Tame class field theory for arithmetic schemes

by Alexander Schmidt

final version, November 6, 2004

The aim of global class field theory is the description of abelian extensions of a finitely generated field k in terms of its arithmetic invariants. The solution of this problem in the case of fields of dimension 1 was one of the major achievements of number theory in the first part of the previous century.

In the 1980s, mainly due to K. Kato and S. Saito [K-S3], a generalization to higher dimensional fields has been found. The description of the abelian extensions is given in terms of a generalized idèle class group, whose rather involved definition is based on Milnor K-sheaves. However, if one restricts attention to unramified extensions (with respect to a regular, projective model X of k), then class field theory has a nice geometric description using the Chow group CH_{0}(X) of zero-cycles modulo rational equivalence (see [K-S1], [Sa]).

In the case of positive characteristic, a description of a similar geometric flavor of the tamely ramified abelian extensions of k (w.r.t. a finite set D_{1}, …, D_{r} of prime divisors of X) was given in [S-S]. The objective of this paper is to prove the following result in the mixed characteristic case.

Theorem 1 Let X be a regular connected scheme, flat and proper over Spec(ℤ) such that its generic fibre X ⊗_{ℤ} ℚ is projective over ℚ. Let D be a divisor on X whose vertical irreducible components are normal schemes. Then there exists a natural isomorphism of finite abelian groups

rec_{X,D} : CH_{0}(X, D) ⟶^{∼} π̃_{1}^{t}(X, D)^{ab} .

Let us explain the ingredients of the theorem. First of all, π̃_{1}^{t}(X, D)^{ab} is the abelianized modified tame fundamental group, which classifies finite abelian étale coverings of X − supp(D) which are at most tamely ramified along D (cf. section 4) and in which every real point splits completely. CH_{0}(X, D) is the relative Chow group of zero-cycles (cf. section 1). It is a quotient of the group of zero-cycles on X − supp(D) by an equivalence relation which is given by the divisors of functions satisfying a condition with respect to D. This equivalence relation is in general finer than rational equivalence. Finally, the map rec_{X,D} is uniquely determined by the property that the class of a closed point x ∈ X − supp(D) is sent to its Frobenius automorphism in π̃_{1}^{t}(X, D)^{ab}.

Remarks. 1. In many cases (e.g. if X is semi-stable), the condition in the theorem on the vertical components of D is void. Furthermore, this condition can be weakened (see theorem 6.5).

2. If D is zero, Theorem 1 reduces to the “unramified class field theory” for arithmetic schemes of K. Kato and S. Saito [K-S1], [Sa] (which we use in the proof).

3. The finiteness of the group π̃_{1}^{t}(X, D)^{ab} was shown in [S1]. Furthermore (loc.cit.) this group depends only on the scheme U = X − supp(D). Hence it is desirable to give a formulation of the tame class field theory of U solely in terms of U. We conjecture that the above theorem is true with the relative Chow group replaced by h_{0}(U), the 0th singular homology group of U, which was considered in [S2]. Indeed, there exists a surjective reciprocity map and the conjecture is that this surjection is an isomorphism. At the moment this is known if dim X = 1 or if D is zero.

4. If dim X = 1, then CH_{0}(X, D) is isomorphic to the ray class group of the number field k(X) with modulus 𝔪_{D} where 𝔪_{D} is the (square-free) product of the points in D. In this case, rec is the reciprocity homomorphism of classical (one-dimensional) class field theory.

I want to thank M. Spieß for many remarks and comments he made during our joint work [S-S], which was the starting point of present paper. I am grateful to U. Jannsen for several helpful discussions and in particular for calling my attention to the existence of [Po]. Hearty thanks go to M. Levine from whom I learned much about relative K-groups. Finally, I want to thank D. Grayson for his comments on a preliminary version of this paper. A large parts of this work was written while I was a Heisenberg fellow of the Deutsche Forschungsgemeinschaft and I want to thank this institution for their financial support.

Notational Convention: Throughout this paper, given a scheme Y, we denote by Y^{+} the disjoint union of its irreducible components and by Ỹ the normalization in its total ring of fractions.

1 Relative Chow groups

We start by collecting some facts on relative K-groups. A more detailed discussion and proofs of the results mentioned below are found in [Le], §1. Following Quillen [Qu], the K-groups of an exact category C are defined by

K_{i}C = π_{i+1}(KC, ∗), KC = BQC.

Here B means geometric realization, Q is Quillen’s construction [Qu] and π_{i}(−, ∗) is the i-th homotopy group of a pointed topological space.

For a noetherian scheme X let M_{X} be the category of coherent O_{X}-sheaves and G_{i}X = K_{i}M_{X}. Assume that X is regular and let D = D_{1} + ⋯ + D_{r} be a sum of distinct irreducible divisors on X. Let Y := supp(D) = D_{1} ∪ ⋯ ∪ D_{r} ⊂ X, hence Y^{+} = D_{1} ∪. · · · ∪. D_{r}. Let M_{X|Y} be the full subcategory in M_{X} consisting of coherent O_{X}-sheaves F with Tor_{i}^{O_{X}} (F, O_{Y}) = 0 for i > 0. Since every coherent O_{X}-sheaf is the quotient of a vector bundle, the resolution theorem shows that the inclusion M_{X|Y} → M_{X} induces an isomorphism on K-groups. The natural functor j_{Y^{+}}^{∗} : M_{X|Y} → M_{Y^{+}} is exact and we define the relative G-groups G_{∗}(X, D) as the homotopy groups of the homotopy fibre of j_{Y^{+}}^{∗} : KM_{X|Y} → KM_{Y^{+}}. If f : X′ → X is a flat morphism inducing a flat morphism f_{|Y′^{+}} : Y′^{+} → Y^{+}, then we obtain a natural homomorphism

i^{∗} : G_{∗}(X, D) ⟶ G_{∗}(X′, D′)

(see [Co], [Le] for the technical details). Recall [Qu] the usual Quillen spectral sequence for X

(1)  E_{1}^{pq}(X) = ⨁_{x∈X^{p}} K_{−p−q}(k(x)) ⇒ G_{−p−q}(X)

which is associated to the filtration by codimension of support on M_{X}

M_{X}^{i} = {F ∈ M_{X} | codim_{X}supp(F) ≥ i}.

As is well known, the Chow group of codimension p cycles on X and the term E_{2}^{p,−p} of the above spectral sequence are naturally isomorphic. Our next aim is to construct a natural spectral sequence converging to G_{∗}(X, D). The relative Chow group of codimension p cycles will be the term E_{2}^{p,−p} of that spectral sequence by definition.

Let M_{X|Y}^{i} ⊂ M_{X|Y} be the full subcategory of coherent sheaves F in M_{X|Y} with codim_{X}supp(F) ≥ i. Let G^{i}(X, D) be the homotopy fibre of BQj_{Y^{+}}^{∗} : BQM_{X|Y}^{i} → BQM_{Y^{+}}^{i} and G_{p}^{i}(X, D) = π_{p+1}(G^{i}(X, D)). For i < k, let M_{X|Y}^{i/k} be the direct limit

M_{X|Y}^{i/k} = lim_{⟶}_{Z⊂X Z intersecting Y properly codim_{X}Z≥k} M_{X−Z|Y −Z}^{i}.

and let M_{Y^{+}}^{i/k} be the direct limit

M_{Y^{+}}^{i/k} = lim_{⟶}_{\begin{matrix}Z⊂Y^{+} \\ codim_{Y ^{+}} Z≥k \end{matrix}} M_{Y^{+}−Z}^{i}.

Let G^{i/k}(X, D) be the homotopy fibre of BQM_{X|Y}^{i/k} → BQM_{Y^{+}}^{i/k} and put

G_{p}^{i/k}(X, D) = π_{p+1}(G^{i/k}(X, D)). By [Le], 1.6, we obtain a spectral sequence

(2)  E_{1}^{pq}(X|Y) ⟹ G_{−p−q}(X),

in which the E_{1}-terms are given by

E_{1}^{pq}(X|Y) = \{ \begin{matrix}K_{−p−q}(M_{X|Y}^{p/p+1}), & −p − q > 0, p ≤ dim X \\ K̄_{0}(M_{X|Y}^{p/p+1}), & −p − q = 0 \\ 0 & otherwise. \end{matrix}

Here K̄_{0}(M_{X|Y}^{p/p+1}) := im (K_{0}(M_{X|Y}^{p}) ⟶ K_{0}(M_{X|Y}^{p/p+1})). The natural functor

M_{X|Y}^{p/p+1} → M_{X}^{p/p+1} induces a map from the E_{1}-terms of this spectral sequence (2) to that of the usual Quillen spectral sequence (1) which is compatible with differentials. Furthermore, one obtains a spectral sequence

(3)  E_{1}^{pq}(X, D) ⟹ G_{−p−q}(X, D),

in which the E_{1}-terms are given by

E_{1}^{pq}(X, D) = \{ \begin{matrix}G_{−p−q}^{p/p+1}(X, D), & −p − q > 0, p ≤ dim X \\ Ḡ_{0}^{p/p+1}(X, D), & −p − q = 0 \\ 0 & otherwise. \end{matrix}

Here Ḡ_{0}^{p/p+1}(X, D) = im (G_{0}^{p}(X, D) → G_{0}^{p/p+1}(X, D)). The filtration on G_{∗}(X, D) is the “topological” filtration:

F^{p}G_{∗}(X, D) = im (G_{∗}^{p}(X, D) ⟶ G_{∗}(X, D)) .

Definition 1.1 We call the group

CH^{p}(X, D) = E_{2}^{p,−p}(X, D)

the relative Chow group of codimension p-cycles. If X is equidimensional of dimension d, we call the group CH_{p}(X, D) = E_{2}^{d−p,−d+p}(X, D) the relative Chow group of p-cycles.

Remark: If D is empty, then

CH^{p}(X, ∅) = coker( ⨁_{x∈X^{p−1}} k(x)^{×} → ⨁_{x∈X^{p}} ℤ)

is the usual group of codimension p cycles on X modulo rational equivalence.

Finally, we mention the existence of a long exact sequence of E_{1}-terms:

⋯ → E_{1}^{p,−p−2}(Y^{+}) → E_{1}^{p,−p−1}(X, D) → E_{1}^{p,−p−1}(X|Y) → E_{1}^{p,−p−1}(Y^{+})

compatible with the differentials. If the maps G_{0}^{p}(X, D) → G_{0}^{p/p+1}(X, D) and K_{0}(M_{X|Y}^{p}) → K_{0}(M_{X|Y}^{p/p+1}) are surjective, we can extend this sequence to the right up to E_{1}^{p,−p}(Y^{+}).

2 Explicit description of CH_{0}(X, D)

The rather implicit definition of the relative Chow groups in the last section was given in order to obtain good functorial properties. In this section we will give a more explicit description of the group CH_{0}(X, D) as the group of zero cycles on X − supp(D) modulo relations. These relation come via the divisor map from families of rational functions on curves in X − supp(D) which satisfy an additional condition. Informally speaking, the condition is that the product of the values of these functions at points of D must be 1.

We assume from now on that X is an excellent regular noetherian scheme. Then the closed subscheme Y = supp(D) is locally principal, locally defined by a non-zero divisor. A coherent sheaf F is in M_{X|Y} if and only if it is without I_{Y} -torsion and for F ∈ M_{X|Y} the support supp(F) has proper intersection with Y. If Z is a closed reduced subscheme, then O_{Z} is in M_{X|Y} if and only if Z intersects Y properly. For 0 ≤ p ≤ dim(X), we consider the (abelian) categories

M_{X:Y}^{p} − coherent O_{X}-sheaves whose support is contained in a subscheme of codimension ≥ p which has proper intersection with Y.

We have a natural inclusion M_{X|Y}^{p} → M_{X:Y}^{p}.

Lemma 2.1 The natural map QM_{X|Y}^{p} → QM_{X:Y}^{p} is a homotopy equivalence for all p.

