On descent theory and main conjectures

in non-commutative Iwasawa theory

David Burns and Otmar Venjakob

Abstract. We develop an explicit descent theory in the context of Whitehead groups of non-commutative Iwasawa algebras. We apply this theory to describe the precise connection between main conjectures of non-commutative Iwasawa theory (in the spirit of Coates, Fukaya, Kato, Sujatha and Venjakob) and the equivariant Tamagawa number conjecture. The latter result is a converse to a theorem of Fukaya and Kato and also provides an important means of both deriving explicit consequences of the main conjecture and proving special cases of the equivariant Tamagawa number conjecture.

1991 Mathematics Subject Classification: Primary 11G40; Secondary 11R65 19A31 19B28

Introduction

There has been much interest in the study of non-commutative Iwasawa theory over the last few years. Nevertheless, there is still no satisfactory understanding of the explicit consequences for Hasse-Weil L-functions that are implied by a ‘main conjecture’ of the kind formulated by Coates, Fukaya, Kato, Sujatha and the second named author in [13]. Indeed, whilst explicit consequences of such a conjecture for the values (at s = 1) of twisted Hasse-Weil L-functions have been studied by Coates et al in [13], by Kato in [20] and by Dokchister and Dokchister in [16], all of these consequences become trivial whenever the L-functions vanish at s = 1. Further, the conjecture of Birch and Swinnerton-Dyer implies that these L-functions should vanish whenever the relevant component of the Mordell-Weil group has strictly positive rank and by a recent result of Mazur and Rubin [22], which is itself equivalent to a special case of an earlier result of Nekovář [24, Th. 10.7.17], this should often be the case. It is therefore of interest to understand what a main conjecture of the kind formulated in [13] predicts concerning the values of derivatives of Hasse-Weil L-functions at s = 1. In this article we take the first step towards developing such a theory by describing a general formalism of descent in non-commutative Iwasawa theory. In a subsequent article it will be shown that the results proved here can be combined with techniques developed by the first named author in [7] to derive from the main conjecture of non-commutative Iwasawa theory a variety of explicit (and highly non-trivial) congruence relations between values of derivatives of twisted Hasse-Weil L-functions. In another direction, in [8] the results of this article play a key role in the first verification of the equivariant Tamagawa number conjecture (for certain Tate motives) for a wide class of non-abelian extensions of number fields and in the proof of a long-standing conjecture of Chinburg. However, as preparation for the above applications, we must first develop several aspects of the theory that appear themselves to be of some independent interest. These include proving a natural Weierstrass Preparation Theorem for Whitehead groups of Iwasawa algebras , defining a canonical ‘characteristic series’ for torsion modules over (localised) Iwasawa algebras, satisfactorily resolving the descent problem in non-commutative Iwasawa theory and formulating a main conjecture in the spirit of Coates et al that deals with interpolation properties of the ‘leading terms at Artin representations’ (in the sense introduced in [11]) of analytic p-adic L-functions. In a little more detail, the main contents of this article is as follows. In §1 we recall some useful preliminaries concerning localisation of Iwasawa algebras, K-theory, virtual objects and derived categories. In §2 we state the main K-theoretical results that are proved in this article. In §3 we define a suitable notion of μ-invariant and in §4 we combine this notion with a result of Schneider and the second named author from [26] and the formalism developed by Fukaya and Kato in [17] to define canonical ‘characteristic series’ in non-commutative Iwasawa theory (this construction extends the notion of ‘algebraic p-adic L-functions’ introduced by the first named author in [6] and hence also refines the notion of ‘Akashi series’ introduced by Coates, Schneider and Sujatha in [12]). As a first application of these characteristic series we use them in §5 to prove an explicit formula for the ‘leading terms at Artin representations ’ of elements of Whitehead groups of non-commutative Iwasawa algebras: this result provides a suitable ‘descent formalism’ in non-commutative Iwasawa theory and in particular plays a crucial role in proving the arithmetic results discussed in the remainder of the article. In §6 we present a result of Kato that allows reduction to a convenient special class of extensions when formulating main conjectures and, in particular, shows that the main result of §5 is indeed a satisfactory resolution of the descent problem in the context of non-commutative Iwasawa theory. In §7 we formulate explicit main conjectures of non-commutative Iwasawa theory for both Tate motives and (certain) critical motives. The approach here is finer than that of [13] since we consider interpolation properties for leading terms of analytic p-adic Lfunctions. In §8 we combine the descent formalism described in §5 with the main results of our earlier article [11] to prove that, under suitable hypotheses, the main conjectures formulated in §7 imply the relevant special cases of the equivariant Tamagawa number conjecture formulated by Flach and the first named author in [9, Conj. 4(iv)]. These results are both a converse to the result of Fukaya and Kato in [17] which asserts that, under suitable hypotheses, the ‘non-commutative Tamagawa number conjecture’ of loc. cit. implies the main conjecture of Coates et al [13] and can also be used to derive explicit consequences of the main conjecture. Finally, in several appendices, we review relevant aspects of the algebraic formalism of localized K_{1}-groups and Bockstein homomorphisms and clarify certain normalizations used in [11]. We are very grateful indeed to Kazuya Kato for his generous help and encouragement, to Manuel Breuning for advice regarding determinant functors, to Peter Schneider for valuable comments on an earlier draft and to Jan Nekovář for many stimulating discussions.

Much of this article was written when the first named author was a Visiting Professor at the Institute of Mathematics of the University of Paris 6 and he is extremely grateful to the Institute for this opportunity.

Part I: K-theory

1. Preliminaries

1.1. Iwasawa algebras. We fix a prime p. For any compact p-adic Lie group G we write Λ(G) and Ω(G) for the ‘Iwasawa algebras’ \underleftarrow{lim}_{U} ℤ_{p}[G/U] and \underleftarrow{lim}_{U} 𝔽_{p}[G/U] where U runs over all open normal subgroups of G and the limits are taken with respect to the natural projection maps ℤ_{p}[G/U] → ℤ_{p}[G/U′] and 𝔽_{p}[G/U] → 𝔽_{p}[G/U′] for U ⊆ U′. The rings Λ(G) and Ω(G) are both noetherian and, if G has no element of order p, they are also regular in the sense that their (left and right) global dimensions are finite. We write Q(G) for the total quotient ring of Λ(G). If O is any subring of ℚ_{p}^{c} that contains ℤ_{p}, then we set Λ_{O}(G) := O ⊗_{ℤ_{p}} Λ(G) and write Q_{O}(G) for its total quotient ring. We assume throughout that the following condition is satisfied

• G has a closed normal subgroup H for which the quotient group Γ := G/H is

isomorphic (topologically) to the additive group of ℤ_{p}. We write π_{Γ} : G → Γ for the natural projection and fix a topological generator γ of Γ. We use γ to identify Λ(Γ) with the power series ring ℤ_{p}[[T]] in an indeterminate T (via the identification T = γ − 1). We recall from [13, §2-§3] that there are canonical left and right denominator sets S_{G,H} and S_{G,H}^{∗} of Λ(G) where

S_{G,H} := {λ ∈ Λ(G) : Λ(G)/(Λ(G) · λ) is a finitely generated Λ(H)-module}

and S_{G,H}^{∗} := ⋃_{i≥0} p^{i}S_{G,H}. When G and H are clear from context we usually abbreviate S_{G,H} to S. We also write 𝔐_{S}(G) and 𝔐_{S^{∗}}(G) for the categories of finitely generated Λ(G)-modules M with Λ(G)_{S} ⊗_{Λ(G)} M = 0 and Λ(G)_{S^{∗}} ⊗_{Λ(G)} M = 0 respectively. For any ℤ_{p}-module M we write M_{tor} for its ℤ_{p}-torsion submodule and set M_{tf} := M/M_{tor}. We recall from [13, Prop. 2.3] that a finitely generated Λ(G)-module M belongs to 𝔐_{S}(G), resp. 𝔐_{S^{∗}}(G), if and only if it is a finitely generated Λ(H)-module (by restriction), resp. when M_{tf} belongs to 𝔐_{S}(G). This means in particular that 𝔐_{S^{∗}}(G) coincides with the category 𝔐_{H}(G) introduced in loc. cit.

1.2. K-groups. For any ring homomorphism R → R′ we write K_{0}(R, R′) for the associated relative algebraic K_{0}-group. We recall that this group is generated by symbols of the form (P, λ, Q) where P and Q are finitely generated projective (left) R-modules and λ is an isomorphism of R′-modules R′ ⊗_{R} P → R′ ⊗_{R} Q (for more details see [29, p. 215]). For any ring homomorphisms R → R′ → R′′ there is a natural commutative diagram of long exact sequences

(1)  \begin{matrix}K_{1}(R) & ⟶ & K_{1}(R′) & ⟶^{∂_{R,R′}} & K_{0}(R, R′) & ⟶ & K_{0}(R) \\ ∥ & & ↓ & & ↓ & & ∥ \\ K_{1}(R) & ⟶ & K_{1}(R′′) & ⟶^{∂_{R,R′′}} & K_{0}(R, R′′) & ⟶ & K_{0}(R). \end{matrix}

If G has no element of order p and Σ denotes either S or S^{∗}, then Λ(G) is a noetherian regular ring and K_{0}(Λ(G), Λ(G)_{Σ}) can be identified with the Grothendieck group K_{0}(𝔐_{Σ}(G)) of the category 𝔐_{Σ}(G). To be precise we normalise this isomorphism as follows: if g = s^{−1}h with s ∈ Σ and h ∈ Λ(G)∩Λ(G)_{Σ}^{×}, then the element (Λ(G), r_{g}, Λ(G)) of K_{0}(Λ(G), Λ(G)_{Σ}) corresponds to [cok(r_{h})] − [cok(r_{s})] in K_{0}(𝔐_{Σ}(G)) where [X] is the element of K_{0}(𝔐_{Σ}(G)) associated to an object X of 𝔐_{Σ}(G) and r_{g}, r_{h} and r_{s} denote the automorphisms of Λ(G)_{Σ} that are induced by right multiplication by g, h and s respectively. In particular, with respect to this isomorphism, the upper row of (1) with R = Λ(G) and R′ = Λ(G)_{S^{∗}} identifies with the exact sequence of [13, (24)]. If R = Λ(G) and R′ = Λ(G)_{S^{∗}}, resp. R = ℤ_{p}[G] and R′ = ℚ_{p}^{c}[G] for a finite group G, then we usually abbreviate the connecting homomorphism ∂_{R,R′} in diagram (1) to ∂_{G}, resp. ∂\overline{G}.

1.3. Virtual objects. We let P_{0} denote the Picard category with unique object 1_{P_{0}} and Aut_{P_{0}}(1_{P_{0}}) = 0. For any associative unital ring R we also write V (R) for the Picard category of virtual objects associated to the category of finitely generated projective Rmodules and we fix a unit object 1_{R} in V (R). For any homomorphism of such rings R → R′ we then define V (R, R′) to be the fibre product category in the diagram

V (R, R′) ⟶ P_{0}
↓  ↓^{F_{2}}
V (R) ⟶^{F_{1}} V (R′)

where F_{2} is the (monoidal) functor sending 1_{P_{0}} to 1_{R′} and F_{1}(L) = R′ ⊗_{R} L for each object L of V (R). We regard the canonical isomorphism

(2)  π_{0}(V (R, R′)) ≅ K_{0}(R, R′)

of [3, Lem. 5.1] (and [9, Prop. 2.5]) as an identification. In particular, for each object L of V (R) and each morphism μ : F_{1}(L) → 1_{R′} in V (R′) we write [L, μ] for the associated element of K_{0}(R, R′).

1.4. Euler characteristics. For any ring R we write D(R) for the derived category of R-modules. If R is noetherian, then we also write D^{fg,−}(R), resp. D^{fg}(R) for the full triangulated subcategory of D(R) comprising complexes that are isomorphic to a bounded above, resp. bounded, complex of finitely generated R-modules and we let D^{p}(R) denote the full subcategory of D^{fg}(R) comprising complexes that are isomorphic to an object of the category C^{p}(R) of bounded complexes of finitely generated projective R-modules. If Σ denotes either S or S^{∗}, then we write D_{Σ}^{p}(Λ(G)) for the full triangulated subcategory of D^{p}(Λ(G)) comprising those complexes C such that Λ(G)_{Σ} ⊗_{Λ(G)} C is acyclic. For each such C we write χ(C) for the inverse of the element of K_{0}(Λ(G), Λ(G)_{S^{∗}}) that corresponds under (2) (with R = Λ(G) and R′ = Λ(G)_{S^{∗}}) to the pair ([P^{•}], ι_{P^{•}}) with [P^{•}] the object of V(Λ(G)) associated to any P^{•} in C^{p}(Λ(G)) isomorphic to C in D^{p}(Λ(G)) and ι_{P^{•}} the morphism in V(Λ(G)_{S^{∗}}) associated to the isomorphism Λ(G)_{S^{∗}} ⊗_{Λ(G)} P^{•} ≅ Λ(G)_{S^{∗}} ˆ⊗_{Λ(G)}C ≅ 0 in D^{p}(Λ(G)_{S^{∗}}). This element χ(C) is the inverse of the Euler characteristic χ_{Λ(G),Λ(G)_{S^{∗}}} (C, t) that is defined in [3, Def. 5.5] with t equal to the isomorphism ⨁_{i∈ℤ} H^{2i}(Λ(G)_{S^{∗}} ⊗_{Λ(G)} C) ≅ 0 ≅ ⨁_{i∈ℤ} H^{2i+1}(Λ(G)_{S^{∗}} ⊗_{Λ(G)} C). (We define χ(C) in terms of the inverse in order to ensure that if G has no element of order p, then the isomorphism K_{0}(Λ(G), Λ(G)_{S^{∗}}) ≅ K_{0}(𝔐_{S^{∗}}(G)) discussed in §1.2 sends χ(C) to ∑_{i∈ℤ}(−1)^{i}[H^{i}(C)].)

1.5. Wedderburn decompositions. We fix an algebraic closure ℚ_{p}^{c} of ℚ_{p}. For any finite group G we write Irr(G) for the set of irreducible finite-dimensional ℚ_{p}^{c}-valued characters of G. Then the Wedderburn decomposition of the (finite dimensional semisimple) ℚ_{p}^{c}-algebra ℚ_{p}^{c}[G] induces a decomposition of its centre

(3)  ζ(ℚ_{p}^{c}[G]) ≅ ∏_{Irr(G)} ℚ_{p}^{c}.

The natural reduced norm map Nrd_{ℚ_{p}^{c}[G]} : K_{1}(ℚ_{p}^{c}[G]) → ζ(ℚ_{p}^{c}[G])^{×} is bijective and we often (and without explicit comment) combine this map with (3) to regard elements of ∏_{Irr(G)} ℚ_{p}^{c,×} as elements of the Whitehead group K_{1}(ℚ_{p}^{c}[G]). In particular, we write ∂_{G} : ∏_{Irr(G)} ℚ_{p}^{c,×} → K_{0}(ℤ_{p}[G], ℚ_{p}^{c}[G]) for the connecting homomorphism of relative Ktheory (normalized as in (1)). For each ρ ∈ Irr(G) we fix a minimal idempotent e_{ρ} in ℚ_{p}^{c}[G] such that the left action of G on ℚ_{p}^{c}[G] given by x ↦ xg^{−1} for g ∈ G induces an isomorphism of (left) ℚ_{p}^{c}[G]-modules e_{ρ}ℚ_{p}^{c}[G] ≅ V_{ρ^{∗}} where V_{ρ^{∗}} ≅ (ℚ_{p}^{c})^{n_{ρ}} is the representation space of the contragredient ρ^{∗} of ρ over ℚ_{p}^{c}. Then for each complex C in D^{p}(ℤ_{p}[G]) the theory of Morita equivalence induces an identification of morphism groups

(4) Mor_{V (ℚ_{p}^{c}[G])}(d_{ℚ_{p}^{c}[G]}(ℚ_{p}^{c}[G] ⊗_{ℤ_{p}[G]}^{𝕃} C), 1_{ℚ_{p}^{c}[G]})

≅ ∏_{Irr(G)} Mor_{V (ℚ^{c})}(d_{ℚ_{p}^{c}}(e_{ρ}ℚ_{p}^{c}[G] ⊗_{ℤ_{p}[G]}^{𝕃} C), 1_{ℚ_{p}^{c}}).

Details of the ‘non-commutative determinants’ d_{ℚ_{p}^{c}[G]}(−) and d_{ℚ_{p}^{c}}(−) that are used here are recalled in Appendix A.

2. Statement of the main results in Part I

The first main result we prove in Part I is the following decomposition theorem for Whitehead groups.

Theorem 2.1. If G has no element of order p, then there is a natural isomorphism of abelian groups

K_{0}(Ω(G)) ⊕ K_{0}(𝔐_{S}(G)) ⊕ im(K_{1}(Λ(G)) → K_{1}(Λ(G)_{S^{∗}})) ≅ K_{1}(Λ(G)_{S^{∗}}).

Our proof of Theorem 2.1 will show that if G = Γ, then the above isomorphism reduces to the assertion that every element of Q(Γ)^{×} can be written uniquely in the from p^{m}du where m is an integer, d is a quotient of distinguished polynomials and u a unit in Λ(Γ) (see Remark 4.2 and §4.3). Theorem 2.1 is therefore a natural generalisation of the classical Weierstrass Preparation Theorem. (For an alternative approach to generalising the latter result see [32]). In the remainder of Part I we apply the decomposition in Theorem 2.1 to help resolve the ‘descent problem’ in non-commutative Iwasawa theory. Before stating our main result in this regard we recall that for each Artin representation ρ : G → GL_{n}(O) the ring homomorphism Λ(G)_{S^{∗}} → M_{n}(Q(Γ)) that sends each element g of G to ρ(g)π_{Γ}(g) induces a homomorphism of groups

(5) Φ_{ρ} : K_{1}(Λ(G)_{S^{∗}}) → K_{1}(M_{n}(Q_{O}(Γ))) ≅ K_{1}(Q_{O}(Γ)) ≅ Q_{O}(Γ)^{×} ≅ Q(O[[T]])^{×}

where the first isomorphism is induced by the theory of Morita equivalence, the second by taking determinants (over Q_{O}(Γ)) and the third by the identification γ − 1 = T. The ‘leading term’ ξ^{∗}(ρ) at ρ of an element ξ of K_{1}(Λ(G)_{S^{∗}}) is then defined to be the leading term at T = 0 of the power series Φ_{ρ}(ξ) (this definition can also be interpreted as a leading term at zero of a p-adic meromorphic function - see [11, Lem. 3.17]). The problem of descent in (non-commutative) Iwasawa theory is then the following: given an element ξ of K_{1}(Λ(G)_{S^{∗}}) and a finite quotient \overline{G} of G, can one use knowledge of the image of ξ under the connecting homomorphism ∂_{G} to give an explicit formula for the image of (ξ^{∗}(ρ))_{ρ∈Irr(\overline{G})} under the connecting homomorphism ∂\overline{G}? This has been known for some time to be an important and delicate problem. To state our result we set

(6)  S̃ := \begin{cases}S, & if G has an element of order p, \\ S^{∗}, & otherwise. \end{cases}

Theorem 2.2. Let \overline{G} be a finite quotient of G. Let ξ be an element of K_{1}(Λ(G)_{S^{∗}}) with ∂_{G}(ξ) = χ(C) where C is a complex that belongs to Dp ; S̃(Λ(G)) and is ‘semisimple at ρ’ for each representation ρ ∈ Irr(\overline{G}) in the sense of [11, Def. 3.11]. Then in K_{0}(ℤ_{p}[\overline{G}], ℚ_{p}^{c}[\overline{G}]) one has

∂\overline{G}((ξ^{∗}(ρ))_{ρ∈Irr(\overline{G})}) = −[d_{ℤ_{p}[\overline{G}]}(ℤ_{p}[\overline{G}] ⊗_{Λ(G)}^{𝕃} C), t(C)\overline{G}]

with t(C)\overline{G} the morphism d_{ℚ_{p}^{c}[\overline{G}]}(ℚ_{p}^{c}[\overline{G}] ⊗_{Λ(G)}^{𝕃} C) → 1_{ℚ_{p}^{c}[\overline{G}]} that corresponds via (4) to

((−1)^{r_{G}(C)(ρ)}t(C_{ρ}))_{ρ∈Irr(\overline{G})} where r_{G}(C)(ρ) is the integer defined in [11, Def. 3.11] and t(C_{ρ}) the morphism d_{ℚ_{p}^{c}}(e_{ρ}ℚ_{p}^{c}[\overline{G}] ⊗_{Λ(G)}^{𝕃} C) → 1_{ℚ_{p}^{c}} defined in [11, Lem. 3.13(iv)].

Remark 2.3. The hypothesis of ‘semisimplicity at ρ’ and the definitions of r_{G}(C)(ρ) and t(C_{ρ}) are recalled explicitly in §5.2.4. However, in the special case that the complex e_{ρ}ℚ_{p}^{c}[\overline{G}] ⊗_{Λ(G)}^{𝕃} C is acyclic one knows that C is automatically semisimple at ρ, that r_{G}(C)(ρ) = 0 and that t(C_{ρ}) is simply the canonical morphism induced by property A.e) of the determinant functor described in Appendix A. In particular, if G = Γ, C = M[0] for a finitely generated torsion Λ(Γ)-module M for which both M^{Γ} and M_{Γ} are finite and ρ is the trivial character, then the equality of Theorem 2.2 is equivalent to the classical descent formula discussed in [34, p. 318, Ex. 13.12]. Upon appropriate specialisation, Theorem 2.2 also recovers the descent formalism proved (in certain commutative cases) by Greither and the first named author in [10, §8] and is therefore related to the earlier (commutative) work of Nekovář in [24, §11].

In §6 we will prove that it suffices to consider main conjectures of non-commutative Iwasawa theory in the case that G has no element of order p. Theorem 2.2 therefore represents a satisfactory resolution of the descent problem in this context. Indeed, in Part II (§6 - §8) of this article we shall combine Theorem 2.2 with the main results of [11] to describe the precise connection between main conjectures of non-commutative Iwasawa theory (in the spirit of Coates et al [13]) and the appropriate case of the equivariant Tamagawa number conjecture. Other important applications of Theorem 2.2 are described in [8].

3. Generalized μ-invariants

The key ingredient in our proof of Theorem 2.1 is the construction of canonical ‘characteristic series’ in non-commutative Iwasawa theory. In this section we prepare for this construction by generalising the classical notion of μ-invariant.

3.1. The definition. In the sequel we write μ_{Γ}(M) for the ‘μ-invariant’ of a finitely generated Λ(Γ)-module M. For each complex C in D^{p}(Λ(Γ)) we also set

μ_{Γ}(C) := ∑_{i∈ℤ}(−1)^{i}μ_{Γ}(H^{i}(C)).

Let ρ : G → GL_{n}(O) be a continuous representation of G and write E_{ρ} ≅ O^{n} for the associated representation module, where O = O_{L} denotes the ring of integers of a finite extension L of ℚ_{p}. We denote the corresponding L-linear representation L ⊗_{O} E_{ρ} by V_{ρ}. We fix a uniformising parameter π of O and denote the residue class field of O by κ. We write ρ̄ for the reduction of ρ modulo π and denote the associated representation space by \overline{E_{ρ}}. For each C in D^{p}(Λ(G)) we set C(ρ^{∗}) := O^{n} ⊗_{ℤ_{p}} C, regarded as an object of D^{p}(Λ_{O}(G)) via the action g(x ⊗_{ℤ_{p}} c^{i}) = ρ^{∗}(g)(x) ⊗_{ℤ_{p}} g(c^{i}) for each g in G, x in O^{n} and c^{i} in C^{i}. We then set

(7)  C_{ρ} := (Λ_{O}(Γ) ⊗_{O} O^{n}) ⊗_{Λ(G)}^{𝕃} C ≅ Λ_{O}(Γ) ⊗_{Λ_{O}(G)}^{𝕃} C(ρ^{∗})

and also

(8)  μ(C, ρ) := μ_{Γ}(C_{ρ}) ∈ ℤ.

3.2. Basic properties.

Lemma 3.1. Fix a continuous representation ρ : G → GL_{n}(O).

(i) If C_{1} → C_{2} → C_{3} → C_{1}[1] is an exact triangle in D_{S^{∗}}^{p}(Λ(G)), then

μ(C_{2}, ρ) = μ(C_{1}, ρ) + μ(C_{3}, ρ).

(ii) If C ∈ D_{S^{∗}}^{p}(Λ(G)) is cohomologically perfect, then

μ(C, ρ) = μ(H(C), ρ)

where H(C) denotes the complex with zero differentials and H(C)^{i} = H^{i}(C) for all i.

