Tamagawa Numbers for Motives with

(Non-Commutative) Coefficients

D. Burns and M. Flach

Received: September 6, 2001 Revised: January 28, 2002

Communicated by Don Blasius

Abstract. Let M be a motive which is defined over a number field and admits an action of a finite dimensional semisimple ℚ-algebra A. We formulate and study a conjecture for the leading coefficient of the Taylor expansion at 0 of the A-equivariant L-function of M. This conjecture simultaneously generalizes and refines the Tamagawa number conjecture of Bloch, Kato, Fontaine, Perrin-Riou et al. and also the central conjectures of classical Galois module theory as developed by Fröhlich, Chinburg, M. Taylor et al. The precise formulation of our conjecture depends upon the choice of an order 𝔄 in A for which there exists a ‘projective 𝔄-structure’ on M. The existence of such a structure is guaranteed if 𝔄 is a maximal order, and also occurs in many natural examples where 𝔄 is non-maximal. In each such case the conjecture with respect to a non-maximal order refines the conjecture with respect to a maximal order. We develop a theory of determinant functors for all orders in A by making use of the category of virtual objects introduced by Deligne.

2000 Mathematics Subject Classification: Primary 11G40; Secondary 11R65 19A31 19B28

1. Introduction

The study of values of L-functions attached to varieties over number fields occupies a prominent place in number theory and has led to some remarkably general conjectures. A seminal step was made by Bloch and Kato who conjecturally described up to sign the leading coefficient at zero of L-functions attached to motives of negative weight [4]. A little later, Fontaine and Perrin-Riou and (independently) Kato used the determinant functor to extend this conjecture to motives of any weight and with commutative coefficients, thereby taking into account the action of endomorphisms of the variety under consideration (cf. [19, 20, 27, 28]). In this article we shall formulate and study a yet more general conjecture which deals with motives with coefficients which need not be commutative and which in the commutative case recovers all of the above conjectures. We remark that the motivation for such a general conjecture is that in many natural cases, ranging from the central conjectures of classical Galois module theory to the recent attempts to develop an Iwasawa theory for elliptic curves which do not possess complex multiplication, it is necessary to consider motives with respect to coefficients which are not commutative. We now fix a motive M which is defined over a number field K and carries an action of a finite dimensional semisimple ℚ-algebra A. The precise formulation of our conjecture depends upon the choice of an order 𝔄 in A for which there exists a ‘projective 𝔄-structure’ on M (as defined in §3.3). We observe that if 𝔄 is any maximal order in A (as in the case considered by Fontaine and Perrin-Riou in [19]), then there always exists a projective 𝔄-structure on M, and in addition that if M arises by base change of a motive through a finite Galois extension L/K and A := ℚ[G] with G := Gal(L/K) (as in the case considered by Kato in [27]), then there exists a projective ℤ[G]-structure on M. In general, we find that if there exists a projective 𝔄-structure on M, then there also exists a projective 𝔄′-structure on M for any order 𝔄 ⊂ 𝔄′ ⊂ A but that the conjecture which we formulate for the pair (M, 𝔄′) is (in general strictly) weaker than that for the pair (M, 𝔄). This observation is important since we shall show that there are several natural examples (such as the case 𝔄 = ℤ[G] described above) in which projective structures exist with respect to orders which are not maximal. The key difficulty encountered when attempting to formulate Tamagawa number conjectures with respect to non-commutative coefficients is the fact that there is no determinant functor over non-commutative rings. In this article we circumvent this difficulty by making systematic use of the notion of ‘categories of virtual objects’ as described by Deligne in [17]. In our approach Tamagawa numbers are then elements of a relative algebraic K-group K_{0}(𝔄, ℝ) and the Tamagawa number conjecture is an identity in this group. The group K_{0}(𝔄, ℝ) is the relative K_{0} which arises from the inclusion of rings 𝔄 → A_{ℝ} := A ⊗_{ℚ} ℝ and hence lies in a natural long exact sequence

K_{1}(𝔄) → K_{1}(A_{ℝ}) → K_{0}(𝔄, ℝ) → K_{0}(𝔄) → K_{0}(A_{ℝ}).

We remark that in the non-equivariant setting originally considered by Bloch and Kato [4] one has 𝔄 = ℤ, A = ℚ and K_{0}(𝔄, ℝ) ≅ ℝ^{×}/ℤ^{×}. This latter quotient identifies with the group of positive real numbers, and hence in this case Tamagawa numbers can be interpreted as volumes. For motives with noncommutative coefficients however, the only way we have at present been able to formulate a conjecture is by use of the group K_{0}(𝔄, ℝ). The basic content of this article is as follows. Various algebraic preliminaries relating to determinant functors, categories of virtual objects and relative algebraic K-theory, which may themselves be of some independent interest, are given in §2. In §3 we recall preliminaries on motives, define the notion of a ‘projective 𝔄-structure’ on M and give several natural examples of this notion. We henceforth assume that 𝔄 is an order in A for which there exists a projective 𝔄-structure on M. In the remainder of §3 we combine the results of §2 with certain standard assumptions on motives to define a canonical element RΩ(M, 𝔄) of K_{0}(𝔄, ℝ). In §4 we review the A-equivariant L-function L(_{A}M, s) of M. This is a meromorphic function of the complex variable s which takes values in the center ζ(A_{ℂ}) of A_{ℂ} := A ⊗_{ℚ} ℂ, and the leading coefficient L^{∗}(_{A}M, 0) in its Taylor expansion at s = 0 belongs to the group of units ζ(A_{ℝ})^{×} of ζ(A_{ℝ}). We also define a canonical ‘extended boundary homomorphism’ δ̂_{𝔄,ℝ}^{1} : ζ(A_{ℝ})^{×} → K_{0}(𝔄, ℝ) which has the property that the composite of δ̂_{𝔄,ℝ}^{1} with the reduced norm map K_{1}(A_{ℝ}) → ζ(A_{ℝ})^{×} is equal to the boundary homomorphism K_{1}(A_{ℝ}) → K_{0}(𝔄, ℝ) which occurs in the above long exact sequence. We then formulate the central conjecture of this article (Conjecture 4) which states that

δ̂_{𝔄,ℝ}^{1}(L^{∗}(_{A}M, 0)) = −RΩ(M, 𝔄) in K_{0}(𝔄, ℝ).

We remark that our use of the map δ̂_{𝔄,ℝ}^{1} in this context is motivated by the central conjectures of classical Galois module theory. In the remainder of §4 we review some of the current evidence for our conjecture, establish its standard functorial properties and also derive several interesting consequences of these functorial properties. Finally, in §5 we use the Artin-Verdier Duality Theorem to investigate the compatibility of our conjecture with the functional equation of L(_{A}M, s). In a sequel to this article [11] we shall give further evidence for our general conjectures by relating them to classical Galois module theory (in particular, to certain much studied conjectures of Chinburg [13, 14]) and by proving them in several nontrivial cases. In particular, we prove the validity of our central conjecture for pairs (M, 𝔄) = (h^{0}(Spec(L)), ℤ[Gal(L/K)]) where K = ℚ and L/ℚ belongs to an infinite family of Galois extensions for which Gal(L/ℚ) is isomorphic to the Quaternion group of order 8. This article together with its sequel [11] subsumes the contents of an earlier preprint of the same title, and also of the preprint [10]. In the preprint [5] the first named author described an earlier approach to formulating Tamagawa number conjectures with respect to non-commutative coefficients, by using the notions of ‘trivialized perfect complex’ and ‘refined Euler characteristic’ (cf. Remark 4 in §2.8 in this regard). However, by making systematic use of virtual objects the approach adopted here seems to be both more flexible and transparent, and in particular allows us to prove the basic properties of our construction in a very natural manner.

We would like to thank P. Deligne for a number of very valuable comments.

2. Determinant functors for orders in semisimple algebras 2.1. Picard categories. This section introduces the natural target categories for our generalized determinant functors. Recall that a groupoid is a nonempty category in which all morphisms are isomorphisms. A Picard category P is a groupoid equipped with a bifunctor (L, M) → L ⊠ M with an associativity constraint [32] and so that all the functors − ⊠ M, M ⊠ − for a fixed object M are autoequivalences of P. In a Picard category there exists a unit object 1_{P}, unique up to unique isomorphism, and for each object M an inverse M^{−1}, unique up to unique isomorphism, with an isomorphism M ⊠ M^{−1} ⟶^{∼} 1_{P}. For a Picard category P define π_{0}(P) to be the group of isomorphism classes of objects of P (with product induced by ⊠), and set π_{1}(P) := Aut_{P}(1_{P}). We shall only have occasion to consider commutative Picard categories in which ⊠ also satisfies a commutativity constraint [32] and for which π_{0}(P) is therefore abelian. The group π_{1}(P) is always abelian. A monoidal functor F : P_{1} → P_{2} between Picard categories induces homomorphisms π_{i}(F) : π_{i}(P_{1}) → π_{i}(P_{2}) for i ∈ {0, 1}, and F is an equivalence of categories if and only if π_{i}(F) is an isomorphism for both i ∈ {0, 1} (by a monoidal functor we mean a strong monoidal functor as defined in [31][Ch. XI.2]).

2.2. The fibre product of categories. Let F_{i} : P_{i} → P_{3}, i ∈ {1, 2} be functors between categories and consider the fibre product category P_{4} := P_{1} ×_{P_{3}} P_{2} [2, Ch. VII, §3]

\begin{matrix}P_{4} & ⟶^{G_{2}} & P_{2} \\ ↓^{G_{1}} & ↓^{F_{2}} \\ P_{1} & ⟶^{F_{1}} & P_{3}. \end{matrix} (1)

Explicitly, P_{4} is the category with objects (L_{1}, L_{2}, λ) with L_{i} ∈ Ob(P_{i}) for i ∈ {1, 2} and λ : F_{1}(L_{1}) ⟶^{∼} F_{2}(L_{2}) an isomorphism in P_{3}, and where morphisms α : (L_{1}, L_{2}, λ) → (L′_{1}, L′_{2}, λ′) are pairs α = (α_{1}, α_{2}) with α_{i} ∈ Hom_{P_{i}}(L_{i}, L′_{i}) so that the diagram

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

in P_{3} commutes. If P_{5} is a category, H_{i} : P_{5} → P_{i} for i ∈ {1, 2} functors and β : F_{1} ∘H_{1} ≅ F_{2} ∘H_{2} a natural isomorphism, then there exists a unique functor H : P_{5} → P_{4} with H_{i} = G_{i} ∘ H for i ∈ {1, 2} and such that β is induced by the natural isomorphism F_{1} ∘ G_{1} ≅ F_{2} ∘ G_{2}. If P_{i} for i ∈ {1, 2, 3} are Picard categories and F_{1} and F_{2} are monoidal functors, then the fibre product category P_{4} is a Picard category with product (L_{1}, L_{2}, λ) ⊠ (L′_{1}, L′_{2}, λ′) = (L_{1} ⊠ L′_{1}, L_{2} ⊠ L′_{2}, λ ⊠ λ′) and the functors G_{i} : P_{4} → P_{i} for i ∈ {1, 2} are both monoidal.

Lemma 1. (Mayer-Vietoris sequence) For a fibre product diagram (1) of Picard categories one has an exact sequence

0 → π_{1}(P_{4}) ⟶^{(π_{1}(G_{1}),π_{1}(G_{2}))} π_{1}(P_{1}) ⊕ π_{1}(P_{2}) ⟶^{π_{1}(F_{1})−π_{1}(F_{2})} π_{1}(P_{3}) ⟶^{δ}

→ π_{0}(P_{4}) ⟶^{(π_{0}(G_{1}),π_{0}(G_{2}))} π_{0}(P_{1}) ⊕ π_{0}(P_{2}) ⟶^{π_{0}(F_{1})−π_{0}(F_{2})} π_{0}(P_{3}).

Proof. The map δ is defined by δ(β) = (1_{P_{1}}, 1_{P_{2}}, β). Given the explicit description of π_{0}(−) and π_{1}(−) it is an elementary computation to establish the exactness of this sequence. A general Mayer-Vietoris sequence for categories with product can be found in [2, Ch. VII, Th. (4.3)]. For Picard categories this sequence specialises to our Lemma (except for the injectivity of the first map). □

2.3. Determinant functors and virtual objects. Let E be an exact category [38, p. 91] and (E, is) the subcategory of all isomorphisms in E. The main example we have in mind is the category PMod(R) of finitely generated projective modules over a (not necessarily commutative) ring R. By a determinant functor we mean a Picard category P together with the following data.

a) A functor [ ] : (E, is) → P.

b) For each short exact sequence

Σ : 0 → E′ → E → E′′ → 0

a morphism [Σ] : [E] ⟶^{∼} [E′] ⊠ [E′′] in P, functorial for isomorphisms of short exact sequences.

c) For each zero object 0 in E an isomorphism

ζ(0) : [0] ⟶^{∼} 1_{P}.

This data is subject to the following axioms.

d) For an isomorphism ϕ : E → E′ and Σ the exact sequence 0 → E → E′ (resp. E → E′ → 0) [ϕ] (resp. [ϕ^{−1}]) is the composite map

[E] ⟶^{[Σ]} [0] ⊠ [E′] ⟶^{ζ(0)⊠id} [E′]

(resp.

[E′] ⟶^{[Σ]} [E] ⊠ [0] ⟶^{ζ(0)⊠id} [E]).

e) For admissible subobjects 0 ⊆ E′′ ⊆ E′ ⊆ E of an object E of E the

diagram

[E]  ⟶  [E′′] ⊠ [E/E′′]
↓  ↓
[E′] ⊠ [E/E′] ⟶ [E′′] ⊠ [E′/E′′] ⊠ [E/E′]

in P commutes.

The terminology here is borrowed from the key example in which E is the category of vector bundles on a scheme, P is the category of line bundles and the functor is taking the highest exterior power (see §2.5 below). However, as was shown by Deligne in [17, §4], there exists a universal determinant functor for any given exact category E. More precisely, there exists a Picard category V (E), called the ‘category of virtual objects’ of E, together with data a)-c) which in addition to d) and e) also satisfies the following universal property.

f) For any Picard category P the category of monoidal functors Hom^{⊠}(V (E), P) is naturally equivalent to the category of determinant functors (E, is) → P.

Although comparatively inexplicit it is this construction which works best for the purposes of this paper. We recall that the category V (E) has a commutativity constraint defined as follows. Let

τ_{E′,E′′} : [E′] ⊠ [E′′] ⟵^{[Σ_{1}]} [E′ ⊕ E′′] ⟶^{[Σ_{2}]} [E′′] ⊠ [E′]

be the isomorphism induced by the short exact sequences

Σ_{1} : 0 → E′ → E′ ⊕ E′′ → E′′ → 0
Σ_{2} : 0 → E′′ → E′ ⊕ E′′ → E′ → 0.

Replacing [Σ] by τ_{E′,E′′} ∘[Σ] yields a datum a), b), c) with values in V (E)^{⊠−op}, the Picard category with product (L, M) ↦ M ⊠ L, and satisfying d),e). By the universal property f) of V (E), this corresponds to a monoidal functor F : V (E) → V (E)^{⊠−op}. Since we have only changed the value of [ ] on short exact sequences, F is the identity on objects and morphisms and so the monoidality of F gives a commutativity constraint on V (E). The proof of the existence of V (E) in [17, §4.2-5] also gives a topological model of V (E) which in turn implies that there are isomorphisms

K_{i}(E) ⟶^{∼} π_{i}(V (E)) (2)

with the algebraic K-groups of the exact category E (see [38]) for i ∈ {0, 1}. An exact functor F : E → E′ induces a datum a),b),c) on E with values in V (E′) and hence by f) a monoidal functor V (F) : V (E) → V (E′). The isomorphism (2) then commutes with the maps induced by F on K_{i}(E) and by V (F) on π_{i}(V (E)) for i ∈ {0, 1}. Moreover, for i = 0 the isomorphism (2) is the map induced by the functor [ ], and for i = 1 the element in K_{1}(E) represented by ϕ ∈ Aut_{E}(P) is sent to [ϕ] ⊠ id([P]^{−1}) under (2).

2.4. Projective modules and extension to the derived category. For a ring R denote by PMod(R) the exact category of finitely generated projective left-R-modules and put V (R) := V (PMod(R)). For a ring homomorphism R → R′ we denote by R′ ⊗_{R} − both the scalar extension functor PMod(R) → PMod(R′) and also the induced functor V (R) → V (R′). It is known that the Whitehead group K_{1}(R) := K_{1}(PMod(R)) of R is generated by automorphisms of objects of PMod(R).

We write D(R) for the derived category of the homotopy category of complexes of R-modules, and D^{p}(R) for the full triangulated subcategory of D(R) which consists of perfect complexes. We say that an R-module X is perfect if the associated complex X[0] belongs to D^{p}(R), and we write D^{p,p}(R) for the full subcategory of D^{p}(R) consisting of those objects for which the cohomology modules are perfect in all degrees. The association X ↦ X[0] gives a full embedding of PMod(R) into D^{p}(R). In what follows we use the term ‘true triangle’ as synonymous for ‘short exact sequence of complexes’. By a ‘true nine term diagram’ we shall mean a commutative diagram of complexes of the form

\begin{matrix}X & ⟶^{u} & Y & ⟶^{v} & Z \\ ^{f}↓ & & ^{g}↓ & & ^{h}↓ \\ X′ & ⟶^{u′} & Y′ & ⟶^{v′} & Z′ \\ ^{f′}↓ & & ^{g′}↓ & & _{h′}↓ \\ X′′ & ⟶^{u′′} & Y′′ & ⟶^{v′′} & Z′′ \end{matrix} (3)

in which all of the rows and columns are true triangles.

Proposition 2.1. The functor [ ] : (PMod(R), is) → V (R) extends to a functor [ ] : (D^{p}(R), is) → V (R). Moreover, for each true triangle

E = E(u, v) : X ⟶^{u} Y ⟶^{v} Z

in which X, Y, Z are objects of D^{p}(R) there exists an isomorphism [E] : [Y] ⟶^{∼} [X] ⊠ [Z] in V (R) which satisfies all of the following conditions:

a) If

X ⟶^{u} Y ⟶^{v} Z
f↓  ^{g}↓  ^{h}↓
X′ ⟶^{u′} Y′ ⟶^{v′} Z′

is a commutative diagram of true triangles and f, g, h are all quasiisomorphisms, then [f] ⊠ [h] ∘ [E(u, v)] ∘ [g]^{−1} = [E(u′, v′)].

b) If u (resp. v) is a quasi-isomorphism, then [E] = [u]^{−1} (resp. [E] = [v]).

c) [ ] commutes with the functors induced by any ring extension R → R′ and for any true triangle E we have R′ ⊗_{R} [E] = [R′ ⊗_{R} E]. d) For any true nine term diagram (3) in which all terms are objects of

D^{p}(R), the diagram

[Y′]  ⟶^{[E(u′,v′)]}  [X′] ⊠ [Z′]
↓^{[E(g,g′)]}  ↓^{[E(f,f′)]⊠[E(h,h′)]}
[Y] ⊠ [Y′′] ⟶^{[E(u,v)]⊠[E(u′′,v′′)]} [X] ⊠ [Z] ⊠ [X′′] ⊠ [Z′′]

in V (R) commutes. (Note that we have suppressed any explicit reference to commutativity constraints in the above diagram).

e) For any object X of D^{p,p}(R) there exists a canonical isomorphism

[X] ⟶^{∼} ⊠_{i∈ℤ}[H^{i}(X)]^{(−1)^{i}} (4)

which is functorial with respect to quasi-isomorphisms.

Proof. This follows directly from [30, Prop. 4, Th. 2] where the same statement is proved for the determinant functor over a commutative ring R (see §2.5 below). Indeed, since the only properties of the determinant functor used in that proof are those listed in [loc. cit., Prop. 1] and all of these properties are satisfied by the functor [ ] (see [17, Lem. 4.8] for nine term diagrams) these arguments apply to give the desired extension of [ ] to D^{p}(R) with properties a)-d). For e) see [30, Rem. b) after Th. 2]. □

Remark 1. As pointed out in [30, Rem. before Prop. 6], it is not possible to construct isomorphisms [E] for all exact triangles E in D^{p}(R) in such a way that the obvious generalisations of properties a)-d) hold. On the subcategory D^{p,p}(R), however, one can at least construct isomorphisms so that a)-c) hold and so that d) holds under further assumptions (for example, that one of Y, X′, Z′ or Y′′ is acyclic, or that X′′ acyclic and Hom_{D^{p}(R)}(X, w) = 0 where w is such that Z[−1] ⟶^{w} X ⟶^{u} Y ⟶^{v} Z is an exact triangle).

2.5. Commutative rings. If R is a commutative ring, then one can consider the Picard category P(R) of graded line bundles on Spec(R) [30]. Recall that a graded line bundle is a pair (L, α) consisting of an invertible (that is, projective rank one) R-module L and a locally constant function α : Spec(R) → ℤ. A homomorphism h : (L, α) → (M, β) is a module homomorphism h : L → M such that α(𝔭) ≠ β(𝔭) implies h_{𝔭} = 0 for all 𝔭 ∈ Spec(R), and P(R) is the category of graded line bundles and isomorphisms of such. The category P(R) is a symmetric monoidal category with tensor product (L, α) ⊗ (M, β) := (L ⊗_{R} M, α + β), the usual associativity constraint, unit object (R, 0), and commutativity constraint

ψ(l ⊗ m) := ψ_{(L,α),(M,β)}(l ⊗ m) = (−1)^{α(𝔭)β(𝔭)}m ⊗ l (5) for local sections l ∈ L_{𝔭} and m ∈ M_{𝔭}. For a finitely generated projective R-module P one defines

Det_{R}(P) := (⋀_{R}^{rank_{R}(P)}P, rank_{R}(P)) ∈ Ob(P(R)).

The functor Det_{R} : (PMod(R), is) → P(R) is equipped with the data b) and c) of §2.3 and satisfies d) and e). Hence by f) there exists a unique monoidal functor

VDet_{R} : V (R) → P(R),

which is also compatible with the commutativity constraints (this is the reason for the choice of signs in (5)). The functor VDet_{R} is an equivalence of categories if and only if the natural maps

K_{0}(R) → Pic(R) × H^{0}(Spec(R), ℤ) (6)

K_{1}(R) → R^{×}

are both bijective. For any commutative ring R, these maps are split surjections by [24, Exp. I, 6.11-6.14; Exp. X, Th. 5.3.2], [43, (1.8)]. They are known to be bijective if, for example, R is either a local ring, a semisimple ring or the ring of integers in a number field.

2.6. Semisimple rings. We recall here some facts about K_{0}(R) and K_{1}(R) for semisimple rings R. For the moment we let F be any field and assume that R is a central simple algebra over F. We fix a finite extension F′/F so that R′ := R ⊗_{F} F′ ≅ M_{n}(F′) and an indecomposable idempotent e of R′. The map V ↦ dim_{F′} e(V ⊗_{F} F′) is additive in V ∈ Ob(PMod(R)) and therefore induces a homomorphism

rr_{R} : K_{0}(R) → ℤ.

This ‘reduced rank’ homomorphism is injective and has image [End_{R}(S) : F]^{\frac{1}{2}} ℤ where here S is the unique simple R-module. Similarly, if ϕ ∈ End_{R}(V), then we set detred(ϕ) := det_{F′}(ϕ ⊗ 1|e(V ⊗_{F} F′)). This is an element of F which is independent of the choices of both F′ and e. Recalling that K_{1}(R) is generated by pairs (V, ϕ) with ϕ ∈ Aut_{R}(V) it is not hard to show that detred induces a homomorphism

nr_{R} : K_{1}(R) → F^{×}

(cf. [15, §45A]). This ‘reduced norm’ homomorphism is in general neither injective nor surjective.

Proposition 2.2. If F is either a local or a global field, then nr_{R} is injective. If F is a local field different from ℝ, then nr_{R} is bijective. If F = ℝ, then im(nr_{R}) = (ℝ^{×})^{2} if R is a matrix algebra over the division ring of real quaternions, and im(nr_{R}) = ℝ^{×} otherwise. Finally, if F is a number field, then

(7) im(nr_{R}) = {f ∈ F^{×} : f_{v} > 0 for all v ∈ S_{A}(F)} where S_{A}(F) denotes the set of places v of F such that F_{v} = ℝ and A ⊗_{F,v} ℝ is a matrix algebra over the division ring of real quaternions.

Proof. See [15, (45.3)] □

If now R is a general semisimple ring, then the above considerations apply to each of the Wedderburn factors of R. The center ζ(R) of R is a product of fields and we obtain maps

rr_{R} : K_{0}(R) → H^{0}(Spec (ζ(R)), ℤ), nr_{R} : K_{1}(R) → ζ(R)^{×}.

If R is finite dimensional over either a local or a global field, then both of these maps are injective.

Lemma 2. If R is any semisimple ring, then the maps rr_{R} and nr_{R} are both induced by a determinant functor (PMod(R), is) → P(ζ(R)).

Proof. We first observe that the target group of rr_{R} (resp. nr_{R}) does indeed coincide with π_{0}(P(ζ(R))) (resp. π_{1}(P(ζ(R)))). To construct a determinant functor it is clearly sufficient to restrict attention to each Wedderburn factor of R. Such a factor is isomorphic to M_{n}(D), say, where D is a division ring with center F. By fixing an exact (Morita) equivalence PMod(M_{n}(D)) → PMod(D), it therefore suffices to construct a determinant functor for D. To this end we suppose that F′/F is a field extension such that D⊗_{F} F′ ≅ M_{d}(F′), that e is an indecomposable idempotent of M_{d}(F′) and that e_{1}, .., e_{d} is an ordered F′-basis of eM_{d}(F′). Any finitely generated projective D-module V is free, and for any D-basis v_{1}, .., v_{r} of V the wedge product b := ⋀ e_{i}v_{j} (with the e_{i} in the fixed ordering) is an F′-basis of Det_{F′}(e(V ⊗_{F} F′)). Since any change of basis v_{i} multiplies b by an element of im(nr_{D}) ⊆ F^{×}, the F -space spanned by b yields a well defined graded F -line bundle. □

This result shows that if the maps rr_{R} and nr_{R} are both injective, then one can dispense with virtual objects and instead use an explicit functor to graded line bundles over ζ(R). However, this approach no longer seems to be possible when one considers orders in non-commutative semisimple algebras, and it is in this setting that the existence of virtual objects will be most useful for us.

2.7. Orders in finite-dimensional ℚ-algebras. Let A be a finitedimensional ℚ-algebra (associative and unital but not necessarily commutative) and put A_{F} := A ⊗_{ℚ} F for any field F of characteristic zero. For brevity we write A_{p} for A_{ℚ_{p}}. Let R be a finitely generated subring of ℚ. We call an R-subalgebra 𝔄 of A an R-order if 𝔄 is a finitely generated R-module and 𝔄 ⊗_{ℤ} ℚ = A. We shall refer to a ℤ-order more simply as an order. For any order 𝔄 we set 𝔄_{p} := 𝔄 ⊗_{ℤ} ℤ_{p}, 𝔄̂ := 𝔄 ⊗_{ℤ} ℤ̂ ≅ ∏_{p} 𝔄_{p} and Â := A ⊗_{ℤ} ℤ̂. The diagram of exact categories and exact (scalar extension) functors

PMod(𝔄) ⟶ PMod(A)
↓  ↓
PMod(𝔄̂) ⟶ PMod(Â)

induces a corresponding diagram of Picard categories and monoidal functors. These diagrams commute up to a natural equivalence of functors. By the universal property of the fibre product category we therefore obtain a monoidal functor

V (𝔄) → V (𝔄̂) ×_{V (Â)} V (A) =: 𝕍(𝔄). (8)

We use the notation 𝕍(𝔄) in an attempt to stress the adelic nature of 𝕍(−).

