ON GALOIS STRUCTURE INVARIANTS ASSOCIATED TO TATE MOTIVES

D. BURNS AND M. FLACH

Abstract. We establish the equivalence of two definitions of invariants measuring the Galois module structure of K-groups of rings of integers in number fields (due to Chinburg et al on the one hand and the authors on the other). We also make some remarks concerning the possibility of yet another such definition via Lichtenbaum complexes.

1. Introduction

Let L/K be a finite Galois extension of number fields of group G, and X a smooth projective variety defined over K. If G is abelian then the present authors have shown how the Galois cohomological methods introduced by Bloch, Kato, Fontaine and Perrin-Riou (cf. [1], [15], [10]) allow one to attach to each motive M = h^{n}(X)(r) (which satisfies certain ‘standard’ conjectures) a canonical element Ω(L/K, M) of Pic(ℤ[G]) (cf. [5]). These elements combine, in a rather complicated way in general, information about the G-structure of lattices in the de Rham, Betti and motivic cohomology spaces attached to M_{L} := h^{n}(X ×_{K} L)(r). The construction of Ω(L/K, M) is unconditional in the case of Tate motives M = ℚ(r), where r can be any integer, and motives M = h^{1}(X)(1) where X/K is an abelian variety for which the Tate-Safarevic group of X ×_{K} L is finite. Moreover, the invariant Ω(L/K, ℚ(0)) (respectively Ω(L/K, ℚ(1))) was shown to be closely related to an invariant Ω(L/K, 3) (resp. Ω(L/K, 1)) defined by Chinburg in [6] (resp. [7]).

More recently Chinburg, Kolster, Pappas and Snaith have introduced, for arbitrary groups G and integers n > 0, an element Ω_{n}(L/K) of the reduced Grothendieck group Cl(ℤ[G]) of the category of finitely generated projective ℤ[G]-modules, which combines information about the ℤ[G]-module structures of K_{2n+1}(O_{L}) and K_{2n}(O_{L}) (cf. [9]). This might be considered as a generalization of Chinburg’s work to higher K-groups, since Ω(L/K, 3) is related to the ℤ[G]-module structure of O_{L}^{×} and Pic(O_{L}). To reflect this analogy one puts Ω_{0}(L/K) := Ω(L/K, 3).

In this note we concentrate on the case that G is abelian so that there is an identification of Cl(ℤ[G]) with Pic(ℤ[G]) induced by det_{ℤ[G]}. It therefore makes sense to compare the classes defined in [9] to those which arise from Tate motives via the approach of [5]. Let ‘#’ denote the involution on Pic(ℤ[G]) induced by changing the G-action on a module by composing with the automorphism g ↦ g^{−1} of G. Our main result is the identity

Ω(L/K, ℚ(−n))^{#} = Ω_{n}(L/K) (1)

for each integer n ≥ 0. The comparison of the two sides in (1) is somewhat easier for n > 0 since then the definitions of both Ω_{n}(L/K) and Ω(L/K, ℚ(−n)) involve perfect complexes naturally arising from étale cohomology. In the case n = 0 on the other hand, the definition of Ω(L/K, 3) is based on complexes representing specific extension classes of ℤ[G]-modules, and these can be recovered from étale cohomology only after some effort.

The second author was partially supported by NSF-grant DMS-9506412 and a Sloan fellowship.

After recalling the definition of our invariant Ω(L/K, ℚ(−n)) in section 2, we therefore treat the two cases n = 0 and n > 0 separately in section 3 and 4, respectively. In the final section 5, we briefly indicate how the expected properties of Lichtenbaum’s complexes Γ(r) for r ≥ 2 naturally give rise to invariants in Cl(ℤ[\frac{1}{2}][G]), and how these invariants relate to Ω_{r−1}(L/K) (and hence to Ω(L/K, ℚ(1 − r))).

Meanwhile, in [3], the construction of Ω(L/K, M) has been generalized from abelian to arbitrary Galois extensions L/K. The methods we develop in this paper also give the identity (1) in the general case.

Acknowledgements: We are very grateful to G. Pappas, B.Kahn and V.Snaith for useful discussions. The second author would also like to thank the IAS, Princeton for its hospitality during the final preparation of this paper.

2. Definition of Ω(L/K, ℚ(r))

We briefly recall from [5] the main steps in the definition of Ω(L/K, M) in the case M = ℚ(r), r ≤ 0, and set up some notation. Given a set S of places of K, the letter S will also stand for the set of places of L lying above those in S. Confusion is avoided because we denote places of K by v and places of L by w. We put S_{p} = S ∪ {v|p}, where p is any prime number, and S_{f} = S \ S_{∞}, where S_{∞} is the set of archimedean places of K (so f will never denote a prime number). From now on we fix a finite set S of places of K containing S_{∞} and all places which ramify in L/K. We put E = ℚ[G], E_{p} = E ⊗ ℚ_{p} = ℚ_{p}[G] and E_{p} = ℤ_{p}[G].

Under the assumption that the motivic cohomology groups of M_{L} are finite dimensional ℚ-spaces a certain invertible (i.e. free rank 1) E-module Ξ(M_{L}) was defined in [5, §1.4]. We recall its definition for M = ℚ(r) with r ≤ 0 in which case the necessary finite-dimensionality is known. If r = 0

Ξ(ℚ(0)_{L}) := det_{E}(O_{L}^{×} ⊗_{ℤ} ℚ)^{#} ⊗_{E} det_{E}^{−1}ℚ ⊗_{E} ⨂_{v∈S_{∞}} det_{E}H^{0}(K_{v}, ℚ[G]),

and if r < 0

Ξ(ℚ(r)_{L}) := det_{E}(K_{1−2r}(O_{L}) ⊗_{ℤ} ℚ)^{#} ⊗_{E} ⨂_{v∈S_{∞}} det_{E}H^{0}(K_{v}, ℚ[G](r)).

Here the notation ^{#} indicates a change of G-action on the same underlying abelian group: gx^{#} := g^{−1}x. The space ℚ[G] should be thought of as the Betti realisation of h^{0}(Spec L), viewed as an étale sheaf of ℚ-vector spaces on Spec K_{v} for v ∈ S_{∞}. Similarly, after a choice of embedding σ : L → K̄, one can identify the p-adic realisation of h^{0}(Spec L) with ℚ_{p}[G] as a (left) G-module. In this identification the (left) action of γ ∈ Gal(K̄/K) is given by

γ(x) := xπ(γ^{−1}), x ∈ ℚ_{p}[G] (2)

where π : Gal(K̄/K) → G is defined by γσ(λ) = σπ(γ)(λ) for all λ ∈ L.

We let O_{L,S} denote the ring of S-integers in L. For a sheaf F on (Spec O_{L,S})_{et}, or more generally for a ℤ_{p} or ℚ_{p}-sheaf, we define the cohomology with compact support as

RΓ_{c}(O_{L,S}, F) := Cone(RΓ(O_{L,S}, F) → ⨁_{w∈S} RΓ(L_{w}, F))[−1]. (3)

Then there are isomorphisms for any prime number p [5, (1.17)]

ϑ_{p} : Ξ(ℚ(r)_{L}) ⊗_{ℚ} ℚ_{p} →^{∼} det_{E_{p}}RΓ_{c}(O_{L,S_{p}}, ℚ_{p}(r)) (4)

which are constructed as follows. Put (−)^{∧} = Hom_{ℚ_{p}}(−, ℚ_{p}) with contragredient G-action so that one has an isomorphism

det_{E_{p}}W^{∧} ≅ det_{E_{p}}^{−1}W^{#} (5)

for any ℚ_{p}[G]-space W. The exact triangle [5][(1.11)]

(6) RΓ_{c}(O_{L,S_{p}}, ℚ_{p}(r)) → RΓ_{f}(L, ℚ_{p}(r)) →

⨁_{v∈S_{p,f}} RΓ_{f}(K_{v}, ℚ_{p}[G](r)) ⊕ ⨁_{v∈S_{∞}} H^{0}(K_{v}, ℚ_{p}[G](r)) →

together with the isomorphism [5][p.73/74]

H_{f}^{i}(L, ℚ_{p}(r)) ≅ \begin{cases}ℚ_{p} & i = r = 0 \\ (K_{1−2r}(O_{L}) ⊗_{ℤ} ℚ_{p})^{∧} & i = 2 \\ 0 & otherwise \end{cases}

and (5) gives an isomorphism

Ξ(ℚ(r)_{L}) ⊗_{ℚ} ℚ_{p} →^{∼} det_{E_{p}}RΓ_{c}(O_{L,S_{p}}, ℚ_{p}(r)) ⊗_{E_{p}} ⨂_{v∈S_{p,f}} det_{E_{p}}RΓ_{f}(K_{v}, ℚ_{p}[G](r)). (7)

Now there are several different ways to identify the determinant of RΓ_{f}(K_{v}, ℚ_{p}[G](r)) with E_{p} and since this is a possible source of confusion we explain the situation here in some detail (see also the remark after [5][(1.16)] to that effect). Recall from [5][(1.5),(1.7)] that the complex RΓ_{f}(K_{v}, ℚ_{p}[G](r)) is naturally quasi-isomorphic

to a complex V_{v} ⟶^{ϕ_{v}} V_{v} where V_{v} is a projective E_{p}-module, and ϕ_{v} ∈ End_{E_{p}}(V_{v}) is moreover ‘semisimple at 0’ in the following sense.

Let R be a semisimple commutative ring, V a finitely generated R-module and ϕ ∈ End_{R}(V). One says that ϕ is semisimple at 0 if the map id_{ϕ} : ker(ϕ) ⊆ V → coker(ϕ) is an isomorphism, or equivalently if V ≅ ker(ϕ) ⊕ D as R[ϕ]-module. In this situation one defines det_{R}^{∗}(ϕ) := det_{R}(ϕ|D) ∈ R^{×} which is independent of the

choice of D. Denote by C the complex V ⟶^{ϕ} V. For any isomorphism of R-modules

V ⟶^{ψ} W let

ψ_{triv} : det_{R}^{−1}V ⊗_{R} det_{R}W ⟶^{∼} R

be the isomorphism obtained by composing det_{R}^{−1}(ψ)⊗1 with the evaluation pairing det_{R}^{−1}W ⊗_{R} det_{R}W ⟶^{∼} R. Here det_{R}^{−1}(V) = Hom_{R}(det_{R}(V), R) and det_{R}^{−1}(ψ) = Hom_{R}(det_{R}(ψ), R)^{−1}.

Lemma 1. Assume ϕ is semisimple at 0. With the notation just introduced there is a commutative diagram

det_{R}(C)  ⟶^{id_{V,triv}} R
↓  ↓^{·det_{R}^{∗}(ϕ)^{−1}}
det_{R}^{−1}H^{0}(C) ⊗_{R} det_{R}H^{1}(C) ⟶^{id_{ϕ,triv}} R

(8)

where the left vertical map is induced by the exact sequence

0 → H^{0}(C) → V ⟶^{ϕ} V → H^{1}(C) → 0

(this is the canonical identification of [16] but with the sign convention of [5][0.2]).

Proof. Immediate from the definitions. □

Coming back to C_{v} = RΓ_{f}(K_{v}, ℚ_{p}[G](r)) over R = E_{p} the isomorphism ϑ_{p} in (4) arises from applying

⨂_{v∈S_{p,f}} id_{V_{v},triv}

to the last term in (7). If r < 0 then RΓ_{f}(K_{v}, ℚ_{p}[G](r)) is acyclic, and the long exact sequence induced by (6) gives isomorphisms

(9)  ⨁_{v∈S_{∞}} H^{0}(K_{v}, ℚ_{p}[G](r)) →^{∼}H_{c}^{1}(O_{L,S_{p}}, ℚ_{p}(r))
(10)  H_{c}^{2}(O_{L,S_{p}}, ℚ_{p}(r)) →^{∼}(K_{1−2r}(O_{L}) ⊗_{ℤ} ℚ_{p})^{∧}.

If we denote by ϑ̃_{p} : Ξ(ℚ(r)_{L}) ⊗_{ℚ} ℚ_{p} →^{∼} det_{E_{p}}RΓ_{c}(O_{L,S_{p}}, ℚ_{p}(r)) the isomorphism induced by (9) and (10) then we have implicitly used the diagonal map in (8) and Lemma 1 implies

ϑ̃_{p} = ϵ_{S_{p}}(r)ϑ_{p} (11)

where (see [5][(1.7)] for the definition of C_{v})

ϵ_{S_{p}}(r) = ∏_{v∈S_{p,f}} det_{E_{p}}^{∗}(ϕ_{v})^{−1} = ∏_{v∈S_{p,f}} det_{E_{p}}(ϕ_{v})^{−1} = ∏_{v∈S_{p,f}}(1 − Nv^{−r}f_{v})^{−1} ∈ E^{×}. (12)

Here

f_{v} ∈ ℚ[G/I_{v}] ≅ ℚ[G]^{I_{v}} ⊆ E (13)

is the Frobenius automorphism at v and I_{v} ⊆ G_{v} ⊆ G are the inertia and decomposition group for a place w|v. For r = 0 the situation is slightly more complicated. We have canonical isomorphisms

(14) H_{f}^{0}(K_{v}, ℚ_{p}[G]) = H_{f}^{0}(L_{w}, ℚ_{p}[G/G_{v}]) ≅ H^{0}(L_{w}, ℚ_{p}[G/G_{v}]) = ℚ_{p}[G/G_{v}]
H_{f}^{1}(K_{v}, ℚ_{p}[G]) ≅ H_{f}^{1}(L_{w}, ℚ_{p}[G/G_{v}]) ≅ Hom(Gal(L_{w}^{ur}/L_{w}), ℚ_{p}[G/G_{v}])
≅ ℚ_{p}[G/G_{v}]

where the last map is evaluation at the Frobenius automorphism. Via these maps we can identify ⨁_{v∈S_{p,f}} H_{f}^{i}(K_{v}, ℚ_{p}[G]) for both i = 0 and i = 1 with the free ℚ_{p}-space Y_{S_{p,f}} ⊗_{ℤ} ℚ_{p} on the set of places of L above those in S_{p,f} (consistently with later notation, Y_{S} will denote the free abelian group on the places above S). So for r = 0 the long exact sequence induced by (6) reads

(15) 0 → ℚ_{p} → ⨁_{v∈S_{∞}} H^{0}(K_{v}, ℚ_{p}[G]) ⊕ (Y_{S_{p,f}} ⊗_{ℤ} ℚ_{p}) → H_{c}^{1}(O_{L,S_{p}}, ℚ_{p}) → 0

→ Y_{S_{p,f}} ⊗_{ℤ} ℚ_{p} → H_{c}^{2}(O_{L,S_{p}}, ℚ_{p}) → (O_{L}^{×} ⊗_{ℤ} ℚ_{p})^{∧} → 0

and we denote by ϑ̃_{p} : Ξ(ℚ(0)_{L}) ⊗_{ℚ} ℚ_{p} →^{∼} det_{E_{p}}RΓ_{c}(O_{L,S_{p}}, ℚ_{p}) the isomorphism induced by (15) together with the identity map on Y_{S_{p,f}} ⊗_{ℤ} ℚ_{p}. From diagram (8) we then deduce equation (11) with the factor ϵ_{S_{p}}(0) = ∏_{v∈S_{p,f}} ϵ_{v}(0) where

ϵ_{v}(0) := det_{E_{p}}^{∗}(ϕ_{v})^{−1} id_{ℚ_{p}[G/G_{v}],triv} ∘ id_{ϕ_{v},triv}^{−1} ∈ E_{p}^{×}.

Lemma 2. Put G^{∨} = Hom(G, ℂ^{×}) and let v be a finite place of K. We introduce the following notation depending on v. For x ∈ ℚ^{×}, resp. e ∈ E, let ^{∗}x ∈ E^{×}, resp. e^{∗} ∈ E, be the element such that χ(^{∗}x) = x, resp. χ(e^{∗}) = 0, if χ(G_{v}) = 1 and χ(^{∗}x) = 1, resp. χ(e^{∗}) = χ(e) otherwise, for all characters χ ∈ G^{∨}. Then with f_{v} as in (13)

ϵ_{v}(0) = ^{∗}|G_{v}/I_{v}| · (1 − f_{v}^{∗})^{−1} ∈ E^{×}. (16)

In particular, ϵ_{v}(0) is independent of p.

Proof. For simplicity we suppose G = G_{v} in this proof. The general case easily follows by inducing from G_{v} to G. We put V = ℚ_{p}[G_{v}] with its natural G = G_{v}-action and with Gal(K̄_{v}/K_{v})-action given by (2). If v ∤ p we set Γ_{v} := Gal(K_{v}^{ur}/K_{v}) ⊇ Γ_{w} := Gal(K_{v}^{ur}/L_{w} ∩ K_{v}^{ur}). Using (2) one verifies that V^{I_{v}} = ℚ_{p}[G_{v}]^{I_{v}} is isomorphic to Coind_{Γ_{w}}^{Γ_{v}}ℚ_{p} as a Γ_{v}-module and that the natural map π^{0} : Coind_{Γ_{w}}^{Γ_{v}}ℚ_{p} → ℚ_{p} is given by

π^{0}( ∑_{g∈G_{v}/I_{v}} λ_{g}g ∑_{i∈I_{v}} i) = λ_{1}.

Consider the following commutative diagram of complexes with vertical differentials

V^{I_{v}} ⟶^{φ_{v}} V^{I_{v}}  V^{I_{v}}  ⟶^{π^{0}}  ℚ_{p}
1−φ_{v} ↓  ^{φ_{v}^{−1}−1}↓  _{δ^{0}}↓  ^{0}↓
V^{I_{v}}  V^{I_{v}} ⟵^{α} Map(Γ_{v}, V^{I_{v}}) ⟶^{π^{1}} Map(Γ_{w}, ℚ_{p})
δ^{1}↓  _{δ^{1}}↓  .

(17)

Here φ_{v} ∈ Γ_{v} is the inverse Frobenius automorphism, the third (resp. fourth) column is the standard continuous cochain complex for Γ_{v} (resp. Γ_{w}) with coefficients in V^{I_{v}} (resp. ℚ_{p}), one has δ^{0}(x) = γ ↦ (γ − 1)(x), α(ψ) = ψ(φ_{v}^{−1}) and π^{1} is given by restricting to Γ_{w} and composing with π^{0}. By Shapiro’s Lemma π^{•} is a quasi-isomorphism and so are the other horizontal maps in (17). By definition [5][(1.7)] the first column in (17) is the complex C_{v} = (V_{v} ⟶^{ϕ_{v}} V_{v}). To compute id_{ℚ_{p},triv} ∘ id_{ϕ_{v},triv}^{−1} we must identify H^{0}(C_{v}) and H^{1}(C_{v}) with ℚ_{p} as indicated in (14), express the map id_{ϕ_{v}} on ℚ_{p}, and take its determinant over E_{p}. The element e := ∑_{g∈G_{v}} g ∈ H^{0}(C_{v}) satisfies π^{0}φ_{v}(e) = π^{0}(e) = 1 ∈ ℚ_{p}. On the other hand, α^{−1}(e) is the cocycle ψ with ψ(φ_{v}^{−1}) = e. If we evaluate ψ at the Frobenius automorphism of L_{w} as in (14) we find

ψ(φ_{v}^{−|G_{v}/I_{v}|}) = (1 + φ_{v}^{−1} + φ_{v}^{−2} + ⋯ + φ_{v}^{−|G_{v}/I_{v}|+1})ψ(φ_{v}^{−1}) = |G_{v}/I_{v}|e

and applying π^{0} gives |G_{v}/I_{v}| ∈ ℚ_{p}. Hence the map id_{ϕ_{v}} on ℚ_{p} is multiplication with |G_{v}/I_{v}| and id_{ℚ_{p},triv} ∘ id_{ϕ_{v},triv}^{−1} is ^{∗}|G_{v}/I_{v}| ∈ E_{p}^{×}. Furthermore, by (2), φ_{v} acts like f_{v} ∈ E^{×} on ℚ_{p}[G_{v}]^{I_{v}} so that det_{E_{p}}^{∗}(ϕ_{v}) = det_{E_{p}}^{∗}(1 − f_{v}) = 1 − f_{v}^{∗}. This finishes the proof for v ∤ p.