(iii) If C ∈ D_{S}^{p}(Λ(G)) is cohomologically perfect, then μ(C, ρ) = 0.

(iv) If U is any closed normal subgroup of G such that U ⊆ H ∩ ker(ρ), then for any C in D^{p}(Λ(G)) we have

μ(C, ρ) = μ(Λ(G/U) ⊗_{Λ(G)}^{𝕃} C, ρ).

Here the first μ-invariant is formed with respect to the group G and the second with respect to G/U.

(v) If U is any open subgroup of G, then for any continuous representation ψ : U → GL_{n}(O) and any C ∈ D^{p}(Λ(G)) one has

μ(C, Ind_{U}^{G}ψ) = μ(Res_{U}^{G}C, ψ)

where the first μ-invariant is formed with respect to G and the second with respect to U. Here Res_{U}^{G} denotes the restriction functor from Λ(G)- to Λ(U)-modules.

Proof. For each D in D_{S^{∗}}^{p}(Λ(G)) all of the Λ(Γ)-modules H^{i}(D_{ρ}) are both finitely generated and torsion. Since μ_{Γ}(−) is additive on exact sequences of finitely generated torsion Λ(Γ)-modules, claim (i) therefore follows from the long exact sequence of cohomology of the exact triangle C_{1,ρ} → C_{2,ρ} → C_{3,ρ} → C_{1,ρ}[1] in D^{p}(Λ(Γ)) that is induced by the given triangle.

If C ≅ H^{i}(C)[i] for some i, then claim (ii) is clear. The general case can then be proved by induction with respect to the cohomological length: indeed, one need only combine claim (i) together with the exact triangles given by (good) truncation. In order to prove claim (iii) it is sufficient by claim (ii) to consider the case C = M[0] with M a Λ(G)-module that is finitely generated over Λ(H). But then H^{i}(C_{ρ}) is a finitely generated ℤ_{p}-module for all i ∈ ℤ and so it is clear that μ(C, ρ) = 0. In the situation of claim (iv) there is a canonical isomorphism of Λ_{O}(G/U)-modules

Λ_{O}(G/U) ⊗_{Λ_{O}(G)}^{𝕃} C(ρ^{∗}) ≅ C(ρ^{∗})_{U} ≅ C_{U}(ρ^{∗}) ≅ (Λ(G/U) ⊗_{Λ(G)}^{𝕃} C)(ρ^{∗}),

from which the claim follows immediately. Similarly, in the situation of claim (v) we have a canonical isomorphism Ind_{U}^{G}((Res_{U}^{G}C)(ψ^{∗})) ≅ C(Ind_{U}^{G}ψ^{∗}) which corresponds to

Λ_{O}(G) ⊗_{Λ_{O}(U)} (O^{n} ⊗_{ℤ_{p}} Res_{U}^{G}C) ≅ (Λ(G) ⊗_{Λ(U)} O^{n}) ⊗_{ℤ_{p}} C,
g ⊗ (a ⊗ c) ↦ (g ⊗ a) ⊗ g(c).

Now we write Γ_{U} for the image of U in Γ under the natural projection and obtain

μ(C, Ind_{U}^{G}ψ) = μ_{Γ}(Λ_{O}(Γ) ⊗_{Λ_{O}(G)}^{𝕃} C(Ind_{U}^{G}ψ^{∗}))
= μ_{Γ}(Λ_{O}(Γ) ⊗_{Λ_{O}(G)}^{𝕃} Ind_{U}^{G}((Res_{U}^{G}C)(ψ^{∗})))
= μ_{Γ}(Λ_{O}(Γ) ⊗_{Λ_{O}(Γ_{U})} (Λ_{O}(Γ_{U}) ⊗_{Λ_{O}(U)}^{𝕃} (Res_{U}^{G}C)(ψ^{∗})))
= μ_{Γ_{U}} (Λ_{O}(Γ_{U}) ⊗_{Λ_{O}(U)}^{𝕃} (Res_{U}^{G}C)(ψ^{∗}))
= μ(Res_{U}^{G}C, ψ)

as had to be shown. □

3.3. Module theory. In order to make a closer examination of the μ-invariant defined in (8) we recall some standard module theory. We write Jac(Λ(G)) for the Jacobson radical of Λ(G) and ∏_{i∈I} A_{i} for the Wedderburn decomposition of the finite dimensional semisimple 𝔽_{p}-algebra A := A(G) := Λ(G)/Jac(Λ(G)) (so I is finite). Let R_{i} = Aa_{i} be a representative for the unique isomorphism class of simple A_{i}-modules with a_{i} some orthogonal primitive idempotent of A(G), always assuming that A_{1} = 𝔽_{p} = R_{1}; the corresponding representations we denote by ψ_{i} : G → GL(R_{i}) for i in I. For each index i we fix an idempotent e_{i} of Λ(G) which is a pre-image of a_{i} under the projection Λ(G) ↠ Λ(G)/Jac(Λ(G)). We consider the projective Λ(G)-modules X_{i} := Λ(G)e_{i} and projective Ω(G)-modules Y_{i} := X_{i}/pX_{i}. They are projective hulls of R_{i} since A(G)⊗_{Λ(G)}X_{i} = A(G)⊗_{Ω(G)}Y_{i} = R_{i}. Every finitely generated projective Λ(G)-module X, resp. Ω(G)-module Y, decomposes in a unique way as a direct sum

X = ⨁_{i∈I} X_{i}^{⟨X,X_{i}⟩}, resp. Y = ⨁_{i∈I} Y_{i}^{⟨Y,Y_{i}⟩},

for suitable natural numbers ⟨X, X_{i}⟩, resp. ⟨Y, Y_{i}⟩. We write l_{R_{i}}(ψ) for the multiplicity of the occurrence of R_{i} in a 𝔽_{p}-linear representation ψ and

χ(G, M) := ∏_{i} |H_{i}(G, M)|^{(−1)^{i}},

if this is finite, for the Euler-Poincaré-characteristic of a Λ(G)-module M.

Lemma 3.2. Let Y be a finitely generated projective Ω(G)-module.

(i) Λ(Γ) ⊗_{Λ(G)} Y is naturally isomorphic to Ω(Γ) ⊗_{Ω(G)} Y = Ω(Γ)^{<Y,Y_{1}>} and thus

χ(G, Y) = p^{⟨Y,Y_{1}⟩}.
(ii) ⟨Y (ψ^{∗}), Y_{1}⟩ = ∑_{i∈I} l_{R_{i}}(ψ) dim_{𝔽_{p}}(End_{Ω(G)}(R_{i}))⟨Y, Y_{i}⟩.

Proof. For each index i the module Ω(Γ) ⊗_{Ω(G)} Y_{i} is isomorphic to Ω(Γ)^{n_{i}} for some natural number n_{i}. Since then 𝔽_{p}^{n_{i}} ≅ 𝔽_{p} ⊗_{Ω(G)} Y_{i} ≅ A_{1} ⊗_{Ω(G)} Y_{i} = a_{1}R_{i} is isomorphic to R_{1} if i = 1 and is zero if i ≠ 1, the first claim follows. Claim (ii) is true because dim_{𝔽_{p}}(End_{Ω(G)}(R_{i}))⟨Y, Y_{i}⟩ = dim_{𝔽_{p}}(Hom_{Ω(G)}(Y, R_{i})) and Hom_{Ω(G)}(Y (ψ_{i}^{∗}), R_{1}) is isomorphic to Hom_{Ω(G)}(Y, R_{i}) (cf. [1, Prop. 4.1, Lem. 4.4]). □

3.4. The regular case. In this section we study the μ-invariants of §3.1 in the case that G has no element of order p.

3.4.1. Pairings. For a field K we write R_{K}(G) for the Grothendieck group of the category of finite-dimensional continuous K-linear representations of G which have finite image. The tensor product induces a structure of rings on both R_{L}(G) and R_{κ}(G) and there exists a canonical surjective homomorphisms of rings R_{L}(G) ↠ R_{κ}(G) that is induced by reducing modulo π any G-stable O-lattice E of an representation V of the above type (cf. [28]).

Proposition 3.3. Assume that G has no element of order p.

(i) If C ∈ D_{S^{∗}}^{p}(Λ(G)), then for each continuous representation ρ : G → GL_{n}(O) one has

μ(C, ρ) = ∑_{i∈ℤ}(−1)^{i}μ_{Γ}(Λ(Γ) ⊗_{Λ(G)}^{𝕃} (\overline{E_{ρ}}^{∗} ⊗_{𝔽_{p}} gr(H^{i}(C)_{tor})[0]))

where \overline{E_{ρ}}^{∗} denotes the contragredient module Hom_{κ}(\overline{E_{ρ}}, κ) while for a ℤ_{p}-module M endowed with the p-adic filtration we denote by gr(M) the associated graded 𝔽_{p}-module.

(ii) The μ-invariant induces a ℤ-bilinear pairing

μ(−, −) : K_{0}(D_{S^{∗}}^{p}(Λ(G))) × R_{L}(G) → ℤ.

This pairing induces a ℤ-bilinear pairing of the form

μ(−, −) : K_{0}(D_{S^{∗}}^{p}(Λ(G))) × R_{κ}(G) → ℤ.

(iii) If C ∈ D_{S^{∗}}^{p}(Λ(G)) and i ∈ I, then the integer μ(C, ψ_{i}) defined by claim (ii) is

divisible by dim_{𝔽_{p}}(End_{Ω(G)}(R_{i})). (iv) If C ∈ D_{S}^{p}(Λ(G)), then μ(C, ψ_{i}) = 0 for all i ∈ I.

Proof. To prove claim (i), we write μ′(C, ρ̄) for the term on the right hand side of the claimed equality. Since μ′(C, ρ̄) = μ′(H(C), ρ̄) by definition and μ(C, ρ) = μ(H(C), ρ) by Lemma 3.1 (ii) we need only consider the case where C ≅ M[0] with M in 𝔐_{S^{∗}}(G). Further, since both μ-invariants are additive on exact triangles (cf. Lemma 3.1(i)), it is actually sufficient to prove the following two special cases (note that M/M_{tor} belongs to 𝔐_{S}(G) for all M in 𝔐_{S∗}(G)): 1.) If M is in 𝔐_{S}(G), then both H^{i}(Λ(Γ) ⊗_{Λ(G)}^{𝕃} (\overline{E_{ρ}} ⊗_{𝔽_{p}} gr(M_{tor})[0])) and H^{i}(A_{ρ}) are finitely generated ℤ_{p}-modules and thus μ(C, ρ) = 0 = μ′(C, ρ̄). 2.) If p^{n}M = 0 for some n, we argue by induction on n. For n = 1 the isomorphism \overline{E_{ρ}}^{∗} ⊗_{𝔽_{p}} gr(M_{tor}) ≅ \overline{E_{ρ}}^{∗} ⊗_{𝔽_{p}} M ≅ \overline{E_{ρ}}^{∗} ⊗_{ℤ_{p}} M implies the equality of the μ-invariants. For n > 1 one uses dévissage and again the additivity of both μ-invariants.

To prove the existence of the first pairing in claim (ii) it suffices to show that μ(C, ρ) depends only on the space V_{ρ}. To this end we assume that E_{ρ′} is another G-stable lattice in V_{ρ} and we have to show that μ(C, ρ) = μ(C, ρ′). By Proposition 3.1(ii) and dévissage we may assume that C ≅ M[0] with pM = 0 and similarly that E_{ρ′}^{∗} ⊆ E_{ρ}^{∗} with πT = 0 for T := E_{ρ}^{∗}/E_{ρ′}^{∗}. In this situation there is an exact sequence

0 → M ⊗_{𝔽_{p}} T → M ⊗_{𝔽_{p}} \overline{E_{ρ′}^{∗}} → M ⊗_{𝔽_{p}} \overline{E_{ρ}^{∗}} → M ⊗_{𝔽_{p}} T → 0

of Λ(G)-modules. The required claim now follows from the known additivity of μinvariants and the fact that the Λ(G)-modules M ⊗_{𝔽_{p}} \overline{E_{ρ}^{∗}} and M ⊗_{𝔽_{p}} \overline{E_{ρ′}^{∗}} are isomorphic to M(ρ^{∗}) and M((ρ′)^{∗}) respectively. The second assertion of claim (ii) then follows from claim (i). To prove claim (iii) we may assume that C ≅ M[0] with M a finitely generated Ω(G)- module. After choosing a finite resolution P of M by finitely generated projective Ω(G)-modules and using the additivity of μ(−, ψ_{i}) on short exact sequences the proof is immediately reduced to the case of a projective Ω(G)-module because μ(P, ψ_{i}) = ∑_{j∈ℤ}(−1)^{j}μ(P^{j}, ψ_{i}). But for every projective Ω(G)-module Y, considered also as a Λ(G)- module, and for each i ∈ I we have μ(Y, ψ_{i}) = ⟨Y (ψ_{i}^{∗}), Y_{1}⟩ = dim_{𝔽_{p}}(End_{Ω(G)}(R_{i}))⟨Y, Y_{i}⟩ by Lemma 3.2(ii). Claim (iv) follows from Lemma 3.1(iii). □

If G has no element of order p, then Proposition 3.3(iii) allows us to define an integer μ_{Λ(G)}^{i}(C) for each complex C in D_{S^{∗}}^{p}(Λ(G)) and each index i in I by setting

μ_{Λ(G)}^{i}(C) := μ(C, ψ_{i}) · dim_{𝔽_{p}}(End_{Ω(G)}(R_{i}))^{−1}.

3.4.2. K-groups. We continue to assume that G has no element of order p and write 𝔇(G) for the category of finitely generated Λ(G)-modules that are annihilated by a power of p. Then, by dévissage and lifting of idempotents, one obtains the following isomorphisms

(9)  K_{0}(𝔇(G)) ≅ K_{0}(Ω(G)) ≅ K_{0}(A(G)) ≅ ℤ^{I}

where the i-th basis vector of the free ℤ-module on the right corresponds to the classes of Y_{i} in K_{0}(𝔇(G)) and K_{0}(Ω(G)). Lemma 3.2(ii) implies that if M belongs to 𝔇(G), then the map in (9) sends the class of M to the vector

(10)  μ(M) := (μ_{Λ(G)}^{i}(M[0]))_{i∈I}.

The proof of the following result is a natural generalization of that given by Kato in [20, Prop. 8.6].

Proposition 3.4. If G has no element of order p, then there are natural isomorphisms

K_{0}(𝔐_{S^{∗}}(G)) ≅ K_{0}(𝔐_{S}(G)) ⊕ K_{0}(Ω(G)),
K_{1}(Λ(G)_{S^{∗}}) ≅ K_{1}(Λ(G)_{S}) ⊕ K_{0}(Ω(G)).

The first of these isomorphisms is induced by the embeddings of categories 𝔐_{S}(G) ⊂ 𝔐_{S^{∗}}(G) and 𝔇(G) ⊂ 𝔐_{S^{∗}}(G) combined with the first isomorphism in (9). The second isomorphism depends on the choice of a splitting of K_{1}(Λ(G)_{S^{∗}}) ↠ K_{0}(𝔇(G)) ≅ ℤ^{I}; once we have fixed an idempotent e_{i} for each i ∈ I a natural choice is induced by sending the i-th basis vector of ℤ^{I} to the class of f_{i} := 1 + (p − 1)e_{i} in K_{1}(Λ(G)_{S^{∗}}).

Proof. We first prove the surjectivity of the homomorphism ∂_{2} in the long exact localisation sequence of K-theory

K_{2}(Λ(G)_{S^{∗}}) ⟶^{∂_{2}} K_{1}(Ω(G)_{S}) → K_{1}(Λ(G)_{S}) → K_{1}(Λ(G)_{S^{∗}}) ⟶^{∂_{1}} K_{0}(Ω(G)_{S}).

But, since Ω(G)_{S} is semi-local, the natural homomorphism Ω(G)_{S}^{∗} ↠ K_{1}(Ω(G)_{S}) is surjective and hence K_{1}(Ω(G)_{S}) is generated by the image of S. The surjectivity of ∂_{2} thus follows from the fact that for each f ∈ S one has ∂_{2}({f, p}) = [f] ∈ K_{1}(Ω(G)_{S}), where {f, p} denotes the symbol of f and p in K_{2}(Λ(G)_{S^{∗}}) (indeed, the latter equality is proved by the argument of [18, Prop. 5]). From the above exact sequence we therefore obtain an exact sequence

(11)  0 → K_{1}(Λ(G)_{S}) → K_{1}(Λ(G)_{S^{∗}}) ⟶^{∂_{1}} K_{0}(Ω(G)_{S}).

We next consider the composite map

(12)  ℤ^{I} → K_{1}(Λ(G)_{S^{∗}}) ⟶^{∂_{1}} K_{0}(Ω(G)_{S}) ⟶^{α} K_{0}(B(G)) ≅ ℤ^{J}.

Here the first map is given by sending the i-th basis vector of ℤ^{I} to the class of f_{i} = 1 + (p − 1)e_{i} (note that Λ(G)/Λ(G)f_{i} ≅ Y_{i}), we set B(G) := Ω(G)_{S}/Jac(Ω(G)_{S}), the canonical map α is injective by [2, Chap. IX, Prop. 1.3] and the index set J parametrizes the isomorphism classes of simple modules over the semisimple Artinian ring B(G). Let N be any closed normal subgroup of G which is both pro-p and open in H. Then it is straightforward to check that (12) factorizes through the composite

(13)  ℤ^{I} ≅ K_{0}(Ω(G/N)) ⟶^{β} K_{0}(Ω(G/N)_{S}) ⟶^{γ} K_{0}(B(G)).

Here the surjective map β comes from the exact localisation sequence and the isomorphism γ is induced from the fact that Ω(G)_{S} ↠ B(G) factors through the composite Ω(G)_{S} ↠ Ω(G/N)_{S} ↠ B(G) by the proof of [13, Lem. 4.3] and the fact that Jac(Ω(G/N)_{S}) is a nilpotent ideal. It follows that the map in (12) is surjective and hence that ∂_{1} is surjective and α is bijective. If 𝔇_{S}(G) denotes the category of finitely generated Λ(G)_{S}-modules which are ℤ_{p}-torsion, then we have shown that the composite map

(14)  K_{0}(𝔇_{S}(G)) ≅ K_{0}(Ω(G)_{S}) ≅ K_{0}(B(G)) = ℤ^{J}

is bijective and that |J| ≤ |I|. By combining (11) with the surjectivity of ∂_{1}, the bijectivity of (14) and the assertion of Lemma 3.5(i) below we obtain an exact sequence

(15)  0 → K_{1}(Λ(G)_{S}) → K_{1}(Λ(G)_{S^{∗}}) ⟶^{∂′_{1}} K_{0}(𝔇(G)) → 0.

Further, it is straightforward to show that, with respect to the isomorphism K_{0}(𝔇(G)) ≅ ℤ^{I} of (9), this sequence is split by the map which sends the i-th basis vector of ℤ^{I} to the class of f_{i} in K_{1}(Λ(G)_{S^{∗}}). This proves the final assertion of Proposition 3.4. We next consider the following diagram with exact rows

(16)  \begin{aligned}0 & ⟶ & im(ι_{S^{∗}}) & ⟶ & K_{1}(Λ(G)_{S}) & ⟶ & K_{0}(𝔐_{S}(G)) & ⟶ & 0 \\ & & ∥ & & ↓ & & ↓^{δ} \\ 0 & ⟶ & im(ι_{S^{∗}}) & ⟶ & K_{1}(Λ(G)_{S^{∗}}) & ⟶ & K_{0}(𝔐_{S^{∗}}(G)) & ⟶ & 0, \end{aligned}

where ι_{S^{∗}} is the natural map K_{1}(Λ(G)) → K_{1}(Λ(G)_{S^{∗}}), δ is induced by the embedding of categories 𝔐_{S}(G) ⊂ 𝔐_{S^{∗}}(G) and [13, Prop. 3.4] implies that each row is indeed exact.

By applying the snake lemma to this diagram and comparing with the sequence (15) we obtain an exact sequence of the form

0 → K_{0}(𝔐_{S}(G)) ⟶^{δ} K_{0}(𝔐_{S^{∗}}(G)) → K_{0}(𝔇(G)) → 0.

The first assertion of Proposition 3.4 now follows because this sequence is split by the homomorphism K_{0}(𝔇(G)) → K_{0}(𝔐_{S^{∗}}(G)) that is induced by the embedding of categories 𝔇(G) ⊂ 𝔐_{S^{∗}}(G). □

Lemma 3.5.

(i) The exact scalar extension functor from Λ(G)-mod to Λ(G)_{S}-mod identifies 𝔇(G) with a full subcategory of 𝔇_{S}(G) and induces an isomorphism K_{0}(𝔇(G)) ≅ K_{0}(𝔇_{S}(G)).

(ii) The natural map ι : K_{0}(𝔇(G)) → K_{0}(𝔐_{S^{∗}}(G)) is injective.

(iii) The natural map K_{0}(Ω(G/N)) → K_{0}(Ω(G/N)_{S}) is bijective.

Proof. The assignment M ↦ (μ_{Λ(G)}^{i}(M[0]))_{i∈I} induces a homomorphism

μ : K_{0}(𝔐_{S^{∗}}(G)) → ℤ^{I}.

Now from (9) and (10) we know that μ∘ι is bijective whilst from Lemma 3.1(iii) we know δ(K_{0}(𝔐_{S}(G))) ⊆ ker(μ) where δ is the homomorphism in diagram (16). This implies that ι is injective (so proving claim (ii)), that μ is surjective and that |I| ≤ |J|. But |J| ≤ |I| (see just after (14)) and so |I| = |J|. Since |I| = |J| the isomorphisms of (9) and (13) combine to imply that the natural map K_{0}(𝔇(G)) → K_{0}(𝔇_{S}(G)) is bijective (proving claim (i)). In a similar way, claim (iii) follows by combining the equality |I| = |J| together with the surjectivity of the map β in (13) and the definition of the index set J (in (12)). □

4. Characteristic series

In this section we associate a canonical ‘characteristic series’ to each complex in Dp ; S̃(Λ(G)). This construction extends the notion of ‘algebraic p-adic L-functions’ introduced in [6] and hence refines the ‘Akashi series’ introduced by Coates, Schneider and Sujatha in [12]. It will also play a key role in our proof of Theorem 2.1 (see in particular the proof of Lemma 5.7).

4.1. The definition. If M is any compact (left) Λ(G)-module, then the completed tensor product

I_{H}^{G}(M) := Λ(G) ˆ⊗_{Λ(H)}Res_{H}^{G}(M)

has a natural structure as a compact Λ(G)-module via multiplication on the left. With respect to this action, one obtains a (well-defined) endomorphism Δ_{γ} of I_{H}^{G}(M) by setting

Δ_{γ}(x ⊗_{Λ(H)} y) := xγ̃^{−1} ⊗_{Λ(H)} γ̃(y)

for each x ∈ Λ(G) and y ∈ M, where γ̃ is any lift of γ through the natural projection G → Γ. It is easily checked that Δ_{γ} is independent of the precise choice of γ̃. Further, if M belongs to 𝔐_{S^{∗}}(G), then [6, Lem. 2.1] implies that

δ_{γ} := id_{I_{H}^{G}(M)} − Δ_{γ}

induces an automorphism of the (finitely generated) Λ(G)_{S^{∗}}-module I_{H}^{G}(M)_{S^{∗}}. Now the ring Λ(G)_{S^{∗}} is both noetherian and regular [17, Prop. 4.3.4] and so K_{1}(Λ(G)_{S^{∗}}) is naturally isomorphic to the group G_{1}(Λ(G)_{S^{∗}}) that is generated (multiplicatively) by symbols ⟨α | M⟩ where α is an automorphism of a finitely generated Λ(G)_{S^{∗}}-module M (cf. [29, Th. 16.11]). For each complex C in D_{S^{∗}}^{p}(Λ(G)) we may therefore define an element of K_{1}(Λ(G)_{S^{∗}}) by setting

char_{G,γ}^{∗}(C) := ∏_{i∈ℤ}⟨δ_{γ} | I_{H}^{G}(H^{i}(C))_{S^{∗}}⟩^{(−1)^{i}}.

For each C in Dp ; S̃(Λ(G)) we also define an ‘equivariant multiplicative μ-invariant’ in im(K_{1}(Λ(G)[\frac{1}{p}]) → K_{1}(Λ(G)_{S^{∗}})) by setting

χ_{G}^{μ}(C) := \begin{cases}⟨∑_{i∈I} p^{μ_{Λ(G)}^{i}(C)}e_{i}|Λ(G)_{S^{∗}}⟩, & if C belongs to Dp ; S̃(Λ(G)) \ D_{S}^{p}(Λ(G)), \\ 1, & if C belongs to D_{S}^{p}(Λ(G)) \end{cases}

where the integer μ_{Λ(G)}^{i}(C) is as defined at the end of §3.4.1 (this definition makes sense because if C belongs to Dp ; S̃(Λ(G)) \ D_{S}^{p}(Λ(G)), then G has no element of order p).

