On the Weil-Étale Topos

of Regular Arithmetic Schemes

M. Flach and B. Morin

Received: March 22, 2010 Revised: February 9, 2011

Communicated by Stephen Lichtenbaum

Abstract. We define and study a Weil-étale topos for any regular, proper scheme X over Spec(ℤ) which has some of the properties suggested by Lichtenbaum for such a topos. In particular, the cohomology with ℝ̃-coefficients has the expected relation to ζ(X , s) at s = 0 if the Hasse-Weil L-functions L(h^{i}(X_{ℚ}), s) have the expected meromorphic continuation and functional equation. If X has characteristic p the cohomology with ℤ-coefficients also has the expected relation to ζ(X , s) and our cohomology groups recover those previously studied by Lichtenbaum and Geisser.

2000 Mathematics Subject Classification: Primary: 14F20, 11S40, Secondary: 11G40, 18F10

1. Introduction

In [30] Lichtenbaum suggested the existence of Weil-étale cohomology groups for arithmetic schemes X (i.e. separated schemes of finite type over Spec(ℤ)) which are related to the zeta-function ζ(X , s) of X as follows.

a) The compact support cohomology groups H_{c}^{i}(X_{W} , ℝ̃) are finite dimensional vector spaces over ℝ, vanish for almost all i and satisfy

∑_{i∈ℤ}(−1)^{i} dim_{ℝ} H_{c}^{i}(X_{W} , ℝ̃) = 0.

b) The function ζ(X , s) has a meromorphic continuation to s = 0 and

ord_{s=0} ζ(X , s) = ∑_{i∈ℤ}(−1)^{i} · i · dim_{ℝ} H_{c}^{i}(X_{W} , ℝ̃).

c) There exists a canonical class θ ∈ H^{1}(X_{W} , ℝ̃) so that the sequence

⋯ ⟶^{∪θ} H_{c}^{i}(X_{W} , ℝ̃) ⟶^{∪θ} H_{c}^{i+1}(X_{W} , ℝ̃) ⟶^{∪θ} ⋯

is exact. d) The compact support cohomology groups H_{c}^{i}(X_{W} , ℤ) are finitely gen-

erated over ℤ and vanish for almost all i.

e) The natural map from ℤ to ℝ̃-coefficients induces an isomorphism

H_{c}^{i}(X_{W} , ℤ) ⊗_{ℤ} ℝ ⟶^{∼} H_{c}^{i}(X_{W} , ℝ̃).

f) If ζ^{∗}(X , 0) denotes the leading Taylor-coefficient of ζ(X , s) at s = 0

and

λ : ℝ ≅ ⨂_{i∈ℤ} det_{ℝ}H_{c}^{i}(X_{W} , ℝ̃)^{(−1)^{i}}

the isomorphism induced by c) then

ℤ · λ(ζ^{∗}(X , 0)^{−1}) = ⨂_{i∈ℤ} det_{ℤ}H_{c}^{i}(X_{W} , ℤ)^{(−1)^{i}}

where the determinant is understood in the sense of [28] and ℝ̃ is the sheaf associated to the field of real numbers endowed with its standard topology (see Definition 14 below).

If X has finite characteristic these groups are well defined and well understood by work of Lichtenbaum [29] and Geisser [18, 19]. In particular all the above properties a)-f) hold for dim(X) ≤ 2 and in general under resolution of singularities. Lichtenbaum also defined such groups for X = Spec(O_{F}) where F is a number field and showed that a)-f) hold if one artificially redefines H_{c}^{i}(Spec(O_{F})_{W} , ℤ) to be zero for i ≥ 4. In [15] it was then shown that H_{c}^{i}(Spec(O_{F})_{W} , ℤ) as defined by Lichtenbaum does indeed vanish for odd i ≥ 5 but is an abelian group of infinite rank for even i ≥ 4. In any case, in Lichtenbaum’s definition the groups H_{c}^{i}(Spec(O_{F})_{W} , ℤ) and H_{c}^{i}(Spec(O_{F})_{W} , ℝ̃) are defined via an Artin-Verdier type compactification \overline{Spec(O_{F})} of Spec(O_{F}) [1], where however H^{i}(\overline{Spec(O_{F})}_{W} , F) is not the cohomology group of a topos but rather a direct limit of such. The first purpose of this article is to give a definition of a topos \overline{Spec(O_{F})}_{W} which recovers Lichtenbaum’s groups (see section 5 below). This definition was proposed in the second author’s thesis [33] and is a natural modification of Lichtenbaum’s idea which is suggested by a closer look at the étale topos Spec(O_{F})_{et}. In [1] Artin and Verdier defined a topos \overline{X}_{et} for any arithmetic scheme X → Spec(ℤ) so that there are complementary open and closed immersions

X_{et} → \overline{X}_{et} ← Sh(X_{∞})

the sense of topos theory [20]. Here X_{∞} is the topological quotient space X (ℂ)/G_{ℝ} where X (ℂ) is the set of complex points with its standard Euclidean topology and G_{ℝ} = Gal(ℂ/ℝ). If X is an arithmetic scheme and Y denotes either X or \overline{X} we define the Weil-étale topos of Y by

Y_{W} := Y_{et} ×_{\overline{Spec(ℤ)}_{et}} \overline{Spec(ℤ)}_{W} ,

On the Weil-Étale Topos of Regular Arithmetic Schemes 315

a fibre product in the 2-category of topoi. This definition is suggested by the fact that the Weil-étale topos defined by Lichtenbaum for varieties over finite fields is isomorphic to a similar fibre product, as was shown in the second author’s thesis [33] and will be recalled in section 3 below. The work of Geisser [19] shows that Lichtenbaums’s definition is only reasonable (i.e. satisfies a)- f)) for smooth, proper varieties over finite fields. Correspondingly, one can only expect our fibre product definition to be reasonable for proper regular arithmetic schemes. The second purpose of this article is to show that this is indeed the case as far as ℝ̃-coefficients are concerned. Our main result is the following

Theorem 1.1. Let X be a regular scheme, proper over Spec(ℤ).

i) For X = Spec(O_{F}) one has

\overline{Spec(O_{F})}_{W} ≅ \overline{Spec(O_{F})}_{et} ×_{\overline{Spec(ℤ)}_{et}} \overline{Spec(ℤ)}_{W} ,

where \overline{Spec(O_{F})}_{W} is the topos defined in section 5 below, based on Lichtenbaum’s idea of replacing Galois groups by Weil groups. ii) If X → Spec(𝔽_{p}) has characteristic p then our groups agree with those

of Lichtenbaum and Geisser and a)-f) hold for X. iii) If X is flat over Spec(ℤ) and the Hasse-Weil L-functions L(h^{i}(X_{ℚ}), s) of all motives h^{i}(X_{ℚ}) satisfy the expected meromorphic continuation and functional equation. Then a)-c) hold for X.

The assumptions of iii) are satisfied, for example, if X is a regular model of a Shimura curve, or of a self product E × ⋯ × E where E is an elliptic curve, over a totally real field F. Unfortunately, properties d) and e) do not hold with our fibre product definition, even in low degrees, and we also do not expect them to hold with any similar definition (see the remarks in section 9.3). The right definition of Weil-étale cohomology with ℤ-coefficients for schemes of characteristic zero will require a key new idea, as is already apparent for X = Spec(O_{F}). We briefly describe the content of this article. In section 2 we recall preliminaries on sites, topoi and classifying topoi. Section 3 contains the proof that Lichtenbaum’s Weil-étale topos in characteristic p is a fibre product via a method that is different from the one in the second author’s thesis [33]. In section 4 we recall the definition of \overline{X}_{et} and the corresponding compact support cohomology groups H_{c}^{i}(X_{et}, F). In section 5 we define \overline{Spec(O_{F})}_{W} and give the proof of Theorem 1.1 i) (see Proposition 5.5). In section 6 we define \overline{X}_{W}, describe its fibres above all places p ≤ ∞ and its generic point. In section 7 we compute the cohomology of \overline{X}_{W} with ℝ̃-coefficients following Lichtenbaum’s method of studying the Leray spectral sequence from the generic point. This section is the technical heart of this article. In section 8 we compute the compact support cohomology H_{c}^{i}(X_{W} , ℝ̃) via the natural morphism \overline{X}_{W} → \overline{X}_{et} and prove properties a) and c) (see Theorem 8.2). The class θ in c) is defined in subsection 8.3. Section 9 introduces Hasse-Weil L-functions of varieties over ℚ as well as Zetafunctions of arithmetic schemes and contains the proof of Theorem 1.1 ii) (see Theorem 9.2) and of property b) (see Theorem 9.1), thereby concluding the proof Theorem 1.1 iii). In subsection 9.4 we show that property f) for ζ(X , s) is compatible with the Tamagawa number conjecture of Bloch and Kato [4] (or rather of Fontaine and Perrin-Riou [16]) for ∏_{i∈ℤ} L(h^{i}(X_{ℚ}), s)^{(−1)^{i}} at s = 0. In order to do this we need to augment the list of properties a)-f) for Weilétale cohomology with further natural assumptions g)-j), some of which hold in characteristic p, and we need to assume a number of conjectures which are preliminary to the formulation of the Tamagawa number conjecture. Finally, in section 10 we prove some results related to the so called local theorem of invariant cycles in l-adic cohomology, and we formulate analogous conjectures in p-adic cohomology. These results may be of some interest independently of Weil-étale cohomology, and are necessary to establish the equality of vanishing orders

ord_{s=0} ζ(X , s) = ord_{s=0} ∏_{i∈ℤ} L(h^{i}(X_{ℚ}), s)^{(−1)^{i}}

for regular schemes X proper and flat over Spec(ℤ).

Acknowledgements: The first author is supported by grant DMS-0701029 from the National Science Foundation. He would also like to thank Spencer Bloch for a helpful discussion about the material in section 10 and the MPI Bonn for its hospitality during the final preparation of this paper.

2. Preliminaries

In this paper, a topos is a Grothendieck topos over \underline{Set}, and a morphism of topoi is a geometric morphism. A pseudo-commutative diagram of topoi is said to be commutative. We suppress any mention of universes.

2.1. Left exact sites. Recall that a Grothendieck topology J on a category C is said to be sub-canonical if J is coarser than the canonical topology, i.e. if any representable presheaf on C is a sheaf for the topology J. A category C is said to be left exact when finite projective limits exist in C, i.e. when C has a final object and fiber products. A functor between left exact categories is said to be left exact if it commutes with finite projective limits.

Definition 1. A Grothendieck site (C, J) is said to be left exact if C is a left exact category endowed with a subcanonical topology J. A morphism of left exact sites (C′, J′) → (C, J) is a continuous left exact functor C′ → C.

Note that any Grothendieck topos, i.e. any category satisfying Giraud’s axioms, is equivalent to the category of sheaves of sets on a left exact site. Note also that a Grothendieck site (C, J) is left exact if and only if the canonical functor (given in general by Yoneda and sheafification)

y : C ⟶ { \atop (C, J)}

identifies C with a left exact full subcategory of { \atop (C, J)}. The following result is proven in [20] IV.4.9.

Lemma 1. A morphism of left exact sites f^{∗} : (C′, J′) → (C, J) induces a morphism of topoi f : (C̃, J) → ({ \atop C′, J′}). Moreover we have a commutative diagram

(C̃, J) ⟵^{f^{∗}} ({ \atop C′, J′})
↑^{y_{C}}  ↑^{y_{C′}}
C ⟵^{f^{∗}} C′

where the vertical arrows are the fully faithful Yoneda functors.

2.2. The topos T . We denote by T op^{lc} (respectively by T op^{c}) the category of locally compact topological spaces (respectively of compact spaces). A locally compact space is assumed to be Hausdorff. The category T op^{lc} is endowed with the open cover topology J_{op}, which is subcanonical. We denote by T the topos of sheaves of sets on the site (T op^{lc}, J_{op}). The Yoneda functor

y : T op^{lc} ⟶ T

is fully faithful, and T op^{lc} is viewed as a generating full subcategory of T . For any object T of T op^{lc}, T is locally compact hence there exist morphisms

∐ yU_{i} → ∐ yK_{i} → yT

where {U_{i} ⊂ T } is an open covering, and K_{i} is a compact subspace of T. It follows that ∐ yU_{i} → yT is an epimorphism in T , hence so is ∐ yK_{i} → yT. This shows that the category of compact spaces T op^{c} is a generating full subcategory of T . The unique morphism t : T → \underline{Set} has a section s : \underline{Set} → T such that t_{∗} = s^{∗} hence we have three adjoint functors t^{∗}, t_{∗} = s^{∗}, s_{∗}. In particular t_{∗} is exact hence we have H^{n}(T , A) = H^{n}(\underline{Set}, A(∗)) = 0 for any n ≥ 1 and any abelian object A (see Lemma 8 for a generalization of this fact).

2.3. Classifying topoi.

2.3.1. General case. For any topos S and any group object G in S, we denote by B_{G} the category of left G-object in S. Then B_{G} is a topos, as it follows from Giraud’s axioms, and B_{G} is endowed with a canonical morphism B_{G} → S, whose inverse image functor sends an object F of S to F with trivial G-action. If there is a risk of ambiguity, the topos B_{G} is denoted by B_{S}(G). We denote by EG the object of B_{G} given by the action of G on itself by left multiplication. The topos B_{G} is said to be the classifying topos of G since for any topos f : E → S over S, the category \underline{Homtop}_{S} (E, B_{G}) is equivalent to the category of f^{∗}G-torsors in E (see [20] IV. Exercice 5.9).

2.3.2. Examples. Let G be a discrete group, i.e. a group object of the final topos \underline{Set}. Then B\underline{Set}G is the category of left G-sets, and the cohomology groups H^{∗}(B\underline{Set}G, A), where A is an abelian object of B_{G} i.e. a G-module, is precisely the cohomology of the discrete group G. Here B\underline{Set}G is called the small classifying topos of the discrete group G and is denoted by B_{G}^{sm}. If G is a profinite group, the small classifying topos B_{G}^{sm} of the profinite group G is the category of continuous G-sets. Let G be a locally compact topological group. Then G represents a group object of T , where T is defined above. Then B_{G} is the classifying topos of the topological group G, and the cohomology groups H^{∗}(B_{G}, A), where A is an abelian object of B_{G} (e.g. a topological G-module) is the cohomology of the topological group G. If G is not locally compact, then we just need to replace T with the category of sheaves on (T op, J_{op}). Let S be a scheme and let G be a smooth group scheme over S. We denote by S_{Et} the big étale topos of S. Then G represents a group object of S_{Et} and B_{G} is the classifying topos of G. The cohomology groups H^{∗}(B_{G}, A), where A is an abelian object of B_{G} (e.g. an abelian group scheme over S endowed with a G-action) is the étale cohomology of the S-group scheme G.

2.3.3. The local section site. For G any locally compact topological group, we denote by B_{T op^{lc}}G the category of G-equivariant locally compact topological spaces endowed with the local section topology J_{ls} (see [30] section 1). The Yoneda functor yields a canonical fully faithful functor

B_{T op^{lc}}G ⟶ B_{G}.

Then one can show that the local section topology J_{ls} on B_{T op^{lc}}G is the topology induced by the canonical topology of B_{G}. Moreover B_{T op^{lc}}G is a generating family of B_{G}. It follows that the morphism

B_{G} ⟶ { \atop (B_{T op^{lc}}G, J_{ls})}

is an equivalence. In other words the site (B_{T op^{lc}}G, J_{ls}) is a site for the classifying topos B_{G} (see [15] for more details).

2.3.4. The classifying topos of a strict topological pro-group. A locally compact topological pro-group \underline{G} is a pro-object in the category of locally compact topological groups, i.e. a functor I^{op} → Gr(T op^{lc}), where I is a filtered category and Gr(T op^{lc}) is the category of locally compact topological groups. A locally compact topological pro-group \underline{G} is said to be strict if the transition maps G_{j} → G_{i} have local sections. We define the limit of \underline{G} in the 2-category of topoi as follows.

Definition 2. The classifying topos of a strict topological pro-group \underline{G} is defined as

B\underline{G} := \underleftarrow{lim}_{I} B_{G_{i}},

where the the projective limit is computed in the 2-category of topoi.

2.3.5. In order to ease the notations, we will simply denote by T op the category of locally compact spaces. For any locally compact group G, we denote by B_{T op}G the category of locally compact spaces endowed with a continuous Gaction.

2.4. Fiber products of topoi. The class of topoi forms a 2-category. In particular, \underline{Homtop} (E, F) is a category for any pair of topoi E and F. If f, g : E ⇉ F are two objects of \underline{Homtop} (E, F), then a morphism σ : f → g is a natural transformation σ : f_{∗} → g_{∗}. Consider now two morphisms of topoi with the same target f : E → S and g : F → S. For any topos G, we define the category

\underline{Homtop} (G, E) ×_{\underline{Homtop} (G,S)} \underline{Homtop} (G, F)

whose objects are given by triples of the form (a, b, α), where a and b are objects of \underline{Homtop} (G, E) and \underline{Homtop} (G, F) respectively, and

α : f ∘ a ≅ g ∘ b

is an isomorphism in the category \underline{Homtop} (G, S). A fiber product E ×_{S} F in the 2-category of topoi is a topos endowed with canonical projections p_{1} : E ×_{S} F → E, p_{2} : E ×_{S} F → F and an isomorphism α : f ∘ p_{1} ≅ g ∘ p_{2} satisfying the following universal condition. For any topos G the natural functor

\begin{matrix}\underline{Homtop} (G, E ×_{S} F) & ⟶ & \underline{Homtop} (G, E) ×_{\underline{Homtop} (G,S)} \underline{Homtop} (G, F) \\ d & ⟼ & (p_{1} ∘ d, p_{2} ∘ d, α ∘ d_{∗}) \end{matrix}

is an equivalence. It is known that fiber products of topoi always exist (see [25] for example). The universal condition implies that such a fiber product is unique up to equivalence. A product of topoi is a fiber product over the final topos

E × F = E ×\underline{Set} F.

A square of topoi

\begin{matrix}E′ & ⟶ & S′ \\ ↓ & & ↓ \\ E & ⟶ & S \end{matrix}

is said to be a pull-back if it is commutative and if the morphism

E′ ⟶ E ×_{S} S′,

given by the universal condition for the fiber product, is an equivalence. The following examples will be used in this paper. Let f : E → S be a morphism of topoi. For any object X of S, the commutative diagram

(1)  \begin{matrix}E/f^{∗}X & ⟶ & S/X \\ ↓ & & ↓ \\ E & ⟶^{f} & S \end{matrix}

is a pull-back (see [20] IV Proposition 5.11). For any group-object G in S, the commutative diagram

(2)  \begin{matrix}B_{E}(f^{∗}G) & ⟶ & B_{S}(G) \\ ↓ & & ↓ \\ E & ⟶^{f} & S \end{matrix}

is a pull-back. This follows from the fact that B_{S}(G) classifies G-torsors.

3. The Weil-étale topos in characteristic p is a fiber product

For any scheme Y, we denote by Y_{et} the (small) étale topos of Y, i.e. the category of sheaves of sets on the étale site on Y. Let G be a discrete group acting on a scheme Y. An étale sheaf F on Y is G-equivariant if F is endowed with a family of morphisms {φ_{g} : g_{∗}F → F; g ∈ G} satisfying φ_{1_{G}} = Id_{F} and φ_{gh} = φ_{g} ∘ g_{∗}(φ_{h}), for any g, h ∈ G. The category S(G; Y_{et}) of G-equivariant étale sheaves on Y is a topos, as it follows from Giraud’s axioms. The cohomology H^{∗}(S(G; Y_{et}), A), for any G-equivariant abelian étale sheaf on Y, is the equivariant étale cohomology for the action (G, Y). An equivariant map of G-schemes u : X → Y induces a morphism of topoi S(G; X_{et}) → S(G; Y_{et}). Let Y be a scheme separated and of finite type over a field k, let \overline{k}/k be a separable closure and let F be an étale sheaf on Y ⊗_{k} \overline{k}. An action of the Galois group G_{k} on F is said to be continuous when the induced action of the profinite group G_{k} on the discrete set F(U ×_{k} \overline{k}) is continuous, for any U étale and quasi-compact over Y. It is well known that the étale topos Y_{et} is equivalent to the category S(G_{k}, \overline{Y}_{et}) of étale sheaves on \overline{Y} := Y ⊗_{k} \overline{k} endowed with a continuous action of the Galois group G_{k}. Let Y be a separated scheme of finite type over a finite field k = 𝔽_{q}. Let \overline{k}/k be an algebraic closure. Let W_{k} and G_{k} be the Weil group and the Galois group of k respectively. Recall that W_{k} is the discrete subgroup of G_{k} generated by the Frobenius element. The small classifying topos B_{W_{k}}^{sm} is defined as the category of W_{k}-sets, while B_{G_{k}}^{sm} is the category of continuous G_{k}-sets. We denote by Y_{W}^{sm} the Weil-étale topos of the scheme Y, which is defined as follows. We consider the scheme \overline{Y} = Y ⊗_{k} \overline{k} endowed with the action of W_{k}. Then the Weil-étale topos Y_{W}^{sm} is the topos of W_{k}-equivariant sheaves of sets on \overline{Y}. We have a morphism

γ_{Y} : Y_{W}^{sm} := S(W_{k}, \overline{Y}_{et}) ⟶ S(G_{k}, \overline{Y}_{et}) ≅ Y_{et}.

Indeed, consider the functor γ_{Y}^{∗} which takes an étale sheaf F on \overline{Y} endowed with a continuous G_{k}-action to the sheaf F endowed with the induced W_{k}- action via the canonical map W_{k} → G_{k}. Then γ_{Y}^{∗} commutes with arbitrary inductive limits and with projective limits. Hence γ_{Y}^{∗} is the inverse image of a morphism of topoi γ_{Y}. This morphism has been defined and studied by T. Geisser in [18]. Note that the Weil-étale topos of Spec(k) is precisely B_{W_{k}}^{sm} and that the étale topos Spec(k)_{et} is equivalent to B_{G_{k}}^{sm}. In this case the morphism γ_{k} := α : B_{W_{k}}^{sm} → B_{G_{k}}^{sm}, from the Weil-étale topos of Spec(k) to its étale topos is the morphism induced by the canonical map W_{k} → G_{k}. The structure map Y → Spec(k) gives a W_{k}-equivariant morphism of schemes \overline{Y} → Spec(\overline{k}), inducing in turn a morphism Y_{W}^{sm} → B_{W_{k}}^{sm}. This structure map also induces a morphism of étale topos Y_{et} → B_{G_{k}}^{sm}. The diagram

(3)  \begin{matrix}Y_{W}^{sm} & ⟶^{γ_{Y}} & Y_{et} \\ ↓ & & ↓ \\ B_{W_{k}} & ⟶^{α} & B_{G_{k}} \end{matrix}

is commutative, where α is induced by the morphism W_{k} → G_{k}. The aim of this section is to prove that the previous diagram is a pull-back of topoi. Our proof is based on a descent argument. We need some basic facts concerning truncated simplicial topoi. A truncated simplicial topos S_{•} is given by the usual diagram

S_{2} ⇉→ S_{1} ⇉← S_{0}

Given such a truncated simplicial topos S_{•}, we define the category Desc(S_{•}) of objects of S_{0} endowed with a descent data. By [32], the category Desc(S_{•}) is a topos. More precisely, Desc(S_{•}) is the inductive limit of the diagram S_{•} in the 2-category of topoi. The simplest example is the following. Let S be a topos and let X be an object of S. We consider the truncated simplicial topos

(S, X)_{•} : S/(X × X × X) ⇉→ S/(X × X) ⇉← S/X

where these morphisms of topoi are induced by the projections maps (of the form X × X × X → X × X and X × X → X) and by the diagonal map X → X × X. It is well known that, if X covers the final object of S (i.e. X → e_{S} is epimorphic where e_{S} is the final object of S), then the natural morphism

Desc(S, X)_{•} ⟶ S

is an equivalence (see [12] Chapter 4 Example 4.1). In other words S/X → S is an effective descent morphism for any X covering the final object of S.

Lemma 2. Let f : E → S be a morphism of topoi and let X be an object of S covering the final object. The morphism f is an equivalence if and only if the induced morphism

f/X : E/f^{∗}X ⟶ S/X

is an equivalence.

Proof. The condition is clearly necessary. Assume that f/X is an equivalence. We have S/(X × X) = (S/X)/(X × X) and S/(X × X × X) = (S/X)/(X × X × X), for any projection maps X × X → X and X × X × X → X. Hence the triple of morphisms (f/X × X × X, f/X × X, f/X) yields an equivalence of truncated simplicial topoi

f/ : (E, f^{∗}X)_{•} ⟶ (S, X)_{•}

This equivalence induces an equivalence of descent topoi

Desc(f/) : Desc(E, f^{∗}X)_{•} ⟶ Desc(S, X)_{•} such that the following square is commutative

Desc(E, f^{∗}X)_{•} ⟶^{Desc(f/)} Desc(S, X)_{•}
↓  ↓
E  ⟶^{f}  S

This shows that f is an equivalence since the vertical maps are equivalences. □

Theorem 3.1. Let Y be a scheme separated and of finite type over a finite field k. The canonical morphism

Y_{W}^{sm} ⟶ Y_{et} ×_{B_{G_{k}}^{sm}} B_{W_{k}}^{sm}

is an equivalence.

Proof. The morphism

f : Y_{W}^{sm} ⟶ Y_{et} ×_{B_{G_{k}}^{sm}} B_{W_{k}}^{sm}

is defined by the commutative square (3). Let p : Y_{et} ×_{B_{G_{k}}^{sm}} B_{W_{k}}^{sm} → B_{W_{k}}^{sm} be the second projection. Consider the object EW_{k} of B_{W_{k}}^{sm} defined by the action of W_{k} on itself by multiplication, and let p^{∗}EW_{k} be its pull-back in Y_{et}×_{B_{G_{k}}^{sm}} B_{W_{k}}^{sm}. It is enough to show that the morphism

f/p^{∗}EW_{k} : Y_{W}^{sm}/f^{∗}p^{∗}EW_{k} ⟶ (Y_{et} ×_{B_{G_{k}}^{sm}} B_{W_{k}}^{sm})/p^{∗}EW_{k}

is an equivalence. Recall that Y_{W}^{sm} := S_{et}(W_{k}, \overline{Y}) is the topos of W_{k}-equivariant étale sheaves on \overline{Y}. The object f^{∗}p^{∗}EW_{k} is represented by the W_{k}-equivariant étale \overline{Y} -scheme ∐_{W_{k}} \overline{Y} → \overline{Y}. One has the following equivalences

Y_{W}^{sm}/f^{∗}p^{∗}EW_{k} = S(W_{k}, \overline{Y}_{et})/ ∐_{W_{k}} y\overline{Y} ≅ S(W_{k}, ∐_{W_{k}} \overline{Y}_{et}) ≅ \overline{Y}_{et}.

Consider now the localization (Y_{et} ×_{B_{G_{k}}^{sm}} B_{W_{k}}^{sm})/p^{∗}EW_{k}. We have the following canonical equivalences:

(4)  (Y_{et} ×_{B_{G_{k}}^{sm}} B_{W_{k}}^{sm})/p^{∗}EW_{k} ≅ Y_{et} ×_{B_{G_{k}}^{sm}} B_{W_{k}}^{sm} ×_{B_{W_{k}}^{sm}} \underline{Set}

≅ Y_{et} ×_{B_{G_{k}}^{sm}} \underline{Set} (5)

≅ \underleftarrow{lim} (Y_{et} ×_{B_{G_{k′/k}}^{sm}} \underline{Set}) (6)

≅ \underleftarrow{lim} (Y_{et} ×_{B_{G_{k′/k}}^{sm}} (B_{G_{k′/k}}^{sm}/EG_{k′/k})) (7)

≅ \underleftarrow{lim} (Y_{et}/Y′) (8)

≅ \underleftarrow{lim} Y′_{et} (9)

≅ (\underleftarrow{lim} Y′)_{et} = \overline{Y}_{et} (10)

Indeed, (4) follows from the canonical equivalence B_{W_{k}}^{sm}/EW_{k} ≅ \underline{Set}. The inverse limit in (6) is taken over the Galois extensions k′/k. Using the natural equivalence

B_{G_{k}}^{sm} ≅ \underleftarrow{lim} B_{G(k′/k)}^{sm}

(6) follow from the universal property of limits of topoi. For (7) we use again

B_{G(k′/k)}^{sm}/EG(k′/k) ≅ \underline{Set}.

Then (8) follows from the fact that the inverse image of EG(k′/k) in the étale topos Y_{et} is the sheaf represented by the étale Y -scheme Y′ := Y ×_{k} k′. Then (9) is given by ([20] III Proposition 5.4), and (10) is given by ([33] Lemma 8.3), since the schemes Y′ are all quasi-compact and quasi-separated. We obtain a commutative square

(11)  \begin{matrix}\overline{Y}_{et} & ⟶^{Id} & \overline{Y}_{et} \\ ↓ & & ↓ \\ Y_{W}^{sm}/f^{∗}p^{∗}EW_{k} & ⟶^{f/p^{∗}EW_{k}} & (Y_{et} ×_{B_{G_{k}}^{sm}} B_{W_{k}}^{sm})/p^{∗}EW_{k} \end{matrix}

where the vertical maps are the equivalences defined above. It follows that f/p^{∗}EW_{k} is an equivalence, and so is f by Lemma 2. □

Corollary 1. There is a canonical equivalence

Y_{et} ×_{B_{G_{k}}^{sm}} B_{W_{k}} ≅ Y_{W}^{sm} × T

Proof. The Weil group W_{k} is a group of the final topos \underline{Set}. If u : T → \underline{Set} denotes the unique map, then u^{∗}W_{k} is the group object of T represented by the discrete group W_{k}. Hence one has (see the pull-back diagram (2)):

B_{W_{k}}^{sm} × T := B\underline{Set}(W_{k}) × T ≅ B_{T} (yW_{k}) =: B_{W_{k}}.

The previous theorem therefore yields

Y_{et} ×_{B_{G_{k}}^{sm}} B_{W_{k}} ≅ Y_{et} ×_{B_{G_{k}}^{sm}} B_{W_{k}}^{sm} × T ≅ Y_{W}^{sm} × T .
□

Definition 3. We define the big Weil-étale topos of Y as the fiber product

Y_{W} := Y_{et} ×_{B_{G_{k}}^{sm}} B_{W_{k}} ≅ Y_{W}^{sm} × T .

Corollary 2. Let p_{1} : Y_{W} → Y_{W}^{sm} and p_{2} : Y_{W} → T be the projections. Then for any abelian object A′ of Y_{W}, one has

H^{n}(Y_{W} , A′) ≅ H^{n}(Y_{W}^{sm}, p_{1∗}A′).

If A is an abelian object of T , then

H^{n}(Y_{W} , p_{2}^{∗}A) ≅ H^{n}(Y_{W}^{sm}, A(∗)).

Proof. This follows from Corollary 12, using the equivalence Y_{W} ≅ Y_{W}^{sm} × T . □

Define the sheaf ℝ̃ on Y_{W} as p_{2}^{∗}(yℝ), where yℝ is the object of T represented by the standard topological group ℝ. Then we have canonical isomorphisms

H^{n}(Y_{W} , ℝ̃) ≅ H^{n}(Y_{W}^{sm}, ℝ) and H^{n}(Y_{W} , ℤ) ≅ H^{n}(Y_{W}^{sm}, ℤ)

as it follows from the previous corollary.

Corollary 3. Let α : G → H and β : G′ → H be two homomorphisms of group objects in a topos S. If α is an epimorphism then the natural morphism

f : B_{G×_{H}G′} ⟶ B_{G} ×_{B_{H}} B_{G′}

is an equivalence.

Proof. Let e_{S} be the final object in S. The unique map G′ → e_{S} is epimorphic, since the unit of G′ yields a section e_{S} → G′. Therefore, the morphism EG′ → e_{G′} in B_{G′}, where e_{G′} is the final object of B_{G′}, is epimorphic. We denote the second projection by

p : B_{G} ×_{B_{H}} B_{G′} ⟶ B_{G′}

Let K be the kernel of α, so that G/K ≅ H. On the one hand, we have the following canonical equivalences:

(B_{G} ×_{B_{H}} B_{G′})/p^{∗}EG′ ≅ B_{G} ×_{B_{H}} (B_{G′}/EG′)
≅ B_{G} ×_{B_{H}} S
≅ B_{G} ×_{B_{H}} (B_{H}/EH)
≅ B_{G}/α^{∗}EH
≅ B_{G}/(G/K)
≅ B_{K}

Here G/K is endowed with its natural G-action. The second, the third and the last equivalences are given by ([20] IV.5.8), and the fourth equivalence is given by the pull-back diagram (1). On the other hand, we have an exact sequence of group objects in S

1 → K → G ×_{H} G′ → G′ → 1.

Indeed, the kernel of G ×_{H} G′ → G′ is given by

G ×_{H} G′ ×_{G′} e_{S} = G ×_{H} e_{S} = K.