Proposition 2.3. The functor (8) induces an isomorphism on π_{0} and a surjection on π_{1}.

Proof. There is a map of long exact Mayer-Vietoris sequences

⟶ K_{1}(𝔄) ⟶ K_{1}(𝔄̂) ⊕ K_{1}(A) ⟶ K_{1}(Â) ⟶ ...
↓  ↓  ↓
0 ⟶ π_{1}(𝕍(𝔄)) ⟶ π_{1}(V (𝔄̂)) ⊕ π_{1}(V (A)) ⟶ π_{1}(V (Â)) ⟶ ...
... ⟶ K_{0}(𝔄) ⟶ K_{0}(𝔄̂) ⊕ K_{0}(A) ⟶ K_{0}(Â)
↓  ↓  ↓
... ⟶ π_{0}(𝕍(𝔄)) ⟶ π_{0}(V (𝔄̂)) ⊕ π_{0}(V (A)) ⟶ π_{0}(V (Â))

where the top sequence can be found in [15, (42.19)], the bottom sequence arises from Lemma 1 and the vertical maps are the isomorphisms (2) or, in the case of K_{i}(𝔄), the isomorphisms (2) composed with the map induced by (8). The commutativity of the diagram follows from the naturality of (2) and, in the case of the boundary map, from an elementary computation using the explicit description of K_{1}(−) (cf. [15, (38.28), (40.6)]). The statement of the proposition is then an easy consequence of the Five Lemma. □

Remark 2. The functor (8) may fail to be an equivalence of categories even if 𝔄 is commutative. Indeed, the map K_{1}(𝔄) → π_{1}(𝕍(𝔄)) is an isomorphism if and only if the map K_{1}(𝔄) → K_{1}(𝔄̂) ⊕ K_{1}(A) is injective. This injectivity condition fails for 𝔄 = ℤ[G] where G is any finite abelian group for which SK_{1}(ℤ[G]) is nontrivial (see [15, Rem. after (48.8)] for examples of such groups G) because ℤ_{p}[G] is then a product of local rings and therefore SK_{1}(ℤ_{p}[G]) = 0. However, if A is semisimple and 𝔄 is a maximal order, then the functor (8) is an equivalence of Picard categories. Indeed, under these hypotheses one can use the Wedderburn decompositions of A and 𝔄 and the Morita invariance of each functor K_{i}(−) in order to reduce to the case in which A is a division ring. In this case the injectivity of K_{1}(𝔄) → K_{1}(𝔄̂) ⊕ K_{1}(A) is a consequence of [15, (45.15)].

Proposition 2.3 is crucial in what follows because it allows us to work in a Picard category 𝕍(𝔄) which has the ‘correct’ π_{0} and in which objects localize in a similar manner to graded line bundles. Indeed, as the following result shows, it is quite reasonable to regard 𝕍(𝔄) as a generalisation of the category P(𝔄) to orders which need not be commutative.

Proposition 2.4. If 𝔄 is a finite flat commutative ℤ-algebra, then there is a natural equivalence of Picard categories P(𝔄) ⟶^{∼} 𝕍(𝔄).

Proof. We shall show that the natural monoidal functors in the diagram

𝕍(𝔄) = V (𝔄̂) ×_{V (Â)} V (A)
↓^{D̂:=VDet𝔄̂ × VDet_{A}}
P(𝔄) ⟶  P(𝔄̂) ×_{P(Â)} P(A)

(9)

are equivalences of Picard categories. We recall that a ring R is said to be local (resp. semilocal) if R/J(R) is a division ring (resp. is semisimple) where J(R) denotes the Jacobson radical of R.

Lemma 3. a) Suppose R = ∏ R_{ι} is a (possibly infinite) product of semilocal rings R_{ι}. Then the natural map K_{i}(R) → ∏ K_{i}(R_{ι}) is injective for i = 0 and bijective for i = 1.

b) If R = ∏ R_{ι} with each R_{ι} local and commutative, then the functor VDet_{R} is an equivalence.

Proof. For a semilocal ring R_{ι} finitely generated projective modules P_{ι} and P′_{ι} are isomorphic if and only if their classes in K_{0}(R_{ι}) agree (see the discussion after [15, (40.35)]). In addition, for any finitely generated projective R-module P the natural map P → ∏_{ι} P ⊗_{R} R_{ι} is an isomorphism since both sides are additive and the map is an isomorphism for P = R. Hence the isomorphism class of P can be recovered from its image in ∏ K_{0}(R_{ι}), and this implies a) for i = 0. For any ring R we have K_{1}(R) = \underrightarrow{lim}_{n}GL_{n}(R)/E_{n}(R) where E_{n}(R) is the subgroup generated by elementary matrices [15, (40.26)]. If R is semilocal, then the map GL_{n}(R)/E_{n}(R) → K_{1}(R) is an isomorphism for n ≥ 2 [15, (40.31), (40.44)]. In addition, the minimal number of generators needed to express a matrix in E_{n}(R_{ι}) as a product of elementary matrices is bounded, depending on n but not on R_{ι} [15, (40.31)]. Hence E_{n}(∏ R_{ι}) = ∏ E_{n}(R_{ι}) and this implies that

K_{1}(R) = K_{1}(∏ R_{ι}) ≅ \underrightarrow{lim}_{n} ∏ GL_{n}(R_{ι})/E_{n}(R_{ι})

≅ \underrightarrow{lim}_{n} ∏ K_{1}(R_{ι}) ≅ ∏ K_{1}(R_{ι}),

i.e. statement a) for i = 1. We remark now that b) follows from a) by using the decomposition R^{×} = ∏ R_{ι}^{×}, the fact that the functors VDet_{R_{ι}} are equivalences, and the fact that the image of the map

K_{0}(R) → ∏ K_{0}(R_{ι}) = ∏ H^{0}(Spec(R_{ι}), ℤ)

lies in (and is therefore isomorphic to) the subgroup H^{0}(Spec(R), ℤ). □

The functor VDet𝔄̂ in (9) is an equivalence by Lemma 3b) since 𝔄̂ is a finite continuous commutative ℤ̂-algebra and hence a product of local rings. Similarly, VDet_{A} is an equivalence since A is Artinian and commutative and hence a product of local rings. The ring Â is a filtered direct limit of rings 𝔄_{S} := ∏_{p∈S} A_{p} × ∏_{p∉S} 𝔄_{p} for finite sets of primes S. As the ring 𝔄_{S} is likewise a product of local rings the functor VDet_{𝔄_{S}} is also an equivalence. It follows that in the commutative diagram

\underrightarrow{lim}_{S} π_{i}(V (𝔄_{S})) ⟶ π_{i}(V (Â))
↓  ^{π_{i}(VDetÂ)}↓
\underrightarrow{lim}_{S} π_{i}(P(𝔄_{S})) ⟶ π_{i}(P(Â))

the left hand vertical map is an isomorphism. Furthermore, the upper horizontal map is an isomorphism by (2) and [43, Lem. 5.9], and the lower horizontal map is an isomorphism for i = 0 (resp. i = 1) since H^{0}(Spec(Â), ℤ) = \underrightarrow{lim}_{S} H^{0}(Spec(𝔄_{S}), ℤ) by [23, 8.2.11] and the fact that an affine scheme is quasi-compact (resp. since Â^{×} = \underrightarrow{lim}_{S} 𝔄_{S}^{×}). We deduce that VDetÂ, and as a consequence also D̂, is an equivalence of Picard categories. It is known that the map (6) is an isomorphism for R = 𝔄 because 𝔄 is Noetherian of dimension 1 [24, Exp. VI 6.9]. Using Proposition 2.3 we find an isomorphism

π_{0}(P(𝔄)) = Pic(𝔄) × H^{0}(Spec(𝔄), ℤ) ⟵^{∼} K_{0}(𝔄)

⟶^{∼} π_{0}(𝕍(𝔄)) ⟶^{π_{0}(D̂)} π_{0}(P(𝔄̂) ×_{P(Â)} P(A)).

This isomorphism coincides with π_{0} of the lower horizontal functor in (9). Moreover, from the Mayer-Vietoris sequence for the fibre product P(𝔄̂) ×_{P(Â)} P(A) one easily deduces that

π_{1}(P(𝔄)) = 𝔄^{×} ≅ π_{1}(P(𝔄̂) ×_{P(Â)} P(A)).

Hence the lower horizontal functor in (9) is an equivalence, and this finishes the proof of Proposition 2.4. □

This proposition makes it reasonable to think of objects of 𝕍(𝔄) as generalized graded line bundles. We invite the reader to do so when following the rest of this paper.

Lemma 4. Assume that A is semisimple. Then the natural functor

𝕍(𝔄) = V (𝔄̂) ×_{V (Â)} V (A) → ∏_{p} V (𝔄_{p}) ×_{∏_{p} V (A_{p})} V (A)

induces an injection on π_{0} and an isomorphism on π_{1}.

Proof. By using Mayer-Vietoris sequences and the Five Lemma it suffices to show that the maps K_{1}(Â) → ∏_{p} K_{1}(A_{p}) and K_{0}(𝔄̂) → ∏_{p} K_{0}(𝔄_{p}) are injective, and that the map K_{1}(𝔄̂) → ∏_{p} K_{1}(𝔄_{p}) is bijective. For the latter two maps this is immediate from Lemma 3a). Since

K_{1}(Â) = \underrightarrow{lim}_{S}K_{1}(∏_{p∈S} A_{p} × ∏_{p∉S} 𝔄_{p}) ≅ ∏_{p∈S} K_{1}(A_{p}) × ∏_{p∉S} K_{1}(𝔄_{p})

is a limit over finite sets S it therefore suffices to show that the map K_{1}(𝔄_{p}) → K_{1}(A_{p}) is injective for any sufficiently large p. To prove this we may assume that A_{p} is a product of matrix algebras over finite field extensions F of ℚ_{p} and that 𝔄_{p} is the corresponding product of matrix algebras over integer rings O_{F}. By Morita equivalence the required result thus follows from the injectivity of the natural map K_{1}(O_{F}) = O_{F}^{×} → K_{1}(F^{×}) = F^{×}. □

Remark 3. We believe that the assertion of Lemma 4 may well continue to hold without the assumption that A is semisimple, but we have no need for such additional generality in what follows.

2.8. The relative K_{0}. Let P_{0} be the Picard category with unique object 1_{P_{0}} and Aut_{P_{0}}(1_{P_{0}}) = 0. For A and 𝔄 as in §2.7 and an extension field F of ℚ we define 𝕍(𝔄, F) to be the fibre product category in the diagram

𝕍(𝔄, F) := 𝕍(𝔄) ×_{V (A_{F})} P_{0} ⟶ P_{0}
↓  ↓^{F_{2}}
𝕍(𝔄)  ⟶^{F_{1}} V (A_{F})

where here F_{2} is the unique monoidal functor and F_{1}((L, M, λ)) = M ⊗_{A} A_{F} for each object (L, M, λ) of 𝕍(𝔄). We define the category V (𝔄_{p}, ℚ_{p}) := V (𝔄_{p}) ×_{V (A_{p})} P_{0} in a similar manner.

Proposition 2.5. For any field extension F of ℚ one has an isomorphism

π_{0}𝕍(𝔄, F) ⟶^{∼} K_{0}(𝔄, F),

and for any prime p an isomorphism

π_{0}V (𝔄_{p}, ℚ_{p}) ⟶^{∼} K_{0}(𝔄_{p}, ℚ_{p}),

where the respective right hand sides are the relative algebraic K-groups as defined in [44, p. 215].

Proof. We recall that K_{0}(𝔄, F) is an abelian group with generators (X, g, Y), where X and Y are finitely generated projective 𝔄-modules and g : X ⊗_{ℤ} F → Y ⊗_{ℤ} F is an isomorphism of A_{F} -modules. For the defining relations we refer to [44, p.215]. By using these relations one checks that the map

(X, g, Y) ↦ ([X] ⊠ [Y]^{−1}, [g] ⊠ id([Y ⊗_{ℤ} F]^{−1}) ∈ π_{0}𝕍(𝔄, F)

induces a homomorphism c : K_{0}(𝔄, F) → π_{0}𝕍(𝔄, F). This homomorphism fits into a natural map of the relative K-theory exact sequence [44, Th. 15.5] to the Mayer-Vietoris sequence of the fibre product defining 𝕍(𝔄, F)

K_{1}(𝔄) ⟶ K_{1}(A_{F}) ⟶^{δ_{𝔄,F}^{1}} K_{0}(𝔄, F) ⟶^{δ_{𝔄,F}^{0}} K_{0}(𝔄) ⟶ K_{0}(A_{F})
↓  ↓  ^{c}↓  ↓  ↓
π_{1}𝕍(𝔄) ⟶ π_{1}V (A_{F}) ⟶ π_{0}𝕍(𝔄, F) ⟶ π_{0}𝕍(𝔄) ⟶ π_{0}V (A_{F}).

The commutativity of this diagram is easy to check, given the explicit nature of all of the maps involved. For example, δ_{𝔄,F}^{0} ((X, g, Y)) = [X] − [Y], δ_{𝔄,F}^{1} sends the element in K_{1}(A_{F}) represented by an n × n-matrix g to (𝔄^{n}, g, 𝔄^{n}) and the vertical maps are as described above. Given the commutativity of this diagram, the isomorphisms (2) combine with Proposition 2.3 and the Five Lemma to imply that c is bijective. The proof for V (𝔄_{p}, ℚ_{p}) is entirely similar using the long exact relative K-theory sequence

K_{1}(𝔄_{p}) → K_{1}(A_{p}) ⟶^{δ_{𝔄p,ℚp}^{1}} K_{0}(𝔄_{p}, ℚ_{p}) ⟶^{δ_{𝔄p,ℚp}^{0}} K_{0}(𝔄_{p}) → K_{0}(A_{p}). (10)

□

Remark 4. For any isomorphism λ : X → Y of A_{F} -modules we write λ_{Triv} for the isomorphism [λ] ⊠ id([Y]^{−1}) : [X] ⊠ [Y]^{−1} ⟶^{∼} 1_{V (A_{F})}. For any ℤ-graded module or morphism X^{•} we write X^{+}, resp. X^{−}, for the direct sum of X^{i} over all even, resp. odd, indices i. If P^{•} is an object of the category PMod(𝔄)^{•} of bounded complexes of objects of PMod(R) and ψ an A_{F} -equivariant isomorphism from H^{+}(P^{•}) ⊗ F to H^{−}(P^{•}) ⊗ F, then we set

⟨P^{•}, ψ⟩ :=([P^{+}] ⊠ [P^{−}]^{−1}, [H^{+}(P^{•}) ⊗ F] ⊠ [H^{−}(P^{•}) ⊗ F]^{−1}, h; ψ_{Triv})
∈𝕍(𝔄, F),

where here h denotes the composite of the canonical isomorphism

[P^{+} ⊗ F] ⊠ [P^{−} ⊗ F]^{−1} ⟶^{∼} [P^{•} ⊗ F],

the isomorphism (4) for X = P^{•} ⊗ F and the canonical isomorphism

⊠_{i∈ℤ}[H^{i}(P^{•}) ⊗ F]^{(−1)^{i}} ⟶^{∼} [H^{+}(P^{•}) ⊗ F] ⊠ [H^{−}(P^{•}) ⊗ F]^{−1}.

If now Y is any object of D^{p}(𝔄) and ψ is an A_{F} -equivariant isomorphism from H^{+}(Y) ⊗ F to H^{−}(Y) ⊗ F, then the pair (Y, ψ^{−1}) constitutes, in the terminology of [5, §1.2], a ‘trivialized perfect complex (of 𝔄-modules)’. Choose an 𝔄-equivariant quasi-isomorphism ξ : P^{•} → Y with P^{•} an object of PMod(𝔄)^{•}, and write ψ_{ξ} for the composite isomorphism H^{−}(ξ ⊗ F)^{−1} ∘ ψ ∘ H^{+}(ξ ⊗ F) : H^{+}(P^{•}) ⊗ F ⟶^{∼} H^{−}(P^{•}) ⊗ F. Then under the isomorphism π_{0}𝕍(𝔄, F) ⟶^{∼} K_{0}(𝔄, F) of Proposition 2.5, the image of the class of ⟨P^{•}, ψ_{ξ}⟩ in π_{0}(𝕍(𝔄, F)) is equal to the inverse of the ‘refined Euler characteristic’ class χ_{𝔄}(Y, ψ^{−1}) which is defined in [loc. cit., Th. 1.2.1].

In the remainder of this section we recall some useful facts concerning the groups K_{0}(𝔄, F). If F is a field of characteristic 0, then one has a commutative diagram of long exact relative K-theory sequences (cf. [44, Th. 15.5])

\begin{matrix}K_{1}(𝔄) & ⟶ & K_{1}(A_{F}) & ⟶ & K_{0}(𝔄, F) & ⟶ & K_{0}(𝔄) & ⟶ & K_{0}(A_{F}) \\ ∥ & & ↑^{β} & & ↑ & & ∥ & & ↑ \\ K_{1}(𝔄) & ⟶ & K_{1}(A) & ⟶ & K_{0}(𝔄, ℚ) & ⟶ & K_{0}(𝔄) & ⟶ & K_{0}(A). \end{matrix} (11)

The scalar extension morphism β is injective and so, as a consequence of the Five Lemma, this diagram induces an inclusion

K_{0}(𝔄, ℚ) ⊆ K_{0}(𝔄, F). (12)

Furthermore, the map

(X, g, Y) ↦ ∏_{p}(X_{p}, g_{p}, Y_{p}),

where X_{p} := X ⊗_{ℤ} ℤ_{p}, Y_{p} := Y ⊗_{ℤ} ℤ_{p} and g_{p} := g ⊗_{ℚ} ℚ_{p} for each prime p, induces an isomorphism

K_{0}(𝔄, ℚ) ⟶^{∼} ⨁_{p} K_{0}(𝔄_{p}, ℚ_{p}) (13)

where the sum is taken over all primes p (see the discussion following [15, (49.12)]).

2.9. The locally free class group. For any field F of characteristic 0 we define

Cl(𝔄, F) := ker(K_{0}(𝔄, F) → K_{0}(𝔄) → ∏_{p} K_{0}(𝔄_{p}))

and

Cl(𝔄) := ker(K_{0}(𝔄) → ∏_{p} K_{0}(𝔄_{p})).

The motivic invariants that we construct will belong to groups of the form Cl(𝔄, ℝ). The group Cl(𝔄) is the ‘locally free class group’ of 𝔄, as discussed in [15, §49]. We observe that the diagram (11) restricts to give a commutative diagram with exact rows

K_{1}(A_{F}) ⟶^{δ_{𝔄,F}^{1}} Cl(𝔄, F) ⟶^{δ_{𝔄,F}^{0}} Cl(𝔄) ⟶ 0
↑^{β}  ↑  ∥
K_{1}(A) ⟶^{δ_{𝔄,ℚ}^{1}} Cl(𝔄, ℚ) ⟶^{δ_{𝔄,ℚ}^{0}} Cl(𝔄) ⟶ 0,

(14)

and hence that (12) restricts to give an inclusion Cl(𝔄, ℚ) ⊆ Cl(𝔄, F). In addition, the restriction of the isomorphism (13) to Cl(𝔄, ℚ) combines with the exact sequence (10) to induce an isomorphism

Cl(𝔄, ℚ) ⟶^{∼} ⨁_{p} K_{1}(A_{p})/im(K_{1}(𝔄_{p})). (15)

In many cases of interest the maps

K_{1}(A_{p})/im(K_{1}(𝔄_{p})) → K_{0}(𝔄_{p}, ℚ_{p})

are bijective for all p, and hence one has

Cl(𝔄, F) = K_{0}(𝔄, F).

For example, this is the case if 𝔄 is commutative, if 𝔄 = ℤ[G] where G is any finite group [15, Rem. (49.11)(iv)] or if 𝔄 is a maximal order in A [loc. cit., Th. 49.32].

3. Motives

3.1. Motivic structures. We fix a number field K and denote by S_{∞} the set of archimedean places of K. For each σ ∈ Hom(K, ℂ) we write v(σ) for the corresponding element of S_{∞}. We also fix an algebraic closure K̄ of K and let G_{K} denote the Galois group Gal(K̄/K). The category of (pure Chow) motives over K is a ℚ-linear category with a functor to the category of realisations [26] and on which motivic cohomology functors are well defined. As is common in the literature on L-functions we shall treat motives in a formal sense: they are to be regarded as given by their realisations, motivic cohomology and the usual maps between these groups (that is, by a motivic structure in the sense of [20]). For example, if X is a smooth, projective variety over K, n a non-negative integer and r any integer, then M := h^{n}(X)(r) is not in general known to exist as a Chow motive. However, the realisations of M are

H_{dR}(M) := H_{dR}^{n}(X/K) , a filtered K-space, with its natural decreasing filtration {F^{i}H_{dR}^{n}(X/K)}_{i∈ℤ} shifted by r;

H_{l}(M) := H_{et}^{n}(X ×_{K} \overline{K}, ℚ_{l}(r)) , a compatible system of l-adic representations of G_{K};

H_{σ}(M) := H^{n}(σX(ℂ), (2πi)^{r}ℚ) , for each σ ∈ Hom(K, ℂ) a ℚ-Hodge structure over ℝ or ℂ according to whether v(σ) is real or complex. If c denotes complex conjugation, then there is an obvious isomorphism of manifolds σX(ℂ) ⟶^{∼} (c ∘ σ)X(ℂ) which we use to identify H_{σ}(M) with H_{c∘σ}(M) if v = v(σ) is complex. We then denote either of the two Hodge structures by H_{v}(M), and we shall subsequently only make constructions which are independent of this choice. One possible definition of the motivic cohomology of M = h^{n}(X)(r) is

H^{0}(K, M) := \begin{cases}(CH^{r}(X)/CH^{r}(X)_{hom∼0}) ⊗_{ℤ} ℚ, & if n = 2r \\ 0, & if n ≠ 2r \end{cases}

and

H^{1}(K, M) := \begin{cases}(K_{2r−n−1}(X) ⊗_{ℤ} ℚ)^{(r)}, & if 2r − n − 1 ≠ 0 \\ CH^{r}(X)_{hom∼0} ⊗_{ℤ} ℚ, & if 2r − n − 1 = 0. \end{cases}

Here (K_{2r−n−1}(X)⊗_{ℤ} ℚ)^{(r)} is the eigenspace for the k-th Adams operator with eigenvalue k^{r}. One also defines a subspace

H_{f}^{1}(K, M) ⊆ H^{1}(K, M)

consisting of classes which are called ‘finite’ (or ‘integral’) at all nonarchimedean places of K and puts H_{f}^{0}(K, M) := H^{0}(K, M). In the K-theoretical version H_{f}^{1}(K, M) is defined just as H^{1}(K, M) but with (K_{2r−n−1}(X) ⊗_{ℤ} ℚ)^{(r)} replaced by

im((K_{2r−n−1}(X) ⊗_{ℤ} ℚ)^{(r)} → (K_{2r−n−1}(X) ⊗_{ℤ} ℚ)^{(r)})

where X is a regular proper model of X over Spec (O_{K}) (see [41] for the definition if such a model does not exist). The spaces H_{f}^{i}(K, M) are expected to be finite dimensional, but this is not yet known to be true in general. Let A be a finite dimensional semisimple ℚ-algebra. From now on we shall be interested in motives with coefficients in A, i.e. in pairs (M, ϕ) where ϕ : A → End(M) is a ring homomorphism. It suffices here to understand End(M) as endomorphisms of motivic structures. However, in all of the explicit examples considered in [11] A is in fact an algebra of correspondences, i.e. consists of endomorphisms in the category of Chow motives. If M has coefficients in A, then the dual motive M^{∗} has coefficients in A^{op}.

3.2. Basic exact sequences. In this section we recall relevant material from [4] and [20]. We write K_{v} for the completion of K at a place v, and we fix an algebraic closure K̄_{v} of K_{v} and an embedding of K̄ into K̄_{v}. We denote by G_{v} ⊆ G_{K} the corresponding decomposition group and, if v is non-archimedean, by I_{v} ⊂ G_{v} and f_{v} ∈ G_{v}/I_{v} the inertia subgroup and Frobenius automorphism respectively. For v ∈ S_{∞} and an ℝ-Hodge structure H over K_{v} (what we call) the Deligne cohomology of H is by definition the cohomology of the complex

RΓ_{D}(K_{v}, H) := (H^{G_{v}} ⟶^{α_{v}} (H ⊗_{ℝ} K̄_{v})^{G_{v}}/F^{0} ) ,

where here G_{v} acts diagonally on H ⊗_{ℝ} K̄_{v}, and α_{v} is induced from the obvious inclusion H ↪ H ⊗_{ℝ} K̄_{v}. Now if H = H_{v}(M) ⊗_{ℚ} ℝ, then there is a canonical comparison isomorphism

H ⊗_{ℝ} K̄_{v} ≅ H_{dR}(M) ⊗_{K,v} K̄_{v}

which is G_{v}-equivariant (the right hand side having the obvious G_{v} action). It follows that the complex

RΓ_{D}(K, M) := ⨁_{v∈S_{∞}} RΓ_{D}(K_{v}, H_{v}(M) ⊗_{ℚ} ℝ)

can also be written as

(16) ⨁_{v∈S_{∞}} (H_{v}(M) ⊗_{ℚ} ℝ)^{G_{v}} ⟶^{α_{M}} ( ⨁_{v∈S_{∞}} H_{dR}(M) ⊗_{K} K_{v}/F^{0})
= (H_{dR}(M)/F^{0}) ⊗_{ℚ} ℝ.

For an ℝ-vector space W we write W^{∗} for the linear dual Hom_{ℝ}(W, ℝ). If W is an A-module, then we always regard W^{∗} as an A^{op}-module in the natural way.

Conjecture 1. (cf. [20][Prop. III.3.2.5]): There exists a long exact sequence of finite-dimensional A_{ℝ}-spaces

(17) 0 ⟶ H^{0}(K, M) ⊗_{ℚ} ℝ ⟶^{ϵ} ker(α_{M}) ⟶^{r_{B}^{∗}} (H_{f}^{1}(K, M^{∗}(1)) ⊗_{ℚ} ℝ)^{∗}

⟶^{δ} H_{f}^{1}(K, M) ⊗_{ℚ} ℝ ⟶^{r_{B}} coker(α_{M}) ⟶^{ϵ^{∗}} (H^{0}(K, M^{∗}(1)) ⊗_{ℚ} ℝ)^{∗} ⟶ 0

where here ϵ is the cycle class map into singular cohomology; r_{B} is the Beilinson regulator map; and (if both H_{f}^{1}(K, M) and H_{f}^{1}(K, M^{∗}(1)) are non-zero so that M has weight −1, then) δ is a height pairing. Moreover, the ℝ-dual of (17) identifies with the corresponding sequence for M^{∗}(1) where the isomorphisms

ker(α_{M})^{∗} ≅ coker(α_{M^{∗}(1)}), coker(α_{M})^{∗} ≅ ker(α_{M^{∗}(1)})

are constructed in Lemma 18 below.

