Contemporary Mathematics

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-Farrell 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 [15], 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 R is isomorphic to the Witt vectors W_{𝔽_{p^{n}}}): the residue class field is k ≅ 𝔽_{p^{n}}. Let χ be the Frobenius automorphism of R 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^{i}, or equivalently by conjugation by S.

It is known that S_{n} is virtually a pro-p group of virtual cohomological dimension n^{2} and type FP_{∞} over ℤ̂_{p}.

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 profinite RG-modules by C_{R}(S_{n}) (and similarly with R replaced by ℤ̂_{p}). 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} ∈ Cℤ̂_{p}(S_{n} ⋊ Gal).

©0000 (copyright holder)

2000 Mathematics Subject Classification. Primary 55Q52; Secondary 20J06.

We would like to calculate the ring H^{∗}(S_{n}, E_{n,∗})^{Gal}, by which we mean the sum ⨁_{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). Notice that, since Gal has order coprime to p, H^{∗}(S_{n}, E_{n,∗})^{Gal} is actually isomorphic to the invariants of H^{∗}(S_{n}, E_{n,∗}) under the action of Gal. What we will actually do is 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}).

Here G is a finite group that will be defined later, and the generators will be defined in the course of the calculation.

Remark 1.2. 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 [15] 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 [13] and [12].

Theorem 1.3. Let G be a profinite group of finite virtual cohomological dimension over ℤ̂_{p}. 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 Cℤ̂_{p}(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 and we want to define a certain subgroup G of S_{n} of order pn^{2}. This is done in [7], but we need to be precise about some of the details so we give the construction here. Mℚ̂_{p} = ℚ̂_{p} ⊗ℤ̂_{p} M is a division algebra with

center ℚ̂_{p}. It has a valuation ν such that ν(S) = \frac{1}{n}, and S_{n} is the set of elements of valuation 0.

The general theory of division algebras tells us that the field ℚ̂_{p}(ω; ω^{p} = 1) can be embedded in Mℚ̂_{p} and the image is self-centralizing ([11] 28.10, 31.10). Furthermore, by the Skolem-Noether Theorem ([11] 7.21), any two embeddings differ only by conjugation in Mℚ̂_{p}.

THE COHOMOLOGY OF S_{p−1} WITH COEFFICIENTS IN E_{p−1} 3

In fact we can find a pth root of unity ω in ℤ̂_{p}[S] ⊂ M as follows: 1 + S is a pth root of unity modulo p and we can lift it by Hensel’s Lemma ([14] II Prop. 7). For convenience we will regard ℚ̂_{p}(ω) = ℚ̂_{p}(S). It is easy to see that the only elements of Mℚ̂_{p} invariant under conjugation by S are in ℚ̂_{p}(S), so ℚ̂_{p}(ω) is self centralizing.

If ϕ : ℚ̂_{p}(ω) → Mℚ̂_{p} is another embedding then, by the discussion above, there

is a u ∈ Mℚ̂_{p} such that uϕ(x)u^{−1} = x for x ∈ ℚ̂_{p}(ω). But S^{i}U will serve as well as

u, since S is central in ℚ̂_{p}(ω), and we can choose i to give this element valuation 0. Thus any two embeddings of ℤ̂_{p}[ω] in M are conjugate by an element of S_{n} and so any two elements of S_{n} of order p are conjugate in S_{n}.

Now let e ∈ ℤ be a generator of (ℤ/p)^{∗}, so ℤ̂_{p}[ω^{e}] = ℤ̂_{p}[ω] ⊂ M. The Galois automorphism ω ↦ ω^{e} must be realized by conjugation by some v ∈ S_{n}, as has just been shown, i.e. v^{−1}ωv = ω^{e}. Now v^{n} centralizes ℤ̂_{p}[ω], so v^{n} ∈ ℤ̂_{p}[ω]. Clearly v^{n} is invariant under conjugation by v, so v^{n} ∈ ℤ̂_{p}.

Let e′ ∈ ℤ̂_{p} be an nth root of unity such that e′ ≡ v^{n} (mod p). Then e′v^{−n} ≡ 1 (mod p) so has an nth root in ℤ̂_{p} (constructed from the binomial power series). Let v′ = v(e′v^{−n})^{\frac{1}{n}}, so v^{′−1}ωv′ = ω^{e} and also v^{′n} = e′, so v^{′n^{2}} = 1. In fact v′ has order n^{2}, since if it had order \frac{n^{2}}{r} then v^{′ \frac{n}{r}} would be in ℤ̂_{p}, contradicting v^{′−1}ωv′ = ω^{e}.

But R itself contains the n^{2}th roots of unity so, by the Skolem-Noether Theorem again, there is an x ∈ Mℚ̂_{p} such that xv′x^{−1} ∈ R. Let a = xωx^{−1} and b = xv′x^{−1} and define the group G = ⟨a, b⟩ ≅ ℤ/p ⋊ ℤ/n^{2} ≤ S_{n}.

When we want to think of b as an element of R rather than G we will denote it by η.

The reason for the last conjugation in the definition of G is to ensure that our formulas are consistent with those of [8, 9].

Some more work with division algebras will show that any subgroup of S_{n} that is isomorphic to G is, in fact, conjugate to G in S_{n}, but we refer the reader to [7].

The significance of G is that it is a maximal finite subgroup of S_{n} and, up to conjugacy, the only one of order divisible by p. Again we refer the reader to [7].

Lemma 2.1. ([8] ) e′ ≡ e (mod p).