Moreover, G ×_{H} G′ → G′ is epimorphic, since epimorphisms are universal in a topos. We obtain

B_{G×_{H}G′}/f^{∗}p^{∗}EG′ = B_{G×_{H}G′}/(G ×_{H} G′/K)
= B_{K}

and we have a commutative square

(12)  \begin{matrix}B_{K} & ⟶^{Id} & B_{K} \\ ↓ & & ↓ \\ B_{G×_{H}G′}/f^{∗}p^{∗}EG′ & ⟶^{f/p^{∗}EG′} & (B_{G} ×_{B_{H}} B_{G′})/p^{∗}EG′ \end{matrix}

where the vertical maps are the equivalences defined above. Hence f/p^{∗}EG′ is an equivalence. By Lemma 2, f is an equivalence as well, since EG′ → e_{G′} is epimorphic. □

Corollary 4. Let α : G → H and β : G′ → H be two morphisms of locally compact topological groups. If α has local sections then the natural morphism

f : B_{G×_{H}G′} ⟶ B_{G} ×_{B_{H}} B_{G′}

is an equivalence.

Proof. Since α : G → H has local sections, the induced morphism y(G) → y(H) is an epimorphism in T . Hence the result follows from Corollary 3. □

4. Artin-Verdier étale topos of an arithmetic scheme

Let X be a scheme separated and of finite type over Spec(ℤ). We denote by X^{an} the complex analytic variety associated to X ⊗_{ℤ} ℂ, endowed with the standard complex topology. The Galois group G_{ℝ} of ℝ acts on X^{an}. The quotient space X_{∞} := X^{an}/G_{ℝ} is endowed with the quotient topology. We consider the pair

\overline{X} := (X , X_{∞}).

As a set, \overline{X} is the disjoint union X ∐ X_{∞}. The Zariski topology on \overline{X} is defined as follows. An open subset (U, D) of \overline{X} is given by a Zariski open subscheme U ⊂ X and an open subspace D ⊂ U_{∞} for the complex topology. We define the category Et\overline{X} of étale \overline{X} -schemes as follows. An étale \overline{X} -scheme is an arrow f : (U, D) → (X , X_{∞}), where U → X is an étale morphism in the usual sense and D is an open subset of U_{∞}. The map f_{∞} : D → X_{∞} is supposed to be unramified in the sense that f_{∞}(d) ∈ X (ℝ) if and only if d ∈ D ∩ U(ℝ). An étale \overline{X} -scheme \overline{U} is said to be connected (respectively irreducible) if it is connected (respectively irreducible) as a topological space. A morphism (U, D) → (U′, D′) in the category Et\overline{X} is given by a morphism of étale X -schemes U → U′ inducing a map D → D′. The étale topology J_{et} on the category Et\overline{X} is the topology generated by the pretopology for which a covering family is a surjective family. The Artin-Verdier étale site is left exact.

Definition 4. The Artin-Verdier étale topos of \overline{X} is the category of sheaves of sets on the Artin-Verdier étale site:

\overline{X}_{et} := ({ \atop Et\overline{X} , J_{et}}).

Let X and Y be schemes which are separated and of finite type over Spec(ℤ). A map f : X → Y induces a map f̄ : \overline{X} → \overline{Y} (in the obvious sense) and a morphism of topoi f̄_{et} : \overline{X}_{et} → \overline{Y}_{et}. The object yX := y(X , ∅) is a subobject of the final object y\overline{X} of \overline{X}_{et}. This yields an open subtopos

\overline{X}_{et}/y(X , ∅) ↪ \overline{X}_{et}.

We have the following canonical identifications (see [20] III Proposition 5.4):

\overline{X}_{et}/y(X , ∅) ≅ { \atop (Et\overline{X} /_{(X ,∅)}, J_{ind})} ≅ ({ \atop Et_{X} , J_{et}}) = X_{et}

where X_{et} is the usual étale topos of X, and J_{ind} is the topology on Et\overline{X} /(X , ∅) induced by J_{et} on Et\overline{X} via the forgetful functor Et\overline{X} /(X , ∅) → Et\overline{X}. We thus obtain an open embedding

φ : X_{et} ↪ \overline{X}_{et}.

Let Sh(X_{∞}) be the topos of sheaves of sets on the topological space X_{∞}, i.e. the category of étalé spaces on X_{∞}. We consider Sh(X_{∞}) as a site endowed with the canonical topology J_{can}. There is a morphism of left exact sites

\begin{aligned}u_{∞}^{∗} : & (Et\overline{X} , J_{et}) & ⟶ & (Sh(X_{∞}), J_{can}) \\ & (U, D) & ⟼ & D → X_{∞} \end{aligned}

The resulting morphism of topoi

u_{∞} : Sh(X_{∞}) ⟶ \overline{X}_{et}

is precisely the closed complement of the open subtopos X_{et} ↪ \overline{X}_{et} defined above, i.e. we have the following result.

Proposition 4.1. There is an open-closed decomposition of topoi

φ : X_{et} ⟶ \overline{X}_{et} ⟵ Sh(X_{∞}) : u_{∞}

The gluing functor u_{∞}^{∗}ϕ_{∗} can be made more explicit as follows. There is a canonical morphism of topoi

α : Sh(G_{ℝ}, X^{an}) ⟶ X_{et}

where Sh(G_{ℝ}, X^{an}) is the topos of G_{ℝ}-equivariant sheaves on the topological space X^{an}, i.e. the category of G_{ℝ}-equivariant étalé spaces on X^{an}. The map α is defined by the morphism of left exact sites which takes an étale X -scheme U to the G_{ℝ}-equivariant étalé space U^{an} over X^{an} (note that U^{an} → X^{an} is a G_{ℝ}-equivariant local homeomorphism since the morphism U ⊗_{ℤ} ℂ → X ⊗_{ℤ} ℂ is étale and compatible with complex conjugation). The quotient map X^{an} → X^{an}/G_{ℝ} yields another morphism of topoi

(π^{∗}, π_{∗}^{G_{ℝ}}) : Sh(G_{ℝ}, X^{an}) ⟶ Sh(X_{∞}).

Here π : X^{an} → X_{∞} is the quotient map, π^{∗} is the usual inverse image and π_{∗}^{G_{ℝ}}F is the G_{ℝ}-invariant subsheaf of the the direct image π_{∗}F, i.e. for any open U ⊂ X_{∞} one has

π_{∗}^{G_{ℝ}}F(U) := F(π^{−1}U)^{G_{ℝ}}.

Then we have an identification of functors

u_{∞}^{∗}φ_{∗} ≅ π_{∗}^{G_{ℝ}}α^{∗} : X_{et} ⟶ Sh(X_{∞})

Let us consider the category (Sh(X_{∞}) , X_{et}, π_{∗}^{G_{ℝ}}α^{∗}) defined in ([20] IV.9.5.1) by Artin gluing. Recall that an object of this category is a triple (F, E, σ), where F is an object of Sh(X_{∞}), E is an object of X_{et} and σ is a map σ : F → π_{∗}^{G_{ℝ}}α^{∗}E.

Corollary 5. The category \overline{X}_{et} is canonically equivalent to (Sh(X_{∞}) , X_{et}, π_{∗}^{G_{ℝ}}α^{∗}).

Proof. There is a canonical functor

\begin{aligned}Φ : & \overline{X}_{et} & ⟶ & (Sh(X_{∞}) , X_{et}, u_{∞}^{∗}φ_{∗}) \\ & F & ⟼ & (u_{∞}^{∗}F, φ^{∗}F, σ) \end{aligned}

where the morphism

σ : u_{∞}^{∗}F ⟶ u_{∞}^{∗}φ_{∗}(φ^{∗}F)

is induced by the adjunction transformation Id → φ_{∗}φ^{∗}. By ([20] IV.9.5.4.a) the functor Φ is an equivalence of categories, since u_{∞} : Sh(X_{∞}) ↪ \overline{X}_{et} is the closed complement of the open embedding ϕ : X_{et} ↪ \overline{X}_{et}. Hence the result follows from the isomorphism

u_{∞}^{∗}φ_{∗} ≅ π_{∗}^{G_{ℝ}}α^{∗}.
□

Corollary 6. We denote by ∞ the archimedean place of ℚ. The commutative square

(13)  \begin{matrix}Sh(X_{∞}) & ⟶ & Sh(∞) \\ ↓ & & ↓ \\ \overline{X}_{et} & ⟶^{f} & \overline{Spec(ℤ)}_{et} \end{matrix}

is a pull-back, where Sh(∞) = \underline{Set} is the category of sheaves on the one point space.

Proof. The map \overline{X} → \overline{Spec(ℤ)} induces a morphism of étale topos f. Consider the open embedding Spec(ℤ)_{et} ↪ \overline{Spec(ℤ)}_{et}. Its inverse image under the map f is X_{et} ↪ \overline{X}_{et}. The result therefore follows from Proposition 4.1 and ([20] IV Corollaire 9.4.3). □

Proposition 4.2. For any prime number p, we have a pull-back

(14)

(X ⊗_{ℤ} 𝔽_{p})_{et} ⟶ Spec(𝔽_{p})_{et}
↓  ↓
\overline{X}_{et}  ⟶^{f} \overline{Spec(ℤ)}_{et}

Proof. The morphism Spec(𝔽_{p})_{et} → \overline{Spec(ℤ)}_{et} factors through Spec(ℤ)_{et}, hence one is reduced to show that

(X ⊗_{ℤ} 𝔽_{p})_{et} ≅ X_{et} ×_{Spec(ℤ)_{et}} Spec(𝔽_{p})_{et}.

This follows from ([20] IV Corollaire 9.4.3) since Spec(𝔽_{p}) → Spec(ℤ) is a closed embedding. □

4.1. Étale cohomology with compact support. It follows from Corollary 5 that we have the usual sequences of adjoint functors (see [20] IV.14)

φ_{!}, φ^{∗}, φ_{∗} and u_{∞}^{∗}, u_{∞∗}, u_{∞}^{!}.

between the categories of abelian sheaves on \overline{X}_{et}, X_{et} and Sh(X_{∞}). In particular u_{∞∗} is exact and φ^{∗} preserves injective objects since φ_{!} is exact. For any abelian sheaf A on \overline{X}_{et}, one has the exact sequence

(15)  0 → φ_{!}φ^{∗}A → A → u_{∞∗}u_{∞}^{∗}A → 0,

where the morphisms are given by adjunction.

Definition 5. Assume that X is proper over Spec(ℤ) and let A be an abelian sheaf on X_{et}. The étale cohomology with compact support is defined by

H_{c}^{n}(X_{et}, A) := H^{n}(\overline{X}_{et}, φ_{!}A).

Proposition 4.3. Let X be a flat proper scheme over Spec(ℤ). Assume that X is normal and connected. Then the ℝ-vector space H_{c}^{n}(X_{et}, ℝ) is finite dimensional, zero for n large, and we have

H_{c}^{n}(X_{et}, ℝ) = 0 for n = 0
= H^{0}(X_{∞}, ℝ)/ℝ for n = 1
= H^{n−1}(X_{∞}, ℝ) for n ≥ 2

Proof. The exact sequence

0 → φ_{!}ℝ → ℝ → u_{∞∗}ℝ → 0

and the fact that u_{∞∗} is exact give a long exact sequence

0 → H_{c}^{0}(X_{et}, ℝ) → H^{0}(\overline{X}_{et}, ℝ) → H^{0}(X_{∞}, ℝ) → H_{c}^{1}(X_{et}, ℝ) → H^{1}(\overline{X}_{et}, ℝ) →

The inclusion of the generic point of X yields a morphism of topoi

η : (Spec K(X))_{et} ⟶ \overline{X}_{et}.

We have immediately R^{n}η_{∗}ℝ = 0 for any n ≥ 1 since Galois cohomology is torsion and ℝ is uniquely divisible. Moreover, we have η_{∗}ℝ = ℝ. Indeed, the scheme X is normal hence the set of connected components of an étale \overline{X} -scheme \overline{U} is in 1-1 correspondence with the set of connected components of U ×_{X} Spec K(X) , i.e. one has

π_{0}(U ×_{X} Spec K(X)) = π_{0}(U) = π_{0}(\overline{U}).

Therefore the Leray spectral sequence associated to the morphism η gives

H^{n}(\overline{X}_{et}, ℝ) = H^{n}(G_{K(X)}, ℝ).

We obtain H^{0}(\overline{X}_{et}, ℝ) = ℝ and H^{n}(\overline{X}_{et}, ℝ) = 0 for n ≥ 1, and the result follows. □

5. The definition of \overline{Spec(O_{F})}_{W}

Let F be a number field. We consider the Arakelov compactification X̄ = (Spec O_{F} , X_{∞}) of X = Spec O_{F}, where X_{∞} is the finite set of archimedean places of F. Note that this is a special case of the previous section, since X_{∞} is the quotient of X ⊗ ℂ by complex conjugation. We endow X̄ with the Zariski topology described previously. If F̄/F is an algebraic closure and F̄/K/F a finite Galois extension then the relative Weil group W_{K/F} is defined by the extension of topological groups

1 → C_{K} → W_{K/F} → G_{K/F} → 1

corresponding to the fundamental class in H^{2}(G_{K/F} , C_{K}) given by class field theory, where C_{K} is the idèle class group of K. A Weil group of F is then defined as the projective limit W_{F} := \underleftarrow{lim} W_{K/F}, computed in the category of topological groups. Alternatively, let F̄/K/F be a finite Galois extension and let S be a finite set of places of F containing all the places which ramify in K. Then the fundamental class in

H^{2}(G_{K/F} , C_{K}) ≅ H^{2}(G_{K/F} , C_{K,S})

yields a group extension

1 → C_{K,S} → W_{K/F,S} → G_{K/F} → 1

where C_{K,S} is the S-idèle class group of K. Then one has (see [30])

W_{F} := \underleftarrow{lim} W_{K/F} = \underleftarrow{lim} W_{K/F,S},

where K runs over the finite Galois extensions F̄/K/F and S runs over the finite sets of places of F containing all the places which ramify in K. However, the structure of the Weil-?tale topos at the generic point suggests to consider the projective system of topological groups W_{K/F,S} as a strict topological pro-group \underline{W}_{K/F,S} (see Sect. 2.3.4).

5.1. The Weil-étale topos. We choose an algebraic closure F̄/F and a Weil group W_{F}. For any place v of F, we choose an algebraic closure F̄_{v}/F_{v} and an embedding F̄ → F̄_{v} over F. Then we choose a local Weil group W_{F_{v}} and a Weil map θ_{v} : W_{F_{v}} → W_{F} compatible with F̄ → F̄_{v}. Let W_{F_{v}}^{1} be the maximal compact subgroup of W_{F_{v}}. For any closed point v ∈ X̄ (ultrametric or archimedean), we define the Weil group of ”the residue field at v” as follows

W_{k(v)} := W_{F_{v}}/W_{F_{v}}^{1},

while the Galois group of the residue field at v can be defined as G_{k(v)} := G_{F_{v}}/I_{v}. Note that G_{k(v)} is the trivial group for v archimedean. For any v, the Weil map W_{F_{v}} → G_{F_{v}} chosen above induces a morphism W_{k(v)} → G_{k(v)}. Finally, we denote by

q_{v} : W_{F_{v}} ⟶ W_{F_{v}}/W_{F_{v}}^{1} =: W_{k(v)}

the map from the local Weil group W_{F_{v}} to the Weil group of the residue field at v ∈ X̄.

Definition 6. Let TX̄ be the category of objects (Z_{0}, Z_{v}, f_{v}) defined as follows. The topological space Z_{0} is endowed with a continuous W_{F} -action. For any place v of F, Z_{v} is a topological space endowed with a continuous W_{k(v)}-action. The continuous map f_{v} : Z_{v} → Z_{0} is W_{F_{v}}-equivariant, when Z_{v} and Z_{0} are seen as W_{F_{v}}-spaces via the maps θ_{v} : W_{F_{v}} → W_{F} and q_{v} : W_{F_{v}} → W_{k(v)}. Moreover, we require the following facts.

• The spaces Z_{v} are locally compact.

• The map f_{v} is an homeomorphism for almost all places v of F and a continuous injective map for all places.

• The action of W_{F} on Z_{0} factors through W_{K/F}, for some finite Galois

subextension F̄/K/F. A morphism

ϕ : (Z_{0}, Z_{v}, f_{v}) → (Z′_{0}, Z′_{v}, f′_{v})

in the category TX̄ is a continuous W_{F} -equivariant map ϕ : Z_{0} → Z′_{0} inducing a continuous map ϕ_{v} : Z_{v} → Z′_{v} for any place v. Then ϕ_{v} is W_{k(v)}-equivariant. The category TX̄ is endowed with the local section topology J_{ls}, i.e. the topology generated by the pretopology for which a family

{φ_{i} : (Z_{i,0}, Z_{i,v}, f_{i,v}) → (Z_{0}, Z_{v}, f_{v}), i ∈ I}

is a covering family if ∐_{i∈I} Z_{i,v} → Z_{v} has local continuous sections, for any place v.

Lemma 3. The site (TX̄, J_{ls}) is left exact.

Proof. The category TX̄ has fiber products and a final object, hence finite projective limits are representable in TX̄. It remains to show that J_{ls} is subcanonical. This follows easily from the fact that, for any topological group G, the local section topology J_{ls} on B_{T op}G coincides with the open cover topology J_{op}, which is subcanonical. □

Definition 7. We define the Weil-étale topos X̄_{W} as the topos of sheaves of sets on the site defined above:

X̄_{W} := { \atop (TX̄, J_{ls})}.

Proposition 5.1. We have a morphism of topoi

j : B_{W_{F}} ⟶ X̄_{W} .

Proof. By [15] Corollary 2, the site (B_{T op}W_{F} , J_{ls}) is a site for the classifying topos B_{W_{F}}. By [15] Corollary 2, the site (B_{T op}W_{F} , J_{ls}) is a site for B_{W_{F}}. The morphism of left exact sites

\begin{aligned}j^{∗} : & (TX̄, J_{ls}) & ⟶ & (B_{T op}W_{F} , J_{ls}) \\ & (Z_{0}, Z_{v}, f_{v}) & ⟼ & Z_{0} \end{aligned}

induces the morphism of topoi j. □

Proposition 5.2. The morphism of topoi j : B_{W_{F}} → X̄_{W} factors through the classifying topos

B_{\underline{W}_{K/F,S}} := \underleftarrow{lim}B_{W_{K/F,S}}

of the strict topological pro-group \underline{W}_{K/F,S} (see Sect. 2.3.4). The induced morphism i_{0} : B_{\underline{W}_{K/F,S}} → X̄_{W} is an embedding.

Proof. Let (Z_{0}, Z_{v}, f_{v}) be an object of TX̄. The action of W_{F} on Z_{0} factors through W_{K/F}, for some finite Galois sub-extension F̄/K/F. Since W_{K/F} and Z_{0} are both locally compact, this action is given by a continuous morphism

ρ : W_{K/F} ⟶ Aut(Z_{0})

where Aut(Z_{0}) is the homeomorphism group of Z_{0} endowed with the compactopen topology. The kernel of ρ is a closed normal subgroup of W_{K/F} since Aut(Z_{0}) is Hausdorff. Moreover, there exists an open subset V of X̄ such that f_{v} : Z_{v} → Z_{0} is an isomorphism of W_{F_{v}}-spaces for any v ∈ V. Let W̃_{F_{v}}^{1} denote the image of the continuous morphism

W_{F_{v}}^{1} ⟶ W_{F_{v}} ⟶ W_{K/F} ,

endowed with the induced topology. Then W̃_{F_{v}}^{1} is in the kernel of ρ for any v ∈ V. Let N_{V} be the closed normal subgroup of W_{K/F} generated by the subgroups W̃_{F_{v}}^{1} for any v ∈ V. Then ρ induces a continuous morphism

W_{K/F} /N_{V} ⟶ Aut(Z_{0}).

We choose V small enough so that K/F is unramified above V and we set S := X̄ − V. Then we have

N_{V} = ∏_{w|v, v∈V} O_{K_{w}}^{×} ⊆ C_{K} ⊆ W_{K/F} and W_{K/F} /N_{V} = W_{K/F,S}.

Hence the action of W_{F} on Z_{0} factors through W_{K/F,S}, for some finite Galois sub-extension F̄/K/F and some finite set S of places of F containing all the places which ramify in K. The morphism of left exact sites

\begin{aligned}j^{∗} : & (TX̄, J_{ls}) & ⟶ & (B_{T op}W_{F} , J_{ls}) \\ & (Z_{0}, Z_{v}, f_{v}) & ⟼ & Z_{0} \end{aligned} .

therefore induces a morphism

\begin{aligned}i_{0}^{∗} : & (TX̄, J_{ls}) & ⟶ & (\underrightarrow{lim} B_{T op}W_{K/F,S}, J_{ls}) \\ & (Z_{0}, Z_{v}, f_{v}) & ⟼ & Z_{0} \end{aligned}

where (\underrightarrow{lim} B_{T op}W_{K/F,S}, J_{ls}) is the direct limit site. More precisely, \underrightarrow{lim} B_{T op}W_{K/F,S} is the direct limit category endowed with the coarsest topology J such that the functors B_{T op}W_{K/F,S} → \underrightarrow{lim} B_{T op}W_{K/F,S} are all continuous, when B_{T op}W_{K/F,S} is endowed with the local section topology. One can identify \underrightarrow{lim} B_{T op}W_{K/F,S} with a full subcategory of B_{T op}W_{F} and J with the local section topology J_{ls}. By ([20] VI.8.2.3), the direct limit site (\underrightarrow{lim} B_{T op}W_{K/F,S}, J_{ls}) is a site for the projective limit topos B_{\underline{W}_{K/F,S}}. We obtain a morphism of topoi

i_{0} : B_{\underline{W}_{K/F,S}} ⟶ X̄_{W} .

It remains to show that this morphism is an embedding. Let F be an object of B_{\underline{W}_{K/F,S}}. Then i_{0}^{∗}i_{0∗}F is the sheaf associated with the presheaf

\begin{aligned}i_{0}^{p}i_{0∗}F : & \underrightarrow{lim} B_{T op}W_{K/F,S} & ⟶ & \underline{Set} \\ & Z & ⟼ & lim_{Z→i_{0}^{∗}(Y_{0},Y_{v},f_{v})}i_{0∗}F(Y_{0}, Y_{v}, f_{v}) \end{aligned}

where the direct limit is taken over the category of arrows Z → i_{0}^{∗}(Y_{0}, Y_{v}, f_{v}). For any object Z of \underrightarrow{lim} B_{T op}W_{K/F,S}, there exist a finite Galois extension K_{Z}/F and a finite set S_{Z} such that Z is an object of B_{T op}W_{K_{Z}/F,S_{Z}}. Consider the cofinal subcategory I_{Z} of the category of arrows defined above, where I_{Z} consists of the following objects. For any finite set S of places of F such that S_{Z} ⊆ S, we consider the map Z → i_{0}^{∗}(Z_{0}, Z_{v}, f_{v}) with Z_{0} = Z as a W_{F} -space, Z_{v} = Z as a W_{k(v)}-space for any place v not in S and Z_{v} = ∅ for any v ∈ S. We thus have

lim_{Z→i_{0}^{∗}(Y_{0},Y_{v},f_{v})}i_{0∗}F(Y_{0}, Y_{v}, f_{v}) = lim_{I_{Z}} i_{0∗}F(Z_{0}, Z_{v}, f_{v}) = F(Z).

Hence i_{0}^{p}i_{0∗}F is already a sheaf and we have

i_{0}^{∗}i_{0∗}F = i_{0}^{p}i_{0∗}F = F.

This shows that i_{0∗} is fully faithful, i.e. i_{0} is an embedding.

□

Proposition 5.3. There is canonical morphism of topoi

𝔣 : X̄_{W} ⟶ B_{ℝ}.

Proof. We have a commutative diagram of topological groups

(16)  \begin{matrix}W_{F_{v}} & ⟶ & W_{k(v)} \\ ↓ & & ↓ \\ W_{F} & ⟶ & ℝ \end{matrix}

where W_{F} → ℝ is defined as the composition

W_{F} → W_{F}^{ab} ≅ C_{F} → ℝ_{+}^{×} ≅ ℝ.

Hence there is a morphism of left exact sites

(17)  \begin{aligned}𝔣^{∗} : & (B_{T op}ℝ, J_{ls}) & ⟶ & (TX̄, J_{ls}) \\ & Z & ⟼ & (Z, Z, Id_{Z}) \end{aligned}

where Z is seen as W_{F} -space (respectively a W_{k(v)}-space) via the canonical morphism W_{F} → ℝ (respectively via W_{k(v)} → ℝ). The result follows. □

5.2. The morphism from the Weil-étale topos to the Artin-Verdier étale topos. Let X̄ be the Arakelov compactification of the number ring O_{F}. We consider below the Artin-Verdier étale site (EtX̄; J_{et}) and the Artin-Verdier étale topos X̄_{et} of the arithmetic curve X̄.

Proposition 5.4. There exists a morphism of left exact sites

\begin{aligned}γ^{∗} : & (EtX̄; J_{et}) & ⟶ & (TX̄; J_{ls}) \\ & Ū & ⟼ & (U_{0}, U_{v}, f_{v}) \end{aligned} .

The underlying functor γ^{∗} is fully faithful and its essential image consists exactly of objects (U_{0}, U_{v}, f_{v}) of TX̄ where U_{0} is a finite W_{F} -set.

This result is a reformulation of [33] Proposition 4.61 and [33] Proposition 4.62. We give below a sketch of the proof.

Proof. For any étale X̄-scheme Ū, we define an object γ^{∗}(Ū) = (U_{0}, U_{v}, f_{v}) of TX̄ as follows. The scheme Ū ×X̄ Spec F is the spectrum of an étale F -algebra and the Grothendieck-Galois theory shows that this F -algebra is uniquely determined by the finite G_{F} -set

U_{0} := Hom_{Spec F} (Spec F̄, Ū ×X̄ Spec F) = HomX̄(Spec \overline{F} , Ū).

Let v be an ultrametric place of F. The maximal unramified sub-extension of the algebraic closure F̄_{v}/F_{v} yields an algebraic closure of the residue field \overline{k(v)}/k(v). The scheme Ū ×X̄ Spec k(v) is the spectrum of an étale k(v)-algebra, corresponding to the finite G_{k(v)}-set

U_{v} := Hom_{Spec k(v)}(Spec \overline{k(v)}, Ū ×X̄ Spec k(v)) = HomX̄(Spec \overline{k(v)}, Ū)

The chosen F -embedding F̄ → F̄_{v} induces a G_{F_{v}}-equivariant map

f_{v} : U_{v} ⟶ U_{0}.

Consider now an archimedean place v of F. Define

U_{v} := HomX̄(v, Ū) = Ū ×X̄ v

where the map v → X̄ is the closed embedding corresponding to the archimedean place v of F. As above, the F -embedding F̄ → F̄_{v} induces a G_{F_{v}}-equivariant map

f_{v} : U_{v} ⟶ U_{0}.

For any place v of F, the set U_{v} is viewed as a W_{k(v)}-topological space via the morphism W_{k(v)} → G_{k(v)}. Respectively, U_{0} is viewed as a W_{F} -topological space via W_{F} → G_{F}. Then the map f_{v} defined above is W_{F_{v}}-equivariant. We check that the map f_{v} is bijective for almost all valuations and injective for all valuations (see [33] Proposition 4.62). We obtain a functor

γ^{∗} : EtX̄ ⟶ TX̄.

This functor is left exact by construction (i.e. it preserves the final objects and fiber product) and continuous (i.e. it preserves covering families) since a surjective map of discrete sets is a local section cover. The last claim of the proposition follows from Galois theory. □

Corollary 7. There is a morphism of topoi γ : X̄_{W} → X̄_{et}.

Proof. This follows from the fact that a morphism of left exact sites induces a morphism of topoi. □

Definition 8. Let X′ be an open subscheme of X̄. We define

X′_{W} := X̄_{W} /γ^{∗}(yX′).

5.3. Structure of X̄_{W} at the closed points. Let v be a place of F. We consider the Weil group W_{k(v)} and the Galois group G_{k(v)} of the residue field k(v) at v ∈ X̄. Note that for v archimedean one has W_{k(v)} ≅ ℝ and G_{k(v)} = {1}. Consider the big classifying topos B_{W_{k(v)}}, i.e. the category of y(W_{k(v)})-objects in T . We consider also the small classifying topos B_{G_{k(v)}}^{sm}, which is defined as the category of continuous G_{k(v)}-sets. The category of locally compact W_{k(v)}-spaces B_{T op}W_{k(v)} is endowed with the local section topology J_{ls}. Recall that the site (B_{T op}W_{k(v)}, J_{ls}) is a site for the classifying topos B_{W_{k(v)}}. We denote by B_{fSets}G_{k(v)} the category of finite G_{k(v)}-sets endowed with the canonical topology J_{can}. The site (B_{fSets}G_{k(v)}, J_{can}) is a site for the small classifying topos B_{G_{k(v)}}^{sm}. For any place v of F, we have a morphism of left exact sites

\begin{aligned}i_{v}^{∗} : & (TX̄, J_{ls}) & ⟶ & (B_{T op}W_{k(v)}, J_{ls}) \\ & (Z_{0}, Z_{v}, f_{v}) & ⟼ & Z_{v} \end{aligned}

hence a morphism of topoi

i_{v} : B_{W_{k(v)}} ⟶ X̄_{W} .

On the other hand one has morphism of topoi

u_{v} : B_{G_{k(v)}}^{sm} ⟶ X̄_{et}

for any closed point v of X̄. For v ultrametric, this morphism is induced by the closed embedding of schemes

Spec k(v) ⟶ X̄

since the étale topos of Spec k(v) is equivalent to the category B_{G_{k(v)}}^{sm} of continuous G_{k(v)}-sets. Note that this equivalence is induced by the choice of an algebraic closure of k(v) made at the beginning of section 5.1. By Corollary 4.1, there is a closed embedding

Sh(X_{∞}) = ∐_{X_{∞}} \underline{Set} ⟶ X̄_{et}

which yields the closed embedding

u_{v} : B_{G_{k(v)}}^{sm} = \underline{Set} ⟶ X̄_{et}

for any archimedean valuation v of F. In both cases, we have a commutative diagram of left exact sites

(B_{T op}W_{k(v)}, J_{ls}) ⟵^{α_{v}^{∗}} (B_{fSets}G_{k(v)}, J_{can})
↑^{i_{v}^{∗}}  ↑^{u_{v}^{∗}}
(TX̄, J_{ls})  ⟵^{γ^{∗}}  (EtX̄, J_{et})

where u_{v}^{∗}(Ū) is the finite G_{k(v)}-set HomX̄(Spec \overline{k(v)}, Ū) (respectively HomX̄(v, Ū)) for v ultrametric (respectively archimedean). Moreover, α_{v}^{∗} maps a finite G_{k(v)}-set E to the discrete topological space E on which W_{k(v)} acts via W_{k(v)} → G_{k(v)}. This commutative diagram of sites induces a commutative diagram of topoi.

Theorem 5.1. For any closed point v of X̄, the following diagram is a pull-back of topoi.

B_{W_{k(v)}} ⟶^{α_{v}} B_{G_{k(v)}}^{sm}
↓^{i_{v}}  ↓^{u_{v}}
X̄_{W} ⟶^{γ} X̄_{et}

In particular, the morphism i_{v} is a closed embedding.

Proof. We first prove a partial result.

Lemma 4. The morphism i_{v} is an embedding, i.e. i_{v∗} is fully faithful.

Proof. We use below the fact that the full subcategory

W_{k(v)} × T op ↪ B_{T op}W_{k(v)}

is a topologically generating subcategory of the site (B_{T op}W_{k(v)}, J_{ls}). Here W_{k(v)} × T op consists in locally compact topological spaces of the form Z = W_{k(v)} × T on which W_{k(v)} acts by left multiplication on the first factor. In particular, a sheaf F of

B_{W_{k(v)}} = { \atop (B_{T op}W_{k(v)}, J_{ls})}

is completely determined by its values F(W_{k(v)} ×T) on objects of W_{k(v)} ×T op. Let F be an object of B_{W_{k(v)}}. Consider the adjunction map

(18)  i_{v}^{∗} ∘ i_{v∗}F ⟶ F.

The sheaf i_{v}^{∗} ∘ i_{v∗}F is the sheaf on (B_{T op}W_{k(v)}, J_{ls}) associated to the presheaf

Z → lim_{Z→i_{v}^{∗}(Y_{0},Y_{w},f_{w})}i_{v∗}F(Y_{0}, Y_{w}, f_{w}) = lim_{Z→i_{v}^{∗}(Y_{0},Y_{w},f_{w})}F(Y_{v})

where the direct limit is taken over the category of arrows Z → i_{v}^{∗}(Y_{0}, Y_{w}, f_{w}) with (Y_{0}, Y_{w}, f_{w}) an object of TX̄.

Let F̄/K/F be a finite Galois sub-extension, let S be a finite set of closed points of X̄ such that v ∈ S and let Z = W_{k(v)} × T be an object of W_{k(v)} × T op. Consider the object of TX̄

Y(K, S, Z) = (T_{0}, T_{w}, f_{w})

defined as follows. We first define the topological space

T_{0} = W_{K/F,S} ×^{W_{Fv}} Z := (W_{K/F,S} × W_{k(v)} × T)/W_{F_{v}} ≅ (W_{K/F,S}/W_{F_{v}}^{1}) × T

endowed with its natural W_{F} -action. For any w not in S, we consider T_{w} = W_{K/F,S} ×^{W_{Fv}} Z on which W_{k(w)} acts via the map

W_{k(w)} = W_{F_{w}}/W_{F_{w}}^{1} ⟶ W_{K/F,S}.

