Local Units and Circular Index in Abelian p-Group Rings

by Klaus Hoechsmann

Abstract: Let A be a finite abelian group of exponent p^{m} > 1, an odd prime power, and consider the ZZ_{p}-module Δ_{p}^{+}(A) generated by the set {(z − 1) + (z^{−1} − 1)| z ∈ A} in the group ring ZZ_{p}A. We show that the multiplicative group S_{p}(A) = 1 + Δ_{p}^{+}(A) has a ZZ_{p}-basis of the form {w_{p}(z)| 1 ≠ z ∈ A}, where w_{p}(x) is a fixed ZZ_{p}-polynomial. The integral unit group S(A) = (ZZA)^{×} ∩ S_{p}(A) is known to have a canonical subgroup Λ(A) of a similar structure; it lies in the group Ω(A) of all units appearing circular under every character of A. Comparing the two situations, we conclude that the index [Ω(A) : Λ(A)] equals the order of a certain group of ideal classes of ZZA.

1. Introduction.

Let A be a finite abelian group of exponent p^{m} > 1, an odd prime power, C ⊂ A a maximal cyclic subgroup, and consider the ideal Δ_{p}(A) generated by the set {(z−1)| z ∈ A} in the p-adic group ring ZZ_{p}A.

The first part of this paper concerns the multiplicative group 1+Δ_{p}(A), more precisely its subgroup S_{p}(A) of symmetric elements, i.e. those which remain fixed under the ringinvolution defined by z^{⋆} = z^{−1} on the group elements. Of course, the symmetric part Δ_{p}(A)^{+} of Δ_{p}(A) has a basis of the form {(z − 1) + (z^{−1} − 1)}, and therefore S_{p}(A) = 1 + Δ_{p}(A)^{+} as a set is nothing mysterious. The problem is the multiplicative structure of S_{p}(A). Abstractly even that is clear: the group S(M_{p}) of symmetric 1-units in the maximal order M_{p} of Q_{p}A is a free ZZ_{p}-module of the same rank as Δ_{p}^{+}(A) (cf.[3],II.15.5); and so is S_{p}(A), because it has finite index in S(M_{p}).

Our task is to obtain an actual basis, and our first main result can be stated as follows: there exists a ZZ_{p}-polynomial f(x), with f(1) = 0, such that {1+f(z)+f(z^{−1})| 1 ≠ z ∈ A} forms a ZZ_{p}-basis of S_{p}(A).

Let M̃ = End(ZZ/p^{m}ZZ) be the multiplicative monoid of ring endomorphisms induced on ZZ_{p}A by the group endomorphisms A → A given by z ↦ z^{s} for s ∈ ZZ, and let G̃ be its maximal subgroup (i.e. s relatively prime to p). Denote by M and G the homomorphic actions of these objects on the groups Δ_{p}^{+}(A) or S_{p}(A). Then M is generated by G and the Frobenius map π : z ↦ z^{p}, with π^{m} = 0. In fact, M is the direct union of sets Gπ^{k}, where k = 0, …, m − 1; and τπ^{k} = π^{k} for any τ ∈ Γ_{k}, the subgroup of order p^{k} in G. Therefore the monoid ring ZZ_{p}M is a direct sum of ZZ_{p}G-modules, namely:

ZZ_{p}M = ∑_{k=0}^{m−1} π^{k}ZZ_{p}G,  (∗)

where the k-th component is isomorphic to ZZ_{p}(G/Γ_{k}).

In these terms our first theorem says that S_{p}(C) is a free ZZ_{p}M-module of rank 1 (cf. 3.4 below); and that, in general, S_{p}(A) is a direct sum of monogenic ZZ_{p}G-modules W_{p}(K), one for each non-trivial cyclic subgroup of K ⊂ G, and each of them isomorphic to the appropriate component of (∗) (cf. 4.1 below). In other words, S_{p}(A) is ZZ_{p}M-isomorphic to Δ_{p}^{+}(A).

It would be too optimistic to hope for a simple, explicit isomorphism. Anything that allows coefficient reduction modulo p, for instance, is bound to get into trouble, because the group of 1-units in IF_{p}A is not an IF_{p}-module (unless m = 1, in which case the logarithm does it). As for an explicit ZZ_{p}M-generator w_{p}(x) = 1 + f(x) + f(x^{−1}) of S_{p}(C), the most tempting choice, f(x) = x − 1 fails already for p^{m} = 5. If p is regular, the global constructible units of [7] will provide generators for S′_{p}(C), the kernel of the G-norm, and thence for S_{p}(C), by 2.2 below. That is about all that can be said so far.

The proof rests on BoreviVc’s description [1] of the unit group in a p-adic field without p-th roots of 1 (this really constrains us to symmetric units). With ζ denoting a p^{m}-th root of 1, let S_{p}(ζ) stand for symmetric (⋆ : ζ ↦ ζ^{−1}) units ≡ 1 modulo ζ − 1 in ZZ_{p}[ζ]. (Incidentally, the reader should be warned that, in [7], the latter condition was not imposed on S(ζ) = (ZZ[ζ]) × S_{p}(ζ)). In section 2, we first consider the problem of lifting units from S′_{p}(ζ) to S′_{p}(C). The result obtained there allows us to show the triviality of the cohomology group H^{1}(G, S_{p}(C)) and the existence of a ZZ_{p}M-generator for S′_{p}(C). In section 3, the other cohomology gets involved, and requires some background on S′′_{p}(C), the G-fixed part of S_{p}(C). (We are avoiding the standard notation X^{G} for the fixed part, because we use exponents as operators.) Then comes the theorem for cyclic groups. Finally, section 4 generalizes everything from C to A. BoreviVc’s result is essentially an existence theorem, and so therefore is ours. We wind up with a generating polynomial w_{p}(x) whose precise nature is unknown and which is certainly not unique.

Most of the effort will be spent on the case A = C, where repeated use will be made of the fundamental pull-back diagram of ring maps

ZZ_{p}(C) ⟶^{φ} ZZ_{p}[ζ]
π ↓  ↓
ZZ_{p}(C^{p}) ⟶^{ρ} IF_{p}(C^{p}),

(0)

in which ρ denotes reduction of coefficients (cf. [8], §1). It is easy to see that the pull-back property is preserved when we apply the functors S or S′. In each case, it shows that π induces an isomorphism kerφ⟶^{∼} kerρ.

The second half of this paper deals with the integral group ring ZZA ⊂ ZZ_{p}A, and proves a conjecture stated (somewhat implicitly) at the end of [7]. That paper constructs an explicit integral polynomial w(x), such that {w(z)| 1 ≠ z ∈ A} generates a subgroup Λ(A), the “constructible” units, of finite index in S(A) = (ZZA)^{×} ∩ S_{p}(A). Apart from w(z) = w(z^{−1}), the only relation between these elements is that their product over each G-orbit in A equals 1. In particular, Λ(A) is the direct product of ZZG-modules W (K), one for each non-trivial cyclic subgroup K ⊂ A. Denoting by X̂ the p-adic extension X ⊗ ZZ_{p} of a ZZ-module X, this state of affairs can be described by saying that Λ̂(A) is ZZ_{p}M-isomorphic to S′_{p}(A), the correspondence given by w(z) ↦ w′_{p}(z) = w_{p}(z)^{σ−1}, where σ is a generator of G.