To discuss the case v | p we make a slight change of notation and put Γ_{v} := Gal(K̄_{v}/K_{v}), Γ_{w} := Gal(K̄_{v}/L_{w}). Then V ≅ Coind_{Γ_{w}}^{Γ_{v}}ℚ_{p} where the natural map π^{0} : V → ℚ_{p} is π^{0}(∑ λ_{g} g) = λ_{1}. Following [5][(1.3)] we denote by B_{v}^{•}(V) the complex of continuous Γ_{v}-modules

B_{crys} ⊗ V ⟶^{1−Φ⊗1} B_{crys} ⊗ V

which is in fact quasi-isomorphic to V via the inclusion V → B_{crys} ⊗ V in degree 0. Denoting by V → C^{•}(Γ_{v}, V) the standard resolution we find maps of complexes of continuous Γ_{v}-modules

B_{v}^{•}(V) → Tot C^{•}(Γ_{v}, B_{v}^{•}(V)) ⟵^{β} C^{•}(Γ_{v}, V)

with β a quasi-isomorphism. After taking Γ_{v}-invariants β still induces a quasiisomorphism and we find the following diagram in degrees 0 and 1

Crys(V) ⟶  B_{crys} ⊗ V  ⟵  V  ⟶^{π^{0}}  ℚ_{p}
1−Φ↓  _{Δ^{0}}↓  _{δ^{0}}↓  ^{0}↓
Crys(V) ⟶^{ι} Map(Γ_{v}, B_{crys} ⊗ V) ⊕ B_{crys} ⊗ V ⟵^{β^{1}} Map(Γ_{v}, V) ⟶^{π^{1}} Map(Γ_{w}, ℚ_{p})

(18)

where Crys(V) = H^{0}(Γ_{v}, B_{crys} ⊗ V), Δ^{0}(x) = (δ^{0}(x), (1 − Φ ⊗ 1)x), ι(x) = (0, x), β^{1}(x) = (inclusion ∘ x, 0), and δ^{0} is as in (17). Again, according to [5][(1.5)], the first column in (18) is the complex C_{v} = RΓ_{f}(K_{v}, V). C_{v} comes equipped with its natural map into RΓ(K_{v}, V) (this the horizontal map into the third column in (18)). If e := ∑_{g∈G_{v}} g ∈ V we have π^{0}(e) = 1 and the image of e in Crys(V) is 1 ⊗ e. One can find an element λ ∈ W (𝔽̄_{p}) ⊂ B_{crys} (W the ring of Witt vectors) such that (1 − Φ)λ = 1. Then it is easy to see that the 1-cocycle ψ := δ^{0}(λ ⊗ e) takes values in ℚ_{p}(1 ⊗ e) ⊆ V. Indeed, any γ ∈ Γ_{v} acts on W (𝔽̄_{p}) like Φ^{d} for some d ∈ ℤ̂ and acts trivially on e, hence if d ∈ ℕ

(19) (γ − 1)(λ ⊗ e) = ((Φ^{d} − 1)λ) ⊗ e =(1 + Φ + Φ^{2} + ⋯ + Φ^{d−1})(Φ − 1)λ ⊗ e
=(1 + ⋯ + Φ^{d−1})(−1) ⊗ e = −d(1 ⊗ e)

and the general case follows by writing d as a limit of positive integers. The equality

Δ^{0}(λ ⊗ e) = ι(1 ⊗ e) + β^{1}(ψ)

shows that 1 ⊗ e maps to the class of −ψ in H^{1}(Γ_{v}, V). Evaluating the cocycle π^{1}(−ψ) at a lift γ of the Frobenius of L_{w}, as required by (14), and using (19) we find the value to be −π^{0}(−d(1⊗e)) = d where d = [L_{w,0} : ℚ_{p}] with L_{w,0} = W (𝔽̄_{p})∩L_{w}.

The conclusion is that the map ℚ_{p} ≅ H^{0}(C_{v}) ⟶^{id_{ϕv}} H^{1}(C_{v}) ≅ ℚ_{p} is multiplication with d = [L_{w,0} : ℚ_{p}] and that id_{ℚ_{p},triv} ∘ id_{ϕ_{v},triv}^{−1} equals ^{∗}d ∈ E_{p}^{×}.

To compute det_{E_{p}}^{∗}(ϕ_{v}) = det_{E_{p}}^{∗}(1 − Φ), note that Crys(V) is isomorphic to L_{w,0} as a G_{v}-Φ-module via the map L_{w,0} ∋ λ ↦ ∑_{γ∈Γ_{v}/Γ_{w}} γλ ⊗ π(γ^{−1}) where π is as in (2). By the normal basis theorem L_{w,0} is a free ℚ_{p}[G_{v}/I_{v}]-module of rank δ = [K_{v,0} : ℚ_{p}], where K_{v,0} := K_{v} ∩ W (𝔽̄_{p}), Φ is linear with matrix

(\begin{matrix}0 & . . . . . . & 0 & f_{v} \\ 1 & 0 & . . . . . . . & 0 \\ 1 & 0 \\ ... & ... & ⋮ \\ & 1 & 0 \end{matrix})

and 1 − Φ is semisimple at 0. Choosing a decomposition L_{w,0} ≅ ℚ_{p} ⊕ D we find

1 − f_{v}X^{δ} = det_{E_{p}}(1 − Φ · X) = det_{E_{p}}^{∗}(1 − Φ · X)(1 − 1^{∗} · X).

Evaluating at X = 1 we get det_{E_{p}}^{∗}(ϕ_{v}) = ^{∗}δ(1 − f_{v}^{∗}) and so

ϵ_{v}(0) =^{∗} d(^{∗}δ)^{−1}(1 − f_{v}^{∗})^{−1} = ^{∗}|G_{v}/I_{v}|(1 − f_{v}^{∗})^{−1}.
□

For any p^{ν}ℤ_{p}(r) ⊂ ℚ_{p}(r) it was shown in [5, Prop. 1.20] that RΓ_{c}(O_{L,S_{p}}, p^{ν}ℤ_{p}(r)) is a perfect complex of E_{p}-modules such that

I_{p}(r) := det_{E_{p}}RΓ_{c}(O_{L,S_{p}}, p^{ν}ℤ_{p}(r)) ⊂ det_{E_{p}}RΓ_{c}(O_{L,S_{p}}, ℚ_{p}(r))

is independent of ν. Put

Ξ(ℚ(r)_{L})_{ℤ} := ⋂_{p} Ξ(ℚ(r)_{L}) ∩ ϑ_{p}^{−1}I_{p}(r). (20)

According to [5, Prop. 1.42] this is an invertible ℤ[G]-module and so it makes sense to define

Ω(L/K, ℚ(r)) := (Ξ(ℚ(r)_{L})_{ℤ}) ∈ Pic(ℤ[G]). (21)

3. The case n = 0

3.1. Definition of Ω(L/K, 3). Assume S is large enough so that Pic(O_{L,S}) = 0 and put

U_{S} := O_{L,S}^{×}, J_{S} := ∏_{w∈S} L_{w}^{×}, C_{S} := J_{S}/U_{S},

where U_{S} is diagonally embedded into J_{S}. Let Y_{S} be the free abelian group on the set S (of places of L) and X_{S} the kernel of the augmentation map Y_{S} → ℤ. All of these groups are naturally G-modules. In [7] it is shown that there exists a commutative diagram with exact row and columns

\begin{aligned}& & 0 & & 0 & & 0 & & 0 \\ & & ↓ & & ↓ & & ↓ & & ↓ \\ 0 & ⟶ & U_{S} & ⟶ & A & ⟶ & B & ⟶ & X_{S} & ⟶ & 0 \\ & & ↓ & & ↓ & & ↓ & & ↓ \\ 0 & ⟶ & J_{S} & ⟶ & A′ & ⟶ & B′ & ⟶ & Y_{S} & ⟶ & 0 \\ & & ↓ & & ↓ & & ↓ & & ↓ \\ 0 & ⟶ & C_{S} & ⟶ & A′′ & ⟶ & B′′ & ⟶ & ℤ & ⟶ & 0 \\ & & ↓ & & ↓ & & ↓ & & ↓ \\ & & 0 & & 0 & & 0 & & 0 \end{aligned} (22)

where the terms A, A′, A′′, B, B′, B′′ are cohomologically trivial ℤ[G]-modules and A, B are finitely generated. Henceforth we shall for brevity write ‘c.t.’ in place of ‘cohomologically trivial’. The lower (resp. middle) row represents a (Yoneda) extension class in

Ext_{G}^{2}(ℤ, C_{S}) ≅ H^{2}(G, C_{S}) ⟶^{inv_{K}} \frac{1}{|G|}ℤ/ℤ

(resp. Ext_{G}^{2}(Y_{S}, J_{S})). This class is specified to be the unique class c_{L/K} with inv_{K}(c_{L/K}) = \frac{1}{|G|} (called the global canonical class) for the lower row. As to the middle row, one has decompositions of G-modules

Y_{S} ≅ ⨁_{v∈S} Ind_{G_{v}}^{G} ℤ, J_{S} ≅ ⨁_{v∈S}(L ⊗_{K} K_{v})^{×} ≅ ⨁_{v∈S} Ind_{G_{v}}^{G} L_{w(v)}^{×} (23)

where for each place v of K we have chosen a place w(v) of L above v and G_{v} denotes the decomposition group for w(v). Hence

Ext_{G}^{2}(Y_{S}, J_{S}) ≅ ⨁_{(v,v′)∈S×S} Ext_{G}^{2}(Ind_{G_{v}}^{G} ℤ, Ind_{G_{v′}}^{G} L_{w(v′)}^{×})

⊇ ⨁_{v∈S} Ext_{G}^{2}(Ind_{G_{v}}^{G} ℤ, Ind_{G_{v}}^{G} L_{w(v)}^{×})

⊇ ⨁_{v∈S} Ext_{G_{v}}^{2}(ℤ, L_{w(v)}^{×}).

Here this last inclusion can be described in two ways. On the one hand it is induced by the natural map L_{w(v)}^{×} → Res_{G_{v}}^{G} Ind_{G_{v}}^{G} L_{w(v)}^{×} together with the isomorphism

Ext_{G}^{2}(Ind_{G_{v}}^{G} ℤ, Ind_{G_{v}}^{G} L_{w(v)}^{×}) ≅ Ext_{G_{v}}^{2}(ℤ, Res_{G_{v}}^{G} Ind_{G_{v}}^{G} L_{w(v)}^{×})

coming from adjointness. From the point of view of Yoneda extensions this inclusion is given by sending a collection (C_{v})_{v∈S} of Yoneda extensions over (G_{v})_{v∈S} to the Yoneda extension ⨁_{v∈S} Ind_{G_{v}}^{G} C_{v} over G (here each C_{v} is supposed to denote the exact sequence specifying a Yoneda extension). We call ⨁_{v∈S} Ext_{G_{v}}^{2}(ℤ, L_{w(v)}^{×}) the diagonal subgroup of Ext_{G}^{2}(Y_{S}, J_{S}).

The middle row in (22) is then chosen to represent the element c_{L/K}^{loc} of the diagonal subgroup of Ext_{G}^{2}(Y_{S}, J_{S}) given by the sum of the local canonical classes c_{L_{w(v)}/K_{v}} ∈ H^{2}(G_{v}, L_{w(v)}^{×}) ≅ Ext_{G_{v}}^{2}(ℤ, L_{w(v)}^{×}).

Throughout, we shall identify Yoneda-Ext-groups with derived functor Extgroups by choosing an injective resolution of the second variable (see [11][Ch. V.9]). One then has the following

Lemma 3. Suppose 0 → A′_{i} → A_{i} → A′′_{i} → 0 are short exact sequences for i = 1, 2 and 0 → A′′_{1} → E_{n} → ⋯ → E_{1} → A′_{2} → 0 a Yoneda extension with class [E] ∈ Ext^{n}(A′_{2}, A′′_{1}). Then

0 → A′′_{1} → E_{n} → ⋯ → E_{1} → A_{2} → A′′_{2} → 0

represents δ[E] ∈ Ext^{n+1}(A′′_{2}, A′′_{1}) and

0 → A′_{1} → A_{1} → E_{n} → ⋯ → E_{1} → A′_{2} → 0

represents (−1)^{n+1}δ[E] ∈ Ext^{n+1}(A′_{2}, A′_{1}).

Proof. See [11][Ch.V.9], in particular the comments after [loc.cit. (9.7)]. □

The uniqueness properties of (22) can be formulated as a statement in the derived category 𝔇 of ℤ[G]-modules as follows. Denote the complex A → B with A placed in degree 0 by K_{S} and define K′_{S}, K′′_{S} similarly. Then (22) represents an exact triangle

K_{S} → K′_{S} ⟶^{γ} K′′_{S} → (24)

in 𝔇. Recall that, given any complex E with just two nonzero cohomology groups H^{0}(E) and H^{n}(E), n ≥ 1, the complex τ^{≤n}τ^{≥0}E represents a Yoneda extension class

e(E) ∈ Ext_{G}^{n+1}(H^{n}(E), H^{0}(E)) = Hom_{𝔇}(H^{n}(E), H^{0}(E)[n + 1]).

Here τ is the truncation of complexes preserving cohomology in the indicated degrees. τ extends to a functor on the derived category, so in particular it preserves quasi-isomorphisms. As usual, we view a ℤ[G]-module (for example H^{i}(E)) as an object of 𝔇 by placing it in degree 0. Given another complex E′ and isomorphisms θ^{i} : H^{i}(E) ≅ H^{i}(E′) for i ∈ ℤ, we obtain an isomorphism Θ : Ext_{G}^{n+1}(H^{n}(E), H^{0}(E)) ≅ Ext_{G}^{n+1}(H^{n}(E′), H^{0}(E′)). The maps θ^{i} are induced by an isomorphism θ : E → E′ in 𝔇 if and only if Θ(e(E)) = e(E′). This follows from the definition of equivalence of Yoneda extensions.

Lemma 4. Suppose given an exact triangle

E → E′ ⟶^{α} E′′ → (25)

in 𝔇 such that H^{i}(E) = H^{i}(E′) = H^{i}(E′′) = 0 for i ≠ 0, 1 and an isomorphism of H^{0}(E) → H^{0}(E′) → H^{0}(E′′) with U_{S} → J_{S} → C_{S} and of H^{1}(E) → H^{1}(E′) → H^{1}(E′′) with X_{S} → Y_{S} → ℤ such that e(E′) = c_{L/K}^{loc} and e(E′′) = c_{L/K}. Then (25) is isomorphic to (24) in 𝔇.

Proof. The assumptions of the lemma, together with our discussion above, yield a diagram

\begin{aligned}K_{S} & ⟶ & K′_{S} & ⟶^{γ} & K′′_{S} & ⟶ \\ & & ^{β′}↓ & ^{β′′}↓ \\ E & ⟶ & E′ & ⟶^{α} & E′′ & ⟶ \end{aligned} (26)

where β′ and β′′ are isomorphisms in 𝔇 and such that the two diagrams

J_{S}  ⟶^{H^{0}(γ)} C_{S}  Y_{S}  ⟶^{H^{1}(γ)} ℤ
H^{0}(β′)↓  ^{H^{0}(β′′)}↓  ^{H^{1}(β′)}↓  ^{H^{1}(β′′)}↓
H^{0}(E′) ⟶^{H^{0}(α)} H^{0}(E′′)  H^{1}(E′) ⟶^{H^{1}(α)} H^{1}(E′′)

commute. If we can show that there is a unique morphism γ ∈ Hom_{𝔇}(K′_{S}, K′′_{S}) such that H^{0}(γ) and H^{1}(γ) are the maps given in (22), then the diagram (26) must commute and can therefore be completed to a morphism of triangles. This will be an isomorphism, since β′ and β′′ are isomorphisms, hence the lemma.

From the two exact triangles in 𝔇

J_{S} → K′_{S} → Y_{S}[−1] → J_{S}[1]
C_{S} → K′′_{S} → ℤ[−1] → C_{S}[1]

we obtain a commutative diagram of abelian groups with exact rows and columns, which are in fact parts of long exact sequences

Hom_{𝔇}(Y_{S}[−1], C_{S}) ⟶ Hom_{𝔇}(Y_{S}[−1], K′′_{S}) ⟶ Hom_{𝔇}(Y_{S}[−1], ℤ[−1])
↓  ↓  ^{f_{2}}↓
Hom_{𝔇}(K′_{S}, C_{S}) ⟶ Hom_{𝔇}(K′_{S}, K′′_{S}) ⟶ Hom_{𝔇}(K′_{S}, ℤ[−1])
↓  ↓  ↓
Hom_{𝔇}(J_{S}, C_{S}) ⟶^{f_{1}} Hom_{𝔇}(J_{S}, K′′_{S}) ⟶ Hom_{𝔇}(J_{S}, ℤ[−1]).

Since Hom_{𝔇}(J_{S}, ℤ[i]) = Ext_{G}^{i}(J_{S}, ℤ) = 0 for i < 0 the maps f_{1} and f_{2} are isomorphisms. Therefore we obtain an exact sequence

Hom_{𝔇}(Y_{S}[−1], C_{S}) → Hom_{𝔇}(K′_{S}, K′′_{S}) → Hom_{𝔇}(J_{S}, C_{S}) ⊕ Hom_{𝔇}(Y_{S}[−1], ℤ[−1]).

Moreover

Hom_{𝔇}(Y_{S}[−1], C_{S}) = Ext_{G}^{1}(Y_{S}, C_{S}) = ⨁_{v∈S} H^{1}(G_{v}, C_{S}) = 0

since C_{S} is the formation module of a class formation (see Proposition 3.5 below). So this gives the desired uniqueness of γ ∈ Hom_{𝔇}(K′_{S}, K′′_{S}). □

Clearly K_{S} is a perfect complex of ℤ[G]-modules (since every c.t. ℤ[G]-module has a two-step projective resolution), and hence so is any other complex E as in Lemma 4. By definition then

Ω(L/K, 3) := (K_{S}) = (A) − (B) ∈ Cl(ℤ[G]). (27)

If G is abelian then with our parity convention for the determinant of a complex, this is equal to

(28) Ω(L/K, 3) = (det_{ℤ[G]}^{−1}K_{S}) ∈ Pic(ℤ[G]) ≅ Cl(ℤ[G]).

Remark: In [7], Chinburg works with J′_{S} = ∏_{w∈S} L_{w}^{×}×∏_{w∉S} O_{L_{w}}^{×} rather than J_{S} (U_{S} being diagonally embedded). However, since S contains all places which ramify in L/K the natural projection J′_{S} → J_{S} (resp. J′_{S}/U_{S} → C_{S}) induces isomorphisms Ext_{G}^{2}(Y_{S}, J′_{S}) → Ext_{G}^{2}(Y_{S}, J_{S}) (resp. Ext_{G}^{2}(ℤ, J′_{S}/U_{S}) → Ext_{G}^{2}(ℤ, C_{S}) ), and so working with either group will lead to quasi-isomorphic complexes K_{S}, hence to the same Ω(L/K, 3).

3.2. Another description of Ω(L/K, ℚ(0)). In this section we shall realize the class Ω(L/K, ℚ(0))^{#} as the class of a perfect complex Ψ_{S} of ℤ[G]-modules, well defined up to quasi-isomorphism. The construction of Ψ_{S} is valid without the assumption that G is abelian, and so in effect provides a generalization of Ω(L/K, ℚ(0))^{#}. In the next section we apply Lemma 4 to show that Ψ_{S} and K_{S} are in fact quasiisomorphic.

We denote by (−)^{∨} = R Hom_{ℤ}(−, ℚ/ℤ) = Hom_{ℤ}(−, ℚ/ℤ) the Pontryagin dual of any complex of abelian groups, and put X̂ := X ⊗_{ℤ} ℤ̂ for any finitely generated abelian group X. Recall that S always contains the archimedean places and those ramified in L/K.