For any w ∈ S such that w ≠ v, we set T_{w} = ∅, and we define T_{v} = Z. The map f_{w} is the identity for any w not in S and

f_{v} : Z ⟶ W_{K/F,S} ×^{W_{Fv}} Z

is the canonical map. This map f_{v} is continuous and injective. The image of W_{F_{v}}^{1} in W_{K/F,S} is compact, and the spaces T_{0} and T_{w} are locally compact for any place w so that Y(K, S, Z) is an object of TX̄. On the one hand, the functor Z ↦ W_{K/F,S} ×^{W_{Fv}} Z is left adjoint to the forgetful functor B_{T op}W_{K/F,S} → B_{T op}W_{F_{v}}. On the other hand, for any object (Y_{0}, Y_{w}, f_{w}) of TX̄, the action of W_{F} on Z_{0} factors through W_{K/F,S} for some finite Galois extension K/F and some finite set S of places of F. It follows that

{Y(K, S, Z), for K/F Galois, S finite }

yields a cofinal system in the category of arrows Z → i_{v}^{∗}(Y_{0}, Y_{w}, f_{w}) considered above, for any fixed object Z of W_{k(v)} × T op. Hence i_{v}^{∗} ∘ i_{v∗}F is the sheaf on (B_{T op}W_{k(v)}, J_{ls}) associated to the presheaf

\begin{matrix}W_{k(v)} × T op & ⟶ & \underline{Set} \\ Z & ⟼ & lim_{Z→i_{v}^{∗}(Y_{0},Y_{w},f_{w})}F(Y_{v}) = lim_{Z→i_{v}^{∗}Y(K,S,Z)}F(Z) = F(Z) \end{matrix}

Since W_{k(v)}×T op is a topologically generating subcategory of (B_{T op}W_{k(v)}, J_{ls}), the sheaf on B_{W_{k(v)}} associated to this presheaf is F, and the adjunction morphism (18) is an isomorphism. This shows that i_{v∗} is fully faithful, i.e. i_{v} is an embedding. □

Definition 9. Let v be a closed point of X̄. We consider the morphism p_{v} : T → B_{W_{k(v)}} whose inverse image p_{v}^{∗} is the forgetful functor, and we denote by i\overline{v} the composite morphism

i\overline{v} := i_{v} ∘ p_{v} : T ⟶ B_{W_{k(v)}} ⟶ X̄_{W} .

For any object Z = W_{k(v)}×T of the full subcategory W_{k(v)}×T op ↪ B_{T op}W_{k(v)} and for any sheaf F of X̄_{W}, we have

(19) i_{v}^{p}F(Z) = lim_{Z→i_{v}^{∗}(Y_{0},Y_{w},f_{w})}F(Y_{0}, Y_{w}, f_{w}) = lim_{Z→i_{v}^{∗}Y(K,S,Z)}F(Y(K, S, Z))

where we consider the pull-back presheaf i_{v}^{p}F on B_{T op}W_{k(v)}. The morphism p_{v} : T → B_{W_{k(v)}} is induced by the morphism of left exact sites given by the forgetful functor B_{T op}W_{k(v)} → T op. By adjunction, for any space T of T op and any presheaf P on B_{T op}W_{k(v)} we have

p_{v}^{p}P(T) = P(W_{k(v)} × T).

Hence the isomorphism ip ; \overline{v} ≅ p_{v}^{p} ∘ i_{v}^{p} gives

(20)  ip ; \overline{v}F(T) = i_{v}^{p}F(Z) = lim_{Z→i_{v}^{∗}Y(K,S,Z)}F(Y(K, S, Z))

where Z := W_{k(v)} × T. We consider the category of compact spaces T op^{c}. The morphism of sites (T op^{c}, J_{op}) → (T op, J_{op}) induces an equivalence of topoi, hence one can restrict our attention to compact spaces. Let us show that ip ; \overline{v}F restricts to a sheaf on (T op^{c}, J_{op}). Let {T_{i} → T, i ∈ I} be a covering family of (T op^{c}, J_{op}), i.e. a local section cover of compact spaces. One can assume that I is finite, since any covering family of (T op^{c}, J_{op}) can be refined by a finite covering family. For any K/F and any S,

{Y(K, S, W_{k(v)} × T_{i}) → Y(K, S, W_{k(v)} × T)}

is a covering family of (TX̄, J_{ls}). Moreover the fiber product

Y(K, S, W_{k(v)} × T_{i}) ×_{Y(K,S,W_{k(v)}×T)} Y(K, S, W_{k(v)} × T_{j})

computed in the category TX̄, is isomorphic to Y(K, S, W_{k(v)} × T_{ij}), where T_{ij} denotes T_{i} ×_{T} T_{i}. It follows that the diagram of sets

F(Y(K, S, W_{k(v)} × T)) → ∏_{i} F(Y(K, S, W_{k(v)} × T_{i}))

⇉ ∏_{i,j} F(Y(K, S, W_{k(v)} × T_{ij}))

is exact. Passing to the inductive limit over K and S, and using left exactness of filtered inductive limits (i.e. using the fact that filtered inductive limits commute with finite products and equalizers), we obtain an exact diagram of sets

ip ; \overline{v}F(T) → ∏_{i} ip ; \overline{v}F(T_{i}) ⇉ ∏_{i,j} ip ; \overline{v}F(T_{ij}),

as it follows from (20). Hence ip ; \overline{v}F is a sheaf on (T op^{c}, J_{op}). Therefore, for any compact space T, one has

(21)  i∗ ; \overline{v}F(T) = ip ; \overline{v}F(T) = lim_{Z→i_{v}^{∗}Y(K,S,Z)}F(Y(K, S, Z))

where Z = W_{k(v)} × T.

Lemma 5. The family of functors

\{i∗ ; \overline{v} : X̄_{W} → T , v ∈ X̄^{0}\}

is conservative, where X̄^{0} is the set of closed points of X̄.

Proof. Let F be an object of X̄_{W}. We need to show that the adjunction map

(22)  F ⟶ ∏_{v∈X̄^{0}} i_{\overline{v}∗}i∗ ; \overline{v}F.

is injective. For any (Z_{0}, Z_{w}, f_{w}) of TX̄, we have

∏_{v∈X̄^{0}}(i_{\overline{v}∗}i∗ ; \overline{v}F)(Z_{0}, Z_{w}, f_{w}) = ∏_{v∈X̄^{0}} i∗ ; \overline{v}F(Z_{v}).

Note that, in the term on the right hand side of the equality above, Z_{v} is considered as a topological space without any action. For any v, we choose a local section cover of the space Z_{v}:

{T_{v,l} ↪ Z_{v}, l ∈ Λ_{v}}

such that T_{v,l} is a compact subspace of Z_{v} for any index l. Such a local section cover exists since Z_{v} is locally compact. The map

i∗ ; \overline{v}F(Z_{v}) ⟶ ∏_{l∈Λ_{v}} i∗ ; \overline{v}F(T_{v,l}).

is injective since i∗ ; \overline{v}F is a sheaf. It is therefore enough to show that the composite map

κ : F(Z_{0}, Z_{w}, f_{w}) ⟶ ∏_{v∈X̄^{0}} i∗ ; \overline{v}F(Z_{v}) ⟶ ∏_{v∈X̄^{0}, l∈Λ_{v}} i∗ ; \overline{v}F(T_{v,l})

is injective. Let α, β ∈ F(Z_{0}, Z_{w}, f_{w}) be two sections such that κ(α) = κ(β). For any pair (v, l), we consider

κ_{v,l} : F(Z_{0}, Z_{w}, f_{w}) ⟶ ∏_{v∈X̄^{0}, l∈Λ_{v}} i∗ ; \overline{v}F(T_{v,l}) ⟶ i∗ ; \overline{v}F(T_{v,l}).

For any (v, l), we have κ_{v,l}(α) = κ_{v,l}(β) and by (21)

i∗ ; \overline{v}F(T_{v,l}) = \underrightarrow{lim}F(Y(K, S, W_{k(v)} × T_{v,l}))

where the direct limit is taken over the category of arrows

W_{k(v)} × T_{v,l} ⟶ i_{v}^{∗}Y(K, S, W_{k(v)} × T_{v,l}).

The inclusion T_{v,l} ⊆ Z_{v} gives a W_{k(v)}-equivariant continuous map

W_{k(v)} × T_{v,l} ⟶ i_{v}^{∗}(Z_{0}, Z_{w}, f_{w}) = Z_{v}.

Thus for any pair (v, l), there is an object Y(K, S, W_{k(v)}×T_{v,l}) and a morphism

Y(K, S, W_{k(v)} × T_{v,l}) ⟶ (Z_{0}, Z_{w}, f_{w})

in the category TX̄ inducing the previous map

W_{k(v)} × T_{v,l} = i_{v}^{∗}Y(K, S, W_{k(v)} × T_{v,l}) ⟶ i_{v}^{∗}(Z_{0}, Z_{w}, f_{w}) = Z_{v}

and such that α_{|(v,l)} = β_{|(v,l)}, where α_{|(v,l)} (respectively β_{|(v,l)}) denotes the restriction of α (respectively of β) to Y(K, S, W_{k(v)} × T_{v,l}). We obtain a local section cover

{Y(K, S, W_{k(v)} × T_{v,l}) → (Z_{0}, Z_{w}, f_{w}), v ∈ X̄^{0}, l ∈ Λ_{v})}

in the site (TX̄, J_{ls}) such that α_{|(v,l)} = β_{|(v,l)} for any (v, l). It follows that α = β since F is a sheaf. Hence κ is injective and so is the adjunction map (22).

□

A morphism of topoi f is said to be surjective if its inverse image functor f^{∗} is faithful.

Corollary 8. The following morphism is surjective:

(i_{v})_{v∈X̄^{0}} : ∐_{v∈X̄^{0}} B_{W_{k(v)}} ⟶ X̄_{W} .

Proof. The morphism of topoi

(i\overline{v})_{v∈X̄^{0}} : ∐_{v∈X̄^{0}} T ⟶ X̄_{W}

is surjective since its inverse image is faithful by the previous result. But (i\overline{v})_{v∈X̄^{0}} factors through (i_{v})_{v∈X̄^{0}}, hence (i_{v})_{v∈X̄^{0}} is surjective as well. □

Proof of Theorem 5.1. Since the morphism i_{v} is an embedding, we have in fact two embeddings of topoi

B_{W_{k(v)}} ⟶ X̄_{W} ×X̄_{et} B_{G_{k(v)}}^{sm} ⟶ X̄_{W}

where the fiber product X̄_{W} ×X̄_{et} B_{G_{k(v)}}^{sm} is defined as the inverse image γ^{−1}(B_{G_{k(v)}}^{sm}) of the closed sub-topos B_{G_{k(v)}}^{sm} ↪ X̄_{et} under the morphism γ (see [20] IV. Corollaire 9.4.3). Therefore B_{W_{k(v)}} is equivalent to a full subcategory of X̄_{W} ×X̄_{et} B_{G_{k(v)}}^{sm}. This fiber product is the closed complement of the open subtopos Ȳ_{W} ↪ X̄_{W} where Ȳ := X̄ − v (see the next section for the definition of Ȳ_{W}). In other words, the strictly full subcategory X̄_{W} ×X̄_{et} B_{G_{k(v)}}^{sm} of X̄_{W} consists in objects G such that G × γ^{∗}Ȳ is the final object of Ȳ_{W}. It follows that

i_{v∗}F × γ^{∗}Ȳ

is the final object of Ȳ_{W}, for any object F of B_{W_{k(v)}}. We have to prove that B_{W_{k(v)}} is in fact equivalent to X̄_{W} ×X̄_{et} B_{G_{k(v)}}^{sm}. Let G be an object of this fiber product, i.e. an object of X̄_{W} such that G × γ^{∗}Ȳ is the final object. Consider the adjunction map

G ⟶ i_{v∗}i_{v}^{∗}G.

If w is a closed point of X̄ such that w ≠ v, then the morphism i_{w} factors through Ȳ_{W}:

i_{w} : B_{W_{k(w)}} ⟶ Ȳ_{W} ⟶ X̄_{W} .

We denote by i_{Ȳ ,w} : B_{W_{k(v)}} → Ȳ_{W} the induced map. Hence

i_{w}^{∗}G = i_{Ȳ ,w}^{∗}(G × Ȳ)

is the final object of B_{W_{k(w)}}, since G × Ȳ is the final object of Ȳ_{W} and i_{Ȳ ,w}^{∗} is left exact. On the other hand

i_{w}^{∗}i_{v∗}i_{v}^{∗}G = i_{Ȳ ,w}^{∗}(i_{v∗}i_{v}^{∗}G × Ȳ)

is the final object of B_{W_{k(w)}}, since i_{v∗}i_{v}^{∗}G × Ȳ is the final object of Ȳ_{W}. Hence the map

i_{w}^{∗}(G) ⟶ i_{w}^{∗}(i_{v∗}i_{v}^{∗}G)

is an isomorphism for any closed point w ≠ v of X̄. Suppose now that w = v. Then the map

i_{v}^{∗}(G) ⟶ i_{v}^{∗}(i_{v∗}i_{v}^{∗}G) = (i_{v}^{∗}i_{v∗})i_{v}^{∗}G = i_{v}^{∗}G

is an isomorphism by Lemma 4. Hence the morphism

i_{w}^{∗}(G) ⟶ i_{w}^{∗}(i_{v∗}i_{v}^{∗}G)

induced by the adjunction map G → i_{v∗}i_{v}^{∗}G is an isomorphism for any closed point w of X̄. Since the family of functors

{i_{w}^{∗} : X̄_{W} → B_{W_{k(w)}}, w ∈ X̄}

is conservative, the adjunction map G → i_{v∗}i_{v}^{∗}G is an isomorphism for any object G of γ^{−1}(B_{G_{k(v)}}^{sm}). Hence any object of γ^{−1}(B_{G_{k(v)}}^{sm}) is in the essential image of i_{v∗}. This shows that the morphism

B_{W_{k(v)}} ⟶ X̄_{W} ×X̄_{et} B_{G_{k(v)}}^{sm}

is an equivalence (this is a connected embedding). Theorem 5.1 follows. □

We consider the morphism

X̄_{W} = \overline{Spec(O_{F})}_{W} ⟶ \overline{Spec(ℤ)}_{W}

induced by the map \overline{Spec(O_{F})} → \overline{Spec(ℤ)}.

Proposition 5.5. The canonical morphism

δX̄ : X̄_{W} ⟶ X̄_{et} ×_{\overline{Spec(ℤ)}_{et}} \overline{Spec(ℤ)}_{W}

is an equivalence.

Proof. Let X̄′ be the open subscheme of X̄ consisting of the points of X̄ where the map X̄ → \overline{Spec(ℤ)} is étale. Let Y → X̄ be the complementary reduced closed subscheme.

(i) The morphism δX̄ is an equivalence over X̄′ and over Y. Recall from Definition 8 that X̄′_{W} is defined as an open subtopos of X̄_{W}. The canonical morphism

X̄′_{W} → X̄′_{et} ×_{\overline{Spec(ℤ)}_{et}} \overline{Spec(ℤ)}_{W}

is an equivalence. Indeed, the morphism X̄′ → \overline{Spec(ℤ)} is étale hence we have

X̄′_{et} ×_{\overline{Spec(ℤ)}_{et}} \overline{Spec(ℤ)}_{W} ≅ (\overline{Spec(ℤ)}_{et}/X̄′) ×_{\overline{Spec(ℤ)}_{et}} \overline{Spec(ℤ)}_{W}
≅ \overline{Spec(ℤ)}_{et} ×_{\overline{Spec(ℤ)}_{et}} (\overline{Spec(ℤ)}_{W} /γ^{∗}X̄′)
≅ \overline{Spec(ℤ)}_{W} /γ^{∗}X̄′.
≅ X̄′_{W}

Let Y′ be the image of Y in \overline{Spec(ℤ)}, such that Y′ ×\overline{Spec(ℤ)} Spec(ℤ) is given with a structure of reduced closed subscheme of Spec(ℤ). The morphism of étale topoi Y_{et} → \overline{Spec(ℤ)}_{et} factors through Y′_{et}. It follows from Theorem 5.1 that one has

Y_{et} ×_{\overline{Spec(ℤ)}_{et}} \overline{Spec(ℤ)}_{W} ≅ Y_{et} ×_{Y′_{et}} Y′_{et} ×_{\overline{Spec(ℤ)}_{et}} \overline{Spec(ℤ)}_{W}
≅ Y_{et} ×_{Y′_{et}} Y′_{W} .

We have the following equivalences Y_{et} ≅ ∐_{v∈Y} B_{G_{k(v)}}^{sm}, Y′_{et} ≅ ∐_{p∈Y′} B_{G_{k(p)}}^{sm} and Y′_{W} := ∐_{p∈Y′} B_{W_{k(p)}}. We obtain

Y_{et} ×_{\overline{Spec(ℤ)}_{et}} \overline{Spec(ℤ)}_{W} ≅ Y_{et} ×_{Y′_{et}} Y′_{W}

≅ ∐_{v∈Y} B_{G_{k(v)}}^{sm} ×_{(∐_{p∈Y ′} B_{G_{k(p)}}^{sm})} ∐_{p∈Y′} B_{W_{k(p)}}

≅ ∐_{v∈Y} (B_{G_{k(v)}}^{sm} ×_{B_{G_{k(p)}}^{sm}} B_{W_{k(p)}})

≅ ∐_{v∈Y} B_{W_{k(v)}} = Y_{W}

In view of the pull-back square (1), the last equivalence above follows from the fact that

B_{G_{k(v)}}^{sm} ≅ B_{G_{k(p)}}^{sm}/(G_{k(p)}/G_{k(v)}) ⟶ B_{G_{k(p)}}^{sm}

is a localization morphism. (ii) The natural transformation t between the glueing functors. The previous step (i) shows that there is an open-closed decomposition of topoi

j : X̄′_{W} → X̄_{et} ×_{\overline{Spec(ℤ)}_{et}} \overline{Spec(ℤ)}_{W} ← Y_{W} : i

By Theorem 5.1, we have another open-closed decomposition

j : X̄′_{W} → X̄_{W} ← Y_{W} : i

The glueing functors associated to these open-closed decompositions are given by i^{∗}j_{∗} and i^{∗}j_{∗}. The map X̄_{W} → X̄_{et} ×_{\overline{Spec(ℤ)}_{et}} \overline{Spec(ℤ)}_{W} induces a natural transformation

(23)  t : i^{∗}j_{∗} ⟶ i^{∗}j_{∗}.

Indeed, the following commutative diagram

(24)

X̄′_{W} ⟶^{j}  X̄_{W}  ⟵^{i} Y_{W}
↓^{Id}  ↓^{δX̄}  ↓^{Id}
X̄′_{W} ⟶^{j} X̄_{et} ×_{\overline{Spec(ℤ)}_{et}} \overline{Spec(ℤ)}_{W} ⟵^{i} Y_{W}

gives δX̄ ∘i = i and δX̄ ∘j = j. Then the natural transformation (23) is induced by the adjunction transformation δ∗ ; X̄δ_{X̄∗} → Id as follows :

i^{∗}j_{∗} ≅ i^{∗}δ∗ ; X̄δ_{X̄∗}j_{∗} ⟶ i^{∗}j_{∗}.

(iii) The glueing functors are naturally isomorphic. Since the disjoint sum topos Y_{W} = ∐_{v∈S} B_{W_{k(v)}} is given by the direct product of the categories B_{W_{k(v)}}, it is enough to show that the natural transformation

(25)  i_{v}^{∗}j_{∗} ⟶ i_{v}^{∗}j_{∗}

is an isomorphism for any v ∈ Y. Let F be an object of X̄′_{W}. The sheaf i_{v}^{∗} ∘ j_{∗}F (respectively i_{v}^{∗} ∘ j_{∗}F) is the sheaf on (W_{k(v)} × T op, J_{ls}) associated to the presheaf i_{v}^{p} ∘ j_{∗}F (respectively to the presheaf i_{v}^{p} ∘ j_{∗}F). Recall that W_{k(v)} × T op is a topologically generating subcategory of (B_{T op}W_{k(v)}, J_{ls}). It is therefore enough to show that the natural map

(26)  i_{v}^{p} ∘ j_{∗}F → i_{v}^{p} ∘ j_{∗}F,

of presheaves on W_{k(v)} × T op, is an isomorphism. On the one hand, for any object F of X̄′_{W} we have

(27) i_{v}^{p}j_{∗}F(W_{k(v)} × T) = lim_{W_{k(v)}×T →i_{v}^{∗}(Y_{0},Y_{w},f_{w})}F((Y_{0}, Y_{w}, f_{w}) × X̄′)

= \underrightarrow{lim}_{K/F,S}F(Y(K/F, S, W_{k(v)} × T) × X̄) (28)

where (28) is given by (19). See the proof of Lemma 4 for the definition of Y(K/F, S, W_{k(v)} × T). On the other hand, for any object F of X̄′_{W} we have

i_{v}^{p}j_{∗}F(W_{k(v)} × T) = \underrightarrow{lim} j_{∗}F((Z_{0}, Z_{w}, f_{w}) → V ← U)
= \underrightarrow{lim} F((Z_{0}, Z_{w}, f_{w}) ×_{V} U × X̄′)

where the direct limit is taken over the category of arrows

(29)  W_{k(v)} × T → i_{v}^{∗}((Z_{0}, Z_{w}, f_{w}) → V ← U) = Z_{p} ×_{V_{p}} U_{v}.

Here, ((Z_{0}, Z_{w}, f_{w}) → V ← U) is an object of the fiber product site CX̄, i.e. (Z_{0}, Z_{w}, f_{w}), V and U are objects of the sites T\overline{Spec(ℤ)}, Et\overline{Spec(ℤ)} and EtX̄ respectively. Then (Z_{0}, Z_{w}, f_{w}) ×_{V} U is seen as an object of TX̄. Finally, the place p is defined as the image of v ∈ X̄ in \overline{Spec(ℤ)}. We refer to [25] and section 7 for the definition of the site CX̄.

There is a natural functor from the category of arrows of the form (29) to the category of arrows (W_{k(v)} × T) → i_{v}^{∗}(Y_{0}, Y_{w}, f_{w}) sending ((Z_{0}, Z_{w}, f_{w}) → V ← V′) to (Z_{0}, Z_{w}, f_{w}) ×_{V} V′. This provides us with the natural map

(30)  i_{v}^{p}j_{∗}F(W_{k(v)} × T) ⟶ i_{v}^{p}j_{∗}F(W_{k(v)} × T).

In order to show that (30) is an isomorphism, we have to show that the system

W_{k(v)} × T ⟶ i_{v}^{∗}((Z_{0}, Z_{w}, f_{w}) ×_{V} U),

where (Z_{0}, Z_{w}, f_{w}) → V ← U) runs over the class of objects in CX̄, is cofinal in the category of arrows A_{v,T} :

W_{k(v)} × T → i_{v}^{∗}(Y_{0}, Y_{w}, f_{w}).

We know that the system given by the Y(K, S, W_{k(v)} × T)’s is cofinal in A_{v,T}. Here v ∈ S and K/F is unramified outside S. One can choose S large enough so that S contains Y. Let S′ be the image of S in \overline{Spec(ℤ)}. Then K/ℚ is unramified outside S′. If we denote by L/ℚ the Galois closure of K/ℚ (in the fixed algebraic closure \overline{ℚ}/ℚ), then L/ℚ remains unramified outside S′, and L/F is Galois and unramified outside S. Moreover, we have a morphism

Y(L/F, S, W_{k(v)} × T) → Y(K/F, S, W_{k(v)} × T) in A_{v,T} .

Hence one can restrict our attention to the objects of A_{v,T} of the form

W_{k(v)} × T → i_{v}^{∗}Y(K/F, S, W_{k(v)} × T)

where K/ℚ is a Galois extension unramified outside S′. We denote again by p the image of v in \overline{Spec(ℤ)} and we consider the object

Y(K/ℚ, S′, W_{k(p)} × T) → V ← U in CX̄,

where V and U are defined as follows. Using Proposition 5.4, the étale \overline{Spec(ℤ)}- scheme V is given by the G_{ℚ}-set G_{K/ℚ}/I_{p}, with no point over S′ − {p}, and exactly one point over the place p corresponding to the distinguished G_{ℚ_{p}}-orbit of G_{K/ℚ}/I_{p} on which the inertia group I_{p} acts trivially. The étale X̄-scheme U is given by the G_{F} -set G_{K/F} /I_{v}, with no point over S − {v}, and exactly one point over the place v corresponding to the distinguished G_{F_{v}}-orbit of G_{K/F} /I_{v} on which the inertia group I_{v} acts trivially. Finally, we enlarge S so that S is the inverse image of S′ (which is the image of S) along the map X̄ → \overline{Spec(ℤ)}. Then the map U → V is well defined. Assume that one has an identification

(31)  Y(K/F, S, W_{k(v)} × T) = Y(K/ℚ, S′, W_{k(p)} × T) ×_{V} U

in the category TX̄. It would follow that the system of objects

W_{k(v)} × T → i_{v}^{∗}((Z_{0}, Z_{w}, f_{w}) ×_{V} U)

is cofinal in the category A_{v,T}. The map (30) would be an isomorphism for any T and any F, hence (26) would be an isomorphism of presheaves for any F. This would show that the transformation (25) is an isomorphism. Hence the transformation (23) would be an isomorphism as well.

It is therefore enough to show (31). One has

Y(K/F, S, W_{k(v)} × T) = Y(K/F, S, W_{k(v)}) × (T, T, Id_{T})

and

Y(K/ℚ, S′, W_{k(p)} × T) = Y(K/ℚ, S′, W_{k(p)}) × (T, T, Id_{T})

in the category TX̄, hence one can assume that T = ∗ is the point. We have a map in TX̄

(32)  Y(K/F, S, W_{k(v)}) ⟶ Y(K/ℚ, S′, W_{k(p)}) ×_{V} U.

and we need to show that it is an isomorphism. Let w be a point of X̄. If w ∈ S and w ≠ v, then the w-component of both the right hand side and the left hand side in (32) are empty. Assume that w is not in S. Then the w-components of Y(K/F, S, W_{k(v)}), Y(K/ℚ, S′, W_{k(p)}), V and U are the W_{k(w)}- spaces W_{K/F,S}/W_{F_{v}}^{1}, W_{K/ℚ,S′}/W_{ℚ_{p}}^{1}, G_{K/ℚ}/I_{p} and G_{K/F} /I_{v} respectively. But we have an W_{k(w)}-equivariant homeomorphism

W_{K/F,S}/W_{F_{v}}^{1} ≅ (W_{K/ℚ,S′}/W_{ℚ_{p}}^{1}) ×_{(G_{K/ℚ}/I_{p})} (G_{K/F} /I_{v}).

Moreover, the v-component of Y(K/F, S, W_{k(v)}), Y(K/ℚ, S′, W_{k(p)}), V and U are the W_{k(v)}-spaces W_{k(v)}, W_{k(p)}, G_{k(p)}/G_{k(u)} and G_{k(v)}/G_{k(u)}, where u the unique point of U lying over v. But we have an W_{k(v)}-equivariant homeomorphism

W_{k(v)} ≅ W_{k(p)} ×_{(G_{k(p)}/G_{k(u)})} (G_{k(v)}/G_{k(u)}).

This shows that (32) is an isomorphism in TX̄, and (31) follows.

(v) The morphism δX̄ is an equivalence. We consider the glued topoi (Y_{W} , X̄′_{W} , i^{∗}j_{∗}) and (Y_{W} , X̄′_{W} , i^{∗}j_{∗}). Recall that an object of (Y_{W} , X̄′_{W} , i^{∗}j_{∗}) is a triple (E, F, σ) with E ∈ Y_{W}, F ∈ X̄′_{W} and σ : E → i^{∗}j_{∗}F (see [20] IV.9.5.3). There is a canonical functor

\begin{matrix}X̄_{W} & ⟶ & (Y_{W} , X̄′_{W} , i^{∗}j_{∗}) \\ F & ⟼ & (i^{∗}F, j^{∗}F, i^{∗}F → i^{∗}j_{∗}j^{∗}F) \end{matrix}

where i^{∗}F → i^{∗}j_{∗}j^{∗}F is given by adjunction. By ([20] IV Theorem 9.5.4), this functor is an equivalence, and the same is true for the canonical functor

X̄_{et} ×_{\overline{Spec(ℤ)}_{et}} \overline{Spec(ℤ)}_{W} ⟶ (Y_{W} , X̄′_{W} , i^{∗}j_{∗})

Under these identifications, the inverse image functor δ∗ ; X̄ is given by (see diagram (24))

\begin{aligned}δ∗ ; X̄ : & (Y_{W} , X̄′_{W} , i^{∗}j_{∗}) & ⟶ & (Y_{W} , X̄′_{W} , i^{∗}j_{∗}) \\ & (E, F, τ) & ⟼ & (E, F, t_{F} ∘ τ) \end{aligned}

Here t is the transformation defined in step (ii), and t_{F} ∘τ denotes the following composition :

t_{F} ∘ τ : E → i^{∗}j_{∗}F → i^{∗}j_{∗}F.

Since t is an isomorphism of functors, the inverse image functor δ∗ ; X̄ is an equivalence, hence so is the morphism δX̄. □

6. The definition of \overline{X}_{W}

6.1. Let X be a scheme separated and of finite type over Spec(ℤ). Recall the defining site Et\overline{X} of the Artin-Verdier étale topos \overline{X}_{et} from section 4. For any object \overline{U} of Et\overline{X} one has the induced topos

\overline{U}_{et} = \overline{X}_{et}/\overline{U} ≅ { \atop (Et\overline{X} /\overline{U}, J_{ind})}.

Definition 10. For any object \overline{U} of Et\overline{X} we define the Weil-étale topos of \overline{U} as the fiber product

\overline{U}_{W} := \overline{U}_{et} ×_{\overline{Spec(ℤ)}_{et}} \overline{Spec(ℤ)}_{W} .

This topos is defined by a universal property in the 2-category of topoi. As a consequence, it is well defined up to a canonical equivalence. We point out two special cases. If \overline{U} = (X , X_{∞}) = \overline{X} is the final object we obtain the definition of \overline{X}_{W} and if \overline{U} = (X , ∅) we obtain the definition of X_{W}. The topos X_{W} will play no role in this paper but \overline{X}_{W} is our central object of study in case X is proper and regular. Note also that for X = Spec(O_{F}) Definition 10 is consistent with Definition 7 by Proposition 5.5.

Proposition 6.1. The first projection yields a canonical morphism

γ\overline{X} : \overline{X}_{W} ⟶ \overline{X}_{et}.

Proposition 6.2. There is a canonical morphism

𝔣\overline{X} : \overline{X}_{W} ⟶ B_{ℝ}.

Proof. The morphism 𝔣\overline{X} is defined as the composition

\overline{X}_{W} ⟶ \overline{Spec(ℤ)}_{W} ⟶ B_{ℝ}

where the first arrow is the projection and the second is the morphism of Proposition 5.3. □

The structure of the topos \overline{X}_{W} over any closed point of \overline{Spec(ℤ)} is made explicit below. Note that X ⊗_{ℤ} 𝔽_{p} is not assumed to be regular.

Proposition 6.3. Let Spec(𝔽_{p}) be a closed point of Spec(ℤ). Then

\overline{X}_{W} ×_{\overline{Spec(ℤ)}_{et}} Spec(𝔽_{p})_{et} ≅ (X ⊗_{ℤ} 𝔽_{p})_{W} ≅ (X ⊗_{ℤ} 𝔽_{p})_{W}^{sm} × T

where (X ⊗_{ℤ} 𝔽_{p})_{W} denotes the big Weil-étale topos of X ⊗_{ℤ} 𝔽_{p}.

Proof. The result follows from the following equivalences.

\overline{X}_{W} ×_{\overline{Spec(ℤ)}_{et}} Spec(𝔽_{p})_{et} ≅ \overline{X}_{et} ×_{\overline{Spec(ℤ)}_{et}} \overline{Spec(ℤ)}_{W} ×_{\overline{Spec(ℤ)}_{et}} Spec(𝔽_{p})_{et}
≅ \overline{X}_{et} ×_{\overline{Spec(ℤ)}_{et}} B_{W_{𝔽p}}
≅ \overline{X}_{et} ×_{\overline{Spec(ℤ)}_{et}} Spec(𝔽_{p})_{et} ×_{Spec(𝔽_{p})_{et}} B_{W_{𝔽p}}
≅ (X ⊗_{ℤ} 𝔽_{p})_{et} ×_{B_{G_{𝔽p}}^{sm}} B_{W_{𝔽p}}
≅ (X ⊗_{ℤ} 𝔽_{p})_{W}

The second equivalence, the fourth and the last one are given by Theorem 5.1, Proposition 4.2 and Corollary 1 respectively. □

Corollary 9. The closed immersion of schemes (X ⊗_{ℤ} 𝔽_{p}) → X induces a closed embedding of topoi

(X ⊗_{ℤ} 𝔽_{p})_{W} ⟶ X_{W} .

We denote by ∞ the closed point of \overline{Spec(ℤ)} corresponding to the archimedean place of ℚ. This point yields a closed embedding of topoi

\underline{Set} = Sh(∞) ⟶ \overline{Spec(ℤ)}_{et}.

This paper suggests the following definition.

Definition 11. We define the Weil-étale topos of X_{∞} as follows:

X_{∞,W} := Sh(X_{∞}) × B_{ℝ}.

The argument of Proposition 6.3 is also valid for the archimedean fiber.

Proposition 6.4. We have a pull-back square of topoi:

(33)  \begin{matrix}X_{∞,W} & ⟶ & \underline{Set} \\ ↓^{i_{∞}} & & ↓ \\ \overline{X}_{W} & ⟶ & \overline{Spec(ℤ)}_{et} \end{matrix}

In particular i_{∞} is a closed embedding.