Definition 4.1. For each C in Dp ; S̃(Λ(G)) the characteristic series of C is the element

char_{G,γ}(C) := χ_{G}^{μ}(C) · char_{G,γ}^{∗}(C)

of K_{1}(Λ(G)_{S^{∗}}).

Remark 4.2. If G = Γ, then Λ(G)_{S^{∗}} = Q(Γ) and so there is a natural isomorphism ι : K_{1}(Λ(G)_{S^{∗}}) ≅ Q(Γ)^{×}. Further, if M is any finitely generated torsion Λ(Γ)-module, then ι(char_{G,γ}(M[0])) = (1 + T)^{−λ(M)}char_{T} (M) where λ(M) is the Iwasawa λ-invariant of M and char_{T} (M) is the characteristic polynomial of M with respect to the variable T = γ − 1. (For a proof of this fact see [6, Lem. 2.3]).

Remark 4.3. In Proposition 4.7(i) we will prove that if G has no element of order p, then char_{G,γ}(C) is a ‘characteristic element for C’ in the sense of [13, (33)] (and see also Remark 6.2 in this regard). In [6, Th. 4.1] it is proved that this is also true if G has rank one as a p-adic Lie group. In these cases at least, we may therefore regard char_{G,γ}(C) as the canonical ‘algebraic p-adic L-function’ associated to C.

4.2. Basic properties.

Lemma 4.4. If C_{1} → C_{2} → C_{3} → C_{1}[1] is an exact triangle in Dp ; S̃(Λ(G)), then char_{G,γ}(C_{2}) = char_{G,γ}(C_{1})char_{G,γ}(C_{3}).

Proof. If G has an element of order p, then each complex C_{j} belongs to D_{S}^{p}(Λ(G)) and so χ_{G}^{μ}(C_{j}) = 1. If G has no element of order p, then the equality χ_{G}^{μ}(C_{2}) = χ_{G}^{μ}(C_{1})χ_{G}^{μ}(C_{3}) follows from Lemma 3.1(i) and Proposition 3.3(iv). The equality char_{G,γ}^{∗}(C_{2}) = char_{G,γ}^{∗}(C_{1})char_{G,γ}^{∗}(C_{3}) is equivalent to that of [6, Prop. 3.1]. □

Let U be a closed subgroup of H that is normal in G and set G_{1} := G/U, H_{1} := H/U and S_{1} := S_{G_{1},H_{1}}. Then there exists a natural ring homomorphism π_{G_{1}} : Λ(G)_{S^{∗}} → Λ(G_{1})_{S_{1}^{∗}} and hence an induced homomorphism of groups

π_{G_{1},∗} : K_{1}(Λ(G)_{S^{∗}}) → K_{1}(Λ(G_{1})_{S_{1}^{∗}}).

Lemma 4.5. Let G_{1}, H_{1} and S_{1} be as above. Fix C in Dp ; S̃(Λ(G)) and assume that either

C belongs to D_{S}^{p}(Λ(G)) or that G_{1} has no element of order p. Then C_{1} := Λ(G_{1}) ˆ⊗_{Λ(G)}^{𝕃}C belongs to Dp ; S̃_{1}(Λ(G_{1})) and π_{G_{1},∗}(char_{G,γ}(C)) = char_{G_{1},γ}(C_{1}).

Proof. If C belongs to D_{S}^{p}(Λ(G)), then C_{1} belongs to D_{S_{1}}^{p}(Λ(G_{1})) and so π_{G_{1},∗}(χ_{G}^{μ}(C)) = π_{G_{1},∗}(1) = 1 = χ_{G_{1}}^{μ}(C_{1}). If C does not belong to D_{S}^{p}(Λ(G)), then neither G or G_{1} has an element of order p and the equality π_{G_{1},∗}(χ_{G}^{μ}(C)) = χ_{G_{1}}^{μ}(C_{1}) follows from Lemma 3.1(iv) and Proposition 3.3(iv). The equality π_{G_{1},∗}(char_{G,γ}^{∗}(C)) = char_{G_{1},γ}^{∗}(C_{1}) is equivalent to that of [6, Prop. 3.2]. □

In the next result we fix an open subgroup U of G and set H_{U} := H ∩U and Γ_{U} := U/H_{U}. We use the natural isomorphism Γ_{U} ≅ HU/H to regard Γ_{U} as an open subgroup of Γ, we set d_{U} := [Γ : Γ_{U}] and write γ_{U} for the topological generator γ^{d_{U}} of Γ_{U}. We set S_{U} := S_{U,H_{U}} and note that Λ(G), resp. Λ(G)_{S}, resp. Λ(G)_{S^{∗}} is a free Λ(U)-, resp. Λ(U)_{S_{U}} -, resp. Λ(U)_{S_{U}^{∗}} -module, of rank [G : U]. In particular, restriction of scalars gives natural functors D_{S}^{p}(Λ(G)) → D_{S_{U}}^{p} (Λ(U)) and D_{S^{∗}}^{p}(Λ(G)) → D_{S_{U}^{∗}}^{p} (Λ(U)) and a natural homomorphism

res_{U,∗} : K_{1}(Λ(G)_{S^{∗}}) → K_{1}(Λ(U)_{S_{U}^{∗}}).

Lemma 4.6. Let G and U be as above and fix C in Dp ; S̃(Λ(G)). Then C_{1} := res_{U}^{G}C belongs to Dp ; S̃(Λ(G)) and res_{U,∗}(char_{G,γ}(C)) = char_{U,γ_{U}} (C_{1}).

Proof. We prove first that res_{U,∗}(χ_{G}^{μ}(C)) = χ_{U}^{μ}(C_{1}). The complex C belongs to D_{S}^{p}(Λ(G)) if and only if C_{1} belongs to D_{S_{U}}^{p} (Λ(U)) and in this case res_{U,∗}(χ_{G}^{μ}(C)) = 1 = χ_{U}^{μ}(C_{1}). We may therefore assume that C belongs to D_{S^{∗}}^{p}(Λ(G)) \ D_{S}^{p}(Λ(G)) and hence that G (and also U) has no element of order p. By the same argument as used in the proof of Proposition 3.3(iii), we can also assume that C = Y_{a}[0] for some index a in I. Then for each i ∈ I one has μ_{Λ(G)}^{i}(C) = ⟨Y_{a}, Y_{i}⟩ = δ_{ai} and so

res_{U,∗}(χ_{G}^{μ}(C)) = res_{U,∗}(⟨pe_{a}|Λ(G)_{S^{∗}}⟩) = res_{U,∗}(⟨p|Λ(G)_{S^{∗}}e_{a}⟩).

We write {f_{j} : j ∈ J} for the idempotents of Λ(U) that are analogous to the idempotents e_{i} of Λ(G) defined in §3.3 and {X(U)_{j} : j ∈ J}, resp. {Y (U)_{j} : j ∈ J}, for the submodules of Λ(U), resp. Ω(U), analogous to the modules X_{i}, resp. Y_{i}, in §3.3. For each j in J we set m_{j} := ⟨Λ(G)e_{a}, X(U)_{j}⟩ = ⟨Y_{a}, Y (U)_{j}⟩. Then the Λ(U)_{S_{U}^{∗}} -module Λ(G)_{S^{∗}}e_{a} is isomorphic to ⨁_{j∈J}(Λ(U)_{S_{U}^{∗}} f_{j})^{m_{j}} and hence the last displayed expression is equal to ⟨∑_{j∈J} p^{m_{j}}f_{j}|Λ(U)_{S_{U}^{∗}} ⟩ = χ_{U}^{μ}(res_{U}^{G}Y_{a}[0]), as required. It remains to prove that res_{U,∗}(char_{G,γ}^{∗}(C)) = char_{U,γ_{U}}^{∗} (C_{1}). To do this we may assume that C = M[0] for a module M in 𝔐_{S^{∗}}(G) so that char_{G,γ}^{∗}(C) is equal to ⟨δ_{γ} | I_{H}^{G}(M)_{S^{∗}}⟩. Now the Λ(U)_{S_{U}^{∗}} -module I_{H}^{G}(M)_{S^{∗}} is equal to the direct sum ⨁_{i=0}^{i=d_{U} −1} Δ_{γ^{i}}(I_{H_{U}}^{U} (M)_{S_{U}^{∗}}) ≅ ⨁_{i=0}^{i=d_{U} −1} I_{H_{U}}^{U} (M)_{S_{U}^{∗}} where the isomorphism identifies each translate Δ_{γ^{i}}(I_{H_{U}}^{U} (M)_{S_{U}^{∗}}) with I_{H_{U}}^{U} (M)_{S_{U}^{∗}} in the natural way. With respect to this decomposition δ_{γ} is the automorphism given by the d_{U} × d_{U} matrix

(\begin{matrix}id & −id & 0 & . . . . . . . . . & 0 \\ 0 & id & −id & 0 & . . . . . & 0 \\ 0 & 0 & id & −id & 0 & … & 0 \\ . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\ 0 & . . . . . . . . . . . & 0 & id & −id \\ −Δ_{γ^{d_{U}}} & 0 & . . . . . . . . . & 0 & id \end{matrix}) .

Elementary row and column operations show that this automorphism represents the same element of K_{1}(Λ(U)_{S_{U}^{∗}}) as does the automorphism α of ⨁_{i=0}^{i=d_{U} −1} I_{H_{U}}^{U} (M)_{S_{U}^{∗}} that acts as id − Δ_{γ^{d_{U}}} on the last direct summand and as the identity on all other summands. The required result thus follows because, since δ_{γ_{U}} := id−Δ_{γ^{d_{U}}}, the class of α in K_{1}(Λ(U)_{S_{U}^{∗}}) is equal to ⟨δ_{γ_{U}} | I_{H_{U}}^{U} (M)_{S_{U}^{∗}} ⟩ =: char_{U,γ_{U}}^{∗} (C_{1}). □

4.3. The proof of Theorem 2.1. We deduce Theorem 2.1 as a consequence of the following result.

Proposition 4.7. Assume that G has no element of order p.

(i) For each C in D_{S^{∗}}^{p}(Λ(G)) one has ∂_{G}(char_{G,γ}(C)) = ∑_{i∈ℤ}(−1)^{i+1}[H^{i}(C)]. (ii) There exists a (unique) homomorphism χ_{G,γ} from K_{0}(𝔐_{S^{∗}}(G)) to K_{1}(Λ(G)_{S^{∗}})

that simultaneously satisfies the following conditions:-

(a) for each M in 𝔐_{S^{∗}}(G) one has χ_{G,γ}([M]) = char_{G,γ}(M[1]);

(b) χ_{G,γ} is right inverse to ∂_{G};

(c) χ_{G,γ} respects the isomorphisms of Proposition 3.4;

(d) Let U be a closed subgroup of H that is normal in G and such that \overline{G} := G/U has no element of order p. Set \overline{H} := H/U and \overline{S} := S_{\overline{G},\overline{H}}. Then there is a commutative diagram

K_{0}(𝔐_{S^{∗}}(G)) ⟶^{χ_{G,γ}} K_{1}(Λ(G)_{S^{∗}})
↓  ↓
K_{0}(𝔐\overline{S}^{∗}(\overline{G})) ⟶^{χ_{\overline{G},γ}} K_{1}(Λ(\overline{G})\overline{S}^{∗})

where the vertical arrows are the natural homomorphisms.

Remark 4.8. Proposition 4.7(i) shows that char_{G,γ}(C) is a ‘characteristic element for C’ in the sense of [13, (33)]. The surjectivity of ∂_{G} (which follows directly from Proposition 4.7(ii)(b)) was first proved in [13, Prop. 3.4].

The proof of Proposition 4.7 will be the subject of §4.4. However, we now show that it implies Theorem 2.1. To do this we consider the map

ι_{G} : K_{0}(Ω(G)) ⊕ K_{0}(𝔐_{S}(G)) ⊕ im(K_{1}(Λ(G)) → K_{1}(Λ(G)_{S^{∗}})) → K_{1}(Λ(G)_{S^{∗}})

which for each M in Ω(G), N in 𝔐_{S}(G) and u in im(K_{1}(Λ(G)) → K_{1}(Λ(G)_{S^{∗}})) satisfies ι_{G}(([M], [N], u)) = char_{G,γ}(M[1])char_{G,γ}(N[1])u. Then Proposition 4.7(ii) implies that ι_{G} is a well-defined homomorphism which, upon restriction to the summand K_{0}(Ω(G))⊕ K_{0}(𝔐_{S}(G)), gives a right inverse to the composite K_{1}(Λ(G)_{S^{∗}}) → K_{0}(𝔐_{S^{∗}}(G)) → K_{0}(Ω(G)) ⊕ K_{0}(𝔐_{S}(G)) where the first arrow is ∂_{G} and the second is the isomorphism of Proposition 3.4. The exactness of the lower row of (16) thus implies that ι_{G} is bijective. This completes the proof of Theorem 2.1.

Remark 4.9. The characteristic series for M in 𝔐_{S^{∗}}(G) and hence also the splitting χ_{G,γ} of ∂_{G} can of course be defined just in terms of modules instead of derived categories. Thus for the proof of Theorem 2.1 one can probably avoid the use of derived categories. However, since our applications involve complexes we prefer to use this language from the outset.

4.4. The proof of Proposition 4.7. In addition to proving Proposition 4.7 we shall also translate Definition 4.1 into the language of localized K_{1}-groups introduced by Fukaya and Kato in [17]. We therefore use the notation of Appendix A.

4.4.1. S-acyclic complexes. In [26, Prop. 2.2, Rem. 2.3] Schneider and the second named author have proved that for each bounded complex P of projective Λ(G)-modules in D_{S}^{p}(Λ(G)) there exists an exact sequence of complexes in D^{p}(Λ(G))

(17)  0 → I_{H}^{G}(P) ⟶^{δ} I_{H}^{G}(P) ⟶^{π} P → 0

where in each degree i the morphism δ, respectively π, is equal to the homomorphism δ_{γ} : I_{H}^{G}(P^{i}) → I_{H}^{G}(P^{i}), respectively to the natural projection I_{H}^{G}(P^{i}) → P^{i} . We may therefore define a trivialization

t_{S}(P) : 1_{Λ(G)} → d_{Λ(G)}(I_{H}^{G}(P))d_{Λ(G)}(I_{H}^{G}(P))^{−1} → d_{Λ(G)}(P)

where the first arrow is induced by the identity map on I_{H}^{G}(P) and the second by applying property A.d) to the exact sequence (17). By using property A.g) of the functor d_{Λ(G)} we then extend this definition to obtain for any object C of D_{S}^{p}(Λ(G)) a canonical morphism

t_{S}(C) : 1_{Λ(G)} → d_{Λ(G)}(C).

We remark that this morphism is analogous to those that arise naturally in the context of varieties over finite fields (cf. [19, Lem. 3.5.8], [5, §3.2]). In the following result we use the homomorphism ch_{Λ(G),Σ_{C}} defined in Appendix A.

Lemma 4.10. For each C in D_{S}^{p}(Λ(G)) one has ch_{Λ(G),Σ_{C}} ([C, t_{S}(C)]) = char_{G,γ}^{∗}(C).

Proof. One has ch_{Λ(G),Σ_{C}} ([C, t_{S}(C)]) = θ_{C,t_{S}(C)} where the latter element is as defined in Appendix A. To compute θ_{C,t_{S}(C)} explicitly we set Q := Λ(G)_{S^{∗}}, C_{Q} := Q ⊗_{Λ(G)} C, H(C)_{Q} := Q ⊗_{Λ(G)} H(C) and H(C)_{H,Q} := Q ⊗_{Λ(H)} H(C) and consider the diagram

1_{Q} ⟶^{α_{1}} d_{Q}(H(C)_{H,Q})d_{Q}(H(C)_{H,Q})^{−1} ⟶^{α_{2}} d_{Q}(H(C)_{Q}) ⟶^{α_{3}} d_{Q}(C_{Q})
α_{5}↓  ∥  ∥
1_{Q} ⟶^{α_{4}} d_{Q}(H(C)_{H,Q})d_{Q}(H(C)_{H,Q})^{−1} ⟶^{α_{2}} d_{Q}(H(C)_{Q}) ⟶^{α_{3}} d_{Q}(C_{Q}).

In this diagram α_{1} is induced by the identity map on H(C)_{H,Q}, α_{2} results from applying property A.d) to (17) with C = H(C), α_{3} is property A.h), α_{4} is induced by d_{Q}(H(δ_{γ})) and α_{5} is defined so that the first square commutes. The upper row of the diagram is equal to the morphism 1_{Q} → d_{Q}(C_{Q}) induced by t_{S}(C) whilst the lower row agrees with the morphism 1_{Q} → d_{Q}(C_{Q}) induced by the acyclicity of C_{Q}. From the commutativity of the diagram we thus deduce that the element θ_{C,t_{S}(C)} of K_{1}(Q) is represented by α_{5}. On the other hand, a comparison of the maps α_{1} and α_{4} shows that α_{5} represents the same element of K_{1}(Q) as does the morphism

d_{Q}(H(C)_{H,Q}) ≅ ∏_{i∈ℤ} d_{Q}(H^{i}(C)_{H,Q})^{(−1)^{i}} → ∏_{i∈ℤ} d_{Q}(H^{i}(C)_{H,Q})^{(−1)^{i}} ≅ d_{Q}(H(C)_{H,Q})

where the first and third maps use property A.h) and the second map is ∏_{i∈ℤ} d_{Q}(H^{i}(δ_{γ}))^{(−1)^{i}}. From here we deduce the required equality

θ_{H(C),t_{S}(H(C))} = ∏_{i∈ℤ}⟨δ_{γ} | H^{i}(C)_{H,Q}⟩^{(−1)^{i}} ∈ K_{1}(Q).

□

4.4.2. p-torsion complexes. In this subsection we assume that G has no element of order p (so that D^{p}(Λ(G)) identifies with D^{fg}(Λ(G))). If T is a bounded complex of finitely generated Ω(G)-modules, then there is a bounded complex of finitely generated projective Ω(G)-modules P̄ that is isomorphic in D(Ω(G)) (and hence also in D^{p}(Λ(G))) to T. Also, following the discussion of §3.3, in each degree i there is a finitely generated projective Λ(G)-module P^{i} and an exact sequence of Λ(G)- modules

(18)  0 → P^{i} ⟶^{p} P^{i} → P̄^{i} → 0.

We may therefore define a morphism

t(T) : 1_{Λ(G)} → ∏_{i∈ℤ}(d_{Λ(G)}(P^{i})d_{Λ(G)}(P^{i})^{−1})^{(−1)^{i}} →

∏_{i∈ℤ} d_{Λ(G)}(P̄^{i})^{(−1)^{i}} = d_{Λ(G)}(P̄) → d_{Λ(G)}(T)

where the first arrow is induced by the identity map on each module P^{i}, the second by applying property A.d) to each of the sequences (18) and the last by the given quasiisomorphism P̄ ≅ T. If now C is any bounded complex of modules in 𝔇(G), then there exists a finite length filtration of C by complexes

(19)  0 = C_{d} ⊂ C_{d−1} ⊂ ⋯ C_{1} ⊂ C_{0} = C

so that each quotient complex T_{i} := C_{i}/C_{i+1} belongs to D^{p}(Ω(G)) (and hence to D^{p}(Λ(G))). This gives an identification d_{Λ(G)}(C) = ∏_{0≤i<d} d_{Λ(G)}(T_{i}) and, with respect to this identification, we set

t(C) := ∏_{0≤i<d} t(T_{i}).

This definition is easily checked to be independent of the choice of filtration (19) and, for each i, of isomorphism P̄ ≅ T_{i} and resolution (18) used to define t(T_{i}).

Lemma 4.11. If G has no element of order p and C is any bounded complex of modules in 𝔇(G), then in K_{1}(Λ(G)_{S^{∗}}) one has ch_{Λ(G),Σ_{S^{∗}}} ([C, t(C)]) = χ_{G}^{μ}(C).

Proof. This follows from the definition of ch_{Λ(G),Σ_{S^{∗}}} and the fact that there is a resolution

(18) of the form 0 → Λ(G)^{n} ⟶^{d^{j}} Λ(G)^{n} → P̄^{j} → 0 where n = ∑_{i=1}^{i=c}⟨P̄^{j}, X̄_{i}⟩ and d^{j} is given with respect to the canonical basis by the diagonal matrix with entries f_{i}^{⟨P̄^{j},X̄_{i}⟩}, 1 ≤ i ≤ n, where the natural numbers ⟨P̄^{j}, X̄_{i}⟩ are defined via the decomposition P̄^{j} ≅ ⨁_{i=1}^{i=c} X̄_{i}^{⟨P̄^{j},X̄_{i}⟩}. The fact that the μ-invariants give the correct multiplicities is the same as for [1, Prop. 4.8]. □

4.4.3. The proof of Proposition 4.7. For each complex C in Σ_{S^{∗}} we write H(C)_{tor} and H(C)_{tf} for the complexes with H(C)_{tor}^{i} = H^{i}(C)_{tor} and H(C)_{tf}^{i} = H^{i}(C)_{tf} in each degree i and in which all differentials are zero. There is a tautological exact sequence of complexes 0 → H(C)_{tor} → H(C) → H(C)_{tf} → 0 and hence an equality χ(C) = χ(H(C)) = χ(H(C)_{tor}) + χ(H(C)_{tf}) in K_{0}(𝔐_{S^{∗}}(G)). From Definition 4.1 it is also clear that χ_{G}^{μ}(C) = χ_{G}^{μ}(H(C)_{tor}) and char_{G,γ}^{∗}(C) = char_{G,γ}^{∗}(H(C)_{tf}). Claim (i) therefore follows upon combining Lemmas 4.10 and 4.11 (with C replaced by H(C)_{tf} and H(C)_{tor} respectively) with the following fact: there is a commutative diagram of homomorphisms of abelian groups

K_{1}(Λ(G), Σ_{S^{∗}}) ⟶^{∂′} K_{0}(Σ_{S^{∗}})
ch^{Λ(G),Σ_{S^{∗}}} ↓  ^{ι}↓
K_{1}(Λ(G)_{S^{∗}}) ⟶^{∂_{G}} K_{0}(𝔐_{S^{∗}}(G))

where ∂′ sends each class [C, a] to −[[C]] (cf. [17, Th. 1.3.15]) and ι sends each class [[C]] to χ(C) = ∑_{i∈ℤ}(−1)^{i}[H^{i}(C)] (cf. [17, §4.3.3]). Regarding claim (ii) we note first that if a homomorphism χ_{G,γ} exists satisfying property (a), then it is automatically unique. Next we note that Lemma 4.4 implies the assignment M ↦ char_{G,γ}(M[1]) for each M in 𝔐_{S^{∗}}(G) induces a well-defined homomorphism χ_{G,γ} : K_{0}(𝔐_{S^{∗}}(G)) → K_{1}(Λ(G)_{S^{∗}}) and claim (i) implies that this homomorphism is a right inverse to ∂_{G}. Further, for each M in 𝔇(G) and N in 𝔐_{S}(G) one has char_{G,γ}(M[1]) = χ_{G,γ}^{μ}(M[1]) ∈ ι(K_{0}(Ω(G))) and char_{G,γ}(N[1]) = char_{G,γ}^{∗}(N[1]) ∈ ι(K_{0}(𝔐_{S}(G))) (where the latter equality follows from Lemma 3.1(iii)) and so property (c) is satisfied. Finally, the commutativity of the diagram in (d) is a direct consequence of Lemma 4.5.

5. Descent theory

In this section we shall consider leading terms of elements of K_{1}(Λ(G)_{S^{∗}}) and in particular prove Theorem 2.2. The approach of this section was initially developed by the first named author in an unpublished early version of the article [6]. We deal first with the case that C is acyclic. In this case the complex ℤ_{p}[\overline{G}] ⊗_{Λ(G)}^{𝕃} C is acyclic so [d_{ℤ_{p}[\overline{G}]}(ℤ_{p}[\overline{G}] ⊗_{Λ(G)}^{𝕃} C), t(C)\overline{G}] is the zero element of K_{0}(ℤ_{p}[\overline{G}], ℚ_{p}^{c}[\overline{G}]) and also ξ belongs to the image of the natural map K_{1}(Λ(G)) → K_{1}(Λ(G)_{S^{∗}}). The equality of Theorem 2.2 is therefore a consequence of the following result.

Lemma 5.1. If u belongs to the image of the natural map λ : K_{1}(Λ(G)) → K_{1}(Λ(G)_{S^{∗}}), then for each finite quotient \overline{G} of G the element (u^{∗}(ρ))_{ρ∈Irr(\overline{G})} belongs to ker(∂\overline{G}).