For each prime number p we set V_{p} := H_{p}(M). Following [8, (1.8)] we shall now construct for each place v a true triangle in D^{p}(A_{p})

0 → RΓ_{f}(K_{v}, V_{p}) → RΓ(K_{v}, V_{p}) → RΓ_{/f}(K_{v}, V_{p}) → 0 (18)

in which all of the terms will be defined as specific complexes, rather than only to within unique isomorphism in D^{p}(A_{p}) as the notation would perhaps suggest (we have however chosen to keep the traditional notation for mnemonic purposes.) For a profinite group Π and a continuous Π-module N we denote by C^{•}(Π, N) the standard complex of continuous cochains. If v ∈ S_{∞}, then we set

RΓ_{f}(K_{v}, V_{p}) := RΓ(K_{v}, V_{p}) := C^{•}(G_{v}, V_{p}).

We also define RΓ_{/f}(K_{v}, V_{p}) := 0 and we take (18) to be the obvious true triangle (with second arrow equal to the identity map). If v ∉ S_{∞} and v ∤ p, then we set

RΓ(K_{v}, V_{p}) := C^{•}(G_{v}, V_{p}),
RΓ_{f}(K_{v}, V_{p}) := C^{•}(G_{v}/I_{v}, V_{p}^{I_{v}}) ⊆ C^{•}(G_{v}, V_{p}).

We define RΓ_{/f}(K_{v}, V_{p}) to be the complex which in each degree i is equal to the quotient of C^{i}(G_{v}, V_{p}) by C^{i}(G_{v}/I_{v}, V_{p}^{I_{v}}) (with the induced differential), and we take (18) to be the tautological true triangle. We observe that there is a canonical quasi-isomorphism

(19)  RΓ_{f}(K_{v}, V_{p}) ⟶^{π}(V_{p}^{I_{v}} ⟶^{1−f_{v}^{−1}} V_{p}^{I_{v}})
=:(V_{p,v} ⟶^{ϕ_{v}} V_{p,v})

where in the latter two complexes the spaces are placed in degrees 0 and 1, and π is equal to the identity map in degree 0 and is induced by evaluating a 1-cocycle at f_{v}^{−1} in degree 1. If now v | p, then by [4, Prop. 1.17] one has an exact sequence of continuous G_{v}-modules

0 → ℚ_{p} → B^{0} ⟶^{β_{1}−β_{2}} B^{1} → 0 (20)

where here B^{0} := B_{cris} × B_{dR}^{+} and B^{1} := B_{cris} × B_{dR} are certain canonical algebras and β_{1}(x, y) = (x, x) and β_{2}(x, y) = (ϕ(x), y) are algebra homomorphisms. We write B^{•} for the complex B^{0} ⟶^{β_{1}−β_{2}} B^{1} where the modules are placed in degrees 0 and 1, and we set

RΓ(K_{v}, V_{p}) := Tot C^{•}(G_{v}, B^{•} ⊗_{ℚ_{p}} V_{p}),
RΓ_{f}(K_{v}, V_{p}) := H^{0}(K_{v}, B^{•} ⊗_{ℚ_{p}} V_{p}).

We observe that, since V_{p} → B^{•} ⊗_{ℚ_{p}} V_{p} is a resolution of V_{p}, the natural map

C^{•}(G_{v}, V_{p}) → RΓ(K_{v}, V_{p}) (21) is a quasi-isomorphism. Also, since RΓ_{f}(K_{v}, V_{p}) is a subcomplex of RΓ(K_{v}, V_{p}) we can define RΓ_{/f}(K_{v}, V_{p}) to be the complex which in each degree i is equal to the quotient of the i-th term of RΓ(K_{v}, V_{p}) by the i-th term of RΓ_{f}(K_{v}, V_{p}) (with the induced differential). With these definitions we take (18) to be the tautological true triangle. Further, using the notation

D_{cris}(V_{p}) := H^{0}(K_{v}, B_{cris} ⊗_{ℚ_{p}} V_{p}),
D_{dR}(V_{p}) := H^{0}(K_{v}, B_{dR} ⊗_{ℚ_{p}} V_{p}),
F^{0}D_{dR}(V_{p}) := H^{0}(K_{v}, B_{dR}^{+} ⊗_{ℚ_{p}} V_{p}),
t_{v}(V_{p}) := D_{dR}(V_{p})/F^{0}D_{dR}(V_{p})

there is a commutative diagram of complexes

0  ⟶  t_{v}(V_{p})
↑  ↑
F^{0}D_{dR}(V_{p})  ⟶^{⊆}  D_{dR}(V_{p})
(0,− id)↓  ↓^{(0,id)}
D_{cris}(V_{p}) ⊕ F^{0}D_{dR}(V_{p}) ⟶^{d} D_{cris}(V_{p}) ⊕ D_{dR}(V_{p})

where d is induced by (β_{1} − β_{2}) ⊗ id_{V_{p}} so that the lower row is RΓ_{f}(K_{v}, V_{p}), and the vertical maps are both quasi-isomorphisms. With t_{v}^{•}(V_{p}) denoting the central complex in the above diagram we obtain a canonical quasi-isomorphism

t_{v}^{•}(V_{p}) ⟶^{∼} t_{v}(V_{p})[−1]

and a canonical true triangle

t_{v}^{•}(V_{p}) → RΓ_{f}(K_{v}, V_{p}) → (V_{p,v} ⟶^{ϕ_{v}} V_{p,v}) (22)

where here (V_{p,v} ⟶^{ϕ_{v}} V_{p,v}) := (D_{cris}(V_{p}) ⟶^{1−φ_{v}} D_{cris}(V_{p})) and the spaces

are placed in degrees 0 and 1. In addition, from Faltings’ fundamental comparison theorem between V_{p} and H_{dR}(M) over K_{v} there exists a canonical A_{p}-equivariant isomorphism

(H_{dR}(M)/F^{0}) ⊗_{ℚ} ℚ_{p} ≅ ⨁_{v|p} t_{v}(V_{p}). (23)

Let now V be any finitely generated (projective) A_{p}-module. If ϕ ∈ End_{A_{p}}(V)

and C denotes the (perfect) complex V ⟶^{ϕ} V (with the modules placed in degrees 0 and 1), then there is an isomorphism in V (A_{p})

[C] = [V] ⊠ [V]^{−1} ≅ 1_{V (A_{p})} (24)

which corresponds to the canonical isomorphism X ⊠ X^{−1} ≅ 1 for any object X in a Picard category. Note however that if ϕ is an automorphism and C is therefore acyclic, then the isomorphism (24) differs from the isomorphism [C] ≅ 1_{V (A_{p})} induced by the quasi-isomorphism C → 0 (cf. also [8, Rem. after (1.16)] in this regard). In the notation which was introduced in Remark 4 after Proposition 2.5 this latter isomorphism is denoted by ϕ_{Triv} whereas the isomorphism (24) is denoted by id_{V,Triv}.

We now fix a finite set S of places of K containing S_{∞} and the places where M has bad reduction. We denote by S_{p} the union of S and the set of places of K above p, and we set S_{p,f} := S_{p} \ S_{∞}. We denote by O_{K,S_{p}} the ring of S_{p}-integers in K and by G_{S_{p}} its étale fundamental group with respect to the previously chosen base point K̄. For any continuous G_{S_{p}}-module N we set

RΓ(O_{K,S_{p}}, N) := C^{•}(G_{S_{p}}, N),
RΓ_{c}(O_{K,S_{p}}, N) := Cone (RΓ(O_{K,S_{p}}, N) → ⨁_{v∈S_{p}} C^{•}(G_{v}, N)) [−1]

where the morphism here is induced by the natural maps G_{v} ⊆ G_{K} → G_{S_{p}}. For N = V_{p} we set

1RΓ_{c}(O_{K,S_{p}}, V_{p}) := Cone (RΓ(O_{K,S_{p}}, V_{p}) → ⨁_{v∈S_{p}} RΓ(K_{v}, V_{p})) [−1],

RΓ_{f}(K, V_{p}) := Cone (RΓ(O_{K,S_{p}}, V_{p}) → ⨁_{v∈S_{p}} RΓ_{/f}(K_{v}, V_{p})) [−1]

where in both cases we have used the morphism (21) for each place v | p. Then there is a natural quasi-isomorphism

RΓ_{c}(O_{K,S_{p}}, V_{p}) ⟶^{∼} _{1}RΓ_{c}(O_{K,S_{p}}, V_{p})

and the maps

RΓ(O_{K,S_{p}}, V_{p}) → ⨁_{v∈S_{p}} RΓ(K_{v}, V_{p}) ↢ ⨁_{v∈S_{p}} RΓ_{f}(K_{v}, V_{p}) (25)

induce a true nine term diagram

⨁_{v∈S_{p}} RΓ_{f}(K_{v}, V_{p})[−1]  ⨁_{v∈S_{p}} RΓ_{f}(K_{v}, V_{p})[−1]
↓  ↓
⨁_{v∈S_{p}} RΓ(K_{v}, V_{p})[−1] ⟶ _{1}RΓ_{c}(O_{K,S_{p}}, V_{p}) ⟶ RΓ(O_{K,S_{p}}, V_{p})
↓  ↓  ∥
⨁_{v∈S_{p}} RΓ_{/f}(K_{v}, V_{p})[−1] ⟶  RΓ_{f}(K, V_{p})  ⟶ RΓ(O_{K,S_{p}}, V_{p}).

(26)

Note that in what follows we will systematically use the lower left numbering to distinguish between different but naturally quasi-isomorphic versions of a complex. When there is no danger of confusion we shall simply drop this numbering and leave implicit the resulting identifications. The complex RΓ_{f}(K, V_{p}) is acyclic outside degrees 0, 1, 2 and 3 and in Lemma 19 below we will define a natural isomorphism in D^{p}(A_{p})

AV_{f} : RΓ_{f}(K, V_{p}) ≅ RΓ_{f}(K, V_{p}^{∗}(1))^{∗}[−3].

Conjecturally therefore, the cohomology of RΓ_{f}(K, V_{p}) is completely described by applying the following to both M and M^{∗}(1).

Conjecture 2. For both i ∈ {0, 1} there exists a canonical A_{p}-equivariant isomorphism

c_{p}^{i}(M) : H_{f}^{i}(K, M) ⊗_{ℚ} ℚ_{p} ⟶^{∼} H^{i}RΓ_{f}(K, V_{p}). (27)

In addition, we recall that for each v ∈ S_{∞} the comparison isomorphism between H_{v}(M) and V_{p} induces an isomorphism in D^{p}(A_{p})

RΓ_{f}(K_{v}, V_{p}) = RΓ(K_{v}, V_{p}) ≅ V_{p}^{G_{v}}[0] ≅ (H_{v}(M)^{G_{v}} ⊗_{ℚ} ℚ_{p})[0]. (28)

3.3. Projective 𝔄-structures. Let M be a motive which is defined over K and admits an action of the finite dimensional semisimple ℚ-algebra A. If 𝔄 is an R-order in A (cf. §2.7) and V is an A-module, then an 𝔄-submodule T of V will be said to be an ‘𝔄-lattice (in V)’ if it is both finitely generated and full (i.e., satisfies V = A ⊗_{𝔄} T).

Definition 1. Let 𝔄 be an R-order in A. An 𝔄-structure T on M is a set {T_{v} : v ∈ S_{∞}} where, for each v ∈ S_{∞}, T_{v} is an 𝔄-lattice in H_{v}(M) and for each prime l ∈ Spec(R) the image T_{l} of T_{v} ⊗_{ℤ} ℤ_{l} under the comparison isomorphism H_{v}(M) ⊗_{ℚ} ℚ_{l} ≅ H_{l}(M) is both independent of v and G_{K}-stable. An 𝔄-structure T on M is projective, resp. free, if each T_{v} is a projective, resp. free, 𝔄-module.

If M is a motive with A-action, then there always exist 𝔄-structures on M. For example, if M = h^{n}(X) for a smooth projective variety X defined over K, then there is an 𝔄-structure h^{n}(X, 𝔄) on M such that, for each v ∈ S_{∞}, h^{n}(X, 𝔄)_{v} is the 𝔄-lattice in H_{v}(M) which is generated by the image of H^{n}(σX(ℂ), ℤ) for an embedding σ : K → ℂ which corresponds to v. However, there need not exist projective 𝔄-structures on M. Indeed, even if there are full projective 𝔄modules T_{v} in each space H_{v}(M), it can occur that none of the corresponding modules T_{l} is G_{K}-stable (this is the case if, for example, M = h^{1}(E) for an elliptic curve E defined over an imaginary quadratic field A for which End_{A}(E) is the maximal order in A and 𝔄 is any proper suborder of End_{A}(E)). Nevertheless, the following examples show that a projective 𝔄-structure on M naturally exists in a variety of interesting cases.

Examples.

a) If 𝔄 is an hereditary R-order, and hence a fortiori if it is a maximal Rorder, then there always exists a projective 𝔄-structure on M (cf. [15, Th. (26.12)]).

b) (‘The Galois case’) Let L/K be a finite Galois extension, and set G := Gal(L/K). If M_{K} is any motive which is defined over K, then the motive M := h^{0}(Spec(L)) ⊗ M_{K} has a natural action of the semisimple algebra ℚ[G] via the first factor. Furthermore, if T_{K} is any ℤ-structure on M_{K}, then H^{0}(Spec(L ⊗_{K,σ} ℂ), ℤ) ⊗_{ℤ} T_{K} is a free ℤ[G]-structure on M. Recall here that for any embedding σ : K → ℂ the scheme Spec(L ⊗_{K,σ} ℂ) naturally identifies with the G-set Σ := {τ ∈ Hom(L, ℂ) : τ|_{K} = σ} and hence that H^{0}(Spec(L ⊗_{K,σ} ℂ), ℤ) = Maps(Σ, ℤ) is a free ℤ[G]-module of rank one.

c) (Cf. [42, §4, Rem. following Cor. 2]). If X is a simple abelian variety defined over K which admits complex multiplication over K by a CMfield A, then the motive h^{1}(X) has a natural A-action. In addition, if the order 𝔄 = End_{K}(X) ⊆ A, consisting of those elements which preserve each lattice H^{1}(σX(ℂ), ℤ), is Gorenstein then each h^{1}(X, 𝔄)_{v} is a projective 𝔄-lattice by [15, (37.13)] and hence there exists a projective 𝔄-structure on M. We note in particular that if X is an elliptic curve, then 𝔄 is automatically Gorenstein as a consequence of [1, 6.3].

d) Continuing the previous example, we assume now that X is an elliptic curve defined over K and so that 𝔄 := End_{K}(X) is an order in an imaginary quadratic field A. Then A is contained in K and one can consider the [K : A]-dimensional abelian variety Y over A which is defined as the Weil restriction of X from K to A. If moreover K/A is an abelian Galois extension and X is isogenous to all of its Galois conjugates over A then T := End_{A}(Y) ⊗ ℚ is an algebra of dimension [K : A] over A and 𝔗 := End_{A}(Y) is an order in T, nonmaximal at primes dividing [K : A][O_{A} : 𝔄]. This is shown in [21][15.1.6] for maximal 𝔄 but the arguments there extend to general 𝔄. The description of End_{A}(Y) in [21][15.1.5] also shows that H^{1}(Y (ℂ), ℤ) is a projective 𝔗-module, and hence that the motive M = h^{1}(Y) over the base field A admits a projective 𝔗-structure.

e) Let N be a prime number and consider the modular curve X = X_{0}(N) defined over K = ℚ. The Hecke algebra A is a finite dimensional commutative semisimple ℚ-algebra consisting of correspondences which act on X, and H^{1}(X(ℂ), ℚ) is known to be a free rank two A-space. If 𝔄 denotes the integral Hecke algebra (i.e., the subring of A which is generated by the Hecke correspondences over ℤ), then 𝔄 is an order in A and Mazur shows in [34, II, (14.2), (16.3), (15.1)] that H^{1}(X(ℂ), ℤ)_{𝔪} is a free module over 𝔄_{𝔪} for all maximal ideals 𝔪 of 𝔄 which do not contain 2. This implies that H^{1}(X(ℂ), ℤ[\frac{1}{2}]) is a projective module over 𝔄[\frac{1}{2}], and hence that there exists a projective 𝔄[\frac{1}{2}]-structure on M.

We shall henceforth assume that 𝔄 is an order (leaving to the reader the obvious modifications which are necessary for R-orders as discussed above) and that we are given a projective 𝔄-structure T on M. We assume that p is a prime number which satisfies neither of the following conditions:

(P1) The motive Res_{ℚ}^{K}M has bad reduction at p. (P2) p − 2 < min{i ≥ 0|F^{i+1}H_{dR}(M) = 0} − max{i ≤ 0|F^{i}H_{dR}(M) =

H_{dR}(M)}.

Then, as explained in [8, 1.5.1], one can use the theory of Fontaine-Laffaille to define complexes RΓ_{f}(K_{v}, T_{p}) for each place v in exactly the same way as for V_{p}. The same arguments which lead to diagram (26) can then be used to derive an analogous diagram in D(𝔄_{p}) in which V_{p} is replaced by T_{p}, and which naturally identifies with (26) after tensoring with ℚ_{p}. For any finite set of primes S we write 𝔄_{S} for the localisation of 𝔄 at the multiplicative set generated by the primes in S. For i ∈ {2, 3} we set H_{f}^{i}(K, M) := H_{f}^{3−i}(K, M^{∗}(1))^{∗} and c_{p}^{i}(M) := H^{i}(AV_{f})^{−1} ∘ (c_{p}^{3−i}(M^{∗}(1))^{∗})^{−1}. We now introduce an additional hypothesis on the pair (M, A).

Coherence hypothesis: For each i ∈ {0, 1, 2, 3} there exists a finitely generated 𝔄-module H_{f}^{i}(K, M; T) and an 𝔄-equivariant map τ^{i} : H_{f}^{i}(K, M; T) → H_{f}^{i}(K, M) such that τ^{i} ⊗_{ℤ} ℚ is an isomorphism. In addition, there exists a finite set S of primes containing all primes satisfying either (P1) or (P2) and such that for each p ∉ S and i ∈ {0, 1, 2, 3} there is a commutative diagram of 𝔄_{p}-modules

H_{f}^{i}(K, M; T) ⊗_{ℤ} ℤ_{p} ⟶^{τ^{i}⊗_{ℤ}ℚ_{p}} H_{f}^{i}(K, M) ⊗_{ℚ} ℚ_{p}
c_{p}^{i}(T)↓  ↓^{c_{p}^{i}(M)}
H^{i}RΓ_{f}(K, T_{p}) ⟶^{⊗_{ℤp} ℚ_{p}} H^{i}RΓ_{f}(K, V_{p})

in which c_{p}^{i}(T) is an isomorphism.

Remark 5. This hypothesis is identical to an assumption on integral structures in motivic cohomology which is made in [8, §1.5], and is independent of the choices of both 𝔄 and T. As we shall see below, the hypothesis is not actually required in order to formulate conjectures on special values of motivic L-functions and is correspondingly not made in either of [4] or [20]. However, under the Coherence hypothesis one can define an invariant in Cl(𝔄) without reference to the L-function of M, and this ties in well with the approach of classical Galois module theory. Indeed, in concrete cases, the 𝔄-module structure of the groups H_{f}^{i}(K, M; T) can be of considerable interest (see for example [8, §1.6]), and this structure can be studied via the conjectures formulated in §4 below.

3.4. Virtual objects attached to motives. Let M be a motive which is defined over K and admits an action of a finite dimensional semisimple ℚalgebra A. In this section we fix an order 𝔄 in A and a projective 𝔄-structure T on M (always assuming that such a structure exists). We shall henceforth use the following notational convention. When referring to the individual triangles in a true nine term diagram with equation number (n) we denote by (n)_{?} with ? equal to ‘top’, ‘bot’, ‘left’, ‘rght’, ‘hor’ or ‘vert’ the top, bottom, left, right, central horizontal and central vertical triangle respectively. We define an object Ξ(M) of V (A) by setting

(29) Ξ(M) :=[H_{f}^{0}(K, M)] ⊠ [H_{f}^{1}(K, M)]^{−1} ⊠ [H_{f}^{1}(K, M^{∗}(1))^{∗}]
⊠[H_{f}^{0}(K, M^{∗}(1))^{∗}]^{−1} ⊠ ⊠_{v∈S_{∞}}[H_{v}(M)^{G_{v}}]^{−1} ⊠ [H_{dR}(M)/F^{0}].

Note that this is the inverse of the space Ξ used in both [8] and [27] (because our choice of normalisation for the virtual object associated to a perfect complex is the inverse of that of [8, (0.2)]). Applying the functor [ ] to the isomorphisms (27), (28), (23), the isomorphisms (24), (19) or the triangle (22) for all v ∈ S_{p,f} and finally to the triangle (26)_{vert}, we obtain for each prime p an isomorphism in V (A_{p})

ϑ_{p}(M, S) : A_{p} ⊗_{A} Ξ(M) ⟶^{∼} [RΓ_{c}(O_{K,S_{p}}, V_{p})] ≅ A_{p} ⊗_{𝔄_{p}} [RΓ_{c}(O_{K,S_{p}}, T_{p})]

which we shall also abbreviate as ϑ_{p}(M) or even ϑ_{p} if there is no danger of confusion. We note here that RΓ_{c}(O_{K,S_{p}}, T_{p}) is a perfect complex of 𝔄_{p}-modules by [18, Th. 5.1], and hence we obtain an object

Ξ(M, T_{p}, S) := ([RΓ_{c}(O_{K,S_{p}}, T_{p})], Ξ(M), ϑ_{p})

of V (𝔄_{p}) ×_{V (A_{p})} V (A).

Lemma 5. For another choice T′ of projective 𝔄-structure on M and another choice of the finite set of places S′ the objects Ξ(M, T_{p}, S) and Ξ(M, T′_{p}, S′) are isomorphic in V (𝔄_{p}) ×_{V (A_{p})} V (A).

Proof. By embedding S and S′ into the union S ∪ S′ we can assume that S ⊆ S′, and by induction we can then reduce to the case that S′ = S ∪ {w} and w ∤ p. For any continuous G_{S_{p}}-module N one has a commutative diagram of complexes

C^{•}(G_{S_{p}}, N) ⟶  ⨁_{v∈S_{p}} C^{•}(G_{v}, N)
↓  ↓
C^{•}(G_{S′_{p}}, N) ⟶^{r} C^{•}(G_{w}, N)/C^{•}(G_{w}/I_{w}, N) ⊕ ⨁_{v∈S_{p}} C^{•}(G_{v}, N)
∥  ↑
C^{•}(G_{S′_{p}}, N) ⟶  ⨁_{v∈S′_{p}} C^{•}(G_{v}, N)

which induces a quasi-isomorphism RΓ_{c}(O_{K,S_{p}}, N) ⟶^{∼} Cone(r)[−1] and a true triangle

RΓ_{f}(K_{w}, N)[−1] → RΓ_{c}(O_{K,S′_{p}}, N) → Cone(r)[−1] (30)

where RΓ_{f}(K_{w}, N) = C^{•}(G_{w}/I_{w}, N) is naturally quasi-isomorphic to

N ⟶^{1−f_{v}^{−1}} N (31)

(compare [36, Chap. II, Prop. 2.3d]). For N = T_{p} the true triangle (30) lies in D(𝔄_{p}). In conjunction with isomorphisms of the form (24), it therefore induces an isomorphism ι : [RΓ_{c}(O_{K,S_{p}}, T_{p})] ⟶^{∼} [RΓ_{c}(O_{K,S′_{p}}, T_{p})] in V (𝔄_{p}). We have a natural map from diagram (25) to the diagram

RΓ(O_{K,S′_{p}}, V_{p}) → ⨁_{v∈S_{p}} RΓ(K_{v}, V_{p}) ⊕ RΓ_{/f}(K_{w}, V_{p}) ← ⨁_{v∈S_{p}} RΓ_{f}(K_{v}, V_{p}). (32)

We now denote by T9(S), resp. T9, the diagram (26), resp. the true nine term diagram which is induced by (32). Then we obtain a map ϕ : T9(S) → T9 which restricts to give quasi-isomorphisms on all terms in the central column. In a similar way, there is a map ψ : T9(S′) → T9 which is moreover a termwise surjection. The kernel of ψ is naturally quasi-isomorphic to a sum of complexes (31) and hence naturally trivialized by isomorphisms of the form (24). Since the same trivializations are used in the construction of ϑ_{p}(M, S′), we have (A_{p}⊗_{𝔄_{p}}ι)∘ϑ_{p}(M, S) = ϑ_{p}(M, S′). Hence the pair (ι, id) defines an isomorphism

(ι, id) : ([RΓ_{c}(O_{K,S_{p}}, T_{p})], Ξ(M), ϑ_{p}) ⟶^{∼} ([RΓ_{c}(O_{K,S′_{p}}, T_{p})], Ξ(M), ϑ′_{p})

in the category V (𝔄_{p}) ×_{V (A_{p})} V (A). Replacing T_{p} by p^{n}T_{p} ⊆ T_{p} ∩ T′_{p} we can assume that T_{p} ⊆ T′_{p}. Then there is a true triangle of perfect complexes of 𝔄_{p}-modules

RΓ_{c}(O_{K,S_{p}}, T_{p}) → RΓ_{c}(O_{K,S_{p}}, T′_{p}) → RΓ_{c}(O_{K,S_{p}}, T′_{p}/T_{p}). (33)

Since T′_{p}/T_{p} is finite RΓ_{c}(O_{K,S_{p}}, T′_{p}/T_{p}) ⊗_{ℤ_{p}} ℚ_{p} is acyclic and hence there is a canonical isomorphism τ_{ℚ} : [RΓ_{c}(O_{K,S_{p}}, T′_{p}/T_{p}) ⊗_{ℤ_{p}} ℚ_{p}] ≅ 1_{V (A_{p})}. By [18, Th. 5.1] the class of (RΓ_{c}(O_{K,S_{p}}, T′_{p}/T_{p}), 1_{V (𝔄_{p})}, τ_{ℚ}) in π_{0}(V (𝔄_{p}, ℚ_{p})) ≅ K_{0}(𝔄_{p}, ℚ_{p}) is 0. Upon unraveling the definition of V (𝔄_{p}, ℚ_{p}) this means that τ_{ℚ} is induced by an isomorphism τ : [RΓ_{c}(O_{K,S_{p}}, T′_{p}/T_{p})] ≅ 1_{V (𝔄_{p})}. Hence the isomorphism induced by the triangle (33)

ι : [RΓ_{c}(O_{K,S_{p}}, T′_{p})] ≅[RΓ_{c}(O_{K,S_{p}}, T_{p})] ⊠ [RΓ_{c}(O_{K,S_{p}}, T′_{p}/T_{p})]
⟶^{id⊠τ}[RΓ_{c}(O_{K,S_{p}}, T_{p})]

is part of an isomorphism

(ι, id) : ([RΓ_{c}(O_{K,S_{p}}, T′_{p})], Ξ(M), ϑ_{p}) ⟶^{∼} ([RΓ_{c}(O_{K,S_{p}}, T_{p})], Ξ(M), ϑ_{p})

in the category V (𝔄_{p}) ×_{V (A_{p})} V (A). □

By taking the product over all primes p we now obtain an object

Ξ(M, T, S)_{ℤ} := (∏_{p} [RΓ_{c}(O_{K,S_{p}}, T_{p})], Ξ(M), ∏_{p} ϑ_{p})

of the fibre product category ∏_{p} V (𝔄_{p}) ×_{∏_{p} V (A_{p})} V (A).

Lemma 6. If the Coherence hypothesis is satisfied, then Ξ(M, T, S)_{ℤ} is isomorphic to the image of an object of 𝕍(𝔄) under the functor of Lemma 4.

