On Equivariant Dedekind Zeta-Functions at s = 1

Dedicated to Professor Andrei Suslin

Manuel Breuning and David Burns

Received: November 19, 2009 Revised: April 26, 2010

Abstract. We study a refinement of an explicit conjecture of Tate concerning the values at s = 1 of Artin L-functions. We reinterpret this refinement in terms of Tamagawa number conjectures and then use this connection to obtain some important (and unconditional) evidence for our conjecture.

2010 Mathematics Subject Classification: 11R42, 11R33

Keywords and Phrases: Artin L-functions, equivariant zeta functions, leading terms

1. Introduction and statement of main results

This article studies a refinement of a conjecture of Tate concerning the values at s = 1 of Artin L-functions. We recall that Tate’s conjecture was originally formulated in [26, Chap. I, Conj. 8.2] as an analogue of (Tate’s reformulation of) the main conjecture of Stark on the leading terms at s = 0 of Artin Lfunctions and that the precise form of the ‘regulators’ and ‘periods’ that Tate introduced in this context were natural generalisations of earlier constructions of Serre in [24]. The refinement of Tate’s conjecture that we study here was formulated by the present authors in [5, Conj. 3.3] and predicts an explicit formula for the leading term at s = 1 of the zeta-function of a finite Galois extension of number fields L/K in terms of the Euler characteristic of a certain perfect complex of Gal(L/K)-modules (see (3) for a statement of this formula). In comparison to Tate’s conjecture, this refinement predicts not only that the quotient by Tate’s regulator of the leading term at s = 1 of the Artin L-function of a complex character χ of Gal(L/K) is an algebraic number but also that as χ varies these algebraic numbers should be related by certain types of integral congruence relations. We further recall that [5, Conj. 3.3] is also known to imply the ‘Ω(1)-Conjecture’ that was formulated by Chinburg in [13].

In the sequel we write ℚ(1)_{L} for the motive h^{0}(Spec L)(1), considered as defined over K and endowed with the natural action of the group ring ℚ[Gal(L/K)]. We recall that the ‘equivariant Tamagawa number conjecture’ applies in particular to pairs of the form (ℚ(1)_{L}, ℤ[Gal(L/K)]) and was formulated by Flach and the second named author in [9] as a natural refinement of the seminal ‘Tamagawa number conjecture’ of Bloch and Kato [3]. The main technical result of the present article is then the following

Theorem 1.1. Let L be a finite complex Galois extension of ℚ. If Leopoldt’s Conjecture is valid for L, then [5, Conj. 3.3] is equivalent to the equivariant Tamagawa number conjecture of [9, Conj. 4] for the pair (ℚ(1)_{L}, ℤ[Gal(L/ℚ)]).

Corollary 1.2. If Leopoldt’s Conjecture is valid for every number field, then for every Galois extension of number fields L/K the conjecture [5, Conj. 3.3] is equivalent to the conjecture [9, Conj. 4] for the pair (ℚ(1)_{L}, ℤ[Gal(L/K)]).

These results connect the explicit leading term formula of [5, Conj. 3.3] to a range of interesting results and conjectures. For example, [9, Conj. 4(iv)] is known to be a consequence of the ‘main conjecture of non-commutative Iwasawa theory’ that is formulated by Fukaya and Kato in [18, Conj. 2.3.2] and also of the ‘main conjecture of non-commutative Iwasawa theory for Tate motives’ that is formulated by Venjakob and the second named author in [12, Conj. 7.1]. Corollary 1.2 therefore allows one to regard the study of the explicit conjecture [5, Conj. 3.3] as an attempt to provide supporting evidence for these more general conjectures. Indeed, when taken in conjunction with the philosophy described by Huber and Kings in [19, §3.3] and by Fukaya and Kato in [18, §2.3.5], Corollary 1.2 suggests that, despite its comparatively elementary nature, [5, Conj. 3.3] may well play a particularly important role in the context of the very general conjecture of Fukaya and Kato. In addition to the above consequences, our proof of Theorem 1.1 also answers an explicit question posed by Flach and the second named author in [7] (see Remark 5.1) and combines with previous work to give new evidence in support of the conjectures made in [5] including the following unconditional results.

Corollary 1.3. If L is abelian over ℚ, and K is any subfield of L, then both [5, Conj. 3.3] and [5, Conj. 4.1] are valid for the extension L/K.

Corollary 1.4. There exists a natural infinite family of quaternion extensions L/ℚ with the property that, if K is any subfield of L, then both [5, Conj. 3.3] and [5, Conj. 4.1] are valid for the extension L/K.

On Equivariant Dedekind Zeta-Functions at s = 1 121

We recall (from [5, Prop. 4.4(i)]) that [5, Conj. 4.1] is a natural refinement of the ‘main conjecture of Stark at s = 0’. For details of connections between [5, Conj. 3.3 and Conj. 4.1] and other interesting conjectures of Chinburg, of Gruenberg, Ritter and Weiss and of Solomon see [5, Prop. 3.6 and Prop. 4.4] and the recent thesis of Jones [20]. The main contents of this article is as follows. In §2 we recall the explicit statement of [5, Conj. 3.3] and in §3 we review (and clarify) certain constructions in étale cohomology that are made in [8]. In §4 we make a detailed analysis of the p-adic completion of the perfect complex that occurs in [5, Conj. 3.3]. In §5 we prove Theorem 1.1 and in §6 we use Theorem 1.1 to prove Corollaries 1.2, 1.3 and 1.4. Acknowledgements. It is a pleasure for us to thank Werner Bley and Matthias Flach and also the referee of [5] for some very useful comments about this paper. In addition, the first author would like to thank the Isaac Newton Institute where part of this paper was written.

2. The explicit leading term conjecture

In this section we quickly review [5, Conj. 3.3]. To do this it is necessary to summarise some background about K-theory and homological algebra.

2.1. K-theory. Let R be an integral domain of characteristic 0, E an extension of the field of fractions of R, and G a finite group. We denote the relative algebraic K-group associated to the ring homomorphism R[G] → E[G] by K_{0}(R[G], E); a description of K_{0}(R[G], E) in terms of generators and relations is given in [25, p. 215]. The group K_{0}(R[G], E) is functorial in the pair (R, E) and also sits inside a long exact sequence of relative K-theory. In this paper we will use the homomorphisms ∂_{R[G],E}^{1} : K_{1}(E[G]) → K_{0}(R[G], E) and ∂_{R[G],E}^{0} : K_{0}(R[G], E) → K_{0}(R[G]) from the latter sequence. Let Z(E[G])^{×} denote the multiplicative group of the centre of the finite dimensional semisimple E-algebra E[G]. The reduced norm induces a homomorphism nr : K_{1}(E[G]) → Z(E[G])^{×} and we denote its image by Z(E[G])^{×+}. In this paper E will always be either ℝ or ℂ_{p} for some prime number p. In both cases the map nr is injective and hence we can use it to identify K_{1}(E[G]) and Z(E[G])^{×+}. In particular we will consider ∂_{R[G],E}^{1} as a map Z(E[G])^{×+} → K_{0}(R[G], E). If E = ℂ_{p} then Z(E[G])^{×+} = Z(E[G])^{×}. For every prime p and embedding j : ℝ → ℂ_{p} there are induced homomorphisms j_{∗} : K_{0}(ℤ[G], ℝ) → K_{0}(ℤ_{p}[G], ℂ_{p}) and j_{∗} : Z(ℝ[G])^{×} → Z(ℂ_{p}[G])^{×}. In [5, §2.1.2] it is shown that there exists a (unique) homomorphism ˆ∂_{G}^{1} : Z(ℝ[G])^{×} → K_{0}(ℤ[G], ℝ) which coincides with ∂_{ℤ[G],ℝ}^{1} on Z(ℝ[G])^{×+} and is such that for every prime p and embedding j : ℝ → ℂ_{p} one has j_{∗} ∘ ˆ∂_{G}^{1} = ∂_{ℤ_{p}[G],ℂ_{p}}^{1} ∘ j_{∗} : Z(ℝ[G])^{×} → K_{0}(ℤ_{p}[G], ℂ_{p}).

2.2. Homological algebra. For our homological algebra constructions in this paper we use the same notations and sign conventions as in [5]. So in particular by a complex we mean a cochain complex of left R-modules for a ring R, we use the phrase ‘distinguished triangle’ in the sense specified in [5, §2.2.1] and by a perfect complex we mean a complex that in the derived category D(R) is isomorphic to a bounded complex of finitely generated projective left R-modules. The full triangulated subcategory of D(R) consisting of the perfect complexes will be denoted by D^{perf}(R). Now let R, E and G be as in §2.1. For any object C of D(R[G]) we write H^{ev}(C) and H^{od}(C) for the direct sums ⊕_{i even}H^{i}(C) and ⊕_{i odd}H^{i}(C) where i runs over all even and all odd integers respectively. A trivialisation t (over E) of a complex C in D^{perf}(R[G]) is an isomorphism of E[G]-modules of the form t : H^{ev}(C) ⊗_{R} E ⟶^{≅} H^{od}(C) ⊗_{R} E. We write χ_{R[G],E}(C, t) for the Euler characteristic in K_{0}(R[G], E) defined in [4, Definition 5.5]. To simplify notation in the sequel we write χ_{G} for χ_{ℤ[G],ℝ}. We shall interpret certain complexes in the derived category in terms of Yoneda extension classes as in [8, p. 1353]. To be specific, for any complex E that is acyclic outside degrees 0 and n ≥ 1 we associate the class in Ext_{R}^{n+1}(H^{n}(E), H^{0}(E)) given by the truncated complex E′ := τ^{≤n}τ^{≥0}E with the induced maps H^{0}(E) ⟶^{≅} H^{0}(E′) → (E′)^{0} and (E′)^{n} → H^{n}(E′) ⟶^{≅} H^{n}(E) considered as a Yoneda extension.

2.3. Notation for number fields. Let L be a number field. We write O_{L} for the ring of integers of L and S(L) for the set of all places of L. For any place w ∈ S(L) we denote the completion of L at w by L_{w}. For a non-archimedean place w we write O_{w} for the ring of integers of L_{w}, 𝔪_{w} for the maximal ideal of O_{w} and U_{L_{w}}^{(1)} for the group 1 + 𝔪_{w} of principal units in L_{w}. If L is an extension of K and v ∈ S(K) then S_{v}(L) is the set of all places of L above v. We also use the notation S_{f}(L) and S_{∞}(L) for the sets of all nonarchimedean and archimedean places, S_{ℝ}(L) for the set of real archimedean places and S_{ℂ}(L) for the set of complex archimedean places. From now on let L/K be a Galois extension of number fields with Galois group G. For w in S(L) we let G_{w} denote the decomposition group of w. For any place v in S(K) we set L_{v} := ∏_{w∈S_{v}(L)} L_{w} and (if v ∈ S_{f}(K)) O_{L,v} := ∏_{w∈S_{v}(L)} O_{w} and 𝔪_{L,v} := ∏_{w∈S_{v}(L)} 𝔪_{w}. Note that L_{v}, O_{L,v} and 𝔪_{L,v} are G-modules in an obvious way. Let S be a finite subset of S(K). The G-stable set of places of L that lie above a place in S will also be denoted by S. This should not cause any confusion because places of K will be called v and places of L will be called w. For a finite subset S of S(K) which contains all archimedean places we let O_{L,S} be the ring of S-integers in L. Note that O_{L,S} is a G-module and that if S = S_{∞}(K), then O_{L} = O_{L,S}.

2.4. The conjecture. Let L/K be a Galois extension of number fields with Galois group G. Let S be a finite subset of S(K) which contains all archimedean places and all places which ramify in L/K and is such that Pic(O_{L,S}) = 0. In [5, Lemma 2.7(ii)] it is shown that the leading term ζ_{L/K,S}^{∗}(1) at s = 1 of the S-truncated zeta-function of L/K belongs to Z(ℝ[G])^{×+}. In this subsection we recall the explicit conjectural description of ˆ∂_{G}^{1}(ζ_{L/K,S}^{∗}(1)) formulated in [5, Conj. 3.3]. For each v ∈ S_{∞}(K) we let exp : L_{v} → L_{v}^{×} denote the product of the (real or complex) exponential maps L_{w} → L_{w}^{×} for w ∈ S_{v}(L). If v ∈ S_{f}(K), then for sufficiently large i the exponential map exp : 𝔪_{L,v}^{i} → L_{v}^{×} is the product of the p-adic exponential maps 𝔪_{w}^{i} → L_{w}^{×} for w ∈ S_{v}(L).

To state [5, Conj. 3.3] we need to choose certain lattices. For each v ∈ S_{f} := S ∩ S_{f}(K), with residue characteristic p, we choose a full projective ℤ_{p}[G]- lattice L_{v} ⊆ O_{L,v} which is contained in a sufficiently large power of 𝔪_{L,v} to ensure that the exponential map is defined on L_{v}. Let L be the full projective ℤ[G]-sublattice of O_{L} which has p-adic completions

(1)  L ⊗_{ℤ} ℤ_{p} = ( ∏_{v∈S_{p}(K)\S} O_{L,v}) × ( ∏_{v∈S_{p}(K)∩S} L_{v}).