Proof. The result follows from the following equivalences.

\overline{X}_{W} ×_{\overline{Spec(ℤ)}_{et}} \underline{Set} ≅ \overline{X}_{et} ×_{\overline{Spec(ℤ)}_{et}} \overline{Spec(ℤ)}_{W} ×_{\overline{Spec(ℤ)}_{et}} \underline{Set}
≅ \overline{X}_{et} ×_{\overline{Spec(ℤ)}_{et}} B_{ℝ}
≅ \overline{X}_{et} ×_{\overline{Spec(ℤ)}_{et}} \underline{Set} ×\underline{Set} B_{ℝ}
≅ Sh(X_{∞}) ×\underline{Set} B_{ℝ}
≅ Sh(X_{∞}) × B_{ℝ}
= X_{∞,W}

Indeed, the second (respectively the fourth) equivalence above is given by Theorem 5.1 (respectively by Corollary 6). □

6.2. We assume here that X is irreducible and flat over Spec(ℤ). Let us study the structure of \overline{X}_{W} at the generic point of X. We denote by K(X) the function field of the irreducible scheme X. Let \overline{K(X)}/K(X) be an algebraic closure. The algebraic closure \overline{ℚ}/ℚ is taken as a sub-extension of \overline{K(X)}/ℚ. Then we have a continuous morphism G_{K(X)} → G_{ℚ}.

Definition 12. Let X be an irreducible scheme which is flat, separated and of finite type over Spec(ℤ). We consider the locally compact topological group

W_{K(X)} := G_{K(X)} ×_{G_{ℚ}} W_{ℚ}

defined as a fiber product in the category of topological groups.

If K(X)/ℚ is a number field, then W_{K(X)} = G_{K(X)} ×_{G_{ℚ}} W_{ℚ} coincides with the Weil group of K(X) defined in Sect. 5.

Proposition 6.5. Let X be an irreducible scheme which is flat, separated and of finite type over Spec(ℤ). There is a canonical morphism j\overline{X} : B_{W_{K(X)}} → \overline{X}_{W}.

Proof. The continuous morphism W_{K(X)} → G_{K(X)} induces a morphism

B_{W_{K(X)}} → B_{G_{K(X)}} → B_{G_{K(X)}}^{sm}.

Here the second map is the canonical morphism from the big classifying topos of G_{K(X)} to its small classifying topos, whose inverse image sends a continuous G_{K(X)}-set E to the sheaf represented by the discrete G_{K(X)}-space E (see [15] Section 7). The generic point of the irreducible scheme X and the previous choice of the algebraic closure \overline{K(X)}/K(X) yield an embedding B_{G_{K(X)}}^{sm} ↪ \overline{X}_{et}. We obtain a morphism

B_{W_{K(X)}} ⟶ \overline{X}_{et}.

On the other hand we have maps

B_{W_{K(X)}} ⟶ B_{W_{ℚ}} ⟶ \overline{Spec(ℤ)}_{W}

and a commutative diagram

B_{W_{K(X)}} ⟶ \overline{Spec(ℤ)}_{W}
↓  ↓
\overline{X}_{et} ⟶ \overline{Spec(ℤ)}_{et}

The result therefore follows from the very definition of \overline{X}_{W}. □

Unfortunately the morphism j\overline{X} is not an embedding. The structure of \overline{X}_{W} at the generic point is more subtle, as it is shown below. We assume again that X is irreducible, flat, separated and of finite type over Spec(ℤ). The generic point Spec(ℚ) → \overline{Spec(ℤ)} and \overline{ℚ}/ℚ induce an embedding

B_{G_{ℚ}}^{sm} ≅ Spec(ℚ)_{et} ↪ \overline{Spec(ℤ)}_{et}.

The corresponding subtopos of \overline{Spec(ℤ)}_{W} is the classifying topos of the topological pro-group (see Proposition 5.2)

\underline{W}_{K/ℚ,S} := \{W_{K/ℚ,S}, for \overline{ℚ}/K/ℚ finite Galois and S finite\}.

Recall that we have

B_{\underline{W}_{K/ℚ,S}} := \underleftarrow{lim} B_{W_{K/ℚ,S}}

where the projective limit is understood in the 2-category of topoi. In other words, there is a pull-back

B_{\underline{W}_{K/ℚ,S}} ⟶ B_{G_{ℚ}}^{sm}
↓^{i_{0}}  ↓^{u_{0}}
\overline{Spec(ℤ)}_{W} ⟶ \overline{Spec(ℤ)}_{et}

The generic point of the irreducible scheme X and an algebraic closure \overline{K(X)}/K(X) yield an embedding B_{G_{K(X)}}^{sm} ↪ \overline{X}_{et}. We obtain

\overline{X}_{W} ×_{\overline{X}_{et}} B_{G_{K(X)}}^{sm} = \overline{Spec(ℤ)}_{W} ×_{\overline{Spec(ℤ)}_{et}} \overline{X}_{et} ×_{\overline{X}_{et}} B_{G_{K(X)}}^{sm}
≅ \overline{Spec(ℤ)}_{W} ×_{\overline{Spec(ℤ)}_{et}} B_{G_{K(X)}}^{sm}
≅ \overline{Spec(ℤ)}_{W} ×_{\overline{Spec(ℤ)}_{et}} B_{G_{ℚ}}^{sm} ×_{B_{G_{ℚ}}^{sm}} B_{G_{K(X)}}^{sm}
≅ B_{\underline{W}_{K/ℚ,S}} ×_{B_{G_{ℚ}}^{sm}} B_{G_{K(X)}}^{sm}

The small classifying topos B_{G_{K(X)}}^{sm} is the projective limit \underleftarrow{lim} B_{G_{L/K(X)}}^{sm} where

L/K(X) runs over the finite Galois sub-extension of \overline{K(X)}/K(X). For such L we set L′ := L ∩ \overline{ℚ}. Then the same is true for B_{G_{ℚ}}^{sm}, i.e. we have B_{G_{ℚ}}^{sm} = \underleftarrow{lim} B_{G_{L′/ℚ}}^{sm}. Since projective limits commute between themselves, we have

B_{G_{K(X)}}^{sm} ×_{B_{G_{ℚ}}^{sm}} B_{\underline{W}_{K/ℚ,S}} = \underleftarrow{lim} B_{G_{L/K(X)}}^{sm} ×_{\underleftarrow{lim} B_{G_{L′/ℚ}}^{sm}} \underleftarrow{lim} B_{W_{L′/ℚ,S}}
= \underleftarrow{lim}_{L,S} (B_{G_{L/K(X)}}^{sm} ×_{B_{G_{L′/ℚ}}^{sm}} B_{W_{L′/ℚ,S}})

By Corollary 4, the fiber product B_{G_{L/K(X)}}^{sm} ×_{B_{G_{L′/ℚ}}^{sm}} B_{W_{L′/ℚ,S}} is equivalent to the classifying topos of the topological group G_{L/K(X)} ×_{G_{L′/ℚ}} W_{L′/ℚ,S} where the fiber product is in turn computed in the category of topological groups. Note that W_{L′/ℚ,S} → G_{L′/ℚ} has local sections since G_{L′/ℚ} is profinite (see [15] Proposition 2.1).

Definition 13. Let \overline{K(X)}/L/K(X) be a finite Galois sub-extension and let S be a finite set of places of ℚ containing all the places which ramify in L′ = L∩\overline{ℚ}. We consider the locally compact topological group

W_{L/K(X),S} := G_{L/K(X)} ×_{G_{L′/ℚ}} W_{L′/ℚ,S}

defined as a fiber product in the category of topological groups.

We have obtained the following result.

Proposition 6.6. Let X be an irreducible scheme which is flat, separated and of finite type over Spec(ℤ). We have a pull-back square of topoi

\underleftarrow{lim}_{L,S} B_{W_{L/K(X),S}} ⟶ B_{G_{K(X)}}^{sm}
↓  ↓
\overline{X}_{W}  ⟶ \overline{X}_{et}

where the vertical arrows are embedding.

7. Cohomology of \overline{X}_{W} with ℝ̃-coefficients

The fiber product topos \overline{X}_{W}, as defined in section 6, is equivalent to the category of sheaves on a site (C\overline{X} , J\overline{X}) lying in a non-commutative diagram of sites

(C\overline{X} , J\overline{X}) ⟵ (T\overline{Spec(ℤ)}, J_{ls})
↑  ↑^{γ^{∗}}
(Et\overline{X} , J_{et}) ⟵^{f^{∗}} (Et\overline{Spec(ℤ)}, J_{et})

The site (C\overline{X} , J\overline{X}) is defined as follows (see [25]). The category C\overline{X} is the category of pairs of morphisms U → V ← Z, where U is an object of Et\overline{X}, V is an object of Et\overline{Spec(ℤ)} and Z is an object of T\overline{Spec(ℤ)}. The map U → V (respectively Z → V) is understood as a morphism U → f^{∗}V in Et\overline{X} (respectively as a morphism Z → γ^{∗}V in T\overline{X}). The topology J\overline{X} is generated by the covering families

{(U_{i} → V_{i} ← Z_{i}) → (U → V ← Z), i ∈ I}

of the following types:

(a) U_{i} = U, V_{i} = V and {Z_{i} → Z} is a covering family.

(b) Z_{i} = Z, V_{i} = V and {U_{i} → U} is a covering family.

(c) {(U′ → V′ ← Z′) → (U → V ← Z)} with U′ = U, and Z′ → Z is obtained by base change from the map V′ → V of Et\overline{Spec(ℤ)}.

(d) {(U′ → V′ ← Z′) → (U → V ← Z)} with Z′ = Z, and U′ → U is obtained by base change from the map V′ → V of Et\overline{Spec(ℤ)}. Then (C\overline{X} , J\overline{X}) is a defining site for the fiber product topos \overline{X}_{W}. The topology J\overline{X} is not subcanonical.

Definition 14. For any T -topos t : E → T , we define the sheaf of continuous real valued functions on E as follows:

ℝ̃ := t^{∗}(yℝ)

Here yℝ is the abelian object of T represented by the standard topological group ℝ.

For an irreducible scheme X which is flat, separated and of finite type over Spec(ℤ), we consider the morphism j\overline{X} : B_{W_{K(X)}} → \overline{X}_{W} defined in Proposition 6.5.

Proposition 7.1. Let X be an irreducible scheme which is flat, separated and of finite type over Spec(ℤ). We have R^{n}j_{\overline{X} ,∗}ℝ̃ = 0 for any n ≥ 1.

Proof. Recall that the morphism j\overline{X} is defined by the following commutative diagram of topoi.

B_{W_{K(X)}} ⟶^{b} \overline{Spec(ℤ)}_{W}
a↓  ^{γ}↓
\overline{X}_{et}  ⟶^{f} \overline{Spec(ℤ)}_{et}

The site (B_{T op}W_{K(X)}, J_{ls}) is a defining site for B_{W_{K(X)}}, and we denote by a^{∗}, b^{∗}, γ^{∗} and f^{∗} the morphism of sites inducing the morphism of topos a, b, γ and f. The morphism j\overline{X} : B_{W_{K(X)}} → \overline{X}_{W} is induced by the morphism of sites :

\begin{matrix}C\overline{X} & ⟶ & B_{T op}W_{K(X)} \\ (U → V ← Z) & ⟼ & a^{∗}U ×_{a^{∗}f^{∗}V} b^{∗}Z \end{matrix}

Note that one has an identification a^{∗}U×_{a^{∗}f^{∗}V} b^{∗}Z = a^{∗}U×_{b^{∗}γ^{∗}V} b^{∗}Z. Consider the object of T\overline{Spec(ℤ)} whose components are all given by the action of W_{ℚ} on W_{ℚ}/W_{ℚ}^{1} ≅ ℝ:

(ℝ, ℝ, Id_{ℝ}) = 𝔣^{∗}Eℝ.

Recall that W_{ℚ}^{1} denotes the maximal compact subgroup of W_{ℚ}. Then 𝔣^{∗}Eℝ is a covering of the final object in T\overline{Spec(ℤ)} for the local section topology, hence

𝔣∗ ; \overline{X} Eℝ = (∗ → ∗ ← 𝔣^{∗}Eℝ) ⟶ (∗ → ∗ ← ∗)

is a covering of the final object of C\overline{X} for the topology J\overline{X}. The sheaf R^{n}j_{\overline{X} ,∗}ℝ̃ is the sheaf on (C\overline{X} , J\overline{X}) associated to the presheaf

\begin{aligned}P^{n}j_{\overline{X} ,∗}ℝ̃ : & C\overline{X} & ⟶ & Ab \\ & (U → V ← Z) & ⟼ & H^{n}(B_{W_{K(X)}}, a^{∗}U ×_{a^{∗}f^{∗}V} b^{∗}Z, ℝ̃) \end{aligned}

Since the object 𝔣∗ ; \overline{X} Eℝ defined above covers the final object of C\overline{X}, we can restrict our attention to the slice category C\overline{X} /𝔣∗ ; \overline{X} Eℝ. Let (U → V ← Z) be an object of C\overline{X} /𝔣∗ ; \overline{X} Eℝ, i.e. (U → V ← Z) is given with a map Z → 𝔣∗ ; \overline{X} Eℝ in T\overline{Spec(ℤ)}. We obtain a morphism

a^{∗}U ×_{a^{∗}f^{∗}V} b^{∗}Z ⟶ W_{K(X)}/W_{K(X)}^{1}

in the category B_{T op}W_{K(X)}, where the homogeneous space (W_{K(X)}/W_{K(X)}^{1}) ≅ ℝ is seen as an object of B_{T op}W_{K(X)}. Here W_{K(X)}^{1} denotes the maximal compact subgroup of W_{K(X)}. On the other hand the continuous morphism

W_{K(X)} ⟶ W_{K(X)}/W_{K(X)}^{1} = ℝ

has a global continuous section. This gives an isomorphism in T

y(W_{K(X)}/W_{K(X)}^{1}) = yW_{K(X)}/yW_{K(X)}^{1} and a canonical equivalence

B_{W_{K(X)}}/y(W_{K(X)}/W_{K(X)}^{1}) ≅ B_{W_{K(X)}}/(yW_{K(X)}/yW_{K(X)}^{1}) ≅ B_{W_{K(X)}^{1}}.

Under this equivalence the object represented by

α : a^{∗}U ×_{a^{∗}f^{∗}V} b^{∗}Z ⟶ (W_{K(X)}/W_{K(X)}^{1}) ≅ ℝ

corresponds to the object of B_{W_{K(X)}^{1}} represented by the subspace

α^{−1}(0) ⊂ a^{∗}U ×_{a^{∗}f^{∗}V} b^{∗}Z

endowed with the induced continuous action of W_{K(X)}^{1}. Thus one has

H^{n}(B_{W_{K(X)}}, a^{∗}U ×_{a^{∗}f^{∗}V} b^{∗}Z, ℝ̃) = H^{n}(B_{W_{K(X)}^{1}}, α^{−1}(0), ℝ̃).

Therefore it is enough to prove

(34)  H^{n}(B_{W_{K(X)}^{1}}, Z, ℝ̃) := H^{n}(B_{W_{K(X)}^{1}}/yZ, ℝ̃ × yZ) = 0,

for any object Z of B_{T op}W_{K(X)}^{1} and any n ≥ 1. We have two canonical equivalences

B_{W_{K(X)}^{1}}/EW_{K(X)}^{1} ≅ T and (B_{W_{K(X)}^{1}}/yZ)/(EW_{K(X)}^{1} × yZ) ≅ T /yZ.

We obtain a pull-back square

T /yZ ⟶^{l′} T
h′↓  ^{h}↓
B_{W_{K(X)}^{1}}/yZ ⟶^{l} B_{W_{K(X)}^{1}}

where all the maps are localization morphisms (local homeomorphisms of topoi in the modern language). It follows easily that this pull-back square satisfies the Beck-Chevalley condition

h^{∗}l_{∗} ≅ l′_{∗}h^{′∗}.

Moreover the functor h^{′∗}, being a localization functor, preserves injective abelian objects. We obtain

(35)  h^{∗}R^{n}(l_{∗})A ≅ R^{n}(l′_{∗})h^{′∗}A

for any abelian object A of B_{W_{K(X)}^{1}}/yZ and any n ≥ 0. The forgetful functor h^{∗} takes an object F of T endowed with an action of yW_{K(X)}^{1} to F. Hence R^{n}(l_{∗})A is the object R^{n}(l′_{∗})A endowed with the induced yW_{K(X)}^{1}-action.

Lemma 6. We have R^{n}(l′_{∗})(ℝ̃ × yZ) = 0 for any n ≥ 1.

Proof. We consider the morphism l′ : T /yZ → T . The sheaf R^{n}(l′_{∗})(ℝ̃ × yZ) on T = (T op^{lc}, J_{ls}) is the sheaf associated to the presheaf

\begin{aligned}P^{n}(l′_{∗})(ℝ̃ × yZ) : & T op^{lc} & ⟶ & Ab \\ & T & ⟼ & H^{n}(T /y(Z × T), ℝ̃ × y(Z × T)) \end{aligned}

It is enough to show that H^{n}(T /yT′, ℝ̃ × yT′) = 0 for any locally compact topological space T′ = Z × T. By ([20] IV.4.10.5) we have a canonical isomorphism

H^{n}(T /yT′, ℝ̃ × yT′) = H^{n}(T′, C^{0}(T′, ℝ))

where the right hand side is the usual sheaf cohomology of the paracompact space T′ with values in the sheaf C^{0}(T′, ℝ) of continuous real valued functions on T′. It is well known that the sheaf C^{0}(T′, ℝ) is fine, hence acyclic for the global section functor. The Lemma follows. □

Therefore the sheaf

h^{∗}R^{n}(l_{∗})(ℝ̃ × yZ) ≅ R^{n}(l′_{∗})h^{′∗}(ℝ̃ × yZ)

vanishes for any n ≥ 1, hence so does R^{n}(l_{∗})(ℝ̃ × yZ). The spectral sequence

H^{p}(B_{W_{K(X)}^{1}}, R^{q}(l_{∗})(ℝ̃ × yZ)) ⇒ H^{p+q}(B_{W_{K(X)}^{1}}/yZ, ℝ̃ × yZ)

degenerates and yields an isomorphism

H^{n}(B_{W_{K(X)}^{1}}, l_{∗}(ℝ̃ × yZ)) ≅ H^{n}(B_{W_{K(X)}^{1}}/yZ, ℝ̃ × yZ)

for any n ≥ 0. The sheaf l_{∗}(ℝ̃ × yZ) is given by the object l′_{∗}(ℝ̃ × yZ) of T endowed with the induced action of yW_{K(X)}^{1}, as it follows from (35). Furthermore, one has

l′_{∗}(ℝ̃ × yZ) = l′_{∗}l^{′∗}(ℝ̃) = \underline{Hom}_{T} (yZ, ℝ̃)

where the right hand side is the internal Hom-object in T (see [20] IV Corollaire 10.8). The sheaf \underline{Hom}_{T} (yZ, ℝ̃) is represented by the abelian topological group \underline{Hom}_{T op}(Z, ℝ) of continuous maps from Z to ℝ endowed with the compactopen topology, since Z is locally compact. The compact-open topology on \underline{Hom}_{T op}(Z, ℝ) is the topology of uniform convergence on compact sets, since ℝ is a metric space. The real vector space \underline{Hom}_{T op}(Z, ℝ) is locally convex, Hausdorff and complete (see [6] X.16. Corollaire 3). Note that the action of W_{K(X)}^{1} on \underline{Hom}_{T op}(Z, ℝ) is induced by the action on Z, and that the group W_{K(X)}^{1} is compact. By [15] Corollary 8, one has

H^{n}(B_{W_{K(X)}^{1}}, \underline{Hom}_{T op}(Z, ℝ)) = 0.

In summary, for any locally compact topological space Z with a continuous action of W_{K(X)}^{1} and any n ≥ 1, one has

H^{n}(B_{W_{K(X)}^{1}}/yZ, ℝ̃ × yZ) = H^{n}(B_{W_{K(X)}^{1}}, l_{∗}(ℝ̃ × yZ))

= H^{n}(B_{W_{K(X)}^{1}}, \underline{Hom}_{T op}(Z, ℝ))

= 0

Hence (34) holds and R^{n}(j_{\overline{X} ,∗})ℝ̃ = 0 for any n ≥ 1. □

Lemma 7. Let X be an irreducible scheme which is flat, separated and of finite type over Spec(ℤ). If X is normal, then the adjunction map

𝔣∗ ; \overline{X} ℝ̃ ⟶ j_{\overline{X} ∗} j∗ ; \overline{X} 𝔣∗ ; \overline{X} ℝ̃ ≅ j_{\overline{X} ∗} ℝ̃

is an isomorphism.

Proof. Firstly, we need to restrict the site C\overline{X}. The class of connected étale \overline{X} -schemes (respectively of connected étale \overline{Spec(ℤ)}-schemes) is a topologically generating family for the étale site of \overline{X} (respectively of \overline{Spec(ℤ)}). It follows easily that the subcategory C′ ; \overline{X} ⊂ C\overline{X}, consisting in objects (U → V ← Z) of C\overline{X} such that U and V are both connected, is a topologically generating family. Then we endow the full subcategory C′ ; \overline{X} with the induced topology via the natural fully faithful functor

C′ ; \overline{X} ↪ C\overline{X} .

Then C′ ; \overline{X} is a defining site for the topos \overline{X}_{W}. The composite map 𝔣\overline{X} ∘j\overline{X} : B_{W_{K(X)}} → \overline{X}_{W} → B_{ℝ} is induced by the morphism of topological groups W_{K(X)} → ℝ. The canonical isomorphism j∗ ; \overline{X} 𝔣∗ ; \overline{X} ℝ̃ ≅ ℝ̃ induces

j_{\overline{X} ∗} j∗ ; \overline{X} 𝔣∗ ; \overline{X} ℝ̃ ≅ j_{\overline{X} ∗} ℝ̃.

On the one hand 𝔣∗ ; \overline{X} ℝ̃ is the sheaf associated to the abelian presheaf

\begin{aligned}𝔣p ; \overline{X} ℝ̃ : & C′ ; \overline{X} & ⟶ & Ab \\ & (U → V ← Z) & ⟼ & Hom_{C′ ; \overline{X}} ((U → V ← Z), (∗ → ∗ ← 𝔣^{∗}ℝ̃)) \end{aligned}

where 𝔣^{∗}ℝ̃ denotes the object (ℝ, ℝ, Id) of T\overline{Spec(ℤ)} (with trivial action of the Weil groups on ℝ). For any object (U → V ← Z) of C′ ; \overline{X} with Z = (Z_{0}, Z_{v}, f_{v}), one has

𝔣p ; \overline{X} ℝ̃(U → V ← Z) = Hom_{C′ ; \overline{X}} ((U → V ← Z), (∗ → ∗ ← 𝔣^{∗}ℝ̃))
= Hom_{T\overline{Spec(ℤ)}}(Z, 𝔣^{∗}ℝ̃)
= Hom_{B_{T op}W_{ℚ}}(Z_{0}, ℝ)
= Hom_{T op}(Z_{0}/W_{ℚ}, ℝ).

One the other hand, the morphism j_{X} is induced by the continuous functor:

\begin{matrix}C′ ; \overline{X} & ⟶ & B_{T op}W_{K(X)} \\ (U → V ← Z) & ⟼ & a^{∗}U ×_{a^{∗}f^{∗}V} b^{∗}Z \end{matrix}

Hence the direct image j_{\overline{X} ∗} is given by

j_{\overline{X} ∗}ℝ̃(U → V ← Z) = Hom_{B_{T op}W_{K(X)}}(a^{∗}U ×_{a^{∗}f^{∗}V} b^{∗}Z, ℝ).

Here the topological group ℝ is given with the trivial action of W_{K(X)}. We set

U_{0} := a^{∗}U and V_{0} := a^{∗}f^{∗}V.

Note that U_{0} (respectively V_{0}) is given by the finite G_{K(X)}-set (respectively the finite G_{ℚ}-set) corresponding, via Galois theory, to the étale K(X)-scheme U ×_{X} Spec K(X) (respectively to the étale ℚ-scheme V ⊗ ℚ). Here, U_{0} (respectively V_{0}) is considered as a finite set on which W_{K(X)} acts via W_{K(X)} → G_{K(X)} (respectively via W_{K(X)} → G_{ℚ}). Finally, W_{K(X)} acts on the space Z_{0} := b^{∗}Z via W_{K(X)} → W_{ℚ}. Since W_{K(X)} acts trivially on ℝ, one has

j_{\overline{X} ∗}ℝ̃(U → V ← Z) = Hom_{B_{T op}W_{K(X)}}(a^{∗}U ×_{a^{∗}f^{∗}V} b^{∗}Z, ℝ)
= Hom_{T op}((U_{0} ×_{V_{0}} Z_{0})/W_{K(X)}, ℝ).

(i) The map 𝔣∗ ; \overline{X} ℝ̃ → j_{\overline{X} ∗} ℝ̃ is a monomorphism. The morphism 𝔣∗ ; \overline{X} ℝ̃ → j_{\overline{X} ∗} ℝ̃ is given by adjunction. It is induced by the morphism of presheaves on C′ ; \overline{X} given by the functorial map

(36)  Hom_{T op}(Z_{0}/W_{ℚ}, ℝ) ⟶ Hom_{T op}((U_{0} ×_{V_{0}} Z_{0})/W_{K(X)}, ℝ)

which is in turn induced by the continuous map

(37)  (U_{0} ×_{V_{0}} Z_{0})/W_{K(X)} ⟶ Z_{0}/W_{ℚ}.

Let (U → V ← Z) be an object of C′ ; \overline{X}. Hence U and V are both connected. Since U and V are both normal, the schemes U ×_{X} Spec(K(X)) and V ×\overline{Spec(ℤ)} Spec(ℚ) are connected as well. By Galois theory, the Galois groups G_{K(X)} and G_{ℚ} act transitively on U_{0} and V_{0} respectively. Hence the Weil groups W_{K(X)} and W_{ℚ} act transitively on U_{0} and V_{0} respectively. We have maps of compactified schemes

U → \overline{X} → \overline{Spec(ℤ)} and U → V → \overline{Spec(ℤ)}.

We consider the subfield L(U) of K(U) consisting in elements of K(U) that are algebraic over ℚ, i.e. we set

L(U) := K(U) ∩ \overline{ℚ}.

Note that U is normal and connected, hence irreducible, so that its function field K(U) is well defined. We consider the arithmetic curve \overline{Spec(O_{L(U)})}. Since U is normal, we have a canonical map

U ⟶ \overline{Spec(O_{L(U)})}.

We denote by U′ the (open) image of U in \overline{Spec(O_{L(U)})}. Then we have a factorization

U → U′ → V → \overline{Spec(ℤ)}

since V → \overline{Spec(ℤ)} is étale. The group W_{K(X)} acts transitively on U_{0}, hence the choice of a base point u_{0} ∈ U_{0} induces an isomorphism of W_{K(X)}-sets :

α : U_{0} ≅ W_{K(X)}/W_{K(U)} = G_{K(X)}/G_{K(U)}.

We fix such a base point u_{0} ∈ U_{0}. Then one has a 1-1 correspondence

\begin{matrix}(U_{0} × Z_{0})/W_{K(X)} & ⟶ & Z_{0}/W_{K(U)} = Z_{0}/W_{L(U)} \\ (u_{0}, z) & ⟼ & z \end{matrix}

where the equality Z_{0}/W_{K(U)} = Z_{0}/W_{L(U)} follows from the fact that the image of W_{K(U)} in W_{ℚ} is precisely W_{L(U)}. We consider now the commutative diagram of topological spaces

(U_{0} ×_{V_{0}} Z_{0})/W_{K(X)} ⟶^{(37)} Z_{0}/W_{ℚ}
i↓  ^{s}↑
(U_{0} × Z_{0})/W_{K(X)} ⟶_{α}^{≅} Z_{0}/W_{L(U)}

where i is injective and s is surjective. We denote by v_{0} the image of u_{0} ∈ U_{0} in V_{0} (note that U → V induces a map U_{0} → V_{0}). Let \overline{z} ∈ Z_{0}/W_{ℚ}. There exists w ∈ W_{ℚ} such that w.z goes to v_{0} under the W_{ℚ}-equivariant map Z_{0} → V_{0}, since W_{ℚ} acts transitively on V_{0}. Then one has \overline{w.z} = \overline{z}, (u_{0}, w.z) ∈ U_{0} ×_{V_{0}} Z_{0}, and

s ∘ α ∘ i\overline{(u_{0}, w.z)} = s ∘ α\overline{(u_{0}, w.z)} = s(\overline{w.z}) = \overline{z}

where \overline{∗} stands for the orbit of some point ∗ under some group action. Using the previous commutative diagram, this shows that the map (37) is surjective whenever U and V are both connected. Hence the map (36) is injective for any object (U → V ← Z) of C′ ; \overline{X}. In other words the morphism of presheaves on C′ ; \overline{X}

𝔣p ; \overline{X} ℝ̃ ⟶ j_{\overline{X} ∗} ℝ̃

is injective. Since the associated sheaf functor is exact, the morphism of sheaves

𝔣∗ ; \overline{X} ℝ̃ ⟶ j_{\overline{X} ∗} ℝ̃

is injective. (ii) The map 𝔣∗ ; \overline{X} ℝ̃ → j_{\overline{X} ∗} ℝ̃ is an epimorphism. One has

𝔣∗ ; \overline{X} ℝ̃(U → ∗ ← Z) = Hom_{\overline{X}_{W}} (εU × εZ; 𝔣∗ ; \overline{X} ℝ̃).

Here we denote by ∗ the final object Et\overline{Spec(ℤ)}. Moreover, we denote by εU (respectively by εZ) the sheaf on the topos \overline{X}_{W} associated to the presheaf represented by (U → ∗ ← ∗) (respectively by (∗ → ∗ ← Z)), where ∗ stands for the final object of the corresponding site. Finally, the product εU × εZ is computed in the topos \overline{X}_{W}. By adjunction, we have

𝔣∗ ; \overline{X} ℝ̃(U → ∗ ← Z) = Hom_{\overline{X}_{W}} (εU × εZ; 𝔣∗ ; \overline{X} ℝ̃)

= Hom_{\overline{X}_{W} /εU}(εU × εZ, εU × 𝔣∗ ; \overline{X} ℝ̃)

where we consider the slice topos \overline{X}_{W} /εU. On the other hand we have

\overline{X}_{W} /εU = (\overline{X}_{et} ×_{\overline{Spec(ℤ)}_{et}} \overline{Spec(ℤ)}_{W})/εU
≅ (\overline{X}_{et}/yU) ×_{\overline{Spec(ℤ)}_{et}} \overline{Spec(ℤ)}_{W}
≅ U_{et} ×_{\overline{Spec(ℤ)}_{et}} \overline{Spec(ℤ)}_{W}
=: U_{W}
≅ U_{et} ×_{U′_{et}} U′_{et} ×_{\overline{Spec(ℤ)}_{et}} \overline{Spec(ℤ)}_{W}
≅ U_{et} ×_{U′_{et}} U′_{W}

where U′ ⊆ \overline{Spec(O_{L(U)})} is defined as in the proof of step (i). The last equivalence above is given by Proposition 5.5. Hence the fiber product site C_{U} for U_{et} ×_{U′_{et}} U′_{W} given by the sites Et_{U}, Et_{U′} and T_{U′}, is a defining site for the topos U_{W}. Then the object εU × εZ of

\overline{X}_{W} /εU = U_{W} ≅ U_{et} ×_{U′_{et}} U′_{W}

is the sheaf associated to the presheaf on C_{U} represented by (∗ → ∗ ← Z) where Z = (Z_{0}, Z_{v}, f_{v}) is seen as an object of T_{U′} by restricting the group of operators on Z_{0} and Z_{v} for any place v of ℚ. Moreover, the object εU × 𝔣∗ ; \overline{X} ℝ̃ of \overline{X}_{W} /εU = U_{W} is precisely 𝔣_{U}^{∗} ℝ̃. It is the sheaf associated to the presheaf 𝔣_{U}^{p} ℝ̃ on C_{U} represented by (∗ → ∗ ← 𝔣_{U′}^{∗} ℝ̃). Therefore, we have

𝔣_{U}^{p} ℝ̃(∗ → ∗ ← Z) = Hom_{C\overline{X}} ((∗ → ∗ ← Z), (∗ → ∗ ← 𝔣_{U′}^{∗} ℝ̃))
= Hom_{T_{U′}} (Z, 𝔣_{U′}^{∗} ℝ̃)
= Hom_{B_{T op}W_{L(U)}}(Z_{0}, ℝ)
= Hom_{T op}(Z_{0}/W_{L(U)}, ℝ)

By the universal property of the associated sheaf functor, we obtain a map from

𝔣_{U}^{p} ℝ̃(∗ → ∗ ← Z) = Hom_{T op}(Z_{0}/W_{L(U)}, ℝ)

to the set

𝔣_{U}^{∗} ℝ̃(∗ → ∗ ← Z) = Hom_{U_{W}} (εU × εZ, 𝔣_{U}^{∗} ℝ̃)

= Hom_{\overline{X}_{W} /εU}(εU × εZ, εU × 𝔣∗ ; \overline{X} ℝ̃)

= Hom_{\overline{X}_{W}} (εU × εZ, 𝔣∗ ; \overline{X} ℝ̃)

= 𝔣∗ ; \overline{X} ℝ̃(U → ∗ ← Z)

Composing this map 𝔣_{U}^{p} ℝ̃(∗ → ∗ ← Z) → 𝔣∗ ; \overline{X} ℝ̃(U → ∗ ← Z) with

(38)  𝔣∗ ; \overline{X} ℝ̃(U → ∗ ← Z) ⟶ j_{\overline{X} ∗} ℝ̃(U → ∗ ← Z),

we obtain the natural bijective map from

𝔣_{U}^{p} ℝ̃(∗ → ∗ ← Z) = Hom_{T op}(Z_{0}/W_{L(U)}, ℝ) to the set

j_{\overline{X} ∗} ℝ̃(U → ∗ ← Z) = Hom_{T op}(Z_{0}/W_{L(U)}, ℝ).