Proof. Assume that S is a finite set of primes as in the Coherence hypothesis and also containing all primes p at which 𝔄_{p} is not a maximal ℤ_{p}-order in A_{p}. Then 𝔄_{S} is a (left) regular ring [15, Th. (26.12)], and so any finitely generated 𝔄_{S}-module is of finite projective dimension. As in [8, (1.24)] there exists a full 𝔄_{S}-sublattice D_{dR} of H_{dR}(M) so that for p ∉ S the isomorphism (23) is induced by an isomorphism

(D_{dR}/F^{0}) ⊗_{ℤ} ℤ_{p} ≅ ⨁_{v|p} D_{cr,v}(T_{p})/F^{0} (34)

where here D_{cr,v}(−) is an integral version of the functor D_{cris}(−) for K_{v} [8, p. 82]. We define an object Ξ_{S} of V (𝔄_{S}) by setting

Ξ_{S} := Ξ_{S}(M, T, S) :=[H_{f}^{0}(K, M; T)_{S}] ⊠ [H_{f}^{1}(K, M; T)_{S}]^{−1} ⊠ [H_{f}^{2}(K, M; T)_{S}]
⊠[H_{f}^{3}(K, M; T)_{S}]^{−1} ⊠ ⊠_{v∈S_{∞}}[(T_{v})_{S}^{G_{v}}]^{−1} ⊠ [D_{dR}/F^{0}].

We set 𝔄̂′ := ∏_{p∉S} 𝔄_{p}. Then the finite product decomposition 𝔄̂ ≅ ∏_{p∈S} 𝔄_{p} × 𝔄̂′ induces a decomposition V (𝔄̂) ≅ ∏_{p∈S} V (𝔄_{p}) × V (𝔄̂′), and via this we define an object

Ξ′ := ((∏_{p∈S} [RΓ_{c}(O_{K,S_{p}}, T_{p})], 𝔄̂′ ⊗_{𝔄_{S}} Ξ_{S}), A ⊗_{𝔄_{S}} Ξ_{S}, ∏_{p∈S} ϑ_{p} × id_{Ξ_{S}})

of 𝕍(𝔄). Under the Coherence hypothesis, there exists a natural isomorphism A ⊗_{𝔄_{S}} Ξ_{S}(M, T, S) ⟶^{∼} Ξ(M). The image of Ξ′ under the functor of Lemma 4 is isomorphic to Ξ(M, T, S)_{ℤ} because for each p ∉ S the isomorphism ϑ_{p} is induced by an isomorphism

ϑ_{p}^{T} : 𝔄_{p} ⊗_{𝔄_{S}} Ξ_{S}(M, T, S) ⟶^{∼} [RΓ_{c}(O_{K,S_{p}}, T_{p})]

in V (𝔄_{p}) (see [8] for more details). This finishes the proof of Lemma 6. □

Lemma 6, Lemma 5 and Lemma 4 now combine to imply that

Ξ(M)_{ℤ} := Ξ(M, T, S)_{ℤ}

is an object of 𝕍(𝔄) which is independent to within isomorphism in 𝕍(𝔄) of the choices of both S and T. The conjectural exact sequence (17) combines with (16) to induce an isomorphism in V (A_{ℝ})

ϑ_{∞} : A_{ℝ} ⊗_{A} Ξ(M) ≅ 1_{V (A_{ℝ})}.

Under the Coherence hypothesis, we therefore obtain an object

(Ξ(M)_{ℤ}, ϑ_{∞}) := (∏_{p} [RΓ_{c}(O_{K,S_{p}}, T_{p})], Ξ(M), ∏_{p} ϑ_{p}; ϑ_{∞})

of 𝕍(𝔄, ℝ). We let RΩ(M, 𝔄) denote the class of this element in π_{0}(𝕍(𝔄, ℝ)) ≅ K_{0}(𝔄, ℝ).

Lemma 7. RΩ(M, 𝔄) ∈ Cl(𝔄, ℝ).

Proof. We need to show that the class of RΓ_{c}(O_{K,S_{p}}, T_{p}) in K_{0}(𝔄_{p}) vanishes, and this follows as an easy consequence of results in [18]. More precisely, if Γ denotes the image of G_{S} in Aut(T_{p}) and ℤ_{p}[[Γ]] the profinite group algebra of Γ, then [18, Prop. 5.1] shows that there exists a bounded complex P_{•} of finitely generated projective ℤ_{p}[[Γ]]-modules and an isomorphism RΓ_{c}(O_{K,S_{p}}, N) ≅ Hom_{ℤ_{p}[[Γ]]}(P_{•}, N) in D(𝔄_{p}) for any continuous, profinite or discrete, 𝔄_{p}[Γ]- module N. If Σ is a set of representatives for the isomorphism classes of simple ℤ_{p}[[Γ]]-modules and P_{I} → I a projective hull for each I ∈ Σ, then we have isomorphisms of ℤ_{p}[[Γ]]-modules P_{i} ≅ ∏_{I∈Σ} P_{I}^{n_{I,i}} for some integers n_{I,i} (note that Σ is finite since Γ contains a pro-p group of finite index). The 𝔄_{p}-module N_{I} := Hom_{ℤ_{p}[[Γ]]}(P_{I}, N) is a direct summand of N, and hence is projective if N is projective. We write cl_{𝔄_{p}}(X), resp. cl_{ℤ_{p}}(Y), for the class in K_{0}(𝔄_{p}) of any perfect complex of 𝔄_{p}-modules, resp. for the class in the Grothendieck group K_{0}(ℤ_{p}, ℚ_{p}) ≅ ℤ of the category of finite ℤ_{p}-modules of any bounded complex of finite ℤ_{p}-modules Y. Then if either Λ = 𝔄_{p}, or if Λ = ℤ_{p} and N is finite, there is an identity

(35) cl_{Λ}(RΓ_{c}(O_{K,S_{p}}, N)) = ∑_{I∈Σ} (∑_{i∈ℤ}(−1)^{i}n_{I,i}) cl_{Λ}(N_{I}).

Assume now that N ∈ Σ. Then N_{I} = 0 for each I ∈ Σ with I ≠ N, and hence (35) with Λ = ℤ_{p} implies that

∑_{i∈ℤ}(−1)^{i}n_{N,i} · cl_{ℤ_{p}}(N) = cl_{ℤ_{p}}(RΓ_{c}(O_{K,S_{p}}, N))

= ∑_{i∈ℤ}(−1)^{i} cl_{ℤ_{p}}(H_{c}^{i}(O_{K,S_{p}}, N)) = 0,

where the last equality follows from Tate’s formula for the Euler characteristic of a finite G_{S}-module. Since cl_{ℤ_{p}}(N) ≠ 0, it follows that ∑_{i∈ℤ}(−1)^{i}n_{I,i} = 0 for all I ∈ Σ. From (35) with Λ = 𝔄_{p} and N = T_{p} we now deduce that cl_{𝔄_{p}}(RΓ_{c}(O_{K,S_{p}}, T_{p})) = 0, as required. □

3.5. Functorialities. Let ρ : 𝔄 → 𝔅 be a homomorphism between orders 𝔄 and 𝔅 in finite-dimensional, semisimple ℚ-algebras A and B respectively. We denote by ρ_{ℚ} : A → B the induced homomorphism of algebras. For any field F of characteristic 0, the scalar extension functor 𝔅 ⊗_{𝔄} − induces a natural homomorphism

ρ_{∗} : K_{0}(𝔄, F) → K_{0}(𝔅, F)

which sends the class of (X, g, Y) to that of (𝔅 ⊗_{𝔄} X, 1 ⊗ g, 𝔅 ⊗_{𝔄} Y). If 𝔅 is a projective 𝔄-module via ρ, then there also exists a homomorphism in the reverse direction

ρ^{∗} : K_{0}(𝔅, F) → K_{0}(𝔄, F)

which is simply induced by restriction of scalars. If 𝔄 is commutative and 𝔅 = M_{n}(𝔄) is a matrix algebra over 𝔄, then we set

e := (\begin{matrix}1 & 0 & … & 0 \\ 0 & 0 & … & 0 \\ . & . & . & . & . & . & . & . & . & . \\ 0 & & . . . . . & & & 0 \end{matrix}) ∈ 𝔅. (36)

In this case the exact functor V ↦ Im(e) ⊂ V induces an equivalence of exact categories μ : PMod(𝔅) → PMod(𝔄) and hence also an isomorphism

μ_{∗} : K_{0}(𝔅, F) ⟶^{∼} K_{0}(𝔄, F). (37)

If M is a motive over K with A-action, then we define B ⊗_{A} M to be the motive over K with B-action which occurs as the largest direct factor of B ⊗_{ℚ} M upon which the left action of A on M and the right action of A on B coincide. Here B ⊗_{ℚ} M is the direct sum of [B : ℚ] copies of M (see [16, 2.1]). With this definition one has

H(B ⊗_{A} M) ≅ B ⊗_{A} H(M) (38)

for H(−) equal to any of the functors H_{v}(−),H_{v}(−)^{G_{v}}, H_{dR}(−), F^{n}H_{dR}(−), H_{l}(−), H_{l}(−)^{I_{v}}, H_{f}^{0}(K, −) or H_{f}^{1}(K, −). If now T is a projective 𝔄-structure in M (as defined in §3.3), then each 𝔅 ⊗_{𝔄} T_{v} is a projective 𝔅-module and hence a lattice in B ⊗_{A} H_{v}(M) ≅ H_{v}(B ⊗_{A} M). It follows that if M admits a projective 𝔄-structure, then B ⊗_{A} M admits a projective 𝔅-structure. If M is a motive over K with B-action, then it can be regarded as a motive with A-action via ρ_{ℚ}. Assuming that 𝔅 is a projective 𝔄-module via ρ, any projective 𝔅-lattice T_{v} in H_{v}(M) is also a projective 𝔄-lattice (via ρ). Hence, if in this case M admits a projective 𝔅-structure, then it also admits a projective 𝔄-structure.

Suppose now that A is commutative, that B = M_{n}(A) for a natural number n and that M is a motive over K with B-action. Since the category of motives is pseudo-abelian (i.e., contains images of idempotents) eM is a motive with A-action. Also, if T_{v} is a projective M_{n}(𝔄)-lattice in H_{v}(M), then eT_{v} is a projective 𝔄-lattice in eH_{v}(M) = H_{v}(eM). Hence, if M admits a projective M_{n}(𝔄)-structure, then eM admits a projective 𝔄-structure.

Theorem 3.1. a) If M admits a projective 𝔄-structure, then B ⊗_{A} M admits a projective 𝔅-structure and

ρ_{∗}(RΩ(M, 𝔄)) = RΩ(B ⊗_{A} M, 𝔅).

b) If M admits a projective 𝔅-structure and 𝔅 is a projective 𝔄-module via ρ, then M admits a projective 𝔄-structure (via ρ_{ℚ}) and

ρ^{∗}(RΩ(M, 𝔅)) = RΩ(M, 𝔄).

c) If 𝔄 is commutative and M admits a projective M_{n}(𝔄)-structure, then eM admits a projective 𝔄-structure and

μ_{∗}(RΩ(M, M_{n}(𝔄))) = RΩ(eM, 𝔄).

Proof. In case a) the exact functor B ⊗_{A} − : PMod(A) → PMod(B) induces a monoidal functor B ⊗_{A} − : V (A) → V (B) and hence a natural isomorphism [B ⊗_{A} −] ≅ B ⊗_{A} [−]. Together with (38) this yields an isomorphism of virtual B-modules

B ⊗_{A} Ξ(M) ≅ Ξ(B ⊗_{A} M). (39)

The map ϑ_{p} is induced by the A-equivariant isomorphisms and exact sequences (28), (27), (19), (22), (23), (24) for all v ∈ S_{p,f} and (26)_{vert}, all of which transform into the corresponding isomorphisms and exact sequences for B⊗_{A}M when tensored over A with B: this follows from the canonical isomorphisms (38) together with ‘projection formula’ isomorphisms of the type

B_{p} ⊗_{A_{p}} RΓ_{?}(X, H_{p}(M)) ≅ RΓ_{?}(X, B_{p} ⊗_{A_{p}} H_{p}(M)) (40)

for (?, X) equal to any of the pairs (c, O_{K,S_{p}}), (f, K) or (ét, K_{v}). Hence, if ϑ_{p}^{B⊗_{A}M} denotes the isomorphism ϑ_{p} for the B-equivariant motive B ⊗_{A} M, then one has a commutative diagram

B ⊗_{A} Ξ(M) ⟶^{1⊗ϑ_{p}} B_{p} ⊗_{A_{p}} RΓ_{c}(O_{K,S_{p}}, H_{p}(M))
(39)↓  ^{(40)}↓
Ξ(B ⊗_{A} M) ⟶^{ϑ_{p}^{B⊗_{A}M}} RΓ_{c}(O_{K,S_{p}}, B_{p} ⊗_{A_{p}} H_{p}(M)).

Moreover, the isomorphism (40) for the pair (?, X) = (c, O_{K,S_{p}}) is induced by an isomorphism

𝔅_{p} ⊗_{𝔄_{p}} RΓ_{c}(O_{K,S_{p}}, T_{p}) ≅ RΓ_{c}(O_{K,S_{p}}, 𝔅_{p} ⊗_{𝔄_{p}} T_{p})

where T_{p} is a projective 𝔄_{p}-lattice in H_{p}(M) [18, Prop. 4.2]. We deduce that there exists an isomorphism in 𝕍(𝔅)

Ξ(B ⊗_{A} M)_{ℤ} ≅ 𝔅 ⊗_{𝔄} Ξ(M)_{ℤ}.

The map ϑ_{∞} is induced by the A-equivariant exact sequence (17) which, as a consequence of (38), is transformed into the corresponding B-equivariant exact sequence for B ⊗_{A} M when one applies B ⊗_{A} −. So the map ϑ_{∞} for B ⊗_{A} M, which we denote by ϑ_{∞}^{B⊗_{A}M}, is equal to the composite

Ξ(B ⊗_{A} M) ⊗_{ℚ} ℝ ≅ B_{ℝ} ⊗_{A_{ℝ}} (Ξ(M) ⊗_{ℚ} ℝ) ⟶^{1⊗ϑ_{∞}} B_{ℝ} ⊗_{A_{ℝ}} 1_{V (A_{ℝ})} ≅ 1_{V (B_{ℝ})}.

Hence one has

ρ_{∗}(RΩ(M, 𝔄)) = ρ_{∗}((Ξ(M)_{ℤ}, ϑ_{∞})) = (𝔅 ⊗_{𝔄} Ξ(M)_{ℤ}, 1 ⊗ ϑ_{∞})

= (Ξ(B ⊗_{A} M)_{ℤ}, ϑ_{∞}^{B⊗_{A}M}) = RΩ(B ⊗_{A} M, 𝔅).

This proves a). We now simply observe that the proofs of b) and c) follow along exactly the same lines with the role of the functor B ⊗_{A} − being played by the exact functor Res_{A}^{B} : PMod(B) → PMod(A) which is restriction of scalars in b) and restriction of scalars and passage to the direct summand cut out by the idempotent e in c). The analogues of the isomorphisms (38) and (40) for the functor Res_{A}^{B} are in both of these cases obvious. □

4. L-functions

4.1. Equivariant L-factors and ϵ-factors. Let A be a finite-dimensional semisimple ℚ-algebra and W a pseudo-abelian, ℂ-linear category. We define W_{A} to be the category of A-modules in W. Thus the objects of W_{A} are pairs (V, A → End_{W} (V)), and morphisms in W_{A} are morphisms in W which commute with the A-actions. We fix a maximal set i(A) of non-conjugate indecomposable idempotents in A_{ℂ}. More concretely, if

A_{ℂ} ≅ ∏_{i=1}^{r} M_{n_{i}}(ℂ),

then we can take i(A) = {e_{1}, ⋯, e_{r}} where e_{i} is the matrix (36) of size n_{i} in the i-th factor, and 0 in all others factors. The functors V ↦ (im(e_{i}))_{1≤i≤r} and (V_{i})_{1≤i≤r} ↦ ∏_{i=1}^{r} V_{i} ⊗_{ℂ} ℂ^{n_{i}} set up an equivalence of pseudo-abelian categories

W_{A} ≅ ∏_{i=1}^{r} W. (41)

If C is a set and ϵ : Ob(W) → C is any map which is constant on isomorphism classes, then we get a well defined induced map

ϵ : Ob(W_{A}) → C^{i(A)} V ↦ ϵ(im(e_{i})) (42) which does not depend on the choice of i(A) and is constant on isomorphism classes. We now suppose given a motive M over K with an action of A. For any nonarchimedean, resp. archimedean, place v of K we let W_{v} be the category of complex representations of the Weil-Deligne group, resp. of the Weil group, of K_{v} [45]. In order to apply the preceding considerations to W_{v} we need the following

Conjecture 3. (Compatibility) For any finite place v of K, any rational prime l ∤ v and any embedding τ : ℚ_{l} → ℂ consider the object H_{l}(M) ⊗_{ℚ_{l},τ} ℂ of W_{v,A}. Then the isomorphism class of the Frobenius semisimplification [45, (4.1.3)] of H_{l}(M) ⊗_{ℚ_{l},τ} ℂ in W_{v,A} is independent of the choices of l and τ.

Remark 6. If A is a number field, then this reduces to the compatibility conjecture formulated in [45, (4.2.4)].

Let K(ℂ) be the multiplicative group of meromorphic functions on ℂ. As in [45] one attaches to any V ∈ Ob(W_{v}) an L-factor L_{v}(V, s) ∈ K(ℂ) and an ϵfactor ϵ_{v}(V, s, ψ_{v}, dx_{v}) ∈ K(ℂ) (also depending upon a choice of Haar measure dx_{v} on K_{v} and of an additive character ψ_{v} : K_{v} → ℂ^{×}). Assuming Conjecture 3 we use (42) to associate to the pair (M, A) equivariant L-factors L_{v}(_{A}M, s) and equivariant ϵ-factors ϵ_{v}(_{A}M, s, ψ_{v}, dx_{v}) in K(ℂ)^{i(A)}, and we view these as meromorphic functions with values in

ℂ^{i(A)} ≅ ζ(A_{ℂ}) ≅ ζ(A) ⊗_{ℚ} ℂ ≅ ∏_{σ∈Hom(ζ(A),ℂ)} ℂ. (43)

We then define

ϵ(_{A}M, s) := ∏_{v} ϵ_{v}(_{A}M, s, ψ_{v}, dx_{v})
Λ(_{A}M, s) := ∏_{v} L_{v}(_{A}M, s)

where the products are taken over all places v of K and ψ_{v}, dx_{v} are chosen as in [45, (3.5)]. We also set

L_{∞}(_{A}M, s) := ∏_{v∈S_{∞}} L_{v}(_{A}M, s)

and for any finite set S of places of K

L_{S}(_{A}M, s) := ∏_{v∉S} L_{v}(_{A}M, s).

We usually abbreviate L_{S_{∞}}(_{A}M, s) to L(_{A}M, s). We observe that the product for L(_{A}M, s) converges in a half plane Re(s) >> 0 and that in the product for ϵ(_{A}M, s) almost all of the terms are equal to 1. If there is no danger of confusion we shall often suppress the dependence on A and so write L(M, s) etc.

Remark 7. Following [16, Rem. 2.12] one can define the L-factors L_{v}(_{A}M, s) in a more direct way than the above, and this allows one to assume a slightly weaker compatibility than that of Conjecture 3. To be more precise, for each finite place v of K of residue characteristic p and any prime l ≠ p, one considers the A_{l}-module V_{l} := H_{l}(M)^{I_{v}} together with its action of the Frobenius automorphism f_{v} ∈ End_{A_{l}}(V_{l}). Under the assumption that

P_{v}(H_{l}(_{A}M), X) := detred_{A_{l}}(1 − f_{v}^{−1} · X|V_{l}) ∈ ζ(A_{l})[X].

belongs to ζ(A)[X] and is independent of the choice of l, one can define L_{v}(_{A}M, s) to be equal to P_{v}(H_{l}(_{A}M), N v^{−s}) ∈ ζ(A_{ℂ}) for each s ∈ ℂ. We observe that the above assumption on P_{v}(H_{l}(_{A}M), X) is a consequence of Conjecture 3, and conversely that it implies Conjecture 3 if H_{l}(M) is unramified at v.

Lemma 8. If s is real, then L_{v}(_{A}M, s), ϵ(_{A}M, s) and L_{S}(_{A}M, s) all belong to ζ(A) ⊗_{ℚ} ℝ ≅ ζ(A_{ℝ}).

Proof. For any α ∈ ζ(A_{ℂ}) we denote by α_{σ} its σ-component under the isomorphism (43). If c denotes complex conjugation the isomorphism (43) identifies ζ(A) ⊗_{ℚ} ℝ = ζ(A_{ℝ}) ⊂ ζ(A_{ℂ}) with the set {(α_{σ})|α_{c∘σ} = c(α_{σ})}. If v is non-archimedean and s is real, then L_{v}(_{A}M, s) belongs to this set because P_{v}(H_{l}(_{A}M), X) has coefficients in ζ(A). Since the action of c is continuous, the same is therefore true for L_{S}(_{A}M, s). If v is archimedean and s is real, then L_{v}(_{A}M, s)_{σ} ∈ ℝ since the Γ-function is real valued for real arguments. On the other hand, if V ∈ Ob(W_{v}) arises from a ℝ-Hodge structure then there is an isomorphism V^{c} ≅ V in W_{v} given by complex conjugation of the coefficients. Finally for any v, V ∈ Ob(W_{v}) and s ∈ ℝ one has ϵ(V^{c}, s, ψ_{v}^{c}, dx_{v}) = ϵ(V, s, ψ_{v}, dx_{v})^{c} by [45, 3.6]. If V = V_{σ} arises from M and σ ∈ Hom(ζ(A), ℂ) we have V_{σ}^{c} = V_{c∘σ} and

ϵ(_{A}M, s)_{σ}^{c} := ∏_{v} ϵ(V_{σ}, s, ψ_{v}, dx_{v})^{c} = ∏_{v} ϵ(V_{σ}^{c}, s, ψ_{v}^{c}, dx_{v}) = ϵ(_{A}M, s)_{c∘σ}

where this last equality follows because ψ_{v}^{c} and dx_{v} also satisfy the conditions of [45, (3.5)]. □

4.2. The extended boundary homomorphism. Recall that the reduced norm homomorphism nr_{A_{ℝ}} : K_{1}(A_{ℝ}) → ζ(A_{ℝ})^{×} is injective but not in general surjective (cf. Proposition 2.2). In this section we define a canonical homomorphism ζ(A_{ℝ})^{×} → Cl(𝔄, ℝ) which upon restriction to im(nr_{A_{ℝ}}) is equal to the composite δ_{𝔄,ℝ}^{1} ∘ nr_{A_{ℝ}}^{−1}, where here δ_{𝔄,ℝ}^{1} is the homomorphism K_{1}(A_{ℝ}) → Cl(𝔄, ℝ) which occurs in diagram (14). This construction plays a key role in the formulation of conjectures in the next section.

Lemma 9. There exists a canonical homomorphism

δ̂_{𝔄,ℝ}^{1} : ζ(A_{ℝ})^{×} → Cl(𝔄, ℝ)

which satisfies δ̂_{𝔄,ℝ}^{1}(nr_{A_{ℝ}}(x)) = δ_{𝔄,ℝ}^{1}(x) for each x ∈ K_{1}(A_{ℝ}).

Proof. In conjunction with the equality (7), the Weak Approximation Theorem guarantees that for each y ∈ ζ(A_{ℝ})^{×} there exists an element λ of ζ(A)^{×} such that λy ∈ im(nr_{A_{ℝ}}). For each prime p we also view λ as an element of ζ(A_{p})^{×} = im(nr_{A_{p}}), and we then set

δ̂_{𝔄,ℝ}^{1}(y) := δ_{𝔄,ℝ}^{1}(nr_{A_{ℝ}}^{−1}(λy)) − ∑_{p} δ_{𝔄_{p},ℚ_{p}}^{1}(nr_{A_{p}}^{−1}(λ)) ∈ Cl(𝔄, ℝ). (44)

Here we view K_{1}(A_{p})/ im(K_{1}(𝔄_{p})) as a subgroup of Cl(𝔄, ℝ) via the isomorphism (15) and the inclusion (12). The sum is taken over all primes p but is finite since for almost all p both λ ∈ ζ(𝔄_{p})^{×} and nr_{A_{p}}^{−1}(ζ(𝔄_{p})^{×}) is contained in the image of K_{1}(𝔄_{p}). If λ′ is any other element of ζ(A)^{×} such that λ′y ∈ im(nr_{A_{ℝ}}), then (7) implies that λ/λ′ ∈ im(nr_{A}). Hence, if δ̂_{𝔄,ℝ}^{1}(y)′ is the element (44) formed with respect to λ′ rather than λ, then

δ̂_{𝔄,ℝ}^{1}(y) − δ̂_{𝔄,ℝ}^{1}(y)′ = δ_{𝔄,ℝ}^{1}(nr_{A_{ℝ}}^{−1}(λ/λ′)) − ∑_{p} δ_{𝔄_{p},ℚ_{p}}^{1}(nr_{A_{p}}^{−1}(λ/λ′))

and this difference is zero since both terms on the right hand side are equal to δ_{𝔄,ℚ}^{1}(nr_{A}^{−1}(λ/λ′)). It now only remains to check that the assignment y ↦ δ̂_{𝔄,ℝ}^{1}(y) is a homomorphism, and this is easy to verify directly. □

4.3. The main conjectures. We can now formulate the central conjecture of this paper. This conjecture is a generalisation to non-commutative coefficients of [8, Conj. 4] (which in turn generalized the central conjectures of [4, 20, 27]).

Conjecture 4. Let M be a motive which carries an action of the finitedimensional semisimple ℚ-algebra A, and let 𝔄 be any order in A for which M admits a projective 𝔄-structure. Assume that (M, A) satisfies the Coherence hypothesis.

(i) L(_{A}M, s) can be analytically continued to s = 0. (ii) Regarding ord_{s=0}L(_{A}M, s) as a locally constant function on Spec(ζ(A_{ℂ}))

one has

ord_{s=0}L(_{A}M, s) = rr_{A}(H_{f}^{1}(K, M^{∗}(1))^{∗}) − rr_{A}(H_{f}^{0}(K, M^{∗}(1))^{∗})

where the map rr_{A} is as defined in §2.6. (iii) (Rationality) Set

L^{∗}(_{A}M, 0) := lim_{s→0} s^{−ord_{s=0}L(_{A}M,s)}L(_{A}M, s) ∈ ζ(A_{ℝ})^{×},
L(M, 𝔄) :=δ̂_{𝔄,ℝ}^{1}(L^{∗}(_{A}M, 0)) ∈ Cl(𝔄, ℝ)

and

T Ω(M, 𝔄) := L(M, 𝔄) + RΩ(M, 𝔄) ∈ Cl(𝔄, ℝ).

Then T Ω(M, 𝔄) ∈ Cl(𝔄, ℚ).

(iv) (Integrality) T Ω(M, 𝔄) = 0.

Remark 8. It is possible to formulate an equivalent conjecture without assuming the Coherence hypothesis (which was required to define RΩ(M, 𝔄)). To do this, we note that the pair (Ξ(M), ϑ_{∞}) represents an object of V (A, ℝ) := V (A) ×_{V (A_{ℝ})} P_{0}, and we consider the Mayer-Vietoris sequence for this fibre product

⋯ → K_{1}(A_{ℝ}) ⟶^{δ} π_{0}(V (A, ℝ)) → K_{0}(A) → K_{0}(A_{ℝ}) → ⋯

Just as in the proof of Lemma 9, we let λ ∈ ζ(A)^{×} be any element such that λL^{∗}(_{A}M, 0) belongs to im(nr_{A_{ℝ}}). Then Conjecture 4(iii) is equivalent to

Conjecture 5. In π_{0}(V (A, ℝ)) one has [Ξ(M), ϑ_{∞}]+δ(nr_{A_{ℝ}}^{−1}(λL^{∗}(_{A}M, 0))) = 0.