We set L_{S} := ∏_{v∈S} L_{v} and L_{S} := ∏_{v∈S} L_{v} (where L_{v} := L_{v} for each v ∈ S_{∞}(K)) and we let exp_{S} denote the map L_{S} → L_{S}^{×} that is induced by the product of the respective exponential maps. We also write Δ_{S} for the natural diagonal embedding from L^{×} to L_{S}^{×}. Following the notation of [23, Chap. VIII] we write I_{L} for the group of idèles of L and regard L^{×} as embedded diagonally in I_{L}. The idèle class group is C_{L} := I_{L}/L^{×} and the S-idèle class group is C_{S}(L) := I_{L}/(L^{×}U_{L,S}), where U_{L,S} := ∏_{w∈S}{1} × ∏_{w∉S} O_{w}^{×}. We remark that since Pic(O_{L,S}) = 0, the natural map L_{S}^{×} → C_{S}(L) is surjective with kernel Δ_{S}(O_{L,S}^{×}). There is also a

canonical invariant isomorphism inv_{L/K,S} : H^{2}(G, C_{S}(L)) ⟶^{≅} \frac{1}{|G|}ℤ/ℤ and we

write e_{S}^{glob} for the element of Ext_{ℤ[G]}^{2}(ℤ, C_{S}(L)) = H^{2}(G, C_{S}(L)) that is sent by inv_{L/K,S} to \frac{1}{|G|}. Let E_{S} be a complex in D(ℤ[G]) which corresponds (in the sense of the last paragraph of §2.2) to e_{S}^{glob}. Then by [5, Lemma 2.4] there is a unique morphism α_{S} : L_{S}[0] ⊕ L[−1] → E_{S} in D(ℤ[G]) for which H^{0}(α_{S}) is the composite L_{S} ⟶^{exp_{S}} L_{S}^{×} → C_{S}(L) and H^{1}(α_{S}) is the restriction of the trace map tr_{L/ℚ} : L → ℚ to L. Let E_{S}(L) be any complex which lies in a distinguished triangle in D(ℤ[G]) of the form

(2)  L_{S}[0] ⊕ L[−1] ⟶^{α_{S}} E_{S} ⟶^{β_{S}} E_{S}(L) ⟶^{γ_{S}} .

To describe the cohomology of E_{S}(L) we set L_{∞} := ∏_{w∈S_{∞}(L)} L_{w} and write L_{∞}^{0} for the kernel of the map L_{∞} → ℝ defined by (l_{w})_{w∈S_{∞}(L)} ↦ ∑_{w∈S_{∞}(L)} tr_{L_{w}/ℝ}(l_{w}). We write exp_{∞} for the product of the exponential maps L_{∞} → L_{∞}^{×}, Δ_{∞} for the diagonal embedding L^{×} → L_{∞}^{×} and log_{∞}(O_{L}^{×}) for the full sublattice of L_{∞}^{0} comprising elements x of L_{∞} with exp_{∞}(x) ∈ Δ_{∞}(O_{L}^{×}). In [5, Lemma 3.1] it is shown that E_{S}(L) is a perfect complex of G-modules, that E_{S}(L) ⊗ ℚ is acyclic outside degrees −1 and 0, that H^{−1}(γ_{S}) induces an identification of H^{−1}(E_{S}(L)) with {x ∈ L_{S} : exp_{S}(x) ∈ Δ_{S}(O_{L}^{×})} and that H^{0}(γ_{S}) induces an identification of H^{0}(E_{S}(L)) ⊗ ℚ with ker(tr_{L/ℚ}). In addition, the projection L_{S} → L_{∞} induces an isomorphism of ℚ[G]-modules from {x ∈ L_{S} : exp_{S}(x) ∈ Δ_{S}(O_{L}^{×})}⊗ℚ to log_{∞}(O_{L}^{×})⊗ℚ. With these identifications the isomorphism ker(tr_{L/ℚ})⊗_{ℚ} ℝ ⟶^{≅} L_{∞}^{0} = log_{∞}(O_{L}^{×})⊗ℝ which is obtained by

restricting the natural isomorphism L ⊗_{ℚ} ℝ ⟶^{≅} L_{∞} to ker(tr_{L/ℚ}) ⊗_{ℚ} ℝ gives a

trivialisation μ_{L} : H^{0}(E_{S}(L)) ⊗ ℝ ⟶^{≅} H^{−1}(E_{S}(L)) ⊗ ℝ of E_{S}(L). In [5, Conj.

3.3] it is conjectured that

(3)  ˆ∂_{G}^{1}(ζ_{L/K,S}^{∗}(1)) = −χ_{G}(E_{S}(L), μ_{L}).

For a discussion of the basic properties of this conjecture see [5, §3]. In particular for a proof of the fact that this conjecture refines Tate’s conjecture [26, Chap. I, Conj. 8.2] see [5, Prop. 3.6(i)].

3. Preliminaries concerning étale cohomology

To relate the conjectural equality (3) to [9, Conj. 4] we will use constructions in étale cohomology that are made in [8]. However, to do this certain aspects of the exposition in [8] require clarification and so in this section we review the relevant parts of these constructions. We fix L/K and S as in §2.4 but for simplicity we also assume henceforth that S contains at least one non-archimedean place. For each w ∈ S(L) we denote the algebraic closure of L in L_{w} by L_{w}^{h}. For w ∈ S_{f}(L) we let O_{w}^{h} be the ring of integers in L_{w}^{h}; note that O_{w}^{h} is the henselization of (the localization of) O_{L} at w (compare [21, Chap. I, Exam. 4.10(a)]) and that L_{w}^{h} is the field of fractions of O_{w}^{h}. Similarly, for a place v ∈ S(K) we define K_{v}^{h} as the algebraic closure of K in K_{v}. The inclusions O_{K,S} ⊂ K_{v}^{h} ⊂ K_{v} induce canonical maps g_{v}^{h} : Spec K_{v}^{h} → Spec O_{K,S}, f_{v} : Spec K_{v} → Spec K_{v}^{h} and g_{v} = g_{v}^{h} ∘ f_{v} : Spec K_{v} → Spec O_{K,S}.

3.1. General conventions. Let X be any scheme and F an étale sheaf on X, i.e. a sheaf on the étale site X_{et}. If Y is an étale X-scheme then we denote by RΓ(Y, F) the complex in the derived category D(ℤ) which is obtained by applying the right derived functor of the section functor Γ(Y, −) to the sheaf F; thus RΓ(Y, F) is defined up to canonical isomorphism in D(ℤ). If Y = Spec R for some commutative ring R, then we will write RΓ(R, F) for RΓ(Spec R, F) and H^{i}(R, F) for the cohomology groups H^{i}(RΓ(R, F)). Now let v ∈ S(K), w ∈ S_{v}(L) and let F be an étale sheaf on Spec K_{v}^{h}. The G_{w}-action on Spec L_{w}^{h} induces a G_{w}-action on the sections Γ(Spec L_{w}^{h}, F) and hence the complex RΓ(L_{w}^{h}, F) naturally lies in D(ℤ[G_{w}]). Similarly, if F is an étale sheaf on Spec O_{K,S}, then RΓ(O_{L,S}, F) belongs to D(ℤ[G]). Finally for v ∈ S(K) and F an étale sheaf on Spec O_{K,S} we can consider ⨁_{w∈S_{v}(L)} RΓ(L_{w}^{h}, (g_{v}^{h})^{∗}F) as a complex in D(ℤ[G]). This is possible because there is a canonical isomorphism

⨁_{w∈S_{v}(L)} RΓ(L_{w}^{h}, (g_{v}^{h})^{∗}F) ≅ RΓ( ∐_{w∈S_{v}(L)} Spec L_{w}^{h}, (g_{v}^{h})^{∗}F),

and ∐_{w∈S_{v}(L)} Spec L_{w}^{h} is a Galois covering of Spec K_{v}^{h} with group G. Of course the same is true with L_{w}^{h} and g_{v}^{h} replaced by L_{w} and g_{v} respectively.

3.2. Local cohomology. Let v be a place of K and w ∈ S_{v}(L). Recall that f_{v} : Spec K_{v} → Spec K_{v}^{h} corresponds to the inclusion K_{v}^{h} → K_{v}. For any étale sheaf F on Spec K_{v}^{h} the canonical map RΓ(L_{w}^{h}, F) ⟶ RΓ(L_{w}, f_{v}^{∗}F) is an isomorphism in D(ℤ[G_{w}]). Indeed, if \overline{L_{w}} is an algebraic closure of L_{w} and \overline{L_{w}^{h}} is the algebraic closure of L_{w}^{h} in \overline{L_{w}}, then the restriction map gives an isomorphism Gal(\overline{L_{w}}/K_{v}) ⟶^{≅} Gal(\overline{L_{w}^{h}}/K_{v}^{h}). Thus, upon identifying étale cohomology and Galois cohomology the claimed isomorphism follows. If F = 𝔾_{m} on (Spec K_{v}^{h})_{et}, then f_{v}^{∗}𝔾_{m} is not isomorphic to the sheaf 𝔾_{m} on (Spec K_{v})_{et}. However the complexes RΓ(L_{w}^{h}, 𝔾_{m}) ≅ RΓ(L_{w}, f_{v}^{∗}𝔾_{m}) and RΓ(L_{w}, 𝔾_{m}) are related as follows.

Lemma 3.1. There is a distinguished triangle in D(ℤ[G_{w}])

RΓ(L_{w}^{h}, 𝔾_{m}) ⟶ RΓ(L_{w}, 𝔾_{m}) ⟶ (L_{w}^{×}/(L_{w}^{h})^{×})[0] ⟶,

whose cohomology sequence in degree 0 identifies with the canonical short exact sequence 0 ⟶ (L_{w}^{h})^{×} ⟶ L_{w}^{×} ⟶ L_{w}^{×}/(L_{w}^{h})^{×} ⟶ 0. The G_{w}-module L_{w}^{×}/(L_{w}^{h})^{×} is uniquely divisible and hence cohomologically trivial.

Proof. There is a canonical injection f_{v}^{∗}𝔾_{m} → 𝔾_{m} of sheaves on (Spec K_{v})_{et} such that the sequence

0 ⟶ f_{v}^{∗}𝔾_{m} ⟶ 𝔾_{m} ⟶ 𝔾_{m}/f_{v}^{∗}𝔾_{m} ⟶ 0

corresponds to the exact sequence 0 → \overline{L_{w}^{h}}^{×} → \overline{L_{w}}^{×} → \overline{L_{w}}^{×}/\overline{L_{w}^{h}}^{×} → 0 of Gal(\overline{L_{w}}/K_{v})-modules. Now \overline{L_{w}}^{×}/\overline{L_{w}^{h}}^{×} is uniquely divisible. Also, the isomorphism Gal(\overline{L_{w}}/K_{v}) ≅ Gal(\overline{L_{w}^{h}}/K_{v}^{h}) combines with Hilbert’s Theorem 90 to imply H^{0}(Gal(\overline{L_{w}}/L_{w}), \overline{L_{w}}^{×}/\overline{L_{w}^{h}}^{×}) = L_{w}^{×}/(L_{w}^{h})^{×} as G_{w}-modules. It follows that L_{w}^{×}/(L_{w}^{h})^{×} is uniquely divisible and hence cohomologically trivial (as a G_{w}-module). In addition, by applying RΓ(L_{w}, −) to the displayed exact sequence we obtain the claimed distinguished triangle. □

Lemma 3.2. There are canonical isomorphisms of G_{w}-modules

H^{i}(L_{w}, 𝔾_{m}) ≅ \{ \begin{matrix}L_{w}^{×} & if i = 0, \\ 0 & if i = 1, \\ Br(L_{w}) & if i = 2. \end{matrix}

If w is non-archimedean then H^{i}(L_{w}, 𝔾_{m}) = 0 for i ≥ 3 and the local invariant isomorphism gives a canonical identification Br(L_{w}) ≅ ℚ/ℤ. With respect to this identification the class of RΓ(L_{w}, 𝔾_{m}) in Ext_{ℤ[G_{w}]}^{3}(ℚ/ℤ, L_{w}^{×}) ≅ H^{2}(G_{w}, L_{w}^{×}) is the local canonical class.

Proof. This is [8, Prop. 3.5.(a)]. □

3.3. Cohomology with compact support. For any étale sheaf F on Spec O_{K,S} we define the complex RΓ_{c}(O_{L,S}, F) in D(ℤ[G]) by

(4) RΓ_{c}(O_{L,S}, F) := cone (RΓ(O_{L,S}, F) → ⨁_{w∈S} RΓ(L_{w}^{h}, (g_{v(w)}^{h})^{∗}F)) [−1],

where, for every w ∈ S, v(w) denotes the place of K below w. Thus this complex lies in a distinguished triangle

(5) RΓ_{c}(O_{L,S}, F) ⟶ RΓ(O_{L,S}, F) ⟶ ⨁_{w∈S} RΓ(L_{w}^{h}, (g_{v(w)}^{h})^{∗}F) ⟶ .

In [8, (3)] a complex RΓ_{c}(O_{L,S}, F) is defined just as in (4) but with L_{w}^{h} and g_{v(w)}^{h} replaced by L_{w} and g_{v(w)} respectively. However, the observation made at the beginning of §3.2 ensures that this definition coincides with that given above.

