ON PRIMITIVE AND REALISABLE CLASSES

A. Agboola

Version of January 11, 2000

Abstract. Let S be a scheme, and let G be a finite, flat, commutative group scheme over S. In this paper we show that (subject to a mild technical assumption) every primitive class in Pic(G) is realisable. This gives an affirmative answer to a question of Waterhouse. We also discuss applications to locally free classgroups and to Selmer groups of abelian varieties.

Introduction

Let F be a number field with ring of integers O_{F}, and suppose that G/O_{F} is a finite, flat, commutative group scheme of exponent N. Then G = Spec(𝔅), where 𝔅 is an O_{F} -Hopf algebra. Let H^{1}(O_{F} , G) denote flat cohomology of Spec(O_{F}) with coefficients in G, and write G^{D} = Spec(𝔄) for the Cartier dual of G. The group H^{1}(O_{F} , G) parametrises isomorphism classes of twisted forms of 𝔅. It may be shown that each twisted form ℭ of 𝔅 is a locally free 𝔄-module, and so determines a class (ℭ) in the Picard group Pic(G^{D}) of G^{D}.

In recent years, a large amount of work has been done concerning the 𝔄-module structure of twisted forms of 𝔅. The initial motivation for this work was the study of the Galois module structure of rings of integers : in many cases, the twisted form ℭ may be viewed as an order in the ring of integers of some (in general wildly ramified) extension of F. An important aspect of this theory is the class invariant homomorphism ψ which is defined by

ψ : H^{1}(O_{F} , G) ⟶ Pic(G^{D})
ℭ ⟼ (ℭ)(𝔅)^{−1}.  (1)

Typeset by AMS-TEX 2 A. AGBOOLA

1991 Mathematics Subject Classification. 11G, 11R, 14K.

Key words and phrases. Class invariants, classgroups, realisable classes, primitive classes, group schemes, twisted forms.

Partially supported by NSF grant no. DMS 9700937.

This homomorphism was first introduced by W. Waterhouse (see [Wa]), and it provides a measure of the 𝔄-module structure of twisted forms of 𝔅. For example, in many cases of arithmetic interest, one can show that 𝔅 is 𝔄-globally free. If this holds, then it follows that ℭ is a globally free 𝔄-module if and only if it lies in the kernel of ψ.

In this note, we shall be concerned with describing the image of ψ. In order to explain our result, we first give two definitions. Let p_{i} : G^{D} × G^{D} → G^{D} (i = 1, 2) denote projection onto the ith factor, and write m : G^{D} × G^{D} → G^{D} for the multiplication map on G^{D}. Each of these maps induce corresponding pullback homomorphisms p_{i}^{∗} : Pic(G^{D}) → Pic(G^{D} × G^{D}) and m^{∗} : Pic(G^{D}) → Pic(G^{D} × G^{D}) on Picard groups. We shall say that an element in Pic(G^{D}) is primitive if it lies in the kernel of the homomorphism m^{∗} − p_{1}^{∗} − p_{2}^{∗}. We shall say that an element in Pic(G^{D}) is realisable if it lies in the image of the homomorphism ψ.

It was shown by Childs and Magid (see [CM]) that every realisable class is primitive, and Waterhouse has raised the question as to whether the converse is true. If the converse does hold, then this gives an analogue for finite group schemes of a standard result concerning extensions of abelian varieties by commutative, connected linear groups (see [S], Chapter VII, no. 15, Theorem 5). The purpose of this note is to point out that Waterhouse’s question does indeed have an affirmative answer in almost all cases of arithmetic interest: it follows by combining the work of Childs and Magid with a vanishing theorem of Breen.

We remark that Waterhouse’s question has been considered by a number of other authors (see e.g. [By1], [C], [C1], [C2], [CP]). Also, by extending earlier work of L. McCulloh on relative abelian Galois module structure (see [M]), N. Byott has given a precise description of the realisable classes of Pic(G^{D}) in terms of certain resolvend maps (see [By2]). However, it does not seem to be easy to use this description to determine whether or not every primitive class is realisable. It would be interesting to obtain an answer to Waterhouse’s question in terms of Byott’s description.