Proof. Let O be the valuation ring of a finite extension L of ℚ_{p} such that all representations can be realised over O. If v ∈ K_{1}(Λ(G)) with u = λ(v), then u^{∗}(ρ) = u(ρ) = λ(v)(ρ) ∈ O^{×} for all ρ ∈ Irr(\overline{G}). Thus by functoriality of K-theory and the fact that the canonical map K_{1}(Λ_{O}(Γ)) → K_{1}(O) is equal to the ‘evaluation at 0’ map Λ_{O}(Γ)^{×} → O^{×}, the image of v in K_{1}(ℤ_{p}[\overline{G}]) under the natural projection is mapped to (u^{∗}(ρ))_{ρ∈Irr(\overline{G})} ∈ K_{1}(L[\overline{G}]). □

5.1. Reduction to S-acyclicity. We now reduce the general case of Theorem 2.2 to the case that C belongs to D_{S}^{p}(Λ(G)). In this subsection we therefore assume that G has no element of order p.

Lemma 5.2. If G has no element of order p, then it is enough to prove Theorem 2.2 in the case that C is acyclic outside at most one degree.

Proof. We assume that the result of Theorem 2.2 is true for all complexes that are acyclic outside at most one degree. To deduce Theorem 2.2 in the general case we use induction on the number of non-zero cohomology groups of C (which we assume to be at least two). We let n denote the largest integer m for which H^{m}(C) is non-zero. We set C_{1} := H^{n}(C)[−n] and write C_{2} for the truncation of C in degrees less than n (which has fewer non-zero cohomology group than does C). Then there is an exact triangle in D_{S^{∗}}^{p}(Λ(G)) of the form

(20)  C_{1} → C → C_{2} → C_{1}[1]

and the assumption that C is semisimple at ρ implies that C_{1} and C_{2} are also semisimple at ρ (for any ρ in Irr(\overline{G})). Let ξ be an element such that ∂_{G}(ξ) = χ(C). If ξ_{1} is such that ∂_{G}(ξ_{1}) = χ(C_{1}), then ξ_{2} := ξξ_{1}^{−1} satisfies ∂_{G}(ξ_{2}) = ∂_{G}(ξ) − ∂_{G}(ξ_{1}) = χ(C) − χ(C_{1}) = χ(C_{2}), where the last equality follows from (20). Hence, by the inductive hypothesis, one has

(21) ∂\overline{G}((ξ^{∗}(ρ))_{ρ} = ∂\overline{G}((ξ_{1}^{∗}(ρ))_{ρ}) + ∂\overline{G}((ξ_{2}^{∗}(ρ))_{ρ})

= −[d_{ℤ_{p}[\overline{G}]}(C_{1,\overline{G}}), t(C_{1})\overline{G}] − [d_{ℤ_{p}[\overline{G}]}(C_{2,\overline{G}}), t(C_{2})\overline{G}].

where C_{i,\overline{G}} := ℤ_{p}[\overline{G}] ⊗_{Λ(G)}^{𝕃} C_{i} for i = 1, 2. Now if we set C\overline{G} := ℤ_{p}[\overline{G}] ⊗_{Λ(G)}^{𝕃} C, then (20) induces an exact triangle in D^{p}(ℤ_{p}[\overline{G}])

(22)  C_{1,\overline{G}} → C\overline{G} → C_{2,\overline{G}} → C_{1,\overline{G}}[1]

and, with respect to this triangle, the trivialisations t(C_{1})\overline{G}, t(C)\overline{G} and t(C_{2})\overline{G} satisfy the ‘additivity criterion’ of [3, Cor. 6.6]. Indeed, for each ρ in Irr(\overline{G}) the exact triangle (20) combines with the definition (7) of each of the complexes C_{1,ρ}, C_{ρ} and C_{2,ρ} to induce an exact triangle

ℚ_{p}^{c} ⊗_{Λ_{O}(Γ)}^{𝕃} C_{1,ρ} → ℚ_{p}^{c} ⊗_{Λ_{O}(Γ)}^{𝕃} C_{ρ} → ℚ_{p}^{c} ⊗_{Λ_{O}(Γ)}^{𝕃} C_{2,ρ} → ℚ_{p}^{c} ⊗_{Λ_{O}(Γ)}^{𝕃} C_{1,ρ}[1]

and the cohomology sequence of this triangle gives an equality r_{G}(C)(ρ) = r_{G}(C_{1})(ρ) + r_{G}(C_{2})(ρ) and a short exact sequence of complexes

0 → H_{bock}(△(C_{1,ρ}, γ)) → H_{bock}(△(C_{ρ}, γ)) → H_{bock}(△(C_{2,ρ}, γ)) → 0.

Here we use the notation of Appendix B and write △(C_{ρ}, γ) for the triangle

(23)  ℚ_{p}^{c} ⊗_{O}^{𝕃} C_{ρ} ⟶^{θ_{γ,ρ}} ℚ_{p}^{c} ⊗_{O}^{𝕃} C_{ρ} → ℚ_{p}^{c} ⊗_{Λ_{O}(Γ)}^{𝕃} C_{ρ} → ℚ_{p}^{c} ⊗_{O}^{𝕃} C_{ρ}[1]

where θ_{γ,ρ} is induced by multiplication by γ − 1, and we use similar notation for C_{1} and C_{2}. This means that the criterion of [3, Cor. 6.6] is satisfied if one takes (in the notation of loc. cit.) Σ to be ℚ_{p}^{c}[\overline{G}], P ⟶^{a} Q ⟶^{b} R ⟶^{c} P [1] to be the exact triangle (22) (so ker(H^{ev}a_{Σ}) = ker(H^{od}a_{Σ}) = 0) and the trivialisations t_{P}, t_{Q} and t_{R} to be induced by (−1)^{r_{G}(C_{1})(ρ)}t(C_{1,ρ}), (−1)^{r_{G}(C)(ρ)}t(C_{ρ}) and (−1)^{r_{G}(C_{2})(ρ)}t(C_{2,ρ}) respectively. From [3, Cor. 6.6] we therefore deduce that the last element in (21) is indeed equal to −[d_{ℤ_{p}[\overline{G}]}(C\overline{G}), t(C)\overline{G}], as required. □

Taking account of Lemmas 5.1 and 5.2 we now assume that C is acyclic outside precisely one degree. To be specific, we assume that C = M[0] with M in 𝔐_{S^{∗}}(Λ(G)). Then there is an exact triangle of the form

(24)  M_{tor}[0] → M[0] → M_{tf}[0] → M_{tor}[1]

where M_{tor} belongs to 𝔇(G) and M_{tf} to 𝔐_{S}(Λ(G)). In this case one has t(M[0])\overline{G} = t(M_{tf}[0])\overline{G} and so (by another application of [3, Cor. 6.6])

(25) [d_{ℤ_{p}[\overline{G}]}(ℤ_{p}[\overline{G}] ⊗_{Λ(G)}^{𝕃} M[0]), t(M[0])\overline{G}] =

[d_{ℤ_{p}[\overline{G}]}(ℤ_{p}[\overline{G}] ⊗_{Λ(G)}^{𝕃} M_{tor}[0]), can] + [d_{ℤ_{p}[\overline{G}]}(ℤ_{p}[\overline{G}] ⊗_{Λ(G)}^{𝕃} M_{tf}[0]), t(M[0])\overline{G}] with can the canonical morphism d_{ℚ_{p}[\overline{G}]}(ℚ_{p}[\overline{G}] ⊗_{Λ(G)}^{𝕃} M_{tor}[0]) = d_{ℚ_{p}[\overline{G}]}(0) → 1_{ℚ_{p}[\overline{G}]}.

Lemma 5.3. Let N be an object of 𝔇(G). If ξ is any element of K_{1}(Λ(G)_{S^{∗}}) with ∂_{G}(ξ) = χ(N[0]), then for any finite quotient \overline{G} of G one has

∂\overline{G}((ξ^{∗}(ρ))_{ρ}) = −[d_{ℤ_{p}[\overline{G}]}(ℤ_{p}[\overline{G}] ⊗_{Λ(G)}^{𝕃} N[0]), can].

Proof. An easy reduction (using dévissage and the additivity of Euler characteristics on exact sequences in 𝔇(G)) allows us to assume that N is an object of 𝔇(G) which lies in an exact sequence of Λ(G)-modules of the form

(26)  0 → Q ⟶^{d} P → N → 0

where Q and P are both finitely generated and projective. (Indeed, it is actually enough to consider the case that Q = P = Λ(G)e_{i} for an idempotent e_{i} as in §3.3 and with d equal to multiplication by p.) To proceed we identify the subgroup K_{0}(𝔇(G)) of K_{0}(𝔐_{S^{∗}}(G)) with the group K_{0}(Λ(G), Λ(G)[\frac{1}{p}]). To be compatible with the normalisations used in §1.2 we must fix this isomorphism so that for every exact sequence (26) the element [N] of K_{0}(𝔇(G)) corresponds to the element (Q, d′, P) of K_{0}(Λ(G), Λ(G)[\frac{1}{p}]) with d′ := Λ(G)[\frac{1}{p}] ⊗_{Λ(G)} d. Now since Λ(G)[\frac{1}{p}] ⊗_{Λ(G)} N = 0 the localisation sequence of K-theory implies that any element ξ as above belongs to the image of K_{1}(Λ(G)[\frac{1}{p}]) in K_{1}(Λ(G)_{S^{∗}}). This implies in particular that ξ^{∗}(ρ) = ξ(ρ) for all ρ in Irr(\overline{G}). The natural commutative diagram of connecting homomorphisms

K_{1}(Λ(G)[\frac{1}{p}]) ⟶ K_{0}(Λ(G), Λ(G)[\frac{1}{p}])
↓  ↓
K_{1}(ℚ_{p}[\overline{G}]) ⟶^{∂\overline{G}} K_{0}(ℤ_{p}[\overline{G}], ℚ_{p}[\overline{G}])

also then implies that ∂\overline{G}((ξ^{∗}(ρ))_{ρ}) = (Q\overline{G}, d′ ; \overline{G}, P\overline{G}) with Q\overline{G} := ℤ_{p}[\overline{G}] ⊗_{Λ(G)} Q, P\overline{G} := ℤ_{p}[\overline{G}] ⊗_{Λ(G)} P and d′ ; \overline{G} := ℚ_{p}[\overline{G}] ⊗_{Λ(G)} d. Hence, with respect to the isomorphism (2) (with R = ℤ_{p}[\overline{G}] and R′ = ℚ_{p}[\overline{G}]), one has

(27)  ∂\overline{G}((ξ^{∗}(ρ))_{ρ}) = [d_{ℤ_{p}[\overline{G}]}(Q\overline{G})d_{ℤ_{p}[\overline{G}]}(P\overline{G})^{−1}, τ]

with τ equal to the composite morphism

d_{ℚ_{p}[\overline{G}]}(ℚ_{p} ⊗_{ℤ_{p}} Q\overline{G})d_{ℚ_{p}[\overline{G}]}(ℚ_{p} ⊗_{ℤ_{p}} P\overline{G})^{−1} →

d_{ℚ_{p}[\overline{G}]}(ℚ_{p} ⊗_{ℤ_{p}} P\overline{G})d_{ℚ_{p}[\overline{G}]}(ℚ_{p} ⊗_{ℤ_{p}} P\overline{G})^{−1} = 1_{ℚ_{p}[\overline{G}]}

where the first arrow is induced by d_{ℚ_{p}[\overline{G}]}(d′ ; \overline{G}). Finally we note that the (image under ℤ_{p}[\overline{G}] ⊗_{Λ(G)} − of the) sequence (26) induces an isomorphism in D^{p}(ℤ_{p}[\overline{G}]) between

ℤ_{p}[\overline{G}] ⊗_{Λ(G)}^{𝕃} N[0] and the complex Q\overline{G} ⟶^{d\overline{G}} P\overline{G} where the first term is placed in degree −1 and d\overline{G} := ℤ_{p}[\overline{G}] ⊗_{Λ(G)} d and this implies that the element on the right hand side of (27) is the inverse of [d_{ℤ_{p}[\overline{G}]}(ℤ_{p}[\overline{G}] ⊗_{Λ(G)}^{𝕃} N[0]), can], as required. □

Lemmas 5.1, 5.2 and 5.3 combine with (24) and (25) to reduce the proof of Theorem 2.2 to consideration of complexes in D_{S}^{p}(Λ(G)). In the remainder of §5 we shall therefore assume that C belongs to D_{S}^{p}(Λ(G)).

5.2. Equivariant twists. In this subsection we introduce the algebraic formalism that is key to a proper understanding of descent.

5.2.1. The definition. We fix an open normal subgroup U of G and set \overline{G} := G/U. We write

Δ\overline{G} : Λ(G) → Λ(\overline{G} × G) ≅ ℤ_{p}[\overline{G}] ⊗_{ℤ_{p}} Λ(G)

for the (flat) ring homomorphism which sends each element σ of G to \overline{σ} ⊗ σ where \overline{σ} is the image of σ in \overline{G}. Then for each Λ(G)-module M the induced Λ(\overline{G} × G)-module Λ(\overline{G} × G) ⊗_{Λ(G),Δ\overline{G}} M can be identified with the module

tw\overline{G}(M) := ℤ_{p}[\overline{G}] ⊗_{ℤ_{p}} M

upon which \overline{G} acts via left multiplication and each σ ∈ G acts by sending x ⊗ y to x\overline{σ}^{−1} ⊗ σ(y). This construction extends to give an exact functor C ↦ tw\overline{G}(C) from D^{p}(Λ(G)) to D^{p}(Λ(\overline{G} × G)) and for each such C we set

tw\overline{G}(C)_{H} := Λ(\overline{G} × Γ) ⊗_{Λ(\overline{G}×G)}^{𝕃} tw\overline{G}(C) ∈ D^{p}(Λ(\overline{G} × Γ)).

5.2.2. Base change. For each s ∈ Λ(G) we write r_{s} and r_{Δ\overline{G}(s)} for the endomorphisms of Λ(G) and Λ(\overline{G} × G) given by right multiplication by s and Δ\overline{G}(s) respectively. Then cok(r_{Δ\overline{G}(s)}) is isomorphic as a Λ(\overline{G} × G)-module to tw\overline{G}(cok(r_{s})) and so is finitely generated over Λ(\overline{G} × H) if cok(r_{s}) is finitely generated over Λ(H). This implies that Δ\overline{G}(S^{∗}) ⊆ S_{1}^{∗} where S := S_{G,H} and S_{1} := S_{\overline{G}×G,\overline{G}×H} and so Δ\overline{G} induces a ring homomorphism

(28)  Λ(G)_{S^{∗}} → Λ(\overline{G} × G)_{S_{1}^{∗}} → Λ(\overline{G} × Γ)_{S_{2}^{∗}} = Q(\overline{G} × Γ)

where S_{2} := S_{\overline{G}×Γ,\overline{G}}, the second arrow is the natural projection and the equality is because \overline{G} × Γ has rank one (as a p-adic Lie group). These maps induce a group homomorphism

π_{\overline{G}×Γ} : K_{1}(Λ(G)_{S^{∗}}) → K_{1}(Q(\overline{G} × Γ))

which forms the upper row of a natural commutative diagram of connecting homomorphisms

(29)

K_{1}(Λ(G)_{S^{∗}}) ⟶ K_{1}(Λ(\overline{G} × G)_{S_{1}^{∗}}) ⟶ K_{1}(Λ(\overline{G} × Γ)_{S_{2}^{∗}})
↓  ↓  ↓
K_{0}(Λ(G), Λ(G)_{S^{∗}}) ⟶ K_{0}(Λ(\overline{G} × G), S_{1}^{∗}) ⟶ K_{0}(Λ(\overline{G} × Γ), S_{2}^{∗})

where we write K_{0}(Λ(\overline{G} × G), S_{1}^{∗}) and K_{0}(Λ(\overline{G} × Γ), S_{2}^{∗}) for K_{0}(Λ(\overline{G} × G), Λ(\overline{G} × G)_{S_{1}^{∗}}) and K_{0}(Λ(\overline{G} × Γ), Λ(\overline{G} × Γ)_{S_{2}^{∗}}) respectively.

5.2.3. Reduced norms. We set R := Λ(\overline{G} × Γ). Then the algebra Q(R) identifies with the group ring Q(Γ)[\overline{G}] and, with respect to this identification, one has

(30)  ζ(Q(R)) ⊂ ζ(Q^{c}(R)) = ∏_{ρ∈Irr(\overline{G})} Q^{c}(Γ)

where Q^{c}(R) := ℚ_{p}^{c} ⊗_{ℚ_{p}} Q(R) and Q^{c}(Γ) := ℚ_{p}^{c} ⊗_{ℚ_{p}} Q(Γ). We write x = (x_{ρ})_{ρ} for the corresponding decomposition of each element x of ζ(Q^{c}(R)).

In the next result we write Nrd_{Q(R)} : K_{1}(Q(R)) → ζ(Q^{c}(R))^{×} for the reduced norm map of the semisimple algebra Q(R) and use the homomorphism Φ_{ρ} and Ore set S̃ defined in (5) and (6) respectively.

Lemma 5.4. For each ξ in K_{1}(Λ(G)S̃) one has Nrd_{Q(R)}(π_{\overline{G}×Γ}(ξ)) = (Φ_{ρ}(ξ))_{ρ∈Irr(\overline{G})}.

Proof. It suffices to prove that, with respect to the decomposition (30), one has Φ_{ρ}(ξ) = Nrd_{Q(R)}(π_{\overline{G}×Γ}(ξ))_{ρ} for each fixed ρ in Irr(\overline{G}). Further, since [13, Prop. 4.2, Th. 4.4] implies that the natural map Λ(G)× ; S̃ → K_{1}(Λ(G)S̃) is surjective, it is enough to verify this for all elements ξ of the form ⟨r_{s} | Λ(G)S̃⟩ with s ∈ Λ(G) ∩ Λ(G)× ; S̃. To do this we fix a finite dimensional ℚ_{p}^{c}-space V_{ρ} that corresponds to ρ and write V_{ρ^{∗}} for the space Hom_{ℚ_{p}^{c}}(V_{ρ}, ℚ_{p}^{c}) that corresponds to ρ^{∗}. Then for each x = ∑_{δ∈\overline{G}} c_{δ}δ in Q(Γ)[\overline{G}]^{×} = Q(R)^{×} the argument of Ritter and Weiss in [25, §3] shows that

(31)  Nrd_{Q(R)}(⟨r_{x} | Q(R)⟩)_{ρ} = det_{Q^{c}(Γ)}(α_{x})

where α_{x} is the automorphism of V_{ρ^{∗}} ⊗_{ℚ_{p}^{c}} Q^{c}(Γ) given by ∑_{δ∈\overline{G}} δ^{−1} ⊗ μ(c_{δ}) with μ(c_{δ}) denoting multiplication by c_{δ}. Now the matrix of the action of δ^{−1} on V_{ρ^{∗}} (with respect to a fixed ℚ_{p}^{c}-basis) is the transpose of the matrix of the action of δ on V_{ρ} (with respect to the dual ℚ_{p}^{c}-basis). Using this fact, and an explication of the role of Morita equivalence in (5), one finds that Φ_{ρ}(s) = det_{Q^{c}(Γ)}(α_{Δ\overline{G}(s)}) for each s ∈ Λ(G) ∩ Λ(G)× ; S̃. Since π_{\overline{G}×Γ}(⟨r_{s} | Λ(G)S̃⟩) = ⟨r_{Δ\overline{G}(s)} | Q(R)⟩ the claimed result is therefore a consequence of the description (31). □

5.2.4. Semisimplicity. There are natural isomorphisms in D^{p}(ℤ_{p}[\overline{G}]) of the form

ℤ_{p} ⊗_{Λ(Γ)}^{𝕃} tw\overline{G}(C)_{H} ≅ ℤ_{p} ⊗_{Λ(G)}^{𝕃} tw\overline{G}(C) ≅ ℤ_{p}[\overline{G}] ⊗_{Λ(G)}^{𝕃} C

and hence an exact triangle in D(Λ(\overline{G} × Γ)) of the form

△(tw\overline{G}(C), γ) : tw\overline{G}(C)_{H} ⟶^{θ_{γ}} tw\overline{G}(C)_{H} → ℤ_{p}[\overline{G}] ⊗_{Λ(G)}^{𝕃} C → tw\overline{G}(C)_{H}[1]

where θ_{γ} is induced by multiplication by γ − id ∈ Λ(Γ) on Λ(\overline{G} × Γ). In the next result we use the terminology and notation of Appendix B. For each ℚ_{p}[\overline{G}]- module M we also define a ℚ_{p}^{c}-module M^{ρ} := Hom_{ℚ_{p}^{c}[\overline{G}]}(V_{ρ}, ℚ_{p}^{c} ⊗_{ℤ_{p}} M).

Lemma 5.5.

(i) The image of △(tw\overline{G}(C), γ) under the (exact) functor e_{ρ}ℚ_{p}^{c}[\overline{G}] ⊗_{ℤ_{p}[\overline{G}]} − is naturally isomorphic to the exact triangle △(C_{ρ}, γ) defined in (23).

(ii) For each ρ in Irr(\overline{G}) one has

r_{G}(C)(ρ) = ∑_{i∈ℤ}(−1)^{i+1}dim_{ℚ_{p}^{c}}(H^{i}(tw\overline{G}(C)_{H})^{Γ,ρ}).

(iii) The morphism θ_{γ} is semisimple if and only if C is semisimple at ρ (in the sense

of [11, Def. 3.11]) for every ρ in Irr(\overline{G}). (iv) If θ_{γ} is semisimple, then

r_{G}(C)(ρ) = ∑_{i∈ℤ}(−1)^{i+1}i · dim_{ℚ_{p}^{c}}(H^{i}(ℤ_{p}[\overline{G}] ⊗_{Λ(G)}^{𝕃} C)^{ρ})

for every ρ in Irr(\overline{G}) and, with respect to the decomposition (4), one has

β_{ℚ_{p}⊗△(tw\overline{G}(C),γ)} = (t(C_{ρ}))_{ρ∈Irr(\overline{G})}

where ℚ_{p}⊗△(tw\overline{G}(C), γ) is the exact triangle in D^{p}(Λ(\overline{G}×Γ)[\frac{1}{p}]) that is obtained from △(tw\overline{G}(C), γ) by scalar extension.

Proof. For every Λ(G)-module P there is a natural isomorphism of Λ(\overline{G} × Γ)-modules Λ(\overline{G} × Γ) ⊗_{Λ(\overline{G}×G)} (ℤ_{p}[\overline{G}] ⊗_{ℤ_{p}} P) ≅ Λ(Γ) ⊗_{Λ(G)} (ℤ_{p}[\overline{G}] ⊗_{ℤ_{p}} P) where the action of \overline{G} on the second module is just on ℤ_{p}[\overline{G}] (from the left). This fact gives rise to natural isomorphisms in D(Λ_{ℚ_{p}^{c}}(Γ)) of the form

(32)  e_{ρ}ℚ_{p}^{c}[\overline{G}] ⊗_{ℤ_{p}[\overline{G}]} tw\overline{G}(C)_{H} ≅ Λ_{ℚ_{p}^{c}}(Γ) ⊗_{Λ_{ℚ^{c}_{p}} (G)}^{𝕃} (e_{ρ}ℚ_{p}^{c}[\overline{G}] ⊗_{ℤ_{p}} C)
≅ Λ_{ℚ_{p}^{c}}(Γ) ⊗_{Λ_{ℚ^{c}_{p}} (G)}^{𝕃} (V_{ρ∗} ⊗_{ℤ_{p}} C)
≅ ℚ_{p}^{c} ⊗_{O} C_{ρ}.

We now set C\overline{G} := ℤ_{p}[\overline{G}]⊗_{Λ(G)}^{𝕃}C. Then claim (i) follows upon combining the isomorphism (32) together with the natural isomorphism in D(Λ_{ℚ_{p}^{c}}(Γ))

e_{ρ}ℚ_{p}^{c}[\overline{G}] ⊗_{ℤ_{p}[\overline{G}]} C\overline{G} ≅ V_{ρ^{∗}} ⊗_{Λ(G)}^{𝕃} C ≅ ℚ_{p}^{c} ⊗_{Λ_{O}(Γ)}^{𝕃} C_{ρ}

that is induced by [7, Lem. 3.13(i)]. Claim (ii) follows by combining the equality

r_{G}(C)(ρ) = ∑_{i∈ℤ}(−1)^{i+1} dim_{ℚ_{p}^{c}}(H^{i}(ℚ_{p}^{c} ⊗_{O} C_{ρ})^{Γ})

of [11, Lem. 3.13(ii)] with the isomorphisms of Λ_{ℚ_{p}^{c}}(Γ)-modules

H^{i}(ℚ_{p}^{c} ⊗_{O} C_{ρ}) ≅ H^{i}(e_{ρ}ℚ_{p}^{c}[\overline{G}] ⊗_{ℤ_{p}[\overline{G}]} tw\overline{G}(C)_{H}) ≅ H^{i}(tw\overline{G}(C)_{H})^{ρ}

that are induced by (32). Next we note that claim (i) implies θ_{γ} is semisimple if and only if θ_{γ,ρ} is semisimple for every ρ in Irr(\overline{G}). Claim (iii) thus follows immediately from the definition of ‘semisimplicity at ρ’ (in terms of θ_{γ,ρ}). In each degree i the exact triangle △(tw\overline{G}(C), γ) induces a short exact sequence

0 → H^{i}(tw\overline{G}(C)_{H})_{Γ} → H^{i}(C\overline{G}) → H^{i+1}(tw\overline{G}(C)_{H})^{Γ} → 0.