3.3.1. The complex RΓ_{c}(O_{L,S}, 𝔾_{m}). We define a G-module C_{S}^{h}(L) in the same way as C_{S}(L) is defined in §2.4 but with L_{w} replaced by L_{w}^{h} for each w ∈ S(L) and O_{w} replaced by O_{w}^{h} for each w ∈ S_{f}(L). Then, since we assume Pic(O_{L,S}) = 0, the natural map ∏_{w∈S}(L_{w}^{h})^{×} → C_{S}^{h}(L) is surjective with kernel O_{L,S}^{×}.

Lemma 3.3. There are canonical isomorphisms of G-modules

H^{i}(RΓ_{c}(O_{L,S}, 𝔾_{m})) ≅ \{ \begin{matrix}C_{S}^{h}(L) & if i = 1, \\ ℚ/ℤ & if i = 3, \\ 0 & otherwise. \end{matrix}

Proof. We first note that there are canonical isomorphisms of G-modules

H^{i}(O_{L,S}, 𝔾_{m}) ≅ \{ \begin{matrix}O_{L,S}^{×} & if i = 0, \\ 0 & if i = 1, \\ ker ( Br(L) → ⨁_{w∉S} Br(L_{w})) & if i = 2, \\ ⨁_{w∈S_{ℝ}(L)} H^{i}(L_{w}, 𝔾_{m}) & if i ≥ 3, \end{matrix}

(cf. [22, Chap. II, Prop. 2.1, Rem. 2.2] and recall that Pic(O_{L,S}) = 0 and S_{f} ≠ ∅). Now, for every w ∈ S one has (g_{v(w)}^{h})^{∗}𝔾_{m} = 𝔾_{m} on (Spec K_{v(w)}^{h})_{et} because K_{v(w)}^{h} is an algebraic extension of K. The cohomology sequence of the distinguished triangle (5) with F = 𝔾_{m} thus combines with Lemmas 3.1 and 3.2 and the above displayed isomorphisms to give exact sequences

0 → H^{0}(RΓ_{c}(O_{L,S}, 𝔾_{m})) → O_{L,S}^{×} → ⨁_{w∈S}(L_{w}^{h})^{×}

→ H^{1}(RΓ_{c}(O_{L,S}, 𝔾_{m})) → 0

and

0 → H^{2}(RΓ_{c}(O_{L,S}, 𝔾_{m})) → ker ( Br(L) → ⨁_{w∉S} Br(L_{w}))

→ ⨁_{w∈S} Br(L_{w}) → H^{3}(RΓ_{c}(O_{L,S}, 𝔾_{m})) → 0

and an equality H^{i}(RΓ_{c}(O_{L,S}, 𝔾_{m})) = 0 for each i ≥ 4. All maps here are the canonical ones, thus for i = 0 and i = 1 the claimed description follows immediately and for i = 2 and i = 3 it follows by using the canonical exact sequence 0 → Br(L) → ⨁_{w∈S(L)} Br(L_{w}) → ℚ/ℤ → 0. □

3.3.2. The complex R̂Γ_{c}(O_{L,S}, 𝔾_{m}). Recall that for every w ∈ S there is a canonical map g_{v(w)} : Spec K_{v(w)} → Spec O_{K,S} of schemes and an inclusion g_{v(w)}^{∗}𝔾_{m} → 𝔾_{m} of étale sheaves on Spec K_{v(w)}. Thus we can consider the composite morphism

RΓ(O_{L,S}, 𝔾_{m}) ⟶ ⨁_{w∈S} RΓ(L_{w}, g_{v(w)}^{∗}𝔾_{m}) ⟶ ⨁_{w∈S} RΓ(L_{w}, 𝔾_{m})

in D(ℤ[G]). We then define the complex R̂Γ_{c}(O_{L,S}, 𝔾_{m}) by setting

R̂Γ_{c}(O_{L,S}, 𝔾_{m}) := cone (RΓ(O_{L,S}, 𝔾_{m}) ⟶ ⨁_{w∈S} RΓ(L_{w}, 𝔾_{m})) [−1].

Lemma 3.4. There are canonical isomorphisms of G-modules

H^{i}(R̂Γ_{c}(O_{L,S}, 𝔾_{m})) ≅ \{ \begin{matrix}C_{S}(L) & if i = 1, \\ ℚ/ℤ & if i = 3, \\ 0 & otherwise. \end{matrix}

The class of R̂Γ_{c}(O_{L,S}, 𝔾_{m})[1] in Ext_{ℤ[G]}^{3}(ℚ/ℤ, C_{S}(L)) ≅ H^{2}(G, C_{S}(L)) is the global canonical class.

Proof. The computation of the cohomology is similar to the proof of Lemma 3.3, except that the role of (5) is now played by the distinguished triangle

(6)  R̂Γ_{c}(O_{L,S}, 𝔾_{m}) ⟶ RΓ(O_{L,S}, 𝔾_{m}) ⟶ ⨁_{w∈S} RΓ(L_{w}, 𝔾_{m}) ⟶

that is induced by the definition of R̂Γ_{c}(O_{L,S}, 𝔾_{m}). In degree 1 we also use the fact that, since Pic(O_{L,S}) = 0, C_{S}(L) is canonically isomorphic to the cokernel of the diagonal embedding O_{L,S}^{×} → ∏_{w∈S} L_{w}^{×}. For the extension class see [8, Prop. 3.5(b)] (but note that the result and proof in [8] apply to R̂Γ_{c}(O_{L,S}, 𝔾_{m}) rather than to RΓ_{c}(O_{L,S}, 𝔾_{m}) as incorrectly stated in loc. cit.). □

Lemma 3.5. There is a distinguished triangle in D(ℤ[G])

RΓ_{c}(O_{L,S}, 𝔾_{m}) ⟶ R̂Γ_{c}(O_{L,S}, 𝔾_{m}) ⟶ ⨁_{w∈S}(L_{w}^{×}/(L_{w}^{h})^{×})[−1] ⟶

which on cohomology in degree 1 induces the canonical exact sequence

0 → C_{S}^{h}(L) → C_{S}(L) → ∏_{w∈S} L_{w}^{×}/(L_{w}^{h})^{×} → 0

and on cohomology in degree 3 induces the identity map ℚ/ℤ ⟶^{=} ℚ/ℤ.

Proof. This follows upon combining the distinguished triangle in Lemma 3.1 for each w ∈ S with the distinguished triangle (5) with F = 𝔾_{m} and the distinguished triangle (6). □

4. Pro-p-completion

Let L/K be a Galois extension of number fields, G = Gal(L/K), and S a set of places of K as in §2.4. We will assume throughout this section that L is totally complex. We fix a prime number p and also assume henceforth that S contains all places of residue characteristic p. As in §2.4 we choose lattices L_{v} for v ∈ S_{f} and define L by (1). We fix an algebraic closure \overline{K} of K containing L and write K_{S} for the maximal extension of K inside \overline{K} which is unramified outside S. For each natural number n we write μ_{p^{n}} for the group of p^{n}-th roots of unity in \overline{K} and let ℤ_{p}(1) denote the continuous Gal(K_{S}/K)-module \underleftarrow{lim}_{n} μ_{p^{n}} where the limit is taken with respect to p-th power maps. In this section we relate E_{S}(L) ⊗ ℤ_{p} to the explicit complex RΓ_{c}(O_{L,S}, ℤ_{p}(1)) that is defined in [9, p. 522]. For convenience we often abbreviate RΓ_{c}(O_{L,S}, ℤ_{p}(1)) to RΓ_{c}(ℤ_{p}(1)). For any abelian group A and natural number m we write A_{[m]} for the kernel of the endomorphism given by multiplication by m. For each natural number n we consider the ℤ/p^{n}[G]-module ∏_{w∈S_{∞}(L)}(L_{w}^{×})_{[p^{n}]} ⊂ L_{∞}^{×}. We then define a

ℤ_{p}[G]-module by setting L(1)_{p} := \underleftarrow{lim}_{n} (∏_{w∈S_{∞}(L)}(L_{w}^{×})_{[p^{n}]}) where the transition morphisms are the p-th power maps. We set L_{p} := ∏_{w∈S_{p}(L)} L_{w} and note that L_{p} := ∏_{v∈S_{p}(K)} L_{v} is a full projective ℤ_{p}[G]-sublattice of L_{p}. We write

λ_{p} for the natural localization map O_{L}^{×} ⊗ ℤ_{p} → ∏_{w∈S_{p}(L)} U_{L_{w}}^{(1)}. Recall that Leopoldt’s Conjecture for the field L and prime number p is the statement that λ_{p} is injective. With these notations we can now describe the cohomology of the complex RΓ_{c}(O_{L,S}, ℤ_{p}(1)) ⊗_{ℤ_{p}} ℚ_{p}.

Lemma 4.1. If λ_{p} is injective (as predicted by Leopoldt’s Conjecture for the field L and prime p), then there are canonical isomorphisms

H^{i}(RΓ_{c}(O_{L,S}, ℤ_{p}(1))) ⊗_{ℤ_{p}} ℚ_{p} ≅ \begin{cases}L(1)_{p} ⊗_{ℤ_{p}} ℚ_{p} & if i = 1, \\ cok(λ_{p}) ⊗_{ℤ_{p}} ℚ_{p} & if i = 2, \\ ℚ_{p} & if i = 3, \\ 0 & otherwise. \end{cases}

Before proving Lemma 4.1 we first state the main result of this section and introduce some further notation.

Proposition 4.2. There is a distinguished triangle in D^{perf}(ℤ_{p}[G]) of the form

(7)  L_{p}[0] ⊕ L_{p}[−1] ⟶ RΓ_{c}(O_{L,S}, ℤ_{p}(1))[2] ⟶ E_{S}(L) ⊗ ℤ_{p} ⟶ .

Now assume that λ_{p} is injective (as predicted by Leopoldt’s Conjecture for the field L and prime p). With respect to the isomorphisms in Lemma 4.1 and the description of the cohomology groups H^{i}(E_{S}(L)) ⊗ ℚ given in §2.4, the image under − ⊗_{ℤ_{p}} ℚ_{p} of the cohomology sequence of (7) is equal to

[figure: 0; L(1)_{p} ⊗_{ℤ_{p}} ℚ_{p}; θ_{1}; H^{−1}(E_{S}(L)) ⊗ ℚ_{p}; θ_{2}; L_{p}; _{exp_{p}}; cok(λ_{p}) ⊗_{ℤ_{p}} ℚ_{p}; 0; H^{0}(E_{S}(L)) ⊗ ℚ_{p}; ⊂; L_{p}; _{tr_{Lp/ℚp}}; ℚ_{p}; 0]

(8)

where θ_{1} sends an element (r_{w} ·\{exp(2π\sqrt{−1}/p^{n})\}_{n≥0})_{w∈S_{∞}(L)} of L(1)_{p} ⊗_{ℤ_{p}} ℚ_{p} to the element (r_{w} · 2π\sqrt{−1})_{w∈S_{∞}(L)} of ker(exp_{∞}) ⊗ ℚ_{p} ⊂ H^{−1}(E_{S}(L)) ⊗ ℚ_{p} and θ_{2} is induced by the projection L_{S} → L_{p}.

In the proofs of Lemma 4.1 and Proposition 4.2 we will need the complex RΓ_{c}(μ_{p^{n}}) := RΓ_{c}(O_{L,S}, μ_{p^{n}}) for each natural number n. This complex can be considered in two different ways. On the one hand, since μ_{p^{n}} is a continuous Gal(K_{S}/K)-module, we can consider RΓ_{c}(μ_{p^{n}}) as the concrete complex of ℤ/p^{n}[G]-modules that is constructed using continuous cochains in [9, p. 522]. On the other hand, there is a natural étale sheaf μ_{p^{n}} on Spec O_{K,S} and we can consider the cohomology with compact support as defined in §3.3. However this will not cause any confusion because it can be shown that these two possible definitions of RΓ_{c}(μ_{p^{n}}) agree (up to canonical isomorphism), and whenever it is necessary to distinguish between these two constructions of RΓ_{c}(μ_{p^{n}}) we will emphasize which one we are using.

Proof of Lemma 4.1. Recall that the complex RΓ_{c}(ℤ_{p}(1)) defined in [9, p. 522] is equal to \underleftarrow{lim}_{n} RΓ_{c}(μ_{p^{n}}), where RΓ_{c}(μ_{p^{n}}) denotes the complex constructed using continuous cochains and the transition morphisms are induced by the p-th power map μ_{p^{n+1}} → μ_{p^{n}}. From the exact sequence 0 → μ_{p^{n}} → 𝔾_{m} ⟶^{p^{n}} 𝔾_{m} → 0 of étale sheaves on Spec O_{K,S} we obtain the distinguished triangle

(9)  RΓ_{c}(μ_{p^{n}}) ⟶^{θ} RΓ_{c}(O_{L,S}, 𝔾_{m}) ⟶^{p^{n}} RΓ_{c}(O_{L,S}, 𝔾_{m}) ⟶

in D(ℤ[G]). To compute the modules H^{i}(RΓ_{c}(ℤ_{p}(1))) explicitly we combine the cohomology sequence of (9) with the identifications of Lemma 3.3 and then pass to the inverse limit over n. In particular, since each module L_{w}^{×}/(L_{w}^{h})^{×} is uniquely divisible (by Lemma 3.1), one obtains in this way canonical isomorphisms

(10)  H^{i}(RΓ_{c}(ℤ_{p}(1))) ≅ \begin{cases}\underleftarrow{lim}_{n} C_{S}(L)_{[p^{n}]} & if i = 1, \\ \underleftarrow{lim}_{n} C_{S}(L)/p^{n} & if i = 2, \\ ℤ_{p} & if i = 3, \\ 0 & otherwise. \end{cases}

To describe this cohomology more explicitly we use the natural exact sequence of finite G-modules

(11) 0 → (O_{L,S}^{×})_{[p^{n}]} ⟶^{Δ_{S}} ∏_{w∈S}(L_{w}^{×})_{[p^{n}]} → C_{S}(L)_{[p^{n}]}

→ O_{L,S}^{×}/p^{n} ⟶^{Δ_{S}/p^{n}} ∏_{w∈S} L_{w}^{×}/p^{n} → C_{S}(L)/p^{n} → 0.