It is clear that this conjecture does not involve Ξ(M)_{ℤ}. Further, as a consequence of the definition of δ and the definition of isomorphism in the category V (A, ℝ), Conjecture 5 implies the existence of an isomorphism in V (A)

ϑ^{(λ)} : Ξ(M) ≅ 1_{V (A)}

which maps to − nr_{A_{ℝ}}^{−1}(λL^{∗}(_{A}M, 0)) ∘ ϑ_{∞} in V (A_{ℝ}). Since the map K_{1}(A) ≅ π_{1}(V (A)) → π_{1}(V (A_{ℝ})) ≅ K_{1}(A_{ℝ}) is injective, the isomorphism ϑ^{(λ)} is unique. One can therefore define an object

ξ(M, 𝔄_{p}, λ) := ([RΓ_{c}(O_{K,S_{p}}, T_{p})], (ϑ^{(λ)} ⊗ ℚ_{p}) ∘ ϑ_{p}^{−1})

of V (𝔄_{p}, ℚ_{p}) and formulate the following

Conjecture 6. Assuming Conjecture 5, the class

T Ω(M, 𝔄_{p}) := [ξ(M, 𝔄_{p}, λ)] − δ_{𝔄_{p},ℚ_{p}}^{1}(nr_{A_{p}}^{−1}(λ))

vanishes in π_{0}(V (𝔄_{p}, ℚ_{p})) ≅ K_{0}(𝔄_{p}, ℚ_{p}).

Under the Coherence hypothesis (and Conjecture 5) one can show that T Ω(M, 𝔄_{p}) is equal to the p-component of the element T Ω(M, 𝔄) of K_{0}(𝔄, ℚ) under the decomposition (13). This implies that, under the Coherence hypothesis, Conjecture 6 is valid for all but finitely many primes p, and that its validity for all p is equivalent to the validity of Conjecture 4(iv).

Remark 9. In this remark we assume that 𝔄 is commutative. Then Proposition 2.4 implies that the Picard category 𝕍(𝔄) is equivalent to the category P(𝔄) of graded invertible 𝔄-modules. Hence one can work with the graded determinant functor and the category P(𝔄) to formulate conjectures which are equivalent to those of Conjecture 4. This is the approach taken in [8], and also in [20] and [28], except that in each of these references ordinary rather than graded determinants are used.

We recall that, for any commutative ring R, an isomorphism in P(R) of the form

Det_{R}(⨁_{i∈I} P_{i}) ≅ ⨂_{i∈I} Det_{R}(P_{i}) (45)

is well defined because one can define an isomorphism for any given ordering of I and the isomorphisms so obtained are compatible with reordering I in the same way on both sides. (This is a consequence of standard coherence theorems for symmetric monoidal categories [32]). However, if one ignores rank data, then (45) depends upon an ordering of I. As a consequence, for example, unless an ordering of the set S_{p} is specified the definition of the isomorphism ϑ_{p} in [8, §1.4] is ambiguous to within multiplication by an element η of A_{p}^{×} which corresponds to a locally constant map Spec(A_{p}) → {±1} (see also the remarks in [20, 0.4] or [28, Rem. 3.2.3(3) and 3.2.6] to this effect). It is clear that such ambiguity cannot be permitted in the formulation of Conjecture 4(iv) because in general η ∉ 𝔄_{p}^{×}. By working in P(A) the definition of ϑ_{p} in [8] becomes unambiguous and the same is true for all of the other determinant computations in loc. cit. All computations involving the determinant functor in both loc. cit. and [9], and also in the work [20, 28] of other authors, should therefore be understood to take place in categories of the form P(R).

Remark 10. We quickly review some of the current evidence for Conjecture 4. At the outset, we remark that any proven case of the central conjectures of [4, 20] provides evidence for Conjecture 4 for pairs of the form (M, ℤ) (in this regard see also Remark 11 in §4.5). Moreover, in [11] it is shown that Conjecture 4 implies the central conjecture of Kato’s paper [27] (in all cases to which the latter applies), and that in the context of Tate motives Conjecture 4(iv) refines a number of previously formulated (and much studied) conjectures. For example, if L/K is a finite Galois extension of number fields, then it is shown in [11] that Conjecture 4(iv) for M = h^{0}(Spec(L)) and with 𝔄 equal to ℤ[Gal(L/K)], resp. equal to any maximal order in ℚ[Gal(L/K)] which contains ℤ[Gal(L/K)], is a refinement of the main conjecture formulated by Chinburg in [13], resp. is equivalent to the so called ‘Strong Stark Conjecture’ (that is, [loc. cit., Conj. 2.2]). In this direction, the reader can also consult [6]. We now fix a Galois extension L of ℚ and set G := Gal(L/ℚ). The main result of [25] is equivalent to the validity of Conjecture 4(iv) for pairs (h^{0}(Spec (L))(r), 𝔐_{(2)}) where here G is abelian, 𝔐_{(2)} denotes the maximal ℤ[\frac{1}{2}]-order in ℚ[G] and r is any integer. In addition, the main result of [12] implies that Conjecture 4(iv) is valid for all pairs (h^{0}(Spec (L))(r), ℤ[\frac{1}{2}][G]) with G abelian and r any integer less than 1 (in this regard see also Remark 19 in §5.3). Relaxing the condition that G is abelian, it is also known that Conjecture 4(iv) is valid for the pairs (h^{0}(Spec (L)), ℤ[G]) where L ranges over a natural (infinite) family of fields for which G is isomorphic to the quaternion group of order 8 [11].

The above examples can all be regarded as providing evidence for Conjecture 4 in the setting of Example b) in §3.3 (‘The Galois case’). The equivariant Birch-Swinnerton Dyer conjecture for elliptic curves with CM by the maximal order O of the CM-field, as formulated by Gross in [22], is perhaps the earliest integral equivariant special value conjecture in a setting other than the Galois case. Moreover, the relative algebraic K-group K_{0}(O, ℝ) is introduced in [22] in an ad-hoc manner in order to formulate the conjecture (which can indeed be shown to be equivalent to Conjecture 4(iv) in all relevant cases). Some instances of Gross’ conjecture have been proved by Rubin [40]. However, at present we are unaware of any examples in which Conjecture 4(iv) has been verified in a non-Galois case and with 𝔄 non-maximal.

4.4. Functorialities. In this section we shall discuss the behaviour of the element L(M, 𝔄), and hence (given Theorem 3.1) also of T Ω(M, 𝔄), under the functorialities discussed in §3.5. We let ρ : 𝔄 → 𝔅 be as in §3.5, and we use the notation ρ_{∗} for any of the maps induced by the exact functor 𝔅 ⊗_{𝔄} − : PMod(𝔄) → PMod(𝔅) or its scalar extensions on algebraic K-groups. These maps ρ_{∗} combine to give a map of the localization sequence (11) into the corresponding sequence with 𝔄 replaced by 𝔅. The same holds for the maps ρ^{∗} (resp. μ_{∗}) induced by the functor res_{𝔄}^{𝔅} : PMod(𝔅) → PMod(𝔄) if 𝔅 is a projective 𝔄-module (resp. by the functor μ : PMod(M_{n}(𝔄)) → PMod(𝔄) if 𝔄 is commutative). Our first result describes the functorial properties of the extended boundary homomorphism.

Lemma 10. There exists a homomorphism ρ_{∗} : ζ(A_{ℝ})^{×} → ζ(B_{ℝ})^{×} which fits into a commutative diagram

K_{1}(A_{ℝ}) ⟶^{nr_{A_{ℝ}}} ζ(A_{ℝ})^{×} ⟶^{δ̂_{𝔄,ℝ}^{1}} Cl(𝔄, ℝ)
ρ_{∗}↓  ^{ρ_{∗}}↓  ^{ρ_{∗}}↓
K_{1}(B_{ℝ}) ⟶^{nr_{B_{ℝ}}} ζ(B_{ℝ})^{×} ⟶^{δ̂_{𝔅,ℝ}^{1}} Cl(𝔅, ℝ).

(46)

The analogous statements also hold for both ρ^{∗} and μ_{∗}.

Proof. For any field F of characteristic 0 the ring homomorphism ρ_{F} : A_{F} → B_{F} induces an exact functor B_{F} ⊗_{A_{F}} − : PMod(A_{F}) → PMod(B_{F}) and hence also a group homomorphism ρ_{F,∗} : K_{1}(A_{F}) → K_{1}(B_{F}). Although the reduced norm map nr_{A_{F}} is not in general bijective it identifies ζ(A_{F})^{×} with the sheafification of the presheaf F ↦ K_{1}(A_{F}) for the étale topology on Spec(F). If F̄ is an algebraic closure of F and Γ = Gal(F̄/F), then we have

H^{0}(Γ, K_{1}(AF̄)) ≅ H^{0}(Γ, ζ(AF̄)^{×}) ≅ H^{0}(Γ, (ζ(A_{F}) ⊗_{F} F̄)^{×}) ≅ ζ(A_{F})^{×},

and the map ρ_{F,∗} can be defined on ζ(A_{F})^{×} via this formula. By construction then, the left hand square in (46) commutes (even with ℝ replaced by any field of characteristic 0) and we also have a commutative diagram

\begin{matrix}ζ(A_{F})^{×} & ⟶^{ρ_{F,∗}} & ζ(B_{F})^{×} \\ ^{⊆}↓ & & ↓^{⊆} \\ ζ(A_{E})^{×} & ⟶^{ρ_{E,∗}} & ζ(B_{E})^{×} \end{matrix} (47)

for any fields E ⊇ F ⊇ ℚ. The localisation sequences (11) for A and B form a commutative diagram with the maps induced by ρ. With notation as in the proof of Lemma 9, the commutativity of the left hand square in (46) therefore implies that

ρ_{∗}(δ̂_{𝔄,ℝ}^{1}(y)) = ρ_{∗}(δ_{𝔄,ℝ}^{1}(nr_{A_{ℝ}}^{−1}(λy))) − ∑_{p} ρ_{∗}(δ_{𝔄_{p},ℚ_{p}}^{1}(nr_{A_{p}}^{−1}(λ)))

= δ_{𝔅,ℝ}^{1}(nr_{B_{ℝ}}^{−1}(ρ_{ℝ,∗}(λy))) − ∑_{p} δ_{𝔅_{p},ℚ_{p}}^{1}(nr_{B_{p}}^{−1}(ρ_{ℚ_{p},∗}(λ))).

From the commutativity of (47) with E/F = ℚ_{p}/ℚ and E/F = ℝ/ℚ it is clear that one can use the element ρ_{ℚ,∗}(λ) of ζ(B)^{×} to compute δ̂_{𝔅,ℝ}^{1}(ρ_{ℝ,∗}(y)). It follows that the above displayed formula is equal to δ̂_{𝔅,ℝ}^{1}(ρ_{ℝ,∗}(y)), and hence that the right hand square in (46) commutes. The arguments for ρ^{∗} and μ_{∗} are entirely similar. □

Theorem 4.1. All assertions of Theorem 3.1 remain valid with RΩ(−, −) replaced by either L(−, −) or T Ω(−, −).

Proof. We have a commutative diagram of exact functors

PMod(A_{ℂ}) ⟶^{κ_{A}} ∏_{i(A)} PMod(ℂ)
B⊗_{A}−↓  ^{η}↓
PMod(B_{ℂ}) ⟶^{κ_{B}} ∏_{i(B)} PMod(ℂ)

(48)

where κ_{A} and κ_{B} are given by (41), and η := κ_{B} ∘ (B ⊗_{A} −) ∘ κ_{A}^{−1} is essentially given by an |i(A)|×|i(B)|−matrix N = (n_{i,j}) with non-negative integer entries. Indeed, one checks easily that η sends (V_{i})_{i∈i(A)} ∈ Ob(∏_{i(A)} PMod(ℂ)) to (⊕_{i}V_{i}^{n_{i,j}})_{j∈i(B)}. There exists a similar diagram involving the same matrix N for any pseudo-abelian ℂ-linear category W in place of PMod(ℂ). For an abelian group C and any map ϵ defined as in (42) which is additive for direct sums in W, we therefore obtain a commutative diagram

Ob(W_{A}) ⟶^{ϵ} C^{i(A)}
B⊗_{A}−↓  ^{N}↓
Ob(W_{B}) ⟶^{ϵ} C^{i(B)}.

This observation applies to L_{v}(M, s) and so by taking into account (38) it follows that N(L_{v}(_{A}M, s)) = L_{v}(_{B}(B ⊗_{A} M), s) ∈ ℂ^{i(B)}. Now since ℂ^{i(A)} ⟶^{N} ℂ^{i(B)} is an analytic map it commutes with the Euler product and hence N(L(_{A}M, s)) = L(_{B}(B ⊗_{A} M), s) ∈ ℂ^{i(B)}. By analytic continuation it follows that N(L^{∗}(_{A}M, 0)) = L^{∗}(_{B}(B ⊗_{A} M), 0) ∈ ζ(B_{ℝ})^{×}. We now recall that the map ρ_{∗} defined on ζ(A_{ℂ})^{×} ≅ (ℂ^{×})^{i(A)} in Lemma 10 is compatible with the induced map on K_{1}(A_{ℂ}). The diagram of exact categories (48) then shows that ρ_{∗} is given by the matrix N after making the canonical identification K_{1}(ℂ) ≅ ℂ^{×}. Lemma 10 then implies that ρ_{∗}(L(M, 𝔄)) = L(B ⊗_{A} M, 𝔅), i.e. the precise analogue of Theorem 3.1a). The analogues of b) and c) in Theorem 3.1 follow by exactly the same argument using the maps ρ^{∗} and μ_{∗}. □

4.5. Consequences of functoriality. In terms of the notation of Theorem 3.1, Theorem 4.1 implies that if Conjecture 4(iv) is valid for the pair (M, 𝔄), then it is also valid for the pair (B ⊗_{A} M, 𝔅). In addition, if ρ_{∗} is injective, then the converse is also true. Analogous statements also hold for ρ^{∗}. It is therefore of some interest to know when the maps ρ_{∗} and ρ^{∗} are injective. The next result investigates ker(ρ_{∗}) in the case that ρ is injective.

Lemma 11. Let ι : 𝔄 → 𝔅 denote the inclusion map between orders in finite dimensional semisimple ℚ-algebras A ⊆ B. Assume that ζ(B) ∩ A = ζ(A).

a) The natural map ι_{∗} : Cl(𝔄, ℝ) → Cl(𝔅, ℝ) has finite kernel contained in Cl(𝔄, ℚ). Moreover, ι_{∗}^{−1}(Cl(𝔅, ℚ)) = Cl(𝔄, ℚ).

b) If either 𝔄 is a maximal order, or B is commutative and 𝔅∩A = 𝔄, then ι_{∗} is injective.

c) The group Cl(𝔄, ℚ) is torsion-free if and only if for each prime p the image of the natural map K_{1}(𝔄_{p}) → K_{1}(A_{p}) ≅ ζ(A_{p})^{×} is equal to the group of units of the maximal ℤ_{p}-order in ζ(A_{p}). This condition holds if 𝔄 is a maximal order in A.

d) If 𝔅 is a maximal order, then ker(ι_{∗}) is the torsion subgroup of Cl(𝔄, ℚ).

Proof. For any finite dimensional semisimple ℚ-algebra C and field F of characteristic 0 we set ζ(C_{F})^{×+} := im(nr_{C_{F}}) ⊂ ζ(C_{F})^{×}. The map nr_{C_{F}} induces an isomorphism K_{1}(C_{F}) ≅ ζ(C_{F})^{×+}, and in what follows we regard this as an identification. We will often use the fact that since ζ(B) ∩ A = ζ(A) the natural map K_{1}(A_{F}) → K_{1}(B_{F}) corresponds under the above identifications to the inclusion ζ(A_{F})^{×+} ⊆ ζ(B_{F})^{×+} [15, (45.3)]. We also use the fact that Proposition 2.2 implies an explicit description of ζ(C_{F})^{×+} in terms of positivity conditions at each quaternion component of C. We first prove that ι_{∗}^{−1}(Cl(𝔅, ℚ)) = Cl(𝔄, ℚ). We thus suppose that x is any element of Cl(𝔄, ℝ) for which ι_{∗}(x) ∈ Cl(𝔅, ℚ). The fact that diagram (14) is exact implies that, after possibly adding to x an element of Cl(𝔄, ℚ), we can assume that there exists an element x̃ of K_{1}(A_{ℝ}) = ζ(A_{ℝ})^{×+} such that x = δ_{𝔄,ℝ}^{1}(x̃). Since the image of x̃ in K_{0}(𝔅, ℝ) lies in K_{0}(𝔅, ℚ) diagram (11) (with A replaced by B) implies that x̃ ∈ ζ(B)^{×+}. Now B ∩ A_{ℝ} = A and so ζ(B)^{×+} ∩ ζ(A_{ℝ})^{×+} = ζ(A)^{×+} (as a consequence of Proposition 2.2). Hence x ∈ Cl(𝔄, ℚ) ⊂ Cl(𝔄, ℝ), as required. We note in particular that this implies that ker(ι_{∗}) ⊆ Cl(𝔄, ℚ). We write

ι_{∗,p} : ζ(A_{p})^{×}/ im(K_{1}(𝔄_{p})) → ζ(B_{p})^{×}/ im(K_{1}(𝔅_{p})) (49)

for the natural map which is induced by the inclusion ζ(A_{p}) ⊆ ζ(B_{p}). We observe that the decomposition (15) induces a corresponding decomposition ι_{∗} = ⨁_{p} ι_{∗,p}, and hence that ι_{∗} is injective if and only if each map ι_{∗,p} is injective. We now consider b). If firstly B and hence A are commutative, then the image of K_{1}(𝔄_{p}) in K_{1}(A_{p}) = A_{p}^{×} is isomorphic to 𝔄_{p}^{×} [15, (45.12)] and similarly for B_{p}. Hence if 𝔅_{p} ∩ A_{p} = 𝔄_{p}, then the map ι_{∗,p} is injective, as required. We assume now that 𝔄_{p} is a maximal ℤ_{p}-order in A_{p}. In this case ζ(𝔄_{p}) is the (unique) maximal ℤ_{p}-order in ζ(A_{p}) and the map nr_{A_{p}} induces an identification im(K_{1}(𝔄_{p})) = ζ(𝔄_{p})^{×} ⊂ ζ(A_{p})^{×} by [15, (45.8)]. To prove injectivity of ι_{∗,p} we embed 𝔅_{p} in a maximal ℤ_{p}-order 𝔐_{p} of B_{p}. Then im(K_{1}(𝔅_{p})) ⊆ im(K_{1}(𝔐_{p})) = ζ(𝔐_{p})^{×}. In addition, the intersection C_{p} := ζ(𝔐_{p}) ∩ ζ(A_{p}) is a ℤ_{p}-order in ζ(A_{p}) and is therefore contained in ζ(𝔄_{p}). Hence one has

im(K_{1}(𝔅_{p})) ∩ ζ(A_{p})^{×} ⊆ C_{p}^{×} ⊆ ζ(𝔄_{p})^{×} = im(K_{1}(𝔄_{p}))

and so ι_{∗,p} is indeed injective. This finishes the proof of b). We next prove c) and also the first (and only remaining) assertion of a). We observe that

ker(ι_{∗,p}) = (im(K_{1}(𝔅_{p})) ∩ ζ(A_{p})^{×})/ im(K_{1}(𝔄_{p})),

and that this quotient is finite since its numerator and denominator are both of finite index in the unit group of the maximal ℤ_{p}-order in ζ(A_{p}) (cf. [15, Exer. (45.4)]). In addition if 𝔄_{p} is maximal, then ζ(A_{p}) is a product of local fields and ζ(𝔄_{p}) is the corresponding product of valuation rings, and hence K_{0}(𝔄_{p}, ℚ_{p}) ≅ ζ(A_{p})^{×}/ζ(𝔄_{p})^{×} is torsion free (in fact free abelian of finite rank). Hence in this case ker(ι_{∗,p}) is trivial. This implies that ker(ι_{∗}) is finite (as claimed in a)) since 𝔄_{p} is a maximal ℤ_{p}-order for almost all p. This argument also implies that the torsion subgroup of ζ(A_{p})^{×}/ im(K_{1}(𝔄_{p})) is equal to 𝔐_{p}^{×}/ im(K_{1}(𝔄_{p})) where 𝔐_{p} is the maximal ℤ_{p}-order in ζ(A_{p}). It follows that Cl(𝔄, ℚ) is indeed torsion free if and only if the condition in c) is satisfied. We remark that if 𝔄 is a maximal order, then this condition is satisfied as a consequence of [15, (45.8)]. We observe finally that d) follows immediately upon combining a) and c). □

Remark 11. The original conjecture of Bloch and Kato (as formulated in [4] and reworked in [20]) is equivalent to Conjecture 4(iv) for the pair (M, ℤ). Now for any order 𝔄 the unique homomorphism ℤ → 𝔄 is flat. Hence, if 𝔄 is any order in A for which M admits a projective 𝔄-structure, then Conjecture 4(iv) for the pair (M, 𝔄) implies the conjecture of Bloch and Kato.

Remark 12. From Lemma 11a), it follows that Conjecture 4(iii) for any given pair (M, 𝔄) is equivalent to Conjecture 4(iii) for any pair (B ⊗_{A} M, 𝔅) where 𝔄 ⊆ 𝔅. As a consequence, it suffices to verify Conjecture 4(iii) after an arbitrary extension A ⊆ B of the operating algebra and for any choice of order in B.

Remark 13. In the Galois case (cf. Example b) in §3.3) there is a natural interplay between a change of coefficients and a change of field extension. This situation is described precisely by the following result.

Proposition 4.1. Let M_{K} be a motive over K and L/K a Galois extension with group G so that ℚ[G] acts on M := M_{L} = h^{0}(Spec(L)) ⊗ M_{K}. Let H be a subgroup of G.

a) Let K′ = L^{H} denote the fixed field of H and T Ω(M′, ℤ[H]) the element constructed from the base change M′_{K′} of M_{K} to K′ and the extension L/K′ with group H. Then

ρ_{H}^{G,∗}(T Ω(M, ℤ[G])) = T Ω(M, ℤ[H]) = T Ω(M′, ℤ[H]) (50)

where ρ_{H}^{G} : ℤ[H] → ℤ[G] is the natural inclusion morphism (which is flat).

b) Set Q := G/H, L′ := L^{H} and M_{L′} := h^{0}(Spec(L′)) ⊗ M_{K}. Then

q_{Q,∗}^{G}(T Ω(M, ℤ[G])) = T Ω(ℚ[Q] ⊗_{ℚ[G]} M, ℤ[Q]) = T Ω(M_{L′}, ℤ[Q]) (51)

where q_{Q}^{G} : ℤ[G] → ℤ[Q] is the natural projection.

Proof. After taking into account Theorem 4.1, we need only prove the second equalities of (50) and (51). We observe first that the second equality of (51) is an immediate consequence of the isomorphism ℚ[Q] ⊗_{ℚ[G]} h^{0}(Spec(L)) ≅ h^{0}(Spec(L′)) of motives over K with ℚ[Q]-action. On the other hand, the second equality of (50) is best understood by thinking of M_{K} as arising from a variety X → Spec(K). Then both M and M′ will arise from the same variety X′ = Spec(L) ×_{Spec(K)} X, respectively viewed over K and K′ (and with H-action in both cases). It is well known that the Lfunctions taken over either K or K′ are the same [16, Rem 2.9]. In addition, the groups H_{f}^{i}(−, −) and H_{dR}(−, −) are the same from both points of view since they only depend on the underlying scheme X′. Since also H_{v}(M) = ⨁_{v′|v} H_{v′}(M′) for each v ∈ S_{∞} it follows that Ξ(M) = Ξ(M′). The exact sequence (17) is the same for M and M′. Further, if π : Spec(O_{K′,S_{p}}) → Spec(O_{K,S_{p}}) denotes the natural finite morphism, then π_{∗}(H_{p}(M′)) = H_{p}(M) and so RΓ_{c}(O_{K′,S_{p}}, H_{p}(M′)) ≅ RΓ_{c}(O_{K,S_{p}}, H_{p}(M)). The map ϑ_{p} is therefore the same for both M and M′ and hence Ξ(M)_{ℤ} = Ξ(M′)_{ℤ}. This in turn implies that T Ω(M, ℤ[H]) = T Ω(M′, ℤ[H]), as required. □

4.6. Reduction to the commutative case. In this section we use Theorem 4.1 to prove that Conjecture 4(iii), and also Conjecture 4(iv) for all pairs (M, 𝔄) with 𝔄 a maximal order, can be verified by restricting to motives with commutative coefficients.

Proposition 4.2. a) Conjecture 4(iii) holds for all pairs (M, 𝔄) if it holds for all such pairs with 𝔄 commutative and maximal.

b) Conjecture 4(iv) holds for all pairs (M, 𝔄) where 𝔄 is a maximal order, if it holds for all such pairs with 𝔄 commutative and maximal.

Proof. By remark 12 after Lemma 11 we may assume throughout that 𝔄 is maximal. Consider the Wedderburn decomposition A ≅ ∏_{i=1}^{r} M_{m_{i}}(D_{i}) of A and put F_{i} := ζ(D_{i}). Pick a splitting field E_{i} for each i so that M_{m_{i}}(D_{i}) ⊗_{F_{i}} E_{i} ≅ M_{n_{i}}(E_{i}) with n_{i} = m_{i}\sqrt{[D_{i} : F_{i}]}. Then B = ∏_{i=1}^{r} M_{n_{i}}(E_{i}) contains A and we have ζ(B)∩A = ζ(A). The image of 𝔄 can be embedded into a maximal order 𝔅 in B, and we write ι : 𝔄 → 𝔅 for the corresponding morphism. One has ι_{∗}(T Ω(M, 𝔄)) = T Ω(B⊗_{A}M, 𝔅) by Theorem 4.1, and so Lemma 11 implies that Conjecture 4(iii), resp. (iv), is valid for (M, 𝔄) if and only if it is valid for (B ⊗_{A} M, 𝔅). Now [15, Th. (26.25)] implies that, perhaps after enlarging each field E_{i}, we can assume that 𝔅 = b · 𝔅′ · b^{−1} with 𝔅′ = ∏_{i=1}^{r} M_{n_{i}}(O_{E_{i}}) and b ∈ B^{×}. In this case multiplication by b gives an isomorphism of pairs (B ⊗_{A} M, B) ≅ (B ⊗_{A} M, b · B · b^{−1}) which in turn induces an equality

T Ω(B ⊗_{A} M, 𝔅) = T Ω(B ⊗_{A} M, 𝔅′)

= ∏_{i=1}^{r} T Ω(ϵ_{i}(B ⊗_{A} M), M_{n_{i}}(O_{E_{i}}))

∈ ⨁_{i=1}^{r} K_{0}(M_{n_{i}}(O_{E_{i}}), ℝ) ≅ K_{0}(𝔅′, ℝ)

where ϵ_{i} are the central idempotents of B. From Theorem 4.1 one has

μ_{i,∗}(T Ω(ϵ_{i}(B ⊗_{A} M), M_{n_{i}}(O_{E_{i}}))) = T Ω(e_{i}ϵ_{i}(B ⊗_{A} M), O_{E_{i}})

where here e_{i} is the matrix (36) of size n_{i}, and μ_{i,∗} : K_{0}(M_{n_{i}}(O_{E_{i}}), ℝ) ⟶^{∼} K_{0}(O_{E_{i}}, ℝ) is the associated isomorphism (37). Since also K_{0}(M_{n_{i}}(O_{E_{i}}), ℚ) = μ_{i,∗}^{−1}(K_{0}(O_{E_{i}}, ℚ)), it is clear that Conjecture 4(iii), resp. (iv), is true for (B ⊗_{A} M, 𝔅) if and only if it is true for each pair (e_{i}ϵ_{i}(B ⊗_{A} M), O_{E_{i}}). This finishes the proof of the proposition. □

Remark 14. Let A be a central simple algebra over a number field F with ring of integers O. If 𝔄 is any maximal order in A, then the reduction to commutative coefficients effected by Proposition 4.2b) implies that Conjecture 4(iv) for the pair (M, 𝔄) can only determine L^{∗}(_{A}M, 0) to within multiplication by all elements of O^{×} (inside (F ⊗_{ℚ} ℝ)^{×} = ζ(A_{ℝ})^{×}). This reflects the general fact that if 𝔄 is any maximal order in a finite dimensional semisimple ℚ-algebra A, then δ̂_{𝔄,ℝ}^{1} vanishes on all of ζ(𝔄)^{×} rather than only on ζ(𝔄)^{×} ∩ im(nr_{A_{ℝ}}). (The latter fact follows as an easy consequence of [15, (45.7), (45.8)]).

