculating the invariants under b using the last part of Lemma 4.1 and setting x_{i} = ζ^{−i} ⊗ y_{i}.

The calculation for 𝔽_{p} coefficients is almost identical. □

3. Coefficients in E_{n}

Next we calculate Ĥ^{∗}(⟨a⟩, E_{n,∗}) following the method of Nave [7, 8], which in turn is based on unpublished work of Hopkins and Miller. This is also treated in detail for the prime 3 in [6].

First we need a change of basis.

Lemma 3.1. ([7, 8]) There are elements z, z_{1}, …, z_{n−1} ∈ E_{n,0} such that, where 𝔪 denotes the ideal (p, u_{1}, …, u_{n−1}) in E_{n,0}:

(1) z ≡ cu mod (p, 𝔪^{2}) for some c a unit in R,

(2) z_{i} ≡ c_{i}uu_{i} mod (p, u_{1}, …, u_{i−1}, 𝔪^{2}) for some c_{i} a unit in R.

(3) (1 + a + ⋯ + a^{p−1})z = 0,

(4) b(z) = ηz for η ∈ R a primitive n^{2} root of unity such that η^{p−1} = e (mod p),

(5) (a − 1)z = z_{n−1} and (a − 1)z_{i+1} = z_{i} for 1 ≤ i < n − 1.

It follows from (1) and (2) that E_{n,∗} = R[[z^{−1}z_{1}, …, z^{−1}z_{n−1}]][z^{±1}].

Let V be the R-submodule of E_{n,−2} spanned by {z, z_{1}, …, z_{n−1}}. It follows from (3),(4) and (5) that V is an RG-submodule. Let δ = Π_{i=0}^{p−1}a^{i}(z): then a(δ) = δ and b(δ) = η^{p}δ = eηδ.

Consider the symmetric algebra S[V] ⊂ E_{n,∗}. We claim that, as RG-modules,

(†)  R[V]_{r} = \begin{cases}0 & r odd, \\ δ^{−r′}R ⊕ (proj) & r = 2pr′ ≤ 0, \\ δ^{−r′}V ⊕ (proj) & r = 2(pr′ − 1) ≤ 0, \\ (proj) & otherwise. \end{cases}

and

R[V]_{r−2p} = δR[V]_{r} ⊕ (proj) for r < 0.

Here (proj) indicates a projective summand. We will write this in the condensed form R[V] = B ⊕ (proj), where B = ⨁_{i} δ^{i}(R ⊕ V).

Recall that if G is a finite group of order not divisible by p^{2} and M ∈ C_{R}(G) is projective in C_{R} then the isomorphism class of M is uniquely determined by its reduction modulo p, k ⊗_{R} M. This is true for a cyclic group of order p by the classification of Rℤ/p-lattices, (see [9], [3] 34.31), and this classification extends to C_{R}(ℤ/p). The general case follows by a transfer argument.

Thus we only have to check the claim over kG. But it is true over k⟨a⟩ from the calculation of the symmetric algebra by Almkvist and Fossum [1]. The general case follows because both δ^{−r′}R and δ^{−r′}V are defined over G, and the quotients by them must still be projective over G since this depends only on the restriction to the Sylow p-subgroup. Being projective they force the extension to split, and our claim is proved.

If we invert δ we obtain a dense subset R[V][δ^{−1}] ⊂ E_{n,∗}. As an RG-module this still has the same form B ⊕ (proj), by the second identity in †. In fact this form is preserved by completion:

Proposition 3.2. As a sum of compact modules for RG, E_{n,∗} = B ⊕ (proj).

Proof. Let (z^{−1}z_{1}, …, z^{−1}z_{n−1}) denote the ideal generated (topologically) by the given elements in E_{n,0}. It is easy to check that

E_{n,−2r} = R[V]_{−2r} ⊕ (z^{−1}z_{1}, …, z^{−1}z_{n−1})^{r+1}E_{n,−2r}, r > 0

and also

R[V]_{−2r} = B_{−2r} ⊕ P_{−2r}, r > 0

for some projective P_{−2r}. Thus, for r + pt > 0,

E_{n,−2r} = δ^{t}E_{n,−2(r+pt)}
= δ^{t}R[V]_{−2(r+pt)} ⊕ δ^{t}(z^{−1}z_{1}, …, z^{−1}z_{n−1})^{r+1}E_{n,−2(r+pt)}
= B_{−2r} ⊕ P_{−2(r+pt)} ⊕ (z^{−1}z_{1}, …, z^{−1}z_{n−1})^{r+1}E_{n,−2r}.