It follows that the map (38) is surjective. It remains to show that the map

(39)  𝔣∗ ; \overline{X} ℝ̃(U → V ← Z) ⟶ j_{\overline{X} ∗} ℝ̃(U → V ← Z)

is surjective when V is not necessarily the final object. For any object (U → V ← Z) of C′ ; \overline{X}, we consider the following commutative diagram

𝔣∗ ; \overline{X} ℝ̃(U → ∗ ← Z) ⟶^{(38)} j_{\overline{X} ∗} ℝ̃(U → ∗ ← Z)
↓  ^{p}↓
𝔣∗ ; \overline{X} ℝ̃(U → V ← Z) ⟶^{(39)} j_{\overline{X} ∗} ℝ̃(U → V ← Z)

We have proven above that the map (38) is surjective. The vertical arrow p is the natural map from

j_{\overline{X} ∗} ℝ̃(U → ∗ ← Z) = Hom_{B_{T op}W_{K(X)}}(U_{0} × Z_{0}, ℝ)

to the set

j_{\overline{X} ∗} ℝ̃(U → V ← Z) = Hom_{B_{T op}W_{K(X)}}(U_{0} ×_{V_{0}} Z_{0}, ℝ),

which is surjective as well. Indeed, U_{0} ×_{V_{0}} Z_{0} is an open and closed W_{K(X)}- equivariant subspace of U_{0} × Z_{0}, hence any equivariant continuous map U_{0} ×_{V_{0}} Z_{0} → ℝ extends to an equivariant continuous map U_{0} × Z_{0} → ℝ. It follows immediately from the previous commutative diagram that the map (39) is surjective, for any object (U → V ← Z) of C′ ; \overline{X}. Therefore the morphism of sheaves 𝔣∗ ; \overline{X} ℝ̃ → j_{\overline{X} ∗} ℝ̃ is surjective. □

Theorem 7.1. Let X be an irreducible scheme which is flat, separated and of finite type over Spec(ℤ). If X is normal then the morphism

𝔣∗ ; \overline{X} : H^{n}(B_{ℝ}, ℝ̃) ⟶ H^{n}(\overline{X}_{W} , ℝ̃)

is an isomorphism for any n ≥ 0.

Proof. The Leray spectral sequence

H^{p}(\overline{X}_{W} , R^{q}j_{\overline{X} ∗}ℝ̃) ⇒ H^{p+q}(B_{W_{K(X)}}, ℝ̃)

degenerates by Proposition 7.1. This shows that the canonical morphism

H^{n}(\overline{X}_{W} , ℝ̃) ≅ H^{n}(\overline{X}_{W} , j_{\overline{X} ∗}ℝ̃) ⟶ H^{n}(B_{W_{K(X)}}, ℝ̃)

is an isomorphism, where the first identification is given by Lemma 7. These cohomology groups can be computed using the spectral sequence associated with the extension

1 → W_{K(X)}^{1} → W_{K(X)} → ℝ → 1.

Indeed, localizing along Eℝ, we obtain a pull-back square

(40)  \begin{matrix}B_{W_{K(X)}^{1}} & ⟶^{q′} & T ≅ B_{ℝ}/Eℝ \\ ^{p′}↓ & & ^{p}↓ \\ B_{W_{K(X)}} & ⟶^{q} & B_{ℝ} \end{matrix}

This gives an isomorphism

p^{∗}R^{n}(q_{∗}) ≅ R^{n}(q′_{∗})p^{′∗}

for any n ≥ 0. The argument of the proof of Proposition 7.1 shows that R^{n}(q′_{∗})ℝ̃ = 0 for any n ≥ 1. Hence R^{n}(q_{∗})ℝ̃ = 0 for any n ≥ 1. Moreover q is connected, i.e. q^{∗} is fully faithful, hence we have

q_{∗}ℝ̃ = q_{∗}q^{∗}ℝ̃ = ℝ̃.

Therefore, the Leray spectral sequence given by the morphism q

H^{i}(B_{ℝ}, R^{j}(q_{∗})ℝ̃) ⇒ H^{i+j}(B_{W_{K(X)}}, ℝ̃)

degenerates and yields

H^{n}(B_{ℝ}, ℝ̃) = H^{n}(B_{ℝ}, q_{∗}ℝ̃) ≅ H^{n}(B_{W_{K(X)}}, ℝ̃)

for any n ≥ 0. The result follows. □

8. Compact support Cohomology of X_{W} with ℝ̃-coefficients

Throughout this section, the arithmetic scheme X is supposed to be irreducible, normal, flat, and proper over Spec(ℤ).

8.1. The morphism γ\overline{X} : \overline{X}_{W} → \overline{X}_{et}. Recall the notion of étale \overline{X} -scheme defined in section 4. An étale \overline{X} -scheme is in particular a topological space so that it makes sense to speak of connected étale \overline{X} -schemes. Theorem 7.1 yields the following result.

Corollary 10. For any connected étale \overline{X} -scheme \overline{U}, the morphism

𝔣∗ ; \overline{U} : H^{n}(B_{ℝ}, ℝ̃) ⟶ H^{n}(\overline{U}_{W} , ℝ̃)

is an isomorphism for any n ≥ 0. In particular, one has H^{n}(\overline{U}_{W} , ℝ̃) = ℝ for n = 0, 1 and H^{n}(\overline{U}_{W} , ℝ̃) = 0 for n ≥ 2.

Proof. This is clear from the fact that an étale \overline{X} -scheme \overline{U} = (U, D) is connected if and only if the scheme U is connected, and Theorem 7.1 applies to U. □

Recall that γ\overline{X} : \overline{X}_{W} → \overline{X}_{et} is the projection induced by Definition 10.

Proposition 8.1. The sheaf R^{n}γ_{\overline{X} ∗}(ℝ̃) is the constant étale sheaf on \overline{X} associated to the discrete abelian group ℝ for n = 0, 1 and R^{n}γ_{\overline{X} ∗}(ℝ̃) = 0 for n ≥ 2.

Proof. For any n ≥ 0, R^{n}γ_{\overline{X} ∗}(ℝ̃) is the sheaf associated to the presheaf

\begin{matrix}Et\overline{X} & ⟶ & Ab \\ \overline{U} & ⟼ & H^{n}(\overline{X}_{W} /γ∗ ; \overline{X} \overline{U}, ℝ̃) \end{matrix}

Hence the result follows immediately from Corollary 10, since \overline{X}_{W} /γ∗ ; \overline{X} \overline{U} ≅ \overline{U}_{W}. □

8.2. The morphism γ_{∞} : X_{∞,W} → X_{∞}.

8.2.1. If T is a locally compact space (or any space), one can define the big topos T OP (T) of T as the category of sheaves on the site (T op/T, J_{op}) where J_{op} is the open cover topology. It is well known that the natural morphism T OP (T) → Sh(T) is a cohomological equivalence. The following lemma gives a slight generalization of this result.

Lemma 8. Let T be an object of T op. Let T /_{T} be the big topos of T and let Sh(T) be its small topos. For any topos S, the canonical morphism

f : T /_{T} × S ⟶ Sh(T) × S

has a section s such that s^{∗} ≅ f_{∗}.

Proof. We first observe that one has a canonical equivalence

T OP (T) := { \atop (T op/T, J_{op})} ≅ T /_{T} ,

where T OP (T) is the big topos of the topological space T. In what follows, we shall identify T /_{T} with T OP (T). The morphism f′ : T /_{T} → Sh(T) has a canonical section s′ : Sh(T) → T /_{T}, hence the map

f = (f′, Id_{S}) : T /_{T} × S ⟶ Sh(T) × S

has a section

s := (s′, Id_{S}) : Sh(T) × S ⟶ T /_{T} × S.

Moreover, we have s^{′∗} = f′_{∗} hence a sequence of three adjoint functors

f^{′∗}, f′_{∗} = s^{′∗}, s′_{∗}.

The functor f′_{∗} = s^{′∗} is called restriction and is denoted by Res. The functor f^{′∗} is called prolongement and is denoted by P rol. The category Op(∗) of open sets of the one point space is a defining site for the final topos \underline{Set}. The site (S, J_{can}) and (T /_{T} , J_{can}) can be seen as sites for the topoi S and T /_{T} respectively, where J_{can} denotes the canonical topology. Then the morphisms f′ and s′ are induced by the left exact continuous functors f^{′∗} and s^{′∗} respectively. A site for T /_{T} × S (respectively for Sh(T) × S) is given by the category C of objects of the form (F → ∗ ← S) (respectively by the category C of objects (F → ∗ ← S)). Here S is an object of S, F is an étalé space on T, F is a big sheaf on T and ∗ is the set with one element. The categories C and C both have an initial object (∅ → ∅ ← ∅). The morphism of topoi f is induced by the morphism of sites

\begin{aligned}f^{−1} : & C & ⟶ & C \\ & (F → ∗ ← S) & ⟼ & (P rol(F) → ∗ ← S) \end{aligned}

and the morphism s is induced by the morphism of sites

\begin{aligned}s^{−1} : & C & ⟶ & C \\ & (F → ∗ ← S) & ⟼ & (Res(F) → ∗ ← S) \end{aligned}

Let L be a sheaf of T /_{T} × S. Then one has

f_{∗}L(F → ∗ ← S) = L(P rol(F) → ∗ ← S)

for any object (F → ∗ ← S) of C. On the other hand, s^{∗}L is the sheaf associated with the presheaf

\begin{aligned}s^{p}L : & C & ⟶ & \underline{Set} \\ & (F → ∗ ← S) & ⟼ & \underrightarrow{lim} L(F → ∗ ← S′) \end{aligned}

where the direct limit is taken over the category of arrows

(F → ∗ ← S) ⟶ (Res(F) → ∗ ← S′).

But this category has an initial object given by (P rol(F) → ∗ ← S), since P rol is left adjoint to Res. We obtain

s^{−1}L(F → ∗ ← S) = L(P rol(F) → ∗ ← S).

Hence s^{−1}L is already a sheaf isomorphic to f_{∗}L. This identification is functorial hence gives an isomorphism of functors

f_{∗} ≅ s^{∗}.
□

Corollary 11. Let A be an abelian object of T /_{T} ×S and let A′ be an abelian object of Sh(T) × S. We have the following canonical isomorphisms:

H^{n}(T /_{T} × S, A) ≅ H^{n}(Sh(T) × S, f_{∗}A)
H^{n}(Sh(T) × S, A′) ≅ H^{n}(T /_{T} × S, f^{∗}A′).

Proof. By Lemma 8, the Leray spectral sequence associated with the morphism f : T /_{T} × S → Sh(T) × S degenerates since f_{∗} ≅ s^{∗} is exact. This yields the first isomorphism

H^{n}(T /_{T} × S, A) ≅ H^{n}(Sh(T) × S, f_{∗}A) = H^{n}(Sh(T) × S, s^{∗}A).

Applying this identification to the sheaf f^{∗}A′, we obtain

H^{n}(T /_{T} × S, f^{∗}A′) ≅ H^{n}(Sh(T) × S, A′)

Indeed, we have f ∘ s ≅ Id hence

f_{∗}f^{∗}A′ ≅ s^{∗}f^{∗}A′ ≅ (f ∘ s)^{∗}A′ ≅ A′.

□

Corollary 12. Let S be any topos. We denote by p_{1} : T × S → T and p_{2} : T × S → S the projections. For any abelian object A′ of T × S, one has

H^{n}(T × S, A′) ≅ H^{n}(S, p_{2∗}A′).

For any abelian object A of T , one has

H^{n}(T × S, p_{1}^{∗}A) ≅ H^{n}(S, A(∗)).

Proof. The topos T is the big topos of the one point space {∗} while Sh(∗) = \underline{Set} is the final topos. The map

f : T × S → Sh(∗) × S = \underline{Set} × S = S

is the second projection p_{2}. Hence one has

(41)  H^{n}(T × S, A′) ≅ H^{n}(S, p_{2∗}A′)

as it follows from Corollary 11. There is a pull-back square :

(42)  \begin{matrix}T × S & ⟶^{p_{2}} & S \\ ^{p_{1}}↓ & & ^{e_{S}} ↓ \\ T & ⟶^{e_{T}} & \underline{Set} \end{matrix}

The functor e_{T ∗} has a right adjoint, so that e_{T ∗} commutes with arbitrary inductive limits and in particular with filtered inductive limits. Hence the morphism e_{T} is tidy (see [26] C.3.4.2). It follows that the Beck-Chevalley natural transformation

e_{S}^{∗} ∘ e_{T ∗} ≅ p_{2∗} ∘ p_{1}^{∗}.

is an isomorphism (see [26] C.3.4.10). But the sheaf

p_{2∗}p_{1}^{∗}A = e_{S}^{∗}e_{T ∗}A

is the constant sheaf on S associated with the abelian group A(∗), since e_{T ∗} is the global section functor and e_{S}^{∗} is the constant sheaf functor. Applying (41) to the sheaf p_{1}^{∗}A, we obtain

H^{n}(T × S, p_{1}^{∗}A) ≅ H^{n}(S, p_{2∗}p_{1}^{∗}A) ≅ H^{n}(S, e_{S}^{∗}e_{T ∗}A) = H^{n}(S, A(∗))

for any n ≥ 0. □

Lemma 9. Let U be a contractible topological space and let

q : T × Sh(U) ⟶ T

be the first projection. Then one has

R^{n}(q_{∗})q^{∗}ℝ̃ = 0 for n ≥ 1.

Proof. The sheaf R^{n}(q_{∗})q^{∗}ℝ̃ is the sheaf associated to the presheaf

\begin{aligned}P^{n}(q_{∗})q^{∗}ℝ̃ : & T op & ⟶ & Ab \\ & K & ⟼ & H^{n}(T × Sh(U), q^{∗}yK, q^{∗}ℝ̃) \end{aligned}

Recall that T op denotes the category of locally compact topological spaces. The category T op^{c} of compact spaces is a topologically generating family of the site (T op, J_{ls}). It is therefore enough to show

(43) H^{n}(T × Sh(U), q^{∗}yK, q^{∗}ℝ̃) := H^{n}((T × Sh(U))/_{q^{∗}yK}, q^{∗}ℝ̃ × q^{∗}yK) = 0

for any compact space K and any n ≥ 1. We have immediately

(T × Sh(U))/_{q^{∗}yK} = T /yK × Sh(U).

We denote by q_{K} : T /yK × Sh(U) → T /yK → T the morphism obtained by projection and localization. Equivalently q_{K} is the composition

T /yK × Sh(U) ≅ (T × Sh(U))/_{q^{∗}yK} ⟶ (T × Sh(U)) ⟶ T .

We consider also the map

s : Sh(K) × Sh(U) ⟶ T /yK × Sh(U)

defined in Lemma 8. Then the following identifications

H^{n}(T × Sh(U), q^{∗}yK, ℝ̃) ≅ H^{n}(T /yK × Sh(U), q_{K}^{∗} ℝ̃)
≅ H^{n}(Sh(K) × Sh(U), s^{∗}q_{K}^{∗} ℝ̃)
≅ H^{n}(Sh(K × U), s̃^{∗}q_{K}^{∗} ℝ̃)

are induced by the following composite morphism of topoi

s̃: Sh(K × U) ⟶ Sh(K) × Sh(U) ⟶ T /yK × Sh(U).

Indeed, the first map Sh(K × U) → Sh(K) × Sh(U) is an equivalence since K is compact, and the second map induces an isomorphism on cohomology by Corollary 11. The commutative diagram

(44) \begin{matrix}Sh(K × U) & ⟶ & Sh(K) × Sh(U) & ⟶^{s} & T /yK × Sh(U) \\ ↓^{p_{1}} & & ↓ & & ↓^{q_{K}} \\ Sh(K) & ⟶ & T /_{K} & ⟶ & T \end{matrix}

shows that the sheaf s̃^{∗}q_{K}^{∗} ℝ̃ on the product space K × U is the inverse image of the sheaf C^{0}(K, ℝ), of continuous real functions on K, along the continuous projection p_{1} : K × U → K. In other words, one has

s̃^{∗}q_{K}^{∗} ℝ̃ = p_{1}^{∗}C^{0}(K, ℝ).

Consider the proper map

p_{2} : K × U ⟶ U.

By proper base change, the stalk of the sheaf R^{n}(p_{2∗})p_{1}^{∗}C^{0}(K, ℝ) on U at some point u ∈ U is given by

(R^{n}(p_{2∗})p_{1}^{∗}C^{0}(K, ℝ))_{u} = H^{n}(p_{2}^{−1}(u), p_{1}^{∗}C^{0}(K, ℝ) |_{p_{2}^{−1}(u)}) = H^{n}(K, C^{0}(K, ℝ)).

This group is trivial for any n ≥ 1. Indeed K is compact, in particular paracompact, hence C^{0}(K, ℝ) is fine on K. Thus we have

R^{n}(p_{2∗})p_{1}^{∗}C^{0}(K, ℝ) = 0 for any n ≥ 1.

Applying again proper base change to the proper map K → ∗, we see that p_{2∗}p_{1}^{∗}C^{0}(K, ℝ) is the constant sheaf on U associated with the discrete abelian group

C^{0}(K, ℝ) := H^{0}(K, C^{0}(K, ℝ)).

The Leray spectral sequence associated with the continuous map K × U → U therefore yields

H^{n}(Sh(K × U), p_{1}^{∗}C^{0}(K, ℝ)) ≅ H^{n}(U, p_{2∗}p_{1}^{∗}C^{0}(K, ℝ)) = H^{n}(U, C^{0}(K, ℝ))

for any n ≥ 0. But U is contractible hence H^{n}(U, C^{0}(K, ℝ)) = 0 for n ≥ 1, since sheaf cohomology with constant coefficients of locally contractible spaces coincides with singular cohomology, which is in turn homotopy invariant. We obtain

H^{n}(K × U, p_{1}^{∗}C^{0}(K, ℝ)) = H^{n}(U, C^{0}(K, ℝ)) = 0

for any n ≥ 1. The result follows since we have

H^{n}(T × Sh(U), q^{∗}yK, ℝ̃) ≅ H^{n}(Sh(K × U), s̃^{∗}q_{K}^{∗} ℝ̃)

≅ H^{n}(Sh(K × U), p_{1}^{∗}C^{0}(K, ℝ)) = 0

for any compact space K and any n ≥ 1. □

8.2.2. We still denote by X an irreducible normal scheme which is flat and proper over Spec(ℤ). Recall that X_{∞} is the topological space X^{an}/G_{ℝ}, and that the Weil-étale topos of X_{∞} is defined as follows (see Definition 11):

X_{∞,W} := B_{ℝ} × Sh(X_{∞})

Proposition 8.2. Consider the projection morphism

γ_{∞} : X_{∞,W} = B_{ℝ} × Sh(X_{∞}) ⟶ Sh(X_{∞}).

If ℝ denotes the constant sheaf on X_{∞} associated to the discrete abelian group ℝ we have

R^{n}γ_{∞∗}(ℝ̃) ≅ \begin{cases}ℝ & n = 0, 1 \\ 0 & n ≥ 2 \end{cases}

and

R^{n}γ_{∞∗}(ℤ) ≅ \begin{cases}ℤ & n = 0 \\ 0 & n ≥ 1. \end{cases}

Proof. The sheaf R^{n}γ_{∞∗}(ℝ̃) is the sheaf on the topological space X_{∞} associated to the presheaf

\begin{aligned}P^{n}(γ_{∞∗})(ℝ̃) : & Op(X_{∞}) & ⟶ & Ab \\ & U & ⟼ & H^{n}(B_{ℝ} × Sh(X_{∞}), U, ℝ̃) \end{aligned}

where Op(X_{∞}) is the category of open sets of X_{∞}. One has

H^{n}(B_{ℝ} × Sh(X_{∞}), U, ℝ̃) := H^{n}(B_{ℝ} × Sh(X_{∞})/U, ℝ̃) = H^{n}(B_{ℝ} × Sh(U), ℝ̃).

The family of contractible open subsets U ⊂ X_{∞} forms a topologically generating family of the site (Op(X_{∞}), J_{op}), since X_{∞} is locally contractible. It is therefore enough to compute the groups H^{n}(B_{ℝ} ×Sh(U), ℝ̃) for U contractible. For any contractible open subset U ⊂ X_{∞}, we consider the following pull-back square :

(45)  \begin{matrix}T × Sh(U) & ⟶^{q} & T \\ _{l′}↓ & & ^{l}↓ \\ B_{ℝ} × Sh(U) & ⟶^{p} & B_{ℝ} \end{matrix}

Here the vertical arrows l and l′ are both localization maps (recall that B_{ℝ}/Eℝ ≅ T ), while p and and q are the projections. This yields a canonical isomorphism

l^{∗}(Rp_{∗}) ≅ (Rq_{∗})l^{′∗}.

By Lemma 9, we obtain

l^{∗}(R^{n}p_{∗})ℝ̃ ≅ (R^{n}q_{∗})l^{′∗}ℝ̃ = (R^{n}q_{∗})q^{∗}ℝ̃ = 0

for any n ≥ 1. It follows immediately that (R^{n}p_{∗})ℝ̃ = 0 for n ≥ 1, since l^{∗} : B_{ℝ} → T is the forgetful functor (forget the yℝ-action). The contractible topological space U is connected and locally connected, hence so is the morphism of topoi Sh(U) → \underline{Set}. Since connected and locally connected morphisms are stable under base change (see [26] C.3.3.15), the first projection p : B_{ℝ} × Sh(U) → B_{ℝ} is also connected and locally connected. In particular, p^{∗} is fully faithful hence we have

p_{∗}ℝ̃ := p_{∗}p^{∗}ℝ̃ = ℝ̃

The Leray spectral sequence associated to the morphism p therefore yields

(46)  H^{n}(B_{ℝ} × Sh(U), p^{∗}ℝ̃) ≅ H^{n}(B_{ℝ}, p_{∗}p^{∗}ℝ̃) = H^{n}(B_{ℝ}, ℝ̃).

But one has H^{n}(B_{ℝ}, ℝ̃) = ℝ for n = 0, 1 and H^{n}(B_{ℝ}, ℝ̃) = 0 for n ≥ 2. Hence the sheaf R^{n}γ_{∞∗}(ℝ̃) is the constant sheaf on X_{∞} associated to the discrete abelian group ℝ for n = 0, 1 and R^{n}(γ_{∞∗})ℝ̃ = 0 for n ≥ 2. To compute R^{n}γ_{∞∗}(ℤ) recall that for any group object G in a topos E and any abelian G-object A there is a spectral sequence

H^{p}(H^{q}(E/G^{•}, A)) ⇒ H^{p+q}(B_{G}, A).

Applying this to G = ℝ in E = T × Sh(U) we note that the classifying topos of G is just B_{ℝ} × Sh(U) by [10]. Hence for A = ℤ we obtain a spectral sequence

H^{p}(H^{q}(T /ℝ^{•}×Sh(U), ℤ)) ≅ H^{p}(H^{q}(Sh(ℝ^{•}×U), ℤ)) ⇒ H^{p+q}(B_{ℝ}×Sh(U), ℤ)

where we have again used Corollary 11 and the fact that the spaces ℝ^{q} are locally compact. Now if U is contractible so is ℝ^{q} ×U and H^{q}(Sh(ℝ^{•}×U), ℤ) = ℤ (resp. 0) for q = 0 (resp. q > 0). The spectral sequence degenerates to an isomorphism

H^{p}(B_{ℝ} × Sh(U), ℤ) ≅ H^{p}(C(ℤ)) = \begin{cases}ℤ & p = 0 \\ 0 & p > 0 \end{cases}

where C(ℤ) is the complex associated to the constant simplicial abelian group ℤ which is quasi-isomorphic to ℤ[0]. The sheaf R^{n}γ_{∞∗}(ℤ) is associated to the presheaf U ↦ H^{p}(B_{ℝ} × Sh(U), ℤ) and hence takes the values in the statement of Proposition 8.2. □

By Proposition 8.2 the Leray spectral sequence for γ_{∞} induces a long exact sequence

⋯ → H^{i}(X_{∞}, ℝ) → H^{i}(X_{∞,W} , ℝ̃) → H^{i−1}(X_{∞}, ℝ) → ⋯

which decomposes into a collection of canonical isomorphisms

(47)  H^{i}(X_{∞,W} , ℝ̃) ≅ H^{i}(X_{∞}, ℝ) ⊕ H^{i−1}(X_{∞}, ℝ)

since γ_{∞} is canonically split by the morphism of topoi σ : Sh(X_{∞}) → B_{ℝ} × Sh(X_{∞}) which is the product with Sh(X_{∞}) of the canonical splitting \underline{Set} → T → B_{ℝ} of the canonical projection B_{ℝ} → T → \underline{Set}. Note here that σ^{∗} applied to the adjunction map ℝ = γ_{∞}^{∗}γ_{∞,∗}ℝ̃ → ℝ̃ is an isomorphism ℝ = σ^{∗}ℝ ≅ σ^{∗}ℝ̃ ≅ ℝ.

8.3. The fundamental class. The map 𝔣\overline{X} : \overline{X}_{W} → B_{ℝ} induces an isomorphism

𝔣∗ ; \overline{X} : Hom_{c}(ℝ, ℝ) = H^{1}(B_{ℝ}, ℝ̃) → H^{1}(\overline{X}_{W} , ℝ̃).

Definition 15. The fundamental class is defined as follows:

θ := 𝔣∗ ; \overline{X} (Id_{ℝ}) ∈ H^{1}(\overline{X}_{W} , ℝ̃).

We consider the sheaf ℝ̃ as a ring object on the topos \overline{X}_{W} and we denote by D^{+}(ℝ̃) the derived category of bounded below complexes of ℝ̃-modules on \overline{X}_{W}. For any such complex of ℝ̃-modules M one has (see [20] V.3.5)

H^{n}(\overline{X}_{W} , M) = Ext_{ℤ}^{n}(\overline{X}_{W} , ℤ, M) = Extn ; ℝ̃(\overline{X}_{W} , ℝ̃, M) = Hom_{D^{+}(ℝ̃)}(ℝ̃, M[n])

and we consider θ : ℝ̃ → ℝ̃[1] as a map in D^{+}(ℝ̃). Applying M ⊗L ; ℝ̃ − we get

∪θ : M = M ⊗L ; ℝ̃ ℝ̃ ⟶ M ⊗L ; ℝ̃ ℝ̃[1] = M[1].

Applying Rγ_{\overline{X} ,∗} we obtain

(48)  ∪θ : Rγ_{\overline{X} ,∗}(M) ⟶ Rγ_{\overline{X} ,∗}(M)[1].

Applying in turn RΓ_{\overline{X}_{et}} we get

(49)  ∪θ : RΓ_{\overline{X}_{W}} (M) ⟶ RΓ_{\overline{X}_{W}} (M)[1]

Similarly, for any \overline{U} étale over \overline{X} we have

∪θ\overline{U} : RΓ_{\overline{U}_{W}} (M) ⟶ RΓ_{\overline{U}_{W}} (M)[1]

Taking cohomology we get

(50)  ∪θ : R^{n}γ_{\overline{X} ,∗}(M) ⟶ R^{n+1}γ_{\overline{X} ,∗}(M),
(51)  ∪θ : H^{n}(\overline{X}_{W} , M) ⟶ H^{n+1}(\overline{X}_{W} , M)

and

(52)  ∪θ\overline{U} : H^{n}(\overline{U}_{W} , M) ⟶ H^{n+1}(\overline{U}_{W} , M).

Note that the collection of maps (52), when \overline{U} runs over the category of étale \overline{X} -schemes, gives a map of presheaves inducing the morphism of sheaves (50). Consider now the open-closed decomposition

φ : X_{et} ⟶ \overline{X}_{et} ⟵ Sh(X_{∞}) : u_{∞}

given by Corollary 4.1. The morphism γ\overline{X} : \overline{X}_{W} → \overline{X}_{et} gives pull-back squares

X_{W} ⟶^{γ_{X}} X_{et}
ϕ↓  ^{φ}↓
\overline{X}_{W} ⟶^{γ\overline{X}} \overline{X}_{et}

and

X_{∞,W} ⟶^{γ_{∞}} Sh(X_{∞})
i_{∞}↓  ^{u_{∞}}↓
\overline{X}_{W} ⟶^{γ\overline{X}}  \overline{X}_{et}

The second square is indeed a pull-back, as can be seen from the following commutative diagram:

X_{∞,W} ⟶^{γ_{∞}} Sh(X_{∞}) ⟶ Sh(∞) = \underline{Set}
i_{∞}↓  ^{u_{∞}}↓  ↓
\overline{X}_{W} ⟶^{γ\overline{X}}  \overline{X}_{et} ⟶ \overline{Spec(ℤ)}_{et}

The right hand side square and the total square are both pull-backs by Corollary 6 and Proposition 6.4 respectively. It follows that the left hand side square is a pull-back as well.

Theorem 8.1. There is an isomorphism R^{n}γ_{\overline{X} ∗}(ϕ_{!}ℝ̃) ≅ φ_{!}ℝ̃ for n = 0, 1, and R^{n}γ_{\overline{X} ∗}(ϕ_{!}ℝ̃) = 0 for n ≥ 2. Under these identifications, the morphism

∪θ : R^{0}γ_{\overline{X} ∗}(ϕ_{!}ℝ̃) ⟶ R^{1}γ_{\overline{X} ∗}(ϕ_{!}ℝ̃)

given by cup product with the fundamental class, is the identity of the sheaf φ_{!}ℝ̃.

Proof. We have an exact sequence of abelian sheaves on \overline{X}_{W}:

0 → ϕ_{!}ℝ̃ → ℝ̃ → i_{∞∗}ℝ̃ → 0

Applying the functor Rγ_{\overline{X} ∗}, we obtain an exact sequence of étale sheaves

0 → γ_{\overline{X} ∗}ϕ_{!}ℝ̃ → γ_{\overline{X} ∗}ℝ̃ → γ_{\overline{X} ∗}i_{∞∗}ℝ̃ →

R^{1}γ_{\overline{X} ∗}(ϕ_{!}ℝ̃) → R^{1}γ_{\overline{X} ∗}(ℝ̃) → R^{1}γ_{\overline{X} ∗}(i_{∞∗}ℝ̃) → ...

But we have canonical isomorphisms

(53) R^{n}γ_{\overline{X} ∗}(i_{∞∗}ℝ̃) ≅ R^{n}(γ_{\overline{X} ∗}i_{∞∗})ℝ̃ ≅ R^{n}(u_{∞∗}γ_{∞∗})ℝ̃ ≅ u_{∞∗}R^{n}γ_{∞∗}(ℝ̃)

for any n ≥ 0, since the direct image of a closed embedding of topoi is exact. Therefore, by Proposition 8.1 and Proposition 8.2, we obtain an exact sequence

0 → γ_{\overline{X} ∗}ϕ_{!}ℝ̃ → ℝ → u_{∞∗}ℝ → R^{1}γ_{\overline{X} ∗}(ϕ_{!}ℝ̃) → ℝ → u_{∞∗}ℝ → R^{2}γ_{\overline{X} ∗}(ϕ_{!}ℝ̃) → 0

and R^{n}γ_{\overline{X} ∗}(ϕ_{!}ℝ̃) = 0 for n ≥ 3. The map ℝ → u_{∞∗}ℝ is surjective since u_{∞} is a closed embedding. Hence we have an exact sequence

0 → R^{n}γ_{\overline{X} ∗}(ϕ_{!}ℝ̃) → ℝ → u_{∞∗}ℝ → 0

for n = 0, 1 and R^{n}γ_{\overline{X} ∗}(ϕ_{!}ℝ̃) = 0 for n ≥ 2. The first claim of the theorem follows. For any connected étale \overline{X} -scheme \overline{U}, we have a commutative square of ℝ-vector spaces

H^{0}(\overline{U}_{W} , ℝ̃) ⟶^{∪θ\overline{U}} H^{1}(\overline{U}_{W} , ℝ̃)
↑  ↑
H^{0}(B_{ℝ}, ℝ̃) = ℝ ⟶^{∪Id_{ℝ}} H^{1}(B_{ℝ}, ℝ̃) ≅ ℝ

where the vertical maps are isomorphisms by Corollary 10. The ℝ-linear map

(54)  ∪Id_{ℝ} : H^{0}(B_{ℝ}, ℝ̃) = ℝ ⟶ H^{1}(B_{ℝ}, ℝ̃) = Hom_{cont}(ℝ, ℝ)

sends 1 ∈ ℝ to Id_{ℝ}. Under the identification

H^{1}(B_{ℝ}, ℝ̃) = Hom_{cont}(ℝ, ℝ) ≅ ℝ

which maps f : ℝ → ℝ to f(1), the morphism (54) is the identity of ℝ. Hence the morphism

∪θ\overline{U} : H^{0}(\overline{U}_{W} , ℝ̃) = ℝ ⟶ H^{1}(\overline{U}_{W} , ℝ̃) ≅ ℝ

is just the identity, for any connected étale \overline{X} -scheme \overline{U}. It follows that the morphism of sheaves defined in (50)

(55)  ∪θ : R^{0}γ_{\overline{X} ∗}(ℝ̃) = ℝ̃ ⟶ R^{1}γ_{\overline{X} ∗}(ℝ̃) ≅ ℝ̃

is the identity of the sheaf ℝ̃. The same argument is valid for the sheaf i_{∞∗}ℝ̃. The composite morphism

p : X_{∞,W} = B_{ℝ} × Sh(X_{∞}) ⟶ \overline{X} ⟶ B_{ℝ}

is the first projection. We consider the fundamental class

θ_{∞} := p^{∗}(Id_{ℝ}) = i_{∞}^{∗}(θ) ∈ H^{1}(X_{∞,W} , ℝ̃).

