ON TWISTED FORMS AND RELATIVE ALGEBRAIC K-THEORY

A. AGBOOLA AND D. BURNS

Abstract. This paper introduces a new approach to the study of certain aspects of Galois module theory by combining ideas arising from the study of the Galois structure of torsors of finite group schemes with techniques coming from relative algebraic K-theory.

1. Introduction

Let R be a Dedekind domain with field of fractions F, and suppose that G → Spec(R) is a finite, flat, commutative group scheme. In recent years, there has been some interest in the study of Galois structure invariants attached to torsors of G. The original motivation for this work arose from the study of the Galois module structure of rings of integers: if X → Spec(R) is a G-torsor, then we may write X = Spec(ℭ), and in many cases, the algebra ℭ may be viewed as being an order in the ring of integers of some (in general wildly ramified) field extension of F. The case in which G is a torsion subgroup scheme of an abelian variety is particularly interesting: here the corresponding torsors are obtained by dividing points in the Mordell-Weil group of the abelian variety, and this enables one to relate questions concerning the Galois structure of rings of integers to the arithmetic of abelian varieties. This approach was first introduced by M. J. Taylor (see [31], [30], [13]), and has since been developed in greater generality by several authors (see for example [3], [6], [12], [26]).

The main goal of this paper is to show that combining ideas arising from the study of Galois structure invariants attached to torsors of G with techniques from relative algebraic K-theory yields a natural refinement and reinterpretation of several different aspects of Galois module theory. At the same time, we shall also see that this approach also gives fresh insight into a number of old results. We remark that techniques involving relative algebraic K-theory have already played an essential role in the formulation of an equivariant version of the Tamagawa Number Conjecture of Bloch and Kato, and in the description of its consequences for Galois module theory (see [10], [11]).

We now give an outline of the contents of this paper. In §2, we recall the definition of a fibre product category, and we introduce the notion of a generalised twisted form. This is a generalisation of the usual notion of a twisted form or principal homogeneous space of a Hopf algebra (see Remark 2.4). An example of such an object may be given as follows. Suppose that 𝔄 is a finitely generated R-algebra which spans a semisimple F -algebra A, and let E be an extension of F. Then

Date: Final version, March 23, 2005.

2 A. AGBOOLA AND D. BURNS

a generalised E-twisted 𝔄-form is given by a triple [M, N; ι], where M and N are finitely generated, locally free 𝔄-modules, and ι : M ⊗_{R} E ⟶^{∼} N ⊗_{R} E is an isomorphism of A ⊗_{F} E-modules. We discuss two particular examples of generalised twisted forms that provide motivation for our later work. The first example arises via finite Galois extensions of number fields, while the second is constructed using torsors of finite, flat, commutative group schemes.

It follows from the definition that generalised twisted forms may be classified by appropriate relative algebraic K-groups. (For example, the generalised twisted forms [M, N; ι] described above yield classes in a relative algebraic K-group that we denote by K_{0}(𝔄, E).) In §3, we recall the basic facts concerning relative algebraic K-theory that we use, and we then give a Fröhlich-type ‘Hom-description’ of relative algebraic K-groups of the form K_{0}(𝔄, E). This description may itself be of some independent interest.

Suppose now that F is a number field. In §4, we define the notion of a metric μ on a finitely generated, locally free 𝔄-module M. Following [17], we then define an arithmetic classgroup AC(𝔄) which classifies metrised 𝔄-modules (M, μ). There is a natural homomorphism ∂_{𝔄,F^{c}} from the relative algebraic K-group K_{0}(𝔄, F^{c}) to AC(𝔄) (where F^{c} denotes an algebraic closure of F). Proposition 4.11 implies that if (M, N; ι_{1}) is a generalised F^{c}-twisted 𝔄-form with M and N both locally free 𝔄-modules, then for any metric μ on N, the image of the class (M, N; ι_{1}) under ∂_{𝔄,F^{c}} measures the difference between the elements of AC(𝔄) which are associated to (M, ι_{1}^{∗}(μ)) and (N, μ), where ι_{1}^{∗}(μ) is the metric on M which is obtained from μ via pullback by ι_{1}. This result is of interest since, in many of the examples that occur in Galois module theory, canonical metrics on locally free 𝔄-modules can be shown to be equal to, for instance, the pullback of the trivial metric on 𝔄 by a natural homomorphism.

In §5, we give two applications of the approach described in this paper to the study of Galois structure of rings of integers in tamely ramified extensions. For the first application, we combine our work with results of Bley and the second-named author from [9] and we obtain a natural strengthening of a result of Chinburg, Pappas and Taylor concerning the Hermitian Galois structure of rings of integers. For the second application, we describe how certain natural torsion Galois module invariants first introduced by S. Chase (see [14]) are related to certain ‘equivariant discriminants’ that take values in a relative algebraic K-group.

The remainder of the paper is devoted to describing how the methods we develop may be applied to the study of the Galois structure of torsors in several different contexts. In §6, we discuss reduced resolvends associated to torsors. This notion was first introduced by L. McCulloh in a different setting (see [24]). We use a different approach from McCulloh’s (see [3]), and we give a new characterisation of reduced resolvends as being primitive elements of a certain algebraic group.

We then introduce a natural refinement of the class invariant homomorphism first studied by W. Waterhouse in [33]. Our refined homomorphism takes values in a suitable relative algebraic Kgroup. We show that the homomorphism is injective, and that its image admits a precise functorial description in almost all cases of interest. This extends a result of the first-named author in [2].

ON TWISTED FORMS AND RELATIVE ALGEBRAIC K-THEORY 3

Finally, we explain how the approach described in this paper enables one to refine McCulloh’s results on realisable classes of rings of integers of tame extensions by considering invariants that lie an appropriate relative algebraic K-group, rather than in a locally free classgroup. This gives a relationship between McCulloh’s realisability results, and the work of Chase concerning certain torsion Galois modules. It also yields realisability results in certain arithmetic classgroups. We find that, in general, the collection of realisable classes in the relative algebraic K-group does not form a group.

Acknowledgements. We would like to thank the Mathematics Department of Harvard University for its hospitality while a part of this work was carried out. We are also very grateful to the referee of this paper for a number of very helpful remarks. The research of the first-named author was partially supported by NSF grants DMS-9700937 and DMS-0070449.

Notation. Throughout this paper, all modules are left modules, unless explicitly stated otherwise.

For any field L, we write L^{c} for an algebraic closure of L, and we set Ω_{L} := Gal(L^{c}/L). If L is either a number field or a local field, then O_{L} will denote its ring of integers.

If L is a number field, we write S_{f}(L) (resp. S_{∞}(L)) for the set of finite (resp. infinite) places of L. If v is any place of L, then we write L_{v} for the local completion of L at v. For any O_{L}-module P, we shall usually set P_{v} := P ⊗_{O_{L}} O_{L_{v}}.

If S_{1} ⊂ S_{2} are rings, and if A is any S_{1}-algebra, then we often write A_{S_{2}} for A ⊗_{S_{1}} S_{2}. We also often use similar notation M_{S_{2}} := M ⊗_{S_{1}} S_{2} for any S_{1}-module M.

The symbol ζ(A) denotes the centre of A.

For any isomorphic A-modules M and N, we write Is_{A}(M, N) for the set of A-equivariant isomorphisms M ⟶^{∼} N.

Throughout this paper, R denotes a Dedekind domain with field of fractions F. We write R^{c} for the integral closure of R in F^{c}. We also allow the possibility that R = F, in which case of course R^{c} = F^{c}.

Finally, we remark that if G and H are two groups, then we sometimes use the notation g × h (rather than (g, h)) for an element of the product G × H. □

2. Twisted Forms and Fibre Product Categories

2.1. Fibre product categories. Let F_{i} : P_{i} → P_{3}, i ∈ {1, 2} be functors between categories and consider the fibre product category P_{4} := P_{1} ×_{P_{3}} P_{2}. Explicitly, P_{4} is the category with objects (L_{1}, L_{2}; λ) where L_{i} is an object of P_{i} for i ∈ {1, 2} and λ : F_{1}(L_{1}) ⟶^{∼} F_{2}(L_{2}) is an isomorphism in P_{3}. Morphisms α : (L_{1}, L_{2}; λ) → (L′_{1}, L′_{2}; λ′) in P_{4} are pairs α = (α_{1}, α_{2}) with α_{i} ∈ Hom_{P_{i}}(L_{i}, L′_{i})

4 A. AGBOOLA AND D. BURNS

so that the diagram

F_{1}(L_{1}) ⟶^{F_{1}(α_{1})} F_{1}(L′_{1})
↓_{λ}  ↓_{λ′}
F_{2}(L_{2}) ⟶^{F_{2}(α_{2})} F_{2}(L′_{2})

in P_{3} commutes. Such a morphism α is an isomorphism in P_{4} if and only if the morphisms α_{1} and α_{2} are both isomorphisms. (The reader may consult [7, Chapter VII, §3] for more details concerning fibre product categories.)

2.2. Generalised twisted forms. For any ring Λ we write P_{Λ} for the category of finitely generated projective Λ-modules.

Recall that R is a Dedekind domain, with fraction field F. Suppose that 𝔄 is a finitely generated R-algebra which spans a semisimple F -algebra A. For any extension Λ of R we write P_{𝔄} ×_{Λ} P_{𝔄} for the fibre product category with F_{1} and F_{2} both equal to the scalar extension functor − ⊗_{R} Λ from P_{𝔄} to P_{𝔄⊗_{R}Λ}.

Definition 2.1. We shall refer to an object of the category P_{𝔄} ×_{Λ} P_{𝔄} as a generalised Λ-twisted 𝔄-form. We abbreviate this terminology to generalised twisted form, or even just twisted form, when both 𝔄 and Λ are clear from the context. □

Example 2.2. (Classical) Suppose that F is a number field, and let L/F be any finite extension. We write N for the normal closure of L over F and we let Σ_{F} (L, N) denote the set of distinct F -embeddings of L into N. Set H_{F}^{L} := ∏_{Σ_{F} (L,N)} Z, and write Aut(L/F) for the group of F - automorphisms of L. Then there is a canonical N[Aut(L/F)]-equivariant isomorphism

π_{F}^{L,N} : L ⊗_{F} N ⟶^{∼} H_{F}^{L} ⊗_{Z} N

which is given by mapping each primitive tensor l ⊗ n to the vector (σ(l)n)_{σ}.

Now suppose that S is any subring of F and let G be a subgroup of Aut(L/F). Then for any projective S[G]-submodule L of L such that L ⊗_{S} F = L, the triple (L, H_{F}^{L} ⊗_{Z} S; π_{F}^{L,N}) is a generalised N-twisted S[G]-form.

In special cases there are canonical choices of L. For example, if S = F, then one can take L = L. In addition, if L is a tamely ramified Galois extension of a field K with Gal(L/K) = G and S = O_{F} for any subfield F of K, then one can take L to be any G-stable fractional ideal of O_{L} (see [32]). □

Example 2.3. (Geometrical) Let R be any Dedekind domain, with field of fractions F. (We allow R = F.) Suppose that G → Spec(R) is a finite, flat, commutative group scheme, and let G^{∗} denote its Cartier dual. Write 𝔄 := O_{G^{∗}} and A := 𝔄 ⊗_{R} F. It is shown by Waterhouse in [33, Sections 1 and 2] (see also [22, éxposé VII] or [25]) that there is a canonical isomorphism of groups

H^{1}(Spec(R), G) ≃ Ext^{1}(G^{∗}, G_{m}).

ON TWISTED FORMS AND RELATIVE ALGEBRAIC K-THEORY 5

This implies that given any G-torsor π : X → Spec(R), we can associate to it a canonical commutative extension

1 → G_{m} → G(π) → G^{∗} → 1.  (1)

The scheme G(π) is a G_{m}-torsor over G^{∗}, and we write L_{π} for its associated G^{∗}-bundle.