For each place (resp. finite place) w of L we write L_{w}^{×} ˆ⊗ℤ_{p} (resp. O_{L_{w}}^{×} ˆ⊗ℤ_{p}) for the pro-p-completion of L_{w}^{×} (resp. O_{L_{w}}^{×}). Note that O_{L_{w}}^{×} ˆ⊗ℤ_{p} ≅ U_{L_{w}}^{(1)} if w ∈ S_{p}(L), and that O_{L_{w}}^{×} ˆ⊗ℤ_{p} is finite if w ∈ S_{f}(L) \ S_{p}(L). Hence from the commutative diagram

\xymatrix{O_{L}^{×} ⊗ ℤ_{p} \ar[r] \ar[d]^{⊂} & ∏_{w∈S_{f}} O_{L_{w}}^{×} ˆ⊗ℤ_{p} \ar[d]^{⊂} \\ O_{L,S}^{×} ⊗ ℤ_{p} \ar[r]^{⟵^{lim}_{n} Δ_{S}/p^{n}} & ∏_{w∈S} L_{w}^{×} ˆ⊗ℤ_{p}}

we can deduce that the map \underleftarrow{lim}_{n} Δ_{S}/p^{n} is injective (since λ_{p} : O_{L}^{×} ⊗ ℤ_{p} →

∏_{w∈S_{p}(L)} U_{L_{w}}^{(1)} is injective by assumption), and that cok( \underleftarrow{lim}_{n} Δ_{S}/p^{n})⊗_{ℤ_{p}} ℚ_{p} = cok(λ_{p}) ⊗_{ℤ_{p}} ℚ_{p}. Now the limit \underleftarrow{lim}_{n}(O_{L,S}^{×})_{[p^{n}]} vanishes and one has \underleftarrow{lim}_{n} ∏_{w∈S}(L_{w}^{×})_{[p^{n}]} = \underleftarrow{lim}_{n} ∏_{w∈S_{∞}(L)}(L_{w}^{×})_{[p^{n}]} = L(1)_{p}. By passing to the inverse limit over n the sequence (11) thus induces identifications \underleftarrow{lim}_{n} C_{S}(L)_{[p^{n}]} = L(1)_{p} and \underleftarrow{lim}_{n} C_{S}(L)/p^{n} = cok( \underleftarrow{lim}_{n} Δ_{S}/p^{n}). The explicit description of H^{i}(RΓ_{c}(ℤ_{p}(1)))⊗_{ℤ_{p}} ℚ_{p} given in Lemma 4.1 therefore follows from (10) and the identification cok( \underleftarrow{lim}_{n} Δ_{S}/p^{n}) ⊗_{ℤ_{p}} ℚ_{p} = cok(λ_{p}) ⊗_{ℤ_{p}} ℚ_{p} described above. □

The proof of Proposition 4.2 will occupy the rest of this section. As the first step in this proof we introduce a useful auxiliary complex.

Lemma 4.3. There exists a complex Q in D(ℤ[G]) which corresponds (in the sense of the third paragraph of §2.2) to the extension class e_{S}^{glob} and also possesses all of the following properties.

(i) Q is a complex of ℤ-torsion-free G-modules of the form Q^{−1} → Q^{0} → Q^{1} (where the first term is placed in degree −1).

(ii) The morphism α_{S} used in the distinguished triangle (2) is represented by a morphism of complexes of G-modules α : L_{S}[0] ⊕ L[−1] → Q.

(iii) For each natural number n the complex Q/p^{n} consists of finite projective ℤ/p^{n}[G]-modules.

Proof. At the outset we fix a representative of e_{S}^{glob} of the form A ⟶^{δ} B as in [5, Rem. 3.2] with B a finitely generated projective ℤ[G]-module. We write d^{−1} for the composite of exp_{S} : L_{S} → C_{S}(L) and the inclusion C_{S}(L) ⊂ A. Since cok(exp_{S}) is finite we may choose a finitely generated free ℤ[G]-module F and a homomorphism π : F → A such that the morphism (d^{−1}, π) : L_{S} ⊕ F → A is

surjective. We take Q to be the complex ker((d^{−1}, π)) ⟶^{⊂} L_{S} ⊕ F ⟶^{δ∘(d^{−1},π)} B where the first term is placed in degree −1. Then (d^{−1}, π) restricts to give a surjection ker(δ ∘ (d^{−1}, π)) → C_{S}(L) which induces an identification of H^{0}(Q) with C_{S}(L). Via this identification, the morphism from Q to A → B that is equal to (d^{−1}, π) in degree 0 and to the identity map in degree 1 induces the identity map on cohomology in each degree and so Q represents e_{S}^{glob}. Further, we obtain a morphism α as in claim (ii) by defining α^{0} to be the inclusion

L_{S} ⊂ L_{S} ⊕ F and α^{1} to be any lift L ⟶^{tr′} B of L ⟶^{tr} ℤ through the given surjection B → ℤ. It is easy to see that (L_{S}⊕F)/p^{n} and B/p^{n} are finite and projective as ℤ/p^{n}[G]- modules. So to prove claim (iii) it remains to show that ker((d^{−1}, π))/p^{n} is a finite projective ℤ/p^{n}[G]-module. The proof of [5, Lemma 3.1] shows that ker(L_{S} → C_{S}(L)) is finitely generated, from which we can deduce that ker((d^{−1}, π)) is finitely generated. Since furthermore ker((d^{−1}, π)) is ℤtorsion-free, it follows that ker((d^{−1}, π)) is in fact ℤ-free. But the exact sequence 0 → ker((d^{−1}, π)) → L_{S} ⊕ F → A → 0 implies that the G-module ker((d^{−1}, π)) is cohomologically trivial, and any cohomologically trivial ℤ-free ℤ[G]-module is a projective ℤ[G]-module. From this it immediately follows that ker((d^{−1}, π))/p^{n} is finite and projective as ℤ/p^{n}[G]-module, as required. □

We now fix a complex Q as in Lemma 4.3, and set Q_{lim} := \underleftarrow{lim}_{n} Q/p^{n} where the inverse limit is taken with respect to the natural transition morphisms. To compute the cohomology H^{i}(Q_{lim}) = \underleftarrow{lim}_{n} H^{i}(Q/p^{n}) we use the short exact

sequence 0 → Q ⟶^{p^{n}} Q → Q/p^{n} → 0 together with the identifications H^{0}(Q) = C_{S}(L) and H^{1}(Q) = ℤ to compute the cohomology of Q/p^{n} and then pass to the inverse limit over n. We find that (similar to the proof of Lemma 4.1) H^{−1}(Q_{lim}) = \underleftarrow{lim}_{n} C_{S}(L)_{[p^{n}]}, H^{0}(Q_{lim}) = \underleftarrow{lim}_{n} C_{S}(L)/p^{n}, H^{1}(Q_{lim}) = ℤ_{p}, and H^{i}(Q_{lim}) = 0 otherwise. Hence, if we assume that Leopoldt’s Conjecture is valid for L at the prime p and use the same identifications as in the proof of Lemma 4.1, then we obtain isomorphisms

(12) H^{i}(Q_{lim}) ⊗_{ℤ_{p}} ℚ_{p} ≅ \begin{cases}L(1)_{p} ⊗_{ℤ_{p}} ℚ_{p} & if i = −1, \\ cok(λ_{p}) ⊗_{ℤ_{p}} ℚ_{p} & if i = 0, \\ ℚ_{p} & if i = 1, \\ 0 & otherwise. \end{cases}

Lemma 4.4. There exists an isomorphism Q_{lim} ≅ RΓ_{c}(ℤ_{p}(1))[2] in D(ℤ_{p}[G]). Further, if Leopoldt’s Conjecture is valid for L at the prime p and we use the isomorphisms in Lemma 4.1 and (12) to identify the cohomology groups of RΓ_{c}(ℤ_{p}(1))[2] ⊗_{ℤ_{p}} ℚ_{p} and Q_{lim} ⊗_{ℤ_{p}} ℚ_{p} respectively, then this isomorphism induces the identity map in each degree of cohomology after tensoring with ℚ_{p}.

Proof. Applying RΓ_{c} to the short exact sequence 0 → μ_{p^{n}} → 𝔾_{m} ⟶^{p^{n}} 𝔾_{m} → 0 and combining the resulting distinguished triangle with the triangle of Lemma 3.5 and the fact that each module L_{w}^{×}/(L_{w}^{h})^{×} is uniquely divisible (by Lemma 3.1) one obtains the following commutative diagram of distinguished triangles

\begin{matrix}RΓ_{c}(μ_{p^{n}}) & [figure] & RΓ_{c}(O_{L,S}, 𝔾_{m}) & p^{n} & [figure] & [figure] & RΓ_{c}(O_{L,S}, 𝔾_{m}) & [figure] & [figure] & RΓ_{c}(μ_{p^{n}}) & [figure] & R̂Γ_{c}(O_{L,S}, 𝔾_{m}) & p^{n} & [figure] & R̂Γ_{c}(O_{L,S}, 𝔾_{m}) & [figure] \\ (13) \end{matrix}

Rotating the lower row of (13) (without changing the signs of the maps) gives the distinguished triangle

R̂Γ_{c}(O_{L,S}, 𝔾_{m})[1] ⟶^{p^{n}} R̂Γ_{c}(O_{L,S}, 𝔾_{m})[1] ⟶^{ϱ′_{n}} RΓ_{c}(μ_{p^{n}})[2] → .

It is not difficult to see that one obtains the same identifications for H^{i}(RΓ_{c}(μ_{p^{n}})) (and hence also for H^{i}(RΓ_{c}(ℤ_{p}(1))) = \underleftarrow{lim}_{n} H^{i}(RΓ_{c}(μ_{p^{n}}))) if one computes the cohomology using this distinguished triangle instead of the first row of (13). Let Q̂ denote the complex

Q^{−1} → Q^{0} → Q^{1} → ℚ

where Q^{−1} is placed in degree −1, the first two arrows are the differentials of Q and the third is the natural map Q^{1} → H^{1}(Q) = ℤ ⊂ ℚ. Associated to the

natural short exact sequence 0 → Q̂ ⟶^{p^{n}} Q̂ → Q/p^{n} → 0 is a distinguished triangle

Q̂ ⟶^{p^{n}} Q̂ ⟶^{ϱ_{n}} Q/p^{n} → .

It is easy to see that one obtains the same identifications for H^{i}(Q/p^{n}) (and hence also for H^{i}(Q_{lim}) = \underleftarrow{lim}_{n} H^{i}(Q/p^{n})) if one computes the cohomology

using this distinguished triangle instead of the short exact sequence 0 → Q ⟶^{p^{n}} Q → Q/p^{n} → 0. The second assertion of Lemma 3.4 combines with the fact that Q corresponds to e_{S}^{glob} to imply the existence of an isomorphism ξ : Q̂ ≅ R̂Γ_{c}(O_{L,S}, 𝔾_{m})[1] in D(ℤ[G]) which induces the identity map on each degree of cohomology. We now consider the following diagram in D(ℤ[G])

\begin{matrix}Q̂ & p^{n} & [figure] & ξ & [figure] & Q̂ & ϱ_{n} & [figure] & ξ & [figure] & Q/p^{n} & [figure] & R̂Γ_{c}(O_{L,S}, 𝔾_{m})[1] & p^{n} & [figure] & R̂Γ_{c}(O_{L,S}, 𝔾_{m})[1] & ϱ′_{n} & [figure] & RΓ_{c}(μ_{p^{n}})[2] & [figure] \\ (14) \end{matrix}

Since the left hand square of (14) commutes there exists an isomorphism

ξ_{n} : Q/p^{n} → RΓ_{c}(μ_{p^{n}})[2]

in D(ℤ[G]) that makes the diagram into an isomorphism of distinguished triangles. In fact the isomorphisms ξ_{n} can be chosen to be compatible with the inverse systems over n, i.e. such that for every n the square

\xymatrix{Q/p^{n} \ar[r]^{ξ_{n}} \ar@{->>}[d] & RΓ_{c}(μ_{p^{n}})[2] \ar@{->>}[d] \\ Q/p^{n−1} \ar[r]^{ξ_{n−1}} & RΓ_{c}(μ_{p^{n−1}})[2]}

commutes in D(ℤ[G]). This can be seen for example as follows: if we compute R̂Γ_{c}(O_{L,S}, 𝔾_{m}) and RΓ_{c}(μ_{p^{n}}) using the concrete realisation of all chain complexes given by the Godement resolution of the sheaves (as described, for example, in [21, Chap. III, Rem. 1.20(c)]), then we obtain a short exact sequence

0 → RΓ_{c}(μ_{p^{n}}) → R̂Γ_{c}(O_{L,S}, 𝔾_{m}) ⟶^{p^{n}} R̂Γ_{c}(O_{L,S}, 𝔾_{m}) → 0.