Proposition 3.1. Assume S is large enough so that Pic(O_{L,S}) = 0. Then there exists an exact sequence of ℤ[G]-modules

0 → U_{S} → Ψ_{S}^{0} → Ψ_{S}^{1} → X_{S} → 0 (29)

with the following properties:

• Ψ_{S}^{0}, Ψ_{S}^{1} are finitely generated, cohomologically trivial ℤ[G]-modules.

• Writing Ψ̃_{S} for the complex

Ψ_{S}^{0} → Ψ_{S}^{1} → X_{S} ⊗ ℚ

where the maps are as in (29), there is a map in 𝔇

Ψ̃_{S} → RΓ_{c}(O_{L,S}, ℤ)^{∨}[−3] (30)

inducing an isomorphism on H^{i} for i ≠ 0 and the inclusion U_{S} ↪ Û_{S} for i = 0.

We denote by Ψ_{S} the complex Ψ_{S}^{0} → Ψ_{S}^{1} (which is ℤ[G]-perfect).

Remark. The following two exact triangles in 𝔇 summarize the relationship between Ψ_{S}, Ψ̃_{S} and RΓ_{c}(O_{L,S}, ℤ)^{∨}[−3]:

Ψ̃_{S} → RΓ_{c}(O_{L,S}, ℤ)^{∨}[−3] → Û_{S}/U_{S}[0] → (31)

X_{S} ⊗ ℚ[−2] → Ψ̃_{S} → Ψ_{S} → (32)

Both X_{S} ⊗ ℚ and Û_{S}/U_{S} are uniquely divisible, hence ℚ[G]-modules. As such they are injective, and because ℤ[G] → ℚ[G] is flat, they are also injective ℤ[G]-modules. So for many purposes the differences between Ψ_{S}, Ψ̃_{S} and RΓ_{c}(O_{L,S}, ℤ)^{∨}[−3] are inessential (see for example the proof of Proposition 3.2 below).

Proof. There are natural isomorphisms

H^{i}(RΓ_{c}(O_{L,S}, ℤ)^{∨}[−3]) ≅ \{ \begin{matrix}Û_{S} & i = 0 \\ X_{S} ⊗ ℚ/ℤ & i = 2 \\ 0 & i ≠ 0, 2. \end{matrix} (33)

This follows from Artin-Verdier duality. More precisely, consider sheaves F, F′ on (Spec O_{L,S})_{et} whose restrictions to (Spec L_{w})_{et} we denote by the same letter. The Ext-pairings

R Hom_{O_{L,S}}(F, F′) × R Hom_{O_{L,S}}(ℤ, F) → R Hom_{O_{L,S}}(ℤ, F′)
R Hom_{L_{w}}(F, F′) × R Hom_{L_{w}}(ℤ, F) → R Hom_{L_{w}}(ℤ, F′)

together with (3) define a bilinear pairing

R Hom_{O_{L,S}}(F, F′) × RΓ_{c}(O_{L,S}, F) → RΓ_{c}(O_{L,S}, F′) (34)

which on cohomology induces the pairing from [22][Ch. II, Prop. 2.5.a]. For F′ = 𝔾_{m} there is a canonical trace map H_{c}^{3}(O_{L,S}, 𝔾_{m}) → ℚ/ℤ which lifts to a map RΓ_{c}(O_{L,S}, 𝔾_{m}) → ℚ/ℤ[−3] since ℚ/ℤ is injective. Hence we obtain a map

R Hom_{O_{L,S}}(F, 𝔾_{m}) ⟶^{AV} RΓ_{c}(O_{L,S}, F)^{∨}[−3]. (35)

Similar considerations apply to R̃Γ_{c}(O_{L,S}, F) which is defined as in (3) but with Tate cohomology at each archimedean place w. Denoting by ι : RΓ_{c} → R̃Γ_{c} the natural map and putting F = ℤ we find homomorphisms

RΓ(O_{L,S}, 𝔾_{m}) ⟶^{ÃV} R̃Γ_{c}(O_{L,S}, ℤ)^{∨}[−3] ⟶^{ι^{∨}} RΓ_{c}(O_{L,S}, ℤ)^{∨}[−3]. (36)

By [22][Ch.II, Th. 3.1] the map ÃV induces an isomorphism on H^{i} for i ≠ 0 and the inclusion of U_{S} = H^{0}(O_{L,S}, 𝔾_{m}) into its profinite completion Û_{S} for i = 0. The map H^{i}(ι) : H_{c}^{i} → H̃_{c}^{i} is an isomorphism for i ≥ 2 since Tate cohomology agrees with ordinary cohomology in degrees ≥ 1. So H^{i}(ι^{∨}) is an isomorphism for each i ≤ 1 and this implies (33) for each i ≤ 1. On the other hand, by (3) we have H_{c}^{i} = 0 for i ≤ 0 and an exact sequence

0 → ℤ → ⨁_{w∈S} ℤ → H_{c}^{1}(O_{L,S}, ℤ) → 0

where the first map is the diagonal embedding whose Pontryagin dual is the sum map. This gives the description of (33) for each i ≥ 2.

By our discussion before Lemma 4 the complex

τ^{≥0}τ^{≤2}RΓ_{c}(O_{L,S}, ℤ)^{∨}[−3]

represents an element

(37) e_{L/K} ∈ Ext_{G}^{3}(X_{S} ⊗ ℚ/ℤ, Û_{S}) ≅ Ext_{G}^{3}(X_{S} ⊗ ℚ/ℤ, U_{S}) ≅ Ext_{G}^{2}(X_{S}, U_{S})

where the second (boundary) isomorphism is induced by the short exact sequence obtained by tensoring X_{S} with

0 → ℤ → ℚ → ℚ/ℤ → 0, (38)

and the first isomorphism holds because Ext_{G}^{n}(−, Û_{S}/U_{S}) = 0 for n > 0 (as Û_{S}/U_{S} is injective).

The exact sequence in (29) is then chosen to be a representative of the image of e_{L/K} in Ext_{G}^{2}(X_{S}, U_{S}). The modules Ψ_{S}^{0}, Ψ_{S}^{1} can be chosen finitely generated, because U_{S} and X_{S} are finitely generated over ℤ[G]. They can be chosen c.t. because cup product with e_{L/K} induces a cohomology isomorphism (with a degree shift of 2). This is because τ^{≥1}τ^{≤3}RΓ_{c}(O_{L,S}, ℤ) consists of c.t. G-modules by a standard argument ([5, step 2 in Prop. 1.20] or [14, Prop. and Lemma 2.1], and Lemma 12 below) and the fact that Hom_{ℤ}(−, ℚ/ℤ) preserves cohomological triviality. To prove this last remark note that since Hom_{ℤ}(−, ℚ/ℤ) is exact one can reduce to projective modules by taking a (two-step) projective resolution of a given c.t. module, and then to free modules. But Hom_{ℤ}(⨁ ℤ[G], ℚ/ℤ) ≅ Hom_{ℤ}(ℤ[G], ∏ ℚ/ℤ) is coinduced, and hence certainly c.t..

By Lemma 3 the complex Ψ̃_{S} represents the image of the Yoneda class of (29) under the boundary map in (37). The existence of the map (30) follows from the first isomorphism in (37). This finishes the proof of Proposition 3.1. □

Let S be as in Proposition 3.1 and T a finite set of places disjoint from S. For each v ∈ T let f_{v} ∈ G_{v} ⊆ G be the Frobenius automorphism at v, Y_{v} the free abelian group on the places of L above v with its natural (left) G-action and Ψ_{v} the complex

ℤ[G] ⟶^{1−f_{v}^{−1}} ℤ[G]

with terms in degree 0 and 1. There are isomorphisms of G-modules

H^{i}(Ψ_{v}) ⟶^{ι_{v}^{i}} Y_{v}

for i = 0, 1 defined by ι_{v}^{0}(x) = |G_{v}|^{−1}xw(v) and ι_{v}^{1}(x) = xw(v) for x ∈ ℤ[G] (note that if x is f_{v}-invariant xw(v) is divisible by |G_{v}| in Y_{v}). The discussion before Lemma 1 applies to Ψ_{v} ⊗_{ℤ} ℚ. The map id_{1−f_{v}^{−1}}, when expressed on Y_{v} after using the identifications ι_{v}^{i}, is multiplication with |G_{v}|, hence

id_{Y_{v},triv} ∘ id_{1−f_{v}^{−1},triv}^{−1} = ^{∗}|G_{v}| ∈ E^{×} (39)

with the notation introduced in Lemma 2.

Proposition 3.2. There is an exact triangle in 𝔇

Ψ_{S} → Ψ_{S∪T} → ⨁_{v∈T} Ψ_{v} → . (40)

whose long exact cohomology sequence splits into two short exact sequences

0 → U_{S} → U_{S∪T} ⟶^{v} Y_{T} → 0 (41)

0 → X_{S} → X_{S∪T} → Y_{T} → 0 (42)

in which all maps are the natural ones (v is given by taking valuations). Here we have used the maps ι_{v}^{i} together with (29) to identify the cohomology of (40).

Proof. For v ∈ T with residue field κ(v) we pick w|v with residue field λ(w) and decomposition group G_{v}. Let Γ_{v} (resp. Γ_{w}) be the absolute Galois group of κ(v) (resp. λ(w)) and F_{v} ∈ Γ_{v} the Frobenius automorphism. Then G_{v} ≅ Γ_{v}/Γ_{w}. There are canonical isomorphisms

H^{i}(RΓ(λ(w), ℤ)^{∨}[−2]) ≅ \{ \begin{matrix}ℤ̂ & i = 0 \\ ℚ/ℤ & i = 2 \\ 0 & i ≠ 0, 2. \end{matrix} (43)

For i = 0 this is the Pontryagin-dual of the isomorphism

H^{2}(λ(w), ℤ) ≅ H^{1}(λ(w), ℚ/ℤ) ≅ Hom(Γ_{w}, ℚ/ℤ) ≅ ℚ/ℤ (44)

where the last map is evaluation at the F_{w} := F_{v}^{|G_{v}|} ∈ Γ_{w}, and for i = 2 it is the dual of the isomorphism

H^{0}(λ(w), ℤ) ≅ ℤ. (45)

We shall apply the arguments in the proof of Prop. 3.1 to RΓ(λ(w), ℤ) instead of RΓ_{c}(O_{L,S}, ℤ), except that we can now make this complex much more explicit.

For a profinite group Γ and discrete (left) Γ-module M, denote by C(Γ, M) the group of locally constant maps with (left) Γ-action (γf)(x) = γf(γ^{−1}x). Suppose Γ ≅ ℤ̂ with distinguished generator F. Then for any open subgroup Γ′ = nΓ ⊆ Γ the exact sequence

0 → ℚ/ℤ → C(Γ, ℚ/ℤ) ⟶^{1−F} C(Γ, ℚ/ℤ) → 0 (46)

is a resolution of the trivial module ℚ/ℤ by Γ′-acyclic modules. Put G_{n} = Γ/Γ′. After taking Γ′-invariants we obtain an exact sequence of (left) G_{n}-modules

0 → ℚ/ℤ → C(G_{n}, ℚ/ℤ) ⟶^{1−F} C(G_{n}, ℚ/ℤ) ⟶^{ϵ} ℚ/ℤ → 0 (47)

where ϵ(f) = ∑_{g∈G_{n}} f(g). The complex RΓ(Γ′, ℚ/ℤ) identifies with

C(G_{n}, ℚ/ℤ) ⟶^{1−F} C(G_{n}, ℚ/ℤ).

Lemma 5. The isomorphism H^{1}(Γ′, ℚ/ℤ) ⟶^{ϵ} ℚ/ℤ induced by (47) coincides with the chain of isomorphisms

H^{1}(Γ′, ℚ/ℤ) ≅ Hom_{cont}(Γ′, ℚ/ℤ) ≅ ℚ/ℤ

where the last map is evaluation at F^{n}.

Proof. All rows in the following commutative diagram are resolutions of ℚ/ℤ by Γ′-acyclic modules

ℚ/ℤ ⟶ C(Γ, ℚ/ℤ) ⟶^{1−F} C(Γ, ℚ/ℤ) ⟶  0
∥  ↓^{res}  ↓^{res ∘ (1+F +···+F^{n−1})}
ℚ/ℤ ⟶ C(Γ′, ℚ/ℤ) ⟶^{1−F^{n}} C(Γ′, ℚ/ℤ) ⟶  0
∥  ∥  ^{α}↑  ↑
ℚ/ℤ ⟶ C^{0}(Γ′, ℚ/ℤ) ⟶^{δ^{0}} C^{1}(Γ′, ℚ/ℤ) ⟶^{δ^{1}} C^{2}(Γ′, ℚ/ℤ) ⋯

(48)

The third row is the standard resolution of ℚ/ℤ over Γ′, i.e. C^{i}(Γ′, ℚ/ℤ) = C((Γ′)^{i+1}, ℚ/ℤ) and

(49)  (δ^{0}f)(γ_{0}, γ_{1}) = f(γ_{1}) − f(γ_{0})
(50)  (δ^{1}f)(γ_{0}, γ_{1}, γ_{2}) = f(γ_{1}, γ_{2}) − f(γ_{0}, γ_{2}) + f(γ_{0}, γ_{1}),

and the map α is given by (αf)(γ) = f(F^{−n}γ, γ). Upon taking Γ′-invariants we find the commutative diagram of abelian groups

ℚ/ℤ ⟶ C(G_{n}, ℚ/ℤ) ⟶^{1−F}  C(G_{n}, ℚ/ℤ)
∥  ↓^{res}  ↓^{res ∘ (1+F +···+F^{n−1})=ϵ}
ℚ/ℤ  ℚ/ℤ  ⟶^{0}  ℚ/ℤ
∥  ∥  ^{α}↑
ℚ/ℤ  ℚ/ℤ  ⟶^{0} Hom_{cont}(Γ′, ℚ/ℤ) ≅ ker (δ^{1})^{Γ′}

(51)

where the map res simply becomes evaluation at 1 ∈ G_{n}. If x ∈ ℚ/ℤ and f_{x} ∈ Hom_{cont}(Γ′, ℚ/ℤ) is the unique homomorphism with f_{x}(F^{n}) = x, then f_{x} corresponds to the element ϕ_{x} ∈ ker (δ^{1})^{Γ′} with ϕ_{x}(γ_{0}, γ_{1}) = γ_{0}f_{x}(γ_{0}^{−1}γ_{1}) by the standard correspondence between homogenous and inhomogenous cochains. Hence α(ϕ_{x})(γ) = ϕ_{x}(F^{−n}γ, γ) = f_{x}(F^{n}) = x for any γ ∈ Γ′ which proves the Lemma. □

There is a resolution of the trivial Γ_{w}-module ℤ by Γ_{w}-acyclic discrete Γ_{w}- modules

0 → ℤ → ℚ → C(Γ_{v}, ℚ/ℤ) ⟶^{1−F_{v}} C(Γ_{v}, ℚ/ℤ) → 0 (52)

The complex RΓ(λ(w), ℤ) is obtained by taking Γ_{w}-invariants of this resolution:

ℚ → C(G_{v}, ℚ/ℤ) ⟶^{1−f_{v}} C(G_{v}, ℚ/ℤ) (53)

where f_{v} is the image of F_{v} in G_{v} (this notation is consistent with (13)).

Lemma 6. The complex RΓ(λ(w), ℤ)^{∨}[−2] is naturally quasi-isomorphic to

ℤ̂[G_{v}] ⟶^{1−f_{v}^{−1}} ℤ̂[G_{v}] ⟶^{ϵ} ℤ̂ ⊗ ℚ

where ϵ is as in (47), and the isomorphisms (43) are those induced by the short exact sequence

0 → ℤ̂ ⟶^{Δ} ℤ̂[G_{v}] ⟶^{1−f_{v}^{−1}} ℤ̂[G_{v}] ⟶^{ϵ} ℤ̂ ⊗ ℚ → ℚ/ℤ → 0

where Δ(n) = n ∑_{g∈G_{v}} g.

Proof. The first assertion follows by taking the dual of (53). More precisely, we have isomorphisms of (left) G_{v}-modules

C(G_{v}, ℚ/ℤ)^{∨} = Hom_{ℤ}(Hom_{ℤ}(ℤ[G_{v}], ℚ/ℤ), ℚ/ℤ) ≅ ℤ̂[G_{v}]

where ℤ̂[G_{v}] has its natural left action, and C(G_{v}, ℚ/ℤ) = Hom_{ℤ}(ℤ[G_{v}], ℚ/ℤ) the contragredient action (as opposed to the coinduced action). The Pontryagin-dual of a diagonal map is the sum map. Note that if M is a (left) ℤ[G]-module, the (left) action of G on M^{∨} is given by gx = (g^{∨})^{−1}x, hence the occurrence of f_{v}^{−1}.

The map ℚ/ℤ[−1] → ℤ (representing the extension (38)) in the derived category of discrete Γ_{v}-modules is isomorphic to the map from the first to the second row in the commutative diagram

C(Γ_{v}, ℚ/ℤ) ⟶^{F_{v}−1} C(Γ_{v}, ℚ/ℤ)
∥  ∥
ℚ ⟶^{−1} C(Γ_{v}, ℚ/ℤ) ⟶^{F_{v}−1} C(Γ_{v}, ℚ/ℤ)
∥  ^{−1}↓  ∥
ℚ ⟶ C(Γ_{v}, ℚ/ℤ) ⟶^{1−F_{v}} C(Γ_{v}, ℚ/ℤ).

Here we use the acyclic resolutions (46) of ℚ/ℤ. Note the sign change in the differential of (46) caused by the shift [-1]. The middle row is the mapping cone of ℚ → ℚ/ℤ shifted by [-1] and is quasi-isomorphic to the lower row, which is (52). By Lemma 5 the isomorphism (44) is given by ϵ on C(G_{v}, ℚ/ℤ) whose Pontryagin dual is the map Δ. Similarly, the isomorphism (45) is given by the inclusion ℤ → ℚ in (52) whose Pontryagin dual is the natural projection ℤ̂ ⊗ ℚ → ℚ/ℤ. □

Define Ψ̃_{w} (resp. Ψ_{w}) as the second (resp. third) row in the following commutative diagram (with obvious vertical maps)

\begin{matrix}ℤ̂[G_{v}] & ⟶^{1−f_{v}^{−1}} & ℤ̂[G_{v}] & ⟶^{ϵ} & ℤ̂ ⊗ ℚ \\ ↑ & & ↑ & & ↑ \\ ℤ[G_{v}] & ⟶^{1−f_{v}^{−1}} & ℤ[G_{v}] & ⟶^{ϵ} & ℚ \\ ∥ & & ∥ & & ↓ \\ ℤ[G_{v}] & ⟶^{1−f_{v}^{−1}} & ℤ[G_{v}] & ⟶ & 0. \end{matrix} (54)

Then we can rewrite this diagram in the form of two exact triangles of G_{v}-modules

Ψ̃_{w} → RΓ(λ(w), ℤ)^{∨}[−2] → ℤ̂/ℤ[0] → (55)

ℚ[−2] → Ψ̃_{w} → Ψ_{w} → (56)

analogous to (31) and (32). Put Ψ_{v} = Ψ_{w} ⊗_{ℤ[G_{v}]} ℤ[G] and Ψ̃_{v} = Ψ̃_{w} ⊗_{ℤ[G_{v}]} ℤ[G].

We claim that there is a commutative diagram in 𝔇 where all rows are exact triangles, and where the left (resp. middle, resp. right) vertical maps are as in (31),

(32) (resp. (31), (32) with S replaced by S′ = S ∪ T, resp. (55) and (56)).

RΓ_{c}(O_{L,S}, ℤ)^{∨}[−3] ⟶ RΓ_{c}(O_{L,S′}, ℤ)^{∨}[−3] ⟶ ⨁_{w∈T} RΓ(λ(w), ℤ)^{∨}[−2] ⟶
↑  ↑  ↑
Ψ̃_{S}  ⟶  Ψ̃_{S′}  ⟶  ⨁_{v∈T} Ψ̃_{v}  ⟶
↓  ↓  ↓
Ψ_{S}  ⟶  Ψ_{S′}  ⟶  ⨁_{v∈T} Ψ_{v}  ⟶ .