Let π_{0} : G → Spec(R) denote the trivial G-torsor. Then, over Spec(R^{c}), the G-torsors π and π_{0} become isomorphic, i.e. there is an isomorphism X ⊗_{R} R^{c} ⟶^{∼} G ⊗_{R} R^{c} of schemes with G-action. Hence, via the functoriality of the construction in [33, Section 2], we obtain an 𝔄⊗_{R} R^{c}-equivariant isomorphism

ξ_{π} : L_{π} ⊗_{R} R^{c} ⟶^{∼} 𝔄 ⊗_{R} R^{c}.  (2)

We refer to ξ_{π} as a splitting isomorphism associated to π. Then the triple (L_{π}, 𝔄; ξ_{π}) is a generalised R^{c}-twisted 𝔄-form. □

Remark 2.4. The line bundle L_{π} in Example 2.3 may be described explicitly in the following way (see [33, Section 4]). For any G-torsor π : X → Spec(R) as above, the structure sheaf O_{X} is an O_{G}-comodule, and so it is also an 𝔄-module (see [16, Proposition 1.3]). As an 𝔄-module, O_{X} is locally free of rank one, and so it gives a line bundle M_{π} on G^{∗}. Then it may be shown that L_{π} = M_{π} ⊗ M_{π_{0}}^{−1}. The line bundle M_{π} is a principal homogeneous space, or twisted form of the Hopf algebra 𝔄 in the sense of, for example [31] or [13], and this is the motivation for Definition 2.1 above. □

3. Relative algebraic K-theory

3.1. Basic theory. Let Λ be any extension of R, and let 𝔄 be a finitely generated R-algebra which spans an F -algebra A. In this subsection, we shall recall some basic facts concerning the K-theory of categories of the form P_{𝔄} ×_{Λ} P_{𝔄}. Further details of these results may be found in [29, Chapter 15] and [19, Section 40B].

Definition 3.1. A short exact sequence of objects in P_{𝔄} ×_{Λ} P_{𝔄} is a sequence

0 → (L_{1}, L_{2}; λ) ⟶^{(α_{1},α_{2})} (L′_{1}, L′_{2}; λ′) ⟶^{(α′_{1},α′_{2})} (L′′_{1}, L′′_{2}; λ′′) → 0  (3)

such that (α_{1}, α_{2}) and (α′_{1}, α′_{2}) are morphisms in P_{𝔄} ×_{Λ} P_{𝔄}, and such that

0 → L_{1} ⟶^{α_{1}} L′_{1} ⟶^{α′_{1}} L′′_{1} → 0 and 0 → L_{2} ⟶^{α_{2}} L′_{1} ⟶^{α′_{2}} L′′_{2} → 0

are exact sequences of R-modules. □

Definition 3.2. We write [L_{1}, L_{2}; λ] for the isomorphism class of an object (L_{1}, L_{2}; λ) of P_{𝔄}×_{Λ}P_{𝔄}. We define the relative algebraic K-group K_{0}(𝔄, Λ) to be the abelian group with generators the isomorphism classes of objects of P_{𝔄} ×_{Λ} P_{𝔄} and relations

(i) [L′_{1}, L′_{2}; λ′] = [L_{1}, L_{2}; λ] + [L′′_{1}, L′′_{2}; λ′′] for each exact sequence (3);

(ii) [M_{1}, M_{2}; η ∘ η′] = [M_{1}, M_{3}; η′] + [M_{3}, M_{2}; η].

6 A. AGBOOLA AND D. BURNS

It may be shown that every element of K_{0}(𝔄, Λ) is of the form [L_{1}, L_{2}; λ] (see [29, Lemma 15.6]). However, the natural map from the set of isomorphism classes of objects of P_{𝔄} ×_{Λ} P_{𝔄} to K_{0}(𝔄, Λ) is very far from being injective (for example, the image of the element [𝔄^{n}, 𝔄^{n}; 1_{𝔄_{Λ}}] under this map is zero for every positive integer n). □

For any ring Λ we write K_{0}T (Λ) for the Grothendieck group of the category of Λ-modules which are both finite and of finite projective dimension. Then there are natural isomorphisms

K_{0}(𝔄, F) ⟶^{∼} K_{0}T (𝔄) ⟶^{∼} ⨁_{v} K_{0}T (𝔄_{v})  (4)

where v runs over all finite places of R (see the discussion following (49.12) in [19]). The first isomorphism in (4) is defined in the following way. If M is any 𝔄-module which is both finite and of finite projective dimension, then (since R is a Dedekind domain), there exists an exact sequence of 𝔄-modules of the form

0 → O ⟶^{φ} Q → M → 0

in which P and Q are both finitely generated and projective. The first isomorphism of (4) sends (P, φ, Q) to the class of M.

For any ring S we write K_{i}(S) (i = 0, 1) for the algebraic K-group in degree i of the exact category P_{S}. We recall that if R → Λ is any homomorphism of rings, then there is a long exact sequence of relative algebraic K-theory

K_{1}(𝔄) ⟶^{∂_{𝔄,Λ}^{2}} K_{1}(𝔄_{Λ}) ⟶^{∂_{𝔄,Λ}^{1}} K_{0}(𝔄, Λ) ⟶^{∂_{𝔄,Λ}^{0}} K_{0}(𝔄) → K_{0}(𝔄_{Λ})  (5)

(see [29, Theorem 15.5]). Here the homomorphism ∂_{𝔄,Λ}^{2} is the natural scalar extension morphism. The homomorphism ∂_{𝔄,Λ}^{0} is defined by

∂_{𝔄,Λ}^{0}([L_{1}, L_{2}; λ]) = [L_{1}] − [L_{2}].

In order to describe ∂_{𝔄,Λ}^{1}, we first recall that K_{1}(A_{Λ}) is generated by elements of the form (V, ϕ), where V is a finitely generated free 𝔄_{Λ}-module and ϕ ∈ Is_{𝔄_{Λ}}(V, V). If T is any projective 𝔄submodule of V such that T ⊗_{R} Λ = V, then ∂_{𝔄,Λ}^{1} is defined by setting

∂_{𝔄,Λ}^{1}((V, ϕ)) = [T, T; ϕ].

This definition is independent of the choice of T.

Remark 3.3. Let E be any extension of F. When A_{E} is a semisimple algebra, it is often convenient to compute in K_{1}(A_{E}) by means of the injective ‘reduced norm’ map

nrd_{A_{E}} : K_{1}(A_{E}) → ζ(A_{E})^{×}.

This map sends the element (V, ϕ) to the reduced norm of ϕ, considered as an element of the semisimple E-algebra End_{A_{E}}(V).

If 𝔄 is commutative, then the determinant functors over A_{E} and 𝔄 combine to induce canonical isomorphisms of K_{1}(A_{E})/ im(∂_{𝔄,E}^{2}) with A_{E}^{×}/𝔄^{×} and of ker[K_{0}(𝔄) ⟶ K_{0}(A_{E})] with Pic(𝔄).

ON TWISTED FORMS AND RELATIVE ALGEBRAIC K-THEORY 7

Hence the exact sequence (5) implies that in this case, the group K_{0}(𝔄, E) may be identified with the multiplicative group of invertible 𝔄-modules in A_{E}. □

Proposition 3.4. If E is any extension of F, then the following sequence is exact;

0 → K_{1}(A) ⟶^{f_{1}} K_{1}(A_{E}) ⊕ K_{0}(𝔄, F) ⟶^{f_{2}} K_{0}(𝔄, E) → 0.  (6)

Here f_{1}(x) = (∂_{A,E}^{2}(x), −∂_{𝔄,F}^{1}(x)), and f_{2}(y_{1}, y_{2}) = ∂_{𝔄,E}^{1}(y_{1})+ι_{𝔄,E}(y_{2}), where ι_{𝔄,E} : K_{0}(𝔄, F) → K_{0}(𝔄, E) is the natural scalar extension morphism.

Proof. We first observe that f_{1} is injective because the scalar extension map ∂_{A,E}^{2} is injective. Next, we note that upon comparing the exact sequences (5) with Λ = E and Λ = F, one obtains a commutative diagram:

K_{1}(𝔄) ⟶^{f_{3}} K_{1}(A_{E}) ⟶^{∂_{𝔄,E}^{1}} K_{0}(𝔄, E) ⟶^{f_{4}} K_{0}(𝔄) ⟶^{f_{5}} K_{0}(A_{E})
∥  ^{∂_{A,E}^{2}} ↑  ^{ι_{𝔄,E}} ↑  ∥  ^{ι_{A,E}} ↑
K_{1}(𝔄) ⟶^{f_{6}} K_{1}(A) ⟶^{∂_{𝔄,F}^{1}} K_{0}(𝔄, F) ⟶^{f_{7}} K_{0}(𝔄) ⟶ K_{0}(A).

In this diagram, the morphisms ι_{𝔄,E} and ι_{A,E} are the natural scalar extension morphisms, and are therefore injective. (We remark that in fact the injectivity of ι_{𝔄,E} follows from the injectivity of ∂_{A,E}^{2}, as may be shown by an easy diagram-chasing argument.)

We now show that f_{2} is surjective. Suppose that z ∈ K_{0}(𝔄, E). Then, as f_{5}(f_{4}(z)) = 0 and ι_{A,E} is injective, we may choose z_{1} ∈ K_{0}(𝔄, F) such that f_{7}(z_{1}) = f_{4}(z). We then have that z − ι_{𝔄,E}(z_{1}) = ∂_{𝔄,E}^{1}(z_{2}) for some z_{2} ∈ K_{1}(A_{E}). It is easy to check that f_{2}(z_{2}, z_{1}) = z, and so it follows that f_{2} is surjective.

It now remains to show that Ker(f_{2}) = Im(f_{1}). Suppose that y_{1} ∈ K_{1}(A_{E}) and y_{2} ∈ K_{0}(𝔄, F) are such that

∂_{𝔄,E}^{1}(y_{1}) = ι_{𝔄,E}(y_{2}) = y ∈ K_{0}(𝔄, E),

say. We have to show that there exists y′ ∈ K_{1}(A) such that ∂_{A,E}^{2}(y′) = y_{1} and ∂_{𝔄,F}^{1}(y′) = y_{2}.

Now f_{5}(y) = f_{4}(∂_{𝔄,E}^{1}(y_{1})) = 0, and so it follows that f_{7}(y_{2}) = 0. Hence there exists y_{3} ∈ K_{1}(A) such that ∂_{𝔄,F}^{1}(y_{3}) = y_{2}. We now deduce that f_{3}(y_{4}) = ∂_{A,E}^{2}(y_{3}) − y_{1} for some y_{4} ∈ K_{1}(𝔄). Set y′ = y_{3} − f_{6}(y_{4}). It is easy to show that ∂_{A,E}^{2}(y′) = y_{1} and ∂_{𝔄,F}^{1}(y′) = y_{2}. This completes the proof. □

3.2. Hom-descriptions. Recall that R is a Dedekind domain with field of fractions F. Suppose now that F is a number field, and let Γ be a finite group upon which Ω_{F} acts (possibly trivially). Let 𝔄 be a finitely generated R-subalgebra of F^{c}[Γ] such that there is an equality

𝔄 ⊗_{R} F^{c} = F^{c}[Γ].  (7)

Under this condition, for any extension E of F, we shall give a description of K_{0}(𝔄, E) in terms of idelic-valued functions on the ring R_{Γ} of F^{c}-valued characters of Γ. This description is modeled

8 A. AGBOOLA AND D. BURNS

on the ‘Hom-descriptions’ of class groups introduced by Fröhlich (cf. for example [20, Chapter II]). It will be useful in later sections, and is itself perhaps also of some independent interest.

Let Γ̂ denote the set of F^{c}-valued characters of Γ. The group Ω_{F} acts on R_{Γ} according to the following rule: if χ ∈ Γ̂ and ω ∈ Ω_{F}, then for each γ ∈ Γ one has (ω ∘ χ)(γ) = ω(χ(ω^{−1}(γ))).

We fix an embedding of F^{c} into E^{c} and we view each element of Γ̂ as taking values in E^{c}. For each element a of GL_{n}(A_{E}) we define an element Det(a) of Hom(R_{Γ}, (E^{c})^{×}) in the following way: if T is a representation over F^{c} which has character ϕ, then

Det(a)(ϕ) := det(T (a)).