Then both the top and the bottom row of (14) are canonically isomorphic to the distinguished triangles coming from short exact sequences (i.e. the distinguished triangles which are constructed using mapping cones), and for such distinguished triangles the statement is easy to see. To be able to pass to the inverse limit we must replace the maps ξ_{n} in D(ℤ[G]) by actual maps of complexes. Since both Q/p^{n} and RΓ_{c}(μ_{p^{n}})[2] are cohomologically bounded complexes of ℤ/p^{n}[G]-modules, the natural restriction of scalars homomorphism

(15) Hom_{D(ℤ_{p}[G])}(Q/p^{n}, RΓ_{c}(μ_{p^{n}})[2]) → Hom_{D(ℤ[G])}(Q/p^{n}, RΓ_{c}(μ_{p^{n}})[2])

is bijective (cf. [8, Lemma 17]). Thus for each n the map ξ_{n} : Q/p^{n} → RΓ_{c}(μ_{p^{n}})[2] can be represented as Q/p^{n} ⟵^{∼} T_{n} ⟶^{∼} RΓ_{c}(μ_{p^{n}})[2] where T_{n} is a complex of ℤ_{p}[G]-modules and Q/p^{n} ⟵^{∼} T_{n} and T_{n} ⟶^{∼} RΓ_{c}(μ_{p^{n}})[2] are quasi-isomorphisms of complexes of ℤ_{p}[G]-modules. By choosing a projective resolution we can assume that T_{n} is a bounded above complex of projective ℤ_{p}[G]-modules. There exists a morphism T_{n} → T_{n−1} in D(ℤ_{p}[G]) such that the diagram

\xymatrix{Q/p^{n} \ar@{->>}[d] & T_{n} \ar@{-->}[d] \ar[l]_{∼} \ar[r]^{∼} & RΓ_{c}(μ_{p^{n}})[2] \ar@{->>}[d] \\ Q/p^{n−1} & T_{n−1} \ar[l]_{∼} \ar[r]^{∼} & RΓ_{c}(μ_{p^{n−1}})[2]}

commutes in D(ℤ_{p}[G]). Since T_{n} is a bounded above complex of projective ℤ_{p}[G]-modules, the morphism T_{n} → T_{n−1} in D(ℤ_{p}[G]) can be realised by an actual map of complexes, and the above diagram will commute up to homotopy. The same argument as in [8, p. 1367] shows that after modifying the horizontal maps in this diagram by homotopies we can assume that the diagram is commutative. Finally, we can add suitable acyclic complexes to the T_{n} to guarantee that the maps T_{n} → T_{n−1} are surjective.

To summarise, we have constructed morphisms of inverse systems of complexes of ℤ_{p}[G]-modules (Q/p^{n}) ← (T_{n}) → (RΓ_{c}(μ_{p^{n}})[2]) such that for each n the composite Q/p^{n} ⟵^{∼} T_{n} ⟶^{∼} RΓ_{c}(μ_{p^{n}})[2] considered as a map in D(ℤ[G]) is equal to ξ_{n}. Furthermore the transition maps in each inverse system are surjective. Passing to the inverse limit gives morphisms of complexes of ℤ_{p}[G]-modules

Q_{lim} = \underleftarrow{lim}_{n} Q/p^{n} ⟵ \underleftarrow{lim}_{n} T_{n} ⟶ \underleftarrow{lim}_{n} RΓ_{c}(μ_{p^{n}})[2] = RΓ_{c}(ℤ_{p}(1))[2].

Now [8, Lemma 9] implies that these morphisms are quasi-isomorphisms and that the resulting map Q_{lim} → RΓ_{c}(ℤ_{p}(1))[2] in D(ℤ_{p}[G]) has the required properties. □

We now fix a morphism α as in Lemma 4.3(ii). Then, for each natural number n one has a commutative diagram of morphisms of complexes of G-modules

[figure: L_{S}[0] ⊕ L[−1]; α; p^{n}; Q; β; p^{n}; p^{n}; cone(α); γ; p^{n}; p^{n}; L_{S}[0] ⊕ L[−1]; α; Q; β; cone(α); γ; L_{S}/p^{n}[0] ⊕ L/p^{n}[−1]; α/p^{n}; Q/p^{n}; β/p^{n}; cone(α/p^{n}); γ/p^{n}]

(16)

In this diagram the maps β and γ come from the definition of cone(α) and so the first (and second) row is an explicit representative of the triangle (2). Also, the columns are the short exact sequences which result from the fact that L_{S}, L and all terms of Q (and hence also of cone(α)) are ℤ-torsion-free. Now L_{p} is canonically isomorphic to both \underleftarrow{lim}_{n} L_{S}/p^{n} and \underleftarrow{lim}_{n} L/p^{n}. Furthermore, as cone(α) is a perfect complex of ℤ-torsion-free modules, there is a natural isomorphism cone(α) ⊗ ℤ_{p} ≅ \underleftarrow{lim}_{n} cone(α)/p^{n} in D^{perf}(ℤ_{p}[G]), and clearly \underleftarrow{lim}_{n} cone(α)/p^{n} ≅ \underleftarrow{lim}_{n} cone(α/p^{n}) ≅ cone(\underleftarrow{lim}_{n} α/p^{n}) (where in all cases the limits are taken with respect to the natural transition morphisms). Hence, upon passing to the inverse limit of the lower row of (16), we obtain a distinguished triangle in D^{perf}(ℤ_{p}[G]) of the form

(17) L_{p}[0] ⊕ L_{p}[−1] ⟶^{⟵^{lim}_{n} α/p^{n}} Q_{lim} ⟶^{⟵^{lim}_{n} β/p^{n}} cone(α) ⊗ ℤ_{p} ⟶^{⟵^{lim}_{n} γ/p^{n}} .

The distinguished triangle (17) together with the isomorphism Q_{lim} ≅ RΓ_{c}(ℤ_{p}(1))[2] from Lemma 4.4 show the existence of a triangle of the form (7). It remains to show that if Leopoldt’s Conjecture is valid for L at the prime p and we use the identifications of the cohomology of Q_{lim} given in (12), then after tensoring with ℚ_{p} the long exact sequence of cohomology of the triangle (17) is equal to (8). Now the identifications of the cohomology of the three terms in (17) come from the columns in (16). In particular we have natural isomorphisms H^{i}(L_{p}[0] ⊕ L_{p}[−1]) ≅ \underleftarrow{lim}_{n} H^{i}(L_{S}[0] ⊕ L_{S}[−1])/p^{n} and H^{i}(cone(α) ⊗ ℤ_{p}) ≅ \underleftarrow{lim}_{n} H^{i}(cone(α))/p^{n} for all i, and H^{i}(Q_{lim}) ≅ \underleftarrow{lim}_{n} H^{i}(Q)/p^{n} for i = 0 and i = 1. Therefore by considering the cohomology sequences of the second and third rows in (16), we can easily deduce the explicit description of all maps in (8) except for the map L(1)_{p} ⊗_{ℤ_{p}} ℚ_{p} = H^{−1}(Q_{lim}) ⊗_{ℤ_{p}} ℚ_{p} → H^{−1}(E_{S}(L)) ⊗ ℚ_{p} = log_{∞}(O_{L}^{×}) ⊗ ℚ_{p}. To compute this map we consider the following diagram.

[figure: H^{−1}(Q/p^{n}); H^{−1}(β/p^{n}); H^{−1}(cone(α)/p^{n}); H^{0}(L_{S}[0]⊕L[−1]); H^{0}(α); p^{n}; H^{0}(Q); p^{n}; H^{−1}(cone(α)); H^{−1}(γ); H^{0}(L_{S}[0]⊕L[−1]); H^{0}(α); H^{0}(Q); H^{−1}(cone(α)/p^{n})]

By an easy computation with cochains one shows that if an element of

H^{0}(L_{S}[0] ⊕ L[−1]) lies in the kernel of H^{0}(L_{S}[0] ⊕ L[−1]) ⟶^{p^{n}·H^{0}(α)} H^{0}(Q), then its images under the two maps

H^{0}(L_{S}[0] ⊕ L[−1]) ⟶^{H^{0}(α)} H^{0}(Q) ← H^{−1}(Q/p^{n}) ⟶^{H^{−1}(β/p^{n})} H^{−1}(cone(α)/p^{n})

and

H^{0}(L_{S}[0] ⊕ L[−1]) ⟶^{p^{n}} H^{0}(L_{S}[0] ⊕ L[−1]) ⟵^{H^{−1}(γ)} H^{−1}(cone(α))

→ H^{−1}(cone(α)/p^{n})

coincide (note that the inverse arrows make sense in this context). By considering the elements (r_{w} · 2π\sqrt{−1}/p^{n})_{w∈S_{∞}} ∈ L_{∞} ⊆ L_{S} = H^{0}(L_{S}[0] ⊕ L[−1]) for r_{w} ∈ ℤ we see that the map H^{−1}(Q/p^{n}) → H^{−1}(cone(α)/p^{n}) sends the image of (r_{w} · exp(2π\sqrt{−1}/p^{n}))_{w∈S_{∞}(L)} ∈ (L_{S}^{×})_{[p^{n}]} ⊂ L_{S}^{×} in C_{S}(L)_{[p^{n}]} = H^{−1}(Q/p^{n}) to the image of the element (r_{w}·2π\sqrt{−1})_{w∈S_{∞}(L)} ∈ ker(exp_{∞}) ⊆ H^{−1}(cone(α)) in H^{−1}(cone(α)/p^{n}). Passing to the inverse limit gives the desired description of θ_{1}. This completes the proof of Proposition 4.2. q

5. The proof of Theorem 1.1

In this section we prove Theorem 1.1. Let L/K be a Galois extension of number fields with Galois group G. We define an element of K_{0}(ℤ[G], ℝ) by setting

T Ω(L/K, 1) := ˆ∂_{G}^{1}(ζ_{L/K,S}^{∗}(1)) + χ_{G}(E_{S}(L), μ_{L})

where the terms on the right hand side are as in §2.4. The element T Ω(L/K, 1) depends only upon L/K (see [5, Prop. 3.4]), and the conjectural equality (3) asserts that T Ω(L/K, 1) vanishes. We also recall that [9, Conj. 4(iv)] for the pair (ℚ(1)_{L}, ℤ[G]) asserts the vanishing of an element T Ω(ℚ(1)_{L}, ℤ[G]) of K_{0}(ℤ[G], ℝ) that is defined (unconditionally) in [9, Conj. 4(iii)]. To prove Theorem 1.1 it is therefore enough to prove the following result.

Proposition 5.1. Let L be a complex Galois extension of ℚ and G = Gal(L/ℚ). If Leopoldt’s Conjecture is valid for L and all prime numbers p, then T Ω(L/ℚ, 1) = T Ω(ℚ(1)_{L}, ℤ[G]).

Remark 5.1. Recall that we write ∂_{ℤ[G],ℝ}^{0} for the natural homomorphism of Kgroups K_{0}(ℤ[G], ℝ) → K_{0}(ℤ[G]). The argument of [5, Prop. 3.6(ii)] combines with the equality of Proposition 5.1 to imply that if Leopoldt’s Conjecture is valid, then ∂_{ℤ[G],ℝ}^{0}(T Ω(ℚ(1)_{L}, ℤ[G])) is equal to the element Ω(L/K, 1) of K_{0}(ℤ[G]) defined by Chinburg in [13]. Proposition 5.1 therefore answers the question raised in [7, Question 1.54].

5.1. Preliminaries. From now on let L/ℚ be a complex Galois extension with Galois group G. For each p and each embedding j : ℝ → ℂ_{p} there is an induced homomorphism j_{∗} : K_{0}(ℤ[G], ℝ) → K_{0}(ℤ_{p}[G], ℂ_{p}) and it is known that ⋂_{p,j} ker(j_{∗}) = {0} where p runs over all primes and j over all embeddings ℝ → ℂ_{p} (cf. [5, Lemma 2.1]). To prove Proposition 5.1 it is thus enough to prove that for all p and j one has

(18)  j_{∗}(T Ω(L/ℚ, 1)) = j_{∗}(T Ω(ℚ(1)_{L}, ℤ[G])).