(57)

The first row is the dual of the exact triangle [22, Ch. II, Prop 2.3d)]

RΓ_{c}(O_{L,S′}, ℤ) → RΓ_{c}(O_{L,S}, ℤ) → ⨁_{w∈T} RΓ(λ(w), ℤ) → (58)

and to fill in the other maps we use the following

Lemma 7. Let I be an injective ℤ[G]-module, C a complex of ℤ[G]-modules and n ∈ ℤ.

a) The natural map Hom_{𝔇}(C, I[−n]) → Hom_{G}(H^{n}(C), I) is an isomorphism.

b) There is a short exact sequence

0 → Ext_{G}^{1}(I, H^{n−1}(C)) → Hom_{𝔇}(I[−n], C) → Hom_{G}(I, H^{n}(C)) → 0.

Proof. Part a) follows very easily by computing the set of homotopy classes of maps of complexes C → I[−n] which agrees with Hom_{𝔇}(C, I[−n]) because I is injective. Part b) follows from the spectral sequence

E_{2}^{p,q} = Ext_{G}^{p}(I, H^{q}(C)) ⇒ H^{p+q}R Hom(I, C)

together with the fact that Hom_{𝔇}(I[−n], C) = H^{n}R Hom(I, C) and Ext_{G}^{p}(I, −) = 0 for p ≥ 2 since I is c.t., and hence has a two-step projective resolution. □

The top left square in the diagram (with obvious maps)

\begin{matrix}↑ & & ↑ & & ↑ \\ Û_{S}/U_{S}[0] & ⟶ & Û_{S′}/U_{S′}[0] & ⟶ & ⨁_{v∈T} Ŷ_{v}/Y_{v}[0] & ⟶ \\ ↑ & & ↑ & & ↑ \\ RΓ_{c}(O_{L,S}, ℤ)^{∨}[−3] & ⟶ & RΓ_{c}(O_{L,S′}, ℤ)^{∨}[−3] & ⟶ & ⨁_{w∈T} RΓ(λ(w), ℤ)^{∨}[−2] & ⟶ \\ ↑ & & ↑ & & ↑ \\ Ψ̃_{S} & ⟶ & Ψ̃_{S′} & ⟶ & ⨁_{v∈T} Ψ̃_{v} & ⟶ \end{matrix} (59)

commutes after taking H^{0}, hence commutes in 𝔇 by Lemma 7a). So we get a map of triangles from the first column into the second. To argue similarly for the map from the second to the third column we need:

Lemma 8. The map Û_{S′} → ⨁_{v∈T} Ŷ_{v} induced by

RΓ_{c}(O_{L,S′}, ℤ)^{∨}[−3] → ⨁_{w∈T} RΓ(λ(w), ℤ)^{∨}[−2]

together with (33) and (43) coincides with v ⊗ ℤ̂ (v given by taking valuations), in particular maps U_{S′} to ⨁_{v∈T} Y_{v}.

Proof. Denote the open immersion Spec O_{L,S′} → Spec O_{L,S} by j and let i : Z → Spec O_{L,S} be the closed immersion of the complement. We have two short exact sequences of étale sheaves on Spec O_{L,S}

\begin{aligned}0 & ⟶ & j_{!}j^{∗}ℤ & ⟶ & ℤ & ⟶ & i_{∗}i^{∗}ℤ & ⟶ & 0 \\ & & ^{α}↓ & & ^{β}↓ & & ^{γ}↓ \\ 0 & ⟶ & 𝔾_{m} & ⟶ & j_{∗}𝔾_{m} & ⟶^{val} & i_{∗}i^{∗}ℤ & ⟶ & 0 \end{aligned} (60)

where val is given by taking valuations (we shall discuss below the possible maps α, β, γ). The exact triangle (58) is obtained by applying RΓ_{c} to the top sequence in (60) (see the proof of [22][Prop.2.3d] for this fact). The Artin-Verdier duality map (35) is functorial in F. Hence the morphism i_{∗}i^{∗}ℤ ⟶^{ϵ} j_{!}j^{∗}ℤ[1] induced by the top row in (60), gives a commutative diagram (note i^{∗}ℤ = ℤ)

RΓ_{c}(O_{L,S′}, ℤ)^{∨}[−3] ⟶ ⨁_{w∈T} RΓ(λ(w), ℤ)^{∨}[−2]
↑  ↑
R Hom_{O_{L,S}}(j_{!}j^{∗}ℤ, 𝔾_{m}) ⟶^{b} R Hom_{O_{L,S}}(i_{∗}ℤ, 𝔾_{m})[1],

(61)

where b = R Hom(ϵ, 𝔾_{m})[1] and taking into account the isomorphism

RΓ_{c}(O_{L,S}, i_{∗}ℤ) ⟶^{∼} RΓ(O_{L,S}, i_{∗}ℤ) ⟶^{∼} ⨁_{w∈T} RΓ(λ(w), ℤ). (62)

Consider the composite map

(63) O_{L,S′}^{×} =

Hom_{O_{L,S}}(j_{!}j^{∗}ℤ, 𝔾_{m}) ⟶^{H^{0}(b)} Ext_{O_{L,S}}^{1}(i_{∗}ℤ, 𝔾_{m}) ⟵^{H^{0}(b′)} Hom_{O_{L,S}}(i_{∗}ℤ, i_{∗}ℤ) ≅ ⨁_{w∈T} ℤ

where b′ = R Hom(i_{∗}ℤ, ϵ′), i_{∗}ℤ ⟶^{ϵ′} 𝔾_{m}[1] being given by the lower row in (60). If (60) commutes we have ϵ ∘ α[1] = γ ∘ ϵ′ and hence H^{0}(b)(α) = H^{0}(b′)(γ). Now by adjointness, for any α there is a unique β (and hence a unique γ) making (60) commute. Viewing α (equivalently β) as an element of O_{L,S′}^{×}, the commutativity of (60) shows that γ is given by multiplication with the valuations at w ∈ T of α. Putting this together with the fact that H^{0}(b′) is an isomorphism ([22][Ch. II, Rem. 1.7.(b)]) we conclude that (63) is the valuation map.

It remains to be shown that the identification of the target space of (63) with ⨁_{w∈T} ℤ, combined with (61) induces the identification (43) for i = 0. Inserting the morphism ϵ′ into the F′-variable and putting F = i_{∗}ℤ in (34) we get a commutative diagram of pairings

R Hom_{O_{L,S}}(i_{∗}ℤ, 𝔾_{m}[1]) × RΓ_{c}(O_{L,S}, i_{∗}ℤ) ⟶ RΓ_{c}(O_{L,S}, 𝔾_{m}[1])
↑  ↑^{δ}
R Hom_{O_{L,S}}(i_{∗}ℤ, i_{∗}ℤ) × RΓ_{c}(O_{L,S}, i_{∗}ℤ) ⟶ RΓ_{c}(O_{L,S}, i_{∗}ℤ).

(64)

It suffices to show that δ := RΓ_{c}(ϵ′) is compatible with the trace map which we define for the bottom row by combining (62) with (44). Assume for simplicity T = {w} and put U_{w} = Spec O_{L_{w}} (the general case follows by summing over w ∈ T). There is a commutative diagram of isomorphisms

\begin{matrix}H_{c}^{3}(O_{L,S}, j_{!}𝔾_{m}) & ⟶ & H_{c}^{3}(O_{L,S}, 𝔾_{m}) & ⟵^{H^{2}(δ)} & H_{c}^{2}(O_{L,S}, i_{∗}ℤ) & ⟶ & H^{2}(O_{L,S}, i_{∗}ℤ) \\ ↑ & & ↑ & & ↑ & & ↑ \\ H_{w}^{3}(U_{w}, j_{!}𝔾_{m}) & ⟶ & H_{w}^{3}(U_{w}, 𝔾_{m}) & ⟵ & H_{w}^{2}(U_{w}, i_{∗}ℤ) & ⟶ & H^{2}(U_{w}, i_{∗}ℤ) \end{matrix} (65)

in which all maps are the natural ones. According to [22][p.186] the local trace map can be defined either by the isomorphism

H_{w}^{3}(U_{w}, j_{!}𝔾_{m}) ← H^{2}(L_{w}, 𝔾_{m}) ⟶^{inv} ℚ/ℤ (66)

or by the isomorphism

H^{2}(U_{w}, i_{∗}ℤ) ≅ H^{2}(λ(w), ℤ) ≅ ℚ/ℤ

which arises from (44). By [22][p.216]) the global trace map is defined by (65) and (66). This gives the desired compatibility of δ with the trace map and finishes the proof of Lemma 8. □

Therefore, in (59), we have a nine term commutative diagram with exact columns and two exact rows. So the octahedral axiom implies that the third row in (59), i.e. the second row in (57), is also exact. The same argument gives the exactness of the third row in (57) except that we now have to use Lemma 7b). For example, consider the diagram

\begin{matrix}X_{S} ⊗ ℚ[−2] & ⟶ & Ψ̃_{S} \\ ↓ & & ↓ \\ X_{S′} ⊗ ℚ[−2] & ⟶ & Ψ̃_{S′} \end{matrix} (67)

where the left (resp. right) hand vertical map is the natural one (resp. the one we have just constructed). Applying H^{2} to (67) gives a commutative diagram. Since H^{1}(Ψ̃_{S′}) = 0 we have Hom_{𝔇}(X_{S} ⊗ ℚ[−2], Ψ̃_{S′}) = Hom_{G}(X_{S} ⊗ ℚ, H^{2}(Ψ̃_{S′})) by Lemma 7b), so (67) commutes in 𝔇. This extends to a map of the triangle (32) for S into the triangle (32) for S′.

We have now established that the diagram (57) commutes and has exact rows and this implies Proposition 3.2 (the assertion about v is immediate from Lemma 8 and (54)). □

For a perfect complex of ℤ_{p}[G]-modules P we define P^{∗} = RHom_{ℤ_{p}}(P, ℤ_{p}) with contragredient G-action. Then P^{∗} is again perfect over ℤ_{p}[G] and P^{∗} ⊗_{ℤ_{p}} ℚ_{p} naturally identifies with (P ⊗_{ℤ_{p}} ℚ_{p})^{∧} := Hom_{ℚ_{p}}(P ⊗_{ℤ_{p}} ℚ_{p}, ℚ_{p}). Recall that S_{p} denotes the union of S with the set of places above p.

Proposition 3.3. Let S be as in Prop. 3.1. For any prime number p there exists a quasi-isomorphism

θ_{p} : Ψ_{S_{p}} ⊗_{ℤ} ℤ_{p} →^{∼} RΓ_{c}(O_{L,S_{p}}, ℤ_{p})^{∗}[−2] (68)

such that the sequence (15) is ℚ_{p}-dual to the natural long exact sequence

(69) 0 → O_{L}^{×} ⊗_{ℤ} ℚ_{p} → U_{S_{p}} ⊗_{ℤ} ℚ_{p} → Y_{S_{p,f}} ⊗_{ℤ} ℚ_{p} → 0 → X_{S_{p}} ⊗_{ℤ} ℚ_{p} → Y_{S_{p}} ⊗_{ℤ} ℚ_{p} → ℚ_{p} → 0 when we identify X_{S_{p}}⊗_{ℤ}ℚ_{p} with H_{c}^{1}(O_{L,S_{p}}, ℚ_{p})^{∧} and U_{S_{p}}⊗_{ℤ}ℚ_{p} with H_{c}^{2}(O_{L,S_{p}}, ℚ_{p})^{∧} via θ_{p} ⊗ ℚ_{p}. Here we have also identified ℚ_{p} and Y_{S} ⊗ ℚ_{p} with their respective contragredients by mapping the natural basis to its dual basis.

Proof. Denote by C_{n} (resp. C̃_{n}), the mapping cone of multiplication by p^{n} on Ψ_{S_{p}} (resp. Ψ̃_{S_{p}}), for each positive integer n. The mapping cone of multiplication by p^{n} on RΓ_{c}(O_{L,S_{p}}, ℤ)^{∨}[−3] naturally identifies with RΓ_{c}(O_{L,S_{p}}, ℤ/p^{n}ℤ)^{∨}[−2]. From (31) and (32) we obtain two quasi-isomorphisms

C_{n} ⟵^{ϕ_{1,n}} C̃_{n} ⟶^{ϕ_{2,n}} RΓ_{c}(O_{L,S_{p}}, ℤ/p^{n}ℤ)^{∨}[−2], (70)

compatible with the natural maps C_{n+1} → C_{n} etc. This is because multiplication by p^{n} is an isomorphism on both X_{S_{p}} ⊗ ℚ and Û_{S_{p}}/U_{S_{p}}.

Recall that RΓ_{c}(O_{L,S_{p}}, ℤ_{p}) is a perfect complex of ℤ_{p}[G]-modules. Applying (−)^{∗} = RHom_{ℤ_{p}}(−, ℤ_{p}) to the exact triangle

RΓ_{c}(O_{L,S_{p}}, ℤ_{p}) ⟶^{p^{n}} RΓ_{c}(O_{L,S_{p}}, ℤ_{p}) → RΓ_{c}(O_{L,S_{p}}, ℤ/p^{n}ℤ) →

produces an exact triangle

RΓ_{c}(O_{L,S_{p}}, ℤ/p^{n}ℤ)^{∗} → RΓ_{c}(O_{L,S_{p}}, ℤ_{p})^{∗} ⟶^{p^{n}} RΓ_{c}(O_{L,S_{p}}, ℤ_{p})^{∗} → .

From the short exact sequence 0 → ℤ_{p} → ℚ_{p} → ℚ_{p}/ℤ_{p} → 0 one easily deduces that

RΓ_{c}(O_{L,S_{p}}, ℤ/p^{n}ℤ)^{∗} ≅ RΓ_{c}(O_{L,S_{p}}, ℤ/p^{n}ℤ)^{∨}[−1],

and obtains therefore an exact triangle

RΓ_{c}(O_{L,S_{p}}, ℤ_{p})^{∗} ⟶^{p^{n}} RΓ_{c}(O_{L,S_{p}}, ℤ_{p})^{∗} → RΓ_{c}(O_{L,S_{p}}, ℤ/p^{n}ℤ)^{∨} → .

More concretely then, if we pick a bounded complex of finitely generated projective ℤ_{p}[G]-modules P^{•} quasi-isomorphic to RΓ_{c}(O_{L,S_{p}}, ℤ_{p})^{∗}[−2], the inverse system RΓ_{c}(O_{L,S_{p}}, ℤ/p^{n}ℤ)^{∨}[−2], n ≥ 1, is isomorphic to P^{•}/p^{n}.

We can give a similar description for the system C_{n}. Since Ψ_{S_{p}} is a perfect complex of ℤ[G]-modules, we can choose a quasi-isomorphism Ψ_{S_{p}} ≅ Q^{•} with Q^{•} a bounded complex of finitely generated projective ℤ[G]-modules. The inverse system C_{n} is then quasi-isomorphic to the system of complexes Q^{•}/p^{n}.

The morphisms ϕ_{n} := ϕ_{2,n}ϕ_{1,n}^{−1} : Q^{•}/p^{n} → P^{•}/p^{n} from (70) are quasi-isomorphisms such that the diagram

\begin{aligned}⋯ & ⟶ & Q^{•}/p^{n} & ⟶^{π_{n}^{Q}} & Q^{•}/p^{n−1} & ⟶ & ⋯ \\ & & ^{ϕ_{n}}↓ & & ^{ϕ_{n−1}}↓ \\ ⋯ & ⟶ & P^{•}/p^{n} & ⟶^{π_{n}^{P}} & P^{•}/p^{n−1} & ⟶ & ⋯ \end{aligned} (71)

commutes in the derived category. Since however Q^{•}/p^{n} consists of projective ℤ/p^{n}ℤ[G]-modules and P^{•}/p^{n} of ℤ/p^{n}ℤ[G]-modules we can realize each ϕ_{n} as an actual map of complexes. Moreover, ϕ_{n−1}π_{n}^{Q} will be homotopic to π_{n}^{P} ϕ_{n}, i.e.

ϕ_{n−1}π_{n}^{Q} − π_{n}^{P} ϕ_{n} = dh + hd

for some map h : Q^{•}/p^{n} → P^{•}/p^{n−1}[−1]. Since P^{i}/p^{n} → P^{i}/p^{n−1} is surjective and Q^{i}/p^{n} is a projective ℤ/p^{n}ℤ[G]-module we can lift h to a map h′ : Q^{•}/p^{n} → P^{•}/p^{n}[−1]. If we then replace ϕ_{n} by ϕ_{n} + dh′ + h′d, the diagram (71) will actually be a commutative diagram of maps of complexes. So by induction we may assume that the ϕ_{n} are a map of inverse systems of complexes.

Lemma 9. Let

⋯ → K_{n}^{•} ⟶^{t_{n}} K_{n−1}^{•} → ⋯

be an inverse system of complexes such that each transition map t_{n}^{i} : K_{n}^{i} → K_{n−1}^{i} is surjective, and each cohomology group H^{i}(K_{n}^{•}) is finite. Then there is a canonical isomorphism

H^{i}(\underleftarrow{lim}_{n}K_{n}^{•}) ⟶^{∼} \underleftarrow{lim}_{n}H^{i}(K_{n}^{•})

for each i ∈ ℤ.

Proof. This follows easily by using the argument of [26, (2.2)] or [12, 1.4-6] and the fact that any inverse system of finite abelian groups (F_{n})_{n} satisfies the Mittag-Leffler condition so that \underleftarrow{lim} _{n}^{1}F_{n} = 0. □

This Lemma implies that

Q^{•} ⊗_{ℤ} ℤ_{p} = \underleftarrow{lim}_{n}(Q^{•}/p^{n}) ⟶^{\underleftarrow{lim}_{n}ϕ_{n}} \underleftarrow{lim}_{n}(P^{•}/p^{n}) = P^{•}

is a quasi-isomorphism. But Q^{•} ⊗_{ℤ} ℤ_{p} is of course quasi-isomorphic to Ψ_{S_{p}} ⊗_{ℤ} ℤ_{p} and P^{•} is quasi-isomorphic to RΓ_{c}(O_{L,S_{p}}, ℤ_{p})^{∗}[−2]. So we have constructed the quasi-isomorphism θ_{p} in Proposition 3.3. The fact that θ_{p} identifies the dual of (15) with the natural sequence (69) is a lengthy but straightforward verification (somewhat similar to the proof of Lemma 8) which we leave out for reasons of space. □

The operation P ↦ P^{#} defined in the paragraph before equation (2) induces an involution on Pic(ℤ[G]) which we denote by the same symbol. For e = ∑ λ_{g}g ∈ A[G] we put e^{#} = ∑ λ_{g}g^{−1} where A is any commutative coefficient ring. Note that ^{#} is a functor from G-modules to G-modules and we have ϕ^{#} = ϕ for any Ghomomorphism ϕ. However, if ϕ is multiplication with e then ϕ^{#} is multiplication with e^{#}.

Theorem 3.1. If G is abelian, then Ω(L/K, ℚ(0))^{#} = (Ψ_{S}) ∈ Cl(ℤ[G]).