In particular, if θ_{γ} is semisimple, then θ_{γ,ρ} is semisimple and so by applying the exact functor M ↦ M^{ρ} to this sequence one finds that

dim_{ℚ_{p}^{c}}(H^{i}(C\overline{G})^{ρ}) = dim_{ℚ_{p}^{c}}(H^{i}(tw\overline{G}(C)_{H})^{Γ,ρ}) + dim_{ℚ_{p}^{c}}(H^{i+1}(tw\overline{G}(C)_{H})^{Γ,ρ})

for each integer i and hence that ∑_{i∈ℤ}(−1)^{i+1}i · dim_{ℚ_{p}^{c}}(H^{i}(C\overline{G})^{ρ}) is equal to

∑_{i∈ℤ}(−1)^{i+1}i(dim_{ℚ_{p}^{c}}(H^{i}(tw\overline{G}(C)_{H})^{Γ,ρ}) + dim_{ℚ_{p}^{c}}(H^{i+1}(tw\overline{G}(C)_{H})^{Γ,ρ}))

= ∑_{i∈ℤ}(−1)^{i+1} dim_{ℚ_{p}^{c}}(H^{i}(tw\overline{G}(C)_{H})^{Γ,ρ}).

This proves the explicit formula for r_{G}(C)(ρ) in claim (iv). The explicit description of β_{ℚ_{p}⊗△(tw\overline{G}(C),γ)} in claim (iv) follows from the identification in claim (i) and the fact that t(C_{ρ}) is, by definition, equal to β_{△(C_{ρ},γ)}. □

5.3. Leading terms. We now fix an element ξ and a complex C as in Theorem 2.2. Then Lemma 5.5(iii) implies that the morphism θ_{γ} : tw\overline{G}(C)_{H} → tw\overline{G}(C)_{H} is semisimple and so in each degree i there is a direct sum decomposition of Λ(\overline{G} × Γ)[\frac{1}{p}]-modules

(33)  ℚ_{p} ⊗_{ℤ_{p}} H^{i}(tw\overline{G}(C)_{H}) = D_{0}^{i} ⊕ D_{1}^{i}

where D_{0}^{i} := ℚ_{p} ⊗_{ℤ_{p}} ker(H^{i}(θ_{γ})) = ℚ_{p} ⊗_{ℤ_{p}} H^{i}(tw\overline{G}(C)_{H})^{Γ} and D_{1}^{i} := ℚ_{p} ⊗_{ℤ_{p}} im(H^{i}(θ_{γ})). By assumption, both D_{0}^{i} and D_{1}^{i} are finitely generated (projective) ℚ_{p}[\overline{G}]-modules and H^{i}(θ_{γ}) induces an automorphism of D_{1}^{i}. The proof of the following result will occupy the rest of this section.

Proposition 5.6. ∂\overline{G}(((−1)^{r_{G}(C)(ρ)}ξ^{∗}(ρ))_{ρ∈Irr(\overline{G})}) = ∑_{i∈ℤ}(−1)^{i}∂\overline{G}(⟨H^{i}(θ_{γ}) | D_{1}^{i}⟩).

5.3.1. The descent to Q(R). We write Σ for the subset of R consisting of those elements of Λ(Γ) with non-zero image under the projection Λ(Γ) → ℤ_{p}. This is a multiplicatively closed Ore set in R which consists of central regular elements.

Lemma 5.7. For each integer i we set M^{i} := (I\overline{G}×Γ ; \overline{G}(H^{i}(tw\overline{G}(C)_{H})^{Γ}))_{S^{∗}}. Then the element

y_{ξ} := Nrd_{Q(R)}(π_{\overline{G}×Γ}(ξ) ∏_{i∈ℤ}⟨δ_{γ} | M^{i}⟩^{(−1)^{i+1}})

belongs to ζ(R_{Σ})^{×} ⊆ ζ(Q(R))^{×}.

Proof. We set X := tw\overline{G}(C)_{H}. Then the commutative diagram (29) implies that ∂_{\overline{G}×Γ}(π_{\overline{G}×Γ}(ξ)) = χ(X) in K_{0}(R, Q(R)). But X belongs to D_{S}^{p}(R) and so [6, Th. 4.1(ii)] also implies that ∂_{\overline{G}×Γ}(char_{\overline{G}×Γ,γ}(X)) = χ(X). Hence the upper row of (1) with R′ = Q(R) implies that there exists an element u of K_{1}(R) with

(34)  π_{\overline{G}×Γ}(ξ) = ι_{1}(u)char_{\overline{G}×Γ,γ}(X)

where ι_{1} is the natural homomorphism K_{1}(R) → K_{1}(R_{Σ}) → K_{1}(Q(R)). For each integer i we set N^{i} := (I\overline{G}×Γ ; \overline{G}(H^{i}(tw\overline{G}(C)_{H})))_{S^{∗}}. Then the term ⟨δ_{γ} | N^{i}⟩ occurs in the definition of char_{\overline{G}×Γ,γ}(X) = char_{\overline{G}×Γ,γ}^{∗}(X). Also, from Lemma 5.8 below, the action of δ_{γ} on N^{i} = Q(R)⊗_{ℚ_{p}[\overline{G}]} (ℚ_{p} ⊗_{ℤ_{p}} H^{i}(tw\overline{G}(C)_{H})) restricts to give an automorphism of R_{Σ} ⊗_{ℚ_{p}[\overline{G}]} D_{1}^{i} and so (33) implies that ⟨δ_{γ} | N^{i}⟩ is equal to

⟨δ_{γ} | Q(R) ⊗_{ℚ_{p}[\overline{G}]} D_{0}^{i}⟩⟨δ_{γ} | Q(R) ⊗_{ℚ_{p}[\overline{G}]} D_{1}^{i}⟩

= ⟨δ_{γ} | Q(R) ⊗_{ℚ_{p}[\overline{G}]} D_{0}^{i}⟩ι_{Σ}(⟨δ_{γ} | R_{Σ} ⊗_{ℚ_{p}[\overline{G}]} D_{1}^{i}⟩)

where ι_{Σ} is the natural homomorphism K_{1}(R_{Σ}) → K_{1}(Q(R)). Hence by combining (34) with the definition of char_{\overline{G}×Γ,γ}(X) one finds that

(35) π_{\overline{G}×Γ}(ξ) ∏_{i∈ℤ}⟨δ_{γ} | M^{i}⟩^{(−1)^{i+1}} = π_{\overline{G}×Γ}(ξ) ∏_{i∈ℤ}⟨δ_{γ} | Q(R) ⊗_{ℚ_{p}[\overline{G}]} D_{0}^{i}⟩^{(−1)^{i+1}}

= ι_{1}(u)ι_{Σ}(∏_{i∈ℤ}⟨δ_{γ} | R_{Σ} ⊗_{ℚ_{p}[\overline{G}]} D_{1}^{i}⟩^{(−1)^{i}}) ∈ im(ι_{Σ}).

Now R_{Σ} is finitely generated as a module over the commutative local ring Λ(Γ)_{Σ} and so is itself a semi-local ring (cf. [15, Prop. (5.28)(ii)]). The natural homomorphism R_{Σ}^{×} → K_{1}(R_{Σ}) is thus surjective (by [15, Th. (40.31)]) and so (35) implies that the element π_{\overline{G}×Γ}(ξ) ∏_{i∈ℤ}⟨δ_{γ} | M^{i}⟩^{(−1)^{i+1}} is represented by a pair of the form ⟨r_{y} | Q(R)⟩ with y ∈ R_{Σ}^{×}. Now both y and y^{−1} are of the form zσ^{−1} for suitable elements z ∈ R ∩ Q(R)^{×} and σ ∈ Σ. Thus, to complete the proof of the lemma, it suffices to prove that for all such z and σ both Nrd_{Q(R)}(⟨r_{z} | Q(R)⟩) and Nrd_{Q(R)}(⟨r_{σ^{−1}} | Q(R)⟩) belong to ζ(R_{Σ}). But (31) implies Nrd_{Q(R)}(⟨r_{σ^{−1}} | Q(R)⟩) = (σ^{−d_{ρ}})_{ρ} ∈ ζ(R_{Σ}) with d_{ρ} := dim_{ℚ_{p}^{c}}(V_{ρ}). Also, if ℤ_{p}^{c} is the integral closure of ℤ_{p} in ℚ_{p}^{c} and T_{ρ} is any full ℤ_{p}^{c}-sublattice of V_{ρ}, then the action of \overline{G} on T_{ρ} induces a homomorphism ϱ : R = Λ(Γ)[\overline{G}] → M_{d_{ρ}}(ℤ_{p}^{c} ⊗_{ℤ_{p}} Λ(Γ)) and (31) implies Nrd_{Q(R)}(⟨r_{x} | Q(R)⟩) = (det(ϱ(z)))_{ρ} ∈ (ℚ_{p}^{c} ⊗_{ℤ_{p}} ζ(R))∩ζ(Q(R)) = ℚ_{p} ⊗_{ℤ_{p}} ζ(R) ⊆ ζ(R_{Σ}), as required. □

Lemma 5.8. δ_{γ} induces an automorphism of R_{Σ} ⊗_{ℚ_{p}[\overline{G}]} D_{1}^{i}.

Proof. The argument of [26, Prop. 2.2, Rem. 2.3] gives a short exact sequence

0 → R ⊗_{ℤ_{p}[\overline{G}]} D_{1}^{i} ⟶^{δ_{γ}} R ⊗_{ℤ_{p}[\overline{G}]} D_{1}^{i} → D_{1}^{i} → 0

and so it suffices to show that (D_{1}^{i})_{Σ} = 0. But D_{1}^{i} := ℚ_{p} ⊗_{ℤ_{p}} im(H^{i}(θ_{γ})) and, regarding im(H^{i}(θ_{γ})) as a (finitely generated) module over Λ(Γ) ⊆ R, the decomposition (33) implies that im(H^{i}(θ_{γ}))_{Γ} is finite. This implies that im(H^{i}(θ_{γ})) is a finitely generated torsion Λ(Γ)-module whose characteristic polynomial f(T) is coprime to T. It follows that f(T) is invertible in R_{Σ} and so (D_{1}^{i})_{Σ} = im(H^{i}(θ_{γ}))_{Σ} = 0, as required. □

5.3.2. The proof of Proposition 5.6. From Lemmas 5.4 and 5.7 we know that

(36)  (ξ^{∗}(ρ) ∏_{i∈ℤ}(Nrd_{Q(R)}(⟨δ_{γ} | M^{i}⟩))_{ρ}^{∗}(0)^{(−1)^{i+1}})_{ρ∈Irr(\overline{G})} = π(y_{ξ})

where π is the natural projection ζ(R_{Σ})^{×} → ζ(ℚ_{p}[\overline{G}])^{×}. But if x is in R_{Σ}^{×}, then Nrd_{Q(R)}(⟨r_{x} | Q(R)⟩) belongs to ζ(R_{Σ})^{×} (see the proof of Lemma 5.7) and (31) implies π(Nrd_{Q(R)}(⟨r_{x} | Q(R)⟩)) = (Nrd_{ℚ_{p}[\overline{G}]}(⟨r\overline{x} | ℚ_{p}[\overline{G}]⟩)_{ρ} with \overline{x} the image of x in ℚ_{p}[\overline{G}]^{×}.

Hence (35) implies π(y_{ξ}) = Nrd_{ℚ_{p}[\overline{G}]}(\overline{u}) ∏_{i∈ℤ} Nrd_{ℚ_{p}[\overline{G}]}(⟨H^{i}(θ_{γ}) | D_{1}^{i}⟩)^{(−1)^{i}} where \overline{u} is the image of u under the natural composite homomorphism K_{1}(R) → K_{1}(ℤ_{p}[\overline{G}]) → K_{1}(ℚ_{p}[\overline{G}]). Since ∂\overline{G}(\overline{u}) = 0 one therefore has

(37)  ∂\overline{G}(π(y_{ξ})) = ∑_{i∈ℤ}(−1)^{i}∂\overline{G}(⟨H^{i}(θ_{γ}) | D_{1}^{i}⟩).

The equality of Proposition 5.6 now follows upon substituting (36) into (37) and then using both the explicit formula for r_{G}(C)(ρ) given in Lemma 5.5(ii) and the following result (with M = H^{i}(tw\overline{G}(C)_{H}) for each i).

Lemma 5.9. If M is any finitely generated R-module, then for every ρ in Irr(\overline{G}) one has

(Nrd_{Q(R)}(δ_{γ} | I\overline{G}×Γ ; \overline{G}(M^{Γ})_{S^{∗}}))_{ρ}^{∗}(0) = (−1)^{dim_{ℚ^{c}_{p}} (M^{Γ,ρ})}.

Proof. There are natural isomorphisms of Q^{c}(Γ)-modules of the form

Hom_{ℚ_{p}^{c}[\overline{G}]}(V_{ρ}, I\overline{G}×Γ ; \overline{G}(M^{Γ})_{S^{∗}}) ≅ Hom_{ℚ_{p}^{c}[\overline{G}]}(V_{ρ}, Q^{c}(Γ) ⊗_{ℤ_{p}} M^{Γ}) ≅ M^{Γ,ρ} ⊗_{ℚ_{p}^{c}} Q^{c}(Γ)

under which the induced action of δ_{γ} on the first module corresponds to the endomorphism δ̃_{γ} of the third module that sends m ⊗ x to m ⊗ (γ^{−1} − 1)x. It follows that Nrd_{Q(R)}(δ_{γ} | I\overline{G}×Γ ; \overline{G}(M^{Γ})_{S^{∗}})_{ρ} is equal to

det_{Q^{c}(Γ)}(δ̃_{γ} | M^{Γ,ρ} ⊗_{ℚ_{p}^{c}} Q^{c}(Γ)) = det_{Q^{c}(Γ)}(γ^{−1} − 1 | Q^{c}(Γ))^{dim_{ℚ^{c}_{p}} (M^{Γ,ρ})}

= (−T/(1 + T))^{dim_{ℚ^{c}_{p}} (M^{Γ,ρ})}

where the last equality follows from the fact that γ^{−1} − 1 = (1 − γ)/γ = −T/(1 + T). From this explicit formula it is clear that the first non-zero coefficient of T in the series Nrd_{Q(R)}(δ_{γ} | I\overline{G}×Γ ; \overline{G}(M^{Γ})_{S^{∗}})_{ρ} is equal to (−1)^{dim_{ℚ^{c}_{p}} (M^{Γ,ρ})}. □

5.4. Completion of the proof of Theorem 2.2. We set N := U ∩H. Then in each degree i there is a natural isomorphism of ℤ_{p}[\overline{G}]-modules

H^{i}(tw\overline{G}(C)_{H}) ≅ ℤ_{p}[\overline{G}] ⊗_{Λ(H/N)} H^{i}(Λ(G/N) ⊗_{Λ(G)}^{𝕃} C).

But Λ(G/N) ⊗_{Λ(G)}^{𝕃} C belongs to D_{S_{G/N,H/N}}^{p} (Λ(G/N)) and so each module H^{i}(tw\overline{G}(C)_{H}) is finitely generated over ℤ_{p}[\overline{G}]. This implies that Δ(tw\overline{G}(C), γ) is an exact triangle in D^{p}(ℤ_{p}[\overline{G}]). In view of Lemma 5.5(iv) and Proposition 5.6 we may therefore deduce Theorem 2.2 by applying the following result with G = \overline{G}, R = ℤ_{p} and Δ = Δ(tw\overline{G}(C), γ).

Proposition 5.10. Let G be a finite group, R an integral domain and F the field of fractions of R. Let Δ : C ⟶^{θ} C → D → C[1] be an exact triangle in D^{p}(R[G]). Assume that θ is semisimple and in each degree i fix an F [G][H^{i}(θ)]-equivariant direct complement W^{i} to F ⊗_{R} ker(H^{i}(θ)) in F ⊗_{R} H^{i}(C). Then H^{i}(θ) induces an automorphism of the (finitely generated projective) F [G]-module W^{i}, the element ⟨H^{i}(θ)⟩^{∗} := ⟨H^{i}(θ) | W^{i}⟩ of K_{1}(F [G]) is independent of the choice of W^{i} and in K_{0}(R[G], F [G]) one has

(38)  ∑_{i∈ℤ}(−1)^{i}∂_{G}(⟨H^{i}(θ)⟩^{∗}) = −[d_{R[G]}(D), β_{Δ}]

where ∂_{G} is the connecting homomorphism K_{1}(F [G]) → K_{0}(R[G], F [G]) and β_{Δ} is as defined in (55).

Proof. It is clear that H^{i}(θ) induces an automorphism of W^{i} and straightforward to verify that ⟨H^{i}(θ)⟩^{∗} is independent of the choice of W^{i}. However to prove (38) we replace C by a complex P in C^{p}(R[G]) for which there exists an isomorphism q : P → C in D^{p}(R[G]) and we shall argue by induction on |P | := max{i : P^{i} ≠ 0} − min{j : P^{j} ≠ 0}. To do this we fix a morphism of complexes ϕ : P → P such that q ∘ ϕ = θ ∘ q in D^{p}(R[G]). If |P | = 0, then P = P^{m}[−m] = H^{m}(P) (and ϕ = ϕ^{m} = H^{m}(ϕ)) for some integer

m. In this case D identifies with the mapping cone P^{m} ⟶^{ϕ^{m}} P^{m} of ϕ (so the first term of this complex is placed in degree m − 1) in such a way that the homomorphism H^{m−1}(D) → ker(H^{m}(θ)) → cok(H^{m}(θ)) → H^{m}(D) induced by Δ corresponds to the tautological map τ : ker(ϕ^{m}) → cok(ϕ^{m}). Further, if W is a direct complement to F ⊗_{R}ker(ϕ^{m}) in F ⊗_{R}P^{m}, then ϕ^{m}(W) = W (since ϕ is semisimple) and [d_{R[G]}(D), β_{Δ}] = (P^{m}, ι^{(−1)^{m−1}}, P^{m}) with ι the composite isomorphism

F ⊗_{R} P^{m} = (F ⊗_{R} ker(ϕ^{m})) ⊕ W ⟶^{(F ⊗_{R}τ,H^{m}(ϕ))} (F ⊗_{R} cok(ϕ^{m})) ⊕ W ≅ F ⊗_{R} P^{m}

where the isomorphism is induced by a choice of splitting of the tautological exact sequence 0 → W → F ⊗_{R} P^{m} → F ⊗_{R} cok(ϕ^{m}) → 0 (this description of [d_{R[G]}(D), β_{Δ}] follows, for example, from [3, Th. 6.2]). The equality (38) is therefore valid because −(P^{m}, ι^{(−1)^{m−1}}, P^{m}) = (−1)^{m}∂_{G}(⟨ι | F ⊗_{R} P^{m}⟩) = (−1)^{m}∂_{G}(⟨H^{m}(ϕ) | W ⟩). We now assume that |P | = n > 0 and, to fix notation, that min{j : P^{j} ≠ 0} = 0. We set C_{2} := P and ϕ_{2} = ϕ, write C_{3} for the naive truncation in degree n − 1 of P and set C_{1} := P^{n}[−n]. Then one has a tautological short exact sequence of complexes

(39)  0 → C_{1} → C_{2} → C_{3} → 0.

From the associated long exact cohomology sequence we deduce that H^{i}(C_{3}) = H^{i}(C_{2}) if i < n − 1 and that there are commutative diagrams of exact sequences

(40)  \begin{aligned}0 & ⟶ & H^{n−1}(C_{2}) & ⟶ & H^{n−1}(C_{3}) & ⟶ & B^{n}(C_{2}) & ⟶ & 0 \\ & ^{H^{n−1}(ϕ_{2})}↓ & ^{H^{n−1}(ϕ_{3})}↓ & & ^{ϕ^{n}}↓ \\ 0 & ⟶ & H^{n−1}(C_{2}) & ⟶ & H^{n−1}(C_{3}) & ⟶ & B^{n}(C_{2}) & ⟶ & 0 \end{aligned}

(41)  \begin{aligned}0 & ⟶ & B^{n}(C_{2}) & ⟶ & H^{n}(C_{1}) & ⟶ & H^{n}(C_{2}) & ⟶ & 0 \\ & & ^{ϕ^{n}}↓ & ^{H^{n}(ϕ_{1})}↓ & ^{H^{n}(ϕ_{2})}↓ \\ 0 & ⟶ & B^{n}(C_{2}) & ⟶ & H^{n}(C_{1}) & ⟶ & H^{n}(C_{2}) & ⟶ & 0, \end{aligned}

where B^{n}(C_{2}) denotes the coboundaries of C_{2} in degree n. By mimicking the argument of [6, Lem. 4.4] we may change ϕ by a homotopy in order to assume that, in each degree i, the restriction of ϕ^{i} induces an automorphism of F ⊗_{R} B^{i}(C_{2}). This assumption has two important consequences. Firstly, the above diagrams imply that the morphisms ϕ_{1} and ϕ_{3} of C_{1} and C_{3} that are induced by ϕ are semisimple. Secondly, if we write D_{i} for the mapping cone of ϕ_{i} for i = 1, 2, 3, then (39) induces short exact sequences of the form

0 → D_{1} ⟶ D_{2} ⟶ D_{3} → 0
0 → Z(D_{1}) ⟶ Z(D_{2}) ⟶ Z(D_{3}) → 0
0 → B(D_{1}) ⟶ B(D_{2}) ⟶ B(D_{3}) → 0
0 → H(D_{1}) ⟶ H(D_{2}) ⟶ H(D_{3}) → 0
0 → H_{bock}(Δ_{1}) ⟶ H_{bock}(Δ_{2}) ⟶ H_{bock}(Δ_{3}) → 0.

Here we write B(D_{i}) and Z(D_{i}) for the complexes of coboundaries and cocycles of D_{i} (each with zero differentials) and Δ_{i} for the tautological exact triangle C_{i} ⟶^{ϕ_{i}} C_{i} → D_{i} → C_{i}[1]. Now from the displayed exact sequences (and the definition of each term [D_{i}, β_{Δ_{i}}] in Appendix B) one has an equality

[d_{R[G]}(D_{2}), β_{Δ_{2}}] = [d_{R[G]}(D_{1}), β_{Δ_{1}}] + [d_{R[G]}(D_{3}), β_{Δ_{3}}].

But the inductive hypothesis implies

−[d_{R[G]}(D_{3}), β_{Δ_{3}}] = ∑_{i=0}^{i=n−1}(−1)^{i}∂_{G}(⟨H^{i}(ϕ_{3})⟩^{∗})

and, since |C_{1}| = 0, our earlier argument proves

−[d_{R[G]}(D_{1}), β_{Δ_{1}}] = (−1)^{n}∂_{G}(⟨H^{n}(ϕ_{1})⟩^{∗}).

It is also clear that ⟨H^{i}(ϕ_{3})⟩^{∗} = ⟨H^{i}(ϕ_{2})⟩^{∗} for i < n − 1 whilst (40) and (41) imply ⟨H^{n−1}(ϕ_{2})⟩^{∗} = ⟨H^{n−1}(ϕ_{3})⟩^{∗}⟨ϕ^{n} | F ⊗_{R} B^{n}(C_{2})⟩ and ⟨H^{n}(ϕ_{1})⟩^{∗} = ⟨ϕ^{n} | F ⊗_{R} B^{n}(C_{2})⟩ ⟨H^{n}(ϕ_{2})⟩^{∗} respectively. The claimed description of −[D_{2}, β_{Δ_{2}}] thus follows upon combining the last three displayed equations. □

Remark 5.11. If G is abelian, then Proposition 5.10 can be reinterpreted in terms of graded determinants and in this case has been proved to within a ‘sign ambiguity’ by Kato in [19, Lem. 3.5.8]. (This ambiguity arises because Kato uses ungraded determinants - for more details in this regard see [19, Rem. 3.2.3(3) and 3.2.6(3),(5)] and [9, Rem. 9]).