The proof of this equality will occupy the rest of this section. We fix a prime p and in the sequel assume that Leopoldt’s Conjecture is valid for L and p. We also fix an embedding j : ℝ → ℂ_{p} and often suppress it from our notation; so in particular in a tensor product of the form − ⊗_{ℝ} ℂ_{p} we consider ℂ_{p} as an ℝ-module via j. Just as in §4 we will always assume that S contains all places of residue characteristic p. In the following we will need to use the language of virtual objects. To this end we consider the Picard categories V(ℤ_{p}[G]), V(ℂ_{p}[G]) and V(ℤ_{p}[G], ℂ_{p}[G]) discussed in [4, §5]. We fix a unit object 1_{V(ℂ_{p}[G])} of V(ℂ_{p}[G]) and for each object X of V(ℂ_{p}[G]) we fix an inverse, i.e. an object X^{−1} of V(ℂ_{p}[G]) together with an isomorphism X ⊗ X^{−1} ≅ 1_{V(ℂ_{p}[G])} in V(ℂ_{p}[G]). We also write ι : π_{0}V(ℤ_{p}[G], ℂ_{p}[G]) ≅ K_{0}(ℤ_{p}[G], ℂ_{p}) for the group isomorphism described in [4, Lemma 5.1]. We need to slightly generalise the definition of a trivialised complex and its Euler characteristic. If P is a perfect complex of ℤ_{p}[G]-modules and τ : [H^{ev}(P ⊗_{ℤ_{p}} ℂ_{p})] → [H^{od}(P ⊗_{ℤ_{p}} ℂ_{p})] an isomorphism in V(ℂ_{p}[G]), then we will sometimes call the pair (P, τ) a trivialised complex. Its Euler characteristic χ_{ℤ_{p}[G],ℂ_{p}}(P, τ) is defined as in [4, Definition 5.5] except that [t] is replaced by τ. Clearly any trivialised complex (P, t) as in §2.2 gives rise to the trivialised complex (P, [t]) in the new sense, but in general not every trivialisation τ : [H^{ev}(P ⊗_{ℤ_{p}} ℂ_{p})] → [H^{od}(P ⊗_{ℤ_{p}} ℂ_{p})] of P will be of the form [t] for some isomorphism t : H^{ev}(P ⊗_{ℤ_{p}} ℂ_{p}) → H^{od}(P ⊗_{ℤ_{p}} ℂ_{p}).

5.2. The element j_{∗}(T Ω(L/ℚ, 1)). We set RΓ_{c}(ℤ_{p}(1)) := RΓ_{c}(O_{L,S}, ℤ_{p}(1)) and also H_{c}^{i}(ℂ_{p}(1)) := H^{i}(RΓ_{c}(ℤ_{p}(1)) ⊗_{ℤ_{p}} ℂ_{p}). Furthermore we write H_{c}^{ev}(ℂ_{p}(1)) and H_{c}^{od}(ℂ_{p}(1)) for the direct sums ⊕_{i even}H_{c}^{i}(ℂ_{p}(1)) and ⊕_{i odd}H_{c}^{i}(ℂ_{p}(1)) respectively. We start by defining an isomorphism

ψ : [H_{c}^{ev}(ℂ_{p}(1))] ⊗ [im(θ_{2}) ⊗_{ℚ_{p}} ℂ_{p}] ⟶^{≅} [H_{c}^{od}(ℂ_{p}(1))] ⊗ [im(θ_{2}) ⊗_{ℚ_{p}} ℂ_{p}]

in V(ℂ_{p}[G]) which is induced by the identifications from Lemma 4.1, the exact sequence (8) in Proposition 4.2, and μ_{L}. More precisely, we let ψ be the following composite map.

[H_{c}^{2}(ℂ_{p}(1))] ⊗ [im(θ_{2}) ⊗_{ℚ_{p}} ℂ_{p}]

⟶^{α_{1}} [L_{p} ⊗_{ℚ_{p}} ℂ_{p}]

⟶^{α_{2}} [H^{0}(E_{S}(L)) ⊗ ℂ_{p}] ⊗ [ℂ_{p}]

⟶^{α_{3}} [H^{−1}(E_{S}(L)) ⊗ ℂ_{p}] ⊗ [ℂ_{p}]

⟶^{α_{4}} [L(1)_{p} ⊗_{ℤ_{p}} ℂ_{p}] ⊗ [im(θ_{2}) ⊗_{ℚ_{p}} ℂ_{p}] ⊗ [ℂ_{p}]

⟶^{α_{5}} [H_{c}^{1}(ℂ_{p}(1)) ⊕ H_{c}^{3}(ℂ_{p}(1))] ⊗ [im(θ_{2}) ⊗_{ℚ_{p}} ℂ_{p}].

Here α_{1} is induced by the isomorphism H_{c}^{2}(ℂ_{p}(1)) ≅ cok(λ_{p}) ⊗_{ℤ_{p}} ℂ_{p} and the short exact sequence

(19)  \xymatrix{im(θ_{2}) ⊗_{ℚ_{p}} ℂ_{p} \ar@{^{(}->}[r]^{⊂} & L_{p} ⊗_{ℚ_{p}} ℂ_{p} \ar@{->>}[r]^{^{exp^{p}}} & cok(λ_{p}) ⊗_{ℤ_{p}} ℂ_{p},}

α_{2} and α_{4} are induced by the short exact sequences

(20)  \xymatrix{H^{0}(E_{S}(L)) ⊗ ℂ_{p} \ar@{^{(}->}[r] & L_{p} ⊗_{ℚ_{p}} ℂ_{p} \ar@{->>}[r]^{tr} & ℂ_{p}}

and

(21) \xymatrix{L(1)_{p} ⊗_{ℤ_{p}} ℂ_{p} \ar@{^{(}->}[r]^{θ_{1}⊗_{ℚp} ℂ_{p}} & H^{−1}(E_{S}(L)) ⊗ ℂ_{p} \ar@{->>}[r]^{θ_{2}⊗_{ℚp} ℂ_{p}} & im(θ_{2}) ⊗_{ℚ_{p}} ℂ_{p}}

respectively, α_{3} = [μ_{L} ⊗_{ℝ} ℂ_{p}] ⊗ id, and α_{5} is induced by the isomorphisms H_{c}^{1}(ℂ_{p}(1)) ≅ L(1)_{p} ⊗_{ℤ_{p}} ℂ_{p} and H_{c}^{3}(ℂ_{p}(1)) ≅ ℂ_{p}. Now by the properties of a Picard category there exists a unique isomorphism

ν : [H_{c}^{ev}(ℂ_{p}(1))] ⟶^{≅} [H_{c}^{od}(ℂ_{p}(1))]

in V(ℂ_{p}[G]) such that ψ = ν ⊗ id. We will consider this isomorphism as a trivialisation of the complex RΓ_{c}(ℤ_{p}(1)).

Lemma 5.2. In K_{0}(ℤ_{p}[G], ℂ_{p}) one has

j_{∗}(T Ω(L/ℚ, 1)) = ∂_{ℤ_{p}[G],ℂ_{p}}^{1}(j_{∗}(ζ_{L/ℚ,S}^{∗}(1))) + χ_{ℤ_{p}[G],ℂ_{p}}(RΓ_{c}(ℤ_{p}(1)), ν).

Proof. To simplify the notation we will abbreviate ‘χ_{ℤ_{p}[G],ℂ_{p}}’ to ‘χ_{p}’. It is clear that j_{∗}(ˆ∂_{G}^{1}(ζ_{L/ℚ,S}^{∗}(1))) = ∂_{ℤ_{p}[G],ℂ_{p}}^{1}(j_{∗}(ζ_{L/ℚ,S}^{∗}(1))) (compare §2.1) and also j_{∗}(χ_{ℤ[G],ℝ}(E_{S}(L), μ_{L})) = χ_{p}(E_{S}(L) ⊗ ℤ_{p}, μ_{L} ⊗_{ℝ} ℂ_{p}). Moreover it follows from [4, Prop. 5.6.3] that χ_{p}(RΓ_{c}(ℤ_{p}(1)), ν) = χ_{p}(RΓ_{c}(ℤ_{p}(1))[2], ν). It is thus enough to prove that in K_{0}(ℤ_{p}[G], ℂ_{p}) one has

(22)  χ_{p}(E_{S}(L) ⊗ ℤ_{p}, μ_{L} ⊗_{ℝ} ℂ_{p}) = χ_{p}(RΓ_{c}(ℤ_{p}(1))[2], ν).

To do this we will apply the additivity criterion of [4, Theorem 5.7] to the exact triangle (7) in Proposition 4.2. On the complex L_{p}[0] ⊕ L_{p}[−1] we consider the trivialisation given by the identity map id : L_{p} ⊗_{ℤ_{p}} ℂ_{p} → L_{p} ⊗_{ℤ_{p}} ℂ_{p}, on RΓ_{c}(ℤ_{p}(1))[2] we consider the trivialisation ν, and on E_{S}(L) ⊗ ℤ_{p} we consider the trivialisation μ_{L} ⊗_{ℝ} ℂ_{p}. Note that the additivity criterion in [4] is only stated for trivialisations as defined in §2.2, however it is easy to check that it remains valid for generalised trivialisations as defined in §5.1. In our context, the map a in [4, Theorem 5.7] is the map L_{p}[0] ⊕ L_{p}[−1] → RΓ_{c}(ℤ_{p}(1))[2] in the distinguished triangle (7), and Σ = ℂ_{p}[G]. Therefore ker(H^{ev}a_{Σ}) = im(θ_{2}) ⊗_{ℚ_{p}} ℂ_{p} and ker(H^{od}a_{Σ}) = L_{p}^{0} ⊗_{ℚ_{p}} ℂ_{p} where L_{p}^{0} = ker(tr_{L_{p}/ℚ_{p}} : L_{p} → ℚ_{p}). To apply the additivity criterion we must show that the following diagram commutes in V(ℂ_{p}[G]).

\xymatrix{\begin{matrix}[cok(λ_{p}) ⊗_{ℤ_{p}} ℂ_{p}] \\ ⊗[im(θ_{2}) ⊗_{ℚ_{p}} ℂ_{p}] ⊗ [L_{p}^{0} ⊗_{ℚ_{p}} ℂ_{p}] \end{matrix} \ar[r]^{s^{ev}} \ar[d]^{ν⊗id⊗[−id]} & [L_{p} ⊗_{ℚ_{p}} ℂ_{p}] ⊗ [H^{0}(E_{S}(L)) ⊗ ℂ_{p}] \ar[d]^{id⊗[μ_{L}⊗_{ℝ}ℂ_{p}]} \\ \begin{matrix}[L(1)_{p} ⊗_{ℤ_{p}} ℂ_{p} ⊕ ℂ_{p}] \\ ⊗[im(θ_{2}) ⊗_{ℚ_{p}} ℂ_{p}] ⊗ [L_{p}^{0} ⊗_{ℚ_{p}} ℂ_{p}] \end{matrix} \ar[r]^{s^{od}} & [L_{p} ⊗_{ℚ_{p}} ℂ_{p}] ⊗ [H^{−1}(E_{S}(L)) ⊗ ℂ_{p}]}

Here the horizontal maps are induced by the even respectively odd part of the cohomology sequence (8) after tensoring with ℂ_{p}, i.e. the top horizontal map s^{ev} is induced by the short exact sequence (19) and the isomorphism

(23)  H^{0}(E_{S}(L)) ⊗ ℂ_{p} ≅ L_{p}^{0} ⊗_{ℚ_{p}} ℂ_{p},

and the bottom horizontal map s^{od} is induced by (21) and

(24)  \xymatrix{L_{p}^{0} ⊗_{ℚ_{p}} ℂ_{p} \ar@{^{(}->}[r]^{⊂} & L_{p} ⊗_{ℚ_{p}} ℂ_{p} \ar@{->>}[r]^{tr} & ℂ_{p}.}

To see the commutativity of the above diagram we will show that the automorphism

κ := (id ⊗ [μ_{L} ⊗_{ℝ} ℂ_{p}])^{−1} ∘ (s^{od}) ∘ (ν ⊗ id ⊗ [−id]) ∘ (s^{ev})^{−1}

of [L_{p} ⊗_{ℚ_{p}} ℂ_{p}] ⊗ [H^{0}(E_{S}(L)) ⊗ ℂ_{p}] is the identity map. For this we use the isomorphism

[L_{p} ⊗_{ℚ_{p}} ℂ_{p}] ⊗ [H^{0}(E_{S}(L)) ⊗ ℂ_{p}] ≅ [L_{p}^{0} ⊗_{ℚ_{p}} ℂ_{p}] ⊗ [ℂ_{p}] ⊗ [L_{p}^{0} ⊗_{ℚ_{p}} ℂ_{p}]

which is induced by the short exact sequence (24) and the isomorphism (23). Using ν ⊗ id ⊗ [−id] = ψ ⊗ [−id] and the definition of ψ, it is easy to see that then κ becomes the automorphism of [L_{p}^{0} ⊗_{ℚ_{p}} ℂ_{p}] ⊗ [ℂ_{p}] ⊗ [L_{p}^{0} ⊗_{ℚ_{p}} ℂ_{p}] which is given by (using the obvious abuse of notation)

a ⊗ b ⊗ c ↦ [−id](c) ⊗ b ⊗ a,

i.e. the morphism in V(ℂ_{p}[G]) which swaps the two copies of [L_{p}^{0} ⊗_{ℚ_{p}} ℂ_{p}] composed with the map [−id] on one of the two copies. It now follows from the general properties of a determinant functor (see e.g. [15, §4.9]), that this automorphism (and hence also κ) is the identity morphism as required. The additivity criterion [4, Theorem 5.7] now implies that

χ_{p}(RΓ_{c}(ℤ_{p}(1))[2], ν) = χ_{p}(L_{p}[0] ⊕ L_{p}[−1], id) + χ_{p}(E_{S}(L) ⊗ ℤ_{p}, μ_{L} ⊗_{ℝ} ℂ_{p}).