Proof. Let S be a finite set of places containing the archimedean places and those ramified in L/K and such that Pic(O_{L,S}) = 0. There is a long exact sequence of E-modules

(72) 0 → O_{L}^{×} ⊗_{ℤ} ℚ → U_{S} ⊗_{ℤ} ℚ → Y_{S_{f}} ⊗_{ℤ} ℚ → 0 → X_{S} ⊗_{ℤ} ℚ → Y_{S} ⊗_{ℤ} ℚ → ℚ → 0

where the maps are the natural ones as in (69). If S = S_{p} the sequence (72)⊗ℚ_{p} is identical to (69). The ℚ-dual of (72) induces an isomorphism

(73)  κ(S) : Ξ(ℚ(0)_{L})^{#} ≅ det_{E}(U_{S} ⊗ ℚ) ⊗_{E} det_{E}^{−1}(X_{S} ⊗ ℚ)
= det_{E}^{−1}(Ψ_{S} ⊗_{ℤ} ℚ)

such that for any prime number p one has an identity of maps

ϑ̃_{p} = (det_{E_{p}}^{−1}(θ_{p} ⊗ ℚ_{p}))^{−1} ∘ (κ(S_{p}) ⊗ ℚ_{p}) (74) from Ξ(ℚ(0)_{L})^{#} ⊗_{ℚ} ℚ_{p} to det_{E_{p}}RΓ_{c}(O_{L,S_{p}}, ℚ_{p})^{#} = det_{E_{p}}^{−1}RΓ_{c}(O_{L,S_{p}}, ℚ_{p})^{∗}[−2]. This is because ϑ̃_{p} was induced by (15) which agrees with the dual of (72) ⊗ ℚ_{p} for S_{p} by Proposition 3.3. Put

(75)  λ̃ := κ(S) ∘ κ(S_{p})^{−1} : det_{E}^{−1}(Ψ_{S_{p}} ⊗ ℚ) ⟶^{∼} det_{E}^{−1}(Ψ_{S} ⊗ ℚ),

and let λ_{ℤ} be the isomorphism

det_{ℤ[G]}^{−1}Ψ_{S_{p}} ⟶^{∼} det_{ℤ[G]}^{−1}Ψ_{S} ⊗_{ℤ[G]} ⨂_{v∈T} det_{ℤ[G]}^{−1}(ℤ[G] ⟶^{1−f_{v}^{−1}} ℤ[G]) ⟶^{∼} det_{ℤ[G]}^{−1}Ψ_{S} (76)

where the first map is induced by the triangle (40) with T = S_{p} \ S and the second is 1 ⊗ Hom_{ℤ[G]}(⨂_{v∈T} id_{ℤ[G],triv}, ℤ[G])^{−1} in the notation introduced before Lemma 1 with R = ℤ[G]. Put λ = λ_{ℤ} ⊗ ℚ.

Lemma 10. We have λ̃ = ∏_{v∈T} ϵ_{v}(0)^{#,−1}λ with ϵ_{v}(0) ∈ E^{×} as in (16).

Proof. λ̃ is induced by the natural map from the sequence (72) for S to the sequence (72) for S_{p}, i.e. by the short exact sequences

0 → U_{S} ⊗_{ℤ} ℚ → U_{S_{p}} ⊗_{ℤ} ℚ → Y_{T} ⊗_{ℤ} ℚ → 0, (77)

0 → X_{S} ⊗_{ℤ} ℚ → X_{S_{p}} ⊗_{ℤ} ℚ → Y_{T} ⊗_{ℤ} ℚ → 0 (78)

and the identity map on Y_{T} ⊗_{ℤ} ℚ = ⨁_{v∈T} Y_{v}. By Proposition 3.2 these sequences also arise as the long exact cohomology sequence induced by (40) ⊗ ℚ. This means that λ̃ is likewise a composite as in (76) ⊗ ℚ but with the second map 1 ⊗ Hom_{E}(⨂_{v∈T} id_{E,triv}, E)^{−1} replaced by 1 ⊗ Hom_{E}(⨂_{v∈T} id_{Y_{v},triv}, E)^{−1}. To compare the two maps, note that Lemma 1 gives a commutative diagram

⨂_{v∈T} det_{E}(E ⟶^{1−f_{v}^{−1}} E)  ⟶^{⨂_{v∈T} id_{E,triv}} E
↓  ↓^{· ∏_{v∈T}(1−f_{v}^{−1,∗})^{−1}}
det_{E}^{−1}(Y_{T} ⊗ ℚ) ⊗_{E} det_{E}(Y_{T} ⊗ ℚ) ⟶^{⨂_{v∈T} id_{1−f{−1 \atop v},triv}} E
∥  ↓^{· ∏_{v∈T} β_{v}}
det_{E}^{−1}(Y_{T} ⊗ ℚ) ⊗_{E} det_{E}(Y_{T} ⊗ ℚ) ⟶^{⨂_{v∈T} id_{Yv ,triv}} E

(79)

where β_{v} = id_{Y_{v},triv} ∘ id_{1−f_{v}^{−1},triv}^{−1} ∈ E^{×}. Applying Hom_{E}(−, E)^{−1} to (79) yields

λ̃ = ∏_{v∈T}(1 − f_{v}^{−1,∗})β_{v}^{−1} λ. (80)

Since β_{v} = ^{∗}|G_{v}| by (39), (^{∗}x)^{#} = ^{∗}x and (e^{#})^{∗} = (e^{#})^{∗} in the notation of Lemma 2, Lemma 10 follows. □

For any prime number p, denote the scalar extensions of κ(S), λ̃ and λ to E_{p} by the same letter. Now we can compute, using ϵ_{S_{p}}(0) = ϵ_{S}(0) ∏_{v∈T} ϵ_{v}(0)

ϑ_{p}^{−1}(I_{p}(0)^{#}) = ϵ_{S_{p}}(0)^{#}ϑ̃_{p}^{−1}(I_{p}(0)^{#})  by (11)
= ϵ_{S_{p}}(0)^{#}κ(S_{p})^{−1} ∘ det_{E_{p}}(θ_{p} ⊗ ℚ_{p})^{−1}(I_{p}(0)^{#})  by (74)
= ϵ_{S_{p}}(0)^{#}κ(S_{p})^{−1}(det_{E_{p}}^{−1}(Ψ_{S_{p}} ⊗_{ℤ} ℤ_{p}))  by (68)
= ϵ_{S_{p}}(0)^{#}κ(S)^{−1}λ̃(det_{E_{p}}^{−1}(Ψ_{S_{p}} ⊗_{ℤ} ℤ_{p}))  by (75)
= ϵ_{S}(0)^{#}κ(S)^{−1}λ(det_{E_{p}}^{−1}(Ψ_{S_{p}} ⊗_{ℤ} ℤ_{p}))  by Lemma 10
= ϵ_{S}(0)^{#}κ(S)^{−1}(det_{E_{p}}^{−1}(Ψ_{S} ⊗_{ℤ} ℤ_{p}))  by (76)

and we obtain

Ξ(ℚ(0)_{L})_{ℤ}^{#} = ϵ_{S}(0)^{#}κ(S)^{−1}(det_{ℤ[G]}^{−1}(Ψ_{S})). (81)

Since ϵ_{S}(0)^{#} ∈ E^{×}, taking classes in Pic(ℤ[G]) and recalling our sign convention for the determinant of a complex, this equality implies that

Ω(L/K, ℚ(0))^{#} = (det_{ℤ[G]}^{−1}Ψ_{S}) = (Ψ_{S}) ∈ Cl(ℤ[G]).
□

3.3. Comparison of Ω(L/K, 3) and Ω(L/K, ℚ(0))^{#}. The idea is to construct an exact triangle

Ψ_{S} → Ψ′_{S} → Ψ′′_{S} →

satisfying the assumptions of Lemma 4. Recall that R̃Γ_{c}(O_{L,S}, F) is defined as in (3) but with Tate-cohomology at each archimedean place. If we define

(82) RΓ_{Δ}(L_{w}, F) := Cone(RΓ(L_{w}, F) → RΓ_{Tate}(L_{w}, F))[−1]

for each archimedean place w, the octahedral axiom gives an exact triangle

RΓ_{c}(O_{L,S}, F) → R̃Γ_{c}(O_{L,S}, F) → ⨁_{w∈S_{∞}} RΓ_{Δ}(L_{w}, F) → . (83)

Next consider the diagram with exact rows

\begin{aligned}RΓ(O_{L,S}, 𝔾_{m}) & ⟶^{ÃV} & R̃Γ_{c}(O_{L,S}, ℤ)^{∨}[−3] & ⟶ & Û_{S}/U_{S}[0] \\ & & ↓ & & ∥ \\ Ψ̃_{S} & ⟶ & RΓ_{c}(O_{L,S}, ℤ)^{∨}[−3] & ⟶ & Û_{S}/U_{S}[0] \end{aligned} (84)

where the lower row is (31). By Lemma 7a), the square commutes, so that (84) can be completed to a map of triangles. We now claim that all rows and columns in the following diagram are exact triangles.

⨁_{w∈S_{∞}} RΓ_{Δ}(L_{w}, ℤ)^{∨}[−3]  ⨁_{w∈S_{∞}} RΓ_{Δ}(L_{w}, ℤ)^{∨}[−3]
↓  ↓^{α}
RΓ(O_{L,S}, 𝔾_{m})  ⟶ ⨁_{w∈S} RΓ(L_{w}, 𝔾_{m}) ⟶ RΓ_{c}(O_{L,S}, 𝔾_{m})[1]
↓  ↓  ∥
Ψ̃_{S}  ⟶  Ψ̃′_{S}  ⟶  Ψ̃′′_{S}.

(85)

This is obvious for the middle row and follows for the first column by combining (83) with (84) and using the octahedral axiom. Ψ̃′_{S} is defined to make the last row (or alternatively the middle column) exact and the exactness of the middle column (or last row) then again follows from the octahedral axiom.

Proposition 3.4. Ψ̃′_{S} has a G-equivariant direct sum decomposition

Ψ̃′_{S} ≅ ⨁_{w∈S_{f}} RΓ(L_{w}, 𝔾_{m}) ⊕ ⨁_{w∈S_{∞}} C_{w}.

Proof. An equivalent way of stating the proposition is that the map α in (85) lies in what we have been calling the diagonal subgroup of

Hom_{𝔇}( ⨁_{v∈S_{∞}} Ind_{G_{v}}^{G} RΓ_{Δ}(L_{w(v)}, ℤ)^{∨}[−3], ⨁_{v∈S} Ind_{G_{v}}^{G} RΓ(L_{w(v)}, 𝔾_{m})).

Fix an infinite place w_{0} of L with decomposition group G_{0} and let K′ be the fixed field of G_{0}. Denote by v_{0}, resp. v′_{0}, the place of K, resp. K′, induced by w_{0} and by 𝔇(Γ) the derived category of ℤ[Γ]-modules for any group Γ. What we must show then is that the map which α induces in

Hom_{𝔇(G)}(Ind_{G_{0}}^{G} RΓ_{Δ}(L_{w_{0}}, ℤ)^{∨}[−3], ⨁_{v∈S} Ind_{G_{v}}^{G} RΓ(L_{w(v)}, 𝔾_{m}))

= Hom_{𝔇(G_{0})}(RΓ_{Δ}(L_{w_{0}}, ℤ)^{∨}[−3], ⨁_{v∈S} Res_{G_{0}}^{G} Ind_{G_{v}}^{G} RΓ(L_{w(v)}, 𝔾_{m}))
= Hom_{𝔇(G_{0})}(RΓ_{Δ}(L_{w_{0}}, ℤ)^{∨}[−3], ⨁_{v′∈S} Ind_{G_{v′}}^{G_{0}} RΓ(L_{w(v′)}, 𝔾_{m}))

factors through the summand corresponding to v′_{0}, i.e. RΓ(L_{w_{0}}, 𝔾_{m}). Here the last sum is over all places v′ of K′ lying above a place in S and G_{v′} is the decomposition group inside G_{0}. So we need to show that for each place v′ ≠ v′_{0} of K′ the map

Y := RΓ_{Δ}(L_{w_{0}}, ℤ)^{∨}[−3] ⟶^{β} Ind_{G_{v′}}^{G_{0}} RΓ(L_{w′}, 𝔾_{m}) =: Z

induced by α vanishes in Hom_{𝔇(G_{0})}(Y, Z). Fixing such a choice of v′ we split the discussion into two cases.

Assume first that G_{0} = 1 so that 𝔇(G_{0}) = 𝔇_{ab} is the derived category of abelian groups. Morphisms in this category can be profitably studied via the exact sequence

0 → ∏_{i∈ℤ} Ext^{1}(H^{i}(Y), H^{i−1}(Z)) → Hom_{𝔇_{ab}}(Y, Z) → ∏_{i∈ℤ} Hom(H^{i}(Y), H^{i}(Z)) → 0

which is a consequence of the spectral sequence in [27][III.4.6.10]. It is easy to check that Y has cohomology concentrated in degrees ≥ 3, in fact that

Y = \begin{cases}ℚ/ℤ [−3] & w_{0} complex \\ ℚ/ℤ ⟶^{0} ℚ/ℤ ⟶^{2} ℚ/ℤ ⟶^{0} ⋯ [−3] & w_{0} real. \end{cases}

If w′ is not real one finds that Hom_{𝔇_{ab}}(Y, Z) = 0 using that RΓ(L_{w′}, 𝔾_{m}) has cohomology concentrated in degrees ≤ 2 and that Ext^{1}(ℚ/ℤ, Br(L_{w′})) = 0 since Br(L_{w′}) is an injective abelian group. If w′ is real on the other hand, we pick a λ ∈ L = K′ which is negative at w′ = v′ and positive at w_{0} = v′_{0}, in case w_{0} is real. Such a λ exists because of the independence of the valuations v′ and v′_{0}. In the quadratic extension N = L(\sqrt{λ})/K′ the place v′ ramifies whereas v′_{0} splits. From Lemma 11 below, with L/K replaced by N/L, G by Γ := Gal(N/L) ≅ ℤ/2ℤ and H = G, we find that β = R Hom_{ℤ[Γ]}(ℤ, β′) where

β′ ∈ Hom_{𝔇(Γ)}(Ind_{1}^{Γ} Y, RΓ(N_{w′}, 𝔾_{m})) = Hom_{𝔇_{ab}}(Y, ℂ^{×}[0]) = 0.

Hence β = 0.

If now G_{0} ≅ ℤ/2ℤ, w_{0} is complex and Y ≅ ℚ/ℤ[−3]. We first dispose of the case where v′ is complex. Then

Hom_{𝔇(G_{0})}(Y, Z)
= Hom_{𝔇(G_{0})}(ℚ/ℤ[−3], Ind_{1}^{G_{0}} ℂ^{×}[0]) = Hom_{𝔇_{ab}}(ℚ/ℤ[−3], ℂ^{×}[0]) = 0.

If v′ is not complex we can choose a quadratic extension K′(\sqrt{λ})/K′ in which v′ ramifies or stays inert whereas v′_{0} splits. If L_{w′} ≠ K′_{v′} we also require that the quadratic extension K′_{v′}(\sqrt{λ})/K′_{v′} is isomorphic to L_{w′}/K′_{v′}. The existence of λ again follows from the independence of the valuations v′ and v′_{0}. Since v′_{0} ramifies in L/K′ we have L ≠ K′(\sqrt{λ}) and the field N := L(\sqrt{λ}) has Galois group Γ := G_{0} × G_{1} ≅ (ℤ/2ℤ)^{2} over K′ where G_{1} := Gal(K′(\sqrt{λ})/K′). Again, from Lemma 11 below, applied to the extension N/K′ and the subgroup 1 × G_{1} of Γ, we have β = R Hom_{ℤ[1×G_{1}]}(ℤ, β′) where

β′ ∈ Hom_{𝔇(Γ)}(Ind_{G_{0}×1}^{Γ} ℚ/ℤ[−3], Ind_{Δ}^{Γ} RΓ(N_{w′}, 𝔾_{m})).

Note here that G_{0} × 1 is the decomposition group of v′_{0} in N/K′ and Δ is by definition the decomposition group of v′. Our assumptions imply that Δ∩G_{0} ×1 = {1}. Indeed, if v′ splits in L we have Δ = 1 × G_{1}; otherwise Δ is the diagonal subgroup in G_{0} × G_{1} since then v′ does not split in either L or K′(\sqrt{λ}) and N_{w′} = K′_{v′}(\sqrt{λ}). By standard formulas for induction and restriction [2][III.5.6] we get

Hom_{𝔇(Γ)}(Ind_{G_{0}×1}^{Γ} ℚ/ℤ[−3], Ind_{Δ}^{Γ} RΓ(N_{w′}, 𝔾_{m}))
= Hom_{𝔇(G_{0}×1)}(ℚ/ℤ[−3], Res_{G_{0}×1}^{Γ} Ind_{Δ}^{Γ} RΓ(N_{w′}, 𝔾_{m}))
= Hom_{𝔇(G_{0}×1)}(ℚ/ℤ[−3], Ind_{1}^{G_{0}×1} Res_{1}^{Δ} RΓ(N_{w′}, 𝔾_{m}))
= Hom_{𝔇_{ab}}(ℚ/ℤ[−3], RΓ(N_{w′}, 𝔾_{m})).

The latter group has been shown to be zero in the discussion of the case G_{0} = 1, w′ not real, above. But by our construction of N we have N_{w′}/K′_{v′} ≅ ℂ/ℝ if w′ is archimedean. We deduce that β′ = 0 and therefore β = 0 and this finishes the proof of Proposition 3.4. □

Remark. Proposition 3.4 can partly be proven using the compatibility of local and global Artin-Verdier duality. However, this method seems to be incapable of identifying the map β if G_{0} ≅ ℤ/2ℤ and v′ is a ramified infinite place.

Lemma 11. Let H ◁ G be a normal subgroup and consider the diagram (85) in 𝔇 = 𝔇(G). Then the diagram obtained by applying R Hom_{ℤ[H]}(ℤ, −) to (85) is naturally isomorphic, in the derived category of ℤ[G/H]-modules, to the corresponding diagram formed with respect to the extension L^{H}/K.

Proof. Since Spec O_{L,S} → Spec O_{K,S} is a Galois cover, the identities

R Hom_{ℤ[H]}(ℤ, RΓ(O_{L,S}, F)) = RΓ(O_{L^{H},S}, F)
R Hom_{ℤ[H]}(ℤ, RΓ(L ⊗_{K} K_{v}, F)) = RΓ(L^{H} ⊗_{K} K_{v}, , F)

and hence

R Hom_{ℤ[H]}(ℤ, RΓ_{c}(O_{L,S}, F)) = RΓ_{c}(O_{L^{H},S}, F)

are certainly well known for any étale sheaf F on O_{K,S} [21][III.2.20]. More concretely, each of the complexes in the middle row of (85) can be represented by a bounded below complex C of cohomologically trivial G-modules and

R Hom_{ℤ[H]}(ℤ, C) = C^{H} (86)

gives the corresponding complex over L^{H}. If now v is an archimedean place of K it is still true that C := RΓ_{Tate}(L ⊗_{K} K_{v}, F) can be represented by a complex of c.t. G-modules such that C^{H} = RΓ_{Tate}(L^{H} ⊗_{K} K_{v}, F) but (86) no longer holds since C may be unbounded in both directions. In any case, in both triangles (82) and (83) each complex is represented by c.t. G-modules and taking H-invariants gives the corresponding complex over L^{H}. Moreover, both X = RΓ_{Δ}(L ⊗_{K} K_{v}, F) and X = R̃Γ_{c}(O_{L,S}, F) are bounded above complexes of c.t. G-modules. Since for any c.t. G-module M the natural map M_{H} → M^{H} is an isomorphism, taking H-invariants actually computes the homology of X, i.e. X^{H} = X_{H} = ℤ ⊗_{ℤ[H]}^{L} X. Utilizing the natural isomorphisms

R Hom_{ℤ[H]}(ℤ, Hom(X, ℚ/ℤ)) ≅ Hom(ℤ ⊗_{ℤ[H]}^{L} X, ℚ/ℤ)

and

R Hom_{ℤ[H]}(ℤ, Û_{S}/U_{S}) = (Û_{S}/U_{S})^{H} = Û_{L^{H},S}/U_{L^{H},S}

we find that for all terms in the diagrams (84) and (85) the following is true: Application of R Hom_{ℤ[H]}(ℤ, −) gives the corresponding term over L^{H}. As to the maps between them these are either obvious or constructed from the duality map ÃV in (84). The latter is in turn induced by the G-equivariant pairing of complexes (34) (for R̃Γ_{c}), and the corresponding pairing over L^{H} just arises by taking H invariants, once the complexes are represented by c.t. G-modules. Hence R Hom_{ℤ[H]}(ℤ, ÃV) is the duality map over L^{H}. □

The exact triangle Ψ̃_{S} → Ψ̃′_{S} → Ψ̃′′_{S} → in (85) is essentially our candidate for Lemma 4. Computation of the cohomology of (85) (see Proposition 3.5 below) shows that Ψ̃′′_{S} represents a class in Ext_{G}^{3}(ℚ/ℤ, C_{S}) in a canonical way, and that Ψ̃′_{S} represents a class in Ext_{G}^{3}(Y_{S} ⊗ ℚ/ℤ, J_{S}). For example, H^{2} of the diagram (85) gives a commutative diagram

\begin{aligned}0 & ⟶ & Br(O_{L,S}) & ⟶ & ⨁_{w∈S} Br(L_{w}) & ⟶ & ℚ/ℤ & ⟶ & 0 \\ & & ↓ & & ↓ & & ∥ \\ 0 & ⟶ & X_{S} ⊗ ℚ/ℤ & ⟶ & H^{2}(Ψ̃′_{S}) & ⟶ & ℚ/ℤ & ⟶ & 0 \end{aligned} (87)

which implies that H^{2}(Ψ̃′_{S}) ≅ Y_{S}⊗ℚ/ℤ, and in fact that the lower row is isomorphic to the right hand column of (22) tensored with ℚ/ℤ.