For irregular p and m > 1, it seems rather unlikely that this formal isomorphism can be made natural with respect to the inclusions Λ̂(A) ↪ Ŝ(M) and S′_{p}(A) ↪ S′(M_{p}) into the maximal orders M ⊂ M_{p} of QA ⊂ Q_{p}A. Nevertheless we shall see that the indices of these inclusions are equal, at least as long as p is semi-regular (an empty condition if Vandiver’s Conjecture is correct — cf. [11], p. 159). More precisely, let Ω(ζ) denote the circular units in S(ζ), i.e. those which are power-products of G-conjugates of ζ and 1 − ζ, and let Ω(M) be the corresponding subgroup of S(M). According to section 3 of [7], Λ(A) ↪ Ω(M) is of p-power index, which therefore equals the index of the inclusion Λ̂(A) ↪ Ω̂(M). The main issue in sections 5 and 6 is to show that this is the same as the index of S′_{p}(A) ↪ S′(M_{p}). Semi-regularity of p ensures that Ω̂(M) = Ŝ(M).

The point of all this is to get a handle on yet another index, namely that of Λ(A) in Ω(A) = S(A) ∩ Ω(M), a p-power which (for |A| > p) is trivial if and only if p is regular. This “circular index” c(A) measures the abundance of units of ZZA which appear circular under every character of A without being constructible. Our second main theorem (7.3) says that c(A) equals the order of a certain canonical group of ideal classes of ZZA. For cyclic A, the latter was determined by Ullom [10]. For A of exponent p, the connection between c(A) and ideal classes was (implicitly) proved in [4].

2. Units of Norm One.

In [1], BoreviVc describes the Galois structure of the unit group in certain p-adic fields including, as a very special case, Q_{p}(ζ + ζ^{−1}). For the 1-units (i.e. ≡ 1 modulo the prime ideal of ZZ_{p}[ζ + ζ^{−1}]), his results yield ZZ_{p}G-generators θ, γ with N_{G}θ = 1 and γ^{σ−1} = θ^{e} as the only relations, where e is the order and σ a generator of G. θ is obtained from a suitable prime element Π by Π^{σ−1} = ηθ, with η ∈ ZZ_{p}^{×} of finite order. In fact, one can select such a prime Π_{0} in the field of p-th roots of 1, and for higher fields take any Π whose Galois norm falls on Π_{0} (cf. [1] formulas (8) and (9)). Using Hilbert’s Theorem 90 and the decomposition Q_{p}(ζ + ζ^{−1})^{×} =< Π > ×(ZZ_{p}^{×})tor × S_{p}(ζ), it is easy to see that θ is a ZZ_{p}G-generator of all of S′_{p}(ζ), the elements of G-norm 1.

Since φ : (ZZ_{p}C)^{×} ⟶ ZZ_{p}[ζ]^{×} = U_{p}(ζ) is onto, any symmetric element of U_{p}(ζ)^{σ−1} lies in the φ-image of S′_{p}(C). But we are looking for a generator of this image; it will turn out that θ^{σ−1} does the trick, i.e. that φ(S′_{p}(C)) = θ^{Δ_{p}G}, where Δ_{p} denotes the augmentation kernel of the p-adic group ring.

Consider the crossed homomorphism u_{p} : G ⟶ U_{p}(ζ) given by u_{p}(τ) = Π^{τ−1}. Unfortunately its image is not in S_{p}(ζ) because of the torsion element η occurring in u_{p}(σ) = ηθ. We shall therefore have to restrict u_{p} to a subgroup of G. As usual, the cocycle relation implies

u_{p}(σ^{j}) = ∏_{0<i<j}u_{p}(σ)^{σ^{i}} = η^{j} ∏_{0<i<j}θ^{σ^{i}}.

Since the order of η divides p − 1, we can set j = p − 1 and get

u_{p}(σ^{p−1}) ≡ θ^{p−1} modulo θ^{Δ_{p}G}.

Now, σ^{p−1} generates the maximal p-subgroup Γ of G, and θ^{p−1} still generates S′_{p}(ζ) over ZZ_{p}G. Therefore the group homomorphisms

Γ⟶^{u_{p}}S′_{p}(ζ)/θ^{Δ_{p}G} ⟶ S′_{p}(ζ)/φ(S′_{p}(C))  (1)

are surjective.

To prove the injectivity of (1), and thus identify φ(S′_{p}(C)) as θ^{Δ_{p}G}, we invoke Theorem 1.3 of [7]. For the prime P = (ζ^{1/2} − ζ^{−1/2}) ∈ ZZ[ζ] of the global cyclotomic field and any non-trivial s ∈ G̃, this theorem says that the symmetric unit u(s) = P^{s−1} cannot be lifted to S′_{p}(C).

Proposition 2.1: The composite homomorphism shown in (1) is bijective, and hence

φ(S′_{p}(C)) = θ^{Δ_{p}G} = S′_{p}(ζ)^{Δ_{p}G}.

Proof: We regard ZZ[ζ] as a subring of ZZ_{p}[ζ] and write Π = εP^{2}, where P is the aforementioned prime of ZZ[ζ], and thus P^{2} = (ζ − 1) + (ζ^{−1} − 1) a prime of ZZ_{p}[ζ + ζ^{−1}]. Here ε is a symmetric element of U_{p}(ζ) and therefore ε^{τ−1} ∈ φ(S′_{p}(C)) for any τ ∈ G. Hence if u_{p}(τ) = ε^{τ−1}P^{2(τ−1)} = ε^{τ−1}u(τ)^{2} lies in φ(S′_{p}(C)), then so does u(τ)^{2} and therefore u(τ^{2}). By the theorem cited above, this implies τ^{2} = 1, and for τ ∈ Γ this means that τ = 1.

Looking back at (1), the bijectivity of u_{p} can be interpreted as saying that the first cohomology H^{1}(G, S_{p}(ζ)) = S′_{p}(ζ)/S_{p}(ζ)^{σ−1} is isomorphic to Γ, a result which is in agreement with the the well-known theory of local fields. By contrast, we shall now see that H^{1}(G, S_{p}(C)) is trivial.

Proposition 2.2:

S′_{p}(C) = S_{p}(C)^{σ−1}.