Since clearly χ_{p}(L_{p}[0] ⊕ L_{p}[−1], id) = 0 this completes the proof of (22) and hence of Lemma 5.2. □

5.3. The element j_{∗}(T Ω(ℚ(1)_{L}, ℤ[G])). The motive ℚ(1)_{L} is pure of weight −2. The argument of [10, §2] therefore shows that

(25) j_{∗}(T Ω(ℚ(1)_{L}, ℤ[G])) = ∂_{ℤ_{p}[G],ℂ_{p}}^{1}(j_{∗}(ζ_{L/ℚ,S}^{∗}(1))) + ι(([RΓ_{c}(ℤ_{p}(1))], ω))

with ω the composite morphism

[RΓ_{c}(ℤ_{p}(1)) ⊗_{ℤ_{p}} ℂ_{p}] ⟶^{ϑ̃_{p}⊗_{ℚp} ℂ_{p}} [Ξ(ℚ(1)_{L}) ⊗_{ℚ} ℂ_{p}] ⟶^{ϑ_{∞}⊗_{ℝ}ℂ_{p}} 1_{V(ℂ_{p}[G])}

where ϑ̃_{p} and ϑ_{∞} are as defined in [10, p. 479, resp. p. 477]. Indeed, whilst the argument of [10, §2] is phrased solely in terms of abelian groups G it extends immediately to the general case upon replacing graded determinants by virtual objects and then (25) is the non-abelian generalisation of the equality [10, (11)]. Given the observations of [7, §1.1, §1.3] it is also a straightforward exercise to explicate the space Ξ(ℚ(1)_{L}) and the morphisms ϑ̃_{p} and ϑ_{∞}. To describe the result we introduce further notation. We write Σ(L) for the set of all complex embeddings L → ℂ and consider ⨁_{Σ(L)} ℂ as a G × Gal(ℂ/ℝ)-module where G acts via L and Gal(ℂ/ℝ) acts diagonally. We write H_{B} for the G × Gal(ℂ/ℝ)- submodule ⨁_{Σ(L)} 2π\sqrt{−1} · ℤ of ⨁_{Σ(L)} ℂ and let H_{B}^{+} and ( ⨁_{Σ(L)} ℂ)^{+} denote the G-submodules comprising elements invariant under the action of Gal(ℂ/ℝ). We also set H_{f}^{1} := im(λ_{p}) ⊗_{ℤ_{p}} ℚ_{p}. Then ω is equal to the composite

\begin{aligned}[RΓ_{c}(ℤ_{p}(1)) ⊗_{ℤ_{p}} ℂ_{p}] & ≅ [H_{c}^{1}(ℂ_{p}(1))]^{−1} ⊗ [H_{c}^{2}(ℂ_{p}(1))] ⊗ [H_{c}^{3}(ℂ_{p}(1))]^{−1} \\ & ≅ [H_{c}^{1}(ℂ_{p}(1))]^{−1} ⊗ ([H_{f}^{1} ⊗_{ℚ_{p}} ℂ_{p}] ⊗ [H_{c}^{2}(ℂ_{p}(1))]) \\ & ⊗ [H_{f}^{1} ⊗_{ℚ_{p}} ℂ_{p}]^{−1} ⊗ [H_{c}^{3}(ℂ_{p}(1))]^{−1} \\ & ≅ ([H_{B}^{+} ⊗ ℂ_{p}]^{−1} ⊗ [L ⊗_{ℚ} ℂ_{p}]) \\ & ⊗ ([O_{L}^{×} ⊗ ℂ_{p}]^{−1} ⊗ [ℂ_{p}]^{−1}) \\ & ≅ [ ∏_{S_{∞}(L)} ℂ_{p}] ⊗ [ ∏_{S_{∞}(L)} ℂ_{p}]^{−1} \\ & ≅ 1_{V(ℂ_{p}[G])} \end{aligned} ; (26)

where the maps are defined as follows. The first, second and fifth maps are clear. The third map is induced by the exact sequence

0 → L(1)_{p} ⊗_{ℤ_{p}} ℂ_{p} ⟶^{≅} H_{c}^{1}(ℂ_{p}(1)) ⟶^{0} H_{f}^{1} ⊗_{ℚ_{p}} ℂ_{p} ⟶^{⊂} ∏_{w∈S_{p}(L)} U_{L_{w}}^{(1)} ⊗_{ℤ_{p}} ℂ_{p}

⟶^{π} H_{c}^{2}(ℂ_{p}(1)) → 0 → 0 → H_{c}^{3}(ℂ_{p}(1)) ⟶^{≅} ℂ_{p} → 0,

where π is induced by the identification H_{c}^{2}(ℂ_{p}(1)) ≅ cok(λ_{p}) ⊗_{ℤ_{p}} ℂ_{p} from Lemma 4.1 (this sequence is the cohomology sequence of the distinguished triangle of [10, (3)] with M = ℚ(1)_{L} and A = ℚ[G]), together with the isomorphism L(1)_{p} ≅ H_{B}^{+} ⊗ ℤ_{p} that sends an element (n_{w} · \{exp(2π\sqrt{−1}/p^{n})\}_{n≥0})_{w∈S_{∞}(L)} in L(1)_{p} to the element (n_{w_{σ}} · σ̂(2π\sqrt{−1}))_{σ∈Σ(L)} in H_{B}^{+} ⊗ ℤ_{p} (where w_{σ} denotes the place of L corresponding to σ, and σ̂ : L_{w_{σ}} → ℂ is the unique continuous extension of σ), the isomorphism

(27)  ∏_{w∈S_{p}(L)} U_{L_{w}}^{(1)} ⊗_{ℤ_{p}} ℂ_{p} ≅ ( ∏_{w∈S_{p}(L)} L_{w}) ⊗_{ℚ_{p}} ℂ_{p} = L ⊗_{ℚ} ℂ_{p}

induced by the p-adic logarithm maps U_{L_{w}}^{(1)} → L_{w}, and the isomorphism λ_{p} ⊗_{ℤ_{p}} ℂ_{p} : O_{L}^{×} ⊗ ℂ_{p} ≅ H_{f}^{1} ⊗_{ℚ_{p}} ℂ_{p}. The fourth map is induced by (the image under − ⊗_{ℝ} ℂ_{p} of) the short exact sequence

(28)  \xymatrix{O_{L}^{×} ⊗ ℝ \ar@{^{(}->}[r]^{Reg} & ∏_{S_{∞}(L)} ℝ \ar@{->>}[r] & ℝ}

where Reg : O_{L}^{×} ⊗ ℝ → ∏_{S_{∞}(L)} ℝ denotes the usual regulator map u ⊗ r ↦ r · (2 log|σ_{w}(u)|)_{w∈S_{∞}(L)} (here σ_{w} is a complex embedding of L corresponding to the place w), the natural isomorphism

(29)  ( ⨁_{Σ(L)} ℂ)^{+} ≅ L ⊗_{ℚ} ℝ

and (the image under − ⊗_{ℝ} ℂ_{p} of) the short exact sequence

(30)  \xymatrix{H_{B}^{+} ⊗ ℝ \ar@{^{(}->}[r]^{⊂} & ( ⨁_{Σ(L)} ℂ)^{+} \ar@{->>}[r] & ∏_{S_{∞}(L)} ℝ}

in which the second arrow sends each element (z_{σ})_{σ∈Σ(L)} of ( ⨁_{Σ(L)} ℂ)^{+} to (z_{σ_{w}} + z\overline{σ_{w}})_{w∈S_{∞}(L)} in ∏_{S_{∞}(L)} ℝ (where σ_{w} and \overline{σ_{w}} denote the two complex embeddings of L corresponding to the place w).

5.4. Completion of the proof. Let

ψ′ : [H_{c}^{ev}(ℂ_{p}(1))] ⊗ [H_{f}^{1} ⊗_{ℚ_{p}} ℂ_{p}] ⟶^{≅} [H_{c}^{od}(ℂ_{p}(1))] ⊗ [H_{f}^{1} ⊗_{ℚ_{p}} ℂ_{p}]

denote the composite isomorphism

[H_{c}^{2}(ℂ_{p}(1))] ⊗ [H_{f}^{1} ⊗_{ℚ_{p}} ℂ_{p}] ⟶^{α′_{1}} [L ⊗_{ℚ} ℂ_{p}]

⟶^{α′_{2}} [( ⨁_{Σ(L)} ℂ)^{+} ⊗_{ℝ} ℂ_{p}]

⟶^{α′_{3}} [H_{B}^{+} ⊗ ℂ_{p}] ⊗ [ ∏_{S_{∞}(L)} ℂ_{p}]

⟶^{α′_{4}} [L(1)_{p} ⊗_{ℤ_{p}} ℂ_{p}] ⊗ [O_{L}^{×} ⊗ ℂ_{p}] ⊗ [ℂ_{p}]

⟶^{α′_{5}} [H_{c}^{1}(ℂ_{p}(1)) ⊕ H_{c}^{3}(ℂ_{p}(1))] ⊗ [H_{f}^{1} ⊗_{ℚ_{p}} ℂ_{p}]

where α′_{1} is induced by the short exact sequence

\xymatrix{H_{f}^{1} ⊗_{ℚ_{p}} ℂ_{p} \ar@{^{(}->}[r] & ∏_{w∈S_{p}(L)} U_{L_{w}}^{(1)} ⊗_{ℤ_{p}} ℂ_{p} \ar@{->>}[r] & H_{c}^{2}(ℂ_{p}(1))}

and the isomorphism (27), the map α′_{2} is induced by the isomorphism (29), the map α′_{3} is induced by (the image under − ⊗_{ℝ} ℂ_{p} of) the short exact sequence (30), α′_{4} is induced by (the image under − ⊗_{ℝ} ℂ_{p} of) the short exact sequence (28) and the isomorphism H_{B}^{+} ⊗ ℂ_{p} ≅ L(1)_{p} ⊗_{ℤ_{p}} ℂ_{p}, and α′_{5} is induced by the isomorphisms H_{c}^{1}(ℂ_{p}(1)) ≅ L(1)_{p} ⊗_{ℤ_{p}} ℂ_{p}, H_{c}^{3}(ℂ_{p}(1)) ≅ ℂ_{p} and O_{L}^{×} ⊗ ℂ_{p} ≅ H_{f}^{1} ⊗_{ℚ_{p}} ℂ_{p}.

Let ν′ : [H_{c}^{ev}(ℂ_{p}(1))] ⟶^{≅} [H_{c}^{od}(ℂ_{p}(1))] be the unique isomorphism in V(ℂ_{p}[G]) such that ν′ ⊗ id = ψ′. We recall that the Euler characteristic χ_{ℤ_{p}[G],ℂ_{p}}(RΓ_{c}(ℤ_{p}(1)), ν′) is defined to be ι(([RΓ_{c}(ℤ_{p}(1))], λ)), where λ is the composite isomorphism

[RΓ_{c}(ℂ_{p}(1))] ≅ [H_{c}^{ev}(ℂ_{p}(1))] ⊗ [H_{c}^{od}(ℂ_{p}(1))]^{−1}

⟶^{ν′⊗id} [H_{c}^{od}(ℂ_{p}(1))] ⊗ [H_{c}^{od}(ℂ_{p}(1))]^{−1} ≅ 1_{V(ℂ_{p}[G])}

in V(ℂ_{p}[G]) (compare [4, Definition 5.5]). Now by comparing ω and λ one can show that

(31)  ι(([RΓ_{c}(ℤ_{p}(1))], ω)) = χ_{ℤ_{p}[G],ℂ_{p}}(RΓ_{c}(ℤ_{p}(1)), ν′).

The isomorphism (27) restricts to an isomorphism

φ : H_{f}^{1} ⊗_{ℚ_{p}} ℂ_{p} ≅ im(θ_{2}) ⊗_{ℚ_{p}} ℂ_{p}

of ℂ_{p}[G]-modules and we will show below that the following diagram in V(ℂ_{p}[G]) is commutative.

\xymatrix{[H_{c}^{ev}(ℂ_{p}(1))] ⊗ [H_{f}^{1} ⊗_{ℚ_{p}} ℂ_{p}] \ar[r]^{id⊗[φ]} \ar[d]^{ν′⊗id} & [H_{c}^{ev}(ℂ_{p}(1))] ⊗ [im(θ_{2}) ⊗_{ℚ_{p}} ℂ_{p}] \ar[d]^{ν⊗id} \\ [H_{c}^{od}(ℂ_{p}(1))] ⊗ [H_{f}^{1} ⊗_{ℚ_{p}} ℂ_{p}] \ar[r]^{id⊗[φ]} & [H_{c}^{od}(ℂ_{p}(1))] ⊗ [im(θ_{2}) ⊗_{ℚ_{p}} ℂ_{p}]}

From this diagram it follows that ν = ν′. In view of Lemma 5.2 and equations (25) and (31) this implies the required equality (18) and hence Proposition 5.1. It now only remains to show that the above diagram in V(ℂ_{p}[G]) is commutative. For this we consider the following diagram.