We can then construct a commutative diagram where all rows and columns are exact triangles, and the right hand column is a candidate for Lemma 4:

\begin{matrix}X_{S} ⊗ ℚ[−2] & ⟶ & Ψ̃_{S} & ⟶ & Ψ_{S} & ⟶ \\ ↓ & & ↓ & & ↓ \\ Y_{S} ⊗ ℚ[−2] & ⟶ & Ψ̃′_{S} & ⟶ & Ψ′_{S} & ⟶ \\ ↓ & & ↓ & & ↓ \\ ℚ[−2] & ⟶ & Ψ̃′′_{S} & ⟶ & Ψ′′_{S} & ⟶ \\ ↓ & & ↓ & & ↓ \end{matrix} (88)

Here the top row is (32) and the middle and bottom row are analogous, that is Ψ′_{S} (resp. Ψ′′_{S}) represents the image of e(Ψ̃′_{S}) (resp. e(Ψ̃′′_{S})) in Ext_{G}^{2}(Y_{S}, J_{S}) (resp. Ext_{G}^{2}(ℤ, C_{S})) under the boundary map induced by (38) tensored with Y_{S} (resp. (38)). The existence of (88) follows from the commutativity of the left hand squares in (88) which in turn is a consequence of Lemma 7b).

It remains to compute the precise extension classes represented by Ψ′_{S} and Ψ′′_{S} or, equivalently, Ψ̃′_{S} and Ψ̃′′_{S}.

Proposition 3.5. a) Let K be a local field and L/K a finite Galois extension with group G. Consider the complex of G-modules RΓ(L, 𝔾_{m}). One has H^{i}(L, 𝔾_{m}) = 0 for i ≠ 0, 2 and canonical isomorphisms

H^{0}(L, 𝔾_{m}) ⟵^{∼} L^{×}, H^{2}(L, 𝔾_{m}) ⟶_{∼}^{inv_{L}} ℚ/ℤ,

so that τ^{≤2}RΓ(L, 𝔾_{m}) represents a class r_{L/K} ∈ Ext_{G}^{3}(ℚ/ℤ, L^{×}) ≃ H^{2}(G, L^{×}). Claim: inv_{K}(r_{L/K}) = \frac{1}{|G|}.

b) Let K be a global field and L/K a finite Galois extension with group G, unramified outside a finite set S of places of K containing S_{∞}. Consider the complex of G-modules RΓ_{c}(O_{L,S}, 𝔾_{m}). One has H_{c}^{i}(O_{L,S}, 𝔾_{m}) = 0 for i ≠ 1, 3 and canonical isomorphisms

H_{c}^{1}(O_{L,S}, 𝔾_{m}) ⟵^{∼} C_{S}(L), H_{c}^{3}(O_{L,S}, 𝔾_{m}) ⟶_{∼}^{trace_{L}} ℚ/ℤ,

so that τ^{≥1}τ^{≤3}RΓ_{c}(O_{L,S}, 𝔾_{m}) represents a class r_{L/K} ∈ Ext_{G}^{3}(ℚ/ℤ, C_{S}(L)) ≃ H^{2}(G, C_{S}(L)). Claim: inv_{K}(r_{L/K}) = \frac{1}{|G|}.

Here C_{S}(L) is the group belonging to the S-class formation introduced in [22, Ch.I,§4] and agrees with the group C_{S} = J_{S}/U_{S} used above if Pic(O_{L,S}) = 0.

Proof. The computation of the cohomology of 𝔾_{m} and the construction of the invariant and trace map is standard (see [25, Ch. XIII, Prop. 6], [22, Ch.II,Prop. 2.6]). In the local case the claim essentially follows from [25, Ch. XIII, exercise 2] and the global case follows from the local case. However, we wish to explain this in more detail. In both cases one first shows

Lemma 12. r_{L/K} is the fundamental class of a class formation, i.e. r_{L/K} generates H^{2}(G, L^{×}) (resp. H^{2}(G, C_{S}(L))) and for a tower of extensions L ⊇ L′ ⊇ K (unramified outside S in the global case) one has

(89)  res_{H}^{G}(r_{L/K}) = r_{L/L′}
(90)  inf_{G/H}^{G}(r_{L′/K}) = [L : L′]r_{L/K}

where H = G(L/L′).

Proof. Assume K local. Let

𝔾_{m} ↪ I^{•} (91)

be an injective resolution over (Spec K)_{et}. Then for any finite extension L/K, RΓ(L, 𝔾_{m}) can be identified with H^{0}(L, I^{•}) and therefore consists of cohomologically trivial G-modules. It is acyclic in degrees ≥ 3 for any such L/K, hence a standard argument (cf. [5, step 2 in Prop. 1.20] or [14, Prop. and Lemma 2.1]) shows that τ^{≤2}RΓ(L, 𝔾_{m}) still consists of c.t. G-modules. So in the exact sequence representing r_{L/K}

0 → L^{×} → A → B → C → ℚ/ℤ → 0 (92)

A, B, C are c.t. G-modules, which means that cup product with r_{L/K} induces an isomorphism

Ĥ^{i}(G, ℚ/ℤ) ⟶^{∼} Ĥ^{i+3}(G, L^{×})

for all i ∈ ℤ. The case i = −1 of this isomorphism implies that r_{L/K} is a generator of H^{2}(G, L^{×}). The same argument applies to the global case. Indeed, the complex M^{•} := τ^{≤3}RΓ_{c}(O_{L,S}, 𝔾_{m}) consists of c.t. G-modules and so does τ^{≥1}τ^{≤3}RΓ_{c}(O_{L,S}, 𝔾_{m}) because τ^{≥1} only involves replacing M_{1} by M_{1}/M_{0} and M_{0} by 0 (M_{i} = 0 for i < 0).

Formula (89) follows because (92) can be regarded over the subgroup G(L/L′) of G(L/K) and represents r_{L/L′} since (91) also gives a resolution of 𝔾_{m} over (Spec L′)_{et}. As to (90) the natural map

RΓ(L′, 𝔾_{m}) → RΓ(L, 𝔾_{m})

induced by Spec L → Spec L′ gives a commutative diagram

\begin{aligned}0 & ⟶ & L′^{×} & ⟶ & A′ & ⟶ & B′ & ⟶ & C′ & ⟶ & ℚ/ℤ & ⟶ & 0 \\ & & ↓ & & ↓ & & ↓ & & ↓ & & ↓^{[L′:L]} \\ 0 & ⟶ & L^{×} & ⟶ & A & ⟶ & B & ⟶ & C & ⟶ & ℚ/ℤ & ⟶ & 0 \end{aligned} (93)

The top row represents r_{L′/K} and we get inf_{G/H}^{G}(r_{L′/K}) by regarding the top row as a sequence of G-modules and taking the pushout under the inclusion L′^{×} ↪ L^{×}, which yields a class in Ext_{G}^{3}(ℚ/ℤ, L^{×}) = H^{2}(G, L^{×}). This pushout agrees with the pullback of the bottom row under multiplication by [L : L′] on ℚ/ℤ because (93) is commutative. But the bottom row represents r_{L/K}, hence the formula (90).

The proof in the global case is entirely similar (see [22, Ch.II §3] for the relevant properties of the trace map). □

Lemma 13. If L/K is an unramified extension of local fields then inv_{K}(r_{L/K}) = \frac{1}{|G|}.

Proof. We construct a particular acyclic resolution of 𝔾_{m} on (Spec K)_{et} using [25, XIII, ex. 2]. Let K_{ur}/K be the completion of the maximal unramified extension of K, N/K any finite Galois extension and N_{ur} = NK_{ur}, N_{0} = N ∩ K_{ur} where we consider all fields inside a fixed complete algebraically closed extension of K. Let F ∈ G(N_{ur}/K) be any choice of a Frobenius automorphism and ϕ (resp. f) its image in G(N/K) (resp. G(N_{0}/K)). Consider the commutative diagram of G(N/K)-modules with exact rows

\begin{aligned}1 & ⟶ & N^{×} & ⟶ & (N ⊗_{K} K_{ur})^{×} & ⟶^{1−1⊗F} & (N ⊗_{K} K_{ur})^{×} & ⟶ & ℤ & ⟶ & 0 \\ & & ^{v}↓ & & ṽ↓ & & ṽ↓ & & ∥ \\ 0 & ⟶ & ℤ & ⟶ & C(G(N_{0}/K), ℤ) & ⟶^{1−f^{−1}} & C(G(N_{0}/K), ℤ) & ⟶^{ε} & ℤ & ⟶ & 0. \end{aligned} (94)

Here C(−, ℤ) is the module of ℤ-valued functions with G(N_{0}/K)-action explained before Lemma 5. Setting m = [N_{0} : K] one has a ring isomorphism

N ⊗_{K} K_{ur} ⟶^{∼} ∏_{i=0}^{m−1} N_{ur}, x ⊗ y ↦ (ϕ^{i}(x)y)_{i},

and in this description

ṽ(y_{0}, ..., y_{m−1})(f^{i}) = v(y_{−i})

where v : N_{ur}^{×} → ℤ is the valuation normalized by v(π) = 1 for a uniformizer π. The action of 1 ⊗ F translates into

(1 ⊗ F)(y_{0}, ..., y_{m−1}) = (F ϕ^{−m}(y_{m−1}), F (y_{0}), F (y_{1}), .., F (y_{m−2})).

Here ϕ^{−m} lies in the inertia subgroup of G(N/K), so it can be canonically viewed inside G(N_{ur}/K). The commutativity of (94) and the G(N/K)-equivariance of ṽ follow easily from these descriptions. The bottom row is a standard exact sequence, the exactness of the top row follows from the commutativity and the surjectivity of 1 − F on ker(ṽ) ([25, XIII, Prop. 15]). One has

(N ⊗_{K} K_{ur})^{×} ≅ Ind_{G(N/N_{0})}^{G(N/K)} N_{ur}^{×}

so by [25, X, Prop. 11] this is a c.t. G(N/K)-module.

If N′/N is a finite extension such that N′/K is Galois the diagram (94) naturally maps into the corresponding diagram for N′ where the map on the right hand ℤ is multiplication by [N′ : N]. The map on C(G(N_{0}/K), ℤ) is multiplication by the ramification degree e(N′/N) followed by the natural inclusion C(G(N_{0}/K), ℤ) → C(G(N′_{0}/K), ℤ). So after taking the direct limit over all N the top row gives a resolution

𝔾_{m} ↪ A → B → ℚ → 0 → ⋯ (95)

by acyclic, discrete G(\overline{K}/K)-modules, at least if we also fix an isomorphism of ℚ with the direct limit of the right hand ℤ’s. We do this by prescribing it to be the inclusion ℤ ⊂ ℚ for N = K. In this way we obtain an identification of RΓ(N, 𝔾_{m}) with

(N ⊗_{K} K_{ur})^{×} ⟶^{1−1⊗F} (N ⊗_{K} K_{ur})^{×} ⟶^{w_{N}} ℚ

where w_{N} = \frac{1}{[N:K]} ε ṽ. However, we must still identify the map inv_{N}.

Lemma 14. The map

inv_{N} : ℚ/ im(w_{N}) = ℚ/\frac{1}{[N : K]}ℤ → ℚ/ℤ

is induced by multiplication with −[N : K] on ℚ.

Proof. It suffices to prove this for N = K because of the commutative diagram

(N ⊗_{K} K_{ur})^{×} ⟶^{1−1⊗F} (N ⊗_{K} K_{ur})^{×} ⟶^{w_{N}} ℚ ⟶^{inv_{N}} ℚ/ℤ
↑  ↑  ∥  ↑^{[N:K]}
K_{ur}^{×}  ⟶^{1−1⊗F}  K_{ur}^{×}  ⟶^{w_{K}} ℚ ⟶^{inv_{K}} ℚ/ℤ.

Passing to the limit over all unramified N/K in (94) we get the first two rows of the following commutative diagram of discrete Γ := G(K_{ur}/K)-modules

\begin{matrix}K_{ur}^{×} & ⟶ & (K_{ur} ⊗_{K} K_{ur})^{×} & ⟶^{1−1⊗F} & (K_{ur} ⊗_{K} K_{ur})^{×} & ⟶ & ℚ & ⟶ & 0 \\ ^{v}↓ & & ṽ↓ & & ṽ↓ & & ∥ \\ ℤ & ⟶ & C(Γ, ℤ) & ⟶^{1−F^{−1}} & C(Γ, ℤ) & ⟶ & ℚ & ⟶ & 0 \\ ↓ & & ↓ & & ↓ & & ∥ \\ ℚ & ⟶ & C(Γ, ℚ) & ⟶^{1−F^{−1}} & C(Γ, ℚ) & ⟶^{ϵ} & ℚ & ⟶ & 0 \\ ↓ & & ↓ & & ↓^{−F} & & ∥ \\ ℚ/ℤ & ⟶ & C(Γ, ℚ/ℤ) & ⟶^{1−F} & C(Γ, ℚ/ℤ) & ⟶ & 0 \end{matrix} (96)

where ϵ(f) = n^{−1} ∑_{γ mod nΓ} f(γ) if f factors through Γ/nΓ. All the rows in this diagram are Γ-acyclic resolutions of their left hand terms. The top row has a natural map into (95) inducing the inflation map H^{i}(Γ, K_{ur}^{×}) → H^{i}(K, 𝔾_{m}). This is an isomorphism for i = 2 which is the first step in the construction of inv_{K}. One then uses the chain of isomorphisms

(97) H^{2}(Γ, K_{ur}^{×}) ⟶^{v_{∗}} H^{2}(Γ, ℤ) ⟵^{δ} H^{1}(Γ, ℚ/ℤ) ⟶^{ι} Hom(Γ, ℚ/ℤ) ⟶^{ev} ℚ/ℤ

where the last map is evaluation at F. All of these steps can be made explicit using (96). Upon taking Γ-invariants in (96) we obtain the following commutative diagram of complexes.

RΓ(Γ, K_{ur}^{×}) : K_{ur}^{×} ⟶^{1−F} K_{ur}^{×} ⟶ ℚ
v↓  ↓  ∥
RΓ(Γ, ℤ) :  ℤ ⟶^{0} ℤ ⟶^{⊂} ℚ
↓  ↓  ∥
RΓ(Γ, ℚ) :  ℚ ⟶^{0} ℚ  ℚ
↓  ↓^{−1}  ↓
RΓ(Γ, ℚ/ℤ) : ℚ/ℤ ⟶^{0} ℚ/ℤ ⟶ 0

(98)

By Lemma 5 applied to Γ′ = Γ we have that ι^{−1} ∘ ev^{−1}(x) is indeed the constant function with value x ∈ ℚ/ℤ (the bottom right group in (98)). The remaining isomorphisms in (97) can then be easily identified in (98), and one obtains the desired description of inv_{K}. □

To finish the proof of Lemma 13 we look again at (94) for L = N = N_{0} an unramified extension of K, so that m = [L : K]. By our computation of inv_{L} the class r_{L/K} equals the Yoneda class of the 3-extension of G-modules

0 → L^{×} → (L ⊗_{K} K_{ur})^{×} ⟶^{1−1⊗F} (L ⊗_{K} K_{ur})^{×} ⟶^{w_{L}} ℚ ⟶^{−[L:K]} ℚ/ℤ → 0

where w_{L} = \frac{1}{[L:K]} ε ṽ. So it is easily seen that −r_{L/K} is the image of the element r′_{L/K} ∈ Ext_{G}^{2}(ℤ, L^{×}) = H^{2}(G, L^{×}) represented by the top row of (94), under the boundary map induced by the short exact sequence (38). Here we use again Lemma 3. The pushout r′′_{L/K} ∈ Ext_{G}^{2}(ℤ, ℤ) = H^{2}(G, ℤ) of r′_{L/K} under the map v is represented by the second row in (94) and we shall show that inv_{K}(r′′_{L/K}) = −\frac{1}{m}.

Consider the commutative diagram of G-modules

\begin{aligned}0 & ⟶ & ℤ & ⟶ & C(G, ℤ) & ⟶^{1−f^{−1}} & C(G, ℤ) & ⟶^{ϵ} & ℤ & ⟶ & 0 \\ & & ∥ & & ^{\frac{1}{m} ϵ}↓ & & ^{β}↓ & & ∥ \\ 0 & ⟶ & ℤ & ⟶ & ℚ & ⟶^{(π,0)} & ℚ/ℤ ⊕ ℤ & ⟶ & ℤ & ⟶ & 0 \end{aligned} (99)

where

β(ψ) = ( ∑_{i=0}^{m−1} ψ(f^{i})\frac{i}{m}, ∑_{i=0}^{m−1} ψ(f^{i}))

and the group ℚ/ℤ ⊕ ℤ becomes a G-module by

g(a, b) = (a − ϕ(g)b, b) (100)

where ϕ : G → ℚ/ℤ is the homomorphism such that ϕ(f) = −\frac{1}{m}. In view of Lemma 3 diagram (99) shows that r′′_{L/K} maps to ϕ under the chain of isomorphisms

Ext_{G}^{2}(ℤ, ℤ) ≅ Ext_{G}^{1}(ℤ, ℚ/ℤ) ≅ H^{1}(G, ℚ/ℤ) ≅ Hom(G, ℚ/ℤ).

Note that

0 → ℚ/ℤ → ℚ/ℤ ⊕ ℤ → ℤ → 0

with G-action on the middle term given by (100) represents the 1-extension corresponding to ϕ because (100) is the pushout of the universal 1-extension

0 → I[G] → ℤ[G] → ℤ → 0

under the G-homomorphism Φ : I[G] → ℚ/ℤ given by Φ(g − 1) = ϕ(g). Following through the definition of inv_{K} we conclude that inv_{K}(r′′_{L/K}) = −\frac{1}{m} and this finishes the proof of Lemma 13. □

If now L/K is an arbitrary Galois extension of local fields, there is a (unique) unramified extension N/K with [N : K] = [L : K]. Setting M = NL, we have

inf_{G(L/K)}^{G(M/K)}(r_{L/K}) = [M : L]r_{M/K} = [M : N]r_{M/K} = inf_{G(N/K)}^{G(M/K)}(r_{N/K}) (101)

by Lemma 12. Since invariants are unchanged under inflation we obtain

inv_{K}(r_{L/K}) = inv_{K}(r_{N/K}) = \frac{1}{|G|}

by Lemma 13. This gives part a) of Proposition 3.5.

Given an extension L/K of global fields we can find an extension N/K with [L : K] = [N : K] = [N_{w_{0}} : K_{v_{0}}] for some (non-archimedean) place w_{0} of N, for example a suitable cyclotomic extension. Here v_{0} is the place of K induced by w_{0}. After enlarging S we can assume that N/K is unramified outside S and that v_{0} ∈ S. Enlarging S to S′, say, is unproblematic because the natural map

RΓ_{c}(O_{L,S′}, 𝔾_{m}) → RΓ_{c}(O_{L,S}, 𝔾_{m})

induces an isomorphism on H^{3} [22, Ch.II,§3] and the natural map ρ : C_{S′}(L) → C_{S}(L) on H^{1}. This means that the pushout under ρ of r_{L/K}, defined with S′, is r_{L/K} defined with S. Moreover ρ induces an isomorphism

H^{2}(G, C_{S′}(L)) ⟶^{∼} H^{2}(G, C_{S}(L))

which commutes with inv_{K} (the invariant map for the respective S or S′ class formation), and so we can compute inv_{K}(r_{L/K}) using either S or S′.