Part II: Arithmetic

For any Galois extension of fields F/E we set G_{F/E} := Gal(F/E). For any field E we also fix an algebraic closure E^{c} and abbreviate G_{E^{c}/E} to G_{E}.

6. Field-theoretic preliminaries

We first introduce the class of fields for which the techniques of [13] allow one to formulate a main conjecture of non-commutative Iwasawa theory. We fix an odd prime p and for each number field k we write F_{k} for the set of Galois extensions L of k inside ℚ^{c} which satisfy the following conditions

(i) L contains the cyclotomic ℤ_{p}-extension k^{cyc} of k; (ii) L/k is unramified outside a finite set of places; (iii) G_{L/k} is a compact p-adic Lie group.

If k is totally real, then we also let F_{k}^{+} denote the subset of F_{k} comprising those fields that are totally real. The following result was explained to us by Kazuya Kato. It provides an important general reduction step and also shows that Theorem 2.2 constitutes a satisfactory resolution of the descent problem in the setting of non-commutative Iwasawa theory.

Lemma 6.1. For any number field k and any F in F_{k} there exists a field F′ in F_{k} with F ⊆ F′ and such that G_{F′/k} has no element of order p. If k is totally real and F belongs to F_{k}^{+}, then one can also choose F′ in F_{k}^{+}.

Proof. For any extension E of k we write E(ζ_{p^{∞}}) for the extension of E generated by all p-power roots of unity (in ℚ^{c}). We set F̃ := F (ζ_{p^{∞}}) and choose a p-torsion free open normal subgroup U of V := G_{F̃/k(ζ_{p^{∞}})}. We let L be the extension of k in F̃ that corresponds to U and for each nontrivial p-torsion element σ_{i} of V/U we write L_{i} for the fixed subfield of L by σ_{i}. Then L = L_{i}(a_{i}^{1/p^{n}}) for some a_{i} ∈ L_{i}^{×}. Let a_{ij}, 1 ≤ j ≤ s(i), be all conjugates of a_{i} over F and set L′_{i} denote the field generated over L_{i} by the set {a_{ij}^{1/p^{n}} : 1 ≤ j ≤ s(i), n ≥ 1}. Then F̃L′_{i} is a Galois extension of F that contains L. Furthermore G_{L′_{i}/L_{i}} is isomorphic to a subgroup of ℤ_{p}^{s(i)} by τ ↦ (r(j))_{j} with τ(a_{j}^{1/p^{n}})/a_{j}^{1/p^{n}} = ζ_{p^{n}}^{r(j)}. Let F′ be the composite field of F̃ and L′_{i} for all i. The group G_{F′/k} is a compact p-adic Lie group and we now prove that it has no element of order p. We note first that G_{k(ζ_{p^{∞}})/k} is isomorphic to a subgroup of ℤ_{p}^{×} and hence is p-torsion free by the assumption p ≠ 2. Thus if σ ∈ G_{F′/k} has order p, then the image of σ in G_{F̃/k} is contained in V and so the image of σ in V/U coincides with σ_{i} for some i.

Thus σ fixes all elements of L_{i}. But then the image of σ in G_{L′_{i}/L_{i}} is both p-torsion and also non-trivial (for its restriction to G_{L/L_{i}} is non-trivial). This contradicts the fact that G_{L′_{i}/L_{i}} has no element of order p. Hence G_{F′/k} has no element of order p, as claimed. Lastly we assume that F (and hence k) is totally real. Then F̃ is a CM field with maximal real subfield F̃^{+} equal to the compositum of F and the maximal totally real subfield of k(ζ_{p^{∞}}). Also, by the above construction, the extension F′/F̃ is pro-p. Since p is odd, the group G_{F′/F̃^{+}} therefore contains a unique element of order 2 and the fixed field (F′)^{+} of F′ by this element is totally real, contains F, is Galois over k and such that G_{(F′)^{+}/k} has no element of order p. □

In the remainder of this article we set Γ_{k} := G_{k^{cyc}/k} , H_{L/k} := G_{L/k^{cyc}}, Λ(L/k) := Λ(G_{L/k}) and Ω(L/k) := Ω(G_{L/k}) for each L in F_{k}. We also fix a topological generator γ_{ℚ} of Γ_{ℚ}, set d_{k} := [k ∩ ℚ^{cyc} : ℚ] and write γ_{k} for the topological generator γ_{ℚ}^{d_{k}} of Γ_{k}.

Remark 6.2. If C denotes either F_{k} or F_{k}^{+}, then it is an ordered set (by inclusion). Lemma 6.1 implies that the subset C′ of C comprising those fields F for which G_{F/k} has no element of order p is cofinal. Taking account of the functorial properties of the isomorphism in Theorem 2.1 and of the results in Proposition 4.7(ii) we may therefore deduce the following extensions of these results.

• There is a natural isomorphism of abelian groups

\underleftarrow{lim}_{F ∈C} K_{1}(Λ(F/k)_{S^{∗}}) ≅ \underleftarrow{lim}_{F ∈C} K_{0}(Ω(F/k)) ⊕ \underleftarrow{lim}_{F ∈C} K_{0}(Λ(F/k), Λ(F/k)_{S}) ⊕ \underleftarrow{lim}_{F ∈C} im(λ_{F/k})

where λ_{F/k} is the natural homomorphism K_{1}(Λ(F/k)) → K_{1}(Λ(F/k)_{S^{∗}}) and in each inverse limit the transition maps are induced by the homomorphism Λ(F/k) → Λ(F′/k) for each F′ ⊆ F.

• Let (x_{F})_{F} be an element of \underleftarrow{lim}_{F ∈C} K_{0}(Λ(F/k), Λ(F/k)_{S^{∗}}). Then for each F in C we may define an element char_{G_{F/k},γ_{k}}(x_{F}) of K_{1}(Λ(F/k)_{S^{∗}}) in the following way: we choose F′ in C′ with F ⊆ F′ and let char_{G_{F/k},γ_{k}}(x_{F}) denote the image of char_{G_{F′/k},γ_{k}}(x_{F′}) under the natural projection K_{1}(Λ(F′/k)_{S^{∗}}) → K_{1}(Λ(F/k)_{S^{∗}}). Then Lemma 4.5 implies char_{G_{F/k},γ_{k}}(x_{F}) is independent of the precise choice of F′ and Proposition 4.7 implies ∂_{G_{F/k}}(char_{G_{F/k},γ_{k}}(x_{F})) = x_{F}.

7. Non-commutative main conjectures

In this section we formulate explicit ‘main conjectures of non-commutative Iwasawa theory’ for both Tate motives and (certain) critical motives. In particular, in the setting of elliptic curves, the conjecture we formulate here is finer than that formulated by Coates et al in [13] in that we consider interpolation formulas for the leading terms (rather than values) of p-adic L-functions at Artin representations. Henceforth we will fix an isomorphism of fields j : ℂ ≅ ℂ_{p} and often simply omit it from the notation.

7.1. Tate motives. In this subsection we fix a totally real number field k and formulate a main conjecture for class groups associated to fields in F_{k}^{+}. We therefore fix a field K in F_{k}^{+} and a finite set of places Σ of k that contains the archimedean places and all places that ramify in K/k. For each Artin representation ρ of G_{K/k} we write ρ^{j^{−1}} for the complex representation of G_{K/k} induced by j^{−1} and L_{Σ}(s, ρ^{j^{−1}}) for the Artin L-function of ρ^{j^{−1}} that is truncated by removing the Euler factors attached to places in Σ. To formulate a main conjecture we must multiply the leading term L_{Σ}^{∗}(1, ρ^{j^{−1}}) in the Taylor expansion of L_{Σ}(s, ρ^{j^{−1}}) at s = 1 by an appropriate period. To define this period we let E be any finite degree Galois extension of k with E ⊂ K and G_{K/E} ⊆ ker(ρ). We set E_{∞} := ℝ ⊗_{ℚ} E ≅ ∏_{Hom(E,ℂ)} ℝ and write log_{∞}(O_{E}^{×}) for the inverse image of O_{E}^{×} ↪ E_{∞}^{×} under the (componentwise) exponential map exp_{∞} : E_{∞} → E_{∞}^{×}. Then the Dirichlet Unit Theorem implies that log_{∞}(O_{E}^{×}) is a lattice in the ℝ-space generated by E_{0} := {x ∈ E : Tr_{E/ℚ}(x) = 0} and so there is a canonical isomorphism of ℂ[G_{E/k}]- modules μ_{∞} : ℂ ⊗_{ℤ} log_{∞}(O_{E}^{×}) ≅ ℂ ⊗_{ℚ} E_{0}. In addition, if we write S_{p}(E) for the set of p-adic places of E, then the composite homomorphism

ℤ_{p} ⊗_{ℤ} log_{∞}(O_{E}^{×}) ⟶^{exp_{∞}} ℤ_{p} ⊗_{ℤ} O_{E}^{×}

⟶ ∏_{w∈S_{p}(E)} U_{E_{w}}^{1} ⟶^{(u_{w})_{w}↦(log_{p}(u_{w}))_{w}} ∏_{w∈S_{p}(E)} E_{w} ≅ ℚ_{p} ⊗_{ℚ} E

(where the second arrow is the natural diagonal map) factors through the inclusion ℚ_{p}⊗_{ℚ}E_{0} ⊂ ℚ_{p}⊗_{ℚ}E and hence induces a homomorphism μ_{p} : ℂ_{p}⊗_{ℤ}log_{∞}(O_{E}^{×}) ≅ ℂ_{p}⊗_{ℚ}E_{0} of ℂ_{p}[G_{E/k}]-modules. The resulting period

Ω_{j}(ρ) := det_{ℂ_{p}}(μ_{p} ∘ (ℂ_{p} ⊗_{ℂ,j} μ_{∞})^{−1})^{ρ} ∈ ℂ_{p}

depends upon j and ρ but is independent of the choice of E. We write χ_{cyc} for the cyclotomic character G_{k} → Γ_{k} → ℤ_{p}^{×} and for any Artin representation ρ of G_{K/k} we write ⟨ρ, 1⟩ for the multiplicity with which the trivial representation of G_{K/k} occurs in ρ.

Conjecture 7.1. Assume that G_{K/k} has no element of order p. Then the Galois group X_{Σ}(K) of the maximal pro-p abelian extension of K that is unramified outside Σ belongs to 𝔐_{S^{∗}}(G_{K/k}) and there exists an element ξ of K_{1}(Λ(K/k)_{S^{∗}}) which satisfies both of the following conditions.

(a) At each Artin representation ρ of G_{K/k} one has

ξ(ρ) = (log_{p}(χ_{cyc}(γ_{k}))^{⟨ρ,1⟩}c_{ρ,k})^{−1}Ω_{j}(ρ)L_{Σ}^{∗}(1, ρ^{j^{−1}})^{j}

with c_{ρ,k} := 1 if either ρ is trivial or H_{K/k} ⊄ ker(ρ) and c_{ρ,k} := 1 − ρ(γ_{k}^{−1}) otherwise.

(b) ∂_{G_{K/k}}(ξ) = [X_{Σ}(K)].

Remark 7.2. For each Artin representation ρ of G_{K/k} the ‘(Σ-truncated) p-adic Artin L-function’ of ρ is the unique p-adic meromorphic function L_{p,Σ}(·, ρ) : ℤ_{p} → ℂ_{p} with the property that for each strictly negative integer n and each isomorphism j : ℂ ≅ ℂ_{p} one has L_{p,Σ}(n, ρ) = L_{Σ}(n, (ρ⊗ω^{n−1})^{j^{−1}})^{j} where ω : G_{ℚ} → ℤ_{p}^{×} is the Teichmüller character. Then the ‘p-adic Stark conjecture at s = 1’, as formulated by Serre in [27] and discussed by Tate in [30, Chap. VI, §5], asserts that the term Ω_{j}(ρ)L_{Σ}^{∗}(1, ρ^{j^{−1}})^{j} in Conjecture 7.1(a) is equal to the leading term of L_{p,Σ}(s, ρ) at s = 1. See [11, Rem. 5.3] for more details.

In Conjecture 8.3 we formulate a version of Conjecture 7.1 that does not assume that G_{K/k} has no element of order p.

7.2. Critical motives.

7.2.1. Preliminaries. Let M be a critical motive over ℚ that has good ordinary reduction at p. Then its p-adic realization V = M_{p} has a unique ℚ_{p}-subspace V̂ that is stable under the action of G_{ℚ_{p}} and such that D_{dR}^{0}(V̂) = t_{p}(V) := D_{dR}(V)/D_{dR}^{0}(V). Now let ρ be an Artin representation defined over a number field B and [ρ] the corresponding Artin motive. We fix a p-adic place λ of B, set L := B_{λ} and write O for the valuation ring of L. Then the λ-adic realisation

(42)  W := W_{ρ} := N_{λ} = V ⊗_{ℚ_{p}} [ρ]_{λ}^{∗}

of the motive N := M(ρ^{∗}) := M ⊗ [ρ]^{∗} is an L-adic representation and contains the G_{ℚ_{p}}-subrepresentation Ŵ = V̂ ⊗_{ℚ_{p}} [ρ]_{λ}^{∗}. The algebraic rank of M(ρ^{∗}) is defined as

(43)  r(M)(ρ) := dim_{L}(H_{f}^{1}(ℚ, W_{ρ})) − dim_{L}(H_{f}^{3}(ℚ, W_{ρ})).

Let Σ be a finite set of places of ℚ containing p, ∞ and all places at which M has bad reduction or which ramify in K/ℚ. We fix a field K in F_{ℚ} that is unramified outside Σ. By Υ we denote the set of those primes ℓ ≠ p such that the ramification index of ℓ in K/ℚ is infinite. For a B-motive N over ℚ we denote by Ω_{∞}(N) and Ω_{p}(N) the associated complex and p-adic periods and by R_{p}(N) and R_{∞}(N) the associated complex and p-adic regulators, see again [17] or [11, Th. 6.5]. We recall that Ω_{∞}(N) ≠ 0 if N is critical and that R_{∞}(N) ≠ 0 if the (complex) height pairing of N is non-degenerate. Furthermore, for a ℚ_{p}-linear continuous G_{ℚ_{p}}-representation Z we write Γ(Z) for its Γ-factor (loc. cit.). Finally, for any L-linear continuous representation V and prime number ℓ we define an element of the polynomial ring L[u] by setting

P_{ℓ}(V, u) := P_{L,ℓ}(V, u) := \begin{cases}det_{L}(1 − φ_{ℓ}u|V^{I_{ℓ}}), & if ℓ ≠ p, \\ det_{L}(1 − φ_{p}u|D_{cris}(V)), & if ℓ = p, \end{cases}

where φ_{ℓ} denotes the geometric Frobenius automorphism of ℓ. As shown by Fukaya and Kato in [17, Th. 4.2.26], the behaviour of local ϵ-factors implies that p-adic L-functions can exist only after a suitable extension of scalars. To describe this we must assume that

(44)  \{ \begin{matrix}the maximal absolutely abelian subfield K^{ab,p} of K in which \\ p is unramified is finite. \end{matrix}

Under this hypothesis we let A denote the valuation ring of the completion at any padic place of the field K^{ab,p}. We set Λ_{A}(K/ℚ) := A ⊗_{ℤ_{p}} Λ(K/ℚ) and Λ_{A}(K/ℚ)_{S^{∗}} := A ⊗_{ℤ_{p}} Λ(K/ℚ)_{S^{∗}} and write ∂_{A,G_{K/ℚ}} : K_{1}(Λ_{A}(K/ℚ)_{S^{∗}}) → K_{0}(Λ_{A}(K/ℚ), Λ_{A}(K/ℚ)_{S^{∗}}) for the corresponding connecting homomorphism.

7.2.2. Elliptic curves. We first consider the case of the motive M = h^{1}(E)(1) of an elliptic curve E over ℚ with good ordinary reduction at p with K = ℚ(E(p)) being the extension of ℚ which arises by adjoining the p-power division points and we assume that G_{K/ℚ} does not contain any element of order p. In this situation the formulation of a (refined) main conjecture is very explicit since one can work with the dual X(E_{/K}) of the (p-primary) Selmer group; later we will give another formulation for general critical motives involving Selmer complexes. In the present situation one knows that the condition (44) is satisfied (cf. [13, just before Conj. 5.7]) and also that, if � (E_{/K^{ker(ρ)}}) is finite, then

r(M)(ρ) = dim_{ℂ_{p}}(e_{ρ^{∗}}(ℂ_{p} ⊗_{ℤ} E(K^{ker(ρ)}))).

Upon combining the leading term computations of [11, Th. 6.5] with [17, Th. 4.2.22] and the general approach of [13] we are led to formulate the following conjecture.

Conjecture 7.3. Fix a field K in F_{ℚ} that is unramified outside a finite set of places Σ and is such that G_{K/ℚ} has no element of order p. Then, under the above conditions, the module X(E_{/K}) belongs to 𝔐_{S^{∗}}(G_{K/ℚ}). Further, there exists an element L = L(E) of K_{1}(Λ_{A}(K/ℚ)_{S^{∗}}) which satisfies both of the following conditions:

(a) At each Artin representation ρ of G_{K/ℚ}, the value at T = 0 of T^{−r(M)(ρ)}Φ_{ρ}(L) is equal to

(−1)^{r(M)(ρ)} \frac{L_{B,Υ}^{∗}(M(ρ^{∗}))}{Ω_{∞}(M(ρ^{∗}))R_{∞}(M(ρ^{∗}))} · Ω_{p}(M(ρ^{∗}))R_{p}(M(ρ^{∗})) · \frac{P_{L,p}(Ŵ_{ρ}^{∗}(1), 1)}{P_{L,p}(Ŵ_{ρ}, 1)},

where L_{B,Υ}^{∗}(M(ρ^{∗})) is the leading coefficient at s = 0 of the complex L-function of M(ρ^{∗}), truncated by removing Euler factors for all primes in Υ.

(b) ∂_{A,G_{K/ℚ}}(L) = [Λ_{A}(K/ℚ) ⊗_{Λ(K/ℚ)} X(E_{/K})].

Remark 7.4. The interpolation formula in Conjecture 7.3(a) can of course also be stated in terms of the classical Hasse-Weil L-functions and their twists L(E, ρ^{∗}, s) in the sense of [13, (102)] (which is the same as the L-function attached to the B-motive h^{1}(E) ⊗ [ρ]^{∗}); due to the shift one now has to consider the leading term L^{∗}(E, ρ^{∗}) of L(E, ρ^{∗}, s) at s = 1. Moreover, one can simplify the above expression and make it more explicit. To this end we let u in ℤ_{p} be the unit root of the polynomial 1−a_{p}X +pX^{2} where, as usual, p + 1 − a_{p} = #Ẽ_{p}(𝔽_{p}) with Ẽ_{p} denoting the reduction of E modulo p. Furthermore we write p^{f_{ρ}} for the p-part of the conductor of ρ and ϵ_{p}(ρ) for the local ϵ-factor of ρ at the prime p. Moreover, let d_{+}(ρ) and d_{−}(ρ) denote the dimension of the subspace of [ρ]_{λ} on which complex conjugation acts by +1 and −1, respectively. We denote the periods of E by

Ω_{+}(E) := ∫_{γ^{+}} ω, Ω_{−}(E) := ∫_{γ^{−}} ω

where ω is the Néron differential and γ^{+} and γ^{−} denote a generator for the subspace of H_{1}(E(ℂ), ℤ) on which complex conjugation acts as +1 and −1 respectively. Finally, we write R_{∞}(E, ρ^{∗}) and R_{p}(E, ρ^{∗}) for the complex and p-adic regulators of E twisted by ρ^{∗}. Then the displayed expression in Conjecture 7.3(a) is equal to

(45) (−1)^{dim_{ℂp} (e_{ρ^{∗}} (ℂ_{p}⊗_{ℤ}E(K^{ker(ρ)})))} \frac{L_{R}^{∗}(E, ρ^{∗})}{Ω_{+}(E)^{d_{+}(ρ)}Ω_{−}(E)^{d_{−}(ρ)}R_{∞}(E, ρ^{∗})}

× ϵ_{p}(ρ)u^{−f_{ρ}}R_{p}(E, ρ^{∗})\frac{P_{L,p}([ρ]_{λ}, u^{−1})}{P_{L,p}([ρ]_{λ}^{∗}, up^{−1})}.

Here L_{R}^{∗}(E, ρ^{∗}) is the leading coefficient at s = 1 of the L-function L_{R}(E, ρ^{∗}, s) obtained from the Hasse-Weil L-function of E twisted by ρ^{∗} by removing the Euler factors at p and at all primes ℓ at which the j-invariant j_{E} of E in non-integral. See [17, Rem. 4.2.27] with u = α for the calculation of Ω_{p}(M(ρ^{∗})).

Before stating the next result we recall that the explicit interpolation formula given in the main conjecture of [13, Conj. 5.8] requires minor modification. To be precise, one must interchange all occurrences of ρ and ρ̂ on the right hand side of the equality of [loc. cit., (107)] except for the term ‘e_{p}(ρ)’ (for further details see the footnote at the end of §6.0 in [33]).

Proposition 7.5. Assume the hypotheses of Conjecture 7.3. Assume also that for all Artin representations ρ of G_{K/ℚ} the ‘order of vanishing part’ of the Birch and Swinnerton-Dyer Conjecture for E(ρ^{∗}) holds. Then Conjecture 7.3 implies the ‘main conjecture of non-commutative Iwasawa theory’ of [13, Conj. 5.8] (modified as above).

Proof. In this case Υ is the set comprising the prime p and all prime numbers q with ord_{q}(j_{E}) < 0 (see also [17, 4.5.3] or [33, Rem. 6.5]). In view of Remark 7.4 the only essential difference between the two conjectures is therefore that Conjecture 7.3 involves an interpolation formula for ((r(M)(ρ)!)^{−1} times) the value at T = 0 of the r(M)(ρ)-th derivative of Φ_{ρ}(L) rather than merely for the value at T = 0 of Φ_{ρ}(L) itself as in [13, Conj. 5.8]. (Note that the conjectured value at T = 0 of T^{−r(M)(ρ)}Φ_{ρ}(L) should be the leading term at T = 0 of Φ_{ρ}(L) only if R_{p}(M(ρ^{∗})) ≠ 0.) At the outset we note that r(Φ_{ρ}(L)) ≥ dim_{ℂ_{p}}(e_{ρ^{∗}}(E(K^{ker(ρ)}) ⊗_{ℤ} ℂ_{p})) ≥ 0 because the given interpolation formula has no pole. In particular, L does not have ∞ as its value at any ρ. We also note that the ‘order of vanishing part’ of the Birch and Swinnerton-Dyer Conjecture for E(ρ^{∗}) implies that the order of vanishing of L_{R}(E, ρ^{∗}, s) at s = 1 is equal to dim_{ℂ_{p}}(e_{ρ^{∗}}(ℂ_{p} ⊗_{ℤ} E(K^{ker(ρ)}))). We now assume that e_{ρ^{∗}}(ℂ_{p} ⊗_{ℤ} E(K^{ker(ρ)})) vanishes. Then both R_{p}(M(ρ^{∗})) = 1 and R_{∞}(M(ρ^{∗})) = 1. Also, the leading term L_{R}^{∗}(E, ρ^{∗}) is in this case equal to the value at s = 1 of L_{R}(E, ρ^{∗}, s). Hence, the interpolation formula (45) coincides with that given in [13, Conj. 5.8]. On the other hand, if e_{ρ^{∗}}(ℂ_{p} ⊗_{ℤ} E(K^{ker(ρ)})) ≠ 0, then r(Φ_{ρ}(L)) > 0 and so the value of L at ρ is equal to 0. In addition, in this case the function L_{R}(E, ρ^{∗}, s) vanishes at s = 1 and so the interpolation formula of [13, Conj. 5.8] also implies that the value of L at ρ is equal to 0, as required. □

7.2.3. The general case. We return to the more general case discussed in §7.2.1. We fix a full Galois stable ℤ_{p}-sublattice T of V and define a G_{ℚ_{p}}-stable ℤ_{p}-sublattice of V̂ by setting T̂ := T ∩ V̂ . As before we let 𝕋 denote the Galois representation Λ(K/ℚ) ⊗_{ℤ_{p}} T and set 𝕋̂ := Λ(K/ℚ) ⊗_{ℤ_{p}} T̂ similarly. Then 𝕋̂ is a G_{ℚ_{p}}-stable Λ(K/ℚ)-submodule of 𝕋. For the definition of the Selmer complex SC_{U} := SC_{U}(𝕋̂, 𝕋), which is originally due to Nekovář [24], we refer the reader to either [17, 4.1.2] or [11, (31)].