Proof: Let F ∈ M_{X|Y}^{p} and let Z be the (not necessarily reduced) subscheme associated with the ideal sheaf ann(F). Then F is an O_{Z}-module and O_{Z} ∈ M_{X|Y}^{p}. Therefore the natural inclusion M_{X|Y}^{p} → M_{X:Y}^{p} can be factored as

M_{X|Y}^{p} ⟶^{i} C^{p} ⟶^{j} M_{X:Y}^{p} ,

where C^{p} is the full subcategory of objects in M_{X:Y}^{p} which are O_{Z}-modules for some closed (not necessarily reduced) subscheme Z ⊂ X with O_{Z} ∈ M_{X|Y}^{p}. The category C^{p} is closed under taking subobjects, quotient and finite direct products and is therefore abelian.

Let F be a coherent O_{X}-sheaf whose support is of codimension ≥ p and has proper intersection with Y. Since X is noetherian, the coherent ideal sheaf ann(F) contains a power of its radical. Therefore we find a filtration 0 = F_{0} ⊂ F_{1} ⊂ ⋯ ⊂ F_{n} = F of F such that all subquotients are O_{Z}-modules, where Z = supp(F) as reduced subscheme. Hence all subquotients are in C^{p} and we conclude by devissage that Qj is a homotopy equivalence.

Now assume that F ∈ C^{p}, i.e. F is an O_{Z}-module where Z is a closed subscheme with O_{Z} ∈ M_{X|Y}^{p}. Since X is regular, we find a locally free O_{X}- sheaf F and a surjection F ↠ F. It factors to a surjection

F/I_{Z} −↠^{p} F.

The sheaf F/I_{Z} (being a locally free O_{Z}-sheaf) is without I_{Y} -torsion. Therefore F/I_{Z} and also its subsheaf ker(p) are in M_{X|Y}^{p} and Quillen’s resolution lemma implies that Qi is a homotopy equivalence. □

For a closed subscheme Z ⊂ X, let M_{X}(Z) be the category of coherent O_{X}-sheaves with support in Z. Defining M_{X:Y}^{p/p+1} as the direct limit

M_{X:Y}^{p/p+1} = lim_{⟶}_{W ⊂Z⊂X codim_{X}Z≥p codim_{X}W ≥p+1 W,Z intersect Y properly} M_{X−W} (Z),

we conclude that also the natural map QM_{X|Y}^{p/p+1} → QM_{X:Y}^{p/p+1} is a homotopy equivalence. This allows to calculate some terms of the spectral sequence (2) of the last section.

Proposition 2.2 The map i_{X−Y}^{∗} : M_{X|Y}^{p/p+1} → M_{X−Y}^{p/p+1} induces an isomorphism

K_{0}(M_{X|Y}^{p/p+1}) ⟶^{∼} K_{0}(M_{X−Y}^{p/p+1}) ⟶^{∼} ⨁_{x∈(X−Y)^{p}} ℤ .

For q = 1, 2, we obtain short exact sequences

0 → K_{q}(M_{X|Y}^{p/p+1}) → K_{q}(M_{X−Y}^{p/p+1}) →^{δ} K_{q−1}(M_{Y}^{p/p+1}) → 0
↑^{≀}  ↑^{≀}
⨁_{x∈(X−Y)^{p}}K_{q}(k(x))  ⨁_{y∈Y^{p}} K_{q−1}(k(y)).

Here δ is the composition

K_{q}(M_{X−Y}^{p/p+1}) →^{incl} K_{q}(M_{X}^{p/p+1}) →^{∂} K_{q−1}(M_{X}^{p+1/p+2}) →^{proj} K_{q−1}(M_{Y}^{p/p+1})
↑^{≀}  ↑^{≀}  ↑^{≀}  ↑^{≀}
⨁_{x∈(X−Y)^{p}} K_{q}(k(x))  ⨁_{x∈X^{p}}K_{q}(k(x))  ⨁_{x∈X^{p+1}}K_{q−1}(k(x))  ⨁_{y∈Y^{p}}K_{q−1}(k(y)),

where ∂ is the boundary map of the Quillen spectral sequence.

Proof: First of all, we can replace M_{X|Y}^{p/p+1} by M_{X:Y}^{p/p+1}. The localization theorem shows that

QM_{Y}^{p/p+1} → QM_{X:Y}^{p/p+1} → QM_{X−Y}^{p/p+1}

is a homotopy fibre sequence. This implies a long exact sequence

⋯ → K_{1}(M_{X−Y}^{p/p+1}) →^{δ_{1}} K_{0}(M_{Y}^{p/p+1}) → K_{0}(M_{X:Y}^{p/p+1}) → K_{0}(M_{X−Y}^{p/p+1}).

Let x ∈ (X − Y)^{p} and let Z = \overline{\{x\}} be its closure in X. Then Z has proper intersection with Y and the class of the free rank one O_{Z}-module in K_{0}(M_{X:Y}^{p}) maps to the class of the one-dimensional k(x)-vector space in K_{0}(M_{X−Y}^{p/p+1}). Hence K_{0}(M_{X:Y}^{p/p+1}) → K_{0}(M_{X−Y}^{p/p+1}) is surjective and in order to conclude the proof, it remains to show that the maps δ_{q} : K_{q}(M_{X−Y}^{p/p+1}) → K_{q−1}(M_{Y}^{p/p+1}) are surjective for q ≤ 3.

As is well known, K_{q−1}(M_{Y}^{p/p+1}) = ⨁_{y∈Y^{p}} K_{q−1}(k(y)). Let y ∈ Y^{p} be any point. It suffices to show that any element of α ∈ K_{q−1}(k(y)) (considered as an element of K_{q−1}(M_{Y}^{p/p+1})) is in the image of δ_{q}. Furthermore (q ≤ 3), we may suppose that α = {a_{1}, …, a_{q−1}} is a symbol. Choose a closed integral subscheme Z of codimension p in X which has proper intersection with Y and such that y is a regular point of Z. Let R be the semi-local ring of Y ∩ Z in Z and let F be its quotient field. Since X is excellent, the normalization R̃ of R in F (a semi-local PID) is finite over R. So we obtain the following commutative diagram

K_{q}(M_{X−Y}^{p/p+1}) ⟶ K_{q−1}(M_{Y}^{p+1})
↑  ↑
K_{q}(F)  ⟶ ⨁_{𝔪⊂R} K_{q−1}(R/𝔪)
↑^{≀}  ↑
K_{q}(F)  ⟶ ⨁_{𝔪̃⊂R̃} K_{q−1}(R̃/𝔪̃) .

Since y is a regular point of Z, there is exactly one 𝔪̃_{0} ⊂ R̃ corresponding to y and R̃/m̃_{0} = k(y). Now let π ∈ R̃ be an element which is a local parameter for 𝔪̃_{0} and is congruent to 1 modulo all other 𝔪̃ ⊂ R̃. Then the symbol {a_{1}, …, a_{q−1}, π} ∈ K_{q}(F) defines the required pre-image of α in K_{q}(M_{X−Y}^{p/p+1}).

□

Lemma 2.3 If X is pure of dimension d, then

E_{1}^{d,−d}(X, D) ≅ ⨁_{x∈(X−Y)^{d}} ℤ .

Proof: By definition, E_{1}^{d,−d}(X, D) = im(G_{0}^{d}(X, D) → G_{0}^{d/d+1}(X, D)). By dimension reasons and by proposition 2.2, we have a commutative diagram

G_{0}^{d}(X, D) ⟶^{∼}  K_{0}(M_{X|Y}^{d})
↓  ↓^{α}
G_{0}^{d/d+1}(X, D) ⟶^{∼} K_{0}(M_{X|Y}^{d/d+1}) ⟶^{∼}  ⨁_{x∈(X−Y)^{d}} ℤ .

Furthermore, α is surjective, since it has a section s_{α} which is defined by sending the class of an n-dimensional vector space over k(x), x ∈ (X − Y)^{p} to the class of the free rank-n module over O ¯ {x}. This implies the statement. □

Proposition 2.4 Assume that X is a noetherian, excellent, regular and connected scheme of dimension d. Let D be a divisor on X and let Y = supp(D). Then the group CH_{0}(X, D) is the quotient of ⨁_{x∈(X−Y)^{d}} ℤ by the image of the

group

R_{X,D} =^{df} ker ( ker ( ⨁_{x∈(X−Y)^{d−1}} k(x)^{×} → ⨁_{y∈Y^{d−1}} ℤ) ⟶ ⨁_{z∈(Y^{+})^{d−1}} k(z)^{×})

under the divisor map.

Proof: By definition, we have CH_{0}(X, D) = E_{2}^{d,−d}(X, D) = E_{1}^{d,−d}(X, D)/ im E_{1}^{d−1,d}(X, D). By lemma 2.3, we have a commutative diagram with exact rows

0  → E_{1}^{d,−d}(X, D) →^{∼} E_{1}^{d,−d}(X|Y) →  0
↑  ↑  ↑  ↑
E_{1}^{d−1,−d−1}(Y^{+}) → E_{1}^{d−1,−d}(X, D) → E_{1}^{d−1,−d}(X|Y) → E_{1}^{d−1,−d}(Y^{+})

Therefore, CH_{0}(X, D) is the quotient of E_{1}^{d,−d}(X, D) ≅ ⨁_{x∈(X−Y)^{d}} ℤ by the image of ker(E_{1}^{d−1,−d}(X|Y) → E_{1}^{d−1,−d}(Y^{+})). The statement of the proposition follows from proposition 2.2. □

In order to understand the group CH_{0}(X, D) explicitly, it remains to give a description of the maps

δ_{0} : ⨁_{x∈(X−Y)^{d−1}} k(x)^{×} → ⨁_{y∈Y^{d−1}} ℤ

and

ϕ : ker ( ⨁_{x∈(X−Y)^{d−1}} k(x)^{×} →^{δ_{0}} ⨁_{y∈Y^{d−1}} ℤ) ⟶ ⨁_{z∈(Y^{+})^{d−1}} k(z)^{×}

of the last proposition. The map δ_{0} is determined by the boundary map ∂_{0} : K_{1}(M_{X}^{d−1/d}) → K_{0}(M_{X}^{d/d+1}) of the Quillen spectral sequence and an explicit description of ∂_{0} is given at the beginning of the appendix. To describe ϕ, we need the following lemma which describes the natural map res : E_{1}^{p,−p−q}(X|Y) → E_{1}^{p,−p−q}(Y^{+}) if X is affine and Y = D is irreducible.

Lemma 2.5 If X = Spec(A) is affine and Y is irreducible, defined by a nonzero element r ∈ A, then the following diagram commutes for all p, q ≥ 0

K_{q}(M_{X|Y}^{p/p+1}) ⟶^{can} K_{q}(M_{X−Y}^{p/p+1})
↓^{res}  ↓^{·∪r}
K_{q}(M_{Y}^{p/p+1}) ⟵^{δ_{q}} K_{q+1}(M_{X−Y}^{p/p+1}).

Here res is the natural map induced by the embedding of Y to X, δ_{q} is the com-

posite map K_{q+1}(M_{X−Y}^{p/p+1}) ↪ K_{q+1}(M_{X}^{p/p+1}) →^{∂_{q}} K_{q}(M_{X}^{p+1/p+2}) →^{pr} K_{q}(M_{Y}^{p/p+1})

and · ∪ r is the cup product with the class of r in K_{1}(A[r^{−1}]) = K_{1}(X − Y), induced by the pairing

P(X − Y) × M_{X−Y}^{p/p+1} → M_{X−Y}^{p/p+1}, (P, M) ↦ P ⊗ M.

To describe ϕ, it suffices to understand its z-component for each z ∈ Y^{+}. Therefore we may suppose that X = Spec(A) is affine, local and that D = Y is irreducible, given as the zero locus of an element 0 ≠ r ∈ A. Let y be the closed point of Y. By the lemma (for p = d − 1 and q=1), we obtain the following explicit description of the map

ϕ : ker ( ⨁_{x∈(X−Y)^{d−1}} k(x)^{×} →^{δ_{0}} ℤ) ⟶ k(y)^{×}.

An element a = (a_{x})_{x} ∈ ker ( ⨁_{x∈(X−Y)^{d−1}} k(x)^{×} →^{δ_{0}} ℤ) maps to the image of the symbol {a, r} = ({a_{x}, r})_{x} ∈ ⨁_{x∈(X−Y)^{d−1}} K_{2}(k(x)) under the boundary map

δ_{1} : ⨁_{x∈(X−Y)^{d−1}} K_{2}(k(x)) ⟶ k(y)^{×}.

An explicit description of the boundary map δ_{1} is given at the beginning of the appendix.