Proof. This will be proved by induction on m, the case m = 1 being obvious because |G| is prime to p. If m > 1 we use the following pull-back diagram, which results from (0) by applying the functor S_{p}.

\begin{matrix}S_{p}(C) & ⟶^{φ} & S_{p}(ζ) \\ π ↓ & & ↓ \\ S_{p}(C^{p}) & ⟶^{ρ} & \overline{S}_{p}(C^{p}) \end{matrix}  (2)

For y ∈ S′_{p}(C), we have to find v ∈ S_{p}(C) such that y = v^{σ−1}. By Proposition 2.1, there is an η_{0} ∈ S_{p}(ζ) such that φ(y) = η_{0}^{σ−1}. Taking a φ-preimage y_{0} ∈ S_{p}(C) for η_{0} — this is trivially possible in the p-adic context — and replacing y by y · y_{0}^{1−σ}, we may assume that φ(y) = 1.

Putting u = π(y) ∈ S′_{p}(C^{p}), we now have ρ(u) = 1; and by induction assumption we get w ∈ S_{p}(C^{p}) with u = w^{σ−1}. Note that w̄ = ρ(w) is G-invariant, because ρ(u) = 1.

Suppose

w = 1 + ∑_{1≠c∈C^{p}} a_{c}(c − 1),

with a_{c} = a_{c^{−1}} ∈ ZZ_{p}. Then the G-invariance of w̄ forces the coefficients a_{c} to be constant modulo p on G-orbits. Thus we can write a_{c} = a′_{c} + a′′_{c}p with a′_{c} actually constant on G-orbits. Then

w_{0} = 1 + ∑_{1≠c∈C^{p}} a′_{c}(c − 1) ∈ S′′_{p}(C^{p})

has the same ρ-image as w, and, replacing w by w/w_{0}, we may assume that

u = w^{σ−1}  with  ρ(w) = 1.

The pull-back diagram now yields a v ∈ S_{p}(C) such that π(v) = w and φ(v) = 1. Then π(v^{σ−1}) = u and φ(v^{σ−1}) = 1 show that v^{σ−1} = y, as desired.

Corollary 2.3: The map π : S′_{p}(C) ⟶ S′_{p}(C^{p}) is surjective. Proof: If it were not for the norm-constraint, this would be entirely obvious. Given u ∈ S_{p}(C^{p}), take a w such that u = w^{σ−1}, and now lift w.

We can now take a first step in the direction of a normal basis.

Proposition 2.4: Let ω ∈ S′_{p}(C) be such that φ(ω) generates S′_{p}(ζ)^{σ−1} over ZZ_{p}G. Then ω generates S′_{p}(C) over ZZ_{p}M.

Proof. For m = 1, φ : S′_{p}(C) ⟶ S′_{p}(ζ) is an isomorphism and there is nothing to prove (this is the case Γ = {1} of the discussion surrounding (1)).

Assuming that m ≥ 2, consider the following modified version of the pull-back diagram (2)

\begin{matrix}S′_{p}(C) & ⟶^{φ} & S′_{p}(ζ) \\ π ↓ & & ↓ \\ S′_{p}(C^{p}) & ⟶^{ρ} & \overline{S}_{p}(C^{p}) \end{matrix}  (3)

As a first step, we shall prove that ω^{π} generates S′_{p}(C^{p}). By induction hypothesis, S′_{p}(C^{p}) does have a single ZZ_{p}M-generator, say ω_{1}. Since the left vertical arrow in (3) is surjective by Corollary 2.3, there is a ω̃_{1} ∈ S′_{p}(C) such that ω̃_{1}^{π} = ω_{1}. By Proposition 2.1, φ(ω̃_{1}) = θ^{(σ−1)α} with α ∈ ZZ_{p}G, and ω̃_{1} = ω^{α} · u with u ∈ ker(φ).

At this point, the pull-back property of (3) ensures that π induces an isomorphism ker(φ)⟶^{∼} ker(ρ), and by the p-adic analogue of Kummer’s Lemma (cf. [5]) the latter is S′_{p}(C^{p})^{π−p}. Thus u^{π} = ω_{1}^{β(π−p)}, with β ∈ ZZ_{p}M, and ω_{1} = ω̃_{1}^{π} = ω^{πα} · ω_{1}^{(π−p)β}, in other words, ω^{πα} = ω_{1}^{1−(π−p)β}. Remembering that π^{m} = 0 (on units of augmentation 1), we get for n > m, that (π − p)^{n} ≡ 0 mod p^{n−m}, and hence 1 − (π − p)β is invertible by the geometric series in ZZ_{p}M regarded as a ring of operators on S_{p}(C). Therefore ω_{1} = ω^{παδ} for a suitable δ ∈ ZZ_{p}M, completing our first step.

We still have to show that an arbitrary u ∈ S′_{p}(C) can be written in terms of ω, ZZ_{p}G, and π. Since φ(ω) generates φ(S′_{p}(C)), we may assume that u ∈ ker(φ). Hence u^{π} = ω^{πβ(π−p)} as above (we have merely replaced ω_{1} by the new generator ω^{π} of S′_{p}(C^{p})). Let s_{1} stand for the the trace over the minimal p-subgroup of G and note that π(π−p) = π(π−s_{1}). Putting v = uω^{β(s_{1}−π)} we now have v^{π} = 1 and φ(v) = φ(ω^{β(s_{1}−π)}) ∈ S′_{p}(ζ^{p}). This means that v ∈ S′_{p}(C^{p}), and by Step 1 proves what we want.

Since this point is crucial, let us examime more closely why v must be in S′_{p}(C^{p}) given that v ∈ S′_{p}(C) with v^{π} = 1 and φ(v) = ε ∈ S′_{p}(ζ^{p}). The right vertical arrow of (3) makes ε ≡ 1 modulo the prime ideal of ZZ_{p}[ζ + ζ^{−1}]. Therefore ε ≡ 1 modulo the prime of ZZ_{p}[ζ^{p} + ζ^{−p}], and the fiber product

S′_{p}(C^{p}) ⟶ S′_{p}(ζ^{p})
↓  ↓
S′_{p}(C^{p^{2}}) ⟶ \overline{S}(C^{p^{2}})

produces a v_{1} ∈ S′_{p}(C^{p}) corresponding to the data ε ∈ S′_{p}(ζ^{p}) and 1 ∈ S′_{p}(C^{p^{2}}). The inclusion S′_{p}(C^{p}) ↪ S′_{p}(C) takes v_{1} to an element which “looks like” v under φ and π, hence must equal v.

This finishes the proof of Proposition 2.4.

Remark 2.5: The exactness of

1 ⟶ S′′_{p}(C) ⟶ S_{p}(C) ⟶ S′_{p}(C) ⟶ 1  (4)

shows that the quotient of the first two groups has the same ZZ_{p}M- structure as S′_{p}(C). In particular, S_{p}(C) is generated over ZZ_{p}M by S′′_{p}(C) and the element ω of Proposition 2.4.