Now R[V][δ^{−1}]_{−2r} = B_{−2r} ⊕ \underrightarrow{lim} P_{−2(r+pt)} and E_{n,2r} = B_{−2r} ⊕ \underleftarrow{lim} P_{−2(r+pt)} as t → ∞. As a consequence, if \underrightarrow{lim} P_{−2(r+pt)} = ⊕_{i}Q_{i}, as a sum of indecomposable projective RG-modules then E_{n,2r} = B_{2r} ⊕ Π_{i}Q_{i}. □

We say that x ∈ Ĥ^{r}(−, E_{n,s}) has bidegree |x| = (r, s).

Corollary 3.3. ([7]) The Tate cohomology is given by

Ĥ^{∗}(⟨a⟩, E_{n,∗}) = k[δ^{±1}, ζ^{±1}] ⊗ Λ(ν),

where |δ| = (0, −2p), |ζ| = (2, 0), |α| = (1, −2) and b acts by

b(δ) = eηδ, b(ζ) = eζ, b(ν) = eην.

As a consequence

Ĥ^{∗}(G, E_{n,∗}) = k[Δ^{±1}, β^{±1}] ⊗ Λ(α),
Ĥ^{∗}(G, E_{n,∗})^{Gal} = 𝔽_{p}[Δ^{±1}, β^{±1}] ⊗ Λ(α).

where |Δ| = (−2, 2n), |β| = (2n, 0) and |α| = (1, 2n).

Proof. The first calculation is an easy consequence of 3.2 (we identify δ with its image in Ĥ^{0}(⟨a⟩, E_{n,−2p})).

Ĥ^{∗}(⟨a⟩, E_{n,∗}) = Ĥ^{∗}(⟨a⟩, B)

= ⨁_{r∈ℤ} Ĥ^{∗}(⟨a⟩, δ^{r}(R ⊕ V))

= ⨁_{r∈ℤ}δ^{r}k[ζ^{±1}] ⊗ Λ(ν) (a well-known calculation)

= k[δ^{±1}, ζ^{±1}] ⊗ Λ(ν).

The action of b on δ is from the definition of δ and e and that on ζ was found in the proof of 2.1.

For the action on ν ∈ H^{1}(⟨a⟩, V) it is easy to verify that the quotient map V → V/ rad(V) ≅ kz induces an isomorphism on H^{1}, so H^{1}(⟨a⟩, V) ≅ zH^{1}(⟨a⟩, k) as a ⟨b⟩- module, and this combines the action on z with that found in calculating the action on ζ in 2.1.

Thus Ĥ^{∗}(G, E_{n,∗}) ≅ Ĥ^{∗}(⟨a⟩, E_{n,∗})^{⟨b⟩} and the invariants can be calculated using lemma 4.1 below. They are generated by β = ζ^{n}, Δ = δ^{−n}ζ^{−1} and their inverses and α = δ^{−1}ν.

Finally, notice that c acts on the R-module Ĥ^{r}(G, E_{n,s}) according to the formula c(ℓx) = χ(ℓ)c(x), ℓ ∈ R, x ∈ Ĥ^{r}(G, E_{n,s}). This cohomology group is either k or 0, so the invariants under c are either 𝔽_{p} or 0. Since the generators Δ, β, α can be replaced by any non-zero element of the Ĥ^{r}(G, E_{n,s}) that they appear in, we may assume that they are all invariant under c and hence generate the invariants under c. □

Proof. of 1.1. As before we use Theorem 1.2 to see that Ĥ^{∗}(S_{n}, E_{n,∗})^{Gal} ≅ Ĥ^{∗}(N′, E_{n,∗}).

Recall that, for any short exact sequence of profinite groups of finite virtual cohomological dimension I → J → K with K torsion-free, there is a spectral sequence H^{∗}(K, Ĥ^{∗}(I, M)) ⇒ Ĥ^{∗}(J, M) ([2] X 3 ex. 5).

Apply this to C = ℤ̂_{p}^{n} × ⟨a⟩ to obtain H^{∗}(ℤ̂_{p}^{n}, Ĥ^{∗}(⟨a⟩, E_{n,∗})) ⇒ Ĥ^{∗}(C, E_{n,∗}). If we fix both r and s then Ĥ^{∗}(⟨a⟩, E_{n,r}) is either k or 0 so ℤ̂_{p}^{n}, being a pro-p group, must act trivially. Thus the E_{2}-term is isomorphic to Ĥ^{∗}(⟨a⟩, E_{n,∗}) ⊗ H^{∗}(ℤ̂_{p}^{n}, 𝔽_{p}) ≅ Ĥ^{∗}(⟨a⟩, E_{n,∗}) ⊗ Λ(y_{0}, …, y_{n−1}).