Proof of lemma 2.5: Let us fix some notation: For a category C let its subdivision Sub(C) be the category whose objects are morphisms in C and a morphism from f to g is a commutative diagram

• ⟶^{f} •
↑  ↓
• ⟶^{g} • .

The functor Sub(C) → C which sends an arrow to its target is a homotopy equivalence ([Gr], p.228). Assume that C is an exact category in which every exact sequence splits. Let Inj(C) and Iso(C) be the categories whose objects are those of C and whose morphisms are the admissible monomorphisms and the isomorphisms in C, respectively. Put I = Iso(C) and E = Sub(Inj(C)). Sending

M ↪^{i} N to coker(i) defines a natural functor E → QC. The symmetric monoidal

category I acts on E by P ⊕ (M ↪^{i} N) = (P ⊕ M ↪^{id_{P} ⊕i} P ⊕ N) and we obtain a fibration sequence ([Gr], p.228)

I^{−1}I → I^{−1}E → QC

Furthermore, I^{−1}E is contractible which induces a homotopy equivalence

ψ : BI^{−1}I ⟶^{∼} ΩBQC

(the well-known comparison map between Ext- and Q-construction). For M ∈ Ob(C), we have two natural maps (i_{M})_{!}, (p_{M})^{!} : 0 → M in QC which are associated to the canonical maps i_{M} : 0 ↪ M and p_{M} : M ↠ 0 in C. The map ψ is characterized by sending an object (N, M) of I^{−1}I to the loop

0 ⟶^{(p_{N})_{!}} N ⟵^{(i_{N})^{!}}0 ⟶^{(i_{M})^{!}} M ⟵^{(P_{M})_{!}}0

in BQC.

Next, following [Sn], we construct a categorical description of the boundary map. Let C = M_{X|Y}^{p/p+1} (note that every exact sequence in C splits). Consider the category with the same objects as C and whose morphisms are (not necessarily admissible) injections i : M ↪ N in C whose cokernel is annihilated by r, and let H be its subdivision. The category H is the source of two functors:

1. Let J = Iso(M_{X−Y}^{p/p+1}). Sending the object i : M ↪ N of H to the restriction of N to X − Y yields a natural functor H → J.

2. Sending i : M ↪ N to coker(i) yields a functor H → QM_{Y}^{p/p+1}.

There is a natural I-action on H given by P ⊕(M ↪^{i} N) = (P ⊕M ↪^{id_{P} ⊕i} P ⊕N). We therefore obtain functors

α : I^{−1}H → J^{−1}J, β : I^{−1}H → QM_{Y}^{p/p+1}.

By [Sn], proof of theorem 2.2 (our H corresponds to g^{−1}(0) there), I^{−1}H → I^{−1}J is a homotopy equivalence and I^{−1}J → J^{−1}J is a homotopy equivalence on base point components. Furthermore (loc.cit.), the following diagram commutes for q ≥ 0

π_{q+1}(BI^{−1}H) ⟶^{β_{∗}} π_{q+1}(BQM_{Y}^{p/p+1})
↓^{≀ α_{∗}}  ∥
π_{q+1}(BJ^{−1}J) ⟶^{δ} π_{q+1}(BQM_{Y}^{p/p+1}),

which gives a categorical description of the boundary map δ : K_{q+1}(M_{X−Y}^{p/p+1}) → K_{q}(M_{Y}^{p/p+1}).

For an object M of I we have two morphism r_{M}, r^{M} : (0, 0 = 0) → (0, rM ⊂ M) in I^{−1}H given by the commutative diagrams

\begin{matrix}M & ⟶^{id_{M}} & M \\ _{∪} & & ↓^{id}_{M} \\ rM & ⟶ & M \end{matrix}  and  \begin{matrix}M & ⟶^{id_{M}} & M \\ ↑^{(·r)^{−1}} & ↓^{·r} \\ rM & ⟶ & M & , \end{matrix}

where (·r)^{−1} is the inverse of the isomorphism M →^{·r} rM. Consider the map γ : BI^{−1}I → ΩBI^{−1}H, which is characterized by sending an object (N, M) of I^{−1}I to the loop

(0, 0 = 0) ⟶^{r^{N}} (0, rN ⊂ N)⟵^{r_{N}}(0, 0 = 0) ⟶^{r_{M}} (0, rM ⊂ M)⟵^{r^{M}}(0, 0 = 0)

in BI^{−1}H. Considering the natural image of r ∈ A[r^{−1}]^{×} in the group K_{1}(A[r^{−1}]) = π_{1}(BGL(A[r^{−1}])^{+}) and using the comparison between, + and Ext-construction as described in [Gr], p.224, we see that for q ≥ 0 the induced map π_{q}(BI^{−1}I) ⟶^{γ} π_{q+1}(BI^{−1}H) ⟶^{∼} π_{q+1}(BJ^{−1}J) is the product map

· ∪ r : K_{q}(M_{X|Y}^{p/p+1}) ⟶ K_{q+1}(M_{X−Y}^{p/p+1}).

Furthermore, Ωβ∘γ = Ωres ∘ψ where res : QM_{X|Y}^{p/p+1} → QM_{Y}^{p/p+1} is the natural restriction map. Summing up, we obtain a commutative diagram

ΩBI^{−1}H ⟵^{γ}  BI^{−1}I
↓^{Ωα}  ↓^{Ωres∘ψ}
ΩBJ^{−1}J ⟶^{δ} ΩBQM_{Y}^{p/p+1}.

Passing to homotopy groups, we obtain the statement of lemma 2.5. □

3 Finiteness of CH_{0}(X, D)

The aim of this section is to detect relations in CH_{0}(X, D) and to use these relations to show that CH_{0}(X, D) is finite, when X is an arithmetic scheme. We first recall some well-known facts on the filtration on the K-groups K_{r}(F) for r = 1, 2 of discrete valuation fields. Let (F, v) be a discrete valuation field with residue field k. Let O_{F} be the valuation ring in F and let 𝔪 ⊂ O_{F} be its maximal ideal, thus O_{F} /𝔪 ≅ k. Then one puts U^{0}(F^{×}) = O_{F}^{×} and for i ≥ 1, U^{i}(F^{×}) = ker(O_{F}^{×} → (O_{F} /𝔪^{i})^{×}). As is well known, we have natural isomorphisms

F^{×}/U^{0}(F^{×}) ≅ ℤ , U^{0}(F^{×})/U^{1}(F^{×}) ≅ k^{×}.

For i ≥ 1 the group U^{i}K_{2}(F) is the subgroup generated by symbols {u, x} with u ∈ U^{i}(F^{×}) and x ∈ F^{×}. The group U^{0}K_{2}(F) is the kernel of the tame symbol K_{2}(F) → k^{×}. We have inclusions

K_{2}(F) ⊇ U^{0}K_{2}(F) ⊇ U^{1}K_{2}(F) ⊇ ⋯

and natural isomorphisms ([B-T], prop.4.3,4.5)

K_{2}(F)/U^{0}K_{2}(F) ≅ k^{×} , U^{0}K_{2}(F)/U^{1}K_{2}(F) ≅ K_{2}(k),

If F_{v} and F̂_{v} denote the henselization and the completion of F with respect to v, respectively, we have natural isomorphisms

K_{q}(F)/U^{1}K_{q}(F) ⟶^{∼} K_{q}(F_{v})/U^{1}K_{q}(F_{v}) ⟶^{∼} K_{q}(F̂_{v})/U^{1}K_{q}(F̂_{v})

for q = 1, 2. The usual approximation lemma for a finite set of discrete valuations on a field shows the Lemma 3.1 If v_{1}, …, v_{s} are finitely many different discrete valuations on a field F, then the diagonal maps

F^{×} → ⨁_{i=1}^{s} F_{v_{i}}^{×}/U^{1}(F_{v_{i}}^{×}) and K_{2}(F) → ⨁_{i=1}^{s} K_{2}(F_{v_{i}})/U^{1}K_{2}(F_{v_{i}})

are surjective.

Now let Y be any integral scheme of finite type over Spec(ℤ). We call Y horizontal if the generic point of Spec(ℤ) is in the image of the structural morphism. Otherwise we call Y vertical, in which case it is a scheme of finite type over a finite field. Let Z ⊂ Y be a closed subscheme of positive codimension. Consider the group

SK_{1}(Y, Z) := coker( ⨁_{y∈(Y −Z)_{1}} K_{2}(k(y)) → ⨁_{y∈Y_{0}} K_{1}(k(y) ).

There is a natural surjection SK_{1}(Y, Z) ↠ SK_{1}(Y) which is an isomorphism if Z is empty or has dimension zero. Any finite morphism f : Y_{1} → Y_{2} with f(Y_{1} −Z_{1}) ⊂ Y_{2} −Z_{2} induces a homomorphism f_{∗} : SK_{1}(Y_{1}, Z_{1}) → SK_{1}(Y_{2}, Z_{2}).

If Y is vertical, normal and proper we put 𝔽 = Γ(Y, O_{Y}). Note that 𝔽 is the algebraic closure of the prime field in k(Y) and that Y is a geometrically irreducible variety over 𝔽. The surjective norm maps N : k(y)^{×} → 𝔽^{×} for y ∈ Y_{0} give rise to a surjective norm map ⨁_{y∈Y_{0}} k(y)^{×} → 𝔽^{×}. It can be easily deduced from the special case when Y is a regular proper curve that the last map annihilates the image of the boundary map ⨁_{y∈Y_{1}} K_{2}(k(y)) → ⨁_{y∈Y_{0}} k(y)^{×}. Therefore we obtain a surjective norm map

N : SK_{1}(Y, Z) → 𝔽^{×}.

Proposition 3.2 Let Y be an integral scheme of finite type over Spec(ℤ) and let Z be a proper closed subscheme. Then SK_{1}(Y, Z) is finite. More precisely:

(i) If Y is horizontal or vertical but not proper, then SK_{1}(Y, Z) = 0.

(ii) Assume that Y is vertical and proper. Let Ỹ be the normalization of Y and let ZỸ be preimage of Z in Ỹ. Then the natural map SK_{1}(Ỹ , ZỸ) → SK_{1}(Y, Z) is surjective.

(iii) If Y is vertical, proper and normal and 𝔽 = Γ(Y, O_{Y}), then the norm

map

N : SK_{1}(Y, Z) ⟶^{∼} 𝔽^{×}

is an isomorphism.

(iv) For any y ∈ Y_{0}, the natural homomorphism k(y)^{×} → SK_{1}(Y, Z) is surjective.

Proof: Two special cases of the proposition are classical, namely (i) if Y is the spectrum of a ring of integers in a number field and Z = ∅ (see [BMS]) and (iii) for Y a smooth proper curve over a finite field and Z = ∅ (see [Mo]). We will deduce the general assertion from this two special cases. First we note that the normalization Ỹ → Y is finite. Therefore the norm maps induce a well-defined homomorphism SK_{1}(Ỹ , ZỸ) → SK_{1}(Y, Z), which is surjective, since k(y) is a finite field for any closed point y ∈ Y. Now we proceed in several steps.

1. Let Y be horizontal and irreducible. In order to show the vanishing of SK_{1}(Y, Z), it suffices to show that for every closed point y ∈ Y the image of k(y)^{×} in SK_{1}(Y, Z) is trivial. Let y be an arbitrary closed point. Choose a closed irreducible horizontal subscheme Y′ of Y containing y but not contained in Z. Putting Z′ = Y′ ∩ Z, the image of k(y)^{×} in SK_{1}(Y, Z) is contained in the image of the natural map SK_{1}(Y′, Z′) → SK_{1}(Y, Z). By induction on the dimension, we may assume that Y is of dimension one. In this case we have SK_{1}(Y, Z) = SK_{1}(Y) and Ỹ is the spectrum of the ring of S-integers in a number field. The surjection SK_{1}(Ỹ) → SK_{1}(Y) together with the vanishing of SK_{1}(Ỹ) (see [BMS]) shows that SK_{1}(Y) = 0.

2. If Y is affine and vertical, the same argument as in step 1, but now using Moore’s theorem [Mo], shows the vanishing of SK_{1}(Y, Z).