3. A Normal Basis for S_{p}(C).

Let υ ∈ S_{p}(C) be such that υ^{σ−1} = ω is a ZZ_{p}M-generator of S′_{p}(C). We shall see that υ generates S_{p}(C) as a free ZZ_{p}M-module of rank one. By the remark at the end of the last section, we already know

S_{p}(C) = S′′_{p}(C) · υ^{ZZ_{p}M}.  (5)

To get rid of the first factor, we need two kinds of information on S′′_{p}(C): how it is generated, and how it is related to norms. The next two lemmas will supply this information. Since S′′_{p}(C) = 1 + Δ′′_{p}(C), we shall drop or insert the reference to symmetry in whatever way is notationally most convenient.

Lemma 3.1: The usual power series yields a group isomorphism

log : S′′_{p}(C) ⟶ Δ′′_{p}(C),  (6)

whose inverse is given by the exponential series. Proof. If M_{p} denotes the maximal order in Q_{p}C, the ideal Δ(M_{p}) = M_{p}Δ_{p}(C) is the direct product of the prime ideals of the Wedderburn components. Since Δ′′(M_{p}) is a direct product of copies of pZZ_{p}, and since S′′(M_{p}) = 1 + Δ′′(M_{p}), we have the obvious group homomorphism log : S′′(M_{p}) ⟶ Δ′′(M_{p}) whose inverse is given by exp. When these power series are restricted to elements of S′′_{p}(C) and Δ′′_{p}(C), respectively, the only problem is to verify that their partial sums remain in the group ring.

For 1 ≤ k ≤ m, let C_{k} ⊂ C denote the subgroup of order p^{k}, and put

b_{k} = ∑_{z∈C_{k}}(z − 1) and t_{k} = ∑_{z∈C_{k}} z.

Since all G-orbits of C are of the form C_{k} −C_{k}^{p}, a basis of Δ′′_{p}(C) is formed by the elements b_{k} − b_{k−1}, and hence also by the b_{k} themselves. For t_{k} = b_{k} + p^{k}, we obviously have t_{k}t_{l} = p^{k}t_{l} if l ≥ k.

Every element of Δ′′_{p}(C) is of the form

d = ∑_{k=1}^{m} α_{k}b_{k} = α_{0}p + α_{1}t_{1} + ⋯ + α_{m}t^{m},

and since any product of p of these last summands is divisible by p^{p−1}, we have ord_{p}d^{p} ≥ 2 (at the very least). Now a routine estimate (cf. [9], Ch.IX.2) shows that exp d stays in the group ring, and the exp-log machine works for the G-invariant units of ZZ_{p}C.

Since every b_{k}^{2} is divisible by p, we have log(1 + b_{k}) ≡ b_{k} modulo p, and thus the following result. Corollary 3.2: A basis of S′′_{p}(C) is formed by the elements u_{k} = 1 + b_{k}.

Coming back to our potential generator υ, let υ_{k} be its image under π^{m−k}, for 1 ≤ k ≤ m. This element is left fixed by the subgroup Γ_{k} ⊂ G of order p^{m−k}. Letting N_{k} stand for the norm with respect to G/Γ_{k}, we have

N_{G}υ_{k} = p^{m−k}N_{k}υ_{k}.  (7)

Lemma 3.3: The group S′′_{p}(C)/N_{G}S_{p}(C) has order p ⋯ p^{m−1}. Proof: This is of course a statement about the order of the second cohomology group H^{2}(G, S_{p}(C)), which is equal to the Herbrand quotient because of the triviality of H^{1}. Since S_{p}(C) is of finite index in S(M_{p}), the Herbrand quotients (cf. [9]) of these two groups coincide, and it suffices to work it out for latter.

For a cyclic extension E/F of local fields it is well known that the Galois cohomology groups H^{i}(G, U_{E}) of the units have order e(E/F), the ramification index (cf. [9], IX.3, Lemma 4). For F = Q_{p} and E = Q_{p}(ζ + ζ^{−1}), we have U_{E} = torsion × S_{p}(ζ), and the cohomology decomposes accordingly. For the torsion it has order (p − 1)/2 in both dimensions i = 1 and i = 2; hence for S_{p}(ζ) we get p^{m−1} both times. Applying this to the components of S(M_{p}), we see that H^{1}(G, S(M_{p})) has order ∏_{k} p^{k−1}. With H^{2}(G, S_{p}(ζ^{p})) one has to be careful, because the corresponding Galois group is G/Γ_{1}. By (7) its 2cohomology with respect to G still has order p^{m−1}; and so it goes for all Wedderburn components, giving ∏_{k} p^{m−1} as the order of H^{2}(G, S_{p}(M_{p})), with k still running from 1 to m. The quotient of the two orders is just the number advertised in the statement of the lemma.

Theorem 3.4: Let υ ∈ S_{p}(C) be such that υ^{σ−1} = ω is a ZZ_{p}M-generator of S′_{p}(C). Then the map ZZ_{p}M ⟶ S_{p}(C), given by β ↦ υ^{β}, is an isomorphism of ZZ_{p}M-modules. Proof: We shall prove inductively that S′′_{p}(C) is generated by the elements N_{k}υ_{k}, for k = 1, …, m. The case m = 1 is settled by the triviality of H^{2}.

Consider the matrix whose rows are the coefficients of N_{k}υ_{k} (k counted down from m to 1) in terms of the basis u_{m}, …, u_{1} (in that order). It is an upper triangular matrix, and we have to prove that its diagonal entries d_{m}, …, d_{1} are prime to p. By induction hypothesis (applied to C^{p}), we know this to be the case for all but the first entry d_{m}.

We now make an analogous matrix, using N_{G}υ_{k} for the rows, and get p^{m−k}d_{k} on the diagonal. If d_{m} were not prime to p, this would yield a value > p ⋯ p^{m−1} for the order of S′′_{p}(C)/⟨N_{G}υ_{m}, …, N_{G}υ_{1}⟩, the quotient of G-invariant units modulo the ZZ_{p}- lattice spanned by the norms N_{G}υ_{k}. But this group is H^{2}(G, S_{p}(C)), because N_{G}S_{p}(C) = N_{G}⟨u_{m}, υ_{m}, S_{p}(C^{p})⟩, with N_{G}u_{m} = u_{m}^{|G|} a p-th power. Hence its order is prescribed by the preceding lemma.

We have proved that υ generates S_{p}(C) over ZZ_{p}M. A comparison of ranks proves the isomorphism, Q.E.D.

4. Abelian Groups.

Keeping C and υ as above, let A be an abelian group whose maximal cyclic subgroups have order |C|. Letting x be generator of C, think of υ = υ(x) as a polynomial in x (actually a class modulo x^{p^{m}} − 1 of such). Then υ(z) makes sense for any z ∈ A, and so does

W_{υ}(K) = υ(z)^{ZZ_{p}G} ⊂ S_{p}(A),  (8)

where K = ⟨z⟩ denotes the subgroup generated by z. Since all generators of K are G-conjugate, W_{υ}(K) depends only on K and, of course, υ. By Theorem 3.4, it is ZZ_{p}Gisomorphic to ZZ_{p}(G/Γ_{k}), where p^{k} is the order of K.