This definition depends only upon the character ϕ, and not upon the choice of representation T.

We write J_{f}(F^{c}) for the group of finite ideles of the field F^{c}, and we view F^{×} as being a subgroup of J_{f}(F^{c}) via the natural diagonal embedding. If a is any element of GL_{n}(A ⊗_{F} J_{f}(E)), then the above approach allows one to define an element Det(a) of Hom(R_{Γ}, J_{f}(F^{c})). We set

U_{f}(𝔄) := ∏_{v∈S_{f} (F)} 𝔄_{v}^{×} ⊂ (A ⊗_{F} J_{f}(F))^{×}.

We then define a homomorphism

Δ_{𝔄,E} : Det(A^{×}) → \frac{Hom(R_{Γ}, J_{f}(F^{c}))^{Ω_{F}}}{Det(U_{f}(𝔄))} × Det(A_{E}^{×});  θ ↦ (θ, θ^{−1}).  (8)

Theorem 3.5. Under the above conditions, there is a natural isomorphism

h_{𝔄,E} : K_{0}(𝔄, E) ⟶^{∼} Cok(Δ_{𝔄,E}).  (9)

Proof. From (4) and the isomorphisms of [20, Chapter II, (2.3)], it follows that there is an isomorphism

h_{1} : K_{0}(𝔄, F) ⟶^{∼} \frac{Hom(R_{Γ}, J_{f}(F^{c}))^{Ω_{F}}}{Det(U_{f}(𝔄))} .

Also, the isomorphisms K_{1}(A_{Λ}) ≅ Det(A_{Λ}^{×}) of [loc. cit., Chapter II, Lemma 1.2 and Lemma 1.6] for Λ ∈ {E, F } induce an isomorphism

h_{2} : \frac{K_{1}(A_{E})}{Im(∂_{𝔄,E}^{2})} ⟶^{∼} \frac{Det(A_{E}^{×})}{Det(A^{×})}.

It follows that there is a natural isomorphism

Coker(f_{1}) ≃ Coker(Δ_{𝔄,E}),

where f_{1} is defined in the statement of Theorem 3.4. The desired result now follows from the fact that Theorem 3.4 implies that there is a natural isomorphism

Coker(f_{1}) ⟶^{∼} K_{0}(𝔄, E).

□

ON TWISTED FORMS AND RELATIVE ALGEBRAIC K-THEORY 9

Remark 3.6. An explicit description of the isomorphism h_{𝔄,E} may be given as follows. Suppose then that [X, Y; ξ] is an element of K_{0}(𝔄, E) for which X_{F} and Y_{F} are free A-modules of the same rank (by [28, Lemma 15.6] any element of K_{0}(𝔄, E) is of this shape). Then for any choice of isomorphism θ ∈ Is_{A}(X_{F} , Y_{F}) one has

[X, Y; ξ] = [X, Y; θ_{E}] + [Y, Y; ξ ∘ θ_{E}^{−1}]
= [X, Y; θ_{E}] + ∂_{𝔄,E}^{1}((Y_{E}, ξ ∘ θ_{E}^{−1}))

in K_{0}(𝔄, E). The element h_{𝔄,E}([X, Y; ξ]) of Cok(Δ_{𝔄,E}) is then represented by the pair (h_{1}([X, Y; θ]), h_{2,E}((Y_{E}, ξ ∘ θ_{E}^{−1}))). □

Definition 3.7. A finitely generated 𝔄-module M is said to be locally free if M_{v} is a free 𝔄_{v}- module for all places v in S_{f}(F). It follows easily from this definition that if M is a locally free 𝔄-module, then it is projective and the associated A-module M_{F} is free. □

Remark 3.8. If [X, Y; ξ] is an element of K_{0}(𝔄, E) for which both X and Y are locally free 𝔄-modules, then one can give an explicit representative for h_{𝔄,E}([X, Y; ξ]) as follows.

For any ordered set of d elements \{e^{j} : 1 ≤ j ≤ d\} we write \underline{e^{j}} for the d × 1 column vector with j-th entry equal to e^{j}.

We choose an A-basis {y^{j}} of Y_{F} and, for each v ∈ S_{f}(F), an 𝔄_{v}-basis {y_{v}^{j}} of Y_{v} and an 𝔄_{v}-basis \{x_{v}^{j}\} of X_{v}. Let μ_{v} be the element of GL_{d}(A_{v}) such that \underline{y_{v}^{j}} = μ_{v}\underline{y^{j}}. We choose θ ∈ Is_{A}(X_{F} , Y_{F}). Since {θ^{−1}(y^{j})} is an A-basis of X_{F}, there exists an element λ_{v} of GL_{d}(A_{v}) such that \underline{x_{v}^{j}} = λ_{v}\underline{θ^{−1}(y^{j})}. Finally we let μ ∈ GL_{d}(A_{E}) denote the matrix of ξ ∘ (θ^{−1} ⊗_{F} E) with respect to the A-basis {y^{j}} of Y_{E}. Then h_{𝔄,E}([X, Y; ξ]) is represented by the function

( ∏_{v∈S_{f} (F)} Det(λ_{v}μ_{v}^{−1})) × Det(μ) ∈ Hom(R_{Γ}, J_{f}(Q^{c}))^{Ω_{F}} × Hom(R_{Γ}, (E^{c})^{×}).

This construction will be used in the proof of Proposition 4.11. □

Examples 3.9. Suppose that F is a number field, and let R = O_{F}.

(i) Suppose that 𝔄 = R[G] (see Example 2.2). Then condition (7) is satisfied if we take Γ = G, viewed as a trivial Ω_{F} -module.

(ii) In Example 2.3, set Γ = G(F^{c}), endowed with the natural Ω_{F} -action. Then G ⊗_{R} F^{c} is a finite constant group scheme over Spec(F^{c}), and so its Cartier dual G^{∗}⊗_{R}F^{c} is equal to Spec(F^{c}[Γ]) (see e.g. [34, Section 2.4]). Hence 𝔄 = O_{G^{∗}} satisfies condition (7). Furthermore, in this case the description of Theorem 3.5 may be interpreted more explicitly as follows.

Let J_{f}(A) denote the group of finite ideles of A, i.e. if 𝔐 is the (unique) maximal R-order in A, then J_{f}(A) is the restricted direct product of the groups A_{v}^{×} with respect to the subgroups 𝔐_{v}^{×} for all places v ∈ S_{f}(F). Define a map

Δ′_{𝔄,E} : A^{×} → \frac{J_{f}(A)}{U_{f}(𝔄)} × A_{E}^{×}; a ↦ (a) × a^{−1}.

10 A. AGBOOLA AND D. BURNS

Then, taken in conjunction with Remark 3.3, the result of Theorem 3.5 implies that there is a natural isomorphism

K_{0}(𝔄, E) ⟶^{∼} Cok(Δ′_{𝔄,E}).  (10)
□

4. Metrised structures

In this section we let R denote the ring of algebraic integers in a number field F. We fix an R-order 𝔄 in a finite dimensional F -algebra A, and we continue to assume that the condition (7) is satisfied.

For each ϕ ∈ Γ̂ we write W_{ϕ} for the Wedderburn component of F^{c}[Γ] which corresponds to the contragredient character \overline{ϕ} of ϕ. For any F^{c}[Γ]-module X we then set

X_{ϕ} := ⋀_{F^{c}}^{top}((X ⊗_{F^{c}} W_{ϕ})^{Γ}),

where ‘⋀_{F^{c}}^{top}’ denotes the highest exterior power over F^{c} which is non-zero, and Γ acts diagonally on X ⊗_{F^{c}} W_{ϕ}.

For each v ∈ S_{∞}(F) we fix an embedding σ_{v} : F^{c} → F_{v}^{c} (which induces the place v upon restriction to F), and we also identify F_{v}^{c} with C.

For each complex number z we write \overline{z} for its complex conjugate.

4.1. Metrised 𝔄-modules.

Definition 4.1. Let X be a finitely generated locally-free 𝔄-module. A metric μ on X is a set {μ_{v,ϕ} : v ∈ S_{∞}(F), ϕ ∈ Γ̂} where, for each v ∈ S_{∞}(F) and ϕ ∈ Γ̂, μ_{v,ϕ} is a hermitian metric on the complex line (X ⊗_{R} F^{c})_{ϕ} ⊗_{F^{c},σ_{v}} C. If x ∈ (X ⊗_{R} F^{c})_{ϕ} ⊗_{F^{c},σ_{v}} C, then we shall write μ_{v,ϕ}(x) for the length of x with respect to the metric μ_{v,ϕ}.

A metrised 𝔄-module is a pair (X, μ) consisting of a locally-free 𝔄-module X and a metric μ on X. □

Example 4.2. (Classical) In this example we adopt the notation of Example 2.2. We take 𝔄 to be Z[G], so that Γ = G (viewed as a trivial Ω_{Q}-module). Via the fixed embedding σ_{∞} : Q^{c} → C restricted to the normal closure N of L, we identify Σ_{Q}(L, N) with the set Σ(L) of embeddings of L into C. We remind the reader that if X is a finitely generated, locally free 𝔄-module, then any G-equivariant positive definite hermitian form on X_{C} induces a metric on X in a natural way (see e.g. [17, Definition 2.2]).

We write μ_{C[G]} for the G-equivariant positive definite hermitian form on C[G] which satisfies

μ_{C[G]}(∑_{g∈G} x_{g}g, ∑_{h∈G} y_{h}h) = ∑_{g∈G} x_{g}\overline{y_{g}}.

ON TWISTED FORMS AND RELATIVE ALGEBRAIC K-THEORY 11

(i) There is a G-equivariant positive definite hermitian form μ_{L} on H_{Q}^{L} ⊗_{Z} C which is defined by the rule

μ_{L}( ∑_{σ∈Σ(L)} z′_{σ}, ∑_{τ∈Σ(L)} z_{τ}) = ∑_{σ∈Σ(L)} z′_{σ}\overline{z_{σ}}.

For each ϕ ∈ Ĝ, we write μ_{L,∞,ϕ} for the metric on (H_{Q}^{L} ⊗_{Z} Q^{c})_{ϕ} that is obtained as the highest exterior power of the metric on

(H_{Q}^{L} ⊗_{Z} W_{ϕ})^{G} = (H_{Q}^{L} ⊗_{Z} C) ⊗_{C} (W_{ϕ} ⊗_{Q^{c},σ_{∞}} C)^{G}

which is induced by μ_{L} on H_{Q}^{L} ⊗_{Z} C and by the restriction of μ_{C[G]} on W_{ϕ} ⊗_{Q^{c},σ_{∞}} C.

(ii) There is a G-equivariant positive definite hermitian form h_{L} on C ⊗_{Q} L which is defined by the rule

h_{L}(z_{1} ⊗ m, z_{2} ⊗ n) = z_{1}\overline{z_{2}} ∑_{σ∈Σ(L)} σ(m)\overline{σ(n)}.

We recall that in [17, §5.2] this form is referred to as the ‘Hecke form’.

For each ϕ ∈ Ĝ we write h_{L,∞,ϕ} for the metric on (L ⊗_{Q} Q^{c})_{ϕ} that is obtained as the highest exterior power of the metric on

(L ⊗_{Q} W_{ϕ})^{G} ⊗_{Q^{c},σ_{∞}} C = ((C ⊗_{Q} L) ⊗_{C} (W_{ϕ} ⊗_{Q^{c},σ_{∞}} C))^{G}

which is induced by h_{L} on C ⊗_{Q} L and by the restriction of μ_{C[G]} on W_{ϕ} ⊗_{Q^{c},σ_{∞}} C.

We set h_{L,•} := {h_{L,∞,ϕ} : ϕ ∈ Ĝ}. If L is any full projective Z[G]-sublattice of L, then the pair (L, h_{L,•}) is a metrised Z[G]-module. □

Example 4.3. (Geometrical) In this example we adopt the notation of Example 2.3.