Then the morphism

∪θ : R^{0}γ_{\overline{X} ∗}(i_{∞∗}ℝ̃) ⟶ R^{1}γ_{\overline{X} ∗}(i_{∞∗}ℝ̃)

coincides, via the canonical isomorphism (53), with the morphism u_{∞∗}R^{0}γ_{∞∗}(ℝ̃) → u_{∞∗}R^{1}γ_{∞∗}(ℝ̃) induced by

∪θ_{∞} : R^{0}γ_{∞∗}(ℝ̃) ⟶ R^{1}γ_{∞∗}(ℝ̃).

But for any contractible open subset U ⊂ X_{∞}, one has a commutative square

H^{0}(X_{∞,W} , U, ℝ̃) ⟶^{∪θ_{∞}} H^{1}(X_{∞,W} , U, ℝ̃)
↑  ↑
H^{0}(B_{ℝ}, ℝ̃) = ℝ ⟶^{∪Id_{ℝ}} H^{1}(B_{ℝ}, ℝ̃) ≅ ℝ

where all the maps are isomorphisms, as it follows from (46). Hence the map

∪θ_{∞} : ℝ̃ = γ_{∞∗}(ℝ̃) ⟶ R^{1}γ_{∞∗}(ℝ̃) ≅ ℝ̃

is the identity, and so is the morphism

(56)  ∪θ : R^{0}γ_{\overline{X} ∗}(i_{∞∗}ℝ̃) = u_{∞∗}ℝ̃ ⟶ R^{1}γ_{\overline{X} ∗}(i_{∞∗}ℝ̃) ≅ u_{∞∗}ℝ̃.

The morphism (50) is functorial hence ∪θ gives a morphism of exact sequences from

0 → R^{0}γ_{\overline{X} ∗}(ϕ_{!}ℝ̃) → R^{0}γ_{\overline{X} ∗}(ℝ̃) → R^{0}γ_{\overline{X} ∗}(i_{∞∗}ℝ̃) → 0

to

0 → R^{1}γ_{\overline{X} ∗}(ϕ_{!}ℝ̃) → R^{1}γ_{\overline{X} ∗}(ℝ̃) → R^{1}γ_{\overline{X} ∗}(i_{∞∗}ℝ̃) → 0

But the morphisms (55) and (56) are both given by the identity map, hence so is the morphism

∪θ : R^{0}γ_{\overline{X} ∗}(ϕ_{!}ℝ̃) = φ_{!}ℝ̃ ⟶ R^{1}γ_{\overline{X} ∗}(ϕ_{!}ℝ̃) ≅ φ_{!}ℝ̃.
□

Definition 16. For any abelian sheaf A on X_{W}, the compact support cohomology groups H_{c}^{i}(X_{W} , A) are defined as follows:

H_{c}^{i}(X_{W} , A) := H^{i}(\overline{X}_{W} , ϕ_{!}A)

Theorem 8.2. Assume that X is irreducible, normal, flat and proper over Spec(ℤ). The compact support cohomology groups H_{c}^{i}(X_{W} , ℝ̃) are finite dimensional vector spaces over ℝ, vanish for almost all i and satisfy

∑_{i∈ℤ}(−1)^{i} dim_{ℝ} H_{c}^{i}(X_{W} , ℝ̃) = 0.

Moreover, the complex of ℝ-vector spaces

⋯ ⟶^{∪θ} H_{c}^{i}(X_{W} , ℝ̃) ⟶^{∪θ} H_{c}^{i+1}(X_{W} , ℝ̃) ⟶^{∪θ} ⋯

is acyclic.

Proof. Consider the Leray spectral sequence

H^{p}(\overline{X}_{et}, R^{q}γ_{\overline{X} ∗}(ϕ_{!}ℝ̃)) ⟹ H^{p+q}(\overline{X}_{W} , ϕ_{!}ℝ̃)

given by the morphism γ\overline{X}. This spectral sequence yields

H^{0}(\overline{X}_{W} , ϕ_{!}ℝ̃) = H^{0}(\overline{X}_{et}, φ_{!}ℝ) = 0

and a long exact sequence

0 ⟶ H^{1}(\overline{X}_{et}, R^{0}γ_{\overline{X} ∗}(ϕ_{!}ℝ)) ⟶ H^{1}(\overline{X}_{W} , ϕ_{!}ℝ̃) ⟶ H^{0}(\overline{X}_{et}, R^{1}γ_{\overline{X} ∗}(ϕ_{!}ℝ))
↓^{∪θ}
... ⟶ H^{2}(\overline{X}_{et}, R^{0}γ_{\overline{X} ∗}(ϕ_{!}ℝ)) ⟶ H^{2}(\overline{X}_{W} , ϕ_{!}ℝ̃) ⟶ H^{1}(\overline{X}_{et}, R^{1}γ_{\overline{X} ∗}(ϕ_{!}ℝ))
↓^{∪θ}
... ⟶ H^{3}(\overline{X}_{et}, R^{0}γ_{\overline{X} ∗}(ϕ_{!}ℝ)) ⟶ H^{3}(\overline{X}_{W} , ϕ_{!}ℝ̃) ⟶ H^{2}(\overline{X}_{et}, R^{1}γ_{\overline{X} ∗}(ϕ_{!}ℝ))

... ⟶ H^{4}(\overline{X}_{et}, R^{0}(γ_{\overline{X} ∗})ϕ_{!}ℝ) ⟶  ...

Here the vertical maps ∪θ are given by cup product with the fundamental class. More precisely, the morphism (48)

Rγ_{\overline{X} ,∗}(ϕ_{!}ℝ) ⟶ Rγ_{\overline{X} ,∗}(ϕ_{!}ℝ)[1]

induces a morphism of spectral sequences. This morphism of spectral sequences induces in turn these vertical maps ∪θ. It follows that the composite map

(57) H^{i}(\overline{X}_{et}, R^{0}γ_{\overline{X} ∗}(ϕ_{!}ℝ)) → H^{i}(\overline{X}_{W} , ϕ_{!}ℝ̃) ⟶^{∪θ} H^{i+1}(\overline{X}_{W} , ϕ_{!}ℝ̃)

→ H^{i}(\overline{X}_{et}, R^{1}γ_{\overline{X} ∗}(ϕ_{!}ℝ))

is induced by the isomorphism of sheaves

R^{0}γ_{\overline{X} ∗}(ϕ_{!}ℝ̃) = φ_{!}ℝ ⟶^{∪θ} R^{1}γ_{\overline{X} ∗}(ϕ_{!}ℝ̃) ≅ φ_{!}ℝ.

Hence the map (57) is an isomorphism for any i ≥ 0, by Theorem 8.1. This yields a section to the map

H^{i+1}(\overline{X}_{W} , ϕ_{!}ℝ̃) ⟶ H^{i}(\overline{X}_{et}, R^{1}γ_{\overline{X} ∗}(ϕ_{!}ℝ)).

It follows that the long exact sequence above decomposes into a collection of canonical isomorphisms

(58)  H^{i}(\overline{X}_{W} , ϕ_{!}ℝ̃) ≅ H^{i}(\overline{X}_{et}, R^{0}γ_{\overline{X} ∗}(ϕ_{!}ℝ̃)) ⊕ H^{i−1}(\overline{X}_{et}, R^{1}γ_{\overline{X} ∗}(ϕ_{!}ℝ̃))

≅ H^{i}(\overline{X}_{et}, φ_{!}ℝ) ⊕ H^{i−1}(\overline{X}_{et}, φ_{!}ℝ) (59)

≅ H_{c}^{i}(X_{et}, ℝ) ⊕ H_{c}^{i−1}(X_{et}, ℝ) (60)

for any i ≥ 1. By Proposition 4.3, the ℝ-vector space H_{c}^{i}(X_{et}, ℝ) is finite dimensional and zero for i large. Hence we have

dim_{ℝ} H_{c}^{i}(X_{W} , ℝ̃) = dim_{ℝ} H_{c}^{i}(X_{et}, ℝ) + dim_{ℝ} H_{c}^{i−1}(X_{et}, ℝ)

and

∑_{i∈ℤ}(−1)^{i} dim_{ℝ} H_{c}^{i}(X_{W} , ℝ̃) = 0.

Under the identification (60), the morphism given by cup product with the fundamental class

H_{c}^{i}(X_{W} , ℝ̃) ⟶^{∪θ} H_{c}^{i+1}(X_{W} , ℝ̃)

is obtained by composing the projection with the inclusion as follows:

(61)  H_{c}^{i}(X_{W} , ℝ̃) ↠ H_{c}^{i}(X_{et}, ℝ) ↪ H_{c}^{i+1}(X_{W} , ℝ̃).

It follows immediately from (60) and (61) that the complex of ℝ-vector spaces

⋯ ⟶^{∪θ} H_{c}^{i}(X_{W} , ℝ̃) ⟶^{∪θ} H_{c}^{i+1}(X_{W} , ℝ̃) ⟶^{∪θ} ⋯

is acyclic. □

Proposition 8.3. Assume that X is irreducible, normal, flat and proper over Spec(ℤ). Then one has

∑_{i∈ℤ}(−1)^{i}i dim_{ℝ} H_{c}^{i}(X_{W} , ℝ̃) = ∑_{i∈ℤ}(−1)^{i+1} dim_{ℝ} H_{c}^{i}(X_{et}, ℝ)

= −1 + ∑_{i∈ℤ}(−1)^{i} dim_{ℝ} H^{i}(X_{∞}, ℝ)

= −1 + ∑_{i∈ℤ}(−1)^{i} dim_{ℝ} H^{i}(X^{an}, ℝ)^{+}

Proof. The first equality (respectively the second) follows from (60) (respectively from Proposition 4.3). To prove the third, we consider the morphism of topoi

(π^{∗}, π_{∗}^{G_{ℝ}}) : Sh(G_{ℝ}, X^{an}) → Sh(X_{∞})

given by the quotient map π : X^{an} → X^{an}/G_{ℝ}, where Sh(G_{ℝ}, X^{an}) is the topos of G_{ℝ}-equivariant sheaves on the space X^{an}. The constant sheaf ℝ on X^{an} is endowed with its G_{ℝ}-equivariant structure. For any n ≥ 1, the stalk of R^{n}(π_{∗}^{G_{ℝ}})ℝ at some fixed point x ∈ X (ℝ) ⊂ X^{∞} is the abelian group H^{n}(G_{ℝ}, ℝ), which is zero since ℝ is uniquely divisible. This gives

R^{n}(π_{∗}^{G_{ℝ}})ℝ = 0 for n ≥ 1

and a canonical isomorphism

H^{n}(X_{∞}, ℝ) ≅ H^{n}(Sh(G_{ℝ}, X^{an}), ℝ)

for any n ≥ 0. But the spectral sequence

H^{p}(G_{ℝ}, H^{q}(X^{an}, ℝ)) ⟹ H^{p+q}(Sh(G_{ℝ}, X^{an}), ℝ)

degenerates and gives an isomorphism

H^{n}(Sh(G_{ℝ}, X^{an}), ℝ) ≅ H^{0}(G_{ℝ}, H^{n}(X^{an}, ℝ)) =: H^{n}(X^{an}, ℝ)^{+}

for any n. The result follows. □

9. Relationship to the Zeta-function

9.1. Motivic L-functions. We first recall the expected properties of motivic L-functions [38]. For any smooth proper scheme X/ℚ of pure dimension d and 0 ≤ i ≤ 2d one defines the L-function

L(h^{i}(X), s) = ∏_{p} L_{p}(h^{i}(X), s)

as an Euler product over all primes p where

L_{p}(h^{i}(X), s) = P_{p}(h^{i}(X), p^{−s})^{−1}

and

P_{p}(h^{i}(X), T) = det_{ℚ_{l}}(1 − Frob_{p}^{−1} · T |H^{i}(X_{ℚ̄,et}, ℚ_{l})^{I_{p}})

is a polynomial (conjecturally) with rational coefficients independent of the prime l ≠ p. By [9] this product converges for ℜ(s) > \frac{i}{2} + 1. Set

Γ_{ℝ}(s) = π^{−s/2}Γ(\frac{s}{2}); Γ_{ℂ}(s) = 2(2π)^{−s}Γ(s)

and

L_{∞}(h^{i}(X), s) = ∏_{p<q} Γ_{ℂ}(s − p)^{h^{p,q}} · ∏_{p= \frac{i}{2}} Γ_{ℝ}(s − p)^{h^{p,+}}Γ_{ℝ}(s − p + 1)^{h^{p,−}}

where H^{i}(X(ℂ), ℂ) ≅ ⨁_{p+q=i} H^{p,q} is the Hodge decomposition,

h^{p,q} = dim_{ℂ} H^{p,q}; h^{p,±} = dim_{ℂ}(H^{p,p})^{F_{∞}=±(−1)^{p}}

and F_{∞} is the map induced by complex conjugation on the manifold X(ℂ). Here the product over p = \frac{i}{2} is understood to be empty for odd i. The completed L-function

Λ(h^{i}(X), s) = L_{∞}(h^{i}(X), s)L(h^{i}(X), s)

is expected to meromorphically continue to all s and satisfy a functional equation

(62)  Λ(h^{i}(X), s) = ϵ(h^{i}(X), s)Λ(h^{2d−i}(X), d + 1 − s).

Here ϵ(h^{i}(X), s) is the product of a constant and an exponential function in s, in particular nowhere vanishing.

Lemma 10. Assuming meromorphic continuation and the functional equation we have

ord_{s=0} L(h^{i}(X), s) = \begin{cases}−t + dim_{ℂ} H^{0}(X(ℂ), ℂ)^{F_{∞}=1} & i = 0 \\ dim_{ℂ} H^{i}(X(ℂ), ℂ)^{F_{∞}=1} & i > 0. \end{cases}

where t is the number of connected components of the scheme X.

Proof. For i > 0 the point d + 1 > \frac{2d−i}{2} + 1 lies in the region of absolute convergence of L(h^{2d−i}(X), s) so that L(h^{2d−i}(X), d + 1) ≠ 0. The Gammafunction has no zeros and has simple poles precisely at the non-positive integers. For p + q = 2d − i and p < q we have p < d − \frac{i}{2}, hence Γ_{ℂ}(d + 1 − p) ≠ 0. For p = d − \frac{i}{2} we likewise have Γ_{ℝ}(d + 1 − p) ≠ 0 and Γ_{ℝ}(d + 1 + 1 − p) ≠ 0. Hence L_{∞}(h^{2d−i}(X), d + 1) ≠ 0 and the functional equation shows Λ(h^{i}(X), 0) ≠ 0, i.e.

(63) ord_{s=0} L(h^{i}(X), s) = − ord_{s=0} L_{∞}(h^{i}(X), s)
= ∑_{p<q} h^{p,q} + ∑_{p= \frac{i}{2}} h^{p,±} = dim_{ℂ} H^{i}(X(ℂ), ℂ)^{F_{∞}=1}

where this last identity follows from F_{∞}(H^{p,q}) = H^{q,p} and the sign ± in h^{p,±} is the one for which ±(−1)^{p} = 1. Indeed, Γ_{ℝ}(s − p) (resp. Γ_{ℝ}(s − p + 1)) has a simple pole at s = 0 precisely for even (resp. odd) p. For i = 0 the function

L(h^{0}(X), s) = ζ_{K_{1}}(s) ⋯ ζ_{K_{t}}(s)

is a product of Dedekind Zeta-functions where H^{0}(X, O_{X}) = K_{1} × ⋯ × K_{t} is the ring of global regular functions on X and the K_{i} are number fields. It is classical that ord_{s=1} ζ_{K_{j}}(s) = −1 and therefore

ord_{s=0} Λ(h^{0}(X), s) = ord_{s=1} Λ(h^{0}(X), s) = ∑_{j=1}^{t} ord_{s=1} ζ_{K_{j}}(s) = −t.

Hence (63) holds for i = 0 with −t added to the right hand side.

□

9.2. Zeta-functions. For any separated scheme X of finite type over Spec(ℤ) one defines a Zeta-function

ζ(X , s) := ∏_{x∈X^{cl}} \frac{1}{1 − N(x)^{−s}} = ∏_{p} ζ(X_{𝔽_{p}}, s)

as an Euler product over all closed points. By Grothendieck’s formula [31][Thm. 13.1]

ζ(X_{𝔽_{p}}, s) = ∏_{i=0}^{2 dim(X_{𝔽p})} det_{ℚ_{l}}(1 − Frob_{p}^{−1} · p^{−s}|H_{c}^{i}(X_{𝔽̄_{p},et}, ℚ_{l}))^{(−1)^{i+1}}.

If X_{ℚ} → Spec(ℚ) is smooth and proper of relative dimension d, there will be an open subscheme U ⊆ Spec(ℤ) on which X_{U} → U is smooth and proper. By smooth and proper base change we have for p ∈ U

H_{c}^{i}(X_{𝔽̄_{p},et}, ℚ_{l}) ≅ H^{i}(X_{𝔽̄_{p},et}, ℚ_{l}) ≅ H^{i}(X_{ℚ̄,et}, ℚ_{l}) ≅ H^{i}(X_{ℚ̄,et}, ℚ_{l})^{I_{p}}

and therefore

(64)  ζ(X , s) = ∏_{p∉U} E_{p}(s) ∏_{i=0}^{2d} L(h^{i}(X_{ℚ}), s)^{(−1)^{i}}

where

E_{p}(s) = ∏_{i=0}^{∞} (\frac{det_{ℚ_{l}}(1 − Frob_{p}^{−1} · p^{−s}|H^{i}(X_{ℚ̄,et}, ℚ_{l})^{I_{p}})}{det_{ℚ_{l}}(1 − Frob_{p}^{−1} · p^{−s}|H_{c}^{i}(X_{𝔽̄_{p},et}, ℚ_{l}))})^{(−1)^{i}}

is a rational function in p^{−s}.

Theorem 9.1. Let X be a regular scheme, proper and flat over Spec(ℤ). Assume that the L-functions L(h^{i}(X_{ℚ}), s) can be meromorphically continued and satisfy the functional equation (62). Then

ord_{s=0} ζ(X , s) = ∑_{i∈ℤ}(−1)^{i} · i · dim_{ℝ} H_{c}^{i}(X_{W} , ℝ̃).

Proof. Note that regularity of X implies that X_{ℚ} → Spec(ℚ) is smooth. By Lemma 10 and Proposition 8.3 we have

ord_{s=0} ∏_{i∈ℤ} L(h^{i}(X_{ℚ}), s)^{(−1)^{i}} = − t + ∑_{i∈ℤ}(−1)^{i} dim_{ℂ} H^{i}(X_{ℚ}(ℂ), ℂ)^{F_{∞}=1}
= − t + ∑_{i∈ℤ}(−1)^{i} dim_{ℝ} H^{i}(X^{an}, ℝ)^{F_{∞}=1}
= ∑_{i∈ℤ}(−1)^{i} · i · dim_{ℝ} H_{c}^{i}(X_{W} , ℝ̃)

and in view of (64) it remains to show that ord_{s=0} E_{p}(s) = 0 for all p (or just p ∉ U). This follows from the fact that the Frob_{p}^{−1} eigenvalue 1 (of weight 0) has the same multiplicity on H_{c}^{i}(X_{𝔽̄_{p},et}, ℚ_{l}) = H^{i}(X_{𝔽̄_{p},et}, ℚ_{l}) and on H^{i}(X_{ℚ̄,et}, ℚ_{l})^{I_{p}} by part b) of Theorem 10.1 in the next section. □

Corollary 13. Let F be a totally real number field and X a proper, regular model of a Shimura curve over F, or of E × E × ⋯ × E where E is an elliptic curve over F. Then

ord_{s=0} ζ(X , s) = ∑_{i∈ℤ}(−1)^{i} · i · dim_{ℝ} H_{c}^{i}(X_{W} , ℝ̃).

Proof. For any Shimura curve X, by the now classical results of Eichler, Shimura, Deligne, Carayol and others, L(h^{1}(X), s) is a product of L-functions associated to weight 2 cusp forms for a suitable arithmetic subgroup of PSL_{2}(ℝ) associated to X, hence satisfies (62). It is moreover well known that any curve always has a proper regular model. By the Kuenneth formula we have

h^{i}(E^{d}) ≅ ⨁_{\begin{matrix}i_{0}+i_{1}+i_{2}=d \\ i_{1}+2i_{2}=i \end{matrix}}h^{0}(E)^{⊗i_{0}} ⊗h^{1}(E)^{⊗i_{1}} ⊗h^{2}(E)^{⊗i_{2}} ≅ ⨁_{\begin{matrix}i_{0}+i_{1}+i_{2}=d \\ i_{1}+2i_{2}=i \end{matrix}} h^{1}(E)^{⊗i_{1}}(−i_{2})

and each tensor power h^{1}(E)^{⊗i_{1}} is a direct sum of Tate twists of symmetric powers Sym^{k}h^{1}(E). But for elliptic curves E over totally real fields F the meromorphic continuation and functional equation of L(Sym^{k}h^{1}(E)/F, s) follows from recent deep results of Harris, Taylor, Shin et al (see [5][Cor. 8.8]). We remark that a proper regular model X of E^{d} certainly exists if E has semistable reduction at all primes since then the product singularities of E^{d}, where E is a proper regular model of E, can be resolved [37]. □

Theorem 9.2. Let X be a smooth proper variety over a finite field. Then a)-f) in the introduction hold for X.

Proof. This was proved for X_{W}^{sm} in [18][Thm. 9.1] since one clearly has

H^{i}(X_{W}^{sm}, ℤ) ⊗_{ℤ} ℝ ≅ H^{i}(X_{W}^{sm}, ℝ).

But in view of Corollary 12 (see also Corollary 2 and the remark after it) we have

H^{i}(X_{W} , ℤ) ≅ H^{i}(X_{W}^{sm}, ℤ); H^{i}(X_{W} , ℝ̃) ≅ H^{i}(X_{W}^{sm}, ℝ)

when X_{W} is defined by Definition 10. Note here that our fundamental class θ defined in Definition 15 is different from the class e ∈ H^{1}(X_{W} , ℝ̃) used in [18]. The class e lies in the image of H^{1}(X_{W} , ℤ) and is the pullback of the identity map in

H^{1}(Spec(𝔽_{p})_{W} , ℤ) = Hom_{ℤ}(W_{𝔽_{p}}, ℤ) ≅ Hom_{ℤ}(ℤ, ℤ).

Since the natural map W_{𝔽_{p}} → ℝ sends the Frobenius to log(p), the elements θ and e differ by a factor of log(p). This is consistent with the fact that

(65)  ζ^{∗}(X , 0) = log(p)^{r}Z^{∗}(X , 1)

where Z(X , T) ∈ ℚ(T) is the rational function so that ζ(X , s) = Z(X , p^{−s}) and

Z(X , T) = (1 − T)^{r}Z^{∗}(X , 1) + ...

with r ∈ ℤ and Z^{∗}(X , 1) ≠ 0, ∞. □

9.3. Remarks. We finish this section with some remarks to put our results in perspective.

9.3.1. Cohomology with ℤ-coefficients. If X → Spec(ℤ) is a (proper, flat, regular) arithmetic scheme with a section then RΓ(\overline{Spec(ℤ)}_{W} , ℤ) is a direct summand of RΓ(\overline{X}_{W} , ℤ). Hence by [14] H^{4}(X_{W} , ℤ) will not be a finitely generated abelian group and d) does not hold. Even if one could find a definition of \overline{Spec(ℤ)}_{W} with the expected ℤ-cohomology the definition of X_{W} as a fibre product (Definition 10) will not be the right one. Heuristically this is because one should view the fibre product of topoi as a ”homotopy pullback”, and the ”homotopy fibre” of γ : X_{W} → X_{et} is not independent of X, unlike in the situation over finite fields. Indeed, viewing Rγ_{∗}ℤ as the cohomology of the fibre, Geisser has shown [18] that this complex has cohomology ℤ, ℚ, 0 in degrees 0, 1, ≥ 2, respectively, for any X over Spec(𝔽_{p}). So for any X over Spec(𝔽_{p}) one can view the fibre as the pro-homotopy type of a solenoid.

For X = \overline{Spec(O_{F})} where F is a number field, one expects Rγ_{∗}ℤ to be concentrated in degrees 0 and 2 (see [34][Sec.9]). On the other hand, if

X = ℙ_{Spec(O_{F})}^{1}

has the correct ℤ-cohomology, compatible with the computations of ℝ̃- cohomology in this paper, then H^{4}(X̄_{W} , ℤ) must be a finitely generated group of rank r_{2}, the rank of K_{3}(O_{F}) (see j) in section 9.4.2 below). This can only happen if R^{i}γ_{∗}ℤ is nonzero for i = 3 or i = 4, the most likely scenario being that R^{4}γ_{∗}ℤ is nonzero with global sections H^{0}(X̄_{et}, R^{4}γ_{∗}ℤ) ≅ Hom_{ℤ}(K_{3}(O_{F}), ℚ). Again, this is only a heuristic argument since we have not rigorously defined the homotopy fibre, let alone established any relation between its ℤ-cohomology and Rγ_{∗}ℤ.

9.3.2. Weil-groups of finitely generated fields. The definition of the Weil-étale topos as a fibre product is closely related to the idea, briefly mentioned by Lichtenbaum in the introduction of [30], of defining the Weil-étale topos via Weil-groups for all scheme points x ∈ X, and then gluing into a global topos in the spirit of [30]. This is because the Weil-group of a field k(x) of finite transcendence degree over its prime subfield F would be defined as the fibre product G_{k(x)} ×_{G_{F}} W_{F} (as in Definition 12) and the classifying topos of this group is the fibre product of the classifying topoi of the factors by Corollary 4. The remarks in the previous paragraph would then apply to such a definition as well.

9.3.3. Properties a)-f) for \overline{X}. If X is regular, proper and flat over Spec(ℤ) with generic fibre X of dimension d it follows easily from our results that properties a)-c) hold for \overline{X} where of course

RΓ_{c}(\overline{X}_{W} , ℝ̃) = RΓ(\overline{X}_{W} , ℝ̃)

and

ζ(\overline{X} , s) = ζ(X , s) ∏_{i=0}^{2d} L_{∞}(h^{i}(X), s)^{(−1)^{i}}.

Property d) must also hold for any reasonable definition of RΓ(\overline{X}_{W} , ℤ) as will become clear from our discussion in section 9.4.2 below. This discussion will also show, however, that properties e) and f) will definitely not hold for any definition of RΓ(\overline{X}_{W} , ℤ). This is consistent with the fact that there are no special value conjectures for the completed L-functions Λ(h^{i}(X), s) in the literature.

9.3.4. Non-regular/non-proper schemes. For varieties over finite fields which are not smooth and proper the work of Geisser [19] shows that one has to replace the étale topology by the eh-topology (which allows abstract blow-ups as coverings) in order to define groups H_{c}^{i}(X_{W h}, ℤ) and H_{c}^{i}(X_{W h}, ℝ) which are independent of a choice of compactification of X and which satisfy a)-f) in the introduction (where the index W is replaced by W h). For arithmetic schemes over Spec(ℤ) a similar modification will be necessary, and one also has to assume some strong form of resolution of singularities for arithmetic schemes. We have refrained from trying to incorporate the idea of the eh-topology in this paper since our results (based on the fibre product definition of X_{W}) are only very partial in any case.

9.4. Relation to the Tamagawa number conjecture. In this section we establish the compatibility of the conjectural properties of Weil-étale cohomology, as outlined in the introduction and augmented with some further assumptions below, with the Tamagawa number conjecture of Bloch and Kato.

9.4.1. Statement of the Tamagawa number conjecture. Let X be a proper, flat, regular ℤ-scheme with generic fibre X of dimension d. The original Tamagawa number conjecture of Bloch and Kato [4] concerned the leading Taylor coefficient of L(h^{i}(X), s) at integers s ≥ \frac{i+1}{2} . This was then generalized by Fontaine and Perrin-Riou [16] to a conjecture about the vanishing order and leading coefficient at any integer s. In this paper we are only concerned with s = 0. One defines ”integral motivic cohomology” groups H_{M}^{p}(X_{/ℤ}, ℚ(q)) for example, as

H_{M}^{p}(X_{/ℤ}, ℚ(q)) := im(K_{2q−p}(X)_{ℚ}^{(q)} → K_{2q−p}(X)_{ℚ}^{(q)}),

with K_{j}(X)_{ℚ}^{(q)} the q-th Adams eigenspace of the algebraic K-groups K_{j}(X) ⊗_{ℤ} ℚ. Denote by W^{∗} = Hom_{ℚ}(W, ℚ) the dual ℚ-space and set W_{ℝ} := W ⊗_{ℚ} ℝ.

Conjecture 1. (Vanishing order) The space H_{M}^{2d−i+1}(X_{/ℤ}, ℚ(d + 1)) is finite dimensional and

ord_{s=0} L(h^{i}(X), s)= dim_{ℚ} H_{f}^{1}(h^{i}(X)^{∗}(1))^{∗}− dim_{ℚ} H_{f}^{0}(h^{i}(X)^{∗}(1))^{∗}
= dim_{ℚ} H_{f}^{1}(h^{2d−i}(X)(d + 1))^{∗}− dim_{ℚ} H_{f}^{0}(h^{2d−i}(X)(d + 1))^{∗}
= dim_{ℚ} H_{M}^{2d−i+1}(X_{/ℤ}, ℚ(d + 1))^{∗}

Here we refer to [13] for the notation H_{f}^{i}(M) (not needed in the sequel). Let H_{D}^{p}(X_{/ℝ}, ℝ(q)) denote (real) Deligne cohomology and let

(66)  ρ_{∞}^{i} : H_{M}^{2d−i+1}(X_{/ℤ}, ℚ(d + 1))_{ℝ} →H_{D}^{2d−i+1}(X_{/ℝ}, ℝ(d + 1))

be the Beilinson regulator.

Conjecture 2. (Beilinson) The map ρ_{∞}^{i} is an isomorphism for i ≥ 1 and there is an exact sequence

(67) 0 → H_{M}^{2d+1}(X_{/ℤ}, ℚ(d+1))_{ℝ} ⟶^{ρ_{∞}^{0}} H_{D}^{2d+1}(X_{/ℝ}, ℝ(d+1)) → CH^{0}(X)_{ℝ}^{∗} → 0

for i = 0.

We remark that Deligne cohomology satisfies a duality

(68)  H_{D}^{2d−i+1}(X_{/ℝ}, ℝ(d + 1))^{∗} ≅ H_{D}^{i}(X_{/ℝ}, ℝ) = H^{i}(X(ℂ), ℝ)^{+}

for i ≥ 0 and deduce the well known fact that the vanishing order of L(h^{i}(X), s) predicted by Conjectures 1 and 2 is in accordance with Lemma 10. Another consequence of conjecture 2 is

(69)  H_{M}^{2d−i+1}(X_{/ℤ}, ℚ(d + 1)) = 0

for i ≥ 2d + 1, a particular case of the Beilinson-Soule conjecture. Define the fundamental line

Δ_{f}(h^{i}(X)) = det_{ℚ}^{−1}(H^{i}(X(ℂ), ℚ)^{+}) ⊗_{ℚ} det_{ℚ}H_{M}^{2d−i+1}(X_{/ℤ}, ℚ(d + 1))^{∗}

for i > 0 and

Δ_{f}(h^{0}(X))

= det_{ℚ}CH^{0}(X)_{ℚ} ⊗_{ℚ} det_{ℚ}^{−1}(H^{0}(X(ℂ), ℚ)^{+}) ⊗_{ℚ} det_{ℚ}H_{M}^{2d+1}(X_{/ℤ}, ℚ(d + 1))^{∗}

for i = 0. There is an isomorphism

ϑ_{∞}^{i} : ℝ ≅ Δ_{f}(h^{i}(X))_{ℝ}

induced by (68) and the dual of (66) (resp. (67)) for i > 0 (resp. i = 0). Now fix a prime number l and let U ⊆ Spec(ℤ) an open subscheme on which l is invertible. For any smooth l-adic sheaf V on U and prime p ≠ l define a complex concentrated in degrees 0 and 1

RΓ_{f}(ℚ_{p}, V) = RΓ(𝔽_{p}, i_{p}^{∗}j_{p,∗}V) = V^{I_{p}} ⟶^{1−Frob_{p}^{−1}} V^{I_{p}}

where I_{p} is the inertia subgroup at p and i_{p} : Spec(𝔽_{p}) → Spec(ℤ) and j_{p} : U → Spec(ℤ) are the natural immersions. For p = l define

RΓ_{f}(ℚ_{p}, V) = D_{cris}(V) ⟶^{(1−ϕ,ι)} D_{cris}(V) ⊕ D_{dR}(V)/F^{0}D_{dR}(V)

where D_{cris} and D_{dR} are Fontaine’s functors [16]. In both cases there is a map of complexes

RΓ_{f}(ℚ_{p}, V) → RΓ(ℚ_{p}, V)

and one defines RΓ_{/f}(ℚ_{p}, V) as the mapping cone. The next Lemma shows that the complex RΓ_{f}(ℚ_{p}, V) has a uniform description for p = l and p ≠ l in the case that interests us.