In Section 1 of this note, we recall work of Childs and Magid, and of Breen, and we state our main result. In Section 2, we explain how this result may be used to study the Galois structure of rings of integers of unramified extensions of F and to analyse the structure of certain locally free classgroups. Finally, in Section 3, we apply our earlier results to the case of class invariants arising via abelian varieties, and we explain how to obtain a new description of a certain flat

ON PRIMITIVE AND REALISABLE CLASSES 3

Selmer group in some cases. (In fact, it was this last application that originally aroused our interest in Waterhouse’s question.)

Part of the work described in this paper was carried out while I was visiting the Université de Bordeaux I. I am extremely grateful to the members of the Mathematics Department there for their generous hospitality. I am also very grateful to G. Pappas and R. Schoof for useful conversations, and to L. Breen and L. Childs for helpful remarks via email. I would like to thank the referee for a number of very useful comments.

1. We begin by briefly reviewing certain aspects of the work of Childs and Magid. The reader may find further details in [CM].

Let S be any scheme, and let G be any finite, flat, commutative group scheme over S of exponent N. It is not hard to show, using Yoneda’s lemma, that the primitive elements of Pic(G^{D}) correspond to the natural homomorphisms between the group-valued functors G^{D} and H^{1}(−, 𝔾_{m}). In particular, this implies that the group of primitive classes in Pic(G^{D}) is annihilated by N.

Let P and S denote the categories of abelian presheaves and sheaves respectively for the fpqc topology on S. Childs and Magid show that there is a spectral sequence

Ext_{P}^{p}(G^{D}, H^{q}(−, 𝔾_{m})) ⇒ Ext_{S}^{p+q}(G^{D}, 𝔾_{m}).  (2)

The exact sequence of terms of low degree of (2) yields

0 → Ext_{P}^{1}(G^{D}, 𝔾_{m}) → Ext_{S}^{1}(G^{D}, 𝔾_{m}) → Hom_{P}(G^{D}, H^{1}(−, 𝔾_{m})) → Ext_{P}^{2}(G^{D}, 𝔾_{m}). (3)

Now Waterhouse (see Theorem 2 of [W]) has shown that Ext_{S}^{1}(G^{D}, 𝔾_{m}) is isomorphic to H^{1}(S, G). Hence (3) yields an exact sequence

H^{1}(S, G) → Prim(H^{1}(G^{D}, 𝔾_{m})) → Ext_{P}^{2}(G, 𝔾_{m}),  (4)

where Prim(H^{1}(G^{D}, 𝔾_{m})) denotes the group of primitive classes of H^{1}(G^{D}, 𝔾_{m}).

The first arrow of (4) is the class invariant homomorphism ψ. Hence, (4) implies that the image of ψ is contained in the group of primitive classes of H^{1}(G^{D}, 𝔾_{m}), i.e. that every realisable class is primitive. Furthermore, if Ext_{P}^{2}(G^{D}, 𝔾_{m}) = 0, then it follows from (4) that the image

4 A. AGBOOLA

of ψ consists precisely of the primitive classes, i.e. that every primitive class is realisable. (See Proposition 1.1 of [CM].)

We now give the statement of Breen’s vanishing theorem. Although the following result is not explicitly stated in [Br1] in quite this form, it is a direct consequence of the results described that paper (cf. [Br1], Théorème 3).

Theorem 1.1 (Breen). Let S be any scheme, and let G be a finite, flat commutative group scheme over S. Suppose that the composition series of every geometric fibre of G of residue characteristic 2 does not contain a factor of local-local type. Then Ext_{P}^{2}(G, 𝔾_{m}) = 0. □

Proof. We shall briefy explain how the results of [Br1] imply Theorem 1.1, and we refer the reader to [Br1] for full details. We shall follow the notation of [Br1] as closely as possible in our explanation below.

Suppose first that G is just an ordinary abelian group, and consider the following complex L.(G):

0 → ℤ[G × G] × ℤ[G × G × G] ⟶^{d_{1}} ℤ[G × G] ⟶^{d_{0}} ℤ[G] ⟶^{ε} G → 0.