Proof. Working in S_{n} modulo S^{2} we have a = 1+uS for some u ≢ 0 (mod S). Now

1 + euS = a^{e} = b^{−1}ab = 1 + uη^{−1}Sη = 1 + uη^{n}S = 1 + ue′S. □

As mentioned above, ℚ̂_{p}(a) is self-centralizing in Mℚ̂_{p}, so C = C_{S_{n}}(a) is equal

to the units in ℤ̂_{p}[a]. In additive notation these are of the form ℤ̂_{p}^{n} × ℤ/p × ℤ/n. This is because the p-torsion is known from the construction and the p′-torsion corresponds to the units in the residue class field 𝔽_{p}. To see that the free part has rank n, let U_{1} denote the units congruent to 1 modulo p: then U_{1} is isomorphic to pℤ̂_{p}[a] under the inverse exponential and logarithm power series ([3] 54.2). U_{1} is of finite index in C, so C also has free rank n.

Notice that these maps respect the action of b, so U_{1} ≅ ℤ̂_{p}[a]. Also, since ⟨b⟩ is abelian of order coprime to p, any ℤ̂_{p}⟨b⟩-lattice is a sum of eigenspaces for b, and hence any sublattice of finite index is isomorphic to the original. In particular U_{1} ≅ C/(torsion). Thus C/(torsion) is isomorphic to ℤ̂_{p}[a] as a ℤ̂_{p}⟨b⟩-lattice, so is a sum of rank 1 lattices; one for each nth root of unity on which b acts as multiplication by that root of unity. Finally, since ℤ̂_{p}[a] is free as a ℤ̂_{p}⟨b⟩-module, the map C → C/(torsion) is split over ⟨b⟩. We have shown:

Lemma 2.2. As a ℤ̂_{p}⟨b⟩-module, C ≅ ℤ̂_{p}⟨b⟩ ⊕ ⟨a⟩ ⊕ ℤ/p.
Now N = N_{S_{n}}(⟨a⟩) = ⟨C, b⟩ ≅ (ℤ̂_{p}[a] × ⟨a⟩) ⋊ ⟨b⟩.

Consider N_{S_{n}⋊Gal}(G)/N_{S_{n}}(G). Every conjugate of G in S_{n} ⋊Gal is in S_{n} so, as mentioned above, is conjugate to G in S_{n}. It follows that N_{S_{n}⋊Gal}(G)/N_{S_{n}}(G) ≅ Gal. Choose c ∈ N_{S_{n}⋊Gal}(G) to have image χ ∈ Gal: we may also arrange for c to act trivially on ⟨a⟩ and for ⟨c⟩ to have no pro-p part.

Finally, for similar reasons, N′ = N_{S_{n}⋊Gal}(⟨a⟩) = ⟨N, c⟩.

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

Proposition 2.3. 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 1.3 we find that Ĥ^{∗}(S_{n}, ℤ̂_{p}) ≅ Ĥ^{∗}(N, ℤ̂_{p}). 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), we have 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. □

Remark 2.4. Because c centralizes C we have Ĥ^{∗}(N′, ℤ̂_{p}) ≅ Ĥ^{∗}(N, ℤ̂_{p}) and hence Ĥ^{∗}(S_{n} ⋊ Gal, ℤ̂_{p}) ≅ Ĥ^{∗}(S_{n}, ℤ̂_{p}). Since the action of Gal on S_{n} is via conjugation by S, which is not inner in S_{n}, this is not immediately obvious.

3. Coefficients in E_{n}

Next we calculate Ĥ^{∗}(⟨a⟩, E_{n,∗}) following the method of Nave [8, 9], 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. ([8, 9]) 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,
(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 [10], [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). We will also write Ĥ^{∗}(G, E_{n,∗})^{Gal} for Ĥ^{∗}(⟨G, c⟩, E_{n,∗}) ≅ Ĥ^{∗}(G, E_{n,∗})^{⟨c⟩}. This is not so strange since Ĥ^{∗}(S_{n}, E_{n,∗})^{Gal} ≅ Ĥ^{∗}(⟨S_{n}, c⟩, E_{n,∗}) ≅ Ĥ^{∗}(S_{n}, E_{n,∗})^{⟨c⟩}.

Corollary 3.3. ([8]) 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| = (0, 2pn^{2}), |β| = (2, 2pn) 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 2.3.

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 2.3.

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 Δ = δ^{−n^{2}}, β = ζδ^{−n} 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. We claim that in the former case the invariants are isomorphic to 𝔽_{p} .

To see this let 0 ≠ x ∈ Ĥ^{r}(G, E_{n,s}) be a generator over k. Then c(x) = λx for

some λ ∈ k and x = c^{n}(x) = λ^{\frac{p^{n}−1}{p−1}} x, so λ^{\frac{p^{n}−1}{p−1}} = 1. It follows that λ = ω^{p−1} for some ω ∈ k and the fixed points under c are 𝔽_{p}ω^{−1}x.

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 1.3 to see that Ĥ^{∗}(S_{n}, E_{n,∗})^{Gal} ≅ Ĥ^{∗}(N′, E_{n,∗}).

Recall that, for any short exact sequence I → J → K of profinite groups of finite virtual cohomological dimension 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 we have 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, 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 that Ĥ^{∗}(⟨a⟩, E_{n,∗})^{⟨b⟩} ⊗ Λ_{𝔽_{p}}(x_{0}, …, x_{n−1}), where the x_{i} are as in 2.3.

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 the invariants k[δ^{±1}, ζ^{±1}]^{⟨b⟩} = (k[δ^{±1}] ⊗ k[ζ^{±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 that 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. H acts 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} and similarly for B.

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},

and if C_{B} ⊆ C_{A} then

(A ⊗ B)_{c} = a_{c}(A ⊗ B)^{H}.

Suppose that B is also a graded commutative R-algebra and 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′}. □

References

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

Email address: Peter.Symonds@umist.ac.uk