We claim that this spectral sequences collapses, so that Ĥ^{∗}(C, E_{n,∗}) ≅ Ĥ^{∗}(⟨a⟩, E_{n,∗}) ⊗ Λ_{𝔽_{p}}(y_{0}, …, y_{n−1}). To see this notice that, from the proof of 3.3, that the map E_{n,r} → E_{n,r}/𝔪E_{n,r} ≅ k induces an injection on Ĥ^{∗}(⟨a⟩, −). The corresponding spectral sequence with coefficients k collapses, by the Künneth Theorem, so ours must too.

Now compute the invariants under b using Lemma 4.1. The result is Ĥ^{∗}(⟨a⟩, E_{n,∗})^{⟨b⟩} ⊗ Λ_{𝔽_{p}}(x_{0}, …, x_{n−1}), where the x_{i} are as in 2.1.

Finally, c acts only on the first factor, so taking the invariants under c just replaces Ĥ^{∗}(G, E_{n,∗}) by Ĥ^{∗}(G, E_{n,∗})^{Gal}. □

4. Invariants

The following lemma is elementary, but systematic use of it simplifies the invariant calculations above. For example in the proof of 3.3, first calculate k[δ^{±1}, ζ^{±1}]^{⟨b⟩} = (k[ζ^{±1}]⊗k[Δ^{±1}])^{⟨b⟩} and then (k[δ^{±1}, ζ^{±1}] ⊗ Λ(ν))^{⟨b⟩}.

Lemma 4.1. Let H be a finite abelian group and let R be a commutative integral domain such that |H| is invertible in R and R contains a root of unity of order the exponent of H. Suppose that A and B are two RH-modules such that A is a graded-commutative R algebra and the action of H is compatible with this structure. Let H act on A ⊗_{R} B diagonally.

Let C be the set of isomorphism classes of homomorphisms from H to R^{×}. (This can be identified, perhaps not canonically, with the characters of H.) Then there are decompositions of RH-modules A = ⊕_{c∈C}A_{c} and B = ⊕_{c∈C}B_{c}, where A_{c} = {a ∈ A|ha = c(h)a, h ∈ H} and similarly for B. Let C_{A} = {c ∈ C|A_{c} ≠ 0}.

Suppose that for each c ∈ C_{A} there is a homogeneous element a_{c} ∈ A_{c} that is invertible in A. Then

(A ⊗ B)^{H} = ⨁_{d∈C_{A}} A^{H}a_{d^{−1}} ⊗ B_{d},
(A ⊗ B)_{c} = ⨁_{d∈C_{A}} A^{H}a_{d^{−1}} ⊗ B_{cd}.

Suppose that B is also a graded commutative R-algebra and that H acts compatibly with this structure. Then A ⊗ B is also a graded-commutative R-algebra in the usual way, and H acts as a group of automorphisms.

(1) If, for each c ∈ C_{A} ∩ C_{B}, there is a homogeneous element b_{c} ∈ B_{c} that is invertible

in B, then (A ⊗ B)^{H} is a free A^{H} ⊗ B^{H}-module with basis {a_{c^{−1}} ⊗ b_{c} : c ∈ C′}.

Furthermore if the monomials in c_{1}, …, c_{r} ∈ C_{A} ∩ C_{B} yield all the c ∈ C_{A} ∩ C_{B} then (A ⊗ B)^{H} is generated as a ring by A^{H}, B^{H} and the a_{c_{i}^{−1}} ⊗ b_{c_{i}}.

(2) If B is generated as an R-algebra by d_{1}, …, d_{s}, where d_{i} ∈ B_{c_{d_{i}}} for some c_{d_{i}} ∈ C_{A} ∩ C_{B}, then (A ⊗ B)^{H} is generated as a ring by A^{H} and the a_{c_{d_{i}}^{−1}} ⊗ d_{i}.

If the d_{i} freely generate B as a graded-commutative R-algebra then the a_{c_{d_{i}}^{−1}} ⊗ d_{i}

freely generate (A ⊗ B)^{H} over A^{H}. (So if B = Λ_{R}(d_{1}, …, d_{s}) then (A ⊗ B)^{H} = A^{H} ⊗_{R} Λ_{R}(a_{c_{d_{1}}^{−1}} ⊗ d_{1}, …, a_{c_{ds}^{−1}} ⊗ d_{s}).)

Proof. This is left as an exercise for the reader. Notice that (A ⊗ B)^{H} = ⊕_{c∈C′}A_{c^{−1}} ⊗ B_{c} and A_{c}a_{c′} = A_{cc′}. □

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Department of Mathematics, U.M.I.S.T., P.O. Box 88, Manchester M60 1QD, England

E-mail address: Peter.Symonds@umist.ac.uk