(i) For each v ∈ S_{∞}(F) and ϕ ∈ Γ̂, the space (A ⊗_{F} F^{c})_{ϕ} ⊗_{F^{c},σ_{v}} C identifies naturally with C and so is endowed with a metric μ_{A,v,ϕ} coming from the standard metric on C. The set μ_{A} := {μ_{A,v,ϕ} : v ∈ S_{∞}(F), ϕ ∈ Γ̂} is then a metric (‘the trivial metric’) on 𝔄.

(ii) It is shown in [6, §2] that, for each place v ∈ S_{∞}(F) and character ϕ ∈ Γ̂, the space (L_{π} ⊗_{R} F^{c})_{ϕ} ⊗_{F^{c},σ_{v}} C may be endowed with a canonical ‘Neron metric’ ||·||_{v,ϕ} which is constructed by using canonical splittings of the extension (1). (These canonical splittings are analogous to the canonical splittings of extensions of abelian varieties by tori that may be used to define Néron pairings on abelian varieties (see e.g. [23]).) Let || · ||_{π} denote the family of metrics {|| · ||_{v,ϕ} : v ∈ S_{∞}(F), ϕ ∈ Γ̂}. Then the pair (L_{π}, || · ||_{π}) constitutes a metrised 𝔄-module. □

4.2. Arithmetic class groups. In this subsection we use idelic valued functions on R_{Γ} to define a group which classifies the structure of metrised 𝔄-modules. All of the constructions in this subsection are motivated by those of [17, §3.1, §3.2].

We write |Δ|_{𝔄} for the homomorphism

Det(A^{×}) → \frac{Hom(R_{Γ}, J_{f}(F^{c}))^{Ω_{F}}}{Det(U_{f}(𝔄))} × Hom (R_{Γ}, ∏_{v∈S_{∞}(F)} R_{>0}^{×}); θ ↦ (θ, |θ|)

where |θ| is the homomorphism which sends each ϕ ∈ Γ̂ to the element

∏_{v∈S_{∞}(F)} |σ_{v}(θ(ϕ))|^{−1} ∈ ∏_{v∈S_{∞}(F)} R_{>0}^{×}.

Remark 4.4. We caution the reader that the definition of the homomorphism |θ| given above is slightly different from that used in [17, §3.1]. Our |θ|(ϕ) is equal to |θ|(ϕ)^{−1} in the notation of [17]. This reflects our definitions of Δ_{𝔄,E} in (8) and h_{𝔄,E} in (9). Our normalisation is also consistent with the definition of the Hermitian classgroup given in [21, Chapter II, §5]. □

Definition 4.5. We define the arithmetic classgroup AC(𝔄) of 𝔄 to be the cokernel of |Δ|_{𝔄}. □

Remark 4.6. When F = Q and 𝔄 = Z[Γ] for some finite group Γ, then it is easy to show (taking into account Remark 4.4 above) that AC(𝔄) is isomorphic to the arithmetic classgroup A(𝔄) which is described in [17, Definition 3.2]. □

We now suppose given a metrised 𝔄-module (M, μ). Let M have rank d over 𝔄, and choose an A-basis {m^{j}} of M_{F} and, for each v ∈ S_{f}(F), an 𝔄_{v}-basis {m_{v}^{j}} of M_{v}. Then there exists an element λ_{v} of GL_{d}(A_{v}) such that \underline{m_{v}^{j}} = λ_{v}\underline{m^{j}}.

For each element m of M_{F} we set

r(m) := ∑_{γ∈Γ} γm ⊗ γ ∈ M ⊗_{R} F^{c}[Γ].

For each element w of W_{ϕ} we have

r(m)(1 ⊗ w) ∈ (M ⊗_{R} W_{ϕ})^{Γ}.

Let {w_{ϕ,k} : k} be an F^{c}-basis of W_{ϕ} which is orthonormal with respect to the restriction of μ_{F^{c}[Γ]} to W_{ϕ}. Then the set {r(m^{j})(1 ⊗ w_{ϕ,k}) : j, k} is an F^{c}-basis of (M ⊗_{R} W_{ϕ})^{Γ}, and so the wedge product

⋀_{j} ⋀_{k} r(m^{j})(1 ⊗ w_{ϕ,k})

is an F^{c}-basis of the line (M ⊗_{R} F^{c})_{ϕ}.

Definition 4.7. We define [M, μ] to be the element of AC(𝔄) which is represented by the homomorphism on R_{Γ} which sends each character ϕ ∈ Γ̂ to

∏_{v∈S_{f} (F)} Det(λ_{v})(ϕ)× ∏_{v∈S_{∞}(F)} μ_{v,ϕ} ((⋀_{j} ⋀_{k} r(m^{j})(1 ⊗ w_{ϕ,k})) ⊗_{F^{c},σ_{v}} 1)^{\frac{1}{ϕ(1)}} ∈ J_{f}(F^{c})× ∏_{v∈S_{∞}(F)} R_{>0}^{×}.

It can be shown that [M, μ] is independent of the precise choices of bases {m^{j}}, {m_{v}^{j}} and {w_{ϕ,k}}. □

ON TWISTED FORMS AND RELATIVE ALGEBRAIC K-THEORY 13

4.3. The connection to generalised twisted forms. Let E be any field which contains F^{c} and which is such that, for each v ∈ S_{∞}(F), the fixed embedding σ_{v} : F^{c} → C factors through an embedding σ̃_{v} : E → C.

The map

π_{E} : E^{×} → ∏_{v∈S_{∞}(F)} R_{>0}^{×}; e ↦ ∏_{v∈S_{∞}(F)} |σ̃_{v}(e)|

induces a homomorphism

ε : Det(A_{E}^{×}) ≃ Hom(R_{Γ}, E^{×}) ⟶ Hom(R_{Γ}, ∏_{v∈S_{∞}(F)} R_{>0}^{×}).

Then, via the Hom-descriptions of Theorem 3.5 and Definition 4.5, it is not hard to see that the map

δ : \frac{Hom(R_{Γ}, J_{f}(F^{c}))^{Ω_{F}}}{Det(U_{f}(𝔄))} × Det(A_{E}^{×}) ⟶ \frac{Hom(R_{Γ}, J_{f}(F^{c}))^{Ω_{F}}}{Det(U_{f}(𝔄))} × Hom(R_{Γ}, ∏_{v∈S_{∞}(F)} R_{>0}^{×})

given by x_{1} × x_{2} ↦ x_{1} × ε(x_{2}) induces a homomorphism

∂_{𝔄,E} : K_{0}(𝔄, E) → AC(𝔄).

Lemma 4.8. There is a natural isomorphism

ker(∂_{𝔄,E}) ⟶^{∼} \frac{Hom(R_{Γ}, ker(π_{E}))}{[Hom(R_{Γ}, ker(π_{E})) ∩ Det(A^{×})] ∩ Det(U_{f}(𝔄))}.  (11)

(In the denominator of the right-hand side of (11), the first intersection takes place in

Hom(R_{Γ}, E^{×}) ≃ Det(A_{E}^{×}),

and then the second intersection takes place in Hom(R_{Γ}, J_{f}(F^{c})).)

Proof. Set

N_{1} := \frac{Hom(R_{Γ}, J_{f}(F^{c}))^{Ω_{F}}}{Det(U_{f}(𝔄))} × Det(A_{E}^{×}),

and write N_{2} := δ(N_{1}). Then there is a commutative diagram:

0 ⟶ Hom(R_{Γ}, ker(π_{E})) ⟶ N_{1}  ⟶^{δ} N_{2} ⟶ 0
Δ_{𝔄,E} ↑  ^{|Δ|_{𝔄}} ↑
Det(A^{×})  Det(A^{×}).

Applying the Snake Lemma to this diagram (and taking into account Theorem 3.5 and Definition 4.5) yields the sequence

Hom(R_{Γ}, Ker(π_{E})) → K_{0}(𝔄, E) ⟶^{∂_{𝔄,E}} AC(𝔄) → 0,

and it is easy to check (again using Theorem 3.5) that this implies the desired result. □

Given an explicit element of K_{0}(𝔄, E) we now aim to describe its image under ∂_{𝔄,E}.

Definition 4.9. Let X_{1} and X_{2} be finitely generated locally-free 𝔄-modules and suppose that ξ ∈ Is_{A_{E}}(X_{2,E}, X_{1,E}). For each place v ∈ S_{∞}(F) and character ϕ ∈ Γ̂ we write

ξ_{v,ϕ} : (X_{2} ⊗_{R} F^{c})_{ϕ} ⊗_{F^{c},σ_{v}} C ⟶^{∼} (X_{1} ⊗_{R} F^{c})_{ϕ} ⊗_{F^{c},σ_{v}} C

for the isomorphism of complex lines which is induced by ξ. If h is any metric on X_{1}, then the pullback of h under ξ is defined to be the metric ξ^{∗}(h) on X_{2} which satisfies

ξ^{∗}(h)_{v,ϕ}(z) = h_{v,ϕ}(ξ_{v,ϕ}(z))

for each v ∈ S_{∞}(F), ϕ ∈ Γ̂ and z ∈ (X_{2} ⊗_{R} F^{c})_{ϕ} ⊗_{F^{c},σ_{v}} C. □

Example 4.10. (i) (Classical) An explicit computation shows that, in the notation of Example 4.2, one has (π_{Q}^{L,N})^{∗}(μ_{L,•}) = h_{L,•}.

(ii) (Geometrical) We use the notation of Example 4.3. If ξ_{π} is any isomorphism as in (2), then it follows from [6, Remark 2.3] that ξ_{π} induces an isometry between L_{π} endowed with the Néron metric || · ||_{π} and 𝔄 endowed with the trivial metric. (Note that the map which is denoted by ξ_{π} in [6] is equal to ξ_{π}^{⊗e} in our present notation, where e denotes the exponent of the group scheme G.) Hence we have that (ξ_{π} ⊗_{R^{c}} F^{c})^{∗}(μ_{A}) = || · ||_{π}. □

Proposition 4.11. Let (X, Y; ξ) be any object of P_{𝔄} ×_{E} P_{𝔄} for which X and Y are both locally free 𝔄-modules. Then, for any metric ρ on Y, one has

∂_{𝔄,E}(h_{𝔄,E}([X, Y; ξ])) = [X, ξ^{∗}(ρ)] − [Y, ρ] ∈ AC(𝔄).

Proof. We use the notation introduced in Remark 3.8.

It is clear that ∂_{𝔄,E}(h_{𝔄,E}([X, Y; ξ])) is represented by the homomorphism which sends each character ϕ ∈ Γ̂ to

∏_{v∈S_{f} (F)} Det(λ_{v}μ_{v}^{−1})(ϕ) × ∏_{v∈S_{∞}(F)} |σ̃_{v}(Det(μ)(ϕ))|.  (12)

For each element x of X_{i,ϕ} := (X_{i} ⊗_{R} F^{c})_{ϕ} ⊗_{F^{c}} E (where i = 1, 2) and each place v ∈ S_{∞}(F), we write [x]_{v} for the image of x in X_{i,ϕ} ⊗_{E,σ̃_{v}} C = (X_{i} ⊗_{R} F^{c})_{ϕ} ⊗_{F^{c},σ_{v}} C.

Using this notation, the element [X, ξ^{∗}(ρ)], resp. [Y, ρ], is represented by the homomorphism which sends each character ϕ ∈ Γ̂ to

∏_{v∈S_{f} (F)} Det(λ_{v})(ϕ) × ∏_{v∈S_{∞}(F)} ρ_{v,ϕ} ([⋀_{j} ⋀_{k} r(ξ ∘ ψ(y^{j}))(1 ⊗ w_{ϕ,k})]_{v})^{\frac{1}{ϕ(1)}} ,  (13)

resp.