Taking such a field N and applying (101) we are reduced to the case that G coincides with the decomposition group G_{w_{0}} for some place w_{0}|v_{0} ∈ S. So it remains to find a connection between the local and global r_{L/K} in this case. The natural map

RΓ(L_{w_{0}}, 𝔾_{m}) → RΓ_{c}(O_{L,S}, 𝔾_{m})[1]

gives a commutative diagram

\begin{aligned}0 & ⟶ & L_{w_{0}}^{×} & ⟶ & A′ & ⟶ & B′ & ⟶ & C′ & ⟶ & ℚ/ℤ & ⟶ & 0 \\ & & ↓ & & ↓ & & ↓ & & ↓ & & ↓ \\ 0 & ⟶ & C_{S}(L) & ⟶ & A & ⟶ & B & ⟶ & C & ⟶ & ℚ/ℤ & ⟶ & 0 \end{aligned} (102)

which induces for each integer i a commutative diagram

∪r_{L_{w_{0}} /K_{v_{0}}} :Ĥ^{i}(G_{w_{0}}, ℚ/ℤ) ⟶^{∼} Ĥ^{i+3}(G_{w_{0}}, L_{w_{0}}^{×})
↓  ↓
∪r_{L/K} : Ĥ^{i}(G, ℚ/ℤ) ⟶^{∼} Ĥ^{i+3}(G, C_{S}(L)).

Now take i = −1, so that Ĥ^{−1}(G_{w_{0}}, ℚ/ℤ) ≅ Ĥ^{−1}(G, ℚ/ℤ) ≅ \frac{1}{|G|}ℤ/ℤ. Consider the element \frac{1}{|G_{w_{0}} |} = \frac{1}{|G|} in the top left group. It is mapped to r_{L_{w_{0}} /K_{v_{0}}} in the top right group, and to r_{L/K} in the bottom right group. Since the right hand map for i = −1 commutes with taking invariants we conclude inv_{K}(r_{L/K}) = inv_{K_{v_{0}}} (r_{L_{w_{0}} /K_{v_{0}}} ) = \frac{1}{|G|} by part a) of Proposition 3.5. □

Theorem 3.2. We have Ω(L/K, 3) = Ω(L/K, ℚ(0))^{#}.

Proof. Proposition 3.5b) implies that e(Ψ′′_{S}) = c_{L/K}. As to e(Ψ′_{S}), we know from Proposition 3.4 that e(Ψ̃′_{S}) lies in the diagonal subgroup

⨁_{v∈S} Ext_{G_{v}}^{3}(ℚ/ℤ, L_{w(v)}^{×}) ⊆ Ext_{G}^{3}(Y_{S} ⊗ ℚ/ℤ, J_{S})

and Proposition 3.5a) and Proposition 3.4 imply that the components of c_{L/K}^{loc} and e(Ψ′_{S}) agree for non-archimedean places v. For archimedean places v on the other hand, the group Ext_{G_{v}}^{3}(ℚ/ℤ, L_{w(v)}^{×}) is either trivial or of order 2, the local canonical class c_{L_{w(v)}/K_{v}} being the only non-zero element in the latter case. To show that the class r_{v} ∈ Ext_{G_{v}}^{3}(ℚ/ℤ, L_{w(v)}^{×}) represented by ⨁_{w|v} C_{w} is nontrivial for |G_{v}| = 2 we can argue as follows. As in the proof of Proposition 3.5 the map

⨁_{w|v} C_{w} → RΓ_{c}(O_{L,S}, 𝔾_{m})[1]

gives a commutative diagram

0 ⟶ ⨁_{w|v} L_{w}^{×} ⟶ A′ ⟶ B′ ⟶ C′ ⟶ ⨁_{w|v} ℚ/ℤ ⟶ 0
↓  ↓  ↓  ↓  ↓^{Σ}
0 ⟶ C_{S}(L) ⟶ A ⟶ B ⟶ C ⟶ ℚ/ℤ ⟶ 0

which induces for each integer i a commutative diagram in G-cohomology

\begin{matrix}∪r_{v} : & Ĥ^{i}(G_{v}, ℚ/ℤ) & ⟶ & Ĥ^{i+3}(G_{v}, L_{w(v)}^{×}) \\ ↓^{Σ_{∗}} & & ↓ \\ ∪r_{L/K} : & Ĥ^{i}(G, ℚ/ℤ) & ⟶^{∼} & Ĥ^{i+3}(G, C_{S}(L)). \end{matrix} (103)

The fact that Σ is the sum map is a consequence of (87). We do not know a priori that the top arrow in (103) is an isomorphism but it is easy to check that Σ_{∗} is injective for i = −1. So r_{v} = 1/2 ∪ r_{v} ∈ H^{2}(G_{v}, L_{w(v)}^{×}) cannot be zero.

We conclude that the right hand column in (88) satisfies the assumptions of Lemma 4 and that K_{S} and Ψ_{S} are quasi-isomophic. By Theorem 3.1 and (28) we get Ω(L/K, 3) = Ω(L/K, ℚ(0))^{#}. □

Remark. Theorem 3.2 gives a positive answer to the first half of Question 1.54 in [5] which asks whether the identity Ω(L/K, 3) = Ω(L/K, ℚ(0))^{#} holds. The second half of Question 1.54 (ibid), asking whether Chinburg’s invariant Ω(L/K, 1) equals −Ω(L/K, ℚ(1)), will follow from forthcoming work of the first author [4] in conjunction with Theorem 3.2.

Theorem 3.2 is also consistent with, on the one hand, Chinburg’s conjecture that Ω(L/K, 3) = 0 for abelian G and on the other hand, Conjecture 4 of [5] which predicts the vanishing of Ω(L/K, M) for any motive M.

4. The case n > 0

4.1. Definition of Ω_{n}(L/K). In this subsection we recall the definition of the class Ω_{n}(L/K) from [9]. We fix a finite set S of places as in section 2 and define for any integer r = n + 1 > 1 and any prime number p

K_{p}(r) := Cone(RΓ(O_{L,S_{p}}, ℤ_{p}(r)) → ⨁_{w|∞} RΓ(L_{w}, ℤ_{p}(r))) (104)

Lemma 15. The complex K_{p}(r) is quasi-isomorphic to RΓ(O_{L,S_{p}}, C_{p}(r)) where C_{p}(r) is the cokernel of the natural injection ℤ_{p}(r) → ⨁_{w|∞} i_{w,∗}i_{w}^{∗}ℤ_{p}(r) of sheaves on (Spec O_{L,S_{p}})_{et}. Here i_{w} : Spec L_{w} → Spec O_{L,S_{p}} is the natural map.

Proof. This follows easily from R^{q}i_{w,∗}ℤ_{p}(r) = 0 for q > 0 (see [21, III 1.13]) which implies that RΓ(O_{L,S_{p}}, i_{w,∗}i_{w}^{∗}ℤ_{p}(r)) = RΓ(L_{w}, ℤ_{p}(r)). □

Remark. We have inserted this lemma for the convenience of the reader because the authors of [9] do not directly refer to the complex K_{p}(r) but rather work with the cohomology groups of C_{p}(r).

One shows that K_{p}(r) ≅ RΓ(O_{L,S_{p}}, C_{p}(r)) is ℤ_{p}[G]-perfect (as in [5, Prop. 1.20]), has cohomology concentrated in degrees 0, 1 (using Artin-Verdier duality) and that there are finitely generated ℤ[G]-modules K′_{2r−i} with isomorphisms

τ_{p}^{i} : K′_{2r−1−i} ⊗_{ℤ} ℤ_{p} ≅ H^{i}(K_{p}(r))

for i = 0, 1 and each prime p ([9]). The group K′_{2r−2} is finite and there is a commutative diagram of short exact sequences

\begin{aligned}0 & ⟶ & ⨁_{w|∞} H^{0}(L_{w}, ℚ(r)) & ⟶ & K′_{2r−1} ⊗ ℚ & ⟶ & K_{2r−1}(O_{L}) ⊗ ℚ & ⟶ & 0 \\ & & ↓ & & ^{τ_{p}^{0}⊗ℚ}↓ & & ^{Chern}↓ \\ 0 & ⟶ & ⨁_{w|∞} H^{0}(L_{w}, ℚ_{p}(r)) & ⟶ & H^{0}(K_{p}(r) ⊗ ℚ_{p}) & ⟶ & H^{1}(O_{L,S_{p}}, ℚ_{p}(r)) & ⟶ & 0 \end{aligned} (105)

The top row here is equation (5) in [9], and the bottom row is induced from (104).

From these facts one shows that there is a perfect complex K(r) of ℤ[G]-modules and quasi-isomorphisms

τ_{p} : K(r) ⊗_{ℤ} ℤ_{p} ≅ K_{p}(r) (106)

for each prime number p such that H^{i}(τ_{p}) = τ_{p}^{i}. The argument here is similar to the proof of Lemma 17 below. Moreover, the complex K(r) is specified uniquely up to quasi-isomorphism. By definition then

Ω_{r−1}(L/K) := (K(r)) − (ℤΣ_{∞}) ∈ Cl(ℤ[G]) (107) where Σ_{∞} is the G-set Hom(L, ℂ). If G is abelian, then with our parity convention for the determinant, this is equal to

Ω_{r−1}(L/K) = (det_{ℤ[G]}^{−1}K(r)) − (det_{ℤ[G]}^{−1}ℤΣ_{∞})

(108) = (det_{ℤ[G]}^{−1}K(r)) ∈ Pic(ℤ[G]) ≅ Cl(ℤ[G])

because ℤΣ_{∞} is a free ℤ[G]-module.

Remark. The complex K(r) plays a role similar to that of the complex K_{S} in section 3. Both K_{S} and K(r) depend on S but their classes in Cl(ℤ[G]) do not. But K(r) ⊗_{ℤ} ℚ is also independent of S which reflects the fact that the rank of K_{2r−1}(O_{L,S}) does not vary with S for r > 1, as it does for r = 1.

Another difference is that the class of K_{S}, taken in K_{0}(ℤ[G]), already lies in the direct summand Cl(ℤ[G]) of K_{0}(ℤ[G]). This is not true for K(r) which is the reason for the term ℤΣ_{∞} in (107).

4.2. Comparison with Ω(L/K, ℚ(1 − r))^{#}. We first prove a lemma which is also very useful in other contexts. Define three exact functors from the derived category 𝔇_{p} of ℤ_{p}[G]-modules to itself

(−)^{∨} = R Hom_{ℤ_{p}}(−, ℚ_{p}/ℤ_{p}) = Hom_{ℤ_{p}}(−, ℚ_{p}/ℤ_{p})
(−)^{∧} = R Hom_{ℤ_{p}}(−, ℚ_{p}) = Hom_{ℤ_{p}}(−, ℚ_{p})
(−)^{∗} = R Hom_{ℤ_{p}}(−, ℤ_{p}),

all with the contragredient G-action.

Lemma 16. a) On the full subcategory of 𝔇_{p} consisting of objects X such that H^{i}(X) is finitely generated for each i, we have natural isomorphisms

X ⟶^{∼} X^{∨∨} ⟶^{∼} X^{∨∗}[1] ⟵^{∼} X^{∗∗} (109)

and

X^{∨∧} ⟶^{∼} 0; X^{∧∗} ⟶^{∼} 0. (110)

b) Consider an exact triangle in 𝔇_{p}

X → X ⊗_{ℤ_{p}} ℚ_{p} → Y → (111)

where X has finitely generated cohomology. Then there is a natural isomorphism

Y^{∨} ≅ X^{∗}.

Proof. From the short exact sequence (38), tensored with ℤ_{p}, we obtain an exact triangle

X^{∗} → X^{∧} → X^{∨} → (112) for each object X in 𝔇_{p}. Consider then the 9-term diagram with exact rows and columns

\begin{matrix}X^{∨∗} & ⟶ & X^{∧∗} & ⟶ & X^{∗∗} & ⟶ \\ ↓ & & ↓ & & ↓ \\ X^{∨∧} & ⟶ & X^{∧∧} & ⟶ & X^{∗∧} & ⟶ \\ ↓ & & ^{β}↓ & & ↓ \\ X^{∨∨} & ⟶ & X^{∧∨} & ⟶ & X^{∗∨} & ⟶ \\ ↓ & & ↓ & & ↓ \end{matrix} (113)

Since both ℚ_{p} and ℚ_{p}/ℤ_{p} are injective ℤ_{p}-modules there is an isomorphism

H^{i}(X^{∨∧}) ≅ Hom_{ℤ_{p}}(H^{i}(X)^{∨}, ℚ_{p}).

This group is zero because H^{i}(X)^{∨} is a torsion group, hence X^{∨∧} ⟶^{∼} 0. Similarly, the map H^{i}(β) identifies with the natural map

Hom_{ℤ_{p}}(W, ℚ_{p}) → Hom_{ℤ_{p}}(W, ℚ_{p}/ℤ_{p})

for W = H^{i}(X)^{∧}. But this map is an isomorphism for any finite dimensional ℚ_{p}- space W. Therefore β is a quasi-isomorphism and X^{∧∗} ⟶^{∼} 0. The natural map X → X^{∨∨} is a quasi-isomorphism because M → M^{∨∨} is an isomorphism for any finitely generated ℤ_{p}-module M. Together with (113) this gives a).

Applying the functor (−)^{∗} to (111) we obtain an exact triangle

Y^{∗} → (X ⊗ ℚ_{p})^{∗} → X^{∗} →

where the middle term is acyclic, for example because it can be rewritten as X^{∧∧∗} which is zero by a). Hence we deduce an isomorphism X^{∗} ≅ Y^{∗}[1]. Using finite generation of H^{i}(X) and the long exact cohomology sequence induced by (111) we find that H^{i}(Y) is a torsion group and therefore that Y^{∧} is acyclic. From (112) for Y we then obtain an isomorphism Y^{∨} ≅ Y^{∗}[1] ≅ X^{∗}, i.e. part b). □

The diagram in the following proposition is the key to the comparison of Ω_{r−1}(L/K) and Ω(L/K, ℚ(1 − r)). Recall the definitions of RΓ_{Δ} and R̃Γ_{c} from (82).

Proposition 4.1. There exists a commutative diagram in 𝔇_{p} as follows:

Δ_{p}(r)  ⟶ ⨁_{w|∞} RΓ_{Δ}(L_{w}, ℤ_{p}(1−r))^{∗}[−3] ⟶^{α_{p}} ⨁_{w|∞} RΓ(L_{w}, ℤ_{p}(r))
↓  ↓  ∥
K_{p}(r)[−1]  ⟶  RΓ(O_{L,S_{p}}, ℤ_{p}(r))  ⟶ ⨁_{w|∞} RΓ(L_{w}, ℤ_{p}(r))
↓  ↓
RΓ_{c}(O_{L,S_{p}}, ℤ_{p}(1−r))^{∗}[−3]  RΓ_{c}(O_{L,S_{p}}, ℤ_{p}(1−r))^{∗}[−3]

(114)

Moreover, all rows and columns in this diagram are exact triangles and all terms in the left hand column are ℤ_{p}[G]-perfect .

Proof. If we choose X = R̃Γ_{c}(O_{L,S_{p}}, ℤ_{p}(1−r)) in Lemma 16b) the triangle (111) naturally identifies with the triangle

R̃Γ_{c}(O_{L,S_{p}}, ℤ_{p}(1−r)) → R̃Γ_{c}(O_{L,S_{p}}, ℚ_{p}(1−r)) → R̃Γ_{c}(O_{L,S_{p}}, ℚ_{p}/ℤ_{p}(1−r)) → .

From Artin-Verdier duality and Lemma 16b) we obtain a quasi-isomorphism

RΓ(O_{L,S_{p}}, ℤ_{p}(r)) ≅ R̃Γ_{c}(O_{L,S_{p}}, ℚ_{p}/ℤ_{p}(1 − r))^{∨}[−3]
≅ R̃Γ_{c}(O_{L,S_{p}}, ℤ_{p}(1 − r))^{∗}[−3]

which together with (104) and (83) yields the diagram (114). Since both K_{p}(r) and RΓ_{c}(O_{L,S_{p}}, ℤ_{p}(1 − r)) are ℤ_{p}[G]-perfect (see [5, Prop. 1.20]) the same is true for Δ_{p}(r). □

Proposition 4.2. There exists a perfect complex Δ(r) of ℤ[G]-modules and quasiisomorphisms

σ_{p} : Δ(r) ⊗_{ℤ} ℤ_{p} ≅ Δ_{p}(r)

for each prime p, such that there is a commutative diagram of long exact sequences

0 ⟶ H^{1}(Δ(r) ⊗ ℚ) ⟶ H^{0}(K(r) ⊗ ℚ) ⟶ K_{2r−1}(O_{L}) ⊗ ℚ  ⟶ 0 ⟶ ⋯
H^{1}(σ_{p}⊗ℚ_{p})↓  ^{H^{1}(τ_{p}⊗ℚ_{p})}↓  ^{(10)^{∧}}↓
0 ⟶ H^{1}(Δ_{p}(r) ⊗ ℚ_{p}) ⟶ H^{0}(K_{p}(r) ⊗ ℚ_{p}) ⟶ H_{c}^{2}(O_{L,S_{p}}, ℚ_{p}(1 − r))^{∧} ⟶ 0 ⟶ ⋯

\begin{aligned}⋯ & ⟶ & 0 & ⟶ & ⨁_{w∈S_{∞}} H^{0}(L_{w}, ℚ(r)) & ⟶ & H^{3}(Δ(r) ⊗ ℚ) & ⟶ & 0 \\ & & & & ^{(9)^{∧}}↓ & ^{H^{3}(σ_{p}⊗ℚ_{p})}↓ \\ ⋯ & ⟶ & 0 & ⟶ & H_{c}^{1}(O_{L,S_{p}}, ℚ_{p}(1 − r))^{∧} & ⟶ & H^{3}(Δ_{p}(r) ⊗ ℚ_{p}) & ⟶ & 0 \end{aligned} (115)

where the lower row is induced by the left vertical triangle in (114), tensored with ℚ_{p}.

Proof. Since Δ_{p}(r) has cohomology concentrated in degrees 1, 2, 3, the map α_{p} in (114) induces a cohomology isomorphism in degrees ≥ 4 so that there is an exact triangle

Δ_{p}(r) → τ^{≤4} ⨁_{w|∞} RΓ_{Δ}(L_{w}, ℤ_{p}(1 − r))^{∗}[−3] ⟶^{τ^{≤4}α_{p}} τ^{≤4} ⨁_{w|∞} RΓ(L_{w}, ℤ_{p}(r)) → .

Moreover, if we denote by ℤ(r) the étale sheaf on Spec L_{w} given by the abelian group ℤ on which complex conjugation acts via (−1)^{r} we have natural isomorphisms

RΓ(L_{w}, ℤ_{p}(r)) ≅ RΓ(L_{w}, ℤ(r)) ⊗ ℤ_{p} (116)

and

RΓ_{Δ}(L_{w}, ℤ_{p}(1 − r))^{∗} ≅ RΓ_{Δ}(L_{w}, ℤ(1 − r))^{∗} ⊗ ℤ_{p}

where this last (−)^{∗} means Hom_{ℤ}(−, ℤ). We set

\begin{aligned}A & := τ^{≤4} ⨁_{w|∞} RΓ_{Δ}(L_{w}, ℤ(1 − r))^{∗}[−3] \\ B & := τ^{≤4} ⨁_{w|∞} RΓ(L_{w}, ℤ(r)), \end{aligned} (117)

so that both A and B are bounded complexes of ℤ[G]-modules with finitely generated cohomology and for each i either H^{i}(A) or H^{i}(B) is torsion.