Conjecture 7.6 (General formulation for critical motives). Fix a field K in F_{ℚ} that is unramified outside a finite set of places Σ. Then, under the above conditions, the complex SC_{U} belongs to D_{S^{∗}}^{p}(Λ(K/ℚ)). Further, there exists an element ξ = ξ(U, M) of K_{1}(Λ_{A}(K/ℚ)_{S^{∗}}) which satisfies both of the following conditions:

(a) At each Artin representation ρ : G_{K/ℚ} → GL_{n}(O) for which neither P_{L,p}(Ŵ_{ρ}, 1) or P_{L,p}(W_{ρ}, 1) is equal to 0 the value at T = 0 of T^{−r(M)(ρ)}Φ_{ρ}(ξ) is equal to

(−1)^{r(M)(ρ)} \frac{L_{B,Σ}^{∗}(M(ρ^{∗}))}{Ω_{∞}(M(ρ^{∗}))R_{∞}(M(ρ^{∗}))} · Ω_{p}(M(ρ^{∗}))R_{p}(M(ρ^{∗})) · Γ(V̂)^{−1} · \frac{P_{L,p}(Ŵ_{ρ}^{∗}(1), 1)}{P_{L,p}(Ŵ_{ρ}, 1)},

where L_{B,Σ}^{∗}(M(ρ^{∗})) is the leading coefficient of the complex L-function of M(ρ^{∗}), truncated by removing Euler factors for all primes in Σ \ {∞}.

(b) ∂_{A,G_{K/ℚ}}(ξ) = χ(Λ_{A}(K/ℚ) ⊗_{Λ(K/ℚ)} SC_{U}).

Proposition 7.7. If M = h^{1}(E)(1) and K = ℚ(E(p)) are as in Conjecture 7.3, then Conjecture 7.6 is equivalent to Conjecture 7.3.

Proof. We note first that [17, Prop. 4.3.7] implies the complex SC_{U} belongs to D_{S^{∗}}^{p}(Λ(K/ℚ)) precisely when the module X(E_{/K}) belongs to 𝔐_{H_{K/ℚ}}(G_{K/ℚ}) = 𝔐_{S^{∗}}(G_{K/ℚ}). Also, SC_{U} differs from the complex SC(𝕋̂, 𝕋) in loc. cit. only by local terms which belong to 𝔐_{S^{∗}}(G_{K/ℚ}) (by [17, Prop. 4.3.6]) and have characteristic elements (denoted ζ(ℓ, K/ℚ) in loc. cit.) that correspond to the Euler-factors P_{L,ℓ}(W_{ρ}, s) and whose values P_{L,ℓ}(W_{ρ}, 1) at ρ are neither 0 or ∞ (by [17, Lem. 4.2.23]). To deduce the claimed result from here one need only note that Γ_{ℚ_{p}}(V̂) = 1 in this case and recall (from [17, Prop. 4.3.15-18]) that the class of SC(𝕋̂, 𝕋) in K_{0}(𝔐_{S^{∗}}(G_{K/ℚ})) is equal to [X(E_{/K})]. □

8. Equivariant Tamagawa numbers

Let F/k be a finite Galois extension of number fields. Then for any motive M defined over k the equivariant Tamagawa number conjecture of [9, Conj. 4.1(iv)] asserts the vanishing of an element T Ω(M_{F} , ℤ[G_{F/k}]) of K_{0}(ℤ[G_{F/k}], ℝ[G_{F/k}]) that is constructed from the various realisations and comparison isomorphisms associated to the motive M_{F} := F ⊗_{k} M. Here M_{F} is regarded as defined over k and endowed with a natural left action of ℚ[G_{F/k}] (via the first factor). Now the product over all primes p and all field isomorphisms j : ℂ ≅ ℂ_{p} of the composite homomorphism

j_{∗} : K_{0}(ℤ[G_{F/k}], ℝ[G_{F/k}]) → K_{0}(ℤ[G_{F/k}], ℂ[G_{F/k}])

⟶ K_{0}(ℤ[G_{F/k}], ℂ_{p}[G_{F/k}]) → K_{0}(ℤ_{p}[G_{F/k}], ℂ_{p}[G_{F/k}])

is injective, where the second map is induced by j (cf. [4, Lem. 2.1]). To prove [9, Conj. 4.1(iv)] it therefore suffices to prove that j_{∗}(T Ω(M_{F} , ℤ[G_{F/k}])) = 0 for every such j. This reduction has the further advantage that the element j_{∗}(T Ω(M_{F} , ℤ[G_{F/k}])) can be directly defined without assuming the ‘Coherence Hypothesis’ of [9, §3.3] that is necessary to define T Ω(M_{F} , ℤ[G_{F/k}]) (cf. [9, Rem. 8]). However, even if one assumes the standard compatibility conjectures concerning the definition of Euler factors (cf. [9, Conj. 3]), the definition of j_{∗}(T Ω(M_{F} , ℤ[G_{F/k}])) is in general still conditional, being dependent upon the conjectural existence of a fundamental exact sequence relating the motivic cohomology spaces of M_{F} and its Kummer dual [9, Conj. 1] and of canonical p-adic Chern class isomorphisms [9, Conj. 2]. In particular, since we are assuming here that the element j_{∗}(T Ω(M_{F} , ℤ[G_{F/k}])) is well-defined, the results that we prove in this section will not shed any new light on either of [9, Conj. 1, Conj. 2].

8.1. Tate motives. In this subsection we fix a finite Galois extension F/k of totally real number fields and write ℚ(1)_{F} for the motive h^{0}(Spec F)(1), regarded as defined over k and with coefficients ℚ[G_{F/k}]. We recall that all of the conjectures necessary for the definition of T Ω(ℚ(1)_{F} , ℤ[G_{F/k}]) are known to be valid and hence that j_{∗}(T Ω(ℚ(1)_{F} , ℤ[G_{F/k}])) is defined unconditionally as an element of K_{0}(ℤ_{p}[G_{F/k}], ℂ_{p}[G_{F/k}]). For a discussion of various explicit consequences of the vanishing of T Ω(ℚ(1)_{F} , ℤ[G_{F/k}]) see [4].

Theorem 8.1. Let K be any field which belongs to F_{k}^{+}, contains F and is such that G_{K/k} has no element of order p (such a field K exists by virtue of Lemma 6.1). If K validates Conjecture 7.1 and F validates Leopoldt’s Conjecture (at p), then one has j_{∗}(T Ω(ℚ(1)_{F} , ℤ[G_{F/k}])) = 0.

Proof. For any quotient G of G_{K/k} we write Λ(G)^{#}(1) for the Λ(G)-module Λ(G) endowed with the following action of G_{k}: each σ in G_{k} acts on Λ(G)^{#}(1) as right multiplication by the element χ_{cyc}(σ)σ̄^{−1} where σ̄ denotes the image of σ in G. If K′ is the subfield of K with G_{K′/k} = G and O_{k,Σ} is the subring of k comprising elements that are integral at all places outside Σ then, following Fukaya and Kato [17, §2.1.1] and Nekovář [24], the compact support cohomology complex C_{K′} := RΓ_{c,ét}(O_{k,Σ}, Λ(G)^{#}(1)) is an object of D^{p}(Λ(G)) that lies in a canonical exact triangle in D(Λ(G)) of the form

(46)  C_{K′} → RΓ_{ét}(O_{k,Σ}, Λ(G)^{#}(1)) → ⨁_{v∈Σ} RΓ_{ét}(k_{v}, Λ(G)^{#}(1)) → C_{K′}[1].

In the sequel we use the following facts: there is a natural isomorphism in D^{p}(Λ(G)) of the form

(47)  Λ(G) ⊗_{Λ(K/k)}^{𝕃} C_{K} ≅ C_{K′};

in each degree i there is a natural isomorphism H^{i}(C_{K}) ≅ \underleftarrow{lim}_{K′} H^{i}(C_{K′}) where K′ runs over all finite degree Galois extensions K′/k with K′ ⊆ K and the limit is taken with respect to the natural corestriction maps; for each such K′ there are natural identifications H^{i}(C_{K′}) ≅ H_{c,ét}^{i}(O_{K′,Σ}, ℤ_{p}(1)), H^{i}(RΓ_{ét}(O_{k,Σ}, Λ(G)^{#}(1))) ≅ H_{ét}^{i}(O_{K′,Σ}, ℤ_{p}(1)) and H^{i}(RΓ_{ét}(k_{v}, Λ(G)^{#}(1))) ≅ ⨁_{w|v} H_{ét}^{i}(K′_{w}, ℤ_{p}(1)). In particular, by a standard computation (involving Kummer theory, class field theory and arithmetic duality) one obtains canonical identifications

(48)  H^{i}(C_{K′}) ≅ \begin{cases}ker(λ_{K′}), & i = 1 \\ X_{Σ}(K′), & i = 2 \\ ℤ_{p}, & i = 3 \\ 0, & otherwise, \end{cases}

where λ_{K′} is the diagonal map from O_{K′}[\frac{1}{p}]^{×} ⊗ ℤ_{p} to the direct sum over w ∈ S_{p}(K′) of the pro-p-completion (K′_{w})^{×} ˆ⊗ℤ_{p} of (K′_{w})^{×} and X_{Σ}(K′) is the Galois group of the maximal abelian pro-p extension of K′ that is unramified outside Σ. By passing to the limit over K′ ⊂ K one finds that H^{i}(C_{K}) is acyclic outside degrees 2 and 3 and that its cohomology in degrees 2 and 3 is canonically isomorphic to X_{Σ}(K) and ℤ_{p} respectively. The first claim of Conjecture 7.1 is therefore equivalent to asserting that C_{K} belongs to D_{S^{∗}}^{p}(Λ(K/k)). In addition, since χ(C_{K}) = ∑_{i∈ℤ}(−1)^{i}[H^{i}(C_{K})], Conjecture 7.1(b) asserts that ∂_{G_{K/k}}(ξ) = χ(C_{K}) + [ℤ_{p}] or equivalently, by Proposition 4.7(ii)(a) and (b), that

(49)  ∂_{G_{K/k}}(ξ′) = χ(C_{K})

with ξ′ := char_{G_{K/k},γ_{k}}(ℤ_{p}[0]) · ξ. In the next result we set c_{k} := log_{p}(χ_{cyc}(γ_{k})).

Lemma 8.2. Assume Leopoldt’s Conjecture is valid for F (at p) and fix an Artin representation ρ : G_{K/k} → GL_{n}(O) such that V_{ρ} is an irreducible representation of G_{F/k}.

(i) Conjecture 7.1(a) implies that (ξ′)^{∗}(ρ) = c_{k}^{−⟨ρ,1⟩}Ω_{j}(ρ)L_{Σ}^{∗}(1, ρ^{j^{−1}})^{j}.

(ii) If ρ is non-trivial, then ℚ_{p} ⊗_{Λ(Γ_{k})}^{𝕃} C_{K,ρ} is acyclic and hence C_{K} is semisimple at

ρ, r_{G_{K/k}}(C_{K})(ρ) = 0 and t(C_{K,ρ}) is the canonical morphism. (iii) If ρ is trivial, then ℚ_{p} ⊗_{Λ(Γ)}^{𝕃} C_{K,ρ} is acyclic outside degrees 2 and 3 and its

cohomology in degrees 2 and 3 identifies with ℚ_{p} ⊗_{ℤ_{p}} cok(λ_{k}) and ℚ_{p} respectively. Further, C_{K} is semisimple at ρ, r_{G_{K/k}}(C_{K})(ρ) = 1 and (−1)×t(C_{K,ρ}) is induced by the isomorphism β : ℚ_{p} ⊗_{ℤ_{p}} cok(λ_{k}) → ℚ_{p} that sends each element (x_{v})_{v} of ∏_{v∈S_{p}(k)} k_{v}^{×} to c_{k}^{−1} ∑_{v} log_{p}(N_{v}(x_{v})) with N_{v} the field-theoretic norm k_{v}^{×} → ℚ_{p}^{×}.

Proof. Leopoldt’s Conjecture implies that the determinant Ω_{j}(ρ) and hence also the product (c_{k}^{⟨ρ,1⟩}c_{ρ,k})^{−1}Ω_{j}(ρ)L_{Σ}^{∗}(1, ρ^{j^{−1}})^{j} in Conjecture 7.1(a) is non-zero. The latter conjecture therefore implies that ξ^{∗}(ρ) = ξ(ρ) = (c_{k}^{⟨ρ,1⟩}c_{ρ,k})^{−1}Ω_{j}(ρ)L_{Σ}^{∗}(1, ρ^{j^{−1}})^{j}. Next we observe that claim (i) is a consequence of the equality

(50)  Φ_{ρ}(char_{G_{K/k},γ_{k}}(ℤ_{p}[0])) = \begin{cases}1 − ρ(γ_{k}^{−1})(1 + T)^{−1}, & if H_{K/k} ⊆ ker(ρ), \\ 1, & otherwise. \end{cases}

Indeed, if (50) is true, then char_{G_{K/k},γ_{k}}(ℤ_{p}[0])^{∗}(ρ) = c_{ρ,k} and so claim (i) follows from the obvious equalities (ξ′)^{∗}(ρ) = (char_{G_{K/k},γ_{k}}(ℤ_{p}[0]) · ξ)^{∗}(ρ) = char_{G_{K/k},γ_{k}}(ℤ_{p}[0])^{∗}(ρ)ξ^{∗}(ρ). To prove (50) we regard M_{ρ} := Λ(Γ_{k}) ⊗_{ℤ_{p}} M_{n}(O) as a (Λ_{O}(Γ_{k}), Λ(H_{K/k}))-bimodule, where the (left) action of Λ_{O}(Γ_{k}) is clear and the (right) action of each element h of H_{K/k} is via x ⊗ y ↦ x ⊗ yρ(h). Then the definition of char_{G_{K/k},γ_{k}}(ℤ_{p}[0]) combines with the definition of Φ_{ρ} to imply that

(51) Φ_{ρ}(char_{G_{K/k},γ_{k}}(ℤ_{p}[0]))

= det_{Q(O[[T]])}(id ⊗ id − id ⊗ θ | Q(O[[T]]) ⊗_{Λ_{O}(Γ)} (M_{ρ} ⊗_{Λ(H_{K/k})} ℤ_{p}))

where θ is the endomorphism of M_{ρ} ⊗_{Λ(H_{K/k})} ℤ_{p} = (Λ(Γ_{k}) ⊗_{ℤ_{p}} M_{n}(O)) ⊗_{Λ(H_{K/k})} ℤ_{p} that sends each element (x ⊗ y) ⊗ z to (xγ_{k}^{−1} ⊗ yρ(γ̃_{k}^{−1})) ⊗ z (this recipe is independent of the choice of lift γ̃_{k} of γ_{k} through G_{K/k} → Γ_{k}). Now V_{ρ} is irreducible and H_{K/k} is normal in G_{K/k} and so M_{ρ} ⊗_{Λ(H_{K/k})} ℚ_{p} is either canonically isomorphic to M_{ρ} ⊗_{ℤ_{p}} ℚ_{p} or vanishes depending on whether H_{K/k} ⊆ ker(ρ) or not. Thus, if H_{K/k} ⊄ ker(ρ), then (51) implies Φ_{ρ}(char_{G_{K/k},γ_{k}}(ℤ_{p}[0])) is the determinant of an endomorphism of the zero space and so equal to 1. On the other hand, if H_{K/k} ⊆ ker(ρ), then n = 1 (since Γ_{k} is abelian and V_{ρ} is irreducible) and so (51) implies Φ_{ρ}(char_{G_{K/k},γ_{k}}(ℤ_{p}[0])) is the determinant of the endomorphism of Q(O[[T]]) given by multiplication by 1 − ρ(γ_{k}^{−1})(1 + T)^{−1}. The required equality (50) is therefore clear. To prove claims (ii) and (iii) we note that in each degree i the isomorphism (47) induces an identification H^{i}(ℚ_{p} ⊗_{Λ(Γ_{k})}^{𝕃} C_{K,ρ}) ≅ H^{i}(C_{F})^{ρ}. Further, if Leopoldt’s Conjecture is valid for F, then ker(λ_{F}) vanishes and ℚ_{p} ⊗_{ℤ_{p}} X_{Σ}(F) is a trivial G_{F/k}-module and so the explicit descriptions of (48) with K′ = F imply C_{F} is acyclic outside degrees 2 and 3 and moreover that each space H^{i}(C_{F})^{ρ} vanishes if ρ is non-trivial. This proves the first assertion of claim (ii) and then all remaining assertions of claim (ii) follow immediately from [11, Lem. 3.13]. Also the long exact cohomology sequence of (46) induces an identification H_{c,ét}^{2}(O_{k,Σ}, ℚ_{p}(1)) ≅ cok(λ_{k}) ⊗_{ℤ_{p}} ℚ_{p} and, if we use (47) with K′ = k^{cyc} to identify C_{K,ρ} with C_{k^{cyc}}, then [11, Lem. 3.13] implies that all remaining assertions of claim (iii) will follow if we can show that the given isomorphism β is equal to −1 times the Bockstein homomorphism β_{Δ_{k,c}}^{2} in degree 2 of the canonical exact triangle

Δ_{k,c} : ℚ_{p} ⊗_{ℤ_{p}} RΓ_{c,ét}(O_{k,Σ}, T_{k}) ⟶^{γ_{k}−1} ℚ_{p} ⊗_{ℤ_{p}} RΓ_{c,ét}(O_{k,Σ}, T_{k}) → RΓ_{c,ét}(O_{k,Σ}, ℚ_{p}(1)) →

where T_{k} := Λ(k^{cyc}/k)^{#}(1). Now the argument of [11, §3.2.1] shows that β_{Δ_{k,c}}^{2} is equal to the homomorphism H_{c,ét}^{2}(O_{k,Σ}, ℚ_{p}(1)) → H_{c,ét}^{3}(O_{k,Σ}, ℚ_{p}(1)) induced by taking cupproduct with the element φ_{k} of H_{ét}^{1}(O_{k,Σ}, ℤ_{p}) = Hom_{cts}(M_{Σ}(k), ℤ_{p}) obtained by composing the projection M_{Σ}(k) → Γ_{k} with the continuous homomorphism Γ_{k} → ℤ_{p} that sends γ_{k} to 1. Since cup products commute with corestriction we therefore obtain a commutative diagram

H_{c,ét}^{2}(O_{k,Σ}, ℚ_{p}(1)) ⟶^{β_{Δ_{k,c}}^{2}} H_{c,ét}^{3}(O_{k,Σ}, ℚ_{p}(1))
κ^{2}↓  _{κ^{3}}↓
H_{c,ét}^{2}(ℤ_{Σ′}, ℚ_{p}(1)) ⟶^{d_{k}^{−1}β_{Δ_{ℚ,c}}^{2}} H_{c,ét}^{3}(ℤ_{Σ′}, ℚ_{p}(1))

in which Σ′ is the set of rational places lying below those in Σ, Δ_{ℚ,c} denotes the exact triangle obtained from Δ_{k,c} by replacing k and Σ by ℚ and Σ′ respectively, the vertical arrows are the natural corestriction maps and d_{k} occurs in the lower row because the restriction of φ_{ℚ} ∈ H_{ét}^{1}(O_{ℚ,Σ}, ℤ_{p}) to H_{ét}^{1}(O_{k,Σ}, ℤ_{p}) is equal to φ_{k}^{d_{k}} (since γ_{k} = γ_{ℚ}^{d_{k}}). But, with respect to the canonical identifications H_{c,ét}^{2}(O_{k,Σ}, ℚ_{p}(1)) ≅ cok(λ_{k}) ⊗_{ℤ_{p}} ℚ_{p} and H_{c,ét}^{3}(O_{k,Σ}, ℚ_{p}(1)) ≅ ℚ_{p} (and similarly with k and Σ replaced by ℚ and Σ′), the map κ^{2} is induced by the norm maps N_{v} : k_{v}^{×} → ℚ_{p}^{×} and κ^{3} is the identity map. Thus, since d_{k} × c_{ℚ} = c_{k}, it is enough for us to prove that (−1) × β_{Δ_{ℚ,c}}^{2} is induced by the homomorphism c_{ℚ}^{−1} · log_{p} : ℚ_{p}^{×} → ℚ_{p}. To compute β_{Δ_{ℚ,c}}^{2} explicitly we use the morphism of natural exact triangles

(52)

RΓ_{ét}(ℚ_{p}, T_{ℚ}) ⟶^{γ_{ℚ}−1} RΓ_{ét}(ℚ_{p}, T_{ℚ}) ⟶ RΓ_{ét}(ℚ_{p}, ℤ_{p}(1)) ⟶
↓  ↓  ^{θ}↓
RΓ_{c,ét}(ℤ_{Σ′}, T_{ℚ})[1] ⟶^{γ_{ℚ}−1} RΓ_{c,ét}(ℤ_{Σ′}, T_{ℚ})[1] ⟶ RΓ_{c,ét}(ℤ_{Σ′}, ℤ_{p}(1))[1] ⟶

in which each vertical morphism is induced by the definition of compact support cohomology. Indeed, from the long exact cohomology sequences of the rows in this diagram we obtain a commutative diagram

H_{ét}^{1}(ℚ_{p}, ℚ_{p}(1)) ⟶^{ℚ_{p}⊗_{ℤp} H^{1}(θ)} H_{c,ét}^{2}(ℤ_{Σ′}, ℚ_{p}(1))
β^{1}↓  ↓^{(−1)×β_{Δ_{ℚ,c}}^{2}}
H_{ét}^{2}(ℚ_{p}, ℚ_{p}(1)) ⟶^{ℚ_{p}⊗_{ℤp} H^{2}(θ)} H_{c,ét}^{3}(ℤ_{Σ′}, ℚ_{p}(1)).

Here ℚ_{p} ⊗_{ℤ_{p}} H^{1}(θ) identifies with the natural surjection H_{ét}^{1}(ℚ_{p}, ℚ_{p}(1)) ≅ (ℚ_{p}^{×} ˆ⊗ℤ_{p}) ⊗_{ℤ_{p}} ℚ_{p} → cok(λ_{ℚ})⊗_{ℤ_{p}} ℚ_{p} ≅ H_{c,ét}^{2}(ℤ_{Σ′}, ℚ_{p}(1)), ℚ_{p} ⊗_{ℤ_{p}} H^{2}(θ) is induced by the identifications H_{ét}^{2}(ℚ_{p}, ℚ_{p}(1)) ≅ ℚ_{p} and H_{c,ét}^{3}(ℤ_{Σ′}, ℚ_{p}(1)) ≅ ℚ_{p} and the identity map on ℚ_{p}, β^{1} is the Bockstein homomorphism in degree 1 of the image under ℚ_{p} ⊗_{ℤ_{p}} − of the upper row of (52) and the factor −1 occurs on the right hand vertical arrow because of the 1-shift in the lower row of (52). To complete the proof of claim (iii) it thus suffices to recall that the homomorphism β^{1} is induced by c_{ℚ}^{−1} · log_{p} (for a proof of this fact see, for example, [10, p. 352]). □

Returning to the proof of Theorem 8.1 we now apply Theorem 2.2 to the conjectural equality (49). By taking into account the canonical isomorphism (47) with K′ = F and the explicit descriptions given in Lemma 8.2 we therefore deduce that

(53)  ∂_{G_{F/k}}((c_{k}^{−⟨ρ,1⟩}Ω_{j}(ρ)L_{Σ}^{∗}(1, ρ^{j^{−1}})^{j})_{ρ∈Irr(G_{F/k})}) = −[d_{ℤ_{p}[G_{F/k}]}(C_{F}), β_{∗}]

where β_{∗} is the morphism d_{ℂ_{p}[G_{F/k}]}(ℂ_{p}[G_{F/k}] ⊗_{ℤ_{p}[G_{F/k}]}^{𝕃} C_{F}) → 1_{ℂ_{p}[G_{F/k}]} that is induced by the isomorphism

ℚ_{p} ⊗_{ℤ_{p}} H^{2}(C_{F}) ≅ ℚ_{p} ⊗_{ℤ_{p}} cok(λ_{k}) ⟶^{β} ℚ_{p} ≅ ℚ_{p} ⊗_{ℤ_{p}} H^{3}(C_{F})

coming from Lemma 8.2(ii) and (iii). But the proof of [11, Th. 5.5] shows that (53) is equivalent to an equality of the form [d_{ℤ_{p}[G_{F/k}]}(C_{F}), β′_{∗}] = 0 where β′_{∗} = (β′_{ρ})_{ρ∈Irr(G_{F/k})} under the identification (4) and each β′_{ρ} is the explicit morphism described in [11, (25)]. The fact that (53) implies the vanishing of the element j_{∗}(T Ω(ℚ(1)_{F} , ℤ[G_{F/k}])) then follows directly upon explicitly comparing the definition of T Ω(ℚ(1)_{F} , ℤ[G_{F/k}]) with that of each morphism β′_{ρ}. This therefore completes the proof of Theorem 8.1. □

We end this subsection by noting that the above computations show that the correct generalisation of Conjecture 7.1 (to groups with an element of order p) is the following.