∏_{v∈S_{f} (F)} Det(μ_{v})(ϕ) × ∏_{v∈S_{∞}(F)} ρ_{v,ϕ} ([⋀_{j} ⋀_{k} r(y^{j})(1 ⊗ w_{ϕ,k})]_{v})^{\frac{1}{ϕ(1)}} .  (14)

ON TWISTED FORMS AND RELATIVE ALGEBRAIC K-THEORY 15

For each z = ∑_{γ∈Γ} c_{γ}γ ∈ E[Γ] we set \overline{z} := ∑_{γ∈Γ} c_{γ}γ^{−1}, and we extend this convention to matrices over E[Γ] by applying it to individual entries. Now

ξ ∘ ψ(y^{j}) = ∑_{l} μ_{l,j}y^{l}

where μ_{l,j} is the (l, j)-component of the matrix μ, and so

r(ξ ∘ ψ(y^{j}))(1 ⊗ w_{ϕ,k}) = ∑_{l} r(y^{j})(1 ⊗ \overline{μ_{l,j}}w_{ϕ,k}).

This implies that

⋀_{j} ⋀_{k} r(ξ ∘ ψ(y^{j}))(1 ⊗ w_{ϕ,k}) = Det(\overline{μ})(\overline{ϕ})^{ϕ(1)} · ⋀_{j} ⋀_{k} r(y^{j})(1 ⊗ w_{ϕ,k}).

But Det(\overline{μ})(\overline{ϕ}) = Det(μ(ϕ)), and so for each v ∈ S_{∞}(F) one has

\frac{ρ_{v,ϕ} ([⋀_{j} ⋀_{k} r(ξ ∘ ψ(y^{j}))(1 ⊗ w_{ϕ,k})]_{v})^{\frac{1}{ϕ(1)}}}{ρ_{v,ϕ} ([⋀_{j} ⋀_{k} r(y^{j})(1 ⊗ w_{ϕ,k})]_{v})^{\frac{1}{ϕ(1)}}} = |σ̃_{v}(Det(μ)(ϕ))| ∈ R_{>0}^{×}.

It is now clear that the expression (13) is equal to the product of the expressions (12) and (14), and this immediately implies the claimed equality. □

5. The classical case

In this section we give two applications of our approach in the context of Example 2.2. We thus fix a finite tamely ramified Galois extensions of number fields L/K, with F ⊆ K, and we set G := Gal(L/K). Our interest is in the element

δ_{O_{F} [G]}^{L}(O_{L}) := [O_{L}, H_{F}^{L} ⊗_{Z} O_{F}; π_{F}^{L,N}] ∈ K_{0}(O_{F} [G], N).

5.1. Arithmetic classes. In this subsection we combine Proposition 4.11 with a result of Bley and the second named author in [9] in order to describe an explicit homomorphism which represents the element δ_{Z[G]}^{L}(O_{L}). We then explain how this yields an explicit homomorphism which represents the element [O_{L}, h_{L,•}] of AC(Z[G]). We show that this description of [O_{L}, h_{L,•}] gives a natural refinement of a result of Chinburg, Pappas and Taylor in [17].

Before stating our explicit description of the class δ_{Z[G]}^{L}(O_{L}) we must introduce some notation. For each character ϕ ∈ Ĝ, we write τ(K, ϕ) for the Galois-Gauss sum associated to ϕ, and

we let ϵ(K, ϕ) denote the epsilon constant which arises in the functional equation of the Artin L-function attached to ϕ (see [20, Chapter I, §5]). With respect to the canonical identification ζ(C[G]) = ∏Ĝ C (which is induced by the fixed embedding σ_{∞} : Q^{c} → C) we then set

ϵ_{L/K} := (ϵ(K, \overline{ϕ}))_{ϕ} ∈ ζ(C[G])^{×}.

16 A. AGBOOLA AND D. BURNS

This element is the natural epsilon constant which is associated to the ζ(C[G])-valued equivariant Artin L-function of L/K, and it actually lies in ζ(R[G])^{×}. The last fact implies that we may therefore choose an element

λ = (λ_{ϕ})_{ϕ} ∈ ζ(Q[G])^{×} ⊂ ζ(C[G])^{×}

such that

λϵ_{L/K} ∈ Im(nrd_{R[G]})  (15)

(see [9, §3.1]). We write d_{K} for the absolute discriminant of K.

We write ι : N → C for the embedding induced by restricting σ_{∞} to N, and we use this to view δ_{Z[G]}^{L}(O_{L}) as an element of K_{0}(Z[G], C) (using the inclusion K_{0}(Z[G], N) ⊆ K_{0}(Z[G], C) induced by the functor − ⊗_{N,ι} C).

We can now state the main result of this section.

Theorem 5.1. Let λ be any element of ζ(Q[G])^{×} which satisfies (15). Then the class δ_{Z[G]}^{L}(O_{L}) in K_{0}(Z[G], C) is represented by the homomorphism which sends each character ϕ ∈ Ĝ to

λ_{ϕ}^{−1} ∏_{v∈S_{f} (K)} y_{v}(ϕ) × λ_{ϕ}τ(K, ϕ)|d_{K}|^{\frac{ϕ(1)}{2}} .  (16)

Here y_{v} denotes the ‘unramified characteristic’ function on R_{G} defined in [20, Chapter IV, §1].

We shall derive a description of the class δ_{Z[G]}^{L}(O_{L}) from certain results in [9]. For the reader’s convenience we first recall some notation from loc. cit.

Definition 5.2. (See [9, §3.2]) Set H_{L} := ∏_{Σ(L)} Z. The groups G and Gal(C/R) act on Σ(L), and they endow H_{L} with the structure of G × Gal(C/R)-module. For any Gal(C/R)-module X, write X^{+} and X^{−} for the submodules on which complex conjugation acts by +1 and −1 respectively.

There is a canonical C[G × Gal(C/R)]-equivariant isomorphism

ρ_{L} : L ⊗_{Q} C ⟶^{∼} H_{L} ⊗_{Z} C; l ⊗ z ↦ (σ(l)z)_{σ∈Σ(L)}.

We write π_{L} for the composite R[G]-equivariant isomorphism given by

L ⊗_{Q} R = (L ⊗_{Q} C)^{+}

⟶^{ρ_{L}} (H_{L} ⊗_{Z} C)^{+}

= (H_{L}^{+} ⊗_{Z} R) ⊕ (H_{L}^{−} ⊗_{Z} (i · R))

⟶^{×(1,−i)} (H_{L}^{+} ⊗_{Z} R) ⊕ (H_{L}^{−} ⊗_{Z} R)

= H_{L} ⊗_{Z} R.

We now define the G-equivariant discriminant δ_{L/K}(O_{L}) of O_{L} by setting

δ_{L/K}(O_{L}) := [O_{L}, H_{L}; π_{L}] ∈ K_{0}(Z[G], R).

ON TWISTED FORMS AND RELATIVE ALGEBRAIC K-THEORY 17

In what follows, we shall in fact view δ_{L/K}(O_{L}) as an element of K_{0}(Z[G], C) via the natural inclusion K_{0}(Z[G], R) ⊆ K_{0}(Z[G], C). □

Proof of Theorem 5.1. It follows from results in [9, see especially Lemma 7.4 and Corollary 7.7] that the class δ_{L/K}(O_{L}) is represented in the Hom-description of Theorem 3.5 by the element of Hom(R_{G}, J_{f}(Q^{c}))^{Ω_{Q}} × Hom(R_{G}, C^{×}) which sends each character ϕ ∈ Ĝ to

λ_{ϕ}^{−1} ∏_{v∈S_{f} (K)} y_{v}(ϕ) × λ_{ϕ}ϵ(K, ϕ),

where y_{v} is as defined in the statement of Theorem 5.1.

For each place v ∈ S_{∞}(K), we let G_{v} denote the decomposition group of v in G. If ϕ ∈ Ĝ, we let V_{ϕ} be a representation space for ϕ, and we define a complex number w_{∞}(K, ϕ) by setting

w_{∞}(K, ϕ) := ∏_{v∈S_{∞}(K)} i^{− codim(V_{ϕ}^{Gv})}.

We may identify H_{L} with H_{Q}^{L} via our fixed embedding ι : N → C. By comparing the homomorphisms π_{Q}^{L,N} ⊗_{N,ι} C and π_{L} ⊗_{R} C, it may be shown that the element δ_{L/K}(O_{L}) − δ_{Z[G]}^{L}(O_{L}) of K_{0}(Z[G], C) is represented by the element of Hom(R_{G}, J_{f}(Q^{c}))^{Ω_{Q}} × Hom(R_{G}, C^{×}) which sends each character ϕ ∈ Ĝ to 1 × w_{∞}(K, \overline{ϕ}).

In addition, for each ϕ ∈ Ĝ one has

ϵ(K, ϕ) = τ(K, ϕ)w_{∞}(K, \overline{ϕ})|d_{K}|^{\frac{ϕ(1)}{2}}  (17)

(see [9, (13) and the displayed formula which follows it]).

It now follows that δ_{Z[G]}^{L}(O_{L}) is represented by the homomorphism which sends each character ϕ ∈ Ĝ to

λ_{ϕ}^{−1} ∏_{v∈S_{f} (K)} y_{v}(ϕ) × λ_{ϕ}τ(K, ϕ)|d_{K}|^{\frac{ϕ(1)}{2}} ,

as claimed. □

Corollary 5.3. Let λ be any element of ζ(Q[G])^{×} which satisfies (15). Then the class [O_{L}, h_{L,•}] in AC(Z[G]) is represented by the homomorphism which sends each character ϕ ∈ Ĝ to

λ_{ϕ}^{−1} × |λ_{ϕ}τ(K, ϕ)|(|G|^{[K:Q]}|d_{K}|)^{\frac{ϕ(1)}{2}} .

Proof. We first observe that Example 4.10(i) and Proposition 4.11 imply that

[O_{L}, h_{L,•}] = [H_{Q}^{L}, μ_{L,•}] + ∂_{Z[G],C}(δ_{Z[G]}^{L}(O_{L})) ∈ AC(Z[G]).  (18)

It follows via an explicit computation (using, for example, [17, Lemma 2.3]) that the class [H_{Q}^{L}, μ_{L,•}] is represented by the homomorphism which sends each character ϕ ∈ Ĝ to 1×|G|^{[K:Q] \frac{ϕ(1)}{2}} .

Combining this with (18) and Theorem 5.1, we deduce that [O_{L}, h_{L,•}] is represented by the homomorphism which sends each ϕ ∈ Ĝ to

λ_{ϕ}^{−1} ∏_{v∈S_{f} (K)} y_{v}(ϕ) × |λ_{ϕ}τ(K, ϕ)|(|G|^{[K:Q]}|d_{K}|)^{\frac{ϕ(1)}{2}} .

The desired result now follows from this description upon noting that, as a consequence of [20, Chapter IV, Theorem 29(i)], the homomorphism which sends each ϕ ∈ Ĝ to

( ∏_{v∈S_{f} (K)} y_{v}(ϕ)) × 1

belongs to the image of |Δ|_{𝔄}. □

It is often convenient to replace AC(Z[G]) by weaker classifying groups. As an example of this, we shall now explain how Corollary 5.3 yields a natural refinement of [17, Theorem 5.9].

We first quickly recall the definition of the ‘tame symplectic arithmetic classgroup’ A_{T}^{s} (Z[G]) of Z[G] from [loc. cit., §4.3]. We write R_{G}^{s} for the subgroup of R_{G} which is generated by the irreducible symplectic F^{c}-valued characters of G, and we write Det^{s}(−) for the restriction of the function Det(−) to R_{G}^{s}. Let T denote the maximal abelian tamely ramified extension of Q in Q^{c}, and set Ẑ = \underleftarrow{lim}_{n} Z/nZ. Then we write Det(Ô_{T} [G]^{×}) for the direct limit of Det(Ô_{N}[G]^{×}), where N runs over all finite extensions of Q in T and Ô_{N} denotes the ring of integral adeles Ẑ ⊗ O_{N}.