Remark 15. Non-maximal non-commutative orders 𝔄 arise as natural operating rings in many interesting examples. In general, when attempting to verify Conjecture 4(iv) for any such pair (M, 𝔄) no reduction to commutative coefficients is possible. In [11] we give a detailed discussion of Conjecture 4(iv) for a number of such examples.

5. Kummer duality

We recall that if M is any motive with an action of a semisimple ℚ-algebra A, then the dual motive M^{∗} is naturally endowed with an action of the opposite algebra A^{op}. After fixing an isomorphism A^{∗} ≅ A^{op} of A^{op}-modules [15, (9.8)], we then have a functorial isomorphism of A^{op}-modules

(52)  W^{∗} = Hom_{ℚ}(W, ℚ) ≅ Hom_{ℚ}(A ⊗_{A} W, ℚ)
≅ Hom_{A}(W, Hom_{ℚ}(A^{op}, ℚ)) ≅ Hom_{A}(W, A)

for any A-module W. It follows that if M has a projective 𝔄-structure {T_{v} : v ∈ S_{∞}}, then M^{∗}(1) has a projective 𝔄^{op}-structure {Hom_{𝔄}(T_{v}, 𝔄)(1) : v ∈ S_{∞}}. In this section we shall compare the elements T Ω(M, 𝔄) and T Ω(M^{∗}(1), 𝔄^{op}). This comparison is naturally motivated by the problem of deciding whether Conjecture 4(iv) is compatible with the functional equation of L(_{A}M, s). The comparison result we prove in this section is most conveniently formulated in terms of an element T Ω^{loc}(M, 𝔄) of Cl(𝔄, ℝ) the theory of which is strikingly parallel to that of T Ω(M, 𝔄) but involves no assumptions on the motivic cohomology of M. Indeed, T Ω^{loc}(M, 𝔄) takes the form L^{loc}(M, 𝔄) + RΩ^{loc}(M, 𝔄) where the first term is defined in terms of the equivariant archimedean Euler factors and epsilon factors which are attached to M and M^{∗}(1) and the second term is of an algebraic nature, involving the realisations of M.

5.1. Definition of RΩ^{loc}(M, 𝔄). We first define a virtual A-module

Ξ^{loc}(M) := [H_{dR}(M)] ⊠ [H_{B}(M)]^{−1}

where

H_{B}(M) := ⨁_{σ∈Hom(K,ℂ)} H_{σ}(M).

Recall that for each σ ∈ Hom(K, ℂ) we write v(σ) for the corresponding element of S_{∞}. The action of Gal(ℂ/K_{v(σ)}) on each space H_{σ}(M) induces upon H_{B}(M) an action of Gal(ℂ/ℝ). In addition, by taking the direct sum over the A × Gal(ℂ/K_{v(σ)})-equivariant period isomorphisms

H_{σ}(M) ⊗_{ℚ} ℂ ≅ H_{dR}(M) ⊗_{K,σ} ℂ

one obtains an A × Gal(ℂ/ℝ)-equivariant isomorphism

H_{B}(M) ⊗_{ℚ} ℂ ≅ ⨁_{σ∈Hom(K,ℂ)} H_{dR}(M) ⊗_{K,σ} ℂ = H_{dR}(M) ⊗_{ℚ} ℂ (53)

and after taking Gal(ℂ/ℝ)-invariants this in turn induces an A_{ℝ}-equivariant isomorphism

(H_{B}(M) ⊗_{ℚ} ℂ)^{+} ≅ H_{dR}(M) ⊗_{ℚ} ℝ. (54)

Here and in what follows, for any commutative ring R and R[Gal(ℂ/ℝ)]-module X we write X^{+} and X^{−} for the R-submodules of X upon which complex conjugation acts as multiplication by 1 and −1 respectively. There is also an A-equivariant direct sum decomposition

(H_{B}(M) ⊗_{ℚ} ℂ)^{+} = (H_{B}(M)^{+} ⊗_{ℚ} ℝ) ⊕ (H_{B}(M)^{−} ⊗_{ℚ} ℝ(2πi)^{−1}) (55)

and an isomorphism

H_{B}(M)^{−} ⊗_{ℚ} ℝ(2πi)^{−1} ≅ H_{B}(M)^{−} ⊗_{ℚ} ℝ (56)

which is induced by identifying ℝ(2πi)^{−1} with ℝ by sending (2πi)^{−1} to 1. Let ϵ_{B} (resp. ϵ_{dR}) be the automorphism [ − 1] in π_{1}V (A_{ℝ}) ≅ K_{1}(A_{ℝ}) which is induced by multiplication by -1 on H_{B}(M)^{+} ⊗_{ℚ} ℝ (resp. F^{0}H_{dR}(M) ⊗_{ℚ} ℝ). We write

ϑ_{∞}^{loc} : Ξ^{loc}(M) ⊗_{ℚ} ℝ ≅ 1_{V (A_{ℝ})} (57)

for the isomorphism of virtual A_{ℝ}-modules which is obtained by applying the functor [ ] to (54), (55) and (56) and then multiplying by ϵ_{B}ϵ_{dR}. The reason for the introduction of ϵ_{B}ϵ_{dR} will become clear in the proof of Theorem 5.3 below. As in previous sections, we now fix a finite set S of places of K which contains S_{∞} and all places at which M has bad reduction, and for each rational prime p we set V := V_{p} := H_{p}(M). For a finite group Π and a Π-module N we denote by C_{Tate}^{•}(Π, N) the standard complex computing Tate cohomology. By a slight abuse of notation we also set C_{Tate}^{•}(Π, N) := C^{•}(Π, N) for any infinite profinite group Π. For any continuous G_{S_{p}}-module N we set

R̃Γ_{c}(O_{K,S_{p}}, N) := Cone (C^{•}(G_{S_{p}}, N) → ⨁_{v∈S_{p}} C_{Tate}^{•}(G_{v}, N)) [−1]

and if N = V_{p} is a ℚ_{p}-vector space we define

1R̃Γ_{c}(O_{K,S_{p}}, V_{p}) := Cone (C^{•}(G_{S_{p}}, N) → ⨁_{v∈S_{p,f}} RΓ(K_{v}, V_{p})) [−1]

so that there is a natural quasi-isomorphism

R̃Γ_{c}(O_{K,S_{p}}, V_{p}) → _{1}R̃Γ_{c}(O_{K,S_{p}}, V_{p}). (58)

We fix once and for all an injective resolution 𝔄_{p} → I^{•} of 𝔄_{p}-𝔄_{p}-bimodules, and for any complex N of 𝔄_{p}-modules (which is cohomologically bounded above) we define a complex of 𝔄_{p}^{op}-modules by N^{∗} := Hom_{𝔄_{p}}(N, I^{•}). Note here that, since the natural map 𝔄_{p} → 𝔄_{p} ⊗_{ℤ_{p}} 𝔄_{p}^{op} is flat, each I^{n} is an injective 𝔄_{p}- module and hence that N^{∗} = RHom_{𝔄_{p}}(N, 𝔄_{p}) in D(𝔄_{p}^{op}). We shall moreover assume that I^{0} = A_{p} and that each I^{n} is torsion if n ≥ 1. There is then an isomorphism of complexes of A_{p}-A_{p}-bimodules

I_{ℚ_{p}}^{•} := ℚ_{p} ⊗_{ℤ_{p}} I^{•} ≅ A_{p}[0]. (59)

If N = T = T_{p} ⊂ V_{p} is a projective 𝔄_{p}-lattice, and in that case only, we put T^{∗} = Hom_{𝔄_{p}}(T, 𝔄_{p}). The isomorphism (52) then induces an identification

ℚ_{p} ⊗_{ℤ_{p}} T^{∗}(1) ≅ Hom_{A_{p}}(V, A_{p}(1)) ≅ Hom_{ℚ_{p}}(V, ℚ_{p}(1)) = V^{∗}(1) (60)

of T^{∗}(1) with an 𝔄_{p}^{op}-lattice in V^{∗}(1).

Lemma 12. a) There is a commutative diagram of maps of complexes

RΓ(O_{K,Sp} , V)  ⟶  ⨁_{v∈Sp,f} RΓ(K_{v}, V)  ⟶  ⨁_{v∈Sp,f} RΓ_{/f} (K_{v}, V)
1 ^{AV}↓  ⨁^{AV}_{v} ↓  ⨁^{AV}_{f,v} ↓
1R̃Γ_{c}(O_{K,Sp} , V^{∗}(1))^{∗}[−3] ⟶ ⨁_{v∈Sp,f} RΓ(K_{v}, V^{∗}(1))^{∗}[−2] ⟶ ⨁_{v∈Sp,f} RΓ_{f} (K_{v}, V^{∗}(1))^{∗}[−2]

in which all of the vertical maps are quasi-isomorphisms. Moreover [_{1} AV ], [ AV_{v}] and [ AV_{f,v}] are independent of any choices made in the construction of this diagram.

b) There is a natural quasi-isomorphism

RΓ(O_{K,S_{p}}, T) ⟶^{AV} R̃Γ_{c}(O_{K,S_{p}}, T^{∗}(1))^{∗}[−3] (61)

so that ℚ_{p} ⊗_{ℤ_{p}} AV = ν[−3] ∘_{1} AV where ν is the composite isomorphism

1R̃Γ_{c}(O_{K,S_{p}}, V^{∗}(1))^{∗} = Hom_{ℚ_{p}}(_{1}R̃Γ_{c}(O_{K,S_{p}}, V^{∗}(1)), ℚ_{p})
⟶^{(52)} Hom_{A_{p}}(_{1}R̃Γ_{c}(O_{K,S_{p}}, V^{∗}(1)), A_{p})
⟶^{(59)} Hom_{A_{p}}(_{1}R̃Γ_{c}(O_{K,S_{p}}, V^{∗}(1)), I_{ℚ_{p}}^{•})
⟶^{(58)} Hom_{A_{p}}(R̃Γ_{c}(O_{K,S_{p}}, V^{∗}(1)), I_{ℚ_{p}}^{•})
⟶^{(60)} Hom_{𝔄_{p}}(R̃Γ_{c}(O_{K,S_{p}}, T^{∗}(1)), I^{•}) ⊗_{ℤ_{p}} ℚ_{p}.

Proof. We first define local pairings for places v | p. To do this we continue to use the notation introduced in §3.2. Recall that B^{i} is an algebra for i = 0, 1 and that the differential of B^{•} is a difference of two algebra homomorphisms β_{1} and β_{2} (cf. (20)). There therefore exists a natural morphism of complexes μ : B^{•} ⊗_{ℚ_{p}} B^{•} → B^{•} for which μ^{0} : B^{0} ⊗_{ℚ_{p}} B^{0} → B^{0} is given by multiplication, μ^{1} : (B^{0} ⊗_{ℚ_{p}} B^{1}) ⊕ (B^{1} ⊗_{ℚ_{p}} B^{0}) → B^{1} is defined by μ_{1}(x ⊗ y, y′ ⊗ x′) = β_{2}(x)y + β_{1}(x′)y′, and μ^{2} : B^{1} ⊗_{ℚ_{p}} B^{1} → 0 is the zero map. This morphism induces a commutative diagram of pairings

V ⊗_{ℚ_{p}} V^{∗}(1)  ⟶  ℚ_{p}(1)
↓  ↓
(B^{•} ⊗_{ℚ_{p}} V) ⊗_{ℚ_{p}} (B^{•} ⊗_{ℚ_{p}} V^{∗}(1)) ⟶ B^{•} ⊗_{ℚ_{p}} ℚ_{p}(1)

and also a commutative diagram of cup product pairings

C^{•}(G_{v}, V) × C^{•}(G_{v}, V^{∗}(1))  ⟶^{∪}  C^{•}(G_{v}, ℚ_{p}(1))
↓  ↓
Tot C^{•}(G_{v}, B^{•} ⊗_{ℚp} V) × Tot C^{•}(G_{v}, (B^{•} ⊗_{ℚp} V^{∗}(1))) ⟶ Tot C^{•}(G_{v}, B^{•} ⊗_{ℚp} ℚ_{p}(1)).

We thereby obtain a commutative diagram of local and global cup product pairings

C^{•}(G_{S_{p}}, V) × C^{•}(G_{S_{p}}, V^{∗}(1))  ⟶^{∪} C^{•}(G_{S_{p}}, ℚ_{p}(1))
res^{V} ↓^{× res_{V ^{∗}(1)}}  ↓^{res}^{ℚp(1)}
⨁_{v∈S_{p,f}} RΓ(K_{v}, V) × ⨁_{v∈S_{p,f}} RΓ(K_{v}, V^{∗}(1)) ⟶^{∪} ⨁_{v∈S_{p,f}} RΓ(K_{v}, ℚ_{p}(1))

(62)

and hence an induced pairing on the mapping cone

C^{•}(G_{S_{p}}, V) ×_{1}R̃Γ_{c}(O_{K,S_{p}}, V^{∗}(1)) ⟶^{∪} _{1}R̃Γ_{c}(O_{K,S_{p}}, ℚ_{p}(1)) ⟶^{T̃r} ℚ_{p}[−3] (63)

so that

(64) res_{ℚ_{p}(1)}^{ad}(res_{V} (x) ∪ y) = x ∪ res_{V^{∗}(1)}^{ad}(y)

where here

res_{V}^{ad} : ⨁_{v∈S_{p,f}} RΓ(K_{v}, V) → _{1}R̃Γ_{c}(O_{K,S_{p}}, V)[1]

is the natural map. The morphism T̃r in (63) is chosen to be a lift of the map

1R̃Γ_{c}(O_{K,S_{p}}, ℚ_{p}(1)) ⊇ τ^{≤3}_{1}R̃Γ_{c}(O_{K,S_{p}}, ℚ_{p}(1))

→ H_{c}^{3}(O_{K,S_{p}}, ℚ_{p}(1))[−3] ⟶^{Tr} ℚ_{p}[−3]

(such a lift exists because ℚ_{p} is an injective ℚ_{p}-module). In the diagram of claim a) the map _{1} AV is induced by (63) and the maps AV_{v} by the local cup product pairing composed with T̃r ∘ res_{ℚ_{p}(1)}^{ad}. These maps are quasi-isomorphisms by local and global duality and the compatibility of local and global trace maps [36, Chap. II, §3]. In addition, the commutativity of the left hand square of the diagram in a) is a consequence of (64). The right hand square of the diagram in a) arises as a direct sum of commutative squares over the places in S_{p,f}, and for each such place v the existence of the appropriate square will follow directly if we can show that the complexes RΓ_{f}(K_{v}, V) and RΓ_{f}(K_{v}, V^{∗}(1)) (and not only their cohomology) annihilate each other under the pairing T̃r ∘ res_{ℚ_{p}(1)}^{ad} ∘ ∪ constructed above. To prove the required annihilation property, we consider separately the cases v ∤ p and v | p. If firstly v ∤ p, then RΓ_{f}(K_{v}, V) coincides with the subcomplex C^{•}(G_{v}/I_{v}, V^{I_{v}}) of C^{•}(G_{v}, V). In addition, since H^{2}(G_{v}/I_{v}, ℚ_{p}(1)) = 0, we can certainly choose the lifting T̃r in such a way that it vanishes on the subcomplex

res_{ℚ_{p}(1)}^{ad} ( ⨁_{v∈S_{p,f}} C^{•}(G_{v}/I_{v}, ℚ_{p}(1))) ⊆ _{1}R̃Γ_{c}(O_{K,S_{p}}, ℚ_{p}(1))[1].

If now v | p, then the subcomplex RΓ_{f}(K_{v}, V) = H^{0}(G_{v}, B^{•} ⊗_{ℚ_{p}} V) is concentrated in degrees 0 and 1 and the cup product of x ∈ H^{0}(G_{v}, B^{1} ⊗_{ℚ_{p}} V) and x′ ∈ H^{0}(G_{v}, B^{1} ⊗_{ℚ_{p}} V^{∗}(1)) is given by μ_{2}(x ⊗ x′) = 0. Hence RΓ_{f}(K_{v}, V) and RΓ_{f}(K_{v}, V^{∗}(1)) do indeed annihilate each other. We observe that, for each v ∈ S_{p,f}, the resulting morphism AV_{f,v} : RΓ_{f}(K_{v}, V) → RΓ_{f}(K_{v}, V^{∗}(1))^{∗}[−2] is a quasi-isomorphism as a consequence of [4, Prop. 3.8]. Also, since all of the maps which are induced on cohomology by _{1} AV, AV_{v} and AV_{f,v} are independent of the choice of the lift T̃r, and their sources and targets all belong to D^{p,p}(A_{p}) = D^{p}(A_{p}), Proposition 2.1e) implies the final assertion of claim a). To prove claim b), we argue in a similar way starting with the diagram

C^{•}(G_{S_{p}}, T) × C^{•}(G_{S_{p}}, T^{∗}(1))  ⟶^{∪} C^{•}(G_{S_{p}}, 𝔄_{p}(1))
res^{T} ↓^{× res_{T^{∗}(1)}}  ↓^{res}^{𝔄p(1)}
⨁_{v∈S_{p}} C_{Tate}^{•}(G_{v}, T) × ⨁_{v∈S_{p}} C_{Tate}^{•}(G_{v}, T^{∗}(1)) ⟶^{∪} ⨁_{v∈S_{p}} C_{Tate}^{•}(G_{v}, 𝔄_{p}(1))

(65)

and using a lift T̃r_{𝔄} of the map

R̃Γ_{c}(O_{K,S_{p}}, 𝔄_{p}(1)) ⊇ τ^{≤3}R̃Γ_{c}(O_{K,S_{p}}, 𝔄_{p}(1))

→ H_{c}^{3}(O_{K,S_{p}}, 𝔄_{p}(1))[−3] ⟶^{Tr_{𝔄}} 𝔄_{p}[−3] → I^{•}[−3].

The resulting map AV (as in (61)) is a quasi-isomorphism by [9, Lem. 16]. In addition, there is a natural map from diagram (65) to diagram (62), inducing a commutative diagram

C^{•}(G_{S_{p}}, T) × R̃Γ_{c}(O_{K,S_{p}}, T^{∗}(1)) ⟶ R̃Γ_{c}(O_{K,S_{p}}, 𝔄_{p}(1))) ⟶^{T̃r_{𝔄}} I^{•}[−3]
↓  ↓  ↓
C^{•}(G_{S_{p}}, V) ×_{1}R̃Γ_{c}(O_{K,S_{p}}, V^{∗}(1)) ⟶ _{1}R̃Γ_{c}(O_{K,S_{p}}, ℚ_{p}(1))) ⟶^{T̃r} ℚ_{p}[−3]

where the left vertical arrow involves (60) and the middle and right vertical arrows involve the map A → ℚ which is the image of 1 ∈ A^{op} under the isomorphism A^{op} ≅ A^{∗} = Hom_{ℚ}(A, ℚ) chosen before (52). The second statement in b) then follows easily from this last commutative diagram. □

We now define a virtual A_{p}-module Λ_{p}(S, V_{p}) by setting

Λ_{p}(S, V_{p}) := ⊠_{v∈S_{p,f}} [RΓ(K_{v}, V_{p})]^{−1} ⊠ [ Ind_{K}^{ℚ} V_{p}]^{−1}. (66)

We also define

θ_{p} : A_{p} ⊗_{A} Ξ^{loc}(M) ≅ Λ_{p}(S, V_{p}) (67)

to be the isomorphism in V (A_{p}) which results from composing the isomorphisms obtained by applying [ ] to the canonical A × Gal(ℂ/ℝ)-equivariant comparison isomorphism

H_{B}(M) ⊗_{ℚ} ℚ_{p} ≅ Ind_{K}^{ℚ} V_{p}, (68)

to the A-equivariant (Poincaré duality) exact sequence

0 → (H_{dR}(M^{∗}(1))/F^{0})^{∗} → H_{dR}(M) → H_{dR}(M)/F^{0} → 0, (69)

to (23) for both M and M^{∗}(1), to (18) and the maps AV_{f,v} for each v ∈ S_{p,f}, to (19) for each v ∈ S with v ∤ p and (22) for each v | p, and by then using the isomorphisms (24) for V = V_{p,v} and V = V_{p}^{∗}(1)_{v} and each v ∈ S_{p,f}.

Given a projective 𝔄-structure T on M we define C(K, T_{p}) to be the mapping cone of the composite map

\begin{aligned}RΓ_{c}(O_{K,S_{p}}, T_{p})[−1] & ⟶ & RΓ(O_{K,S_{p}}, T_{p})[−1] \\ AV & ↓ \\ & & R̃Γ_{c}(O_{K,S_{p}}, T_{p}^{∗}(1))^{∗}[−4] & ⟶ & RΓ_{c}(O_{K,S_{p}}, T_{p}^{∗}(1))^{∗}[−4] \end{aligned} (70)

and we set

Λ_{p}(S, T_{p}) := [C(K, T_{p})]. (71)

We next define a canonical isomorphism in V (A_{p})

θ′_{p} : Λ_{p}(S, V_{p}) ⟶^{∼} A_{p} ⊗_{𝔄_{p}} Λ_{p}(S, T_{p}).

To do this we first define _{1}C(K, V_{p}) just as C(K, T_{p}) but using diagram A_{p} ⊗_{𝔄_{p}} (70) with RΓ_{c}(O_{K,S_{p}}, V_{p}) replaced by _{1}RΓ_{c}(O_{K,S_{p}}, V_{p}), and we define _{2}C(K, V_{p}) by also replacing R̃Γ_{c}(O_{K,S_{p}}, V_{p}^{∗}(1)), RΓ_{c}(O_{K,S_{p}}, V_{p}^{∗}(1)) and AV by their respective versions indexed by 1. Then there are natural quasiisomorphisms

A_{p} ⊗_{𝔄_{p}} C(K, T_{p}) ⟶^{∼} _{1}C(K, V_{p}) ⟵^{∼} _{2}C(K, V_{p}) (72)

where we have used the last assertion in Lemma 12b) for the second quasiisomorphism. Setting

2L(S_{p}, V_{p}) :=

Cone (_{1}RΓ_{c}(O_{K,S_{p}}, V_{p}) → RΓ(O_{K,S_{p}}, V_{p}) ⟶^{_{1} AV} _{1}R̃Γ_{c}(O_{K,S_{p}}, V_{p}^{∗}(1))^{∗}[−3])

we obtain a true nine term diagram

\begin{matrix}⨁_{v∈S∞} RΓ(K_{v}, V_{p}^{∗}(1))^{∗}[−4] & & ⨁_{v∈S∞} RΓ(K_{v}, V_{p}^{∗}(1))^{∗}[−4] \\ ↓ & & ↓ \\ _{1}R̃Γ_{c}(O_{K,Sp} , V_{p}^{∗}(1))^{∗}[−4] & ⟶ & _{2}L(S_{p}, V_{p})[−1] & ⟶ & _{1}RΓ_{c}(O_{K,Sp} , V_{p}) \\ ↓ & & ↓ & & ∥ \\ _{1}RΓ_{c}(O_{K,Sp} , V_{p}^{∗}(1))^{∗}[−4] & ⟶ & _{2}C(K, V_{p}) & ⟶ & _{1}RΓ_{c}(O_{K,Sp} , V_{p}). \end{matrix} (73)

There is also a commutative diagram of true triangles

\begin{matrix}RΓ(O_{K,S_{p}}, V_{p})[−1] & ⟶ & _{1}L(S_{p}, V_{p})[−1] & ⟶ & _{1}RΓ_{c}(O_{K,S_{p}}, V_{p}) \\ _{1} ^{AV}↓ & & ^{λ}↓ & & ∥ \\ _{1}R̃Γ_{c}(O_{K,S_{p}}, V_{p}^{∗}(1))^{∗}[−4] & ⟶ & _{2}L(S_{p}, V_{p})[−1] & ⟶ & _{1}RΓ_{c}(O_{K,S_{p}}, V_{p}) \end{matrix} (74)

where the bottom row coincides with the central row in (73) and

1L(S_{p}, V_{p}) := Cone (_{1}RΓ_{c}(O_{K,S_{p}}, V_{p}) → RΓ(O_{K,S_{p}}, V_{p})) .

Lemma 13. Let

E : 0 ⟶ A ⟶^{ι} B ⟶^{π} C ⟶ 0

be any true triangle, and let E′ denote the associated canonical true triangle

0 ⟶ C ⟶ Cone(π) ⟶ B[1] ⟶ 0.

Then there is a natural quasi-isomorphism A[1] ⟶^{q} Cone(π) for which the following diagram commutes

[A[1]] ⟶ [A[1]] ⊠ [C] ⊠ [C]^{−1}
[q]↓  ^{[E]}↓
[ Cone(π)] ⟶^{[E′]}  [C] ⊠ [B[1]].

Proof. This is an immediate consequence of the true nine term diagram

0 ⟶ A[1]  A[1]
↓  ^{q}↓  ↓
C ⟶ Cone(π) ⟶ B[1]
∥  ↓  ↓
C ⟶ Z ⟶ C[1],

where here q(x) = (0, ι(x)) ∈ C ⊕ B[1] = Cone(π) and Z is acyclic. □

By applying Lemma 13 to the short exact sequence given by the central row in (26) we obtain a canonical quasi-isomorphism

L(S_{p}, V_{p}) := ⨁_{v∈S_{p}} RΓ(K_{v}, V_{p}) ⟶^{q} _{1}L(S_{p}, V_{p}). (75)

Upon composing the isomorphisms in V (A_{p}) which are induced by (72), the central column in (73) and the isomorphisms λ^{−1} from (74), q^{−1} from (75),

⨁_{v∈S_{∞}} RΓ(K_{v}, V_{p}^{∗}(1))^{∗} ≅ Ind_{K}^{ℚ} V_{p}^{∗}(1)^{∗+}[0],

(76) (Ind_{K}^{ℚ} V_{p}^{∗}(1))^{∗+} ≅ (Ind_{K}^{ℚ} V_{p}(−1))^{+} ≅ (Ind_{K}^{ℚ} V_{p})^{−}

and

(77) Ind_{K}^{ℚ} V_{p} ≅ (Ind_{K}^{ℚ} V_{p})^{+} ⊕ (Ind_{K}^{ℚ} V_{p})^{−},

we obtain the desired isomorphism

θ′_{p} : Λ_{p}(S, V_{p}) = [L(S_{p,f}, V_{p})]^{−1} ⊠ [ Ind_{K}^{ℚ} V_{p}]^{−1} ≅ A_{p} ⊗_{𝔄_{p}} Λ_{p}(S, T_{p}). (78)

Lemma 14. If p is odd, then the isomorphism θ′_{p} is induced by an isomorphism

⊠_{v∈S_{p,f}} [RΓ(K_{v}, T_{p})]^{−1} ⊠ [ Ind_{K}^{ℚ} T_{p}]^{−1} ≅ Λ_{p}(S, T_{p}). (79)

Proof. For each v ∈ S_{∞} and continuous G_{S_{p}}-module N we define RΓ_{Δ}(K_{v}, N) by the short exact sequence

0 → C^{•}(G_{v}, N) → C_{Tate}^{•}(G_{v}, N) → RΓ_{Δ}(K_{v}, N)[1] → 0

where the second map is the natural inclusion. We then have a true nine term diagram

⨁_{v∈S_{p}} C^{•}(G_{v}, N)[−1] ⟶ RΓ_{c}(O_{K,S_{p}}, N) ⟶ RΓ(O_{K,S_{p}}, N)
↓  ↓  ∥
⨁_{v∈S_{p}} C_{Tate}^{•}(G_{v}, N)[−1] ⟶ R̃Γ_{c}(O_{K,S_{p}}, N) ⟶ RΓ(O_{K,S_{p}}, N)
↓  ↓
⨁_{v∈S_{∞}} RΓ_{Δ}(K_{v}, N)  ⨁_{v∈S_{∞}} RΓ_{Δ}(K_{v}, N).