3. Now assume that Y is vertical, proper and normal and let 𝔽 = Γ(Y, O_{Y}). Then Y is geometrically integral over Spec(𝔽). We first assume that Y is projective. Let H be a geometrically integral hypersurface section not contained in Z (see [Po] for its existence) and Z_{H} = Z ∩ H. We obtain an exact sequence

SK_{1}(H̃, ZH̃) → SK_{1}(Y, Z) → SK_{1}(Y − H, Z − Z_{H}),

where H̃ is the normalization of H and ZH̃ is the preimage of Z_{H} in H̃. Since Y − H is affine, using step 2, the statement that the norm induces an isomorphism N : SK_{1}(Y, Z) ⟶^{∼} 𝔽^{×} can be reduced to the same statement for (H̃, ZH̃). By induction on the dimension, we may suppose that Y is a curve. In this case the statement is just Moore’s theorem [Mo]. This solves the projective case. If Y is proper but not necessarily projective over 𝔽, there exists a birational morphism f : Y′ → Y with Y′ normal and projective. Choose W ⊂ Y containing Z such that f : Y′ − f^{−1}(W) ⟶^{∼} Y − W. Then we obtain a surjective homomorphism SK_{1}(Y′, f^{−1}(W)) → SK_{1}(Y, Z), which shows the statement for (Y, Z).

4. Finally assume that Y is vertical but not proper. In the same manner as in step 3 we may suppose that Y is normal and quasi-projective. Let Ȳ be a projective compactification. Choosing an irreducible hypersurface section H not containing Z but containing at least one point of Ȳ − Y and using step 2, we can proceed by induction on the dimension. Finally, a non-proper curve is affine, showing SK_{1}(Y, Z) = 0.

It remains to show (iv) if Y is vertical and proper. If Y is normal, the statement follows from (iii). In the general situation, let ỹ be a point on the normalization Ỹ of Y projecting to y. The statement follows from the commutative diagram

k(ỹ) −↠ SK_{1}(Ỹ , ZỸ)
↓  ↓ ↓
k(y) → SK_{1}(Y, Z) .

□

Now assume that X is a connected noetherian excellent regular scheme of dimension d, D a divisor on X and Y = supp(D). We arrange the E_{1}-terms E_{1}(X, D), E_{1}(X|Y) and E_{1}(Y^{+}) of the spectral sequences introduced in the last section as a (up to sign convention) double complex in the following way:

0  → E_{1}^{d,−d}(X, D) →^{∼} E_{1}^{d,−d}(X|Y) →  0
↑  ↑  ↑  ↑
E_{1}^{d−1,−d−1}(Y^{+}) → E_{1}^{d−1,−d}(X, D) → E_{1}^{d−1,−d}(X|Y) →^{f} E_{1}^{d−1,−d}(Y^{+})
↑  ↑  ↑^{g}  ↑
E_{1}^{d−2,−d−1}(Y^{+}) → E_{1}^{d−2,−d}(X, D) → E_{1}^{d−2,−d}(X|Y) →^{h} E_{1}^{d−2,−d}(Y^{+})

The rows are exact. The vertical maps are the differentials of the various spectral sequences. In the notation of proposition 2.4, R_{X,D} = ker(f). The zeros in the diagram are due to dimension reasons. A diagram chase shows the

Lemma 3.3 There exists a natural exact sequence

im(f)/im(f ∘ g) ⟶ CH_{0}(X, D) ⟶ E_{2}^{d,−d}(X|Y) ⟶ 0.

In order to analyze the exact sequence of lemma 3.3, we first deduce some approximation results. Note that the maps of the next lemma are not the maps from the above diagram, unless Y is irreducible.

Lemma 3.4 The natural maps

(i) E_{1}^{d−2,−d}(X|Y) ⟶ E_{1}^{d−2,−d}(Y)

(ii) E_{1}^{d−1,−d}(X|Y) ⟶ E_{1}^{d−1,−d}(Y)

(iii) E_{1}^{d−2,−d−1}(X|Y) ⟶ E_{1}^{d−2,−d−1}(Y)

are surjective.

Proof: Let y_{1} ∈ Y^{d−2} and choose a point x ∈ (X − Y)^{d−2} such that y_{1} is a regular point of the closure W of {x} in X. Let F = k(W) be the function field and W′ the normalization of W. Let y′_{1}, … y′_{s} ∈ (W′)^{1} be the finitely many points of (W′)^{1} lying over point in Y^{d−2}. We choose the ordering such that y′_{1} is the unique point over y_{1} (by assumption, y_{1} is a regular point on W). Each y′ ∈ {y′_{1}, …, y′_{s}} defines a discrete valuation on F. Let M be the kernel of the (multi) tame symbol K_{2}(F) → ⨁_{i=1}^{s} k(y′_{i})^{×}. By proposition 2.2, each m ∈ M defines an element in E_{1}^{d−2,−d}(X|Y). We obtain a commutative diagram with exact lines

0 →  M  →  K_{2}(F)  → ⨁_{i=1}^{s} k(y′_{i})^{×} → 0
↓  ↓  ∥
0 → ⨁_{i=1}^{s} K_{2}(k(y′_{i})) → ⨁_{i=1}^{s} K_{2}(F_{y′_{i}})/U^{1}K_{2}(F_{y′_{i}}) → ⨁_{i=1}^{s} k(y′_{i})^{×} → 0 .

By lemma 3.1, the middle vertical arrow is surjective and therefore we find an m ∈ M mapping to an arbitrarily chosen α ∈ K_{2}(k(y_{1})) = K_{2}(k(ỹ_{1})) and to zero in all other components. This shows (i). The proofs of (ii) and (iii) are similar and left to the reader. □

Proposition 3.5 The natural map

E_{2}^{d,−d}(X|Y) ⟶ E_{2}^{d,−d}(X) = CH^{d}(X)

is an isomorphism. If X is flat and of finite type over Spec(ℤ), then both groups are finite.

Proof: By proposition 2.2, K_{1}(M_{X−Y}^{d−1/d}) ⟶^{δ} K_{0}(M_{Y}^{d−1/d}) is surjective. Therefore, any class in CH^{d}(X) can be represented by a cycle with support in X − Y, which shows the surjectivity. Next consider the commutative diagram

E_{1}^{d,−d}(X|Y) ↪^{a} E_{1}^{d,−d}(X) ↠ E_{1}^{d−1,−d−1}(Y)
↑  ↑^{b}  ↑
E_{1}^{d−1,−d}(X|Y) ↪ E_{1}^{d−1,−d}(X) ↠ E_{1}^{d−2,−d−1}(Y)
↑  ↑
E_{1}^{d−2,−d}(X|Y) → E_{1}^{d−2,−d}(X).

The lines and rows are complexes and, by proposition 2.2, the upper line is exact. If α ∈ E_{1}^{d,−d}(X|Y) represents an element in ker(E_{2}^{d,−d}(X|Y) → E_{2}^{d,−d}(X)), we find a β ∈ E_{1}^{d−1,−d}(X) with a(α) = b(β). By lemma 3.4 (iii), the composite map E_{1}^{d−2,−d}(X|Y) → E_{1}^{d−2,−d−1}(Y) is surjective. Therefore, we may choose β ∈ E_{1}^{d,−d}(X − Y) ⊂ E_{1}^{d−1,d}(X). By Proposition 2.2, we obtain β ∈ E_{1}^{d−1,−d}(X|Y), which shows the injectivity. Finally, if X is flat and of finite type over Spec(ℤ) then CH^{d}(X) is finite by [K-S3], theorem 6.1. □

Now we assume that X is of finite type over Spec(ℤ) and let, for i = 1, …, r, Z_{i} = ⋃_{j≠i} D_{j} ∩ D_{i} ⊂ D_{i}. Put Z = ∪. _{i=1}^{r} Z_{i} ⊂ Y^{+}. Let f, g and h be the maps in the diagram before lemma 3.3.

Lemma 3.6 (i) im(f) ⊃ ⨁_{y∈(Y^{+}−Z)_{0}} k(y)^{×}.

(ii) im(h) ⊃ ⨁_{y∈(Y^{+}−Z)_{1}} K_{2}(k(y)).

Proof: This follows from lemma 3.4 (i) and (ii). □

Let Ỹ be the normalization of Y.

Proposition 3.7 We have a natural surjection

SK_{1}(Ỹ) −↠ im(f)/im(f ∘ g).

Proof: First note that Ỹ is also the normalization of Y^{+}. Let ZỸ be the preimage of Z in Ỹ. By lemma 3.6 (ii), we obtain surjections

SK_{1}(Y^{+}, Z) −↠ E_{1}^{d−1,−d}(Y^{+})/im(f ∘ g) −↠ SK_{1}(Y^{+}).

Hence, by proposition 3.2 (iv), and lemma 3.6 (i), the natural inclusion map

im(f)/im(f ∘ g) ⟶ E_{1}^{d−1,−d}(Y^{+})/im(f ∘ g)

is an isomorphism. By proposition 3.2 (ii), the natural map

SK_{1}(Ỹ , ZỸ) ⟶ SK_{1}(Y^{+}, Z)

is surjective. Finally, by proposition 3.2 (iii), SK_{1}(Ỹ , ZỸ) ≅ SK_{1}(Ỹ). □

Using propositions 3.5 and 3.2, lemma 3.3 implies the

Theorem 3.8 Let X be a regular connected scheme of finite type over Spec(ℤ) and let D = D_{1} + ⋯ + D_{r} be a sum of prime divisors on X. Denoting the normalization of D_{i} by D̃_{i} for i = 1, …, r, we have a natural exact sequence

⨁_{i=1}^{r} SK_{1}(D̃_{i}) ⟶ CH_{0}(X, D) ⟶ CH_{0}(X) ⟶ 0.

In particular, if X is flat over Spec(ℤ), the group CH_{0}(X, D) is finite.

Finally, we enlarge the divisor. Let D_{1}, …, D_{r+s} be prime divisors on X, D = D_{1} + ⋯ + D_{r} and D′ = D_{1} + ⋯ + D_{r+s}. Let Y and Y′ be the support of D and D′, respectively, and let CH_{0}^{D′}(X, D) be the quotient of ⨁_{x∈(X−Y′)_{0}}ℤ by

the image of the group

R_{X,D,D′} =^{df} ker ( ker ( ⨁_{x∈X_{1}−Y′} k(x)^{×} → ⨁_{y∈Y′_{0}} ℤ) ⟶ ⨁_{z∈(Y^{+})_{0}} k(z)^{×})

under the divisor map.

Proposition 3.9 Let X be a regular scheme, flat and of finite type over Spec(ℤ) and let D_{1}, …, D_{r+s} be pairwise different prime divisors on X. Let D = D_{1} + ⋯ + D_{r}, D′ = D_{1} + ⋯ + D_{r+s} and denote the normalization of D_{i} by D̃_{i}, i = 1, …, r + s. Then the following holds.

(i) CH_{0}^{D′}(X, D) ⟶^{∼} CH_{0}(X, D).

(ii) We have a natural exact sequence

⨁_{i=r+1}^{r+s} SK_{1}(D̃_{i}) ⟶ CH_{0}(X, D′) ⟶ CH_{0}(X, D) ⟶ 0.

In particular, if D_{r+1}, …, D_{r+s} are horizontal, then CH_{0}(X, D′) ⟶ CH_{0}(X, D) is an isomorphism.

Proof: By the definition of the occurring objects, we have natural maps

CH_{0}(X, D′) ⟶^{p} CH_{0}^{D′}(X, D) ⟶^{q} CH(X, D)

The map ⨁_{i=r+1}^{r+s} SK_{1}(D̃_{i}) ⟶ CH_{0}(X, D′) is the composite map

⨁_{i=r+1}^{r+s} SK_{1}(D̃_{i}) ⟶^{incl} ⨁_{i=1}^{r+s} SK_{1}(D̃_{i}) ⟶ CH_{0}(X, D′)

of the natural inclusion with the map of theorem 3.8. The commutative diagram

⨁_{i=r+1}^{r+s} SK_{1}(D̃_{i})
∩
⨁_{i=1}^{r+s} SK_{1}(D̃_{i}) → CH_{0}(X, D′) → CH_{0}(X) → 0
↓ ↓  ↓^{q∘p}  ∥
⨁_{i=1}^{r} SK_{1}(D̃_{i}) → CH_{0}(X, D) → CH_{0}(X) → 0

shows that the sequence in (ii) is a complex and that q ∘ p is surjective. We consider two copies of the double complex before lemma 3.3: one for (X, D) and one for (X, D′), and the natural map between them. Assume that the class of a zero-cycle ∑ α_{i}P_{i}, P_{i} ∈ X − Y′, is zero in CH_{0}(X, D). By proposition 2.4, ∑ α_{i}P_{i}, is the image of an element a ∈ R_{X,D} under the divisor map. Changing a by the image of an appropriate element in E_{1}^{d−2,−d}(X|Y) (use approximation as in the proof of lemma 3.4), we may suppose that the components of a at all points of Y′ − Y are trivial, i.e.

a ∈ ker ( ker ( ⨁_{x∈(X−Y′)_{1}} k(x)^{×} ⟶ ⨁_{y∈Y_{0}} ℤ) ⟶ ⨁_{z∈Y^{+}} k(z)^{×}).