Definition 5.4. The tame arithmetic symplectic classgroup A_{T}^{s} (Z[G]) of Z[G] is defined to be the cokernel of the map

Det^{s}(Q[G]^{×}) ⟶ \frac{Det^{s}(Ô_{T} [G]^{×}) Hom(R_{G}^{s}, J_{f}(Q^{c}))^{Ω_{Q}}}{Det^{s}(Ô_{T} [G]^{×})} × Hom(R_{G}^{s}, R_{>0}^{×})

which is induced by the diagonal map

Δ^{s} : Det^{s}(Q[G]^{×}) ⟶ Hom(R_{G}^{s}, J_{f}(Q^{c}))^{Ω_{Q}} × Hom(R_{G}^{s}, R_{>0}^{×}); θ ↦ (θ, θ^{−1}).
□

We write

θ_{T}^{s} : Hom(R_{G}^{s}, J_{f}(Q^{c}))^{Ω_{Q}} × Hom(R_{G}^{s}, R_{>0}^{×}) ⟶ A_{T}^{s} (Z[G])

for the natural surjective map, and

π_{T}^{s} : AC(Z[G]) ⟶ A_{T}^{s} (Z[G])

for the homomorphism induced by restricting an element of Hom(R_{G}, J_{f}(Q^{c}))^{Ω_{Q}} × Hom(R_{G}, R_{>0}^{×}) to R_{G}^{s}, and then applying θ_{T}^{s}.

ON TWISTED FORMS AND RELATIVE ALGEBRAIC K-THEORY 19

Definition 5.5. Let Pf(O_{L}) denote the ‘Pfaffian’ element of Hom(R_{G}^{s}, J_{f}(Q^{c})) which is defined (in [loc. cit., Def. 5.8]) as follows: for each ϕ ∈ R_{G}^{s} one has

Pf(O_{L})(ϕ)_{p} = ∏_{𝔭}(−p)^{\frac{1}{2} f_{𝔭}(ϕ,Ind_{I𝔭}^{G} u_{𝔭})}

where here 𝔭 runs over all elements of S_{f}(K) which divide p, f_{𝔭} is the residue class extension degree of 𝔭 in K/Q, I_{𝔭} is the inertia subgroup of any fixed place of L which lies above 𝔭, u_{𝔭} is the augmentation character of I_{𝔭} and (−, −) is the standard inner product on R_{G}. □

We remark that the image of [O_{L}, h_{L,•}] under π_{T}^{s} is equal to the element χ_{T}^{s} (O_{L}, det h_{•}) defined in [17, §4.3]. This implies that the following result is equivalent to [loc. cit., Th. 5.9(a)].

Corollary 5.6. The class π_{T}^{s} ([O_{L}, h_{L,•}]) is represented by the homomorphism which sends each symplectic character ϕ ∈ Ĝ to

w_{∞}(K, ϕ)Pf(O_{L})(ϕ) × (|G|^{[K:Q]}|d_{K}|)^{\frac{ϕ(1)}{2}} .

Proof. Upon computing the image under π_{T}^{s} of the class represented by the explicit homomorphism described in Corollary 5.3, one finds that it suffices to prove the following result: if f denotes the homomorphism which sends each symplectic character ϕ ∈ Ĝ to

λ_{ϕ}w_{∞}(K, ϕ)Pf(O_{L})(ϕ) × |λ_{ϕ}τ(K, ϕ)|^{−1},

then θ_{T}^{s} (f) = 0.

To prove this we first observe that there exists a finite tamely ramified abelian extension N of Q and an element u of Ô_{N}[G]^{×} such that, for each symplectic character ϕ ∈ Ĝ one has

Pf(O_{L})(ϕ) = τ(K, ϕ) · Det(u)(ϕ).

This can be seen, for example, to be a consequence of the argument which proves [17, Proposition 5.11]. Hence one has θ_{T}^{s} (f) = θ_{T}^{s} (f′), where f′ is the homomorphism which sends each symplectic character ϕ ∈ Ĝ to

λ_{ϕ}w_{∞}(K, ϕ)τ(K, ϕ) × |λ_{ϕ}τ(K, ϕ)|^{−1}.

Now the equality (17) combines with the containment (15) to imply that, for each symplectic character ϕ ∈ Ĝ, the real numbers λ_{ϕ} and w_{∞}(K, ϕ)τ(K, ϕ) have the same sign (note that ϕ = \overline{ϕ} if ϕ is symplectic). This implies that the element Φ of Hom(R_{G}^{s}, (Q^{c})^{×})^{Ω_{Q}} which sends each symplectic character ϕ ∈ Ĝ to λ_{ϕ}w_{∞}(K, ϕ)τ(K, ϕ) actually belongs to Det^{s}(Q[G]^{×}). In addition, since for each symplectic ϕ ∈ Ĝ one has w_{∞}(K, ϕ) = ±1, it also implies that λ_{ϕ}w_{∞}(K, ϕ)τ(K, ϕ) = |λ_{ϕ}τ(K, ϕ)|. This in turn implies that f′ = Δ^{s}(Φ), and hence that θ_{T}^{s} (f′) = 0, as required. □

20 A. AGBOOLA AND D. BURNS

5.2. Torsion modules. In this subsection we shall indicate how certain natural torsion Galois module invariants introduced by S. Chase (see [14], [20, Notes to Chapter III]) are related to the element δ_{O_{K}[G]}^{L}(O_{L}).

We first recall that there is a natural O_{L}[G]-equivariant injection

ψ_{L/K} : O_{L} ⊗_{O_{K}} O_{L} ⟶ Map(G, O_{L}); ψ_{L/K}(l_{1} ⊗ l_{2})(g) = g(l_{1})l_{2}.

The cokernel Cok(ψ_{L/K}) of ψ_{L/K} is finite, and is of finite projective dimension as an O_{L}[G]- module because L/K is tamely ramified. Hence Cok(ψ_{L/K}) determines a class in K_{0}T (O_{L}[G]). This class has been studied by Chase and other authors, and it gives information regarding the O_{K}[G]-module structure of O_{L}.

We write

i_{L/K} : K_{0}(O_{K}[G], L) ⟶ K_{0}T (O_{L}[G])

for the composite of the natural scalar extension morphism

K_{0}(O_{K}[G], L) ⟶ K_{0}(O_{L}[G], L); [X, Y; i] ↦ [X ⊗_{O_{K}} O_{L}, Y ⊗_{O_{K}} O_{L}; i]

and the first isomorphism of (4) with 𝔄 = O_{L}[G].

Lemma 5.7. The class of Cok(ψ_{L/K}) in K_{0}T (O_{L}[G]) is equal to the image of δ_{O_{K}[G]}^{L}(O_{L}) under i_{L/K}.

Proof. There is an O_{L}[G]-equivariant isomorphism

θ : Map(G, O_{L}) ⟶^{∼} H_{K}^{L} ⊗_{Z} O_{L}; θ(ϕ) = (ϕ(g))_{g}.

The stated result follows from the definition of isomorphism in P_{O_{L}[G]} ×_{L} P_{O_{L}[G]} (see §2.1) and the commutativity of the following diagram in P_{L[G]}:

L ⊗_{K} L ⟶^{ψ_{L/K}} Map(G, L)
id↓  _{θ}↓
L ⊗_{K} L ⟶^{π_{K}^{L,L}} H_{K}^{L} ⊗_{Z} L

Here π_{K}^{L,L} is as defined in Example 2.2. □

We remark that Chase has given an explicit description of Cok(ψ_{L/K}) as an O_{L}[G]-module (see in particular Theorem 2.15 of [14]). Hence, Lemma 5.7 shows that Chase’s results may be naturally reinterpreted in terms of equivariant discriminants.

6. Torsors

We shall now apply the methods developed in this paper to the ‘geometrical’ setting of Example 2.3.

ON TWISTED FORMS AND RELATIVE ALGEBRAIC K-THEORY 21

6.1. Reduced resolvends. In this subsection, we shall discuss reduced resolvends coming from torsors of finite group schemes. This notion was first introduced by L. McCulloh (see [24], and the references contained therein). Our approach will be rather different from McCulloh’s, and will use ideas that first appeared in [6] and [3].

We use the notation of Example 2.3. Set Γ := G(R^{c}) = G(F^{c}).

Definition 6.1. We define

H(𝔄) := {α ∈ 𝔄_{R^{c}}^{×} | α^{ω} = g_{ω}α for all ω ∈ Ω_{F}, where g_{ω} ∈ Γ},
H(𝔄) := H(𝔄)/Γ,
H(𝔄) := H(𝔄)/(Γ · 𝔄^{×}).

(Note that we allow the possibility that R = F, in which case we have 𝔄 = A.) □

We suppose for the moment that R is a local ring, and we let π : X → Spec(R) be a G-torsor. Then the line bundle L_{π} associated to π (see Example 2.3) is isomorphic to the trivial line bundle on G^{∗}. Let s_{π} : 𝔄 ⟶^{∼} L_{π} be any trivialisation of L_{π}. Recall from Example 2.3 that over Spec(R^{c}), π becomes isomorphic to the trivial torsor π_{0}, i.e. there is an isomorphism i : X ⊗_{R} R^{c} ⟶^{∼} G ⊗_{R} R^{c} of schemes with G-action. This isomorphism is not unique. If i′ is any other such isomorphism, then i^{−1}i′ : G ⊗_{R} R^{c} ⟶^{∼} G ⊗_{R} R^{c} is also an isomorphism of schemes with G-action, and so is given by translation by an element of Γ.

Let ξ_{π} : L_{π} ⊗_{R} R^{c} ⟶^{∼} 𝔄 ⊗_{R} R^{c} be the splitting isomorphism corresponding to i, and consider the map from 𝔄_{R^{c}} to itself defined by

𝔄_{R^{c}} ⟶^{s_{π}⊗_{R}R^{c}} L_{π} ⊗_{R} R^{c} ⟶^{ξ_{π}} 𝔄_{R^{c}}.

This is an isomorphism of 𝔄_{R^{c}}-modules, and so it is given by multiplication by some element r(s_{π}) in 𝔄_{R^{c}}^{×}. We refer to r(s_{π}) as a resolvend of s_{π}. Note that r(s_{π}) depends upon s_{π} as well as upon the choice of ξ_{π}.

If ω ∈ Ω_{F}, then ξ_{π}^{ω} = g_{ω}ξ_{π}, for some g_{ω} ∈ Γ. Since s_{π}^{ω} = s_{π}, we deduce that r(s_{π})^{ω} = g_{ω}r(s_{π}), and so r(s_{π}) ∈ H(𝔄). As i is only well-defined up to translation by an element of Γ, it follows that changing our choice of ξ_{π} alters r(s_{π}) by multiplication by an element of Γ (see also [13, Lemmas 1.2 and 1.3]). Hence the image r(s_{π}) of r(s_{π}) in H(𝔄) depends only upon the trivialisation s_{π}. We refer to r(s_{π}) as the reduced resolvend of s_{π}.

We next observe that changing our choice of trivialisation s_{π} alters r(s_{π}) by multiplication by an element of 𝔄^{×}. It follows that the image of r(s_{π}) in the group H(𝔄) depends only upon the isomorphism class of the G-torsor π.

The following result was first proved by McCulloh in the case in which G is a constant group scheme [24, §1 and §2]. In the generality given below, it is due to Byott [12, Lemma 1.11, Proposition 2.12]. A different proof of this result is given in [3].

Theorem 6.2. Suppose that R is a local ring. Then the map

H^{1}(Spec(R), G) → H(𝔄); π ↦ [r(s_{π})]

is an isomorphism. □

We shall now show that, for any field F, the subgroup H(A) of A_{F^{c}}^{×}/(Γ · A^{×}) has a natural functorial description. In order to do this, we first make the following definition.

Definition 6.3. Let R be any commutative ring, and suppose that F is any contravariant functor from the category of finite, flat, commutative group schemes over Spec(R) to the category of abelian groups.

If G is any finite, flat, commutative group scheme over R, let m : G×G → G denote multiplication on G, and write p_{i} : G × G → G (i = 1, 2) for projection onto the i-th factor. Let m^{∗}, p_{i}^{∗} : F(G) → F(G × G) denote the homomorphisms induced by m and p_{i} respectively. An element x ∈ F(G) is said to be primitive if m^{∗}(x) = p_{1}^{∗}(x) · p_{2}^{∗}(x). (Note in particular that it follows that the subset of primitive elements of F(G) is a group.)