Here

ε[p] = p
d_{0}[p, q] = [p + q] − [p] − [q]
d_{1}[p, q] = [p, q] − [q, p]
d_{1}[p, q, r] = [p + q, r] − [p, q + r] − [p, q] − [q, r],

and we set L_{0}(G) = ℤ[G], L_{1}(G) = ℤ[G × G], L_{2}(G) = ℤ[G × G] × ℤ[G × G × G]. Then (see [G], Exposé VII, 3.5)

H_{0}(L.(G)) = G,
H_{1}(L.(G)) = 0.

Now the above construction is functorial in G (loc. cit.). Hence given any finite, flat commutative group scheme G over S (viewed as an object of P), we may construct a complex L.(G) of abelian presheaves over S exactly as above. (We remark that Breen uses a different complex, denoted

ON PRIMITIVE AND REALISABLE CLASSES 5

by A(G) in [Br1]. We use the complex L.(G) here in order to be able to treat group schemes G of even order (cf. Remarque 3 of [Br1]).)

Now suppose that A is any abelian category which has enough injectives. Let X_{∗} be a complex in A, and let H be any object of A. Then there are two spectral sequences whose abutment is the hypercohomology of X_{∗} with values in H (see [Br2], Sections 4 and 5):

′E_{1}^{i,j} = Ext^{j}(X_{i}, H) ⟹ Ext Ext Ext^{i+j}(X_{∗}, H),
′′E_{2}^{i,j} = Ext^{i}(H_{j}(X_{∗}), H) ⟹ Ext Ext Ext^{i+j}(X_{∗}, H).

We now take A = P, X_{∗} = L.(G), and H = 𝔾_{m}. Since L.(G) is a partial free resolution of G, we have

Ext^{j}(G, 𝔾_{m}) ≃ Ext Ext Ext(L.(G), 𝔾_{m}) 0 ≤ j ≤ 2.

Next, we note that, for any scheme X, we have

Ext^{j}(ℤ[X], 𝔾_{m}) = H^{j}(X, 𝔾_{m}) = 0 (j > 0)

in P, since the global sections functor H ↦ H^{0}(X, H) is exact in P. This implies that the spectral sequence ′E_{i,j} degenerates. Hence its abutment Ext^{i}(G, 𝔾_{m}) (0 ≤ i ≤ 2) is the ith cohomology group of the complex

′E_{1}^{∗,0} : Hom(L_{0}(G), 𝔾_{m}) → Hom(L_{1}(G), 𝔾_{m}) ⟶^{f} Hom(L_{2}(G), 𝔾_{m}).

Thus, to show that Ext^{2}(G, 𝔾_{m}) = 0, we have to show that f is surjective.

For each 0 ≤ i ≤ 2, the presheaf Hom(L_{i}(G), 𝔾_{m}) is represented by a smooth group scheme over S. Hence, via the argument given on p. 345-347 of [Br1], it suffices to show that f is surjective whenever S = Spec(k), where k is an algebraically closed field, i.e. that Ext^{2}(G, 𝔾_{m}) = 0 in this case. If char(k) > 2, this follows immediately from Proposition 2 of [Br1]. On the other hand, if char(k) = 2, then Ext^{2}(ℤ/2ℤ, 𝔾_{m}) = Ext^{2}(μ_{2}, 𝔾_{m}) = 0, while Ext^{2}(α_{2}, 𝔾_{m}) ≠ 0 (see [Br3], Remark 5). Hence it follows via devissage that if the composition series of G does not have a factor of local-local type, then Ext^{2}(G, 𝔾_{m}) = 0. This completes the proof. □

The next result now follows immediately from Theorem 1.1 and the exact sequence (4).

6 A. AGBOOLA

Theorem 1.2. Let S be any scheme, and let G be a finite, flat commutative group scheme over S. Suppose that the composition series of every geometric fibre of G of residue characteristic 2 does not contain a factor of local-local type. Then every primitive class of Pic(G^{D}) is realisable. □

Remark 1.3. If S = Spec(O_{F}) (or more generally, if S = Spec(R), where R is any Dedekind domain), then Pic(G^{D}) may be identified with the locally free classgroup Cl(𝔄) of the Hopf algebra 𝔄. Theorem 1.2 then asserts that (under the given hypotheses on G), every primitive class in Cl(𝔄) is realisable. We remark that most authors have considered Waterhouse’s question in terms of classgroups of Hopf algebras.