Lemma 17. Suppose A and B are bounded complexes of ℤ[G]-modules with finitely generated cohomology such that for each i ∈ ℤ either H^{i}(A) or H^{i}(B) is torsion. Then the natural map

Hom_{𝔇}(A, B) → ⨁_{p} Hom_{𝔇_{p}}(A ⊗ ℤ_{p}, B ⊗ ℤ_{p})

is an isomorphism.

Proof. We first show that for each prime p

Hom_{𝔇}(A, B) ⊗ ℤ_{p} ≅ Hom_{𝔇_{p}}(A ⊗ ℤ_{p}, B ⊗ ℤ_{p}). (118)

We can pick a quasi-isomorphism P ≅ A where P is a complex consisting of finitely generated projective ℤ[G]-modules so that

(119) Hom_{𝔇}(A, B) = H^{0}(R Hom(A, B)) = H^{0}(Hom^{•}(P, B)).

Since B is bounded we have

Hom^{•}(P, B) = ∏_{i∈ℤ} Hom(P^{i}, B^{•+i}) = ⨁_{i∈ℤ} Hom(P^{i}, B^{•+i})

and therefore

Hom^{•}(P, B) ⊗ ℤ_{p} ≅ Hom_{ℤ_{p}[G]}^{•}(P ⊗ ℤ_{p}, B ⊗ ℤ_{p}) (120)

since ℤ → ℤ_{p} is flat so that (118) holds for modules, provided A is finitely generated. Again because of flatness we get

H^{0}(Hom^{•}(P, B)) ⊗ ℤ_{p} = H^{0}(Hom^{•}(P, B) ⊗ ℤ_{p})

which together with (119) and (120) gives (118).

The lemma will follow if we show in addition that

Hom_{𝔇}(A, B) ≅ ⨁_{p} Hom_{𝔇}(A, B) ⊗ ℤ_{p},

i.e. that Hom_{𝔇}(A, B) is a torsion group. But this follows easily from the spectral sequence [27][III.4.6.10]

E_{2}^{p,q} = ∏_{i∈ℤ} Ext_{G}^{p}(H^{i}(A), H^{q+i}(B)) ⇒ H^{p+q}(R Hom(A, B))

together with (119) and the fact that Ext_{G}^{p}(M, −) (resp. Hom_{G}(H^{i}(A), H^{i}(B))) is torsion for finitely generated M and p > 0 (resp. i ∈ ℤ). □

Lemma 17 shows that there is a unique morphism α ∈ Hom_{𝔇}(A, B) mapping to ⊕_{p}τ^{≤4}α_{p}. We define Δ(r) to be the third term in any exact triangle

Δ(r) → A ⟶^{α} B → .

The diagram (115) then follows easily from (105) and (114). □

Theorem 4.1. If G is abelian and r > 1 we have

Ω(L/K, ℚ(1 − r))^{#} = Ω_{r−1}(L/K) ∈ Cl(ℤ[G]).

Proof. The left vertical exact triangle in (114) induces an isomorphism

λ_{p} : det_{E_{p}}^{−1}K_{p}(r) ⊗_{E_{p}} det_{E_{p}}^{−1}Δ_{p}(r) ≅ det_{E_{p}}^{−1}RΓ_{c}(O_{L,S_{p}}, ℤ_{p}(1 − r))^{∗}

≅ det_{E_{p}}RΓ_{c}(O_{L,S_{p}}, ℤ_{p}(1 − r))^{#}. (121)

The top row in (115) gives an isomorphism

κ(r) : Ξ(ℚ(1 − r)_{L})^{#} ≅ det_{E}^{−1}(K(r) ⊗ ℚ) ⊗ det_{E}^{−1}(Δ(r) ⊗ ℚ)

such that

ϑ̃_{p} = (λ_{p} ⊗ ℚ_{p}) ∘ (det^{−1}(τ_{p} ⊗ ℚ_{p}) ⊗ det^{−1}(σ_{p} ⊗ ℚ_{p})) ∘ (κ(r) ⊗ ℚ_{p})

as maps from Ξ(ℚ(1−r)_{L})^{#}⊗ℚ_{p} to det_{E_{p}}RΓ_{c}(O_{L,S_{p}}, ℚ_{p}(1−r))^{#}. This also follows from (115) because ϑ̃_{p} was constructed in section 2 using the maps (9) and (10). By (11) one has

ϑ_{p}^{−1}(I_{p}(1 − r)^{#}) = ϵ_{S_{p}}(1 − r)^{#}ϑ̃_{p}^{−1}(I_{p}(1 − r)^{#})
= ϵ_{S}(1 − r)^{#}ϑ̃_{p}^{−1}(I_{p}(1 − r)^{#})

where the latter equality holds because for v|p (⇒ p|Nv^{r−1}) the factor (1−Nv^{r−1}f_{v}) occurring in (12) is a unit in E_{p} = ℤ_{p}[G]. Hence

ϑ_{p}^{−1}(I_{p}(1 − r)^{#}) = ϵ_{S}(1 − r)^{#}(κ(r) ⊗ ℚ_{p})^{−1}(det_{E_{p}}^{−1}K_{p}(r) ⊗_{E_{p}} det_{E_{p}}^{−1}Δ_{p}(r))

and

Ξ(ℚ(1 − r)_{L})_{ℤ}^{#} = ⋂_{p} Ξ(ℚ(1 − r)_{L})^{#} ∩ ϑ_{p}^{−1}(I_{p}(1 − r)^{#})
= ϵ_{S}(1 − r)^{#}κ(r)^{−1}(det_{ℤ[G]}^{−1}K(r) ⊗_{ℤ[G]} det_{ℤ[G]}^{−1}Δ(r)).

Since ϵ_{S}(1 − r) ∈ E^{×} one deduces

Ω(L/K, ℚ(1 − r))^{#} = Ω_{r−1}(L/K) + (Δ(r)) ∈ Cl(ℤ[G]).

Theorem 4.1 then follows from

Lemma 18. Δ(r) is quasi-isomorphic to a complex of free ℤ[G]-modules, more precisely to

ℤΣ_{∞} → ℤΣ_{∞} → ℤΣ_{∞}

where the first term is in degree 1.

Proof. One first shows that the map α_{p} in (114) respects the direct sum decomposition of its source and target, following the proof of Proposition 3.4. Using the same notation as in that proof, one needs to ascertain the vanishing of the map induced by α_{p} in Hom_{𝔇_{p}(G_{0})}(Y, Z) where

Y := RΓ_{Δ}(L_{w_{0}}, ℤ_{p}(1 − r))^{∗}[−3] = \begin{cases}ℤ_{p}(r − 1)[−3] & w_{0} complex \\ ℤ_{p} ⟶^{δ_{1}} ℤ_{p} ⟶^{δ_{0}} ℤ_{p} ⟶^{δ_{1}} ⋯ [−3] & w_{0} real \end{cases}

and Z = Ind_{G_{v′}}^{G_{0}} Z′ with

Z′ := RΓ(L_{w′}, ℤ_{p}(r)) = \begin{cases}ℤ_{p}(r) & w′ complex \\ ℤ_{p} ⟶^{δ_{0}} ℤ_{p} ⟶^{δ_{1}} ℤ_{p} ⟶^{δ_{0}} ℤ_{p} ⟶^{δ_{1}} ℤ_{p} ⟶^{δ_{0}} ⋯ & w′ real. \end{cases}

Here δ_{i} is multiplication with 1 − (−1)^{r+i} for i = 0, 1 and 𝔇_{p}(Γ) is the derived category of ℤ_{p}[Γ]-modules. Again, one has Hom_{𝔇_{p}(G_{0})}(Y, Z) = 0 if G_{0} = 1 and w′ is complex and one reduces to this case by introducing an appropriate extension N/K′ as in the proof of Proposition 3.4. The analogue of Lemma 11 to be used here is that R Hom_{ℤ_{p}[H]}(ℤ_{p}, −) applied to diagram (114) gives the corresponding diagram formed with respect to the extension L^{H}/K. This is proved along the same lines as Lemma 11. It follows that

Δ(r) ≅ ⨁_{v∈S_{∞}} Ind_{G_{v}}^{G} Δ(r)_{w(v)}

where Δ(r)_{w(v)} sits in a triangle in 𝔇(G_{v})

RΓ_{Δ}(L_{w(v)}, ℤ(1 − r))^{∗}[−3] → RΓ(L_{w(v)}, ℤ(r)) → Δ(r)_{w(v)}[1] → . (122)

If w(v) is real, hence G_{v} = 1, the triangle (122) is simply isomorphic to the short exact sequence of complexes of abelian groups

0 ⟶ 0 ⟶ 0 ⟶ ℤ ⟶^{δ_{1}} ℤ ⟶^{δ_{0}} ℤ ⟶ ⋯
↓  ↓  ↓  ↓  ↓  ↓
ℤ ⟶^{δ_{0}} ℤ ⟶^{δ_{1}} ℤ ⟶^{δ_{0}} ℤ ⟶^{δ_{1}} ℤ ⟶^{δ_{0}} ℤ ⟶ ⋯
↓  ↓  ↓  ↓  ↓
ℤ ⟶^{δ_{0}} ℤ ⟶^{δ_{1}} ℤ ⟶ 0 ⟶ 0 ⟶ ⋯

If w(v) is complex and G_{v} = 1 we have Δ(r)_{w(v)}[1] ≅ ℤ[0] ⊕ ℤ[−2] which can be represented by a complex of abelian groups

ℤ ⊕ ℤ → ℤ ⊕ ℤ → ℤ ⊕ ℤ.

Finally, if w(v) is complex and G_{v} ≅ ℤ/2ℤ the complex Δ(r)_{w(v)}[1] represents an extension class in Ext_{G}^{3}(ℤ(r − 1), ℤ(r)) ≅ Ext_{G}^{3}(ℤ, ℤ(1)) ≅ ℤ/2ℤ. Since we already know that Δ(r)_{w(v)} is ℤ[G_{v}]-perfect this must be the non-trivial class in this group which is represented by a complex

ℤ[G_{v}] → ℤ[G_{v}] → ℤ[G_{v}].

This gives the desired description of Δ(r). □

Remark. The occurrence of the complex Δ(r) corresponds to that of ℤΣ_{∞} in formula (107). Indeed, one has (K(r)) + (Δ(r)) = Ω_{r−1}(L/K) in K_{0}(ℤ[G]) not merely after projection into Cl(ℤ[G]).

5. Lichtenbaum Complexes

In [17] Lichtenbaum has proposed the existence of complexes of étale sheaves Γ(r) for r ≥ 0 and any scheme X which, if X is of finite type over ℤ, should play an important role in the description of arithmetic properties of X. The model case to think of is the sheaf Γ(1) := 𝔾_{m}[−1] and its appearance in the Artin-Verdier duality theorem for regular one-dimensional X and in the formula for the residue of the Zeta-function of such X.

We have seen how both 𝔾_{m} and Γ(0) := ℤ have played a crucial part in our discussion of Ω(L/K, ℚ(0)) and Ω(L/K, 3). In this section we shall indicate briefly how the expected properties of Γ(r) for r ≥ 2 naturally give rise to perfect complexes and therefore to classes related to Ω(L/K, ℚ(1 − r)). The possibility of such constructions was also noted independently, and in fact a little earlier, by Chinburg, Kolster, Pappas and Snaith (cf. remarks to that effect in [9]). In order to simplify our discussion we shall tensor all occurring complexes with Λ := ℤ[\frac{1}{2}] and therefore only obtain classes in Cl(Λ[G]). In this section we need not assume that G is abelian. If Λ[G] ≅ R_{1} ×⋯×R_{m} where each ring R_{i} has no non-trivial idempotents we define the class groups Cl(Λ[G]) to be the kernel of the rank map

K_{0}(Λ[G]) → ℤ^{m}.

Then Cl(Λ[G]) is a direct summand of K_{0}(Λ[G]) and is isomorphic to Pic(Λ[G]) for abelian G.

In the following we shall assume the existence of Γ(r) in the derived category of complexes of sheaves of abelian groups on the small étale site of schemes of the type Spec O_{L,S} considered before. In fact, we assume that Γ(r) on Spec O_{L,S} is the pull back of Γ(r) on Spec ℤ and moreover that the following properties are satisfied.

1. For any integer N which is invertible in O_{L,S}, there is an exact triangle

Γ(r) ⟶^{N} Γ(r) → ℤ/Nℤ(r) → (123)

2. The Chern class maps K_{2r−i}(O_{L,S}) → H^{i}(O_{L,S}, ℤ/Nℤ(r)) factor through ℍ^{i}(O_{L,S}, Γ(r)) where N is any integer as in 1.

3. The hypercohomology groups ℍ^{i}(O_{L,S}, Γ(r)) are finitely generated abelian groups for r ≥ 2 and i ∈ ℤ.

4. Let T be a finite set of places disjoint from S. Then there is an exact triangle in 𝔇 for r ≥ 2

ℝΓ(O_{L,S}, Γ(r)) → ℝΓ(O_{L,S∪T} , Γ(r)) → ⨁_{w∈T} K_{2r−3}(λ(w))[−2] → (124)

where λ(w) is the residue field. Remarks. This is only a subset of the axioms for Γ(r), adapted to our needs

(see for example [22] for a full list). The term K_{2r−3}(λ(w))[−2] should be written ℝΓ(λ(w), Γ(r − 1))[−1] and (124) then represents a purity axiom which also holds for r = 1. The finite generation of ℍ^{i}(X, Γ(r)) is not true in general (as the case X = O_{L,S}, r = 1, i = 3 shows) but one might expect that ℍ^{i}(X, Γ(r)) is finitely generated for all i ∈ ℤ if X is regular of finite type over ℤ and if r is large enough with respect to the dimension of X.

The complex Γ(2) constructed by Lichtenbaum in [18] satisfies the above list of properties (see [14], [19]), so in the case r = 2 our results described below are unconditional.

Proposition 5.1. a) The complex ℝΓ(O_{L,S}, Γ(r)) ⊗_{ℤ} Λ is Λ[G]- perfect for r ≥ 2. b) For each prime p ≠ 2 there is a quasi-isomorphism

ℝΓ(O_{L,S_{p}}, Γ(r)) ⊗_{ℤ} ℤ_{p} ≅ RΓ(O_{L,S_{p}}, ℤ_{p}(r))

which is compatible with the Chern class maps in 2).

Proof. Part a) can be shown along the lines of the proof of Prop. 1.20 in [5]. The complex M := ℝΓ(O_{L,S}, Γ(r)) consists of c.t. G-modules, hence the same is true for M ⊗_{ℤ} Λ. Moreover M ⊗_{ℤ} Λ is acyclic outside degrees 1, 2 as follows easily by combining properties 1) and 3) above (and this is also true if L is replaced by a field between L and K). Therefore M ⊗_{ℤ} Λ is quasi-isomorphic to a bounded complex of c.t. Λ[G]-modules. Perfectness follows because the cohomology of ℝΓ(O_{L,S}, Γ(r)) ⊗_{ℤ} Λ is finitely generated over Λ by property 3).

To demonstrate b) one can copy the proof of Proposition 3.3. For the complex Q^{•} (resp. P^{•}) one takes a bounded complex of finitely generated projective Λ[G]- modules (resp. ℤ_{p}[G]-modules) quasi-isomorphic to ℝΓ(O_{L,S_{p}}, Γ(r)) ⊗_{ℤ} Λ (resp. RΓ(O_{L,S_{p}}, ℤ_{p}(r))). The perfectness of RΓ(O_{L,S_{p}}, ℤ_{p}(r)) over ℤ_{p}[G] for odd primes p follows again as in [5, Prop. 1.20]. Property 1) implies that Q^{•}/p^{n} and P^{•}/p^{n} are quasi-isomorphic, compatibly with Chern class maps by property 2). One shows that these quasi-isomorphisms can be realized as maps of inverse systems of complexes and concludes with Lemma 9. □

Theorem 5.1. The image of Ω_{r−1}(L/K) in Cl(Λ[G]) agrees with the class of ℝΓ(O_{L,S}, Γ(r))⊗_{ℤ}Λ. This class also coincides with the image of Ω(L/K, ℚ(1−r))^{#} in Cl(Λ[G]) if G is abelian.

Remark. We understand that Victor Snaith has obtained results along similar lines in the case r = 2. He is able to avoid inverting 2 and shows that Ω_{1}(L/K) coincides in K_{0}(ℤ[G]) with a class defined from the cohomology of Γ(2).

Proof. By Theorem 4.1 it suffices to prove the assertion concerning Ω_{r−1}(L/K). The triangle (124) and Proposition 5.1b) give a quasi-isomorphism

ℝΓ(O_{L,S}, Γ(r)) ⊗_{ℤ} ℤ_{p} ≅ RΓ(O_{L,S_{p}}, ℤ_{p}(r)) (125)

since for any place w|p, the group K_{2r−3}(λ(w)) is finite of order prime to p [23, §12]. Arguing as in the proof of Lemma 17 one finds a unique map in the derived category of Λ[G]-modules

β : ℝΓ(O_{L,S}, Γ(r)) ⊗_{ℤ} Λ → ⨁_{w|∞} H^{0}(L_{w}, Λ(r)) (126)

inducing the natural map

RΓ(O_{L,S_{p}}, ℤ_{p}(r)) → ⨁_{w|∞} H^{0}(L_{w}, ℤ_{p}(r))

after tensorisation with ℤ_{p} for each odd prime p. Note here that for any odd prime p, and any archimedean place w, the complex RΓ(L_{w}, ℤ_{p}(r)) is quasi-isomorphic to H^{0}(L_{w}, ℤ_{p}(r)) concentrated in degree 0. One then checks that the mapping cone of β satisfies the defining properties of the complex K(r)⊗_{ℤ}Λ (The isomorphisms τ_{p} in (106) are obvious and compatibility with Chern class maps follows from Proposition 5.1b)). The complex K(r) ⊗_{ℤ} Λ is thus quasi-isomorphic to this mapping cone and so its class in Cl(Λ[G]) agrees with

(ℝΓ(O_{L,S}, Γ(r)) ⊗_{ℤ} Λ) − (⨁_{w|∞} H^{0}(L_{w}, Λ(r))).

But each module

⨁_{w|v} H^{0}(L_{w}, Λ(r)) ≅ H^{0}(K_{v}, Λ[G](r))

has trivial class in Cl(Λ[G]) because it is a direct factor of Λ[G]. Hence the theorem. □

Remark. It seems reasonable to speculate that the map β in (126) exists without inverting 2 and indeed has a more natural interpretation as a composite map

ℝΓ(O_{L,S}, Γ(r)) → ⨁_{w|∞} ℝΓ(L_{w}, Γ(r)) → ⨁_{w|∞} RΓ(L_{w}, ℤ(r)) (127)

where the first map is the natural restriction and the last map arises as follows. Recall that Deligne cohomology RΓ_{D}(ℂ, ℤ(r)) of a point (i.e. Spec ℂ) is concentrated in degree 1 and identifies with ℂ/ℤ(r)[−1] (see [24] for more about Deligne cohomology and the Beilinson regulator map). Moreover, on Spec ℂ the object Γ(r) identifies with a complex of abelian groups. One might conjecture that there is a map in the derived category of abelian groups

ρ : Γ(r) → ℂ/ℤ(r)[−1],

factoring the Beilinson regulator map after taking H^{1}:

K_{2r−1}(ℂ) → H^{1}(Γ(r)) ⟶^{H^{1}(ρ)} ℂ/ℤ(r).

The map ρ should also respect the action of complex conjugation, i.e. be a map in the derived category of étale sheaves on Spec ℝ. Composing ρ in the derived category with the obvious map ℂ/ℤ(r)[−1] → ℤ(r) would induce the second map in (127). Finally, one would be led to conjecture that the mapping cone of (127) is cohomologically bounded, hence perfect over ℤ[G] by our standard arguments. So one could take the class of this mapping cone in Cl(ℤ[G]) to define an invariant. However, by adapting the proof of Theorem 5.1 it would then follow that this class coincides with Ω_{r−1}(L/K).

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

Dept. of Math., Caltech, Pasadena CA 91125, USA