Lemma 11. Let V be finite dimensional ℚ_{p}-vector space with a continuous G_{p} := Gal(ℚ̄_{p}/ℚ_{p})-action and such that D_{dR}(V)/F^{0}D_{dR}(V) = 0. Then there is a commutative diagram in the derived category of ℚ_{p}-vector spaces

RΓ_{f}(ℚ_{p}, V) ⟶ RΓ(ℚ_{p}, V)
κ↓  ∥
RΓ(𝔽_{p}, V^{I_{p}}) ⟶ RΓ(ℚ_{p}, V)

where κ is a quasi-isomorphism.

Proof. For a profinite group G and continuous G-module M we denote by C^{∗}(G, M) the standard complex of continuous cochains. There is an exact sequence of continuous G_{p}-modules

0 → V → B^{0}(V) ⟶^{d^{0}} B^{1}(V) → 0

where

B^{0}(V) = B_{cris} ⊗_{ℚ_{p}} V

(with diagonal G_{p}-action),

B^{1}(V) = B_{cris} ⊗_{ℚ_{p}} V ⊕ (B_{dR}/F^{0}B_{dR}) ⊗_{ℚ_{p}} V

and d^{0}(x) = ((1 − ϕ)(x), ι(x)) where ι is induced by the canonical inclusion B_{cris} → B_{dR} (see [16] for more on Fontaine’s rings B_{cris} and B_{dR}). Viewing this sequence as a quasi-isomorphism between V and a two term complex we obtain a quasi-isomorphism

RΓ(ℚ_{p}, V) = RΓ(G_{p}, V) = C^{∗}(G_{p}, V)
≅ Tot (C^{∗}(G_{p}, B^{0}(V)) ⟶^{d^{0∗}} C^{∗}(G_{p}, B^{1}(V)))

where Tot denotes the simple complex associated to a double complex. By definition RΓ_{f}(ℚ_{p}, V) is the subcomplex

H^{0}(G_{p}, B^{0}(V)) ⟶^{d^{0}} H^{0}(G_{p}, B^{1}(V))

of this double complex, i.e. the complex

D_{cris}(V) ⟶^{d^{0}} D_{cris}(V) ⊕ D_{dR}(V)/F^{0}D_{dR}(V).

For any continuous G_{p}-module M there is moreover a quasi-isomorphism

RΓ(G_{p}, M) ≅ RΓ(𝔽_{p}, RΓ(I_{p}, M)) ≅ Tot (C^{∗}(I_{p}, M) ⟶^{1−Frob_{p}^{−1}} C^{∗}(I_{p}, M))

where Frob_{p} ∈ G_{p} is any lift of the Frobenius automorphism in G_{p}/I_{p}, acting simultaneously on I_{p} (by conjugation) and on M. The complex RΓ(𝔽_{p}, H^{0}(I_{p}, M)) is the subcomplex

H^{0}(I_{p}, M) ⟶^{1−Frob_{p}^{−1}} H^{0}(I_{p}, M)

of this double complex. Combining these two constructions, we deduce that RΓ(ℚ_{p}, V) is canonically isomorphic to the total complex of the triple complex

(70)

C^{∗}(I_{p}, B^{0}(V)) ⟶^{d^{0∗}} C^{∗}(I_{p}, B^{1}(V))
1−Frob_{p}^{−1}↓  ^{1−Frob_{p}^{−1}}↓
C^{∗}(I_{p}, B^{0}(V)) ⟶^{d^{0∗}} C^{∗}(I_{p}, B^{1}(V)).

We have an isomorphism

ℚ̂_{p}^{ur} ⊗_{ℚ_{p}} D_{cris}(V) ≅ H^{0}(I_{p}, B^{0}(V)) where ℚ̂_{p}^{ur} is the p-adic completion of the maximal unramified extension of ℚ_{p}. The map ϕ on this space (induced by ϕ on B_{cris}) is semilinear with respect to the automorphism ϕ of ℚ̂_{p}^{ur}. The classical Dieudonné-Manin structure theorem for such semilinear ϕ-modules [27] shows that H^{0}(I_{p}, B^{0}(V)) is a direct sum of simple modules E(\frac{N}{M}) ≅ ℚ̂_{p}^{ur}[ϕ]/(ϕ^{M} − p^{N}), parametrized by non-negative rational numbers \frac{N}{M}, and an explicit computation shows that 1−ϕ is surjective on E(\frac{N}{M}). Hence 1 − ϕ is surjective on H^{0}(I_{p}, B^{0}(V)). By our assumption on V we have

H^{0}(I_{p}, B^{1}(V)) ≅ H^{0}(I_{p}, B^{0}(V)) ⊕ ℚ̂_{p}^{ur} ⊗_{ℚ_{p}} (D_{dr}(V)/F^{0}D_{dR}(V))
≅ H^{0}(I_{p}, B^{0}(V))

and d^{0} is simply the map 1 − ϕ on H^{0}(I_{p}, B^{0}(V)). Hence H^{0}(I_{p}, V) is quasiisomorphic to the complex

H^{0}(I_{p}, B^{0}(V)) ⟶^{d^{0}} H^{0}(I_{p}, B^{1}(V))

and RΓ(𝔽_{p}, H^{0}(I_{p}, V)) is canonically isomorphic to the total complex of the double subcomplex

H^{0}(I_{p}, B^{0}(V)) ⟶^{d^{0}} H^{0}(I_{p}, B^{1}(V))
1−Frob_{p}^{−1}↓  ^{1−Frob_{p}^{−1}}↓
H^{0}(I_{p}, B^{0}(V)) ⟶^{d^{0}} H^{0}(I_{p}, B^{1}(V))

of (70). Now, again using our assumption D_{dR}(V)/F^{0}D_{dR}(V) = 0, this double complex is naturally quasi-isomorphic to RΓ_{f}(ℚ_{p}, V) via the natural map κ (given by the inclusion H^{0}(G_{p}, −) → H^{0}(I_{p}, −)) in the following diagram

D_{cris}(V) ⟶^{1−ϕ} D_{cris}(V)
κ^{0}↓  _{κ^{1}}↓
H^{0}(I_{p}, B^{0}(V)) ⟶^{1−ϕ} H^{0}(I_{p}, B^{0}(V))
1−Frob_{p}^{−1}↓  ^{1−Frob_{p}^{−1}}↓
H^{0}(I_{p}, B^{0}(V)) ⟶^{1−ϕ} H^{0}(I_{p}, B^{0}(V)).

Indeed, the vertical sequences in this diagram are short exact sequences. The space D_{cris}(V) = H^{0}(G_{p}, B^{0}(V)) is the kernel of 1 − Frob_{p}^{−1} on H^{0}(I_{p}, B^{0}(V)) by definition, and 1 − Frob_{p}^{−1} is surjective on H^{0}(I_{p}, B^{0}(V)). This is because there is an isomorphism of Frob_{p}-modules

H^{0}(I_{p}, B^{0}(V)) = ℚ̂_{p}^{ur} ⊗_{ℚ_{p}} D_{cris}(V) ≅ (ℚ̂_{p}^{ur})^{d}

where d = dim_{ℚ_{p}} D_{cris}(V), using again the fact that Frob_{p} acts trivially on D_{cris}(V). Finally, it is well known that 1 − Frob_{p} : ℚ̂_{p}^{ur} → ℚ̂_{p}^{ur} is surjective. This concludes the proof of the Lemma. □

We continue with the notations introduced before Lemma 11. One defines a global complex RΓ_{f}(ℚ, V) as the mapping fibre of

RΓ(U_{et}, V) → ⨁_{p∉U} RΓ_{/f}(ℚ_{p}, V).

Then there is an exact triangle in the derived category of ℚ_{l}-vector spaces

(71)  RΓ_{c}(U_{et}, V) → RΓ_{f}(ℚ, V) → ⨁_{p∉U} RΓ_{f}(ℚ_{p}, V)

where the primes p ∉ U include p = ∞ with the convention RΓ_{f}(ℝ, V) = RΓ(ℝ, V). One can further show that Artin-Verdier duality induces a duality

H_{f}^{i}(ℚ, V) ≅ H_{f}^{3−i}(ℚ, V^{∗}(1))^{∗}.

The index ”f” stands for ”finite” which in this context is synonymous for ”unramified” or ”coming from an integral model”. The following proposition justifies this interpretation of the complex RΓ_{f} in the case of interest in this paper.

Proposition 9.1. Let π : X → Spec(ℤ) be a regular, proper, flat ℤ-scheme and X̄_{et} its Artin-Verdier étale topos. Let U ⊆ Spec(ℤ) be an open subscheme so that π_{U} : X_{U} → U is proper and smooth, let l be a prime number invertible on U and set X_{p} = X ⊗_{ℤ} 𝔽_{p}. For brevity we write X_{∞,et} for Sh(X_{∞}) (see Prop. 4.1). Assume Conjecture 9 in the next section. Then there is an isomorphism of exact triangles in the derived category of ℚ_{l}-vector spaces

RΓ_{c}(X_{U,et}, ℚ_{l}) ⟶  RΓ(X̄_{et}, ℚ_{l})  ⟶ ⨁_{p∉U} RΓ(X_{p,et}, ℚ_{l}) ⟶
↓  ↓  ↓
⨁_{i=0}^{2d} RΓ_{c}(U_{et}, V_{l}^{i})[−i] ⟶ ⨁_{i=0}^{2d} RΓ_{f}(ℚ, V_{l}^{i})[−i] ⟶ ⨁_{p∉U} ⨁_{i=0}^{2d} RΓ_{f}(ℚ_{p}, V_{l}^{i})[−i] ⟶

where V_{l}^{i} := H^{i}(X_{ℚ̄,et}, ℚ_{l}) and the bottom exact triangle is a sum over triangles (71).

Proof. For all p and l (including p = ∞ with a suitable interpretation of the terms) we shall first show that there is a commutative diagram (72)

RΓ(X_{p,et}, ℚ_{l}) ⟶ RΓ(X_{ℚ_{p},et}, ℚ_{l}) ⟶ RΓ_{X_{p}}(X_{ℤ_{p},et}, ℚ_{l})[1] ⟶
α↓  ^{β}↓  ↓
⨁_{i=0}^{2d} RΓ_{f}(ℚ_{p}, V_{l}^{i})[−i] ⟶ ⨁_{i=0}^{2d} RΓ(ℚ_{p}, V_{l}^{i})[−i] ⟶ ⨁_{i=0}^{2d} RΓ_{/f}(ℚ_{p}, V_{l}^{i})[−i] ⟶

where the rows are exact and the vertical maps are quasi-isomorphisms. This then induces a commutative diagram where the vertical maps are quasiisomorphisms (73)

RΓ(X_{U,et}, ℚ_{l}) ⟶ ⨁_{p∉U} RΓ(X_{ℚ_{p},et}, ℚ_{l}) ⟶ ⨁_{p∉U} RΓ_{X_{p}}(X_{ℤ_{p},et}, ℚ_{l})[1]
↓  ↓  ↓
⨁_{i=0}^{2d} RΓ(U_{et}, V_{l}^{i})[−i] ⟶ ⨁_{p∉U} ⨁_{i=0}^{2d} RΓ(ℚ_{p}, V_{l}^{i})[−i] ⟶ ⨁_{p∉U} ⨁_{i=0}^{2d} RΓ_{/f}(ℚ_{p}, V_{l}^{i})[−i].

Indeed, the first commutative square is induced by the commutative diagram

X_{U} ⟵ X_{ℚ_{p}}
↓  ↓
U ⟵ Spec(ℚ_{p})

and a decomposition Rπ_{U,∗}ℚ_{l} ≅ ⨁_{i=0}^{2d} V_{l}^{i}[−i] in the derived category of l-adic sheaves on U, and the second is a sum over p ∉ U of the right hand square in (72). Taking mapping fibres of the composite horizontal maps in (73) we obtain an isomorphism of exact triangles

RΓ(X̄_{et}, ℚ_{l})  ⟶ RΓ(X_{U,et}, ℚ_{l}) ⟶ ⨁_{p∉U} RΓ_{X_{p}}(X_{ℤ_{p},et}, ℚ_{l})[1] ⟶
↓  ↓  ↓
⨁_{i=0}^{2d} RΓ_{f}(ℚ, V_{l}^{i})[−i] ⟶ ⨁_{i=0}^{2d} RΓ(U_{et}, V_{l}^{i})[−i] ⟶ ⨁_{p∉U} ⨁_{i=0}^{2d} RΓ_{/f}(ℚ_{p}, V_{l}^{i})[−i] ⟶

where we use excision to identify the first fibre with RΓ(X̄_{et}, ℚ_{l}). The octahedral axiom then gives the isomorphism of exact triangles in Proposition 9.1, using the fact that the mapping fibre of the top left (resp. bottom left) horizontal map in (73) is RΓ_{c}(X_{U,et}, ℚ_{l}) (resp. ⨁_{i=0}^{2d} RΓ_{c}(U_{et}, V_{l}^{i})[−i]). Concerning (72), for p = ∞ we declare RΓ(X_{p,et}, ℚ_{l}) = RΓ(X_{ℚ_{p},et}, ℚ_{l}) and RΓ_{X_{p}}(X_{ℤ_{p},et}, ℚ_{l}) = 0. This agrees with the convention RΓ_{f}(ℝ, −) = RΓ(ℝ, −) introduced above. For p ≠ ∞ the top exact triangle is simply a localization triangle in étale cohomology since we have RΓ(X_{p,et}, ℚ_{l}) ≅ RΓ(X_{ℤ_{p},et}, ℚ_{l}) by proper base change. It suffices to construct quasi-isomorphisms α and β so that the left hand square in (72) commutes. For brevity we now omit the index et when referring to (continuous l-adic) étale cohomology. The quasi-isomorphism β is induced by the Leray spectral sequence for π_{ℚ_{p}} and a decomposition

(74)  Rπ_{ℚ_{p},∗}ℚ_{l} ≅ ⨁_{i=0}^{2d} V_{l}^{i}[−i]

in the derived category of l-adic sheaves on Spec(ℚ_{p}). The existence of α follows if the composite map

H^{i}(X_{p}, ℚ_{l}) → H^{i}(X_{ℚ_{p}}, ℚ_{l}) ⟶^{H^{i}(β)} H^{0}(ℚ_{p}, V_{l}^{i}) ⊕ H^{1}(ℚ_{p}, V_{l}^{i−1}) ⊕ H^{2}(ℚ_{p}, V_{l}^{i−2})

induces an isomorphism

H^{i}(X_{p}, ℚ_{l}) ≅ H_{f}^{0}(ℚ_{p}, V_{l}^{i}) ⊕ H_{f}^{1}(ℚ_{p}, V_{l}^{i−1}).

We shall show this only referring to the filtration F^{∗} on H^{i}(X_{ℚ_{p}}, ℚ_{l}) induced by the Leray spectral sequence for π_{ℚ_{p}}, not any particular decomposition (74). The Hochschild-Serre spectral sequence for the covering Xℤ̂_{p}^{ur} → X_{ℤ_{p}}, whose group we identify with Gal(𝔽̄_{p}/𝔽_{p}), induces a commutative diagram with exact rows (75)

0 ⟶ H^{1}(𝔽_{p}, H^{i−1}(X𝔽̄_{p}, ℚ_{l})) ⟶ H^{i}(X_{p}, ℚ_{l}) ⟶ H^{0}(𝔽_{p}, H^{i}(X𝔽̄_{p}, ℚ_{l})) ⟶ 0
↓  ↓  ↓
0 ⟶ H^{1}(𝔽_{p}, H^{i−1}(Xℚ̂_{p}^{ur}, ℚ_{l})) ⟶ H^{i}(X_{ℚ_{p}}, ℚ_{l}) ⟶ H^{0}(𝔽_{p}, H^{i}(Xℚ̂_{p}^{ur}, ℚ_{l})) ⟶ 0
↓  ↓  ↓^{γ}
H^{1}(𝔽_{p}, H^{0}(I_{p}, V_{l}^{i−1}))  H^{0}(ℚ_{p}, V_{l}^{i})  H^{0}(𝔽_{p}, H^{0}(I_{p}, V_{l}^{i})).

The left and right composite vertical maps are isomorphisms by Theorem 10.1 b) for l ≠ p (resp. Conjecture 9 for l = p) and the fact that

RΓ(𝔽_{p}, V) ≅ RΓ(𝔽_{p}, W_{0}V)

for any l-adic sheaf V on Spec(𝔽_{p}) where W_{0}V ⊆ V is the generalized Frobenius eigenspace for eigenvalues which are roots of unity (or just for the eigenvalue 1). Note also that

H_{f}^{k}(ℚ_{p}, V_{l}^{i}) = H^{k}(𝔽_{p}, H^{0}(I_{p}, V_{l}^{i}))

for k = 0, 1 and all l and i by Lemma 11 since

D_{dR}(V_{p}^{i}) ≅ H_{dR}^{i}(X_{ℚ_{p}}/ℚ_{p}) = F^{0}H_{dR}^{i}(X_{ℚ_{p}}/ℚ_{p}) ≅ F^{0}D_{dR}(V_{p}^{i}).

The kernel of the map γ in (75) is H^{0}(𝔽_{p}, H^{1}(I_{p}, V_{l}^{i−1})), hence there is a commutative diagram with exact rows

0 ⟶ H^{1}(𝔽_{p}, H^{i−1}(Xℚ̂_{p}^{ur}, ℚ_{l})) ⟶ F^{1}H^{i}(X_{ℚ_{p}}, ℚ_{l}) ⟶ H^{0}(𝔽_{p}, H^{1}(I_{p}, V_{l}^{i−1})) ⟶ 0
↓  ↓  ∥
0 ⟶ H^{1}(𝔽_{p}, H^{0}(I_{p}, V_{l}^{i−1})) ⟶ H^{1}(ℚ_{p}, V_{l}^{i−1}) ⟶ H^{0}(𝔽_{p}, H^{1}(I_{p}, V_{l}^{i−1})) ⟶ 0

which implies that the left vertical isomorphism in (75) fits into a commutative diagram with the natural map F^{1}H^{i}(X_{ℚ_{p}}, ℚ_{l}) → H^{1}(ℚ_{p}, V_{l}^{i−1}). This finishes the proof of the existence of α and of Proposition 9.1.

We remark that for l ≠ p we have W_{0}H^{1}(I_{p}, V_{l}^{i}) = W_{0}((V_{l}^{i})_{I_{p}}(−1)) = 0 and hence isomorphisms

W_{0}H^{i}(X𝔽̄_{p}, ℚ_{l}) ≅ W_{0}H^{i}(Xℚ̂_{p}^{ur}, ℚ_{l}) ≅ W_{0}H^{0}(I_{p}, V_{l}^{i})

which implies that the top left and right, and therefore the top middle vertical maps in (75) are isomorphisms. We conclude that

RΓ(X_{p}, ℚ_{l}) ≅ RΓ(X_{ℚ_{p}}, ℚ_{l})

for l ≠ p like for p = ∞. □

We continue with the statement of the Tamagawa number conjecture. One might view the following conjecture as an l-adic analogue of Beilinson’s conjecture, or as a generalization of Tate’s conjecture.

Conjecture 3. (Bloch-Kato) There are isomorphisms

ρ_{l}^{i} : H_{f}^{2}(ℚ, V_{l}^{i}) ≅ H_{M}^{2d−i+1}(X_{/ℤ}, ℚ(d + 1))_{ℚ_{l}}^{∗}

and H_{f}^{1}(ℚ, V_{l}^{i}) = 0 for any i.

One can show easily that H_{f}^{0}(ℚ, V_{l}^{0}) ≅ Ch^{0}(X)_{ℚ_{l}}, H_{f}^{0}(ℚ, V_{l}^{i}) = 0 for i > 0 and H_{f}^{3}(ℚ, V_{l}^{i}) = 0 so that Conjecture 3 computes the entire cohomology of RΓ_{f}(ℚ, V_{l}^{i}). Together with Artin’s comparison isomorphism

V_{l}^{i} = H^{i}(X_{ℚ̄,et}, ℚ_{l}) ≅ H^{i}(X(ℂ), ℚ)_{ℚ_{l}}

as well as the isomorphisms

ι_{p} : det_{ℚ_{l}}RΓ_{f}(ℚ_{p}, V) ≅ ℚ_{l}

induced by the identity map on (V_{l}^{i})^{I_{p}} and D_{cris}(V_{l}^{i}), Conjecture 3 induces an isomorphism

ϑ_{l}^{i} : Δ_{f}(h^{i}(X))_{ℚ_{l}} ≅ det_{ℚ_{l}}RΓ_{f}(ℚ, V_{l}^{i}) ⊗ det_{ℚ_{l}}RΓ(ℝ, V_{l}^{i}) ≅ det_{ℚ_{l}}RΓ_{c}(U_{et}, V_{l}^{i}).

Conjecture 4. (l-part of the Tamagawa number conjecture) There is an identity of free rank one ℤ_{l}-submodules of det_{ℚ_{l}}RΓ_{c}(U_{et}, V_{l}^{i})

ℤ_{l} · ϑ_{l}^{i} ∘ ϑ_{∞}^{i}(L^{∗}(h^{i}(X), 0)^{−1}) = det_{ℤ_{l}}RΓ_{c}(U_{et}, T_{l}^{i})

for any Galois stable ℤ_{l}-lattice T_{l}^{i} ⊆ V_{l}^{i}.

This conjecture is independent of the choice of the lattice T_{l}^{i} since

(76)  ∏_{i∈ℤ} |H_{c}^{i}(U_{et}, M)|^{(−1)^{i}} = 1

for any finite locally constant sheaf M whose cardinality is invertible on U. The following conjecture allows a reformulation of the Tamagawa number conjecture in terms of the L-function

L_{U}(h^{i}(X), s) = ∏_{p∈U} L_{p}(h^{i}(X), s)

associated to the smooth l-adic sheaf V_{l}^{i} over U. Recall that a two term complex C = (W ⟶^{λ} W ) is called semisimple at 0 if the composite map

H^{0}(C) = ker(λ) ⊆ W → coker(λ) = H^{1}(C)

is an isomorphism. This is always the case, for example, if the complex C is acyclic.

Conjecture 5. (Frobenius-Semisimplicity at the eigenvalue 1) For any prime number p the complex RΓ_{f}(ℚ_{p}, V_{l}^{i}) is semisimple at zero.

Under this conjecture one has a second isomorphism

ι̃_{p} : det_{ℚ_{l}}RΓ_{f}(ℚ_{p}, V_{l}^{i}) ≅ ℚ_{l}

which satisfies

ι_{p} = P_{p}^{∗}(h^{i}(X), 1)^{−1}ι̃_{p} = L_{p}^{∗}(h^{i}(X), 0) log(p)^{r_{i,p}} ι̃_{p}

where r_{i,p} = ord_{T =1} P_{p}(h^{i}(X), T) = − ord_{s=0} L_{p}(h^{i}(X), s) (see [7][Lemma 2]). If RΓ_{f}(ℚ_{p}, V_{l}^{i}) is acyclic then ι̃_{p} is the canonical trivialization of the determinant of an acyclic complex. Using this second isomorphism the Tamagawa number conjecture becomes

(77)  ℤ_{l} · ϑ̃_{l}^{i} ∘ ϑ̃_{∞}^{i}(L_{U}^{∗}(h^{i}(X), 0)^{−1}) = det_{ℤ_{l}}RΓ_{c}(U_{et}, T_{l}^{i})

where

ϑ̃_{l}^{i} = ∏_{p∉U} P_{p}^{∗}(h^{i}(X), 1)ϑ_{l}^{i}, ϑ̃_{∞}^{i} = ∏_{p∉U} log(p)^{r_{i,p}}ϑ_{∞}^{i}.

9.4.2. Further assumptions on Weil-étale cohomology. In order to establish the compatibility of the conjectural picture a)-f) outlined in the introduction with the Tamagawa number conjecture, we need to augment it with a number of further assumptions. Even though a)-f) only refer to cohomology groups we now assume that these groups do indeed arise from a topos X_{W} - different from the one defined in Definition 10 - and that compact support cohomology is defined via an embedding into a proper scheme followed by an Artin-Verdier type compactification X̄_{W} (and is independent of a choice of compactification).

g) For an open subscheme U of an arithmetic scheme X with closed complement Z there is an exact triangle of perfect complexes in the derived category of abelian groups

RΓ_{c}(U_{W} , ℤ) → RΓ_{c}(X_{W} , ℤ) → RΓ_{c}(Z_{W} , ℤ) → .

h) There is a morphism of topoi γ : X_{W} → X_{et} for any arithmetic scheme X (or the Artin-Verdier compactification of such a scheme). Moreover, for any constructible sheaf F on X_{et} the adjunction F → Rγ_{∗}γ^{∗}F is an isomorphism.

If X has finite characteristic then h) holds if one understands the index W as denoting the Weil-eh cohomology of Geisser (see [19][Thm. 5.2b), Thm. 3.6]). In addition, the exact triangle in g) exists for Weil-eh cohomology by [19][Def. 5.4, eq. (4)] but perfectness is only known under resolution of singularities (more precisely, under the assumption R(d) of [19][Def. 2.4], see loc. cit. Lemma 2.7). The following property is a natural extension of property g) to the Artin-Verdier compactification.

i) If X is regular, proper, flat over Spec(ℤ) then there is an exact triangle in the derived category of abelian groups

RΓ_{c}(X_{W} , ℤ) → RΓ(X̄_{W} , ℤ) → RΓ(X_{∞,W} , ℤ) →

and there is an exact triangle

RΓ_{c}(X_{W} , ℝ̃) → RΓ(X̄_{W} , ℝ̃) → RΓ(X_{∞,W} , ℝ̃) →

in the derived category of ℝ-vector spaces, where X_{∞,W} was defined in Definition 11.

Note that

RΓ(X_{∞,W} , ℤ) ≅ RΓ(X_{∞}, γ_{∞∗}(ℤ)) ≅ RΓ(X_{∞}, ℤ)

by Proposition 8.2 and this last complex is isomorphic to the singular complex of the (locally contractible) compact space X_{∞} and is therefore a perfect complex of abelian groups. Since the complex RΓ_{c}(X_{W} , ℤ) is perfect by d) the triangle in i) then implies that RΓ(X̄_{W} , ℤ) is also a perfect complex of abelian groups. Note also that, unlike in the situation g), the triangle for ℝ̃- coefficients is not the scalar extension of the triangle for ℤ-coefficients since neither RΓ(X_{∞,W} , ℤ) nor RΓ(X̄_{W} , ℤ) satisfies property e). One rather has a commutative diagram of long exact sequences

⟶ H^{i+1}(X_{∞,W} , ℝ̃) ⟶ H_{c}^{i+2}(X_{W} , ℝ̃) ⟶  0  ⟶ H^{i+2}(X_{∞,W} , ℝ̃)
↑^{α_{∞}}  ↑^{α}  ↑  ↑
⟶ H^{i+1}(X_{∞,W} , ℤ) ⟶ H_{c}^{i+2}(X_{W} , ℤ) ⟶ H^{i+2}(X̄_{W} , ℤ) ⟶ H^{i+2}(X_{∞,W} , ℤ)

where only α_{ℝ} is an isomorphism by e). Here we assume i ≥ 0 so that H^{i+2}(X̄_{W} , ℝ̃) = 0 by Theorem 7.1. There is a direct sum decomposition

H^{i+1}(X_{∞,W} , ℝ̃) ≅ H^{i+1}(X_{∞}, ℝ) ⊕ H^{i}(X_{∞}, ℝ)

by (47) and an isomorphism

H^{i+1}(X_{∞,W} , ℤ)_{ℝ} ≅ H^{i+1}(X_{∞}, ℤ)_{ℝ} ≅ H^{i+1}(X_{∞}, ℝ).

One therefore obtains a map for i ≥ 0

r_{∞}^{i} : H^{i}(X_{∞}, ℝ) → H^{i+2}(X̄_{W} , ℤ)_{ℝ}

which is an isomorphism for i > 0. Proposition 9.1 and assumption h) yield an isomorphism for i ≥ 0

r_{l}^{i} : H_{f}^{2}(ℚ, V_{l}^{i}) ≅ H^{i+2}(X̄_{et}, ℚ_{l}) ≅ H^{i+2}(X̄_{W} , ℚ_{l}) ≅ H^{i+2}(X̄_{W} , ℤ)_{ℚ_{l}}.

The following is the key requirement on a definition of a Weil-étale topos. j) If X is regular, proper, flat over Spec(ℤ) with generic fibre X of di-

mension d then there are isomorphisms

λ^{i} : H^{i+2}(\overline{X}_{W} , ℤ)_{ℚ} ≅ H_{M}^{2d−i+1}(X_{/ℤ}, ℚ(d + 1))^{∗}

for i ≥ 0 such that λ_{ℝ}^{i} ∘ r_{∞}^{i} = (ρ_{∞}^{i})^{∗} and λ_{ℚ_{l}}^{i} ∘ r_{l}^{i} = ρ_{l}^{i}. This is true for d = i = 0 with Lichtenbaum’s current definition where

H^{2}(\overline{Spec(O_{F})}_{W} , ℤ)_{ℚ} ≅ H_{M}^{1}(Spec(F)_{/ℤ}, ℚ(1))^{∗} = Hom_{ℤ}(O_{F}^{×}, ℚ).

Note that j) together with (69) and h) also implies

H^{i+2}(\overline{X}_{W} , ℤ) = 0

for i > 2d + 1, which is not satisfied by the current definition of \overline{Spec(O_{F})}_{W}.

Proposition 9.2. Suppose there is a definition of Weil-étale cohomology groups for arithmetic schemes satisfying a)-j) except perhaps f) for schemes of characteristic 0. Let X be a proper, smooth variety over ℚ of dimension d which has a proper, regular model over Spec(ℤ) such that Conjectures 1,2,3,5,9 are satisfied. Assume L(h^{i}(X), s) has a meromorphic continuation to s = 0 for all i. Then the Tamagawa number conjecture (Conjecture 4) for the motive

h(X) = ⨁_{i=0}^{2d} h^{i}(X)[−i]

is equivalent to statement f) for any arithmetic scheme X with generic fibre X.

Proof. If X is any arithmetic scheme with generic fibre X then there exists an open subscheme U ⊆ Spec(ℤ) so that π : X_{U} → U is proper and smooth. Let Z be the closed complement of U. Then by g) we have an isomorphism

det_{ℤ}RΓ_{c}(X_{W} , ℤ) ≅ det_{ℤ}RΓ_{c}(X_{U,W} , ℤ) ⊗_{ℤ} det_{ℤ}RΓ_{c}(X_{Z,W} , ℤ)

as well as factorizations

ζ(X , s) = ζ(X_{U}, s)ζ(X_{Z}, s); ζ^{∗}(X , 0) = ζ^{∗}(X_{U}, 0)ζ^{∗}(X_{Z}, 0).

Since we assume f) for X_{Z} = ∐_{p∈Z} X_{p} , statement f) for X is equivalent to statement f) for X_{U}. We now assume that X is the proper regular model of X. For any prime p ∈ Z, the exact sequence

⋯ ⟶^{∪e} H^{i}(X_{p,W} , ℤ)_{ℚ} ⟶^{∪e} H^{i+1}(X_{p,W} , ℤ)_{ℚ} ⟶^{∪e} ⋯,

where e ∈ H^{1}(X_{p,W} , ℤ) is Lichtenbaum’s canonical class, yields a trivialization

(78)  ℚ ≅ det_{ℚ}RΓ(X_{p,W} , ℤ)_{ℚ}.

The exact triangle

RΓ_{c}(X_{U,W} , ℤ) → RΓ(X̄_{W} , ℤ) → ⨁_{p∈Z∪{∞}} RΓ(X_{p,W} , ℤ)

together with assumptions i), j), (78) and the isomorphisms

H^{i}(X_{∞,W} , ℤ)_{ℚ} ≅ H^{i}(X (ℂ), ℚ)^{+}, CH^{0}(X)_{ℚ} ≅ H^{0}(X̄_{W} , ℤ)_{ℚ}

induce an isomorphism

ϑ_{W} : det_{ℚ}RΓ_{c}(X_{U,W} , ℤ)_{ℚ} ≅det_{ℚ}RΓ(X̄_{W} , ℤ)_{ℚ} ⊗ ⨂_{p∈Z∪{∞}} det_{ℚ}^{−1}RΓ(X_{p,W} , ℤ)_{ℚ}

≅ ⨂_{i=0}^{2d} Δ_{f}(h^{i}(X))^{(−1)^{i}}.