2. Suppose now that S = Spec(O_{F}) and that G = ⊕_{i}ℤ/n_{i}ℤ is a constant group scheme over S. Then G^{D} = ⊕_{i}μ_{n_{i}}. If we set Γ = G(F), then 𝔄 = O_{F} Γ, and 𝔅 = Map(Γ, O_{F}). In this situation, the twisted forms of 𝔅 are the rings of integers in unramified Galois algebra extensions of F with Galois group Γ which are unramified at all finite primes of F (see e.g. [BT]).

Theorem 2.1. (a) We have Im(ψ) = Prim(Cl(𝔄)).

(b) Suppose that |Γ| is coprime to | Cl(O_{F})|. If either

(i) F is totally imaginary, or

(ii) F is not totally imaginary and |Γ| is odd,

then Prim(Cl(𝔄)) = 0.

If F is not totally imaginary and |Γ|is even, then 2. Prim(Cl(𝔄)) = 0.

Proof. (a) This follows from Theorem 1.2 and Remark 1.3.

(b) Let C/F be a Galois algebra extension of F with Galois group Γ, and assume that C/F is unramified at all finite places of F. Suppose either that F is totally imaginary, or that F is not totally imaginary and |Γ| is odd. Then classfield theory implies that each Wedderburn component of C is isomorphic to F. Hence the subgroup of realisable classes of Cl(𝔄) is trivial, and so it follows from Theorem 1.2 that Prim(Cl(𝔄)) is also trivial.

On the other hand, if F is not totally imaginary and |Γ| is even, then each Wedderburn component of C is isomorphic either to F or to a quadratic extension of F which is unramified at all finite places of F. This implies that, in this case, the subgroup of realisable classes of Cl(𝔄) is annihilated by 2. Hence we deduce from Theorem 1.2 that 2. Prim(Cl(𝔄)) = 0. □

ON PRIMITIVE AND REALISABLE CLASSES 7

Remark 2.2. Theorem 2.1(b) is rather striking because it gives quite fine information concerning the structure of Cl(𝔄) solely in terms of the classgroup Cl(O_{F}) of the base O_{F}. In some sense, it may be viewed as a sort of analogue of Vandiver’s conjecture for classgroups of integral group rings. We shall illustrate this point via a simple example. (It is possible to construct more complicated examples by using work of Ullom ([U]) on the fine structure of certain classgroups.)

Suppose that Γ is cyclic of prime order p > 2, and let Γ̂ be the group of characters of Γ. Write K = F (ζ_{p}), where ζ_{p} is a primitive pth root of unity, and set Δ = Gal(K/F). Let Map_{Δ}(Γ̂, Cl(O_{K})) denote the group of Δ-equivariant maps from Γ̂ to Cl(O_{K}). If M denotes the maximal O_{F} -order in F Γ, then it follows from the theory of locally free classgroups (see e.g. Chapter 1 of [T]) that Cl(M) ≃ Map_{Δ}(Γ̂, Cl(O_{K})). Let us now assume that Cl(𝔄) ≃ Cl(M). We therefore regard each element of Cl(𝔄) as being an element of Map_{Δ}(Γ̂, Cl(O_{K})).

Next, we recall that for each integer r with 0 ≤ r ≤ p − 1, there is an Adams operator Ψ_{r} on Cl(𝔄) which is defined as follows (see Chapter 9 of [T1]). Suppose that f ∈ Map_{Δ}(Γ̂, Cl(O_{K})). Then Ψ_{r}f(χ) = f(χ^{r}) for each χ ∈ Γ̂. Set

Cl(𝔄)^{(1)} := {f ∈ Cl(𝔄)|Ψ_{r}(f) = f^{r} for all r with 1 ≤ r ≤ p − 1}.

Then it is easy to see that Cl(𝔄)^{(1)} ≃ Hom_{Δ}(Γ̂, Cl(O_{K})) ⊆ Map_{Δ}(Γ̂, Cl(O_{K})).