Theorem 4.1: S_{p}(A) is the direct product of the ZZ_{p}G-modules W_{υ}(K), as K ranges over the non-trivial cyclic subgroups of A. Proof: By [6], we have S_{p}(A) = ∏_{K} S_{p}(K), the product taken inside S_{p}(A). On the other hand, for every K Theorem 3.4 ensures that S_{p}(K) = ∏_{K′} W_{υ}(K′), with K′ running over non-trivial subgroups. Therefore

S_{p}(A) = ∏_{K⊂A} W_{υ}(K),  (9)

and the product must be direct because of the rank count, Q.E.D.

In particular, the units S′_{p}(A) of norm 1 split up into the norm-kernels W′_{υ}(K) = W_{υ}(K)^{σ−1} of the individual components. This yields a generalization of Proposition 2.2.

Corollary 4.2:

S′_{p}(A) = ∏_{K} W′_{υ}(K) = S_{p}(A)^{σ−1},

in particular H^{1}(G, S_{p}(A)) is trivial.

Convention 4.3: Henceforth we fix a generator σ of G, as well as ZZ_{p}M-generators w_{p}(x) = υ and w′_{p}(x) = υ^{σ−1} of S_{p}(C) and S′_{p}(C), respectively. Accordingly, we write W_{p} instead of W_{υ}. Thus we have

S_{p}(A) = ∏_{K⊂A} W_{p}(K)  and  S′_{p}(A) = ∏_{K⊂A} W′_{p}(K).  (10)

For later use, we conclude this section by extending our observations on G-invariant units to the general abelian case.

If M_{p} denotes the maximal order in Q_{p}A, we again have the group homomorphism log : S′′(M_{p}) ⟶ Δ′′(M_{p}) whose inverse is given by exp. We know that these power series converge inside ZZ_{p}A for A = C cyclic, and shall infer that the logarithm still works for general A.

By way of preparation, put

b_{K} = ∑_{z∈K}(z − 1) and t_{K} = ∑_{z∈K} z,

for any non-trivial cyclic subgroup K ⊂ A. Since all G-orbits of A are of the form K −K^{p}, a basis of Δ′′_{p}(A) is formed by the elements b_{K} −b_{K^{p}}, and hence also by the b_{K} themselves. Note that t_{K}^{2} = |K|t_{K}, and that b_{K}^{2} is divisible by |K| because b_{K} = t_{K} − |K|. Thus it follows that the p-th power of any d ∈ Δ′′_{p}(A) vanishes modulo p. This ensures that ord_{p}d^{n} ≥ [n/p] ≥ ord_{p}n for all n ≥ 1. Therefore d^{n}/n stays in ZZ_{p}A, and so does log(1+d).

Proposition 4.4: The usual power series yields a group isomorphism

log : S′′_{p}(A) ⟶ Δ′′_{p}(A).

Proof. The only thing missing is the surjectivity of the log-map. To prove it, we consider the diagram

∏_{K} S′′_{p}(K) ⟶ S′′_{p}(A)
↓  ↓
∏_{K} Δ′′_{p}(K) ⟶ Δ′′_{p}(A)

whose commutativity is due to the homomorphic property of the log. The discussion preceding the statement of this proposition shows that the bottom arrow is surjective, Q.E.D.

Since every b_{K}^{2} is divisible by p, we have log(1 + b_{K}) ≡ b_{K} modulo p, and thus the following result. Corollary 4.5: A basis of S′′_{p}(A) is formed by the elements 1 + b_{K}, as K ranges over the non-trivial cyclic subgroups of A.

5. Comparison with Global Units.

By the index of a module homomorphism f : X ⟶ X′ (denoted ind f), we simply mean the cardinality (possibly ∞) of its cokernel. We shall repeatedly use the following principle on “triangular” maps

X_{1} ⊕ ⋯ ⊕ X_{N} ⟶^{f} X′_{1} ⊕ ⋯ ⊕ X′_{N},

i.e. those with the property that f(X_{i}) ⊂ X′_{1} ⊕ ⋯ ⊕ X′_{i}. Suppose that, for each i, the components X_{i} and X′_{i} are free ZZ_{p}-modules of the same rank. Then the index of f equals the product of the indices of the maps f_{i} : X_{i} ⟶ X′_{i} obtained by applying f followed by projection onto the i-th component. (Think determinants!)

We now turn to units of the integral group ring ZZA, as described in the introduction. Concerning the commutative square of inclusions of ZZ_{p}G-modules,

\begin{matrix}∏_{C} Ŵ (C) = Λ̂(A) & ⟶^{f} & S′_{p}(A) = ∏_{C} W′_{p}(C) \\ h ↓ & & ↓ h_{p} \\ ∏_{K} Ω̂(ζ_{K}) = Ω̂(M) & ⟶^{g} & S′(M_{p}) = ∏_{K} S′_{p}(ζ_{K}) \end{matrix}  (11)

(with C and K running over cylic subgroups and factor groups, respectively), we wish to show that the indices of the two vertical arrows (or equivalently, the two horizontal arrows) are equal.

The inclusion g is diagonal because Ω̂(ζ_{k}) ⊂ S′_{p}(ζ_{k}), for any p^{k}-th root ζ_{k} of 1, and the index e_{p^{k}} = [S′_{p}(ζ_{k}) : Ω̂(ζ_{k})] is a purely arithmetical invariant, trivial if p is regular. On the other hand, since W (C) ⊂ S′_{p}(C), we can make f appear triangular by numbering the C so that |C_{i}| < |C_{j}| implies i < j. Thus

ind g = ∏_{K} e_{|K|}  and  ind f = ∏_{C} ind f_{C},  (12)

where f_{C} is the composite Ŵ (C) ⟶ S′_{p}(C) ⟶ W′_{p}(C) of f and the obvious projection. Since the latter depends on the choice of the polynomial w_{p}(x), so does the map f_{C}, but not its index. In fact, we shall see that

ind f_{C} = e_{|C|},  (13)

and since the cyclic subgroups and factor groups of A are in one-to-one correspondence,

(12) and (13) will imply the desired equality of indices. We state this formally.

Proposition 5.1: ind h = ind h_{p} and ind f = ind g, in diagram (11).

Since the proof depends only on equation (13), it is easily reduced to the cyclic case. Consider the diagram (11) in the special case, where A = C is cyclic of order p^{m}:

\begin{matrix}Ŵ (C) × ⋯ × Ŵ (C^{p^{m−1}}) & ⟶^{f} & W′_{p}(C) × ⋯ × W′_{p}(C^{p^{m−1}}) \\ h ↓ & & ↓ h_{p} \\ Ω̂(ζ) × ⋯ × Ω̂(ζ^{p^{m−1}}) & ⟶^{g} & S′_{p}(ζ) × ⋯ × S′_{p}(ζ^{p^{m−1}}) \end{matrix}  (14)

with ζ = ζ_{m} denoting a p^{m}-th root of 1. In this setting, (12) boils down to

ind g = ∏_{i=0}^{m−1} e_{p^{m−i}}  and  ind f = ∏_{i=0}^{m−1} ind f_{C^{p^{m−i}}} ,  (15)

and (13) easily results by recursive application of the equality ind f = ind g for cyclic groups.