It is sometimes helpful to formulate this definition in terms of Hopf algebras rather than group schemes. This may be done as follows. Each functor F as above naturally corresponds to a covariant functor \overline{F} from the category of finite, flat, commutative and cocommutative Hopf algebras over R to the category of abelian groups. For any such Hopf algebra A, let Δ : A → A ⊗_{R} A denote the comultiplication on A. Define i_{1}, i_{2} : A → A ⊗_{R} A by i_{1}(a) = a ⊗ 1 and i_{2}(a) = 1 ⊗ a for each a ∈ A. Write Δ^{∗}, i_{1}^{∗}, i_{2}^{∗} : \overline{F}(A) → \overline{F}(A ⊗_{R} A) for the homomorphisms induced by Δ, i_{1}, and i_{2} respectively. Then an element y ∈ \overline{F}(A) is primitive if Δ^{∗}(y) = i_{1}^{∗}(y) · i_{2}^{∗}(y). □

Theorem 6.4. The group H(A) is equal to the subgroup of primitive elements of the group A_{F^{c}}^{×}/(Γ· A^{×}).

Proof. Suppose first that a ∈ H(A), with a^{ω} = g_{ω}a for all ω ∈ Ω_{F}. We have that A_{F^{c}} = F^{c}[Γ], and the comultiplication map Δ : A_{F^{c}} → A_{F^{c}} ⊗_{F} A_{F^{c}} is induced by Δ(γ) = γ ⊗ γ for γ ∈ Γ. This implies that, for all ω ∈ Ω,

Δ(a)^{ω} = (g_{ω} ⊗ g_{ω})Δ(a), i_{1}(a)^{ω} = (g_{ω} ⊗ 1)i_{1}(a), i_{2}(a)^{ω} = (1 ⊗ g_{ω})i_{2}(a).

Hence we have that Δ(a)[i_{1}(a)i_{2}(a)]^{−1} ∈ (A_{F^{c}} ⊗_{F} A_{F^{c}})^{Ω_{F}} = A ⊗_{F} A. This implies that the class of a in A_{F^{c}}^{×}/(Γ · A^{×}) is primitive.

Suppose conversely that a ∈ A_{F^{c}}^{×} represents a primitive class in A_{F^{c}}^{×}/(Γ · A^{×}). For each ω ∈ Ω_{F}, write a^{ω} = u_{ω}a. We wish to show that u_{ω} ∈ Γ for all ω.

Since a ∈ A_{F^{c}}^{×} represents a primitive class in A_{F^{c}}^{×}/(Γ · A^{×}), we have

\frac{Δ(a)}{i_{1}(a)i_{2}(a)} = (g ⊗ h)β,

ON TWISTED FORMS AND RELATIVE ALGEBRAIC K-THEORY 23

where g, h ∈ Γ, and β ∈ (A ⊗_{F} A)^{×}. Hence if ω ∈ Ω_{F}, then β^{ω} = β and

[\frac{Δ(a)}{i_{1}(a)i_{2}(a)}]^{ω} = (g^{ω} ⊗ h^{ω})β = (g^{ω}g^{−1} ⊗ h^{ω}h^{−1}) (\frac{Δ(a)}{i_{1}(a)i_{2}(a)}) .  (19)

On the other hand, we also have that

[\frac{Δ(a)}{i_{1}(a)i_{2}(a)}]^{ω} = \frac{Δ(u_{ω})}{i_{1}(u_{ω})i_{2}(u_{ω})} (\frac{Δ(a)}{i_{1}(a)i_{2}(a)}) .  (20)

Hence (19) and (20) imply that

\frac{Δ(u_{ω})}{i_{1}(u_{ω})i_{2}(u_{ω})} = g^{ω}g^{−1} ⊗ h^{ω}h^{−1}.  (21)

Therefore, in order to prove that u_{ω} ∈ Γ, it suffices to prove the following assertion. Suppose that u ∈ A_{F^{c}}^{×} satisfies

\frac{Δ(u)}{i_{1}(u)i_{2}(u)} = e ⊗ f,  (22)

with e, f ∈ Γ. Then u ∈ Γ.

In order to establish this, we argue as follows. Suppose first that u = a_{γ}γ for some γ ∈ Γ. Then

\frac{Δ(u)}{i_{1}(u)i_{2}(u)} = \frac{a_{γ}(γ ⊗ γ)}{a_{γ}^{2}(γ ⊗ γ)} = \frac{1}{a_{γ}} 1_{Γ} ⊗ 1_{Γ},

and so in this case u satisfies (22) if and only if a_{γ} = 1. Hence we may assume that u = ∑_{γ∈Γ} a_{γ}γ with a_{γ_{1}}a_{γ_{2}} ≠ 0 for some γ_{1} ≠ γ_{2}.

Under this assumption, (22) implies that

∑_{γ} a_{γ}γ ⊗ γ = (e ⊗ f) ∑_{β,β′} a_{β}a_{β′}(β ⊗ β′)
= (e ⊗ f)[a_{γ_{1}}^{2}(γ_{1} ⊗ γ_{1}) + a_{γ_{2}}^{2}(γ_{2} ⊗ γ_{2}) + a_{γ_{1}}a_{γ_{2}}(γ_{1} ⊗ γ_{2}) + (other terms)].  (23)

If e = f, then (23) implies that a_{γ_{1}}a_{γ_{2}}(γ_{1} ⊗ γ_{2}) = 0, whence it follows that a_{γ_{1}}a_{γ_{2}} = 0, which is a contradiction. On the other hand, if e ≠ f, then we see from (23) that a_{γ_{1}}^{2} = a_{γ_{2}}^{2} = 0, which is also a contradiction. Hence u ∈ Γ as claimed.

This completes the proof of the result. □

6.2. Class invariant maps. We retain the notation established in the previous section. In this subsection we describe a natural refinement of the class invariant homomorphism introduced by Waterhouse in [33].

Suppose that F is a number field. Then the class invariant homomorphism ψ is defined by

ψ : H^{1}(Spec(R), G) → Pic(G^{∗}); π ↦ (L_{π}).

Remark 6.5. In general ψ is not injective (see e.g. [1], [26], [28]). It is known that the image of ψ is always contained in the subgroup PPic(G^{∗}) of primitive classes of Pic(G^{∗}) (see e.g. [15]). It is shown by the first-named author in [2] that if the composition series of every geometric fibre of G → Spec(R) of residue characteristic 2 does not contain a factor of local-local type, then the image of ψ is in fact equal to PPic(G^{∗}). □

24 A. AGBOOLA AND D. BURNS

Now suppose that π : X → Spec(R) is a G-torsor, and fix a choice of isomorphism ξ_{π} as in (2). For ease of notation, we use the same symbol ξ_{π} for the map L_{π} ⊗_{R} F^{c} ⟶^{∼} A_{F^{c}} induced by (2). In general the element [L_{π}, 𝔄; ξ_{π}] ∈ K_{0}(𝔄, R^{c}) depends upon the choice of ξ_{π}.

Recall that there is a natural morphism

ι_{𝔄,R^{c}} : K_{0}(𝔄, R^{c}) → K_{0}(𝔄, F^{c})

arising via extension of scalars from R to F. The image of ι_{𝔄,R^{c}}([L_{π}, 𝔄; ξ_{π}]) under the isomorphism (10) with E = F^{c} may be described as follows. Choose any trivialisation s_{π} : A ⟶^{∼} L_{π} ⊗_{R} F. For each finite place v of F, choose a trivialisation t_{π,v} : 𝔄_{v} ⟶^{∼} L_{π} ⊗_{R} R_{v}. It is easy to check that the isomorphism

A_{v} ⟶^{t_{π,v}} L_{π} ⊗_{R} F_{v} ⟶^{(s_{π}⊗_{F} F_{v})^{−1}} A_{v}

is given by multiplication by r(t_{π,v}).r(s_{π})^{−1} ∈ A_{v}^{×}. Then Remark 3.8 implies that a representative of the image of ι_{𝔄,R^{c}}([L_{π}, 𝔄; ξ_{π}]) under the isomorphism (10) is given by

∏_{v∈S_{f} (F)}(r(t_{π,v}).r(s_{π})^{−1}) × r(s_{π}) ∈ J_{f}(A) × A_{F^{c}}^{×}.  (24)

Observe that if we change the choice of splitting isomorphism ξ_{π}, the element r(s_{π}) is altered by multiplication by an element of Γ. This motivates the following definition:

Definition 6.6. Let Δ′′_{𝔄,F^{c}} denote the composition of the homomorphism

Δ′_{𝔄,F^{c}} : A^{×} ⟶ \frac{J_{f}(A)}{U_{f}(𝔄)} × A_{F^{c}}^{×}

defined in Example 3.9(ii) with the quotient map

\frac{J_{f}(A)}{U_{f}(A)} × A_{F^{c}}^{×} ⟶ \frac{J_{f}(A)}{U_{f}(A)} × \frac{A_{F^{c}}^{×}}{Γ} .

We define \overline{K}_{0}(𝔄, F^{c}) to be the quotient of K_{0}(𝔄, F^{c}) defined by the isomorphism

\overline{K}_{0}(𝔄, F^{c}) ≃ \frac{\frac{J_{f}(A)}{U_{f}(𝔄)} × \frac{A_{F^{c}}^{×}}{Γ}}{Im(Δ′′_{𝔄,F^{c}})} .  (25)

We write \overline{K}_{0}(𝔄, R^{c}) for the image of K_{0}(𝔄, R^{c}) under the composite of the scalar extension map ι_{𝔄,R^{c}} and the natural projection map K_{0}(𝔄, F^{c}) → \overline{K}_{0}(𝔄, F^{c}). □

Lemma 6.7. The natural projection map K_{0}(𝔄, F^{c}) → \overline{K}_{0}(𝔄, F^{c}) is bijective if and only if the points of Γ = G(F^{c}) are fixed by Ω_{F}.

Proof. It is clear that the stated projection is bijective if and only if Γ ⊆ 𝔄_{v}^{×} for all v ∈ S_{f}(F), i.e. if and only if Γ ⊆ 𝔄^{×}. But this last condition obtains if and only if the points of Γ are fixed by Ω_{F}. □

ON TWISTED FORMS AND RELATIVE ALGEBRAIC K-THEORY 25

It follows from (24) that the class of ι_{𝔄,R^{c}}([L_{π}, 𝔄; ξ_{π}]) in \overline{K}_{0}(𝔄, F^{c}) is represented by

∏_{v∈S_{f} (F)}(r(t_{π,v}).r(s_{π})^{−1}) × r(s_{π}) ∈ J_{f}(A) × H(A).

It is easy to check that this class is independent of the choice of splitting isomorphism ξ_{π} and that it depends only upon the isomorphism class of the G-torsor π : X → Spec(R).

Theorem 6.8. The map

ψ : H^{1}(Spec(R), G) → \overline{K}_{0}(𝔄, R^{c}); π ↦ ι_{𝔄,R^{c}}([L_{π}, 𝔄; ξ_{π}])

is an injective group homomorphism.

Proof. Let us first show that ψ is a group homomorphism. Suppose that π and π′ are G-torsors. Let ξ_{π} and ξ_{π′} denote splitting isomorphisms associated to π and π′ respectively. It follows from the functoriality of Waterhouse’s construction in [33] that there is a canonical isomorphism L_{π} ⊗_{𝔄}L_{π′} ≃ L_{π·π′} and that the map

ξ_{π} ⊗_{𝔄_{R^{c}}} ξ_{π′} : (L_{π} ⊗_{R} R^{c}) ⊗_{𝔄_{R^{c}}} (L_{π′} ⊗_{R} R^{c}) ⟶^{∼} 𝔄_{R^{c}} ⊗_{𝔄_{R^{c}}} 𝔄_{R^{c}} ⟶^{∼} 𝔄_{R^{c}}

is a splitting isomorphism associated to π · π′. Since in K_{0}(𝔄, R^{c}), we have

[L_{π}, 𝔄; ξ_{π}] + [L_{π′}, 𝔄; ξ_{π′}] = [L_{π} ⊗_{𝔄} L_{π′}, 𝔄; ξ_{π} ⊗_{𝔄_{R^{c}}} ξ_{π′}],

(cf. for example [8, proof of Lemma 2.6(i)]), it follows that

ψ(π) + ψ(π′) = ψ(π · π′),

as required.