Let D denote the p-part of Cl(O_{K}), and write ω : Δ → ℤ_{p}^{∗} for the character giving the action of Δ on the pth roots of unity in K. Then we have

Hom_{Δ}(Γ̂, Cl(O_{K})) ≃ Hom_{Δ}(Γ̂, D) ≃ D(ω),

where D(ω) denotes the ω-eigenspace for the action of Δ on D. Hence

Cl(𝔄)^{(1)} ≃ D(ω).

Now consider the following examples:

(a) Suppose that F = ℚ. Then it is a theorem of Rim (see [R] or Theorem 50.2 of [CR]) that Cl(𝔄) ≃ Cl(M). Also, it follows from Herbrand’s theorem (see [W], Proposition 6.16 and Theorem 6.17) that D(ω) = 0. Hence we deduce that Prim(Cl(𝔄)) = Cl(𝔄)^{(1)} = 0. (So in fact this gives another proof of Theorem 2.1(b) in this special case.)

8 A. AGBOOLA

(b) Suppose that F = ℚ(ζ_{p})^{+}, the maximal real subfield of K = ℚ(ζ_{p}), and that p is an irregular prime, i.e. that p divides | Cl(O_{K})|. (There are infinitely many irregular primes.) Suppose further that Vandiver’s conjecture holds, i.e. that p does not divide | Cl(O_{F})|. Then Theorem 2.1(b) implies that Prim(Cl(𝔄)) = 0.

On the other hand, via an argument virtually identical to that given in Theorem 50.2 of [CR], it may be shown that Cl(𝔄) ≃ Cl(M). Hence, since Gal(K/F) is generated by complex conjugation, we have that Cl(𝔄)^{(1)} ≃ D(ω) = D^{−}, where D^{−} denotes the minus-eigenspace for the action of complex conjugation on D. Since D is non-trivial, and p does not divide | Cl(O_{F})|, it follows that D^{−} is non-trivial.

Hence, if Vandiver’s conjecture is true, then for infinitely many p, we have Prim(Cl(𝔄)) = 0, whilst Cl(𝔄)^{(1)} ≠ 0. □

Remark 2.3. Let F be a real quadratic field of odd discriminant and odd classnumber. Then F has at most one quadratic extension L that is unramified at all finite primes. This extension L exists if and only if the norm of a fundamental unit of F is +1 (cf. [FT], Chapter V, 1.14). Hence it follows from a theorem of Brinkhuis (cf. [Bri], Theorem 2.2) that if H denotes Gal(L/F), then O_{L} is a free O_{F} H-module.

Now suppose that Γ is any finite abelian 2-group. Let C/F be any Galois algebra extension of F with Galois group Γ, and assume that C/F is unramified at all finite places of F. Then each Wedderburn component of C is isomorphic either to F or to L. Hence the ring of integers of C is a free O_{F} Γ-module. This implies that the subgroup of realisable classes of O_{F} Γ is trivial, and so it follows from Theorem 1.2 that Prim(Cl(O_{F} Γ)) = 0.

It would be interesting to know whether a similar strengthening of Theorem 2.1(b) holds for other fields F which are not totally imaginary and which have odd classnumber. □

3. We shall now describe the application that led to our interest in these problems. For further details, the reader may consult [A1], [A2], and [AT].

Let A/F be an abelian variety with everywhere good reduction, and let A/O_{F} denote its Néron model. Write A^{D}/O_{F} for the dual abelian scheme of A/O_{F}. Fix a prime p, and let A_{p^{n}} denote the O_{F} -group scheme of p^{n}-torsion on A/O_{F}. Then A_{p^{n}} is a finite, flat, commutative group scheme, and its Cartier dual is A_{p^{n}}^{D}, the O_{F} -group scheme of p^{n}-torsion on A^{D}/O_{F}. We therefore have a class invariant homomorphism

ψ_{n} : H^{1}(O_{F} , A_{p^{n}}) ⟶ Prim(Pic(A_{p^{n}}^{D}))

as described in Section 1. The study of the class invariant homorphism in this context was first introduced by M. J. Taylor in [T2].