(80)

Upon defining

3L(S_{p}, T_{p}) :=

Cone (RΓ_{c}(O_{K,S_{p}}, T_{p}) → RΓ(O_{K,S_{p}}, T_{p}) ⟶^{AV} R̃Γ_{c}(O_{K,S_{p}}, T_{p}^{∗}(1))^{∗}[−3])

there is in addition a true nine term diagram

⨁_{v∈S∞} RΓ_{Δ}(K_{v}, T_{p}^{∗}(1))^{∗}[−4]  ⨁_{v∈S∞} RΓ_{Δ}(K_{v}, T_{p}^{∗}(1))^{∗}[−4]
↓  ↓
R̃Γ_{c}(O_{K,Sp} , T_{p}^{∗}(1))^{∗}[−4] ⟶  _{3}L(S_{p}, T_{p})[−1]  ⟶ RΓ_{c}(O_{K,Sp} , T_{p})
↓  ↓  ∥
RΓ_{c}(O_{K,Sp} , T_{p}^{∗}(1))^{∗}[−4] ⟶  C(K, T_{p})  ⟶ RΓ_{c}(O_{K,Sp} , T_{p})

(81)

in which the left hand column is the dual of the central column in (80) with N = T_{p}^{∗}(1). If p is odd, then all terms in the central column of (81) belong to D^{p}(𝔄_{p}), and RΓ_{Δ}(K_{v}, T_{p}^{∗}(1)) is naturally quasi-isomorphic to RΓ(K_{v}, T_{p}^{∗}(1)). In addition, setting

4L(S_{p}, T_{p}) := Cone (RΓ_{c}(O_{K,S_{p}}, T_{p}) → RΓ(O_{K,S_{p}}, T_{p}))

there exists a commutative diagram of true triangles similar to (74) and quasiisomorphisms

⨁_{v∈S_{p}} RΓ(K_{v}, T_{p}) ⟶^{q} _{4}L(S_{p}, T_{p}) ← _{3}L(S_{p}, T_{p})

which together give (79). □

Remark 16. If p is odd, then the isomorphism (79) allows a more direct definition of Λ_{p}(S, T_{p}) than that given by (71). However, we do not expect the statement of Lemma 14 to hold for p = 2. More concretely, if for example 𝔄 = ℤ, p = 2 and we interpret virtual objects as graded determinants, then the ℤ_{2}-lattices in the ℚ_{2}-line Λ_{p}(S, V_{p}) given respectively by Λ_{p}(S, T_{p}) and ⊠_{v∈S_{p,f}} [RΓ(K_{v}, T_{p})]^{−1} ⊠ [ Ind_{K}^{ℚ} T_{p}]^{−1} may well differ.

We now define

ϑ_{p}^{loc} := ϵ(S, p) ∘ θ′_{p} ∘ θ_{p} : A_{p} ⊗_{A} Ξ^{loc}(M) ≅ A_{p} ⊗_{𝔄_{p}} Λ_{p}(S, T_{p}) (82)

where θ_{p} was defined in (67), θ′_{p} in (78) and where ϵ(S, p) ∈ π_{1}(V (A_{p})) is the automorphism [ − 1] which is induced by multiplication by −1 on ⨁_{v∈S_{p,f}} RΓ_{/f}(K_{v}, V_{p}) (again, the reason for the introduction of ϵ(S, p) will become clear in the proof of Theorem 5.3 below). We then define an object of the category V (𝔄_{p}) ×_{V (A_{p})} V (A) by setting

Ξ^{loc}(M, T_{p}, S) := (Λ_{p}(S, T_{p}), Ξ^{loc}(M), ϑ_{p}^{loc}).

The following result is a natural analogue of Lemmas 5 and 6 for Ξ^{loc}(M, T_{p}, S).

Lemma 15. a) For a different choice of projective 𝔄-structure T′ on M and a different set of places S′ the objects Ξ^{loc}(M, T_{p}, S) and Ξ^{loc}(M, T′_{p}, S′) are isomorphic in V (𝔄_{p}) ×_{V (A_{p})} V (A).

b) Let M be a direct factor of h^{n}(X)(r) for a smooth projective variety X over K. If (M, A) satisfies Conjecture 3, then the object

Ξ^{loc}(M)_{ℤ} := (∏_{p} Λ_{p}(S, T_{p}), Ξ^{loc}(M), ∏_{p} ϑ_{p}^{loc})

of the category ∏_{p} V (𝔄_{p}) ×_{∏_{p} V (A_{p})} V (A) is isomorphic to the image of an object of 𝕍(𝔄) under the functor of Lemma 4.

Proof. Under further assumptions on M both of these claims follow from the proof of Theorem 5.3 given below. For brevity, we shall therefore just sketch a proof here. For a second lattice T′ ⊆ T one has a commutative diagram

RΓ_{c}(O_{K,S_{p}}, T′_{p})[−1] ⟶ RΓ_{c}(O_{K,S_{p}}, T_{p})[−1]
AV↓  ^{AV}↓
RΓ_{c}(O_{K,S_{p}}, (T′_{p})^{∗}(1))^{∗}[−4] ⟶ RΓ_{c}(O_{K,S_{p}}, T_{p}^{∗}(1))^{∗}[−4].

One can then argue just as in the proof of Lemma 5 using [18, Th. 5.1] for both of the modules F := T_{p}/T′_{p} and Hom_{ℤ_{p}}(F, ℚ_{p}/ℤ_{p}(1)) ≅ Ext_{𝔄_{p}}^{1}(F, 𝔄_{p}(1)). For the independence of S it is enough for us to consider the case S′ = S ∪ {w} with w ∉ S_{p}. In this case, the true triangle (18) is induced from a triangle in D^{p}(𝔄_{p})

(T_{p} ⟶^{1−f_{w}^{−1}} T_{p}) → RΓ(K_{w}, T_{p}) → (T_{p}^{∗}(1) ⟶^{1−f_{w}^{−1}} T_{p}^{∗}(1))^{∗} [−2] (83)

and the isomorphism (24) is induced by an isomorphism [T_{p} ⟶^{1−f_{w}^{−1}} T_{p}] ≅ 1_{V (𝔄_{p})}, and similarly for T_{p}^{∗}(1). Moreover, ϵ(S′, p)ϵ(S, p)^{−1} coincides with the

automorphism which is induced by multiplication by −1 on [T_{p}^{∗}(1) ⟶^{1−f_{w}^{−1}} T_{p}^{∗}(1)]^{∗}[−2] and hence lies in the image of π_{1}(V (𝔄_{p})). This suffices to construct an isomorphism between Ξ^{loc}(M, T_{p}, S) and Ξ^{loc}(M, T_{p}, S′). Claim b) is proved by choosing a smooth proper model of X over Spec(O_{K,S}) and then arguing just as in [8, pp. 81-83]. □

Following the last result, we define RΩ^{loc}(M, 𝔄) to be the class of (Ξ^{loc}(M)_{ℤ}, ϑ_{∞}^{loc}) in π_{0}(𝕍(𝔄, ℝ)) ≅ K_{0}(𝔄, ℝ). We observe that, as a consequence of (71), one has

RΩ^{loc}(M, 𝔄) ∈ Cl(𝔄, ℝ).

5.2. Definition of L^{loc}(M, 𝔄). Recall that L_{∞}(_{A}M, s) = ∏_{v∈S_{∞}} L_{v}(_{A}M, s) and Λ(_{A}M, s) = L_{∞}(_{A}M, s)L(_{A}M, s). The following conjecture is standard.

Conjecture 7. There is an identity of meromorphic ζ(A_{ℂ})-valued functions of the complex variable s

Λ(_{A}M, s) = ϵ(_{A}M, s)Λ(_{A^{op}}M^{∗}(1), −s).

Letting ρ ∈ ℤ^{π_{0}(Spec(ζ(A_{ℝ})))} denote the algebraic order at s = 0 of the meromorphic function Λ(_{A^{op}}M^{∗}(1), s), we set

E(_{A}M) := (−1)^{ρ}ϵ(_{A}M, 0)\frac{L_{∞}^{∗}(_{A^{op}}M^{∗}(1), 0)}{L_{∞}^{∗}(_{A}M, 0)} ∈ ζ(A_{ℝ})^{×}

and

L^{loc}(M, 𝔄) := δ̂_{𝔄,ℝ}^{1}(E(_{A}M)) ∈ Cl(𝔄, ℝ).

We then set

T Ω^{loc}(M, 𝔄) := L^{loc}(M, 𝔄) + RΩ^{loc}(M, 𝔄) ∈ Cl(𝔄, ℝ).

The following result can be proved by mimicking the proofs of Theorems 3.1 and 4.1.

Theorem 5.1. All assertions of Theorem 3.1 remain valid with RΩ(−, −) replaced by either RΩ^{loc}(−, −), L^{loc}(−, −) or T Ω^{loc}(−, −). □

We now describe conditions under which T Ω^{loc}(M, 𝔄) can be shown to belong to Cl(𝔄, ℚ).

Theorem 5.2. If Deligne’s conjecture [16, Conj. 6.6] on the nature of rank one motives over ℚ is valid, then T Ω^{loc}(M, 𝔄) ∈ Cl(𝔄, ℚ). More precisely, if (in the notation of the proof of Proposition 4.2b) in §4.6) for each index i ∈ {1, …, r} there exists an integer p_{i}, an E_{i}-valued Dirichlet character χ_{i} and an isomorphism of motives over ℚ with coefficients in E_{i}

⋀_{E_{i}}^{max}e_{i}ϵ_{i}(Res_{ℚ}^{K}(B ⊗_{A} M)) ≅ E_{i}(p_{i})(χ_{i}), (84)

then T Ω^{loc}(M, 𝔄) ∈ Cl(𝔄, ℚ).

Proof. Upon combining the functorial behaviour of T Ω^{loc}(−, −) which is described in Theorem 5.1 together with the arguments used to prove Proposition 4.2 one finds that the containment T Ω^{loc}(M, 𝔄) ∈ Cl(𝔄, ℚ) can be decided by considering the motives which occur on the left hand side of (84). Indeed, it follows that T Ω^{loc}(M, 𝔄) ∈ Cl(𝔄, ℚ) if and only if T Ω^{loc}(e_{i}ϵ_{i}(Res_{ℚ}^{K}(B ⊗_{A} M)), O_{E_{i}}) ∈ K_{0}(O_{E_{i}}, ℚ) for each index i ∈ {1, …, r}. We now fix such an index i and set N := e_{i}ϵ_{i}(Res_{ℚ}^{K}(B ⊗_{A} M)) and E := E_{i}. Following [11, Lemma 1a)], one has T Ω^{loc}(N, O_{E}) ∈ K_{0}(O_{E}, ℚ) if and only if

E(_{E}N)^{−1}ϑ_{∞}^{loc}(Ξ^{loc}(N)) ⊆ E,

where here ϑ_{∞}^{loc} is the isomorphism (57) for the pair (N, E). It therefore suffices to prove the displayed inclusion and to do this we adapt the proof of [16, Th. 5.6]. After fixing E-bases of H_{B}(N) and H_{dR}(N) we let δ(N) ∈ E_{ℂ} denote the corresponding determinant of the isomorphism (53) (with M = N and K = ℚ). We set d^{±} := rank_{E}(H_{B}(N)^{±}). After adjoining E-bases of H_{B}(N)^{+} and H_{B}(N)^{−} the isomorphism (56) (with M = N) implies that ϑ_{∞}^{loc}(Ξ^{loc}(N)) is the E-subspace of E_{ℝ} which is generated by the element (2πi)^{−d^{−}}δ(N), and so we need to prove that

E(_{E}N)^{−1}(2πi)^{−d^{−}}δ(N) ∈ E. (85)

Now from [37, Lem. C.3.7] one has

\frac{L_{∞}^{∗}(_{E}N^{∗}(1), 0)}{L_{∞}^{∗}(_{E}N, 0)}.(2π)^{d^{−}+t} ∈ E

where here t := \frac{1}{2}w(d^{+} + d^{−}) ∈ ℤ with w equal to the weight of N. On the other hand, by assuming that there is an isomorphism of the form (84) Deligne has proved that

ϵ(_{E}N, 0)i^{d^{−}}(2π)^{−t}δ(N)^{−1} ∈ E

([16, second formula on p. 331]). Upon combining the last two displayed containments we obtain (85). □

5.3. The comparison of T Ω(M, 𝔄) and T Ω(M^{∗}(1), 𝔄^{op}). The exact functor P ↦ P^{∗} := Hom_{𝔄^{op}}(P, 𝔄^{op}) induces an equivalence of exact categories PMod(𝔄^{op}) → PMod(𝔄)^{op}. We obtain induced equivalences PMod(A^{op}) → PMod(A)^{op} under scalar extension and also induced equivalences of Picard categories

𝕍(𝔄^{op}) ⟶^{∗} 𝕍(𝔄)^{op} ⟶^{ι} 𝕍(𝔄), 𝕍(𝔄^{op}, ℝ) ⟶^{∗} 𝕍(𝔄, ℝ)^{op} ⟶^{ι} 𝕍(𝔄, ℝ)

where the functor ι sends each morphism to its inverse. We denote each of these composite functors by X ↦ X^{∗} and we use ψ^{∗} to denote the induced isomorphisms on algebraic K-groups. If F is any field of characteristic 0, then the maps ψ^{∗} combine to give an isomorphism of localisation sequences

⋯ ⟶ K_{1}(A_{F}^{op}) ⟶^{δ_{𝔄^{op},F}^{1}} Cl(𝔄^{op}, F) ⟶^{δ_{𝔄^{op},F}^{0}} Cl(𝔄^{op}) ⟶ 0
ψ^{∗}↓  ^{ψ^{∗}}↓  ^{ψ^{∗}}↓
⋯ ⟶ K_{1}(A_{F}) ⟶^{δ_{𝔄,F}^{1}} Cl(𝔄, F) ⟶^{δ_{𝔄,F}^{0}} Cl(𝔄) ⟶ 0.

(86)

Lemma 16. One has

ψ^{∗} ∘ δ̂_{𝔄^{op},ℝ}^{1} = −δ̂_{𝔄,ℝ}^{1}

on ζ(A_{ℝ}^{op})^{×} = ζ(A_{ℝ})^{×}.

Proof. For any field F of characteristic 0 there is a commutative diagram

\begin{matrix}K_{1}(A_{F}^{op}) & ⟶^{ψ^{∗}} & K_{1}(A_{F}) \\ ^{nr}^{A{op \atop F}} ↓ & & ^{nr}^{A_{F}} ↓ \\ ζ(A_{F})^{×} & ⟶^{−1} & ζ(A_{F})^{×}. \end{matrix} (87)

This is a consequence of the fact that if V ∈ Ob(PMod(A_{F}^{op})) and ϕ ∈ Aut_{A_{F}^{op}}(V), then ψ^{∗}(ϕ) = Hom_{F} (ϕ, F)^{−1}. The claimed equality thus follows from the definition of δ̂_{𝔄,ℝ}^{1} in terms of δ_{𝔄,ℝ}^{1} and δ_{𝔄_{p},ℚ_{p}}^{1} by using the commutativity of (86) and (87) (cf. the proof of Lemma 10). □

Theorem 5.3. Assume that Conjectures 1 and 2 and the Coherence hypothesis are valid for both (M, A) and (M^{∗}(1), A^{op}), and also that Conjecture 7 is valid for (M, A). Let 𝔄 be an order in A for which M has a projective 𝔄-structure. Then M^{∗}(1) has a projective 𝔄^{op}-structure and there is an equality

T Ω(M, 𝔄) + ψ^{∗}(T Ω(M^{∗}(1), 𝔄^{op})) = T Ω^{loc}(M, 𝔄) (88)

in K_{0}(𝔄, ℝ).

Corollary 1. Assume that Conjecture 4 is valid for the pair (M, 𝔄). Then Conjecture 4(iii), resp. 4(iv), is valid for the pair (M^{∗}(1), 𝔄^{op}) if and only if T Ω^{loc}(M, 𝔄) ∈ Cl(𝔄, ℚ), resp. T Ω^{loc}(M, 𝔄) = 0.

Proof. This follows as an immediate consequence of Theorem 5.3 and the fact that ψ^{∗} restricts to give an isomorphism Cl(𝔄^{op}, ℚ) ≅ Cl(𝔄, ℚ). □

Remark 17. If 𝔄 is commutative, then 𝔄 = 𝔄^{op} and ψ^{∗} coincides with multiplication by −1 on K_{0}(𝔄, ℝ) and so (88) simplifies to give an equality

T Ω(M, 𝔄) − T Ω(M^{∗}(1), 𝔄) = T Ω^{loc}(M, 𝔄).

To justify the claim that ψ^{∗} = −1 whenever 𝔄 is commutative we first recall that, as a consequence of Propositions 2.5 and 2.4, all elements in K_{0}(𝔄, ℝ) can be represented by pairs ((L, α), g) where (L, α) is a graded invertible 𝔄module and g : A_{ℝ} ⊗_{𝔄} (L, α) ≅ 1_{A_{ℝ}} = (A_{ℝ}, 0) is an isomorphism in P(A_{ℝ}). Since the image of Spec(A_{ℝ}) is dense in Spec(𝔄), this implies that α = 0 and also that g : L ⊗_{ℤ} ℝ ⟶^{∼} A_{ℝ} is an isomorphism of ordinary line bundles. Then ψ^{∗}(L, 0) = (L^{∗}, 0) and

ψ^{∗}((L, 0)_{ℝ} ⟶^{g} (A_{ℝ}, 0)) = ((L^{∗}, 0)_{ℝ} ⟶^{(g^{∗})^{−1}} (𝔄_{ℝ}^{∗}, 0) ≅ (𝔄_{ℝ}, 0))

where this last isomorphism sends the identity map in 𝔄^{∗} = Hom_{𝔄}(𝔄, 𝔄) to

the identity element of 𝔄. Now since (L ⊗_{𝔄} L^{∗})_{ℝ} ⟶^{g⊗(g^{∗})^{−1}} 𝔄_{ℝ} ⊗ 𝔄_{ℝ} = 𝔄_{ℝ} is isomorphic to 𝔄_{ℝ} ⟶^{id} 𝔄_{ℝ} via the evaluation map L ⊗_{𝔄} L^{∗} → 𝔄, it follows that ψ^{∗}((L, 0), g) does indeed represent the inverse of ((L, 0), g) in K_{0}(𝔄, ℝ).

Proof of Theorem 5.3. By applying the (monoidal) functor (−)^{∗} to the object Ξ(M^{∗}(1)) of V (A^{op}) (as defined in (29)) one finds that there is an isomorphism in V (A)

Ξ(M^{∗}(1))^{∗} ≅ [H_{f}^{0}(K, M^{∗}(1))^{∗}] ⊠ [H_{f}^{1}(K, M^{∗}(1))^{∗}]^{−1}

⊠ [H_{f}^{1}(K, M)] ⊠ [H_{f}^{0}(K, M)]^{−1}

⊠ ( ⊠_{v∈S_{∞}} [H_{v}(M^{∗}(1))^{G_{v},∗}]^{−1}) ⊠ [(H_{dR}(M^{∗}(1))/F^{0})^{∗}]

and hence also an isomorphism

(89) Ξ(M) ⊠ Ξ(M^{∗}(1))^{∗} ≅ [ ⨁_{v∈S_{∞}} H_{v}(M)^{G_{v}}]^{−1} ⊠ [(H_{dR}(M)/F^{0})]

⊠ [ ⨁_{v∈S_{∞}} H_{v}(M^{∗}(1))^{G_{v},∗}]^{−1} ⊠ [(H_{dR}(M^{∗}(1))/F^{0})^{∗}].

We now observe that

⨁_{v∈S_{∞}} H_{v}(M)^{G_{v}} = H_{B}(M)^{+} (90)

and that there are natural A-equivariant isomorphisms

H_{B}(M^{∗}(1))^{+∗} ≅ H_{B}(M^{∗}(1))^{∗+} ≅ H_{B}(M)^{−}(−1) ≅ H_{B}(M)^{−} (91)

where the last map is induced by sending each element y ⊗ (2πi)^{−1} to y. After applying [ ] to (90), to the linear dual of (90) for M^{∗}(1), to (91), to the natural isomorphism

H_{B}(M) ≅ H_{B}(M)^{+} ⊕ H_{B}(M)^{−} (92)

and to the Poincaré duality sequence (69), the right hand side of (89) identifies with Ξ^{loc}(M), and hence one obtains an isomorphism of virtual A-modules

ϑ^{P D} : Ξ(M) ⊠ Ξ(M^{∗}(1))^{∗} ≅ Ξ^{loc}(M).

Lemma 17. a) ϑ_{∞}^{loc} ∘ (A_{ℝ} ⊗_{A} ϑ^{P D}) = ϑ_{∞}(M) ⊠ ϑ_{∞}(M^{∗}(1))^{∗}.

b) For each projective 𝔄-structure T on M and each prime p there is a commutative diagram in V (A_{p})

A_{p} ⊗_{A} (Ξ(M) ⊠ Ξ(M^{∗}(1))^{∗})  ⟶^{A_{p}⊗_{A}ϑ^{P D}} A_{p} ⊗_{A} Ξ^{loc}(M)
ϑ_{p}(M)⊠ϑ_{p}(M^{∗}(1))^{∗}↓  ^{ϑ_{p}^{loc}}↓

A_{p} ⊗_{𝔄_{p}} [RΓ_{c}(O_{K,S_{p}}, T_{p})] ⊠ [RΓ_{c}(O_{K,S_{p}}, T_{p}^{∗}(1))^{∗}] ⟶^{A_{p} ⊗_{𝔄p}ϑ_{p}^{AV}} A_{p} ⊗_{𝔄_{p}} Λ_{p}(S, T_{p})

where

ϑ_{p}^{AV} : [RΓ_{c}(O_{K,S_{p}}, T_{p})] ⊠ [RΓ_{c}(O_{K,S_{p}}, T_{p}^{∗}(1))^{∗}] ⟶^{∼} Λ_{p}(S, T_{p})

is the isomorphism in V (𝔄_{p}) which is induced by the definition (71) of Λ_{p}(S, T_{p}).

We assume for the moment that this lemma is true. Then from claim b) we deduce that there is an isomorphism in 𝕍(𝔄)

ϑ_{ℤ}^{P D} : Ξ(M)_{ℤ} ⊠ Ξ(M^{∗}(1))_{ℤ}^{∗} ≅ Ξ^{loc}(M)_{ℤ}.

Taken in conjunction with the equality of claim a), this isomorphism in turn implies that there is an equality

RΩ(M, 𝔄) + ψ^{∗}(RΩ(M^{∗}(1), 𝔄^{op})) = RΩ^{loc}(M, 𝔄) (93)

in K_{0}(𝔄, ℝ). On the other hand, by taking leading coefficients at s = 0 in Conjecture 7 we find that

L^{∗}(_{A}M, 0) = E(_{A}M)L^{∗}(_{A^{op}}M^{∗}(1), 0) in ζ(A_{ℝ})^{×}. By applying δ̂_{𝔄,ℝ}^{1} to this equality and then using Lemma 16 we obtain an equality

L(M, 𝔄) =L^{loc}(M, 𝔄) + δ̂_{𝔄,ℝ}^{1}(L^{∗}(M^{∗}(1), 0))
=L^{loc}(M, 𝔄) − ψ^{∗}(L(M^{∗}(1), 𝔄^{op}))

in K_{0}(𝔄, ℝ). Upon comparing this equality to (93) we finally obtain the formula of Theorem 5.3.

It therefore only remains to prove Lemma 17, and our proof of this result will occupy the rest of this section. Before starting the proof however we introduce another useful convention. For any integer n the symbol (n)^{∗} refers to the equality, isomorphism or exact triangle which is obtained by applying the functor ^{∗} to the displayed formula (n) with M^{∗}(1) in place of M and V_{p}^{∗}(1) in place of V_{p}; (n)^{+} refers to the equality, isomorphism or exact triangle obtained by taking Gal(ℂ/ℝ)-invariants of (n); (n)_{v} indicates that the formula (n) is to be used for all places v in S_{p,f} to which it applies.

Proof of Lemma 17. We begin with the proof of part a).

Lemma 18. (cf. [20][Prop. III.1.1.6 iii)]) With notation as in section 3.2 there are natural isomorphisms of A_{ℝ}-modules

ker(α_{M}) ≅ coker(α_{M^{∗}(1)})^{∗}, coker(α_{M}) ≅ ker(α_{M^{∗}(1)})^{∗}.