This equality will be proved in the next section via its counterpart: ind h = ind h_{p}. But instead of relating these two maps by means of f and g, we shall give an abstract description h : X ⟶ Y, which fits both of them and at the same time nails down their index. For this purpose we shall now describe some of their crucial features.

From section 2, recall the isomorphism u_{p} : Γ ⟶ S′_{p}(ζ)/L′_{p}(ζ), with L′_{p}(ζ) = φ(S′_{p}(C)) = φ(W′_{p}(C)) consisting of those elements of S′_{p}(ζ) which are liftable to S′_{p}(C). Remember also that this is a ZZ_{p}G-module generated by θ^{σ−1}, and that L′_{p}(ζ_{m−i}) is similarly generated by the Γ_{i}-norm ν_{i}(θ^{σ−1}) of this element, for i = 1, …, m − 1 (because of the choice of the prime Π used in section 2 to define θ). We note in particular the ZZ_{p}G-maps

ν_{i} : L′_{p}(ζ) ⟶ L′_{p}(ζ_{m−i})  and  φ : W′_{p}(C) ⟶ L′_{p}(ζ),  (16)

which are surjective and bijective, respectively.

To describe the analogous situation for global cyclotomic units, we have to refer to [7] — a task made awkard by the use of the full group G̃ and a larger module Ω̃(ζ) in that paper. If we set L(ζ) = φ(Ω(C)) ⊂ Ω(ζ), section 1 of [7] provides an isomorphism u : G̃ ⟶ Ω̃(ζ)/L(ζ), where Ω̃(ζ) = P^{ΔG} and u(s) = P^{s−1}, for any s ∈ G̃, as in section 2 above. The exact sequence

1 ⟶ Ω(ζ) ⟶ Ω̃(ζ) ⟶ IF_{p}^{×} ⟶ 1

shows that [Ω̃(ζ) : Ω(ζ)] = p^{m−1} = [G̃ : Γ], and hence the restriction of u to Γ yields an isomorphism u : Γ ⟶ Ω(ζ)/L(ζ) analogous to u_{p}.

By [7, Lemma 2.1], w(ζ^{p^{i}}) = ν_{i}(w(ζ)), and by the subsequent Theorem 2.3, φ maps W (C) isomorphically onto L(ζ). Thus

ν_{i} : L(ζ) ⟶ L(ζ_{m−i})  and  φ : W (C) ⟶ L(ζ),  (17)

are again surjective and bijective, respectively.

The isomorphisms u and u_{p} are mentioned in this context only for the sake of the equation [Ω(ζ) : L(ζ)] = p^{m−1} = [S′_{p}(ζ) : L′_{p}(ζ)], which allows us to transfer our attention from (14) to

\begin{matrix}Ŵ (C) × ⋯ × Ŵ (C^{p^{m−1}}) & ⟶^{f} & W′_{p}(C) × ⋯ × W′_{p}(C^{p^{m−1}}) \\ h ↓ & & ↓ h_{p} \\ L̂(ζ) × ⋯ × L̂(ζ^{p^{m−1}}) & ⟶^{g} & L′_{p}(ζ) × ⋯ × L′_{p}(ζ^{p^{m−1}}) \end{matrix}  (18)

for the proof of index equality.

We are now ready for an abstract rendition of h (or h_{p}).

6. Indices of Filtered G-Modules.

We are interested in injections h : X ↪ Y of ZG-modules for a group G and a coefficient ring Z equal to ZZ or ZZ_{p}. Then obviously

(a) the induced map X̄ = X/X′′ ⟶ Ȳ = Y/Y′′ is injective, and

(b) the sequence 0 ⟶ Y′′/X′′ ⟶ Y/X ⟶ Ȳ /X̄ ⟶ 0 is exact, where the double prime denotes the G-fixed part. As a consequence, the indices are multiplicative, in the sense that

[Y : X] = [Ȳ : X̄] · [Y′′ : X′′],  (19)

provided that [Y : X] is finite. Note also that, if Y is free of Z-torsion, so is Ȳ = Y/Y′′. In fact, if y ∈ Y − Y′′ is such that ay ∈ Y′′, and if σ ∈ G is such that σy ≠ y, then σay = ay implies a(σy − y) = 0.

Filtration. Let G = G_{0} ⊃ G_{1} ⊃ ⋯ ⊃ G_{m} = 1 be a chain of normal subgroups, and consider the corresponding filtration 0 ⊂ X^{(0)} ⊂ ⋯ ⊂ X^{(m)} = X of the G-module X, where now X^{(μ)} denotes the G_{μ}-fixed part.

If h : X ↪ Y is an injection of finite index, we can apply the formula (19) to obtain [Y : X] = [Y^{(m)}/Y^{(m−1)} : X^{(m)}/X^{(m−1)}] · [Y^{(m−1)} : X^{(m−1)}]. Since X^{(m−1)} ↪ Y^{(m−1)} is an injection of G/G_{m−1}-modules, an obvious induction yields

[Y : X] = ∏_{μ=0}^{m} [Y^{(μ)}/Y^{(μ−1)} : X^{(μ)}/X^{(μ−1)}],  (20)

where X^{(−1)} = Y^{(−1)} = 0. We shall refer to the module Y^{(μ)}/Y^{(μ−1)} as the μ-th “slice” of Y, and denote it by Ȳ^{(μ)} for short. Remember that, by (a) above, each slice h^{(μ)} : X̄^{(μ)} ⟶ Ȳ^{(μ)} of h is still injective.

Grading. Now let G be abelian. Starting with a monogenic ZG-module L, define a graded module Y = Y_{m} ⊕ ⋯ ⊕ Y_{1} by putting Y_{μ} = L^{(μ)}. If we now slice this module as above, we get graded slices

Ȳ^{(μ)} = Ȳ_{m}^{(μ)} ⊕ ⋯ ⊕ Ȳ_{μ}^{(μ)},

in which all the components are canonically isomorphic to L̄^{(μ)} = L^{(μ)}/L^{(μ−1)}. Let us abbreviate ZG by R, and its action on L̄^{(μ)} by R̄_{μ}. Then each component of Ȳ^{(μ)} is a free R̄^{(μ)}-module of rank 1.

Taking a cue from the surjectivities of the ν_{i} in (16) and (17), we further assume that the G_{μ}/G_{k}-traces

T_{k,μ} : L^{(k)} ⟶ L^{(μ)}  (21)

are surjective, for any 0 ≤ μ ≤ k ≤ m.