The natural inclusion map A_{p^{n}}^{D} ↪ A_{p^{n+1}}^{D} induces natural homomorphisms H^{1}(O_{F} , A_{p^{n+1}}) → H^{1}(O_{F} , A_{p^{n}}) and Prim(Pic(A_{p^{n+1}}^{D})) → Prim(Pic(A_{p^{n}}^{D})). It may be shown that the following diagram is commutative:

H^{1}(O_{F} , A_{p^{n+1}}) ⟶^{ψ_{n+1}} Prim(Pic(A_{p^{n+1}}^{D}))
↓ ↓
H^{1}(O_{F} , A_{p^{n}}) ⟶^{ψ_{n}} Prim(Pic(A_{p^{n}}^{D})).

Passing to inverse limits yields a homomorphism

ψ_{∞} : \underleftarrow{lim} H^{1}(O_{F} , A_{p^{n}}) ⟶ \underleftarrow{lim} Prim(Pic(A_{p^{n}}^{D})).

We now observe that for all odd primes p (and for all positive integers n), the group schemes A_{p^{n}} satisfy the hypotheses of Theorem 1.2. Furthermore, this is also the case when p = 2 provided that A has ordinary reduction at all places dividing 2. Hence we have the following result.

Theorem 3.1. The homomorphism ψ_{∞} is surjective whenever p is odd. If A has ordinary reduction at all places dividing 2, then ψ_{∞} is also surjective when p = 2.

Remarks 3.2. (1) Theorem 3.1 gives an affirmative answer to Question 1 (ii) of [A1] if in addition the p-primary part of the Tate-Shafarevich group of A/F is assumed to be finite.

(2) Suppose that the kernel of ψ_{∞} is finite. (The author believes that this is the case for all abelian varieties A/F with everywhere good reduction, and for all primes p.) Then Theorem 3.1 implies that there is an isomorphism

\{\underleftarrow{lim} H^{1}(O_{F} , A_{p^{n}})\}/\{a finite group\} ≃ \underleftarrow{lim} Prim(Pic(A_{p^{n}}^{D})).

Hence we obtain a new description of the flat Selmer group \underleftarrow{lim} H^{1}(O_{F} , A_{p^{n}}) (modulo a finite subgroup) in terms of the Galois structure of twisted forms of Hopf orders.

10 A. AGBOOLA

It is shown in [AT] that, (subject to certain technical hypotheses), if A/F is a CM elliptic curve and p is an odd prime of ordinary reduction, then the kernel of ψ_{∞} is indeed finite.

(3) Observe that the kernel of ψ_{∞} is finite if and only if the order of the kernel of ψ_{n} is bounded independently of n. The exact sequence (3) implies that this is equivalent to the assertion that the order of the groupExt_{P}^{1}(A_{p^{n}}^{D}, 𝔾_{m}) is bounded independently of n. It would be very interesting to have an answer to the following question. Suppose that (G_{n})_{n≥0} is any pdivisible group scheme over O_{F}. Under what conditions is the order of the group Ext_{P}^{1}(G_{n}, 𝔾_{m}) bounded independently of n?

I am gateful to R. Schoof for drawing my attention to the following example. Let (ℤ/p^{n}ℤ)_{n≥0} denote the constant p-divisible group scheme over O_{F}, and suppose that p ∤ | Cl(O_{F})|. Then, for each n, the p-primary part of H^{1}(ℤ/p^{n}ℤ, 𝔾_{m}) is trivial, and so it follows that

Ext_{P}^{1}(ℤ/p^{n}ℤ, 𝔾_{m}) = Ext_{S}^{1}(ℤ/p^{n}ℤ, 𝔾_{m}).

Now Theorem 2 of [W] implies that

Ext_{S}^{1}(ℤ/p^{n}ℤ, 𝔾_{m}) ≃ H^{1}(O_{F} , μ_{p^{n}}),

and this last group is of exponent p^{n}. Hence we deduce that the order of Ext_{P}^{1}(ℤ/p^{n}ℤ, 𝔾_{m}) increases with n.