[figure: [H_{c}^{2}(ℂ_{p}(1))] ⊗ [H_{f}^{1} ⊗_{ℚ_{p}} ℂ_{p}]; id⊗[φ]; α′_{1}; [H_{c}^{2}(ℂ_{p}(1))] ⊗ [im(θ_{2}) ⊗_{ℚ_{p}} ℂ_{p}]; α_{1}; [L ⊗_{ℚ} ℂ_{p}]; α′_{3}∘α′_{2}; [L ⊗_{ℚ} ℂ_{p}]; α_{3}∘α_{2}; [H_{B}^{+} ⊗ ℂ_{p}] ⊗ [( ∏_{S_{∞}(L)} ℝ) ⊗_{ℝ} ℂ_{p}]; β_{1}; [H^{−1}(E_{S}(L)) ⊗ ℂ_{p}] ⊗ [ℂ_{p}]; α_{4}; \begin{matrix}[L(1)_{p} ⊗_{ℤ_{p}} ℂ_{p}] \\ ⊗[im(θ_{2}) ⊗_{ℚ_{p}} ℂ_{p}] ⊗ [ℂ_{p}] \end{matrix}; β_{2}; \begin{matrix}[L(1)_{p} ⊗_{ℤ_{p}} ℂ_{p}] \\ ⊗[im(θ_{2}) ⊗_{ℚ_{p}} ℂ_{p}] ⊗ [ℂ_{p}] \end{matrix}; α_{5}; \begin{matrix}[H_{c}^{1}(ℂ_{p}(1)) ⊕ H_{c}^{3}(ℂ_{p}(1))] \\ ⊗[H_{f}^{1} ⊗_{ℚ_{p}} ℂ_{p}] \end{matrix}; id⊗[φ]; \begin{matrix}[H_{c}^{1}(ℂ_{p}(1)) ⊕ H_{c}^{3}(ℂ_{p}(1))] \\ ⊗[im(θ_{2}) ⊗_{ℚ_{p}} ℂ_{p}] \end{matrix}]

Here the maps α_{i} and α′_{i} are as above. The map β_{1} is induced by the isomorphism L(1)_{p} ⊗_{ℤ_{p}} ℂ_{p} ≅ H_{B}^{+} ⊗ ℂ_{p} and the short exact sequence

(32)  \xymatrix{im(θ_{2}) ⊗_{ℚ_{p}} ℂ_{p} \ar@{^{(}->}[r] & ( ∏_{S_{∞}(L)} ℝ) ⊗_{ℝ} ℂ_{p} \ar@{->>}[r] & ℂ_{p}}

which is obtained by applying − ⊗_{ℝ} ℂ_{p} to the short exact sequence (28) and using the identification O_{L}^{×} ⊗ ℂ_{p} ≅ im(θ_{2}) ⊗_{ℚ_{p}} ℂ_{p}, and the map β_{2} is induced by the isomorphisms H_{c}^{1}(ℂ_{p}(1)) ≅ L(1)_{p} ⊗_{ℤ_{p}} ℂ_{p}, H_{c}^{3}(ℂ_{p}(1)) ≅ ℂ_{p} and φ. By definition the composite of the right vertical maps is ψ = ν⊗id. Furthermore it is not difficult to see that β_{2} ∘ β_{1} = α′_{5} ∘ α′_{4}, hence the composite of the left vertical maps is ψ′ = ν′ ⊗ id. Clearly the bottom square is commutative. The isomorphism of short exact sequences

\xymatrix{H_{f}^{1} ⊗_{ℚ_{p}} ℂ_{p} \ar@{^{(}->}[r] \ar[d]^{φ} & ∏_{w∈S_{p}(L)} U_{L_{w}}^{(1)} ⊗_{ℤ_{p}} ℂ_{p} \ar@{->>}[r] \ar[d]^{≅} & H_{c}^{2}(ℂ_{p}(1)) \ar[d]^{≅} \\ im(θ_{2}) ⊗_{ℚ_{p}} ℂ_{p} \ar@{^{(}->}[r] & L ⊗_{ℚ} ℂ_{p} \ar@{->>}[r] & cok(λ_{p}) ⊗_{ℤ_{p}} ℂ_{p}}

implies that the top square is commutative. The commutativity of the middle rectangle follows from the properties of a determinant functor applied to the following commutative diagram of short exact sequences.

[figure: L(1)_{p} ⊗_{ℤ_{p}} ℂ_{p}; θ_{1}⊗_{ℚp} ℂ_{p}; ≅; H^{−1}(E_{S}(L)) ⊗ ℂ_{p}; θ_{2}⊗_{ℚp} ℂ_{p}; im(θ_{2}) ⊗_{ℚ_{p}} ℂ_{p}; H_{B}^{+} ⊗ ℂ_{p}; L ⊗_{ℚ} ℂ_{p}; tr; ( ∏_{S_{∞}(L)} ℝ) ⊗_{ℝ} ℂ_{p}; ℂ_{p}; ℂ_{p}]

Here the top horizontal and right vertical short exact sequences are (21) and (32) respectively. The middle horizontal short exact sequence comes from combining (30) with the isomorphism (29), and the middle vertical short exact sequence comes from combining (20) with the isomorphism μ_{L} ⊗_{ℝ} ℂ_{p}. The commutativity of this diagram is easily checked.

6. The proofs of Corollaries 1.2, 1.3 and 1.4

In this section we use Theorem 1.1 to prove Corollaries 1.2, 1.3 and 1.4.

6.1. The proof of Corollary 1.2. Let F/E be a Galois extension of number fields and set Γ := Gal(F/E). Let L be a totally complex finite Galois extension of ℚ containing F and set G := Gal(L/ℚ). We write π for the natural composite homomorphism K_{0}(ℤ[G], ℝ) → K_{0}(ℤ[Gal(L/E)], ℝ) → K_{0}(ℤ[Γ], ℝ) where the first arrow is restriction and the second projection. Then it is known that π(T Ω(L/ℚ, 1)) = T Ω(F/E, 1) and π(T Ω(ℚ(1)_{L}, ℤ[G])) = T Ω(ℚ(1)_{F} , ℤ[Γ]) (see [5, Prop. 3.5] and [9, Prop. 4.1]). In particular, to prove that T Ω(F/E, 1) = T Ω(ℚ(1)_{F} , ℤ[Γ]) it is enough to prove that T Ω(L/ℚ, 1) = T Ω(ℚ(1)_{L}, ℤ[G]). Given this observation, Corollary 1.2 is an immediate consequence of Theorem 1.1.

6.2. The proof of Corollary 1.3. By the functorial properties of the conjectures (see [5, Prop. 3.5 and Rem. 4.2]) it suffices to consider the case K = ℚ and L totally complex. Since L is abelian over ℚ, Leopoldt’s Conjecture is known to be valid for L and all primes p [6]. In addition, the validity of [9, Conj. 4(iv)] for the pair (ℚ(1)_{L}, ℤ[Gal(L/ℚ)]) has been proved by Flach and the second named author in [11, Cor. 1.2]. (The proof of [11, Cor. 1.2] relies on certain 2-adic results of Flach in [16] and unfortunately the relevant results in [16] are now known to contain errors. However, in [17] Flach has recently provided the necessary corrections so that, in particular, the result of [11, Cor. 1.2] is valid as stated.) Given the validity of [9, Conj. 4(iv)] for (ℚ(1)_{L}, ℤ[Gal(L/ℚ)]), the first assertion of Corollary 1.3 follows immediately from Theorem 1.1. We now assume that [5, Conj. 3.3] is valid for L/ℚ. Then [5, Theorem 5.2] implies that [5, Conj. 4.1] is valid for L/ℚ if and only if [5, Conj. 5.3] is valid for L/ℚ. Also, in [5, Rem. 5.4] it is shown that [5, Conj. 5.3] is equivalent to the earlier conjecture [2, Conj. 4.1]. To prove the second assertion of Corollary 1.3 we therefore need only note that [2, Conj. 4.1] is proved for abelian extensions L/ℚ of odd conductor in [2, Cor. 6.2] and for abelian extensions L/ℚ of arbitrary conductor in [11, Theorem 1.1] (see in particular the discussion at the end of [11, §3.1]). This completes the proof of Corollary 1.3.

Remark 6.1. By using the main result of Bley in [1] one can prove an analogue of Corollary 1.3 for certain classes of abelian extensions of imaginary quadratic fields.

6.3. The proof of Corollary 1.4. Let p, q and r be distinct (odd) rational primes which satisfy p ≡ r ≡ −q ≡ 3 (mod 4) and are such that the Legendre symbols ({p \atop q}) and ({r \atop q}) are both equal to −1. Then if ℓ is any odd prime such that ({ℓ \atop pr}) = −({ℓ \atop q}) = 1 Chinburg has shown that there exists a unique totally complex field L_{p,q,r,ℓ} which contains ℚ(\sqrt{pr}, \sqrt{q}), is Galois over ℚ with group isomorphic to the quaternion group of order 8 and is such that L_{p,q,r,ℓ}/ℚ is ramified precisely at p, q, r, ℓ and infinity (cf. [14, Prop. 4.1.3]). We observe that the primes p = 3, q = 5 and r = 7 satisfy the congruence conditions described above and will now prove that the conjectures [5, Conj. 3.3] and [5, Conj. 4.1] are both valid for any extension of the form L_{3,5,7,ℓ}/K. To do this we set L_{ℓ} := L_{3,5,7,ℓ} and G_{ℓ} := Gal(L_{3,5,7,ℓ}/ℚ). We note first that L_{ℓ}/K is tamely ramified and we recall that for any tamely ramified extension of number fields F/E the element T Ω^{loc}(F/E, 1) that is defined in [5, §5.1.1] vanishes (by [5, Prop. 5.7(i)]) and hence that the conjectures [5, Conj. 3.3] and [5, Conj. 4.1] are equivalent for F/E (by [5, Theorem 5.2]). It therefore suffices for us to prove that [5, Conj. 3.3] is valid for all extensions L_{ℓ}/K. We recall that this is equivalent to asserting that the element T Ω(L_{ℓ}/K, 1) of K_{0}(ℤ[Gal(L_{ℓ}/K)], ℝ) that is defined in [5, §3.2] vanishes. Taking account of the functorial behaviour described in [5, Prop. 3.5(i)] it is therefore enough to prove that each element T Ω(L_{ℓ}/ℚ, 1) vanishes. We claim next that T Ω(L_{ℓ}/ℚ, 1) belongs to the subgroup K_{0}(ℤ[G_{ℓ}], ℚ)_{tor} of K_{0}(ℤ[G_{ℓ}], ℝ). Indeed, since T Ω^{loc}(L_{ℓ}/ℚ, 1) vanishes the equality of [5, Theorem 5.2] implies T Ω(L_{ℓ}/ℚ, 1) = ψ_{G_{ℓ}}^{∗}(T Ω(L_{ℓ}/ℚ, 0)) where ψ_{G_{ℓ}}^{∗} is the involution of K_{0}(ℤ[G_{ℓ}], ℝ) defined in [5, §2.1.4] and T Ω(L_{ℓ}/ℚ, 0) the element of K_{0}(ℤ[G_{ℓ}], ℝ) defined in [5, §4]. Now ψ_{G_{ℓ}}^{∗} preserves the subgroup K_{0}(ℤ[G_{ℓ}], ℚ)_{tor} and from [5, Prop. 4.4(ii)] one knows that T Ω(L_{ℓ}/ℚ, 0) belongs to K_{0}(ℤ[G_{ℓ}], ℚ)_{tor} if the ‘strong Stark conjecture’ of Chinburg is valid for L_{ℓ}/ℚ. It thus suffices to recall that, since every complex character of G_{ℓ} is rational valued, the strong Stark conjecture for L_{ℓ}/ℚ has been proved by Tate in [26, Chap. II]. We write F_{ℓ} for the maximal abelian extension of ℚ in L_{ℓ} (and note that F_{ℓ}/ℚ is biquadratic). Then, since the element T Ω(L_{ℓ}/ℚ, 1) belongs to K_{0}(ℤ[G_{ℓ}], ℚ)_{tor}, the result of [10, Lemma 4] implies T Ω(L_{ℓ}/ℚ, 1) vanishes if it belongs to the kernels of both the natural projection homomorphism q : K_{0}(ℤ[G_{ℓ}], ℝ) → K_{0}(ℤ[Gal(F_{ℓ}/ℚ)], ℝ) and the connecting homomorphism ∂_{ℤ[G_{ℓ}],ℝ}^{0} : K_{0}(ℤ[G_{ℓ}], ℝ) → K_{0}(ℤ[G_{ℓ}]). Now from [5, Prop. 3.6(ii)] one knows that T Ω(L_{ℓ}/ℚ, 1) belongs to ker(∂_{ℤ[G_{ℓ}],ℝ}^{0}) if Chinburg’s ‘Ω_{1}-Conjecture’ [13, Question 3.2] is valid for L_{ℓ}/ℚ. In addition, the equality of [13, (3.2)] shows that the Ω_{1}-Conjecture is valid for L_{ℓ}/ℚ if the ‘Ω_{3}-Conjecture’ [13, Conj. 3.1] and ‘Ω_{2}-Conjecture’ [13, Question 3.1] are both valid for L_{ℓ}/ℚ. But Chinburg proves the Ω_{3}-Conjecture for L_{ℓ}/ℚ in [14] and, since L_{ℓ}/ℚ is tamely ramified, the validity of the Ω_{2}-Conjecture for L_{ℓ}/ℚ follows directly from [13, Theorems 3.2 and 3.3]. At this stage it suffices to prove that T Ω(L_{ℓ}/ℚ, 1) belongs to ker(q). But, by [5, Prop. 3.5(ii)], this is equivalent to asserting that [5, Conj. 3.3] is valid for the extension F_{ℓ}/ℚ and since F_{ℓ}/ℚ is abelian this follows from Corollary 1.3. This completes the proof of Corollary 1.4.

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Manuel Breuning

King’s College London

Department of Mathematics London WC2R 2LS England manuel.breuning@kcl.ac.uk David Burns King’s College London Department of Mathematics London WC2R 2LS England david.burns@kcl.ac.uk