Our task is to study a certain injection h : X ⟶ Y where X = X_{m} ⊕ ⋯ ⊕ X_{1} is a module with components X_{k} isomorphic to Y_{k}. The map h is to be structurally similar to the vertical arrows in (18). It will be defined in terms of the following data:

(i) a set {ω_{m}, ω_{m−1}, …, ω_{1}} of R-generators of X_{m}, X_{m−1}, …, X_{1}, and (ii) a family {φ_{m}, φ_{m−1}, …, φ_{1}} of R-isomorphisms φ_{k} : X_{k} ⟶ Y_{k}. To simplify the notation, we write φ_{k}(ω_{k}) = ω̃_{k}. Whether this is to be considered as an element of the component Y_{k} or some Y_{l} with l > k, will be left to the context (again in the interest of notation). Here is what h does to the generator ω_{k}:

h(ω_{k}) = (ω̃_{k}, ω̃_{k−1}, …, ω̃_{1}, 0, …, 0),  (22)

where the first component lies in Y_{m}, the second one in Y_{m−1}, and the last one in Y_{1}.

Next we come to the maps h^{(μ)} : X̄^{(μ)} ⟶ Ȳ^{(μ)} induced on the graded slices. The index j occurring in (22) is now constrained so that k − j ≥ μ, that is 0 ≤ j ≤ k − μ. The components X̄_{k}^{(μ)} are isomorphic to Ȳ_{k}^{(μ)} via the φ_{k}, so we have free R̄_{μ}-modules of rank m − μ + 1 on both sides. We shall choose bases and display the corresponding matrix for h^{(μ)}.

On the Y -side we simply take (the class of) ω̃_{μ} as a generator in each of the isomorphic components. Now, for each m ≥ k ≥ μ, we have Y_{μ} ⊂ Y_{k}, and therefore ω̃_{μ} = t_{k,μ}ω̃_{k} with a suitable element t_{k,μ} ∈ R. Note that, for 0 ≤ j ≤ k −μ, we have t_{k,μ}ω̃_{k−j} = t_{k,μ}t_{k,k−j}ω̃_{k} = t_{k,k−j}ω̃_{μ}, because R is commutative.

For the k-th component X̄_{k}^{(μ)}, we shall take t_{k,μ}ω_{k} as a generator. Choosing t_{k,k} = 1, we then have

h^{(μ)}(t_{k,μ}ω_{k}) = (ω̃_{μ}, t_{k,k−1}ω̃_{μ}, …, t_{k,μ}ω̃_{μ}, 0, …, 0).  (23)

Up to sign, the determinant of the corresponding matrix is just the product of the entries on the “anti-diagonal”, namely det h^{(μ)} = ∏_{k=μ}^{m} t_{k,μ} ∈ R̄_{μ}.

Because of the stipulated surjectivity of the traces (23), both ω̃_{μ} and T_{k,μ}ω̃_{k} are generators of Y^{(μ)}. Hence there is an element α_{k,μ} ∈ R such that t_{k,μ} = α_{k,μ}T_{k,μ}, and this element acts as an automorphism on Y^{(μ)}. Since G_{μ} acts trivially on Y^{(μ)}, so do all the groups G_{k} with k ≥ μ, and t_{k,μ} = α_{k,μ}[G_{μ} : G_{k}] in R̄_{μ}. The desired determinant thus turns out to be det h^{(μ)} = α_{μ}n_{μ}, with α_{μ} ∈ R̄_{μ} invertible and

n_{μ} = ∏_{k=μ}^{m} [G_{μ} : G_{k}].  (24)

Finally, suppose that L is a free Z-module of finite rank. Then so is L̄^{(μ)}, say of rank r_{μ}. In the setting described above, we can now conclude that

ind h^{(μ)} = r_{μ}^{n_{μ}}.  (25)

This concludes the proof of Proposition 5.1, since the two “models” for L, namely L̂(ζ) and L′_{p}(ζ), are ZZ_{p}G-isomorphic (though not via g).

Remark 6.1: Because of Lemma 2.1 of [7], we have t_{k,μ} = T_{k,μ} for the left vertical arrow of (18). If p is regular, generators on the p-adic side can be chosen so as to enforce the same rule there, simply because f and g are then isomorphisms. However, for irregular p, there is no reason for non-trivial “twists” α_{k,μ} not to occur in the p-adic case. It is probably impossible to replace f and g by isomorphisms compatible with h and h_{p}, and obtain a more natural proof.

7. Kernel Groups.

Let A ⊂ B be orders of the rational group algebra Q(A). It follows easily from the work of Fröhlich [2] that there is an exact sequence

1 ⟶ U(B)/U(A) ⟶ U(B_{p})/U(A_{p}) ⟶ D(A, B) ⟶ 1,

where U stands for units, and D(A, B) = ker(cl(A) ⟶ cl(B)) is the kernel of the natural map between the ideal class groups. Another way of saying this is that U(A) is the kernel of the obvious “difference” map

δ : U(B) × U(A_{p}) ⟶ U(B_{p}),

and that D(A, B) is its cokernel. Splitting off augmentations, we can also get D(A, B) as the cokernel of

δ : E(B) × E(A_{p}) ⟶ E(B_{p}),

where E(A) denotes the kernel of the augmentation map on U(A). If B = M is the maximal order, we shall write D(A) instead of D(A, M). Then D(A) is the cokernel of

∏_{ψ≠1} U(ψ) × E(A_{p}) ⟶ ∏_{ψ≠1} U_{p}(ψ),

where ψ runs through a complete set of rationally inequivalent characters of A, and U(ψ) stands for the units of the ψ-th component of M.

Now suppose that A = ZZA is the group ring, and let us write D(A) instead of D(ZZA), etc. Then, since every ψ maps the augmentation ideal Δ_{p}(A) ⊂ ZZ_{p}A into the prime ideal of the respective cyclotomic field, it will take E_{p}(A) = E(ZZ_{p}A) into the group E_{p}(ψ) ⊂ U_{p}(ψ) of arithmetic 1-units, i.e. those which are ≡ 1 modulo that prime ideal. The next two results are not intended as news but as elucidation.

Lemma 6.1: D(A) is the cokernel of the “difference” of inclusions

δ : ∏_{ψ≠1} E(ψ) × E_{p}(A) ⟶ ∏_{ψ≠1} E_{p}(ψ).

Proof: With every ψ associate the reduction ρ_{ψ} : ZZ[ζ_{ψ}] → IF_{p}, and define B to be the order consisting of all b ∈ M such that ρ_{ψ}(b) = ρ_{1}(b) for all ψ. Then A ⊂ B ⊂ M, and E(B) is the product, over all ψ ≠ 1, of the groups E(ψ), so that D(A, B) is just the cokernel of the map shown in the statement of the lemma. Moreover, D(A, B) is of course a subgroup of D(A, M) = D(A). To see that these groups are in fact equal, we note that the vertical maps induce a surjection on the cokernels of the horizontal ones in the diagram

E(B) × E(A_{p}) ⟶ E(B_{p})
↓  ↓
E(M) × E(A_{p}) ⟶ E(M_{p}),

because U(ψ)/E(ψ) → U_{p}(ψ)/E_{p}(ψ) is always bijective.