Since all (non-zero) components of div(a) = ∑ n_{i}P_{i} are in X − Y′, we conclude a ∈ R_{X,D,D′} ⊂ E_{1}^{d−1,−d}(X|Y′), proving (i). With respect to the map f′ : E_{1}^{d−1,−d}(X|Y′)→E_{1}^{d−1,−d}((Y′)^{+}), we obtain

f′(a) ∈ ⨁_{z∈((Y′)^{+}−Y^{+})_{0}} k(z)^{×} ⊂ E_{1}^{d−1,−d}((Y′)^{+}).

Therefore the class in ⨁_{i=r+1}^{r+s} SK_{1}(D̃_{i}) of any preimage of f′(a) in ⨁_{i=r+1}^{r+s} ⨁_{z∈(D̃_{i})_{0}}k(z)^{×}

maps to the class of ∑ α_{i}P_{i} in CH_{0}(X, D′). This completes the proof. □

4 Tame coverings

Coverings of a regular scheme which are tamely ramified along a normal crossing divisor have been studied in [SGA1], [G-M]. A naive extension of the (valuation theoretic) definition of tame ramification in the normal crossing case proves to be not useful in the general situation. For example, it would not be stable under base change (cf. [S1], Example 1.3). A definition of tameness in the general situation was given in [S1] and it was shown there that it coincides with the previous one in the normal crossing case. Here we restrict our attention to tamely ramified extensions of normal schemes, where one can define tameness using inertia groups.

Let X be a connected, noetherian, normal scheme and let K be its function field. Let L|K be a finite Galois extension and let X_{L} be the normalization of X in L. Furthermore, let U ⊂ X be an open subscheme such that U_{L} → U is étale. Put Y = X − U.

Definition 4.1 We say that U_{L} → U is tamely ramified along Y if the inertia subgroup (cf. [Bo], V,§2.2) of every point y ∈ Y in G(L|K) is of order prime to the characteristic of k(y).

If X is regular, D = D_{1} + ⋯ + D_{r} a divisor with normal crossings on X and Y = supp(D), then U_{L} → U is tamely ramified along Y if and only if L|K is tamely ramified at the discrete valuations v_{1}, …, v_{r} of K corresponding to D_{1}, …, D_{r} (see [S1], prop.1.14).

We say that X_{L} → X is tame along a divisor D, if, putting Y = supp(D) and U = X − Y, the induced morphism U_{L} → U is étale and tame along Y. For any base point ∗ ∈ U = X − Y, the tame fundamental group π_{1}^{t}(X, D, ∗) is the unique quotient of π_{1}(U, ∗) which classifies finite connected étale coverings Ũ of U with at most tame ramification along D.

If X is of finite type over Spec(ℤ), we denote by π̃_{1}^{t}(X, D, ∗)^{ab} the unique quotient of π_{1}^{t}(X, D, ∗) which classifies abelian finite étale coverings of U = X − Y which are at most tamely ramified along Y = supp(D) and in which every real (i.e. ℝ-valued) point of U splits completely. Fundamental groups with respect to different base points are isomorphic, the isomorphism being canonical up to inner automorphisms. As usual, we omit the base point as long as we deal with abelian quotients.

Proposition 4.2 Let X be a connected regular scheme, proper and flat over Spec(ℤ) and let D be a divisor on X. Then for any horizontal prime divisor D′ on X the natural map

π̃_{1}^{t}(X, D + D′)^{ab} ⟶ π̃_{1}^{t}(X, D)^{ab}

is an isomorphism.

Proof: The map in question is obviously surjective. To show injectivity, we first show that every connected cyclic étale covering Ṽ → V := X − (Y ∪ D′) of p-power degree (p an arbitrary prime number) which is tamely ramified along Y ∪D′ extends to an étale covering of U := X −Y. Let K and L be the function fields of V and of Ṽ, respectively. Since D′ is horizontal, the map D′ → Spec(ℤ) is proper and dominant, hence surjective. Thus D′ contains a point of residue characteristic p, which has a trivial inertia group in L|K since Ṽ → V is tame along D′. Hence L|K is unramified at the discrete valuation associated to D′. The theorem on the purity of the branch locus then shows that the normalization Ũ of U in L is étale over U. This shows that π_{1}^{t}(X, D + D′)^{ab} → π_{1}^{t}(X, D)^{ab} is an isomorphism. Finally, we have to deal with the real places. By [Sa], lemma 4.9 (iii), the subset of points in U(ℝ) which split completely in Ũ is (norm) closed and open in U(ℝ). U(ℝ) is a real manifold and V (ℝ) is (norm) dense in U(ℝ). Hence, if all points of V (ℝ) split completely, in Ṽ, then all points of U(ℝ) split completely in Ũ. Therefore the result extends to the modified fundamental groups. □

The following finiteness result is a special case of ([S1], Theorem 2).

Theorem 4.3 Let X be a connected, regular, proper and flat scheme over Spec(ℤ) and let D be a divisor on X. Then the group π̃_{1}^{t}(X, D)^{ab} is finite.

We conclude this section with a lemma which will be needed later on.

Lemma 4.4 Let X be a connected regular scheme, flat and of finite type over Spec(ℤ). Let D = D_{1} + ⋯ + D_{r} be a sum of vertical prime divisors on X (not necessarily with normal crossings) and put U = X − supp(D). Let v_{1}, …, v_{r} be the discrete valuations of the function field k(U) = k(X) which are associated with D_{1}, …, D_{r}. Then a finite abelian étale covering Ũ → U is tamely ramified along D if and only if the extension k(Ũ)|k(U) is tamely ramified at v_{1}, …, v_{r}.

Proof: In order to show the nontrivial implication, assume that k(Ũ)|k(U) is tamely ramified at v_{1}, …, v_{r}. We may assume that Ũ → U is cyclic of prime power order, say of order p^{n}. After reordering, we may assume that D_{1}, …, D_{s} lie over p and that D_{s+1}, …, D_{r} lie over prime numbers different to p. Put V = X − supp(D_{s+1} + ⋯ + D_{r}). Since v_{1}, …, v_{s} are tamely ramified, hence unramified in k(Ũ), the theorem on the purity of the branch locus shows that the normalization Ṽ of V in k(Ũ) lies étale over V. Since every point of supp(D) with residue characteristic p lies on V, this shows that Ũ → U is tamely ramified along D. □

5 Some local computations

Let X be an excellent regular scheme, i : Z ↪ X a closed regular subscheme of pure codimension c, p a prime number with \frac{1}{p} ∈ O_{X} and n ≥ 1 an integer. All cohomology groups occurring in this section are taken with respect to the étale topology. The cup-product with the fundamental class (cf. [De]) cl(Z) ∈ H_{D}^{2c}(X, ℤ/p^{n}ℤ(c)) induces the so-called Gysin map

Gys : i^{∗}(ℤ/p^{n}ℤ) ⟶ Ri^{!}(ℤ/p^{n}ℤ)(c)[2c].

It is a special case of “purity for étale cohomology” (proved by Gabber, cf. [Fu]) that Gys is an isomorphism. The following proposition follows from this result in a straightforward manner.

Proposition 5.1 Let X be an excellent regular scheme, i : Y ↪ X a closed subscheme of codimension ≥ c, p a prime number with \frac{1}{p} ∈ O_{X} and n ≥ 1 an integer. Furthermore, let Y^{sing} be the singular locus of Y and Y^{ns} = Y − Y^{sing}. Then, for any i ∈ ℤ, we have

H_{Y}^{r} (X, ℤ/p^{n}ℤ(i)) = 0 for r < 2c.

If Y is of pure codimension c in X, then we have natural isomorphisms

H_{Y}^{2c}(X, ℤ/p^{n}ℤ(i)) ⟶^{∼} H_{Y^{ns}}^{2c}(X − Y^{sing}, ℤ/p^{n}ℤ(i)) ⟵^{∼} H^{0}(Y^{ns}, ℤ/p^{n}ℤ(i − c)).

The first map is the natural restriction homomorphism and the second one is the cup-product with the fundamental class c_{Y^{ns}} ∈ H_{Y^{ns}}^{2c}(X − Y^{sing}, ℤ/p^{n}ℤ(c)).

Proof: If Y is regular, this follows from purity. In order to deal with the general case, we proceed by (decreasing) induction on the codimension and use the long exact sequence

⋯ → H_{Y^{sing}}^{r}(X, ℤ/p^{n}ℤ(i)) → H_{Y}^{r} (X, ℤ/p^{n}ℤ(i))
→ H_{Y −Y^{sing}}^{r}(X − Y^{sing}, ℤ/p^{n}ℤ(i)) → · · · .
□

Now let D = D_{1} +⋯+D_{r} be the sum of distinct vertical irreducible divisors on X (i.e. the D_{i}’s are schemes over fields), and we put Y = supp(D). Let p be a prime number and let Y_{p} be the inverse image of {p} ⊂ Spec(ℤ) under the natural projection Y → Spec(ℤ). An abelian étale covering of X − Y of p-power degree which is tame along Y extends to an étale covering of X − (Y − Y_{p}). Since all D_{i} are vertical, tameness of such a covering along Y − Y_{p} is a void condition. Therefore we have a natural isomorphism π_{1}^{t}(X, D)^{ab}(p) ⟶^{∼} π_{1}(X − (Y − Y_{p}))^{ab}(p). Using excision and proposition 5.1, we have

H_{Y −Y_{p}}^{2}(X, ℚ_{p}/ℤ_{p}) = H_{Y −Y_{p}}^{2}(X−Y_{p}, ℚ_{p}/ℤ_{p}) = H^{0}(Y −(Y_{p}∪Y^{sing}), ℚ_{p}/ℤ_{p}(−1))

and therefore an exact sequence

H^{0}(X − (Y_{p} ∪ Y^{sing}), ℚ_{p}/ℤ_{p}(−1))^{∨} → π_{1}^{t}(X, D)^{ab}(p) → π_{1}(X)^{ab}(p) → 0.

For a field K, we use the notation

H^{0}(K, ℚ/ℤ(−1)) = lim_{⟶}_{n} μ_{n}(K)^{∨},

where ∨ denotes Pontryagin dual. In particular, the char(K)-part of the above group is trivial. We obtain the

Lemma 5.2 Let D = D_{1} + ⋯ + D_{r} be the sum of distinct vertical irreducible divisors on an excellent regular scheme X. Then we obtain a natural exact sequence

⨁_{i=1}^{r} H^{0}(k(D_{i}), ℚ/ℤ(−1))^{∨} ⟶ π_{1}^{t}(X, D)^{ab} ⟶ π_{1}(X)^{ab} ⟶ 0.

Next we introduce a technical condition.

Definition 5.3 Let Y be an integral scheme of finite type over a finite field 𝔽. We say that Y satisfies condition (μ) if for every natural number n, prime to the characteristic of 𝔽, the natural map

H^{0}(Y, ℤ/nℤ(−1)) ⟶ H^{0}(k(Y), ℤ/nℤ(−1))

is an isomorphism.

Condition (μ) is satisfied if Y is normal. If Y is proper, condition (μ) is equivalent to the condition that Γ(Y, O_{Y}) coincides with the algebraic closure of 𝔽 in k(Y).

Example. If p is a prime number, p ≡ 3 mod 4, then Y = Spec(𝔽_{p}[x, y]/x^{2} + y^{2}) does not satisfy condition (μ). Indeed, k(Y) is the rational function field in one variable over 𝔽_{p^{2}}.