By assumption j) there is a commutative diagram of isomorphisms

ℝ ⟶^{γ}  det_{ℝ}RΓ_{c}(X_{U,W} , ℤ)_{ℝ}
∥  ↓^{ϑ_{W,ℝ}}
ℝ ⟶^{⊗_{i}(ϑ̃_{∞}^{i})^{(−1)^{i}}} ⨂_{i=0}^{2d} Δ_{f}(h^{i}(X))_{ℝ}^{(−1)^{i}}

where γ is induced by c). The power of log(p) in ϑ̃ appears for the same reason as in the proof of Theorem 9.2. Similarly, j) implies that for any prime l ∈ Z we have a commutative diagram of isomorphisms

(79)

det_{ℚ_{l}}RΓ_{c}(X_{U,W} , ℤ)_{ℚ_{l}} ⟶  det_{ℚ_{l}}RΓ_{c}(X_{U,et}, ℚ_{l})
↓^{ϑ_{W,ℚ_{l}}}  ↓
⨂_{i=0}^{2d} Δ_{f}(h^{i}(X))_{ℚ_{l}}^{(−1)^{i}} ⟶^{⊗_{i}(ϑ̃_{l}^{i})^{(−1)^{i}}} ⨂_{i=0}^{2d} det_{ℚ_{l}}^{(−1)^{i}}RΓ_{c}(U_{et}, V_{l}^{i})

where the top isomorphism is induced by an isomorphism

RΓ_{c}(X_{U,W} , ℤ) ⊗_{ℤ} ℤ_{l} ≅ RΓ_{c}(X_{U,et}, ℤ_{l})

(coming from assumption h) together with the finite generation of the cohomology of RΓ_{c}(X_{U,W} , ℤ) claimed in g)), while the right vertical isomorphism is induced by the isomorphism

RΓ_{c}(X_{U,et}, ℤ_{l}) ≅ RΓ_{c}(U_{et}, Rπ_{∗}ℤ_{l})

and

det_{ℤ_{l}}RΓ_{c}(U_{et}, Rπ_{∗}ℤ_{l}) ≅ ⨂_{i=0}^{2d} det_{ℤ_{l}}^{(−1)^{i}}RΓ_{c}(U_{et}, L_{l}^{i}) = ⨂_{i=0}^{2d} det_{ℤ_{l}}^{(−1)^{i}}RΓ_{c}(U_{et}, T_{l}^{i})

where L_{l}^{i} := R^{i}π_{∗}ℤ_{l} and T_{l}^{i} ⊆ V_{l}^{i} is the torsion free part of L_{l}^{i}. Note that we have an exact sequence of locally constant ℤ_{l}-sheaves on U

0 → L_{l,tor}^{i} → L_{l}^{i} → T_{l}^{i} → 0

and an identity det_{ℤ_{l}}RΓ_{c}(U_{et}, T_{l}^{i}) = det_{ℤ_{l}}RΓ_{c}(U_{et}, L_{l}^{i}) of invertible ℤ_{l}- submodules of det_{ℚ_{l}}RΓ_{c}(U_{et}, V_{l}^{i}) by (76). As discussed above statement f) for X_{U} is equivalent to statement f) for X_{U′} for U′ ⊂ U, hence we can always assume that a given prime l is not in U. If we know statement f) for X_{U} then the image under γ of

ζ^{∗}(X_{U}, 0)^{−1} = ∏_{i=0}^{2d} L_{U}^{∗}(h^{i}(X), 0)^{(−1)^{i+1}}

generates the natural invertible ℤ_{l}-submodule

RΓ_{c}(X_{U,W} , ℤ) ⊗_{ℤ} ℤ_{l} ≅ ⨂_{i=0}^{2d} det_{ℤ_{l}}^{(−1)^{i}}RΓ_{c}(U_{et}, T_{l}^{i})

in (79) (see the discussion in the previous paragraph). Hence we obtain the Tamagawa number conjecture in the form (77) for h(X). Conversely, knowing the Tamagawa number conjecture for h(X), we obtain the l-primary part of statement f) for X_{U\{l}} which is equivalent to the l-primary part of statement f) for X_{U}. Varying l we obtain f) for X_{U}. Here by l-primary parts, we mean that for any perfect complex of abelian groups C, such as RΓ_{c}(X_{U,W} , ℤ), an element b ∈ det_{ℤ}(C) ⊗ ℚ is a generator of det_{ℤ}(C) if and only if the image of b in det_{ℤ}(C) ⊗ ℚ_{l} is a generator of det_{ℤ}(C) ⊗ ℤ_{l} for all primes l.

□

10. On the local theorem of invariant cycles

Let R be a complete discrete valuation ring with quotient field K and finite residue field k of characteristic p. Set S = Spec(R), η = Spec(K), s = Spec(k). Let S̄ = (S̄, s̄, η̄) be the normalization of S in a separable closure K̄ of K and denote by I ⊆ G := Gal(K̄/K) the inertia subgroup.

10.1. l-adic cohomology for p ≠ l. In this section l is a prime different from p. The following lemma might be well known as a consequence of de Jong’s theorem on alterations [11], and also of Deligne’s work [9] in case char(K) = p. We shall only need it for X_{η} → Spec(K) proper and smooth.

Lemma 12. Let X_{η} → Spec(K) be separated and of finite type. Then the G-representation H^{i}(Xη̄, ℚ_{l}) has a (unique) G-invariant weight filtration

⋯ ⊆ W_{j}H^{i}(Xη̄, ℚ_{l}) ⊆ W_{j+1}H^{i}(Xη̄, ℚ_{l}) ⊆ ⋯

in the sense of [9][Prop.-Def. 1.7.5], i.e. if F ∈ G is any lift of a geometric Frobenius element in Gal(k̄/k) then the eigenvalues of F on gr_{j}^{W} H^{i}(Xη̄, ℚ_{l}) are Weil numbers of weight j ∈ ℤ with respect to |k|. The same is true for the Grepresentation H_{c}^{i}(Xη̄, ℚ_{l}). One has W_{−1}H^{i}(Xη̄, ℚ_{l}) = W_{−1}H_{c}^{i}(Xη̄, ℚ_{l}) = 0.

Proof. By [9][Prop.-Def. 1.7.5] it suffices to show that all eigenvalues α of F on H^{i}(Xη̄, ℚ_{l}) are Weil numbers of some weight j = j(α) ∈ ℤ. In doing so, one may pass to an open subgroup G′ ⊆ G, i.e. replace X_{η} by its base change to a finite extension K′/K, since an algebraic number α is a Weil number with respect to |k| if and only if α^{[k′:k]} is a Weil number with respect to |k′|. One can now argue exactly as in the proof of [2][Prop. 6.3.2] to which we refer for more details. If X_{η} is the generic fibre of a proper, strictly semistable scheme, then the vanishing cycle spectral sequence computed by Rapoport and Zink [36][Satz 2.10]

(80)  E_{1}^{−r,i+r} = ⨁_{q≥0,r+q≥0} H^{i−r−2q}(Y^{(r+2q)}, ℚ_{l})(−r − q) ⇒ H^{i}(Xη̄, ℚ_{l})

together with the Weil conjectures for the smooth proper schemes Y^{(i)} give the statement (and moreover the weight filtration on H^{i}(Xη̄, ℚ_{l}) is the filtration induced by the spectral sequence). If X_{η} is only smooth and proper then by de Jong’s theorem [2][Thm. 1.4.1] there is a generically finite, flat X′_{η′} → X_{η} where X′_{η′} is strictly semistable. Hence H^{i}(Xη̄, ℚ_{l}) is a direct summand of the G′-representation H^{i}(X′ ; η̄, ℚ_{l}) for which the statement holds. Then one can use induction on the dimension together with the long exact localization sequence to prove the statement for H_{c}^{i}(Xη̄, ℚ_{l}) for any separated X_{η} of finite type. Another application of de Jong’s theorem is necessary here to assure that a regular open subscheme U ⊆ X_{η} has a finite cover U′ → U which is open in a proper regular K-scheme. For X_{η} smooth over K, Poincare duality then implies the statement for H^{i}(Xη̄, ℚ_{l}) and for general X one uses a hypercovering argument. In this proof, starting with (80), all occurring F -eigenvalues have non-negative weight, i.e. we have W_{−1}H^{i}(Xη̄, ℚ_{l}) = W_{−1}H_{c}^{i}(Xη̄, ℚ_{l}) = 0. □

Let f : X → S be a proper, flat, generically smooth morphism of relative dimension d. For 0 ≤ i ≤ 2d one defines the specialization morphism

(81)  sp : H^{i}(Xs̄, ℚ_{l}) → H^{i}(Xη̄, ℚ_{l})^{I}

as the composite

(82)  H^{i}(Xs̄, ℚ_{l}) ≅ H^{i}(X′, ℚ_{l}) → H^{i}(X′_{η}, ℚ_{l}) → H^{i}(Xη̄, ℚ_{l})

where X′ is the base change of X to a strict Henselization of S at s̄ and the first isomorphism is proper base change. The map sp is G-equivariant and respects the weight filtration.

Theorem 10.1. If X is regular then the following hold.

a) The map

H^{i}(Xs̄, ℚ_{l}) = W_{i}H^{i}(Xs̄, ℚ_{l}) → W_{i}H^{i}(Xη̄, ℚ_{l})^{I}

induced by sp is surjective for all i.

b) The map

(83)  W_{1}H^{i}(Xs̄, ℚ_{l}) → W_{1}H^{i}(Xη̄, ℚ_{l})^{I}

induced by sp is an isomorphism for all i, and the zero map for i > d.

c) The map sp is an isomorphism for i = 0, 1.

d) If W_{i}H^{i}(Xη̄, ℚ_{l})^{I} = H^{i}(Xη̄, ℚ_{l})^{I} for all i then the map

W_{i−1}H^{i}(Xs̄, ℚ_{l}) → W_{i−1}H^{i}(Xη̄, ℚ_{l})^{I}

induced by sp is an isomorphism for all i.

Remarks. a) By part a) of the theorem, the assumption of part d) is equivalent to the surjectivity of the map sp for all i, a statement which is called the local theorem on invariant cycles. It is known to hold if R is the local ring of a smooth curve over k by [9][Lemma 3.6.2], see also [9][Thm. 3.6.1], but it is only conjectured in mixed characteristic. Unconditionally, we were only able to prove the weak statement in b) rather than the full conclusion of d). Part c) is probably well known and follows, for example, from b) and results of [22][Exposé IX] on Neron models which assure that W_{1}H^{1}(Xη̄, ℚ_{l})^{I} = H^{1}(Xη̄, ℚ_{l})^{I}.

b) It is easy to construct examples where sp is not injective for i ≥ 2. For example if X is the blowup of a proper smooth relative curve over S in a closed point, then H^{2}(Xs̄, ℚ_{l}) will have an extra summand ℚ_{l}(−1) corresponding to the exceptional divisor which gives a new irreducible component of Xs̄.

c) If X arises by base change from a regular, proper, flat scheme X → Spec(ℤ) part b) of the theorem implies

ord_{s=n} ζ(X , s) = ord_{s=n} ∏_{i=0}^{2d} L(h^{i}(X_{ℚ}), s)^{(−1)^{i}}

for integers n ≤ 0 (and for n = \frac{1}{2}) where ζ(X , s) is the Zeta-function of the arithmetic scheme X and L(h^{i}(X_{ℚ}), s) is the L-function of the motive h^{i}(X_{ℚ}) defined by Serre [38]. Indeed the former (resp. latter) is an Euler product of characteristic polynomials of Frobenius on H^{i}(X ⊗ 𝔽̄_{p}, ℚ_{l}) (resp. H^{i}(X ⊗ ℚ̄_{p}, ℚ_{l})^{I_{p}}). These are equal for almost all primes and at the finitely many (bad reduction) primes where they might differ, part b) assures that the vanishing order at s ≤ 0 of both factors, which equals (−1)^{i+1} times the multiplicity of the Frobenius-eigenvalue p^{n} (of weight 2n), is the same.

d) Regularity of X is a key assumption in the theorem. The map sp will be an isomorphism for i = 0 if X is only normal but for i = 1 normality is not even sufficient for surjectivity of sp on W_{0}, as the following example of de Jeu [8] shows. If E is an elliptic curve over ℚ given by a projective Weierstrass equation

Y^{2}Z = X^{3} + AXZ^{2} + BZ^{3}

with A, B ∈ ℚ then for any u ∈ ℚ^{×} the curve

Y^{2}Z = X^{3} + u^{4}AXZ^{2} + u^{6}BZ^{3}

is isomorphic to E, and if u^{4}A, u^{6}B ∈ ℤ this equation defines a normal scheme E, proper and flat over Spec(ℤ), inside ℙ_{ℤ}^{2}. Indeed, the affine coordinate ring of the complement of the zero section (X : Y : Z) = (0 : 1 : 0) is R = ℤ[x, y]/(y^{2} − x^{3} − u^{4}Ax − u^{6}B) and hence a complete intersection. So R is normal if and only if all local rings A_{𝔭} for primes 𝔭 of height ≤ 1 are regular. If 𝔭 maps to the generic point of Spec(ℤ) this is clear because E is a smooth curve over ℚ. If 𝔭 maps to (p) for some prime number p, then 𝔭 = R · p since R · p is already a prime ideal as the equation y^{2} − x^{3} − u^{4}Ax − u^{6}B remains irreducible modulo p. Hence 𝔭 is principal and A_{𝔭} is a DVR. The generic point of the zero section maps to the generic point of Spec(ℤ), hence E is normal.

If we now pick u in addition to be a multiple of some prime p where E has split multiplicative reduction, then Es̄ is a cuspidal cubic curve and therefore

0 = H^{1}(Es̄, ℚ_{l}) → H^{1}(Eη̄, ℚ_{l})^{I} = W_{0}H^{1}(Eη̄, ℚ_{l})^{I} ≅ ℚ_{l}

is not surjective. However, the condition that X is locally factorial (all local rings are UFDs) lies between normality and regularity and is sufficient to ensure that our proof of c) given below goes through. Regularity is only used for the isomorphism Pic(X) ≅ Cl(X) and for [35][Thm. 6.4.1] via normality.

Proof. Since the statement of Theorem 10.1 only depends on the base change of f to the strict Henselization of S at s̄ we may assume that S is strictly Henselian. Note that regularity is preserved by this base change by [31][I,3.17 c)]. For a) we follow Deligne’s proof of [9][Thm. 3.6.1], replacing duality for the essentially smooth morphism X → Spec(k) by duality for the morphism f combined with purity for the regular schemes X, proved by Thomason and Gabber (see [17]), and S, proved by Grothendieck in [21][I,Thm. 5.1]. The same arguments as in loc. cit. lead to the commutative diagram with exact rows and columns

(84) H_{Xs̄}^{i+1}(X, ℚ_{l}) ; ↑ ; 0 ⟶ H^{i−1}(Xη̄, ℚ_{l})_{I}(−1) ⟶ H^{i}(X_{η}, ℚ_{l}) ⟶ H^{i}(Xη̄, ℚ_{l})^{I} ⟶ 0 ; ↑  ↑^{sp} ; H^{i}(X, ℚ_{l}) ⟶^{∼} H^{i}(Xs̄, ℚ_{l}) ; ↑ ; H_{Xs̄}^{i}(X, ℚ_{l})

and after application of the exact functor W_{i} to a diagram

W_{i}H_{Xs̄}^{i+1}(X, ℚ_{l})
↑
W_{i}H^{i}(X_{η}, ℚ_{l}) ⟶ W_{i}H^{i}(Xη̄, ℚ_{l})^{I} ⟶ 0
↑  ↑^{sp}
W_{i}H^{i}(X, ℚ_{l}) ⟶^{∼} W_{i}H^{i}(Xs̄, ℚ_{l})

so that it remains to show that

(85)  W_{i}H_{Xs̄}^{i+1}(X, ℚ_{l}) = 0

for all i. The vertical long exact sequence in (84) arises by applying the (exact) global section functor Γ(S, −) to the exact triangle

Rf_{∗}R\underline{Hom}_{X}(i_{∗}ℚ_{l}, ℚ_{l}) → Rf_{∗}ℚ_{l} → Rf_{∗}Rj_{∗}ℚ_{l}

where i : Xs̄ → X and j : X_{η} → X are the inclusions. By purity for X [17][§8] we have ℚ_{l} ≅ Rf^{!}ℚ_{l}(−d)[−2d] and the (sheafified) adjunction between Rf^{!} and Rf_{!} gives

Rf_{∗}R\underline{Hom}_{X}(i_{∗}ℚ_{l}, ℚ_{l}) ≅ R\underline{Hom}_{S}(Rf_{!}i_{∗}ℚ_{l}, ℚ_{l}(−d))[−2d]
≅ R\underline{Hom}_{S}(i_{s,∗}Rf_{s,∗}ℚ_{l}, ℚ_{l}(−d))[−2d]
≅ i_{s,∗}R\underline{Hom}_{s}(Rf_{s,∗}ℚ_{l}, Ri_{s}^{!}ℚ_{l}(−d))[−2d]
≅ i_{s,∗}R\underline{Hom}_{s}(Rf_{s,∗}ℚ_{l}, ℚ_{l})(−d − 1)[−2d − 2]

where i_{s} : s → S is the closed immersion and f_{s} : X_{s} → s the base change of f. Here we have also used Rf_{∗} = Rf_{!} (f proper) as well as the sheafified adjunction between i_{s,!} = i_{s,∗} and i_{s}^{!}, and purity for S. This last complex has cohomology in degree i + 1 given by

Hom_{ℚ_{l}}(H^{2d+2−i−1}(Xs̄, ℚ_{l}), ℚ_{l})(−d − 1)

which has weights greater or equal to 2(d + 1) − (2d + 2 − i − 1) = i + 1 since W_{k}H^{k}(Xs̄, ℚ_{l}) = H^{k}(Xs̄, ℚ_{l}) by [9][Cor. 3.3.8]. This finishes the proof of a).

Concerning b), we apply the exact functor W_{1} to the diagram (84) and obtain a commutative diagram

W_{1}H^{i}(X_{η}, ℚ_{l}) ⟶^{β} W_{1}H^{i}(Xη̄, ℚ_{l})^{I}
↑^{α}  ↑^{sp}
W_{1}H^{i}(X, ℚ_{l}) ⟶^{∼} W_{1}H^{i}(Xs̄, ℚ_{l}).

For i ≥ 2 the map α is an isomorphism since

W_{1}H_{Xs̄}^{j}(X, ℚ_{l}) ⊆ W_{j−1}H_{Xs̄}^{j}(X, ℚ_{l}) = 0

for j = i, i + 1 by (85). For i = 0, 1 we already have

H_{Xs̄}^{i}(X, ℚ_{l}) ≅ Hom_{ℚ_{l}}(H^{2d+2−i}(Xs̄, ℚ_{l}), ℚ_{l})(−d − 1) = 0

before applying W_{1} and the map α is also an isomorphism. For any i the map β is an isomorphism since

W_{1}(H^{i−1}(Xη̄, ℚ_{l})_{I}(−1)) = W_{−1}(H^{i−1}(Xη̄, ℚ_{l})_{I})(−1) = 0

by Lemma 12. Hence the map induced by sp on W_{1} is also an isomorphism. For i > d both sides of (83) vanish. Indeed, the weights of H^{i}(Xs̄, ℚ_{l}) are greater or equal to 2(i − d) ≥ 2 by [9][Cor. 3.3.4] and the same is true for H^{i}(Xη̄, ℚ_{l}) as follows from Poincare duality and the fact that the weights on H^{i}(Xη̄, ℚ_{l}) are ≤ 2i for i < d. This in turn can be read off from the spectral sequence (80) in the strictly semistable case and follows in general from de Jong’s theorem. Hence

W_{1}H^{i}(Xs̄, ℚ_{l}) = W_{1}H^{i}(Xη̄, ℚ_{l})^{I} = 0

for i > d and we have finished the proof of b). Concerning d), we apply the exact functor W_{i−1} to the diagram (84) and obtain a commutative diagram

W_{i−1}H^{i}(X_{η}, ℚ_{l}) ⟶^{β} W_{i−1}H^{i}(Xη̄, ℚ_{l})^{I}
↑^{α}  ↑^{sp}
W_{i−1}H^{i}(X, ℚ_{l}) ⟶^{∼} W_{i−1}H^{i}(Xs̄, ℚ_{l})

where α is an isomorphism for the same reason as in the proof of b) and β is an isomorphism since

W_{i−1}(H^{i−1}(Xη̄, ℚ_{l})_{I}(−1)) = W_{i−3}(H^{i−1}(Xη̄, ℚ_{l})_{I})(−1)

is dual to

H^{2d−i+1}(Xη̄, ℚ_{l})^{I}(d + 1)/W_{2d−i+2}(H^{2d−i+1}(Xη̄, ℚ_{l})^{I})(d + 1)

which vanishes by the assumption in d). Concerning c), the case i = 0 follows from b) since W_{1}H^{0}(Xs̄, ℚ_{l}) = H^{0}(Xs̄, ℚ_{l}) and W_{1}H^{0}(Xη̄, ℚ_{l})^{I} = H^{0}(Xη̄, ℚ_{l})^{I}. The case i = 1 can be deduced from b) and [22][Exposé IX] or from results of Raynaud on the Picard functor [35]. We give the details of this last argument because the method, essentially using motivic cohomology, might be of some interest. The short exact sequence 0 → μ_{l^{ν}} → 𝔾_{m} ⟶^{l^{ν}} 𝔾_{m} → 0 of sheaves on X_{et} induces an isomorphism

(86)  R^{1}f_{∗}μ_{l^{ν}} ≅ (R^{1}f_{∗}𝔾_{m})_{l^{ν}}

of sheaves on S_{et} since (f_{∗}𝔾_{m})/l^{ν} = 0. Indeed, the stalks H^{0}(Y, O_{Y}^{×}) = ∏_{i} R_{i}^{×} and H^{0}(Yη̄, O_{Yη̄}^{×}) = ∏_{i}(L_{i} ⊗_{K} K̄)^{×} of f_{∗}𝔾_{m} are l-divisible since S is strictly Henselian. Here

X → Y = ∐_{i} Spec(R_{i}) → S

is the Stein factorization and L_{i} is the fraction field of R_{i}. The Leray spectral sequence for f gives an exact sequence

0 → H^{1}(S, f_{∗}𝔾_{m}) → H^{1}(X, 𝔾_{m}) → H^{0}(S, R^{1}f_{∗}𝔾_{m}) → H^{2}(S, f_{∗}𝔾_{m})

and H^{i}(S, f_{∗}𝔾_{m}) = 0 for i = 1, 2. Indeed, H^{1}(S, f_{∗}𝔾_{m}) = Pic(Y) = 0 (resp. H^{2}(S, f_{∗}𝔾_{m}) = Br(Y) = 0) since Y is the disjoint union of spectra of local (resp. strictly Henselian local) rings. Hence

(87)  Pic(X) ≅ H^{1}(X, 𝔾_{m}) ≅ H^{0}(S, R^{1}f_{∗}𝔾_{m}).

A similar argument for f_{η} shows

(88)  Pic(X_{η}) ≅ H^{1}(X_{η}, 𝔾_{m}) ≅ H^{0}(η, R^{1}f_{∗}𝔾_{m}).

We have a commutative diagram

H^{1}(Xs̄, μ_{l^{ν}}) ⟶^{∼} H^{0}(S, R^{1}f_{∗}μ_{l^{ν}}) ⟶^{∼} H^{0}(S, R^{1}f_{∗}𝔾_{m})_{l^{ν}} ⟶^{∼} Pic(X)_{l^{ν}}
sp↓  ↓  ↓  ↓
H^{1}(Xη̄, μ_{l^{ν}})^{I} ⟶^{∼} H^{0}(η, R^{1}f_{∗}μ_{l^{ν}}) ⟶^{∼} H^{0}(η, R^{1}f_{∗}𝔾_{m})_{l^{ν}} ⟶^{∼} Pic(X_{η})_{l^{ν}}

where the isomorphisms in the top row are given by proper base change, (86) and (87) and in the bottom row by an elementary stalk computation, (86) and (88). Passing to the inverse limit over ν we are reduced to studying the map

(89)  \underleftarrow{lim}_{ν} Pic(X)_{l^{ν}} =: T_{l} Pic(X) → T_{l} Pic(X_{η}) := \underleftarrow{lim}_{ν} Pic(X_{η})_{l^{ν}}

and the proof of c) for i = 1 is then finished by the following Lemma. □

Lemma 13. The map (89) is injective with finite cokernel.

Proof. Since X is regular and X_{η} is an open subscheme the map

Pic(X) = Cl(X) → Cl(X_{η}) = Pic(X_{η})

is surjective and its kernel K is the subgroup of Cl(X) generated by divisors supported in the closed subscheme Xs̄ ⊂ X, hence is a finitely generated abelian group [23][II.6]. By the snake lemma we obtain an exact sequence

(90)  0 = T_{l}K → T_{l} Pic(X) → T_{l} Pic(X_{η}) → K̂ ⟶^{ρ} P̂ic(X)

where Â = \underleftarrow{lim}_{ν} A/l^{ν} denotes the l-completion of an abelian group A. Let Pic^{0}(X) ⊆ Pic(X) be the subgroup defined in [35][3.2 d)], i.e. the kernel of the map

(91)  Pic(X) = P (S) → (P/P^{0})(s̄) × (P/P^{0})(η̄)

where P = Pic_{X/S} is the relative Picard functor of f [35][1.2] and P^{0} is the connected component of P restricted to schemes over s̄ (resp. η). Note that over a field P is represented by a group scheme, locally of finite type, hence has a well defined connected component. By [35][Thm. 3.2.1] the target group in (91) - the product of the Neron-Severi groups of the geometric fibres - is finitely generated, hence so is Pic(X)/ Pic^{0}(X). By [35][Thm. 6.4.1] - and this is the key fact in the proof- the group K∩Pic^{0}(X) is finite. In the notation of loc. cit. we have K = E(S) by Prop. 6.1.3 and Pic^{0}(X) = P^{0}(S) ⊆ P^{τ} (S). Hence the kernel of K → Pic(X)/ Pic^{0}(X) is finite and since both groups are finitely generated, so is the kernel on their l-completions. But this means that the map ρ in (90) has finite kernel which proves the Lemma. □

10.2. p-adic cohomology. In this section we assume that K has characteristic 0 and for simplicity also that k = 𝔽_{p}. For l = p one still has the specialisation map

(92)  sp : H^{i}(Xs̄, ℚ_{p}) → H^{i}(Xη̄, ℚ_{p})^{I}

since proper base change holds for arbitrary torsion sheaves. However, it is well known that p-adic étale cohomology of varieties in characteristic p only captures the slope 0 part of the full p-adic cohomology, which is Berthelot’s rigid cohomology H_{rig}^{i}(X_{s}/k) (for proper X_{s} this follows from [3, Thm. 1.1] and [24, Prop. 3.28, Lemma 5.6]). Here the slope 0 part V^{slope 0} of a finite dimensional ℚ_{p}-vector space V with an endomorphism ϕ is the maximal subspace on which the eigenvalues of ϕ are p-adic units. One knows that the eigenvalues of ϕ on H_{rig}^{i}(X_{s}/k) are Weil numbers, and a proof similar to that of Lemma 12 shows that the same is true for D_{pst}(H^{i}(Xη̄, ℚ_{p})), and hence for

D_{cris}(H^{i}(Xη̄, ℚ_{p})) = D_{st}(H^{i}(Xη̄, ℚ_{p}))^{N=0} = D_{pst}(H^{i}(Xη̄, ℚ_{p}))^{I,N=0}.

Therefore one deduces weight filtrations on both spaces. In analogy with the l-adic situation one might make the following conjecture.

Conjecture 6. Let X → S be proper, flat and generically smooth. Then there is a ϕ-equivariant specialization map

H_{rig}^{i}(X_{s}/k) ⟶^{sp′} D_{cris}(H^{i}(Xη̄, ℚ_{p}))

and a commutative diagram of Gal(k̄/k)-modules

H^{i}(Xs̄, ℚ_{p})  ⟶^{sp}  H^{i}(Xη̄, ℚ_{p})^{I}
λ_{s}↓  ^{λ_{η}} ↓
H_{rig}^{i}(X_{s}/k) ⊗_{ℚ_{p}} ℚ̂_{p}^{ur} ⟶^{sp′⊗1} D_{cris}(H^{i}(Xη̄, ℚ_{p})) ⊗_{ℚ_{p}} ℚ̂_{p}^{ur}

where ℚ̂_{p}^{ur} is the p-adic completion of the maximal unramified extension of ℚ_{p}. Moreover, the vertical maps induce isomorphisms

λ_{s} : H^{i}(Xs̄, ℚ_{p}) ≅ (H_{rig}^{i}(X_{s}/k) ⊗_{ℚ_{p}} ℚ̂_{p}^{ur})^{ϕ⊗ϕ=1} ≅ H_{rig}^{i}(X_{s}/k)^{slope 0}

and

λ_{η} : H^{i}(Xη̄, ℚ_{p})^{I} ≅ (D_{cris}(H^{i}(Xη̄, ℚ_{p})) ⊗_{ℚ_{p}} ℚ̂_{p}^{ur})^{ϕ⊗ϕ=1}

≅ D_{cris}(H^{i}(Xη̄, ℚ_{p}))^{slope 0}. (93)

Note here that for any ϕ-module D the Gal(k̄/k)-module (D ⊗_{ℚ_{p}} ℚ̂_{p}^{ur})^{ϕ⊗ϕ=1} can also be viewed as a ϕ-module (via the action of ϕ ⊗ 1) and as such is noncanonically isomorphic to D^{slope 0}. Moreover the action of Frob_{p}^{−1} ∈ Gal(k̄/k) coincides with that of 1 ⊗ ϕ^{−1} = ϕ ⊗ 1 = ϕ. The p-adic analogue of Theorem 10.1 (replacing a) by the conjectural local theorem on invariant cycles) would be the following conjecture.

Conjecture 7. Assume that X is moreover regular. Then the following hold. a) The map sp′ is surjective.

b) The map

W_{i−1}H_{rig}^{i}(X_{s}/k) ⟶^{sp′} W_{i−1}D_{cris}(H^{i}(Xη̄, ℚ_{p}))

induced by sp′ is an isomorphism.

c) The map sp′ is an isomorphism for i = 0, 1.

Combining both conjectures we deduce the following statement for p-adic étale cohomology.

Conjecture 8. If X is regular then the map

W_{i−1}H^{i}(Xs̄, ℚ_{p}) ⟶^{sp} W_{i−1}H^{i}(Xη̄, ℚ_{p})^{I}

induced by sp is an isomorphism.

Here we deduce the weight filtrations on H^{i}(Xs̄, ℚ_{p}) and H^{i}(Xη̄, ℚ_{p})^{I} from Conjecture 6 via the injectivity of the maps λ_{s} and λ_{η}. For the applications in this paper we only need this isomorphism on W_{0} (or in fact on the still smaller generalized eigenspace for the eigenvalue 1). For reference we record this statement separately.

Conjecture 9. If X is regular then the map

W_{0}H^{i}(Xs̄, ℚ_{p}) ⟶^{sp} W_{0}H^{i}(Xη̄, ℚ_{p})^{I}

induced by sp is an isomorphism, where W_{0} is the sum of generalized ϕeigenspaces for eigenvalues which are roots of unity.

Again, if Conjecture 6 holds the maps λ_{s} and λ_{η} are injective and it suffices to establish an isomorphism

W_{0}H_{rig}^{i}(X_{s}/k) ≅ W_{0}D_{cris}(H^{i}(Xη̄, ℚ_{p})).

We do not know how to establish Conjecture 6 or Conjecture 9 in general, since it seems difficult to make use of the regularity assumption. In case X has semistable reduction, however, it seems plausible that one can avoid any reference to rigid cohomology and establish a commutative diagram of Gal(k̄/k)- modules

H^{i}(Xs̄, ℚ_{p})  ⟶^{sp}  H^{i}(Xη̄, ℚ_{p})^{I}
λ̃_{s}↓  ^{λ_{η}} ↓
(H_{HK}^{i}(X_{s}/k)^{N=0}) ⊗_{ℚ_{p}} ℚ̂_{p}^{ur} ⟶^{c′⊗1} D_{cris}(H^{i}(Xη̄, ℚ_{p})) ⊗_{ℚ_{p}} ℚ̂_{p}^{ur}

where H_{HK}^{i}(X_{s}/k) is Hyodo-Kato cohomology. Contrary to what the notation suggests this cohomology theory not only depends on X_{s}/k but on the scheme X/S. Building on work of Fontaine-Messing, Bloch-Kato, Hyodo-Kato, and Kato-Messing, Tsuji [39] proved that there is an isomorphism of (ϕ, N)-modules

H_{HK}^{i}(X_{s}/k) ⟶^{c} D_{st}(H^{i}(Xη̄, ℚ_{p}))

and hence an isomorphism of ϕ-modules

H_{HK}^{i}(X_{s}/k)^{N=0} ⟶^{c′} D_{st}(H^{i}(Xη̄, ℚ_{p}))^{N=0} = D_{cris}(H^{i}(Xη̄, ℚ_{p})).

In addition to the commutative diagram it would then be enough to show that λ̃_{s} and λ_{η} are injective. We refrain from giving more details since in this paper Conjecture 9 is only used in the proof of Proposition 9.2 (via Proposition 9.1) which already needs to assume a host of other, much deeper conjectures that we are unable to prove.

Remark 1. (Added in Proof) Conjectures 6, 7 c), 8 and 9 have meanwhile been proven in the Caltech thesis of Yitao Wu [40]. In fact Wu shows that

H^{i}(Xs̄, ℚ_{p}) ⟶^{sp} H^{i}(Xη̄, ℚ_{p})^{I}

is an isomorphism for regular X. Conjectures 7 a) and b) however, still seem out of reach.

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M. Flach

Department of Mathematics Caltech Pasadena CA 91125 USA flach@caltech.edu B. Morin Department of Mathematics Caltech Pasadena CA 91125 USA 400

Documenta Mathematica 17 (2012)