Proof. For any ℚ-space W and field of characteristic zero F we set W_{F} := W ⊗_{F} ℚ. There is a commutative diagram of A_{ℝ} × Gal(ℂ/ℝ)-modules

\begin{matrix}H_{B}(M)_{ℝ} & ⟶^{α_{M,ℂ}} & H_{dR}(M)_{ℂ}/F^{0} \\ ↑ & & ↑ \\ F^{0}H_{dR}(M)_{ℂ} ⊕ H_{B}(M)_{ℝ} & ⟶ & H_{dR}(M)_{ℂ} \\ ∥ & & ↓^{(53)} \\ F^{0}H_{dR}(M)_{ℂ} ⊕ H_{B}(M)_{ℝ} & ⟶ & H_{B}(M)_{ℂ} \\ ↓ & & ↓ \\ F^{0}H_{dR}(M)_{ℂ} & ⟶ & H_{B}(M)_{ℂ}/H_{B}(M)_{ℝ} \\ ^{β_{1}}↓ & & ^{β_{2}}↓ \\ (H_{dR}(M^{∗}(1))_{ℂ}/F^{0})^{∗} & ⟶^{α_{M^{∗}(1),ℂ}^{∗}} & H_{B}(M^{∗}(1))_{ℝ}^{∗} \end{matrix} (94)

where all arrows other than β_{1}, β_{2} are natural projections, inclusions or sum maps, possibly combined with the comparison isomorphism (53). The maps β_{1} and β_{2} arise as follows. There is a perfect duality of ℝ-vector spaces

H_{B}(M)_{ℂ} × H_{B}(M^{∗}(1))_{ℂ} ⟶ H_{B}(ℚ(1))_{ℂ}
↓≅  ↓≅
H_{dR}(M)_{ℂ} × H_{dR}(M^{∗}(1))_{ℂ} ⟶ H_{dR}(ℚ(1))_{ℂ} ⟶^{τ} ℝ

where the vertical isomorphisms are given by (53) for M,M^{∗}(1) and ℚ(1), and τ is the ℝ-linear splitting of the inclusion

ℝ = H_{dR}(ℚ(1))_{ℝ} ⊂ H_{dR}(ℚ(1))_{ℂ} = ℂ

with kernel H_{B}(ℚ(1))_{ℝ} = 2πi · ℝ. One verifies that H_{B}(M)_{ℝ} is the orthogonal complement of H_{B}(M^{∗}(1))_{ℝ} under this pairing, and it is also well known that F^{0}H_{dR}(M) is the orthogonal complement of F^{0}H_{dR}(M^{∗}(1)). Hence we obtain the isomorphisms β_{i}. Viewing the rows of (94) as complexes concentrated in degrees zero and one, an easy inspection shows that all rows are quasiisomorphic. The same is then true for the diagram (94)^{+} whose top (resp. bottom) row coincides with RΓ_{D}(K, M) (resp. RΓ_{D}(K, M^{∗}(1))^{∗}[−1]). This proves the Lemma. □

We shall next establish existence of the following commutative diagram in V (A_{ℝ})

\begin{matrix}A_{ℝ} ⊗_{A} (Ξ(M) ⊠ Ξ(M^{∗}(1))^{∗}) & ⟶^{A_{ℝ}⊗_{A}ϑ^{P D}} & A_{ℝ} ⊗_{A} Ξ^{loc}(M) \\ ^{A_{ℝ}⊗_{A}(89)}↓^{A_{ℝ}⊗_{A}(90)} & & ∥ \\ \begin{matrix}[H_{B}(M)_{ℝ}^{+}]^{−1} ⊠ [(H_{dR}(M)/F^{0})_{ℝ}] ⊠ \\ [H_{B}(M^{∗}(1))_{ℝ}^{∗,+}]^{−1} ⊠ [(H_{dR}(M^{∗}(1))/F^{0})_{ℝ}^{∗}] \end{matrix} & ⟶^{γ_{1}} & [H_{B}(M)_{ℝ}]^{−1} ⊠ [H_{dR}(M)_{ℝ}] \\ ↓^{[β_{1}],[β_{2}]} & & ^{(55)}↓^{(56)} \\ \begin{matrix}[H_{B}(M)_{ℝ}^{+}]^{−1} ⊠ [(H_{dR}(M)/F^{0})_{ℝ}] ⊠ \\ [(H_{B}(M)_{ℂ})^{+}/H_{B}(M)_{ℝ}^{+}]^{−1} ⊠ [F^{0}H_{dR}(M)_{ℝ}] \end{matrix} & ⟶ & [(H_{B}(M)_{ℂ})^{+}]^{−1} ⊠ [H_{dR}(M)_{ℝ}] \\ ^{β_{3}}↓ & & ^{ϵ_{B}} ↓^{ϵ_{dR}} \\ \begin{matrix}[F^{0}H_{dR}(M)_{ℝ}]^{−1} ⊠ [H_{B}(M)_{ℝ}^{+}]^{−1} ⊠ [H_{dR}(M)_{ℝ}] \\ ⊠ [F^{0}H_{dR}(M)_{ℝ}] ⊠ [H_{B}(M)_{ℝ}^{+}] ⊠ [(H_{B}(M)_{ℂ})^{+}]^{−1} \end{matrix} & ⟶^{γ_{2}} & [(H_{B}(M)_{ℂ})^{+}]^{−1} ⊠ [H_{dR}(M)_{ℝ}] \\ ^{β_{4}}↓ & & ^{(54)}↓ \\ 1_{V (A_{ℝ})} & & 1_{V (A_{ℝ})}. \end{matrix} (95)

The first square in (95) is commutative by the definition of ϑ^{P D} if we define γ_{1} to be induced by equations A_{ℝ} ⊗_{A} (91), A_{ℝ} ⊗_{A} (92) and A_{ℝ} ⊗_{A} (69). The second square in (95) is the ⊠-product of the two commutative squares

\begin{matrix}\begin{matrix}[H_{B}(M)_{ℝ}^{+}]^{−1} ⊠ \\ [H_{B}(M^{∗}(1))_{ℝ}^{∗,+}]^{−1} \end{matrix} & ⟶^{A_{ℝ}⊗\{(91),(92)\}} & [H_{B}(M)_{ℝ}]^{−1} \\ ↓^{[β_{2}]} & & ^{(55)}↓^{(56)} \\ \begin{matrix}[H_{B}(M)_{ℝ}^{+}]^{−1} ⊠ \\ [(H_{B}(M)_{ℂ})^{+}/H_{B}(M)_{ℝ}^{+}]^{−1} \end{matrix} & ⟶ & [(H_{B}(M)_{ℂ})^{+}]^{−1} \end{matrix} (96)

and

\begin{matrix}[(H_{dR}(M)/F^{0})_{ℝ}] ⊠ \\ [(H_{dR}(M^{∗}(1))/F^{0})_{ℝ}^{∗}] \end{matrix} ⟶^{A_{ℝ}⊗_{A}(69)} [H_{dR}(M)_{ℝ}]
↓^{[β_{1}]}  ∥
\begin{matrix}[(H_{dR}(M)/F^{0})_{ℝ}] ⊠ \\ [F^{0}H_{dR}(M)_{ℝ}] \end{matrix} ⟶ [H_{dR}(M)_{ℝ}].

In both of those squares the bottom horizontal maps are induced by the obvious short exact sequences. The square (96) commutes since the identification H_{B}(M)_{ℝ}^{−} ≅ H_{B}(M)^{−} ⊗ (2πi)^{−1}ℝ used in A_{ℝ}⊗_{A}(91) is inverse to that used in (56). Concerning the third square in (95), the map β_{3} is the ⊠-product of the isomorphism

[RΓ_{D}(K, M)]^{−1} = [H_{B}(M)_{ℝ}^{+}]^{−1} ⊠ [(H_{dR}(M)/F^{0})_{ℝ}] ⟶^{∼}

[F^{0}H_{dR}(M)_{ℝ}]^{−1} ⊠ [H_{B}(M)_{ℝ}^{+}]^{−1} ⊠ [H_{dR}(M)_{ℝ}]

induced by the quasi-isomorphism between the first and second row in (94)^{+} and a similar isomorphism induced by the quasi-isomorphism between the fourth and third row in (94)^{+}. The map γ_{2} is induced by canceling mutually inverse terms in the first and second row of its source term. Effectively then, the third square in (95) is the ⊠-product of the squares

\begin{matrix}\begin{matrix}1_{V (A_{ℝ})} ⊠ \\ [F^{0}H_{dR}(M)_{ℝ}] \end{matrix} & ⟶ & [F^{0}H_{dR}(M)_{ℝ}] \\ ↓ & & ^{ϵ_{dR}}↓ \\ \begin{matrix}[F^{0}H_{dR}(M)_{ℝ}]^{−1} ⊠ [F^{0}H_{dR}(M)_{ℝ}] \\ [F^{0}H_{dR}(M)_{ℝ}] \end{matrix} & ⟶ & 1_{V (A_{ℝ})} ⊠ [F^{0}H_{dR}(M)_{ℝ}] \end{matrix} (97)

and

\begin{matrix}\begin{matrix}[H_{B}(M)_{ℝ}^{+}]^{−1} \\ ⊠ 1_{V (A_{ℝ})} \end{matrix} & ⟶ & [(H_{B}(M)_{ℝ}^{+}]^{−1} \\ ↓ & & ^{ϵ_{B}} ↓ \\ \begin{matrix}[H_{B}(M)_{ℝ}^{+}]^{−1} \\ ⊠ [H_{B}(M)_{ℝ}^{+}] ⊠ [H_{B}(M)_{ℝ}^{+}]^{−1} \end{matrix} & ⟶ & 1_{V (A_{ℝ})} ⊠ [(H_{B}(M)_{ℝ}^{+}]^{−1} \end{matrix} (98)

with (the identity maps on) [(H_{dR}(M)/F^{0})_{ℝ}] and [(H_{B}(M)_{ℂ})^{+}/H_{B}(M)_{ℝ}^{+}]^{−1}. In both diagrams (97) and (98) the left and bottom arrows are respectively induced by the two different ways to parenthesize the lower left term, and we have written this term so that the positions of its factors roughly match with their position in (95). We refer to [17][(4.1.1)] for the commutativity of (97) and (98), with the particular correcting factors ϵ_{dR} and ϵ_{B} given in [17][(4.9)]. Finally, the map β_{4} in (95) is induced by the quasi-isomorphism between the second and third row in (94)^{+}, and the commutativity of the bottom square in (95) simply follows from the identity (53)^{+} = (54). We now observe that the right hand vertical map in (95) coincides with ϑ_{∞}^{loc} by definition, and that the left hand vertical map

A_{ℝ} ⊗_{A} (Ξ(M) ⊠ Ξ(M^{∗}(1))^{∗})

⟶^{∼} [RΓ_{D}(K, M)]^{−1} ⊠ [RΓ_{D}(K, M^{∗}(1))^{∗}[−1]] ⟶^{(94)^{+}} 1_{V (A_{ℝ})}

does indeed coincide with ϑ_{∞}(M) ⊠ ϑ_{∞}(M^{∗}(1))^{∗} (using the second condition in Conjecture 1). Lemma 17a) then follows from the commutativity of diagram (95).

We now consider claim b) of Lemma 17. To further shorten notation we henceforth write RΓ_{?} for RΓ_{?}(O_{K,S_{p}}, V_{p}), RΓ_{f} for RΓ_{f}(K, V_{p}), R̃Γ_{c} for R̃Γ_{c}(O_{K,S_{p}}, V_{p}) and L_{?}(S) for ⨁_{v∈S} RΓ_{?}(K_{v}, V_{p}). In addition we use RΓ_{?}^{∗} as an abbreviation for RΓ_{?}(O_{K,S_{p}}, V_{p}^{∗}(1))^{∗}, and also introduce similar abbreviations RΓ_{f}^{∗}, R̃Γ_{c}^{∗} and L_{?}(S)^{∗}.

We shall first establish the existence of the following commutative diagram in V (A_{p})

A_{p} ⊗_{A} (Ξ(M) ⊠ Ξ(M^{∗}(1))^{∗})  ⟶^{Ap⊗AϑP D}  A_{p} ⊗_{A} Ξ^{loc}(M)
α1↓  ^{θp}↓
\begin{matrix}[RΓ_{f}] ⊠ [L_{f} (S_{p,f})]^{−1} ⊠ [L(S_{∞})]^{−1} \\ ⊠ & [RΓ_{f}^{∗}] ⊠ [L_{f} (S_{p,f})^{∗}]^{−1} ⊠ [L(S_{∞})^{∗}]^{−1} \end{matrix}  ⟶^{β1}  \begin{matrix}[L(S_{p,f})]^{−1} ⊠ [ Ind V_{p}]^{−1} \\ =: Λ(S, V_{p}) \end{matrix}

(26)bot⊠↓^{(100)bot∘[α2]}  ∥
\begin{matrix}[L_{/f} (S_{p,f})]^{−1} ⊠ [RΓ] ⊠ [L_{f} (S_{p,f})]^{−1} ⊠ [L(S_{∞})]^{−1} \\ ⊠ & [L_{f} (S_{p,f})^{∗}] ⊠ [_{1}R̃Γ_{c}^{∗}] ⊠ [L_{f} (S_{p,f})^{∗}]^{−1} ⊠ [L(S_{∞})^{∗}]^{−1} \end{matrix}  ⟶^{β2}  [L(S_{p,f})]^{−1} ⊠ [ Ind V_{p}]^{−1}
(18)v∈Sp ↓  ^{(76) (77)}↓^{ϵ(S,p)}
[L(S_{p})]^{−1} ⊠ [RΓ]⊠[_{1}R̃Γ_{c}^{∗}[−4]] ⊠ [L(S_{∞})^{∗}]^{−1}  ⟶^{1 AVTriv} [L(S_{p})]^{−1} ⊠ [L(S_{∞})^{∗}]^{−1}
(26)hor⊠↓^{(73)left}  ^{(73)vert}↓^{∘[λ]∘[q]}
[_{1}RΓ_{c}] ⊠ [_{1}RΓ_{c}^{∗}[−4]]  ⟶^{(73)bot}  [_{2}C(K, V_{p})]
α3↓  ^{(72)}↓
[RΓ_{c}] ⊠ [RΓ_{c}^{∗}[−4]]  ⟶^{ϑ{AV \atop p}}  A_{p} ⊗_{𝔄p} Λ_{p}(S, T_{p}).

(99)

The maps α_{1}, α_{2}, α_{3}, β_{1}, β_{2} which occur in this diagram will all be defined in the course of our proof that the diagram is commutative. Concerning the first square, we recall that the construction of ϑ^{P D} involves the set of equations

ϑ^{P D}: (29) (29)^{∗} AV^{mot} (69) (91) (92)

where here we denote by AV^{mot} the collection of four isomorphisms

[H_{f}^{i}(K, M)]^{−1} ⊠ [H_{f}^{i}(K, M)] ≅ 1, [H_{f}^{i}(K, M^{∗}(1))^{∗}]^{−1} ⊠ [H_{f}^{i}(K, M^{∗}(1))^{∗}] ≅ 1

with i ∈ {0, 1}. The construction of θ_{p} involves the following equations (with v running through the set S_{p,f}):

\begin{aligned}θ_{p} : & (68) & (69) & (18)_{v} & AV_{f,v} & (19)_{v∤p} & (22)_{v|p} & (23) & (24)_{v} \\ & & & & & (19)_{v∤p}^{∗} & (22)_{v|p}^{∗} & (23)^{∗} & (24)_{v}^{∗} \end{aligned}

Note now that when listing all of the equations that are involved in the composition κ := θ_{p} ∘ (A_{p} ⊗_{A} ϑ^{P D}), the equations (91) and (92) are transformed into (76) and (77) respectively as they are ‘conjugated’ by (68), and that (68) is in turn equivalent to a combination of (28)^{+} and (28)^{∗,+} for v ∈ S_{∞}. The isomorphism AV^{mot} is a combination of AV_{f}, (27) and (27)^{∗}. Finally note that the isomorphism induced by (69) is used in ϑ^{P D} whilst its inverse is used in θ_{p}, so that (69) does not in fact occur in the composition κ. In summary, we find that the isomorphism κ involves the sets of equations

\begin{aligned}α_{1} : & (27) & (28)_{v|∞} & (29) & (19)_{v∤p} & (22)_{v|p} & (23) & (24)_{v} \\ & (27)^{∗} & (28)_{v|∞}^{∗} & (29)^{∗} & (19)_{v∤p}^{∗} & (22)_{v|p}^{∗} & (23)^{∗} & (24)_{v}^{∗} \end{aligned}

and

β_{1} : (18)_{v} AV_{f} AV_{f,v} (76) (77).

If we then define α_{1} (resp. β_{1}) as the isomorphism induced by the set of equations carrying the label α_{1} (resp. β_{1}) we have κ = β_{1} ∘ α_{1}, i.e. commutativity of the first square in (99). In order to consider the second square in (99) we first define the map AV_{f}.

Lemma 19. There exists a commutative diagram of true triangles

⨁_{v∈Sp,f} RΓ_{/f} (K_{v}, V_{p})[−1] ⟶  RΓ_{f} (K, V_{p})  ⟶  RΓ(O_{K,Sp} , V_{p})
⊕ AVf,v [−1]↓  ^{AV}_{f} ↓  _{1} ^{AV}↓
⨁_{v∈Sp,f} RΓ_{f} (K_{v}, V_{p}^{∗}(1))^{∗}[−3] ⟶ _{1}RΓ_{f} (K, V_{p}^{∗}(1))^{∗}[−3] ⟶ _{1}R̃Γ_{c}(O_{K,Sp} , V_{p}^{∗}(1))^{∗}[−3],

(100)

in which the upper row coincides with (26)_{bot}, all of the vertical maps are quasi-isomorphisms, and there exists a natural quasi-isomorphism

RΓ_{f}(K, V_{p}^{∗}(1))^{∗}[−3] ⟶^{α_{2}} _{1}RΓ_{f}(K, V_{p}^{∗}(1))^{∗}[−3].

Proof. In view of Lemma 12a) it suffices to show that the mapping cone of the lower composite map in Lemma 12a) is naturally quasi-isomorphic to RΓ_{f}(K, V_{p}^{∗}(1))^{∗}[−2]. Indeed, if this is true, then (100) is simply induced by taking the mapping cones of the composite horizontal maps in the diagram of Lemma 12a). We observe that there is a map from diagram (25) into the diagram

RΓ(O_{K,S_{p}}, V_{p}) → ⨁_{v∈S_{p,f}} RΓ(K_{v}, V_{p}) ↢ ⨁_{v∈S_{p,f}} RΓ_{f}(K_{v}, V_{p}) (101)

and hence an induced map of the true nine term diagram (26) into the true nine term diagram that is induced by (101). In particular on the central columns we obtain a map which coincides with the map between the second and third column in the following true nine term diagram

⨁_{v∈S_{∞}} RΓ(K_{v}, V_{p})[−1] ⟶ ⨁_{v∈S_{p}} RΓ_{f}(K_{v}, V_{p})[−1] ⟶ ⨁_{v∈S_{p,f}} RΓ_{f}(K_{v}, V_{p})[−1]
∥  ↓  ↓
⨁_{v∈S_{∞}} RΓ(K_{v}, V_{p})[−1] ⟶ _{1}RΓ_{c}(O_{K,S_{p}}, V_{p}) ⟶ _{1}R̃Γ_{c}(O_{K,S_{p}}, V_{p})
↓  ↓
RΓ_{f}(K, V_{p})  RΓ_{f}(K, V_{p}).

(102)

Upon replacing V_{p} by V_{p}^{∗}(1), dualizing the third column of this diagram and shifting by [-4] we obtain a true triangle

RΓ_{f}^{∗}[−4] → _{1}R̃Γ_{c}^{∗}[−4] → L_{f}(S_{p,f})^{∗}[−3]. (103)

The map α_{2} then arises as the map q of Lemma 13 when the latter is applied to the triangle (103). □

The diagram (100) induces a commutative diagram

[RΓ_{f}] ⊠ [_{1}RΓ_{f}^{∗}]  ⟶^{[ AV_{f} ]_{Triv}}  1_{V (A_{p})}
(26)_{bot}⊠↓^{(100)_{bot}}  ∥

\begin{matrix}[L_{/f}(S_{p,f})]^{−1} ⊠ [RΓ] \\ ⊠ & [L_{f}(S_{p,f})^{∗}] ⊠ [_{1}R̃Γ_{c}^{∗}] \end{matrix} [⊕AV_{f,v} ]_{Triv}⊠[_{1} AV ]_{Triv} ⟶ 1_{V (A_{p})}.

(104)

The second square in (99) is obtained by taking the ⊠-product of the rows in (104) with the isomorphism

\begin{matrix}[L_{f}(S_{p,f})]^{−1} ⊠ [L(S_{∞})]^{−1} \\ ⊠[L_{f}(S_{p,f})^{∗}]^{−1} ⊠ [L(S_{∞})^{∗}]^{−1} \end{matrix} ⟶^{β′_{2}} [L(S_{p,f})]^{−1} ⊠ [ Ind V_{p}]^{−1},

where here β′_{2} involves the equations

β′_{2} : (18)_{v} AV_{f,v} (76) (77).

In order to establish the third square in (99) we begin with the commutative diagram

\begin{matrix}[L(S_{∞})]^{−1} ⊠ [L(S_{∞})^{∗}]^{−1} & ⟶^{(76) (77)} & [ Ind V_{p}] \\ ↓ & & ^{(76) (77)}↓ \\ [L(S_{∞})]^{−1} ⊠ [L(S_{∞})^{∗}]^{−1} & & [L(S_{∞})]^{−1} ⊠ [L(S_{∞})^{∗}]^{−1}. \end{matrix} (105)

Next we consider the diagram

\begin{matrix}\begin{matrix}[L_{/f}(S_{p,f})]^{−1} ⊠ [L_{f}(S_{p,f})]^{−1} \\ ⊠ & [L_{f}(S_{p,f})^{∗}] ⊠ [L_{f}(S_{p,f})^{∗}]^{−1} \end{matrix} & ⟶^{β′′_{2}} & [L(S_{p,f})]^{−1} \\ ^{[α_{2}]}↓ & & ↓^{ϵ(S,p)} \\ [L(S_{p,f})]^{−1} & & [L(S_{p,f})]^{−1}. \end{matrix} (106)

Here β′′_{2} is induced by applying AV_{f,v}, (18)_{v} to the two right hand terms in the top left item and by trivializing the two left hand terms via another application of AV_{f,v}. For the map α_{2} on the other hand one applies (18)_{v} to the upper two terms and trivializes the lower two terms. The commutativity of (106) then follows from [17, (4.1.1)] where the particular correcting factor ϵ(S, p) is as computed in [loc. cit., 4.9]. Indeed, upon writing diagram [loc. cit., (4.1.1)] with X = [L_{/f}(S_{p,f})], taking ⊠-product with [L_{f}(S_{p,f})]^{−1} and using AV_{f,v} twice we obtain (106). The third square in (99) is obtained by taking the ⊠-product of the diagrams (105) and (106) and then taking the ⊠-product of both rows of the resulting diagram with the isomorphism

[RΓ] ⊠ [_{1}R̃Γ_{c}^{∗}] ⟶^{_{1} AV_{Triv}} 1_{V (A_{p})}.

We now consider the fourth square in (99). We observe first that there is a commutative diagram in V (A_{p})

[L(S_{p})]^{−1} ⊠ [RΓ] ⊠ [RΓ]^{−1} ⊠ [L(S_{∞})^{∗}]^{−1} ⟶^{τ} [L(S_{p})]^{−1} ⊠ [L(S_{∞})^{∗}]^{−1}
(26)_{hor}↓  ^{[q]}↓
[_{1}RΓ_{c}] ⊠ [RΓ]^{−1} ⊠ [L(S_{∞})^{∗}]^{−1}  ⟶^{(74)_{top}} [_{1}L(S_{p})]^{−1} ⊠ [L(S_{∞})^{∗}]^{−1}
[_{1} ^{AV}^{]}↓  ^{[λ]}↓
[_{1}RΓ_{c}] ⊠ [_{ ˜ 1}RΓ_{c}^{∗}[−4]] ⊠ [L(S_{∞})^{∗}]^{−1}  ⟶^{(73)_{hor}} [_{2}L(S_{p})]^{−1} ⊠ [L(S_{∞})^{∗}]^{−1}
↓^{(73)_{left}}  ^{(73)_{vert}}↓
[_{1}RΓ_{c}] ⊠ [_{1}RΓ_{c}^{∗}[−4]]  ⟶^{(73)_{bot}}  [_{2}C(K, V_{p})]

in which the lower square is induced by the true nine term diagram (73), the central square by (74) and the upper square by Lemma 13. The map τ is the canonical isomorphism [RΓ] ⊠ [RΓ]^{−1} ≅ 1_{V (A_{p})}. Upon replacing both occurrences of [RΓ]^{−1} in this diagram by [_{1}R̃Γ_{c}^{∗}[−4]] and τ by _{1} AV_{Triv}, the upper square is still commutative and the resulting total square gives the fourth square in (99).

Finally, in the bottom square in (99) the map α_{3} is defined by replacing complexes by their versions indexed by 1 in exactly the same order as in (72). The commutativity of this square is then clear.

It is clear that the right vertical map of (99) is equal to ϑ_{p}^{loc}. Hence, having now established the commutativity of (99), we shall prove Lemma 17b) if we can show that the left vertical map in (99) is equal to ϑ_{p}(M) ⊠ ϑ_{p}(M^{∗}(1))^{∗}. In view of the definition of α_{1} it therefore suffices for us to show that

(107) (26)_{vert} = (26)_{bot} ∘ (18)_{v∈S_{p}} ∘ (26)_{hor}

and also

(108) (26)_{vert}^{∗} = [α_{2}] ∘ (100)_{bot} ∘ (73)_{left}.

The identity (107) coincides with the identity in V (A_{p}) induced by the true nine term diagram (26) since (26)_{left} coincides with the sum of (18) over v ∈ S_{p}. On the other hand, the identity (108) is a consequence of the following commutative diagram

[_{1}RΓ_{c}^{∗}]  ⟶^{(26){∗ \atop vert}}  [RΓ_{f}^{∗}] ⊠ [L_{f} (S_{p})^{∗}]^{−1}
(73)left↓  ∥
[_{1}R̃Γ_{c}^{∗}] ⊠ [L(S_{∞})^{∗}]^{−1}  ⟶^{(103)} [RΓ_{f}^{∗}] ⊠ [L_{f} (S_{p,f})^{∗}]^{−1} ⊠ [L(S_{∞})^{∗}]^{−1}
(100)bot↓  ∥
[_{1}RΓ_{f}^{∗}] ⊠ [L_{f} (S_{p,f})^{∗}]^{−1} ⊠ [L(S_{∞})^{∗}]^{−1}  ⟶^{[α2]} [RΓ_{f}^{∗}] ⊠ [L_{f} (S_{p,f})^{∗}]^{−1} ⊠ [L(S_{∞})^{∗}]^{−1}

where here the first square is induced by the dual of diagram (102) with V_{p} replaced by V_{p}^{∗}(1) and the second square results from applying Lemma 13 to (103) and then taking the ⊠-product of each vertex of the resulting square with [L_{f}(S_{p,f})^{∗}]^{−1}. □

In view of Corollary 1 we are naturally led to make the following

Conjecture 8. T Ω^{loc}(M, 𝔄) = 0.

This conjecture is itself of some independent interest, and will be considered in greater detail elsewhere. We therefore restrict ourselves here to a few brief remarks concerning the Galois case. We fix a finite Galois extension of number fields L/K and set G := Gal(L/K).

Remark 18. In this remark we assume that G is abelian. If M is any motive which is defined over K, then Conjecture 8 for the pair (h^{0}(Spec (L))⊗M, ℤ[G]) can be interpreted in terms of the ‘local epsilon conjecture’ formulated by Kato in [29]. In particular, [loc. cit., Th. 4.1] can be used to verify (at least modulo the ‘sign ambiguities’ discussed in Remark 9 of §4.3) that if K = ℚ, then for all integers r Conjecture 8 is valid for the pair (h^{0}(Spec (L))(r), ℤ[\frac{1}{2}][G]). (Details of this deduction will be given elsewhere.) When combined with Corollary 1 and the main result of [12] (cf. Remark 10 in §4.3) this implies (again modulo the same sign ambiguities as above) that T Ω(h^{0}(Spec (L))(r), ℤ[\frac{1}{2}][G]) = 0 for all integers r.

Remark 19. In [7] it is shown that (for any G)

δ_{ℤ[G],ℝ}^{0}(T Ω^{loc}(h^{0}(Spec(L))(1), ℤ[G])) = Ω(L/K, 2) − w(L/K)

where here Ω(L/K, 2) and w(L/K) are respectively equal to the ‘second Chinburg invariant’ and the ‘Cassou-Noguès-Fröhlich root number class’ as defined in [14]. This implies that Conjecture 8 is compatible with the conjectures formulated by Chinburg in loc. cit. For a further discussion of connections between Theorem 5.3 and the extensive existing theory concerning the conjectures of Chinburg, the reader can consult [11].

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D. Burns

Dept. of Math.

King’s College London Strand London WC2R 2LS England david.burns@kcl.ac.uk M. Flach Dept. of Math. Caltech Pasadena CA 91125, USA flach@its.caltech.edu