Conjecture 8.3. Fix a totally real number field k and a field K in F_{k}^{+} that is unramified outside a finite set of places Σ. Then C_{K} belongs to D_{S^{∗}}^{p}(Λ(K/k)). Further, there exists an element ξ′ of K_{1}(Λ(K/k)_{S^{∗}}) which satisfies both of the following conditions.

(a) At each Artin representation ρ of G_{K/k} one has

ξ′(ρ) = log_{p}(χ_{cyc}(γ_{k}))^{−⟨ρ,1⟩}Ω_{j}(ρ)L_{Σ}^{∗}(1, ρ^{j^{−1}})^{j}.

(b) ∂_{G_{K/k}}(ξ′) = χ(C_{K}).

This conjecture is compatible with that formulated (in the case that G_{K/k} has rank one) by Ritter and Weiss in [25, §4]. For further details see [8].

8.2. Critical Motives. In this subsection we assume the notation and hypotheses of Conjecture 7.6 and fix a subfield F of K that is both Galois and of finite degree over ℚ. We set 𝕋̂_{F} := Λ(F/ℚ) ⊗_{ℤ_{p}} T̂ ≅ Λ(F/ℚ) ⊗_{Λ(K/ℚ)} 𝕋̂. We write Z = Z_{ρ} and Z̃ = Z̃_{ρ} for the Kummer duals W_{ρ}^{∗}(1) and Ŵ_{ρ}^{∗}(1) of W_{ρ} and Ŵ_{ρ} respectively; finally we set W̃ = W̃_{ρ} := W_{ρ}/Ŵ_{ρ}. In terms of the notation of [11] we consider the following assumption on W_{ρ}.

Assumption (W): For each ρ in Irr(G_{F/ℚ}) the space W = W_{ρ} satisfies all of the following conditions:-

(A_{1}) P_{ℓ}(W, 1)P_{ℓ}(Z, 1) ≠ 0 for all primes ℓ ≠ p, (B_{1}) P_{p}(W, 1)P_{p}(Z, 1) ≠ 0, (C_{1}) P_{p}(W̃ , 1)P_{p}(Z̃, 1) ≠ 0 and (D_{2}) H_{f}^{0}(ℚ, W) = H_{f}^{0}(ℚ, Z) = 0.

In the following result we write ι_{A} : K_{0}(ℤ_{p}[G_{F/ℚ}], ℂ_{p}[G_{F/ℚ}]) → K_{0}(A[G_{F/ℚ}], ℂ_{p}[G_{F/ℚ}]) for the canonical homomorphism obtained by regarding A as a subring of ℂ_{p}. We also recall that [9, Conj. 4(iii)] (which is a natural equivariant version of the Deligne-Beilinson Conjecture) for the motive M_{F}, regarded as defined over ℚ and with an action of ℚ[G_{F/ℚ}], implies that the element j_{∗}(T Ω(M_{F} , ℤ[G_{F/ℚ}])) belongs to the image of the natural map K_{0}(ℤ_{p}[G_{F/ℚ}], ℚ_{p}[G_{F/ℚ}]) → K_{0}(ℤ_{p}[G_{F/ℚ}], ℂ_{p}[G_{F/ℚ}]).

Theorem 8.4. Assume that • Assumption (W) is valid;

• the complex SC_{U} is semi-simple at all ρ in Irr(G_{F/ℚ});

• an ϵ-isomorphism

ϵ_{p,ℤ_{p}[G_{F/ℚ}]}(𝕋̂_{F}) : 1_{ℤ_{p}[G_{F/ℚ}]} → d_{ℤ_{p}[G_{F/ℚ}]}(RΓ(ℚ_{p}, 𝕋̂_{F}))d_{ℤ_{p}[G_{F/ℚ}]}(𝕋̂_{F})

in the sense of [17, Conj. 3.4.3] exists;

• Conjecture 7.6 is valid for the motive M and the extension K/ℚ. Then ι_{A}(j_{∗}(T Ω(M_{F} , ℤ[G_{F/ℚ}]))) vanishes. Further, if j_{∗}(T Ω(M_{F} , ℤ[G_{F/ℚ}])) belongs to the image of the natural map K_{0}(ℤ_{p}[G_{F/ℚ}], ℚ_{p}[G_{F/ℚ}]) → K_{0}(ℤ_{p}[G_{F/ℚ}], ℂ_{p}[G_{F/ℚ}]), then j_{∗}(T Ω(M_{F} , ℤ[G_{F/ℚ}])) vanishes.

Proof. We fix an element ξ as in Conjecture 7.6. Since SC_{U} is semisimple at each ρ in Irr(G_{F/ℚ}), the obvious analogue of Theorem 2.2 with A in place of ℤ_{p} combines with Conjecture 7.6(b) to imply that

∂_{G_{F/ℚ}}((ξ^{∗}(ρ))_{ρ∈Irr(G_{F/ℚ})}) = −ι_{A}([d_{ℤ_{p}[G_{F/ℚ}]}(SC_{U}(𝕋̂_{F} , 𝕋_{F})), t(SC_{U}(𝕋̂_{F} , 𝕋_{F}))_{G_{F/ℚ}}]).

After unwinding the identification (2), this means that there exists a morphism in V (A[G_{F/ℚ}])

ψ : 1_{A[G_{F/ℚ}]} → d_{A[G_{F/ℚ}]}(A[G_{F/ℚ}] ⊗_{ℤ_{p}[G_{F/ℚ}]} SC_{U}(𝕋̂_{F} , 𝕋_{F}))

such that

(ξ^{∗}(ρ)^{−1})_{ρ∈Irr(G_{F/ℚ})} = t(SC_{U}(𝕋̂_{F} , 𝕋_{F}))_{G_{F/ℚ}} ∘ ψ_{ℂ_{p}[G_{F/ℚ}]}

∈ Aut_{V (ℂ_{p}[G_{F/ℚ}])}(1_{ℂ_{p}[G_{F/ℚ}]}) ≅ K_{1}(ℂ_{p}[G_{F/ℚ}])

under the identification (4). After recalling the explicit definition of the morphism t(SC_{U}(𝕋̂_{F} , 𝕋_{F}))_{G_{F/ℚ}} given in Theorem 2.2 and then taking inverses we obtain a morphism in V (A[G_{F/ℚ}])

ψ^{−1} : 1_{A[G_{F/ℚ}]} → d_{A[G_{F/ℚ}]}(A[G_{F/ℚ}] ⊗_{ℤ_{p}[G_{F/ℚ}]} SC_{U}(𝕋̂_{F} , 𝕋_{F}))^{−1}

such that

(−1)^{r_{G}(SC_{U})(ρ)}ξ^{∗}(ρ) = t(SC_{U}(ρ^{∗}))^{−1} ∘ ψ^{−1}(ρ) ∈ Aut_{V (ℂ_{p})}(1_{ℂ_{p}}) ≅ ℂ_{p}^{×}

for all ρ in Irr(G_{F/ℚ}). Here we write ψ^{−1}(ρ) for the ρ-component of the morphism 1_{ℂ_{p}[G_{F/ℚ}]} → d_{ℂ_{p}[G_{F/ℚ}]}(ℂ_{p}[G_{F/ℚ}] ⊗_{ℤ_{p}[G_{F/ℚ}]} SC_{U}(𝕋̂_{F} , 𝕋_{F}))^{−1} that is induced by ψ^{−1}. This is equivalent to asserting the existence of a morphism in V (A[G_{F/ℚ}])

ψ′ : 1_{A[G_{F/ℚ}]} → d_{A[G_{F/ℚ}]}(A[G_{F/ℚ}] ⊗_{ℤ_{p}[G_{F/ℚ}]} RΓ_{c}(U, 𝕋_{F}))^{−1}

such that for all ρ in Irr(G_{F/ℚ}) the composite morphism

(54) 1_{ℂ_{p}} ⟶^{ψ′(ρ)_{ℂp}} d_{L}(RΓ_{c}(U, W_{ρ}))_{ℂ_{p}}^{−1} ⟶^{β(ρ)ϵ(𝕋̂)^{−1}(ρ)} d_{L}(SC_{U}(Ŵ_{ρ}, W_{ρ}))_{ℂ_{p}}^{−1}

⟶^{t(SC_{U} (ρ^{∗}))_{ℂp}^{−1}} 1_{ℂ_{p}}

corresponds to (−1)^{r_{G}(SC_{U})(ρ)}ξ^{∗}(ρ). In this displayed expression we write ϵ(𝕋̂)(ρ) for V_{ρ^{∗}} ⊗_{ℤ_{p}[G_{F/ℚ}]} ϵ_{p,ℤ_{p}[G_{F/ℚ}]}(𝕋̂_{F}) and β(ρ) for V_{ρ^{∗}} ⊗_{ℤ_{p}[G_{F/ℚ}]} (ℤ_{p}[G_{F/ℚ}]⊗_{Λ(G)} β) ≅ V_{ρ^{∗}} ⊗_{Λ(G)} β with β the morphism d_{Λ}(𝕋^{+})Λ̃ ≅ d_{Λ}(𝕋̂)Λ̃ that is defined in [11, (35)], and all underlying identifications are as explained in [11, §6]. Now the hypothesis that SC_{U} is semisimple at ρ combines with the assumption (W), the duality isomorphism H_{f}^{3}(ℚ, W) ≅ H_{f}^{0}(ℚ, Z) and the results of [11, Lem. 6.7 and Lem. 3.13(ii)] to imply that the algebraic rank r(M)(ρ) defined in (43) is equal to r_{G}(SC_{U})(ρ), that [11, Condition (F)] is satisfied and that the value at T = 0 of T^{−r(M)(ρ)}Φ_{ρ}(ξ) is equal to the leading term ξ^{∗}(ρ). Conjecture 7.6(a) therefore gives an explicit formula for ξ^{∗}(ρ). Taking this formula into account, one can compare the composite morphism (54) to the first displayed morphism after [11, Lem. 6.8]. After unwinding the proof of [11, Th. 6.5] (for which we use assumption (W)) this comparison shows that

ψ′(ρ) = ϑ_{λ}(M(ρ^{∗}))_{ℂ_{p}} ∘ ζ_{K}(M(ρ^{∗}))_{ℂ_{p}}

for all ρ in Irr(G_{F/ℚ}), where ϑ_{λ}(M(ρ^{∗}))_{ℂ_{p}} and ζ_{K}(M(ρ^{∗}))_{ℂ_{p}} are the morphisms that occur in [11, Conj. 4.1]. Finally we note that the validity of the last displayed equality (for all ρ in Irr(G_{F/ℚ})) is equivalent to asserting that the element ι_{A}(j_{∗}(T Ω(M_{F} , ℤ[G_{F/ℚ}]))) vanishes (by the very definition of the latter element). This proves the first claim of the theorem. The second claim of Theorem 8.4 will now follow if we can show that the natural composite homomorphism

K_{0}(ℤ_{p}[G_{F/ℚ}], ℚ_{p}[G_{F/ℚ}]) → K_{0}(ℤ_{p}[G_{F/ℚ}], ℂ_{p}[G_{F/ℚ}]) ⟶^{ι_{A}} K_{0}(A[G_{F/ℚ}], ℂ_{p}[G_{F/ℚ}])

is injective. To do this we write F for the field of fractions of A (so F ⊂ ℂ_{p}). Then, since the natural scalar extension map K_{1}(F [G_{F/ℚ}]) → K_{1}(ℂ_{p}[G_{F/ℚ}]) is injective, the exact commutative diagram (1) with R = A[G_{F/ℚ}], R′ = F [G_{F/ℚ}] and R′′ = ℂ_{p}[G_{F/ℚ}] implies that the natural homomorphism K_{0}(A[G_{F/ℚ}], F [G_{F/ℚ}]) → K_{0}(A[G_{F/ℚ}], ℂ_{p}[G_{F/ℚ}]) is also injective. It therefore suffices to prove that the natural homomorphism K_{0}(ℤ_{p}[G_{F/ℚ}], ℚ_{p}[G_{F/ℚ}]) → K_{0}(A[G_{F/ℚ}], F [G_{F/ℚ}]) is injective. But, since A/ℤ_{p} is unramified, this is an immediate consequence of a result of M. Taylor [31, Chap. 8, Th. 1.1]. Indeed, one need only note that the groups K_{0}(ℤ_{p}[G_{F/ℚ}], ℚ_{p}[G_{F/ℚ}]) and K_{0}(A[G_{F/ℚ}], F [G_{F/ℚ}]) are naturally isomorphic to the Grothendieck groups K_{0}T (ℤ_{p}[G_{F/ℚ}]) and K_{0}T (A[G_{F/ℚ}]) which occur in loc. cit. □

Remark 8.5. If M = h^{1}(A) for an abelian variety A that has good ordinary reduction at p and is such that the Tate-Shafarevich group � (A_{/F}) of A over F is finite, then the vanishing of j_{∗}(T Ω(M_{F} , ℤ[G_{F/ℚ}])) implies the ‘p-part’ of a Birch and Swinnerton-Dyer type formula (see, for example, [33, §3.1]). However, Conjecture 7.6 does not itself imply that � (A_{/F}) is finite.

Remark 8.6. Explicit consequences of Conjecture 7.6 for the values (at s = 1) of twisted Hasse-Weil L-functions have been described by Coates et al in [13], by Kato in [20] and by Dokchister and Dokchister in [16]. However, all of the consequences described in [13, 16, 20] become trivial when the L-functions vanish. One of the key advantages of Theorem 8.4 is that in many of these cases it can be combined with the approach of [7] to show that Conjecture 7.6 implies a variety of explicit (and highly non-trivial) congruence relations between values of derivatives of twisted Hasse-Weil L-functions. Such explicit (conjectural) congruences will be considered elsewhere.

Remark 8.7. Following Theorem 8.4 it is of some interest to study elements in K-theory of the form T Ω(h^{1}(E_{/F})(1), ℤ[G_{F/ℚ}]) with E an elliptic curve over ℚ and F/ℚ a finite non-abelian Galois extension. The study of such elements is however still very much in its infancy. Indeed, the only explicit computation that we are currently aware of is the following. Let E be the elliptic curve y^{2} + y = x^{3} − x^{2} − 10x − 20 (this is the curve 11A1 in the sense of Cremona [14]). Then, with F equal to the splitting field of the polynomial x^{3} − 4x − 1, the group G_{F/ℚ} is dihedral of order 6 and Navilarekallu [23] has proved numerically that if � (E_{/F}) is trivial, then the element T Ω(h^{1}(E_{/F})(1), ℤ[G_{F/ℚ}]) vanishes.

Appendix A. Determinant functors

In this appendix we recall the formalism of determinant functors introduced by Fukaya and Kato in [17] and used in [11] (see also [33]) For any ring R we write B(R) for the category of bounded complexes of (left) R-modules, C(R) for the category of bounded complexes of finitely generated (left) R-modules, P (R) for the category of finitely generated projective (left) R-modules, C^{p}(R) for the category of bounded (cohomological) complexes of finitely generated projective (left) R-modules. By D^{p}(R) we denote the category of perfect complexes as full triangulated subcategory of the derived category D^{b}(R) of B(R). We write (C^{p}(R), quasi) and (D^{p}(R), is) for the subcategory of quasi-isomorphisms of C^{p}(R) and isomorphisms of D^{p}(R), respectively. For each complex C = (C^{•}, d_{C}^{•}) and each integer r we define the r-fold shift C[r] of C by setting C[r]^{i} = C^{i+r} and d_{C[r]}^{i} = (−1)^{r}d_{C}^{i+r} for each integer i. We first recall that for any (associative unital) ring R there exists a Picard category C_{R} and a determinant functor d_{R} : (C^{p}(R), quasi) → C_{R} with the following properties (for objects C, C′ and C′′ of C^{p}(R))

A.d) ^{1} If 0 → C′ → C → C′′ → 0 is a short exact sequence of complexes, then there is a canonical isomorphism d_{R}(C) ≅ d_{R}(C′)d_{R}(C′′). which we take as an identification.

A.e) If C is acyclic, then the quasi-isomorphism 0 → C induces a canonical isomor-

phism 1_{R} → d_{R}(C). A.f) For any integer r one has d_{R}(C[r]) = d_{R}(C)^{(−1)^{r}}. A.g) the functor d_{R} factorizes over the image of C^{p}(R) in D^{p}(R) and extends

(uniquely up to unique isomorphisms) to (D^{p}(R), is). A.h) For each C in D^{b}(R) we write H(C) for the complex which has H(C)^{i} = H^{i}(C)

in each degree i and in which all differentials are 0. If H(C) belongs to D^{p}(R) (in which case one says that C is cohomologically perfect), then C belongs to D^{p}(R) and there are canonical isomorphisms

1The listing starts with d) to be compatible with the notation of [33].

d_{R}(C) ≅ d_{R}(H(C)) ≅ ∏_{i∈ℤ} d_{R}(H^{i}(C))^{(−1)^{i}}.

(For an explicit description of the first isomorphism see [21, §3] or [3, Rem. 3.2].) A.i) If R′ is any further ring and Y an (R′, R)-bimodule that is both finitely generated

and projective as an R′-module, then the functor Y ⊗_{R}− : P (R) → P (R′) extends to a commutative diagram

(D^{p}(R), is) ⟶^{d_{R}} C_{R}
Y ⊗_{R}^{𝕃}−↓  ↓^{Y ⊗_{R}−}
(D^{p}(R′), is) ⟶^{d_{R′}} C_{R′}.

In particular, if R → R′ is a ring homomorphism and C is in D^{p}(R), then we often write d_{R}(C)_{R′} in place of R′ ⊗_{R} d_{R}(C).

In [17] a localized K_{1}-group was defined for any full subcategory Σ of C^{p}(R) which satisfies the following four conditions:

(i) 0 ∈ Σ, (ii) if C, C′ are in C^{p}(R) and C is quasi-isomorphic to C′, then C ∈ Σ ⇔ C′ ∈ Σ, (iii) if C ∈ Σ, then also C[n] ∈ Σ for all n ∈ ℤ, (iv′) if C′ and C′′ belong to Σ, then C′ ⊕ C′′ belongs to Σ.

Definition A.1. (Fukaya-Kato) Assume that Σ satisfies (i), (ii), (iii) and (iv′). The localized K_{1}-group K_{1}(R, Σ) is defined to be the (multiplicatively written) abelian group which has as generators symbols of the form [C, a] for each C ∈ Σ and morphism a : 1_{R} → d_{R}(C) in C_{R} and the following relations:

(0) [0, id_{1_{R}}] = 1,

(1) [C′, d_{R}(f) ∘ a] = [C, a] if f : C → C′ is an quasi-isomorphism with C (and thus C′) in Σ,

(2) if 0 → C′ → C → C′′ → 0 is an exact sequence in Σ, then

[C, a] = [C′, a′] · [C′′, a′′]

where a is the composite of a′ · a′′ with the isomorphism induced by property A.d),

(3) [C[1], a^{−1}] = [C, a]^{−1}.

We now assume given a left denominator set S of R and let R_{S} := S^{−1}R denote the corresponding localization and Σ_{S} the full subcategory of C^{p}(R) consisting of all complexes C such that R_{S} ⊗_{R} C is acyclic. For any C in Σ_{S} and any morphism a : 1_{R} → d_{R}(C) in C_{R} we write θ_{C,a} for the element of K_{1}(R_{S}) which corresponds under the canonical isomorphism K_{1}(R_{S}) ≅ Aut_{C_{R_{S}}} (1_{R_{S}}) to the composite

1_{R_{S}} ⟶ d_{R_{S}}(R_{S} ⊗_{R} C) → 1_{R_{S}}

where the first arrow is induced by a and the second by the fact that R_{S} ⊗_{R} C is acyclic. Then it can be shown that the assignment [C, a] ↦ θ_{C,a} induces an isomorphism of groups

ch_{R,Σ_{S}} : K_{1}(R, Σ_{S}) ≅ K_{1}(R_{S})

(cf. [17, Prop. 1.3.7]). Hence, if Σ is any subcategory of Σ_{S} we also obtain a composite homomorphism

ch_{R,Σ} : K_{1}(R, Σ) → K_{1}(R, Σ_{S}) ≅ K_{1}(R_{S}).

In particular, we often use this construction in the following case: C ∈ Σ_{S} and Σ is equal to smallest full subcategory Σ_{C} of C^{p}(R) that contains C and also satisfies the conditions (i), (ii), (iii) and (iv′) that are described above.

Appendix B. Bockstein homomorphisms

Let A be a noetherian regular ring and assume given an exact triangle in D^{p}(A)

Δ : C ⟶^{θ} C → D → C[1].

For each integer i we define the Bockstein homomorphism in degree i of Δ to be the composite homomorphism

β_{Δ}^{i} : H^{i}(D) → ker(H^{i+1}(θ)) → H^{i+1}(C) → cok(H^{i+1}(θ)) → H^{i+1}(D)

where the first and fourth maps occur in the long exact sequence of cohomology of Δ and the second and third are tautological and write

H_{bock}(Δ) : ⋯ ⟶^{β_{Δ}^{i−1}} H^{i}(D) ⟶^{β_{Δ}^{i}} H^{i+1}(D) ⟶^{β_{Δ}^{i+1}} ⋯

for the associated complex (with H^{i}(D) is placed in degree i). The morphism θ is said to be ‘semisimple’ if the tautological map ker(H^{i}(θ)) → cok(H^{i}(θ)) is bijective in each degree i. This condition is equivalent to asserting the acyclicity of H_{bock}(Δ). Hence, if true, there is a composite morphism in V (A) of the form

(55)  β_{Δ} : d_{A}(D) → d_{A}(H(D)) → d_{A}(H_{bock}(Δ)) → 1_{V (A)}

where the first map is as in A.h), the second is the obvious map (induced by the fact that the complexes H(D) and H_{bock}(Δ) agree termwise) and the third is induced by property A.e) and the fact that H_{bock}(Δ) is acyclic.

Appendix C. Sign conventions in [11]

Due to a difference of conventions, which unfortunately had not been noticed by the authors, the following sign conflict has arisen: in [17] for a discrete valuation ring O with field of fractions L an element c ∈ O \ {0} ⊆ L^{×} corresponds to the class [O →^{c} O, id] in K_{1}(L), where the complex is concentrated in degree 0 and 1 (while it is implicitly concentrated in degrees −1 and 0 in [11, Rem. 2.4]). With this convention, Fukaya and Kato must define the connecting homomorphism as [C, a] ↦ −[[C]] in order to ensure that the connecting homomorphism L^{×} = K_{1}(L) → K_{0}(Σ_{O\{0}}) ≅ ℤ coincides with the valuation ord_{L} (cf. [17, Rem. 1.3.16]). It follows that in [11, Rem. 2.4] the correct formula is

ord_{L}(c) = −length_{O}(A),

if we identify A with the complex A[0]. For the same reason, the signs in [11, Prop. 3.19] are incorrect, the correct versions being

χ_{add}(G, C(ρ^{∗})) = −ord_{L}(L^{∗}(ρ))

and

χ_{mult}(G, C(ρ^{∗})) = |L^{∗}(ρ)|_{p}^{[L:ℚ_{p}]}.

44 David Burns and Otmar Venjakob

We note also that L := [C, a] ∈ K_{1}(Λ, Σ_{S^{∗}}) is a characteristic element of −[[C]] = [[C[1]]] (rather than of [[C]]) in K_{0}(Σ_{S^{∗}}) due to the normalisation of the connecting homomorphism in [17]. For a similar reason we have to add a sign in the formulae of [loc. cit., (38) and (39)] to obtain the corrected versions

(56)  L_{U,β} := L_{U,β}(M) : 1_{Λ} → d_{Λ}(SC_{U}(𝕋̂, 𝕋))^{−1}

and

(57)  L_{β} := L_{β}(M) : 1_{Λ} → d_{Λ}(SC(𝕋̂, 𝕋))^{−1}

Also in the following convention we need a shift by one: we write L_{U,β} and L_{β} for the elements [SC_{U}[1], L_{U,β}] and [SC[1], L_{β}] of K_{1}(Λ(G), Σ_{SC_{U}}) and K_{1}(Λ(G), Σ_{SC}) respectively. Finally, in the first displayed formula after [11, Lem. 6.8] one has to replace t(SC_{U}(ρ^{∗}))L̃

by t(SC_{U}(ρ^{∗})[1])L̃ = t(SC_{U}(ρ^{∗}))−1 ; L̃.

The authors would like to apologise for these oversights.

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King’s College London, Dept. of Mathematics, London WC2R2LS, United Kingdom.

University of Heidelberg, Mathematisches Institut, 69120 Heidelberg, Germany.