Now let ℤ_{p}^{h} be the henselization of ℤ at a prime p. Let X → Spec(ℤ_{p}^{h}) be a regular, proper scheme of finite type and X_{η} and X_{s} its generic and special fibre. Let D = D_{1} + ⋯ + D_{r} be the sum of distinct vertical irreducible divisors on X and Y = supp(D). Local class field theory induces a natural map

τ : ⨁_{x∈(X_{η})_{0}} k(x)^{×} ⟶ ⨁_{x∈(X_{η})_{0}} π_{1}(x)^{ab} ⟶ π_{1}^{t}(X, D)^{ab}

and our next aim is to detect a natural subgroup of its kernel. This will be the key point in the proof of the existence of the global reciprocity map in the next section. Its exactly here, where we have to impose the above condition (μ) on the divisors D_{i}. We consider the group

U_{D}^{t}(X) =^{df} ker ( ker ( ⨁_{(X_{η})_{0}} k(x)^{×} → ⨁_{y∈(X_{s})_{0}} ℤ) ⟶^{ϕ} ⨁_{y∈(Y^{+})_{0}} k(y)^{×}),

where ϕ is the restriction of the natural map E_{1}^{d−1,−d}(X|Y) → E_{1}^{d−1,d}(Y^{+}), d = dim X, of the spectral sequence for relative G-theory from section 1 to the subgroup of those elements which have nontrivial components only at points in the generic fibre. U_{D}^{t}(X) is nothing else but the group R_{X,D,D′} defined before proposition 3.9 with D′ the sum of all vertical divisors of X. A point x ∈ (X_{η})_{0} specializes to a unique point y ∈ (X_{s})_{0}. Therefore, putting

U_{D}^{t}(y) = ker ( ker ( ⨁_{\begin{matrix}x∈(X_{η})_{0} \\ x→y \end{matrix}} k(x)^{×} → ℤ) ⟶^{ϕ} ⨁_{D_{i}∋y} k(y)^{×}),

we obtain a decomposition

U_{D}^{t}(X) = ⨁_{y∈(X_{s})_{0}} U_{D}^{t}(y).

Proposition 5.4 Let X → Spec(ℤ_{p}^{h}) be a regular, proper scheme of finite type and let D = D_{1} + ⋯ + D_{r} be the sum of distinct vertical irreducible divisors on X. If D_{1}, …, D_{r} satisfy condition (μ), then the natural map

τ : ⨁_{x∈(X_{η})_{0}} k(x)^{×} ⟶ π_{1}^{t}(X, D)^{ab}

annihilates the subgroup U_{D}^{t}(X).

Proof: We consider the l-part for all prime numbers l separately. The easiest case is l = p. Indeed, in the commutative diagram (cf. A.3)

⨁_{x∈(X_{η})_{0}} k(x)^{×} ⟶ SK_{1}(X_{η}) ⟶^{τ} π_{1}(X_{η})^{ab} ⟶ π_{1}^{t}(X, D)^{ab}
↓  ↓^{∂}  ↓^{sp}  ↓
⨁_{y∈(X_{s})_{0}} ℤ ⟶ SK_{0}(X_{s}) ⟶^{ϕ} π_{1}(X_{s})^{ab} ⟵^{∼}  π_{1}(X)^{ab} ,

the right vertical arrow is an isomorphism on p parts (by lemma 5.2).

For the prime-to-p part, we first note that it suffices to consider U_{D}^{t}(y) for all y ∈ (X_{s})_{0}. We fix once and for all such a y and renumber the D_{i} such that D_{i} ∋ y for i = 1, … s and D_{i} ∌ y for i = s + 1, …, r. Now let n be a natural number with (n, p) = 1. Let x_{1}, …, x_{m} ∈ (X_{η})_{0} be points that specialize to y and let W be the closure of {x_{1}, …, x_{m}} in X. W is a one dimensional, local henselian scheme with finite residue field. In particular, H^{2}(W, ℤ/nℤ) = 0 and we obtain a short exact sequence

0 ⟶ H^{1}(W, ℤ/nℤ) ⟶ ⨁_{j=1}^{m} H^{1}(k(x_{i}), ℤ/nℤ) ⟶^{δ} H_{y}^{2}(W, ℤ/nℤ) ⟶ 0.

Now consider the commutative diagram

⨁_{j=1}^{r} H^{0}(k(D_{j}), ℤ/nℤ(−1))  ⟶^{∼}  H_{Y^{ns}}^{2}(X − Y^{sing}, ℤ/nℤ)
∪ ^{α}  ↑^{≀}
⨁_{j=1}^{r} H^{0}(D_{i}, ℤ/nℤ(−1))  \underrightarrow{∑(?∪c_{D_{j}})}  H_{Y}^{2} (X, ℤ/nℤ)
↓^{⊕_{j=1}^{s} i_{j}^{∗}}  ↓_{i^{∗}}
⨁_{j=1}^{s} H^{0}(y, ℤ/nℤ(−1)) \underrightarrow{∑(?∪i_{j}^{∗}c_{D_{j}})}  H_{y}^{2}(W, ℤ/nℤ)

Here i : (W, y) ↪ (X, Y) and i_{j} : y ↪ D_{j}, j = 1, …, s, are the inclusion morphisms and c_{D_{j}} ∈ H_{D_{j}}^{2}(X, ℤ/nℤ(1)), j = 1, …, r, is the fundamental class of the divisor D_{j} in X ([De], 2.1). The horizontal cup-products are defined in [De], 1.2. Since the D_{j} satisfy condition (μ), the inclusion map α is an isomorphism, and hence the same is true for the middle horizontal arrow ∑(? ∪ c_{D_{j}}).

Dualizing, and using the well-known duality theorems for Galois cohomology of finite and of one-dimensional henselian fields, we obtain a commutative diagram

ker ( ⨁_{i=1}^{m} H^{1}(k(x_{i}), ℤ/nℤ(1)) → ℤ/nℤ)  ⟶ ⨁_{i=1}^{m} H^{1}(k(x_{i}), ℤ/nℤ(1))
↓^{≀}  ↓^{≀}
⨁_{j=1}^{s} H^{1}(k(y), ℤ/nℤ(1)) ⟵ H_{y}^{2}(W, ℤ/nℤ)^{∨} ⟶ H^{1}(W − y, ℤ/nℤ)^{∨}
↓  ↓  ↓
⨁_{j=1}^{s} H^{0}(D_{j}, ℤ/nℤ(−1))^{∨} ⟵^{∼} H_{Y}^{2} (X, ℤ/nℤ)^{∨} ⟶ H^{1}(X − Y, ℤ/nℤ)^{∨} .

In order to show the proposition, it therefore remains to show the commutativity of the following diagram

ker ( ⨁_{i=1}^{s} k(x_{i})^{×} ⟶ ℤ) ⟶ ker ( ⨁_{i=1}^{m} H^{1}(k(x_{i}), ℤ/nℤ(1)) → ℤ/nℤ)
↓^{ϕ}  ↓^{ψ}
⨁_{j=1}^{s} k(y)^{×}  ⟶  ⨁_{j=1}^{s} H^{1}(k(y), ℤ/nℤ(1)),

in which the horizontal maps are induced by the respective Kummer sequences, ϕ is the map from the definition of U_{D}^{t}(y) and ψ is the map induced by the last diagram. We gave a description of ϕ in section 2. It remains to describe ψ.

Choose a a local parameter π_{j} defining D_{j} in a neighborhood of y in X. Then π_{j} defines an element in H^{0}(W − y, 𝔾_{m}) and, denoting its image in H^{1}(W − y, ℤ/nℤ(1)) by π̄_{j}, the pull-back of the fundamental class i_{j}^{∗}(c_{D_{j}}) ∈ H_{y}^{2}(W, ℤ/n(1)) is the image of π̄_{j} under the natural boundary map H^{1}(W − y, ℤ/nℤ(1)) → H_{y}^{2}(W, ℤ/nℤ(1)) (see [De], 2.1.1.1). We therefore obtain a commutative diagram

H^{0}(y, ℤ/nℤ(−1)) ⟶^{?∪i_{j}^{∗}c_{D_{j}}}  H_{y}^{2}(W, ℤ/nℤ)
↑^{≀}  ↑^{δ}
H^{0}(W, ℤ/nℤ(−1)) ⟶^{?∪π̄_{j}} H^{1}(W − y, ℤ/nℤ)

Next we recall the boundary map in étale homology (see [J-S], [B-O])

b : H^{i}(W − y, ℤ/nℤ(j)) ⟶ H^{i−1}(y, ℤ/nℤ(j − 1)),

which, denoting the structure map by f : W → Spec(ℤ_{p}^{h}), is defined by

H^{i}(W − y, ℤ/nℤ(j)) = H^{i}(W − y, f^{!}ℤ/nℤ(j))

⟶^{δ} H^{i+1}(y, i^{!}f^{!}ℤ/nℤ(j)) = H^{i−1}(y, ℤ/nℤ(j − 1)).

We have the following commutative diagram relating b with the symbol map h : K_{∗}(F) → H^{∗}(F, ℤ/nℤ(∗)), ∗ = 1, 2, from K-theory to Galois cohomology

⨁_{i=1}^{m} K_{2}(k(x_{i})) ⟶^{h} H^{2}(W − y, ℤ/nℤ(2))
↓^{∂}  ↓^{b}
k(y)^{×}  ⟶^{h} H^{1}(k(y), ℤ/n(1)),

where ∂ is the boundary map of the Quillen spectral sequence for W (see [Ka], Lemma 1.4 (1) for the case when W is regular, the general case follows easily from that). Furthermore, we have the following commutative diagram, relating the Tate-duality for W − y with that of y

H^{2}(W − y, ℤ/nℤ(2)) ≅ H^{0}(W − y, ℤ/nℤ(−1))^{∨}
↓^{b}  ↓^{can}
H^{1}(y, ℤ/nℤ(1))  ≅ H^{0}(y, ℤ/nℤ(−1))^{∨} .

Here can is the dual to the natural map

H^{0}(y, ℤ/nℤ(−1)) ≅ H^{0}(W, ℤ/nℤ(−1)) → H^{0}(W − y, ℤ/nℤ(−1)).

Summing up, the j-component of ψ is the restriction of the composite map

⨁_{i=1}^{m} H^{1}(k(x_{i}), ℤ/nℤ(1)) ⟶^{?∪π̄_{j}} ⨁_{i=1}^{m} H^{2}(k(x_{i}), ℤ/nℤ(2)) ⟶^{b} H^{1}(k(y), ℤ/nℤ(1))

to the subgroup ker ( ⨁_{i=1}^{m} H^{1}(k(x_{i}), ℤ/nℤ(1)) → ℤ/nℤ), and the required dia-

gram commutes. □

6 Proof of the main theorem

Now we are going to prove our main theorem. We change the notation for better compatibility with the notation of section A.4 and use the letter 𝔛 (instead of X) for the scheme in question. We assume for the rest of this section that 𝔛 is a d-dimensional connected regular, proper and flat scheme over Spec(ℤ) such that X_{η} = 𝔛 ⊗_{ℤ} ℚ is projective over Spec(ℚ). Let, as in section A.4, k be the algebraic closure of ℚ in the function field of X_{η} and let S = Spec(O_{k}). The structural morphism 𝔛 → Spec(ℤ) factors through S and X_{η} is geometrically irreducible as a variety over k. Let S_{f} be the set of closed points of S and let S_{∞} be the set of archimedean places of the number field k. For v ∈ S_{f} let Y_{v} = 𝔛 ⊗_{S} v be the special fibre of 𝔛 over v. For v ∈ S_{∞} ∪ S_{f} let k_{v} be the algebraic closure of k in the completion of k at v and X_{v} = X_{η} ⊗_{k} k_{v}.

Let 𝔇 = 𝔇_{1} + ⋯ + 𝔇_{r} be a sum of vertical divisors on 𝔛 and Z = supp(𝔇). We choose a sufficiently small open subscheme U ⊂ S such that 𝔛_{U} is disjoint to Z. For v ∈ S_{f}, we denote the base change of 𝔛, 𝔇, Z to Spec(O_{v}) by 𝔛_{v}, 𝔇_{v}, Z_{v}. Let P_{U} denote the set of places of k (including the archimedean ones) which are not in U_{0} and let P_{U}^{a} and P_{U}^{f} be the subset of archimedean and nonarchimedean places in P_{U}, respectively. Recall the idèle group (section A.4)

I(𝔛/U) = ( ∏_{v∈P_{U}} SK_{1}(X_{v})) × ( ⨁_{v∈U_{0}} CH_{0}(Y_{v})) .