To show that ψ is injective, we argue as follows. If we compose the natural injection H^{1}(Spec(R), G) → H^{1}(F, G) with the isomorphism H^{1}(F, G) → H(A) afforded by Theorem 6.2, then we obtain an injection H^{1}(Spec(R), G) → H(A). It follows easily from the definitions that this injection is the same as the homomorphism obtained by composing ψ with the inclusion \overline{K}_{0}(𝔄, R^{c}) ⊆ \overline{K}_{0}(𝔄, F^{c}) and the natural projection

\overline{K}_{0}(𝔄, F^{c}) → \frac{A_{F^{c}}^{×}}{Γ · A^{×}} .

This implies that ψ is injective. □

Remark 6.9. Let ϕ ∈ Γ̂. Then for each γ ∈ Γ and v ∈ S_{∞}(F), the complex number σ_{v}(ϕ(γ)) is a root of unity and hence has absolute value 1. This implies that the morphism

∂_{𝔄,F^{c}} : K_{0}(𝔄, F^{c}) → AC(𝔄)

defined in §4.3 induces a morphism \overline{K}_{0}(𝔄, F^{c}) → AC(𝔄). Upon composing this morphism with ψ we obtain a morphism

ψ̂ : H^{1}(Spec(R), G) → AC(𝔄) ≃ P̂ic(G^{∗})

26 A. AGBOOLA AND D. BURNS

which was first introduced and studied by the first-named author and G. Pappas in [6]. In turn, upon composing ψ̂ with the natural surjection AC(𝔄) → Pic(G^{∗}) we obtain the homomorphism

ψ : H^{1}(Spec(R), G) → Pic(G^{∗})

which was introduced by Waterhouse in [33]. □

Write P \overline{K}_{0}(𝔄, R^{c}) for the subgroup of primitive elements of \overline{K}_{0}(𝔄, R^{c}).

Theorem 6.10. (i) The image of ψ lies in P \overline{K}_{0}(𝔄, R^{c}).

(ii) Suppose that the composition series of every geometric fibre of G → Spec(R) of residue characteristic 2 does not contain a factor of local-local type. Then the image of ψ is equal to P \overline{K}_{0}(𝔄, R^{c}).

Proof. (i) Recall from Example 2.3 that we have a canonical isomorphism

Ext^{1}(G^{∗}, G_{m}) ≃ H^{1}(Spec(R), G).

Composing this isomorphism with ψ gives a homomorphism

κ_{G^{∗}} : Ext^{1}(G^{∗}, G_{m}) → \overline{K}_{0}(𝔄, R^{c})

whose image is the same as that of ψ. It follows from the functoriality of Waterhouse’s construction in [33] that the homomorphism κ_{G^{∗}} is functorial in G^{∗}.

Let m : G^{∗} × G^{∗} → G^{∗} denote multiplication on G^{∗}, and write p_{i} : G^{∗} × G^{∗} → G^{∗} (i = 1, 2) for projection onto the i-th factor. Let

m^{∗}, p_{i}^{∗} : Ext^{1}(G^{∗}, G_{m}) → Ext^{1}(G^{∗} × G^{∗}, G_{m})

be the homomorphisms induced by m and p_{i}. Then it follows from standard properties of the functor Ext^{1} that m^{∗}(x) = p_{1}^{∗}(x) + p_{2}^{∗}(x) for all x ∈ Ext^{1}(G^{∗}, G_{m}) (see e.g. [27, Proposition 1 on p. 163, and p. 182]). Since κ_{G^{∗}} is functorial in G^{∗}, this implies that κ_{G^{∗}}(x) is a primitive element of \overline{K}_{0}(𝔄, R^{c}).

(ii) It is shown in [3] that there is a natural isomorphism Ker(ψ) ≃ H(𝔄). From [2, Theorem 1.3], we know that, under the given hypotheses, we have that Im(ψ) = PPic(G^{∗}). Since ψ is injective, this implies that there is an exact sequence

0 → H(𝔄) → Im(ψ) → PPic(G^{∗}) → 0.  (26)

Next, we observe that upon comparing the exact sequences (5) with Λ = R^{c} and Λ = F^{c}, we obtain a commutative diagram of short exact sequences

0 ⟶ cok[K_{1}(𝔄) → K_{1}(𝔄_{R^{c}})] ⟶ K_{0}(𝔄, R^{c}) ⟶ ker[K_{0}(𝔄) → K_{0}(𝔄_{R^{c}})] ⟶ 0
↓  ^{ι_{𝔄,R^{c}}} ↓  ^{⊆}↓
0 ⟶ cok[K_{1}(𝔄) → K_{1}(A_{F^{c}})] ⟶ K_{0}(𝔄, F^{c}) ⟶ ker[K_{0}(𝔄) → K_{0}(A_{F^{c}})] ⟶ 0.

ON TWISTED FORMS AND RELATIVE ALGEBRAIC K-THEORY 27

Now, since 𝔄 is commutative, the second and fourth terms of the lower sequence can be identified with A_{F^{c}}^{×}/𝔄^{×} and Pic(𝔄) respectively (cf. the end of Remark 3.3). Further, the image of the natural composite morphism

K_{1}(𝔄_{R^{c}}) → K_{1}(A_{F^{c}}) → cok[K_{1}(𝔄) → K_{1}(A_{F^{c}})] ≅ A_{F^{c}}^{×}/𝔄^{×}

is equal to 𝔄_{R^{c}}^{×}/𝔄^{×}. Upon identifying Pic(𝔄) with Pic(G^{∗}) the above diagram implies that there is an exact sequence

1 → \frac{𝔄_{R^{c}}^{×}}{𝔄^{×}} → ι_{𝔄,R^{c}}(K_{0}(𝔄, R^{c})) → Pic(G^{∗}).

It is easy to see that this in turn induces an exact sequence

1 → \frac{𝔄_{R^{c}}^{×}}{Γ · 𝔄^{×}} → \overline{K}_{0}(𝔄, R^{c}) → Pic(G^{∗}).  (27)

Since Im(ψ) ⊆ P \overline{K}_{0}(𝔄, R^{c}) it follows from (26) that the natural map P \overline{K}_{0}(𝔄, R^{c}) → PPic(G^{∗}) is surjective. Hence Theorem 6.4 implies that (27) induces an exact sequence

0 → H(𝔄) → P \overline{K}_{0}(𝔄, R^{c}) → PPic(G^{∗}) → 0.  (28)

We now see from (26) and (28) that Im(ψ) = P \overline{K}_{0}(𝔄, R^{c}), as asserted. □

6.3. Rings of integers. In this subsection we shall briefly explain how McCulloh’s results on realisable classes of tame extensions in [24] can be naturally lifted to the corresponding relative algebraic K-group. Proposition 5.7 implies that such realisability results in the relative algebraic K-group are related to those of Chase in [14].

Suppose that F is a number field with ring of integers R. Let G be any finite, constant abelian group scheme over Spec(R), and set Γ := G(F^{c}). Then in this case, we have G = Spec(𝔅) and G^{∗} = Spec(𝔄), where

𝔅 = Map(Γ, R), 𝔄 = R[Γ].

Write G_{F} and G_{F}^{∗} for the generic fibres of G and G^{∗} respectively. Note that, since G_{F} is a constant group scheme over Spec(F), Lemma 6.7 implies that \overline{K}_{0}(𝔄, F^{c}) is canonically isomorphic to K_{0}(𝔄, F^{c}).

Now suppose that π : X → Spec(F) is a G_{F} -torsor, and write L_{π} for its associated line bundle on G_{F}^{∗}. Then X = Spec(C_{π}), where C_{π} is Galois algebra extension of F with Galois group Γ.

Definition 6.11. We say that π is tame if it is trivialised by a tamely ramified extension of F. Equivalently, π is tame if and only if the extension C_{π}/F is tamely ramified.

We write H_{t}^{1}(Spec(F), G_{F}) for the subgroup of H^{1}(Spec(F), G_{F}) consisting of isomorphism classes of tame G_{F} -torsors. □

Let O_{π} denote the integral closure of R in C_{π}. If π is tame, then Noether’s theorem (see e.g. [20, Chapter I, §3]) implies that O_{π} is a locally free, rank one 𝔄-module, and so it yields a line

28 A. AGBOOLA AND D. BURNS

bundle on G^{∗} which we shall denote by M_{π}. Let π_{0} : G_{F} → Spec(F) denote the trivial G_{F} -torsor, and set

\overline{L}_{π} := M_{π} ⊗ M_{π_{0}}^{−1}.

It follows from the definitions that there is a natural isomorphism \overline{L}_{π} ⊗_{R} F ≃ L_{π}. If we make this identification, then any choice of splitting isomorphism ξ_{π} for π gives a class [\overline{L}_{π}, 𝔄; ξ_{π}] in K_{0}(𝔄, F^{c}). This class is independent of the choice of ξ_{π}. Hence we obtain a map

ψ_{t} : H_{t}^{1}(Spec(F), G_{F}) → K_{0}(𝔄, F^{c});  π ↦ [\overline{L}_{π}, 𝔄; ξ_{π}].  (29)

If F is a number field, then a representative in J_{f}(A) × A_{F^{c}}^{×} of ψ_{t}(π) may be given exactly as in the previous section. It may be shown via an argument very similar to that given in Theorem 6.8 that the map ψ_{t} is injective. However, in general, ψ_{t} is not a group homomorphism (see Remark 6.13 below).

Definition 6.12. We say that an element of K_{0}(𝔄, F^{c}) is realisable if it lies in the image of ψ_{t}. □

The general strategy for describing the set of realisable classes in K_{0}(𝔄, F^{c}) is as follows. Let

j : J_{f}(A) × [A_{F^{c}}^{×}/Γ] → K_{0}(𝔄, F^{c})

denote the obvious quotient map afforded by (25) (together with Lemma 6.7), and write H(𝔸(A)) for the restricted direct product of the groups H(A_{v}) with respect to the subgroups H(𝔄_{v}). Then the obvious natural maps A_{v}^{×} → H(A_{v}) induce a homomorphism

rag : J_{f}(A) → H(𝔸(A)).

Suppose that \underline{c} := c_{fin} × c_{∞} ∈ J_{f}(A) × [A_{F^{c}}^{×}/Γ] satisfies j(\underline{c}) ∈ Im(ψ_{t}). McCulloh shows that the idele rag(c_{fin}) admits a special type of decomposition (see [24, Theorem 5.6]). It is a straightforward exercise to use the results of [24] to characterise the image of ψ_{t} in a manner similar to that given in [24, Theorem 6.7]. We omit the details in order to keep this paper to a reasonable length.

Remarks 6.13. (i) McCulloh has shown that the image of the map obtained by composing ψ_{t} with the quotient map K_{0}(𝔄, F^{c}) → Pic(G^{∗}) is a subgroup of Pic(G^{∗}) (see [24, Corollary 6.20]). On the other hand, by using the description of the image of ψ_{t} mentioned above, it may be shown that in general, Im(ψ_{t}) is not a group. This situation is very similar to that which obtains in the case of realisable classes in classgroups of sheaves discussed in [4, Remark 2.10(iii)].

(ii) By composing ψ_{t} with the natural morphism ∂_{𝔄,F^{c}} : K_{0}(𝔄, F^{c}) → AC(𝔄) defined in §4.3, we obtain a map

ψ̂_{t} : H_{t}^{1}(Spec(F), G_{F}) → AC(𝔄).

The results of [24] may also be used to give a description of Im(ψ̂_{t}). It follows from Proposition 4.11 that such a description may be viewed as a realisability result for certain metrised 𝔄-modules (cf. Remark 6.9). □

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Dept. of Math., University of California at Santa Barbara, CA 93106, U.S.A.

E-mail address: agboola@math.ucsb.edu

Dept. of Math., King’s College London, Strand, London WC2R 2LS, England

E-mail address: david.burns@kcl.ac.uk