Since each E_{p}(ψ) is a ZZ_{p}-module, and p is odd, we obtain the following

Corollary 6.2: D(A) is a finite p-group, and

1 ⟶ S(M)/S(A) ⟶ S(M_{p})/S(A_{p}) ⟶ D^{+}(A) ⟶ 1  (26)

is exact, where D^{+}(A) denotes the ⋆-invariant part.

Replacing S by S′ in the middle term of (26), we get a subgroup D^{♯}(A) ⊂ D^{+}(A), which is of special arithmetical interest. The two groups are obviously related by the exact sequence

Σ(A) : 1 ⟶ D^{♯}(A) ⟶ D^{+}(A) ⟶ Θ(A) ⟶ 1,  (27)

whose right hand term Θ(A) = N_{G}S(M_{p})/N_{G}S(A_{p}) will be examined in the next section. It essentially measures the complexity of the subgroup lattice of A, and is trivial if and only if A is cyclic.

The other term D^{♯}(A), being the cokernel of Ŝ(M)/Ŝ(A) ⟶ S′(M_{p})/S′(A_{p}), is trivial if p is regular and (for semi-regular p) only then. The point of the present exercise is to relate its order to the circular index c(A) = [Ω(A) : Λ(A)], which by Theorem 3.5 of [7] is known to be a p-power, trivial if and only if p is regular or |A| = p.

Indeed, if p is semi-regular, we have Ω̂(A) = Ŝ(A) and hence the exact sequence

1 ⟶ Ω̂(A)/Λ̂(A) ⟶ Ω̂(M)/Λ̂(A) ⟶ S′_{p}(M)/S′_{p}(A) ⟶ D^{♯}(A) ⟶ 1.  (28)

By Proposition 5.1, the two middle terms have equal orders and therefore so do the end terms. This is our second main result. Theorem 7.3: If p is semi-regular, the circular index c(A) = [Ω(A) : Λ(A)] equals the order of the group D^{♯}(A).

8. The Character Index.

For clarity of notation, we start in the abstract context X ↪ Y of (multiplicative) G-modules for a cyclic group G with generator σ. With N_{G} denoting the G-norm and Y′ its kernel, the exact sequence

1 ⟶ Y′/X′ ⟶ Y/X ⟶ N_{G}Y/N_{G}X ⟶ 1

shows that [Y : X] = [Y′ : X′] · [N_{G}Y : N_{G}X]. The analogue with (σ − 1) instead of N_{G} makes this also equal to [Y′′ : X′′] · [Y^{σ−1} : X^{σ−1}], where the double prime denotes Fix_{G}. Equating indices of the composites X^{σ−1} ⟶ X′ ⟶ Y′ and X^{σ−1} ⟶ Y^{σ−1} ⟶ Y′, we get h^{1}(X)[Y′ : X′] = [Y^{σ−1} : X^{σ−1}]h^{1}(Y), where h^{1} means the order of the first cohomology group (over G or its maximal p-subgroup, it matters not if X, Y are ZZ_{p}-lattices). Putting everything together we finally obtain

[N_{G}Y : N_{G}X] = \frac{h^{1}(X)}{h^{1}(Y)} · [Y′′ : X′′].  (29)

Letting Y = S(M_{p}) and X = S(A_{p}), we know by 4.2 that h^{1}(X) = 1. By the proof of 3.3, we have h^{1}(S_{p}(ζ)) = p^{m−1} and hence h^{1}(Y) = ∏_{K} p^{−1}|K|, where K runs over the non-trivial cyclic factor-groups of A. Thus (29) turns into

[N_{G}S(M_{p}) : N_{G}S(A_{p})] = [S′′(M_{p}) : S′′(A_{p})] ∏_{K} p|K|^{−1}.  (30)

Definition 8.1: Let I be the set of non-trivial cyclic subgroups of A, and J be the analogous set for the dual group of A. Define a matrix (e_{ij}) on I × J by setting e_{ij} = 0 or = 1, depending on whether j does or does not annihilate i. The character index χ(A) is defined to be the absolute value of its determinant.

Proposition 8.2:

[S′′(M_{p}) : S′′(A_{p})] = χ(A) ∏_{K} p^{−1}|K|.

Proof: We know from 4.4 that the logarithm makes the index in question equal to [Δ′′(M_{p}) : Δ′′(A_{p})]. Interpreting the K’s in the formula as referring to cyclic subgroups (rather than factor-groups), let b_{K} be the basis elements of Δ′′(A_{p}) described in section 4.

Picking a generator ψ for every j ∈ J, we clearly have

ψ(b_{K}) = \{ \begin{matrix}|K|, & if ψ(K) ≠ 1; \\ 0, & otherwise. \end{matrix}

This yields an I × J-matrix representing the injection Δ′′(A_{p}) ⟶ Δ′′(M_{p}): as a basis for Δ′′(M_{p}) = ∏_{J} pZZ_{p} we take the J-vectors with p in the j-th position and 0 elsewhere, while for Δ′′(A_{p}) we choose the elements b_{K}. Therefore our matrix is (p^{−1}|K|e_{ij}), whence the result.

In view of (29), the proposition immediately yields the following formula:

|Θ(A)| = [N_{G}S(M_{p}) : N_{G}S(A_{p})] = χ(A).  (31)

Proposition 8.3: χ(A) = 1 if and only if A is cyclic. Proof: If A itself is cyclic, the incidence matrix defining χ(A) is triangular with 1’s on the diagonal, and hence χ(A) = 1.

Conversely, any group epimorphism A ⟶ B clearly entails a surjection M_{A} ⟶ M_{B}, which is just a projection onto a subset of the Wedderburn components, and is therefore compatible with all relevant functors. Hence the defining sequence (26) (together with its variant for S′_{p}) leads to an epimorphism Σ(A) ⟶ Σ(B) of exact sequences, in particular to the divisibility of χ(A) by χ(B). If A is not cyclic, B can be taken to be non-cyclic of order p^{2}, in which case χ(B) = p by Proposition 3.2 of [4].

For the record, we give the correct version of the relevant formula of [4, loc. cit.], which got mangled by a misplaced bracket. For an elementary abelian p-group B with n + 1 independent generators, the argument given in [4] shows that the value of log_{p} χ(B) should be

\frac{1}{2}[n(1 + p + ⋯ + p^{n}) + (1 − p) + ⋯ + (1 − p^{n})].  (32)

Acknowledgements.

The author is indebted to NSERC of Canada for financial support and to Jürgen Ritter of Universtät Augsburg for many stimulating conversations and helpful suggestions.

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University of British Columbia Vancouver, B.C. Canada.