For v ∈ P_{U} we endow the fields k_{v} with the restriction of the natural topology of the completion of k at v to k_{v}. This induces a natural topology on SK_{1}(X_{v}) for v ∈ P_{U} (cf. section A), and (giving CH_{0}(Y_{v}) the discrete topology for v ∈ U_{0}), on I(𝔛/U). Furthermore, recall that the idèle class group C(𝔛/U) is defined as the quotient of I(𝔛/U) by the image of the natural diagonal map SK_{1}(X_{η}) → I(𝔛/U).

Definition 6.1 We define U_{𝔇}^{t}C(𝔛/U) as the image of the natural composite map

( ⨁_{v∈P_{U}^{a}} ⨁_{x∈(X_{v})_{0}} k(x)^{×} ⊕ ⨁_{v∈P_{U}^{f}} U_{𝔇}^{t}(𝔛_{v})) → I(𝔛/U) → C(𝔛/U).

For v ∈ P_{U}^{f}, the group U_{𝔇}^{t}(𝔛_{v}) contains the subgroup ⨁_{x∈(X_{v})_{0}} U^{1}(k(x)^{×}) (principal units with respect to the structure of k(x) as a discrete valuation field). Therefore U_{𝔇}^{t}C(𝔛/U) is open in C(𝔛/U).

Theorem 6.2 Let 𝔛′ be the normalization of 𝔛 in a finite cyclic extension of its function field such that 𝔛′_{U}/𝔛_{U} is étale. Then 𝔛′/𝔛 is at most tamely ramified along 𝔇 and every real point of 𝔛 − Z splits completely in 𝔛′ if and only if the associated character

χ_{𝔛′} : C(𝔛/U) → ℚ/ℤ

annihilates the subgroup U_{𝔇}^{t}C(𝔛/U).

Proof: Firstly we assume that 𝔛′/𝔛 is étale over 𝔛 − Z, at most tamely ramified along 𝔇 and that every real point of 𝔛−Z splits completely in 𝔛′. The reciprocity map τ : C(𝔛/U) → π_{1}(𝔛_{U})^{ab} of [Sa] (cf. section A.4) is constructed using the reciprocity maps k(x)^{×} → π_{1}(x)^{ab} of the fields k(x) for x ∈ (X_{v})_{0}, v ∈ P_{U}. The results of section A.2 show that the image of

⨁_{v∈P_{U}^{a}} ⨁_{x∈(X_{v})_{0}} k(x)^{×}

in C(𝔛/U) is annihilated by χ_{𝔛′}. For v ∈ P_{U}^{f}, we have a commutative diagram

⨁_{x∈(X_{v})_{0}} k(x)^{×} ⟶ π_{1}^{t}(𝔛_{v}, 𝔇_{v})^{ab}
↓  ↓
C(𝔛/U)  ⟶^{τ} π_{1}^{t}(𝔛, 𝔇)^{ab}

By proposition 5.4, χ_{𝔛′} annihilates the image of U_{𝔇_{v}}^{t}(𝔛_{v}) in C(𝔛/U). Therefore χ_{𝔛′} annihilates U_{𝔇}^{t}C(𝔛/U).

Now assume that χ_{𝔛′} annihilates U_{𝔇}^{t}C(𝔛/U). We may assume that 𝔛′ → 𝔛 is cyclic of prime power degree, say of order p^{α}. Let v ∈ P_{U}^{f} and x ∈ (X_{v})_{0}. Let y ∈ Y_{v} be the unique point to which x specializes. The natural map k(x)^{×} → U_{𝔇}^{t}C(𝔛/U) annihilates the subgroup U^{1}k(x)^{×} or U^{0}k(x)^{×} of k(x)^{×} if y ∈ Z_{v} or y ∉ Z_{v}, respectively. By local class field theory, the character χ_{𝔛′} induces a cyclic field extension of the henselian field k(x) which is tamely ramified or unramified, respectively. In order to show that 𝔛′/𝔛 is étale over 𝔛 − Z, it suffices (purity of the branch locus) to show that it is unramified at every prime divisor E not contained in Z. Since 𝔛′_{U}/𝔛_{U} is étale, we may assume that E ⊂ Y_{v} for some v ∈ P_{U}^{f}. The branch locus is closed, and so it suffices to find a closed unramified point on E. This is easy, as for any closed point y ∈ Z_{v} which is a regular point of the reduced subscheme Y_{v,red} of Y_{v}, the required property follows from lemma A.4. It remains to show the tameness along Z. By lemma 4.4, it suffices to show that 𝔛′/𝔛 is tamely ramified at all generic points of Z. Tameness at points of residue characteristic different to p is immediate. Let E be a prime divisor contained in Z ∩ Y_{v} for an v ∈ P_{U}, v|p. We will show that 𝔛′/𝔛 is unramified along E. Let y by a closed regular point of E. For every point x ∈ (X_{v})_{0} that specializes to y, k(x) is a henselian field of characteristic zero and of residue field characteristic p. Since χ_{𝔛′} annihilates U_{𝔇}^{t}C(𝔛/U), the associated character on k(x)^{×} annihilates U^{1}k(x)^{×}. It is therefore tamely ramified and of p-power order, hence unramified. Using lemma A.4, we conclude that 𝔛′/𝔛 is étale at y, and therefore also at the generic point of E. Hence 𝔛′/𝔛 is tame along Z. The remaining assertion concerning the real points follows in a straightforward manner from the results of section A.2. This concludes the proof. □

Theorems 6.2, 4.3 and the results of section A.4 imply the

Corollary 6.3 Putting C_{𝔇}^{t}(𝔛/U) =^{df} C(𝔛/U)/U_{𝔇}^{t}(𝔛/U), we obtain a surjective map

τ′ : C_{𝔇}^{t}(𝔛/U) −↠ π̃_{1}^{t}(𝔛, 𝔇)^{ab}.

For all n ∈ ℕ the map

τ′/n : C_{𝔇}^{t}(𝔛/U)/n ⟶^{∼} π̃_{1}^{t}(𝔛, 𝔇)^{ab}/n

is an isomorphism of finite abelian groups.

For the proof of our main theorem, we need the following approximation result:

Lemma 6.4 Assume that 𝔇 is the sum of the irreducible components of the fibres over the points in P_{U}^{f}. Consider the diagonal map

diag_{2} : ⨁_{x∈(X_{η})_{1}} K_{2}(k(x)) ⟶ ⨁_{v∈P_{U}^{f}} ⨁_{x∈(X_{v})_{1}} K_{2}(k(x))

and the boundary map

δ_{loc} : ⨁_{v∈P_{U}^{f}} ⨁_{x∈(X_{v})_{1}} K_{2}(k(x)) ⟶ ⨁_{v∈P_{U}^{f}} ⨁_{x∈(X_{v})_{0}} k(x)^{×} .

Let c ∈ ⨁_{v∈P_{U}^{f}} ⨁_{x∈(X_{v})_{1}} K_{2}(k(x)) be given. Then there exist an e ∈ ⨁_{x∈(X_{η})_{1}} K_{2}(k(x)) with

δ_{loc}(c − diag_{2}(e)) ∈ ⨁_{v∈P_{U}^{f}} ⨁_{y∈(Y_{v})_{0}} U_{𝔇}^{t}(y) ⊂ ⨁_{v∈P_{U}^{f}} ⨁_{x∈(X_{v})_{0}} k(x)^{×}.

Proof: First we choose a finite set of points T ⊂ (X_{η})_{1} such that c_{x} = 0, for all x ∈ (X_{v})_{1}, v ∈ P_{U}^{f}, mapping to X_{1} − T. Then we choose local parameters z_{1}, …, z_{r} defining 𝔇_{1}, …, 𝔇_{r} in an affine neighborhood of \overline{T} ∩ Z. Now let x ∈ T and let W be the closure of {x} in 𝔛. For v ∈ P_{U}^{f} we consider those 𝔇_{i}, 1 ≤ i ≤ r, which are in Y_{v}. The irreducible components of the pre-image of W ∩ 𝔇_{i} in the normalization W̃ of W induce discrete valuations w_{1}, …, w_{s} on k(x). The restriction of all these valuations to k is v. The natural maps k(x) ⊗_{k} k_{v} → k(x)_{w_{i}} for i = 1, …, s, induce a partition of the set {w_{1}, …, w_{s}} with respect to the ‘underlying’ point in (X_{v})_{1}. Applying this construction all v ∈ P_{U}^{f} we end up with a finite number of valuations of k(x) for each x ∈ T.

Now we put e_{x} = 0 for x ∉ T and we choose e_{x} ∈ K_{2}(k(x)) sufficiently close to c_{x_{i}} (x_{i} ∈ (X_{v})_{1} mapping to x) with respect to the valuations constructed above. By near we mean near in the sense that we approximate the entries of the symbols. Then the image of diag_{2}(e) − c under the boundary map

⨁_{v∈P_{U}^{f}} ⨁_{x∈(X_{v})_{1}} K_{2}(k(x)) ⟶ ⨁_{v∈P_{U}^{f}} ⨁_{y∈(Y_{v})_{1}} k(y)^{×}

is zero and the same holds for the images of the elements (diag_{2}(e) − c) ∪ z_{i}, i = 1, …, r, under the map

⨁_{v∈P_{U}^{f}} ⨁_{x∈(X_{v})_{1}} K_{3}(k(x)) ⟶ ⨁_{v∈P_{U}^{f}} ⨁_{y∈(Y_{v})_{1}} K_{2}(k(y)).

Consider the complex

⨁_{\begin{matrix}v∈P_{U}^{f} \\ x∈(X_{v})_{1} \end{matrix}} K_{j+2}(k(x)) ⟶ ⨁_{\begin{matrix}v∈P_{U}^{f} \\ x∈(X_{v})_{0} \end{matrix}} K_{j+1}(k(x))⊕ ⨁_{\begin{matrix}v∈P_{U}^{f} \\ x∈(Y_{v})_{1} \end{matrix}} K_{j+1}(k(y))

⟶ ⨁_{\begin{matrix}v∈P_{U}^{f} \\ x∈(Y_{v})_{0} \end{matrix}} K_{j}(k(y)).

For j = 0 we conclude that δ_{loc}(diag_{2}(e) − c) is in the kernel of the boundary map

⨁_{v∈P_{U}^{f}} ⨁_{x∈(X_{v})_{0}} k(x)^{×} ⟶ ⨁_{v∈P_{U}^{f}} ⨁_{y∈(Y_{v})_{0}} ℤ.

For j = 1 and 1 ≤ i ≤ r, δ_{loc}(diag_{2}(e) − c) ∪ z_{i} is in the kernel of the map

⨁_{v∈P_{U}^{f}} ⨁_{x∈(X_{v})_{0}} K_{2}(k(x)) ⟶ ⨁_{v∈P_{U}^{f}} ⨁_{y∈(Y_{v})_{0}} k(y)^{×} ⟶^{pr} ⨁_{y∈(Z_{i})_{0}} k(y)^{×}.

By lemma 2.5 this implies that δ_{loc}(diag_{2}(e) − c) is in the kernel of

ϕ : ker ( ⨁_{v∈P_{U}^{f}} ⨁_{x∈(X_{v})_{0}} k(x)^{×} ⟶ ⨁_{v∈P_{U}^{f}} ⨁_{y∈(Y_{v})_{0}} ℤ) ⟶ ⨁_{i=1}^{r} ⨁_{y∈(Z_{i})_{0}} k(y)^{×}.

By definition, this means that δ_{loc}(diag_{2}(e) − c) ∈ ⨁_{v∈P_{U}^{f}} ⨁_{y∈(Y_{v})_{0}} U_{𝔇}^{t}(y). □

Now we are going to prove the following theorem (which is slightly sharper than theorem 1 of the introduction).