This suggests that, in general, perhaps the exponent of Ext_{P}^{1}(G_{n}, 𝔾_{m}) remains bounded if the p-divisible group (G_{n})_{n≥0} does not have a subquotient of twisted constant type. We remark that if A/O_{F} is any abelian scheme, then the p-divisible group scheme (A_{p^{n}})_{n≥0} does not have a subquotient of twisted constant type (see [MW], Chapter 3, Section 7).

References

[A1] A. Agboola, A geometric description of the class invariant homomorphism, Journal de la Théorie des Nombres de Bordeaux 6 (1994), 273-280.

[A2] A. Agboola, Iwasawa theory of elliptic curves and Galois module structure, Duke Math. J. 71 (1993), 441-462.

[AT] A. Agboola, M. J. Taylor, Class invariants of Mordell-Weil groups, Crelle 447 (1994), 23-61.

[Br1] L. Breen, Une théorème d’annulation pour certains Ext^{i} de faisceaux abéliens, Ann. scient. Éc. Norm. Sup. 5 (1975), 339-352.

[Br2] L. Breen, Extensions of abelian sheaves and Eilenberg-Mac Lane algebras, Invent. Math. 9 (1969), 1249-1253.

[Br3] L. Breen, On a non-trivial higher extension of representable abelian sheaves, AMS Bull. 75 (1969), 1249- [Bri] J. Brinkhuis, On the Galois module structure over CM-fields, Manuscripta Math. 75 (1992), 333-347.

[By1] N. Byott, Picard invariants of Galois algebras over a class of Hopf orders, preprint (1997).

[By2] N. Byott, Tame realisable classes over Hopf orders, J. Algebra 210 (1998), 284-316.

[BT] N. Byott, M. J. Taylor, Hopf orders and Galois module structure, In: Group rings and classgroups, K. W. Roggenkamp, M. J. Taylor (eds), Birkhauser, 1992, pp. 153-210.

[C] S. Caenepeel, Kummer theory for monogenic Larson orders, In: Rings, Hopf algebras and Brauer groups. Lecture Notes Pure Appl. Math 197, S. Caenepeel (ed) et. al., Marcel Dekker, 1998, pp. 85-102.

[C1] L. N. Childs, The group of unramified Kummer extensions of prime degree, Proc. London Math. Soc. 35 (1977), 407-422.

[C2] L. N. Childs, Abelian Galois extensions of rings containing roots of unity, Ill. J. Math. 15 (1971), 273-280.

[CM] L. Childs, A. Magid, The Picard invariant of a principal homogeneous space, J. Pure and Appl. Alg. 4 (1974), 273-286.

[CP] M. P. Cox Paul, The image of the Picard invariant map for Hopf Galois extensions, Ph.D. Thesis, SUNY Albany (1994).

[CR] C. W. Curtis and I. Reiner, Methods of Representation Theory, Volume II, Wiley, 1987.

[FT] A. Fröhlich and M. J. Taylor, Algebraic Number Theory, Cambridge University Press, 1991.

[G] A. Grothendieck et al., Groupes de monodromie en géometrie algébrique (SGA 7), Lecture Notes in Mathematics 288 (1972), Springer Verlag.

[MW] B. Mazur and A. Wiles, Class fields of abelian extensions of ℚ, Invent. Math. 76 (1984), 179-330.

[M] L. R. McCulloh, Galois module structure of abelian extensions, Crelle 375/376 (1987), 259-306.

[R] D. S. Rim, Modules over finite groups, Ann. Math. 69 (1959), 700-712.

[T1] M. J. Taylor, Classgroups of Group Rings, Cambridge University Press, 1984.

[T2] M. J. Taylor, Mordell-Weil groups and the Galois module structure of rings of integers, Ill. J. Math. 32 (1988), 428-452.

[U] S. V. Ullom, Fine structure of class groups of cyclic p-groups, J. Algebra 49 (1977), 112-124.

[W] L. C. Washington, Introduction to Cyclotomic Fields, Springer, 1997.

[Wa] W. Waterhouse, Principal homogeneous spaces and group scheme extensions, AMS Transactions 153 (1971), 181-189.

Department of Mathematics, University of California, Santa Barbara, CA 93106, USA E-mail address: agboola@math.ucsb.edu