Theorem 6.5 Let 𝔛 be a connected regular, proper and flat scheme over Spec(ℤ) such that X = 𝔛⊗_{ℤ} ℚ is projective over Spec(ℚ). Let 𝔇 be a divisor on 𝔛 whose vertical components satisfy condition (μ) of definition 5.3 and put Z = supp(𝔇). Then the map

r : ⨁_{x∈(𝔛−Z)_{0}} ℤ ⟶ π̃_{1}^{t}(𝔛, 𝔇)^{ab}; 1_{x} ↦ Frob_{x}

induces an isomorphism of finite abelian groups

rec_{𝔛,𝔇} : CH_{0}(𝔛, 𝔇) → π̃_{1}^{t}(𝔛, 𝔇)^{ab}.

Proof: By propositions 3.9(ii) and 4.2, we may assume that 𝔇 is vertical. We proceed in several steps.

Step 1. Existence of rec: We consider the possibly larger divisor 𝔇′ which is the sum of all vertical prime divisors lying over points in P_{U}^{f}. With Z′ = supp(𝔇′) we have 𝔛 − Z′ = 𝔛_{U}. We show that the natural map

r′ : ⨁_{x∈(𝔛−Z′)_{0}} ℤ ⟶ C_{𝔇}^{t}(𝔛/U)

factors through CH_{0}^{𝔇′}(𝔛, 𝔇). Then proposition 3.9 (i) and corollary 6.3 imply the existence of

rec : CH_{0}(𝔛, 𝔇) ≅ CH_{0}^{𝔇′}(𝔛, 𝔇) ⟶ C_{𝔇}^{t}(𝔛/U) ⟶^{τ′} π̃_{1}^{t}(𝔛, 𝔇)^{ab}.

It remains to show that r′ annihilates the image of

R_{𝔛,𝔇,𝔇′} = ker ( ker ( ⨁_{x∈(𝔛−Z′)_{1}} k(x)^{×} → ⨁_{y∈Z′_{0}} ℤ) ⟶ ⨁_{z∈(Z^{+})_{0}} k(z)^{×})

under the divisor map. We have a decomposition

(𝔛 − Z′)_{1} = (X_{η})_{0} ∪ ⋃_{v∈U_{0}}(Y_{v})_{1}.

For any point x ∈ (Y_{v})_{1}, v ∈ U_{0}, the image of k(x)^{×} in C_{𝔇}^{t}(𝔛/U) is trivial by definition. We obtain an anti-commutative diagram

⨁_{x∈(𝔛−Z′)_{1}} k(x)^{×}  ⟶^{div} ⨁_{x∈(𝔛−Z′)_{0}} ℤ
↓^{ψ}  ↓_{r′}
⨁_{v∈P_{U}^{f}} ⨁_{y∈(Y_{v})_{0}} ( ⨁_{\begin{matrix}x∈(X_{v})_{0} \\ x→y \end{matrix}} k(x)^{×})/U_{𝔇}^{t}(y) ⟶ C_{𝔇}^{t}(𝔛, U)

in which ψ is the diagonal map on k(x)^{×} for x ∈ (X_{η})_{0} and the zero map for x ∈ (Y_{v})_{1}, v ∈ U_{0}. Hence ψ, and therefore also r′ ∘ div annihilates R_{𝔛,𝔇,𝔇′}, which shows the existence of rec and, moreover, the factorization

rec : CH_{0}(𝔛, 𝔇) ⟶^{α} C_{𝔇}^{t}(𝔛/U) ⟶^{τ′} π̃_{1}^{t}(𝔛, 𝔇)^{ab}.

Second Step: τ′ is an isomorphism of finite abelian groups: By corollary 6.3 it suffices to show that C_{𝔇}^{t}(𝔛/U) is finite. The image of diagonal map SK_{1}(X_{η}) → ∏_{v∈P_{U}} SK_{1}(X_{v}) is dense, and U_{𝔇}^{t}C(𝔛/U) is an open subgroup of C(𝔛/U). Therefore, every element in C_{𝔇}^{t}(𝔛/U) can be represented by an element in ⨁_{v∈U_{0}} CH_{0}(Y_{v}) which shows that α is surjective. Finally, the finiteness of CH_{0}(𝔛, 𝔇) shows the statement of step 2.

Last step: α is an isomorphism: We show that α′ : CH_{0}^{𝔇′}(𝔛, 𝔇) → C_{𝔇}^{t}(𝔛/U) is injective, which finishes the proof in view of proposition 3.9 (i). Consider the homomorphisms

div_{gen} :  ⨁_{x∈(X_{η})_{0}} k(x)^{×} ⟶ ⨁_{x∈(𝔛−Z)_{0}} ℤ ,
div_{sp} :  ⨁_{v∈U_{0}} ⨁_{y∈(Y_{v})_{1}} k(y)^{×} ⟶ ⨁_{x∈(𝔛−Z)_{0}} ℤ ,
δ_{loc} : ⨁_{v∈P_{U}^{f}} ⨁_{x∈(X_{v})_{1}} K_{2}(k(x)) ⟶ ⨁_{v∈P_{U}^{f}} ⨁_{x∈(X_{v})_{0}} k(x)^{×} ,
δ_{gl} :  ⨁_{x∈(X_{η})_{1}} K_{2}(k(x)) ⟶ ⨁_{x∈(X_{η})_{0}} k(x)^{×} ,

which are induced by the boundary maps of the respective Quillen spectral sequences. Further, consider the diagonal maps

diag_{1} :  ⨁_{x∈(X_{η})_{0}} k(x)^{×}  ⟶ ⨁_{v∈P_{U}^{f}} ⨁_{x∈(X_{v})_{0}} k(x)^{×} ,
diag_{2} : ⨁_{x∈(X_{η})_{1}} K_{2}(k(x)) ⟶ ⨁_{v∈P_{U}^{f}} ⨁_{x∈(X_{v})_{1}} K_{2}(k(x)).

Note that diag_{1} ∘ δ_{gl} = δ_{loc} ∘ diag_{2}. Further note that

⨁_{v∈P_{U}^{f}} ⨁_{x∈(X_{v})_{0}} k(x)^{×} = ⨁_{v∈P_{U}^{f}} ⨁_{y∈(Y_{v})_{0}} ⨁_{\begin{matrix}x∈(X_{v})_{0} \\ x→y \end{matrix}} k(x)^{×} .

and, with respect to this identity,

⨁_{v∈P_{U}^{f}} R_{𝔛_{v},𝔇_{v},𝔇′_{v}} = ⨁_{v∈P_{U}^{f}} ⨁_{y∈(Y_{v})_{0}} U_{𝔇}^{t}(y).

where R_{𝔛_{v},𝔇_{v},𝔇′_{v}} is the group defined before proposition 3.9 for the triple (𝔛_{v}, 𝔇_{v}, 𝔇′_{v}).

Now let ∑ n_{i}P_{i} ∈ ⨁_{x∈(𝔛−Z′)_{0}} ℤ be a zero-cycle which represents an element in

ker(CH_{0}^{𝔇′}(𝔛, 𝔇) → C_{𝔇}^{t}(𝔛/U)). By definition, there exist elements

a ∈ ⨁_{x∈(X_{η})_{0}} k(x)^{×}, b ∈ ⨁_{v∈U_{0}} ⨁_{y∈(Y_{v})_{1}} k(y)^{×}; c ∈ ⨁_{v∈P_{U}^{f}} ⨁_{x∈(X_{v})_{1}} K_{2}(k(x)),

such that the following conditions (1) and (2) hold.

(1)  ∑ n_{i}P_{i} = div_{gen}(a) + div_{sp}(b),

(2)  diag_{1}(a) − δ_{loc}(c) ∈ ⨁_{v∈P_{U}^{f}} ⨁_{y∈(Y_{v})_{0}} U_{𝔇}^{t}(y).

If already diag_{1}(a) would be in ⨁_{v∈P_{U}^{f}} ⨁_{y∈(Y_{v})_{0}} U_{𝔇}^{t}(y), then div_{gen}(a) and

hence also div_{gen}(a) + div_{sp}(b) would be a relation in CH_{0}^{𝔇′}(𝔛, 𝔇) showing that the class of ∑ α_{i}P_{i} is trivial in CH_{0}^{𝔇′}(𝔛, 𝔇). We will achieve this by replacing a by a − δ_{gl}(e), where e is a suitable chosen element in ⨁_{x∈(X_{η})_{1}} K_{2}(k(x)). In view of the complex

⨁_{x∈𝔛^{d−2}} K_{2}(k(x)) ⟶ ⨁_{x∈𝔛^{d−1}} k(x)^{×} ⟶ ⨁_{x∈𝔛^{d}} ℤ

this does not change div_{gen}(a) modulo the image of div_{sp}. Now we construct e. By lemma 6.4 applied to 𝔇′, we find e such that

δ_{loc}(diag_{2}(e) − c) ∈ ⨁_{v∈P_{U}^{f}} ⨁_{y∈(Y_{v})_{0}} U_{𝔇′}^{t}(y) ⊂ ⨁_{v∈P_{U}^{f}} ⨁_{y∈(Y_{v})_{0}} U_{𝔇}^{t}(y)

Setting, a′ = a − δ_{gl}(e), we obtain

diag_{1}(a′) = diag_{1}(a) − diag_{1}(δ_{gl}(e))
= diag_{1}(a) − δ_{loc}(c) + δ_{loc}(c − diag_{2}(e)) ∈ ⨁_{v∈P_{U}^{f}} R_{𝔛_{v},𝔇_{v},𝔇′_{v}}.

Therefore a′ ∈ R_{𝔛,𝔇,𝔇′} and we conclude that the class of ∑ α_{i}P_{i} is zero in CH_{0}^{𝔇′}(𝔛, 𝔇). □

The following variant of theorem 6.5 describes the full group π_{1}^{t}(𝔛, 𝔇)^{ab} and not its quotient π̃_{1}^{t}(𝔛, 𝔇)^{ab}. Of course, this yields nothing new if 𝔛 has no real points. We define a modified relative Chow group by allowing only such relations div(f) of functions f which are positive at every real-valued point. More precisely, for a point x ∈ 𝔛_{1} we put

k(x)_{+}^{×} = {f ∈ k(x)^{×} | ι(f) > 0 for every embedding ι : k(x) → ℝ}.

If the global field k(x) is of positive characteristic or a totally imaginary number field, then k(x)_{+}^{×} = k(x)^{×}.

Definition 6.6 The group C̃H_{0}(𝔛, 𝔇) is the quotient of ⨁_{x∈(𝔛−Z)_{0}} ℤ by the image

of

R̃_{𝔛,𝔇} = ker ( ker ( ⨁_{x∈(𝔛−Z)_{1}} k(x)_{+}^{×} → ⨁_{y∈Z_{0}} ℤ) ⟶ ⨁_{z∈Z_{0}^{+}} k(z)^{×})

under the divisor map.

A slight modification of the proof of theorem 6.5, which we leave to the reader, shows the

Theorem 6.7 Let 𝔛 be a regular connected scheme, flat and proper over Spec(ℤ) such that its generic fibre 𝔛 ⊗_{ℤ} ℚ is projective over ℚ. Let 𝔇 be a divisor on 𝔛 whose vertical components satisfy condition (μ). Then there exists a natural isomorphism of finite abelian groups

r̃ec_{𝔛,𝔇} : C̃H_{0}(𝔛, 𝔇) ⟶^{∼} π_{1}^{t}(𝔛, 𝔇)^{ab} .

A Appendix: Results from class field theory

For reference, to fix notation and to deduce some corollaries, we recall several results from the higher dimensional local and global class field theory (mainly due to the work of K. Kato and S. Saito). Let X be a connected noetherian scheme. For j ∈ ℕ, let

X_{j} = \{x ∈ X | dim (\overline{\{x\}}) = j\} , X^{j} = \{x ∈ X | codim_{X} (\overline{\{x\}}) = j\} .

For an integer i ≥ 0, the localization theory in Quillen K-theory on X gives rise to a homomorphism

(1)  ∂_{i} : ⨁_{y∈X_{1}} K_{i+1}(k(y)) ⟶ ⨁_{x∈X_{0}} K_{i}(k(x)).

We define SK_{i}(X) to be the cokernel of this map ∂_{i}. For i = 0, 1, theorems 2.2 and 2.6 of [Sn] imply the following explicit descriptions of ∂_{i}: The restriction of ∂_{i} to the y-component K_{i+1}(k(y)) is the composite of the following homomorphisms:

K_{i+1}(k(y)) ⟶^{∂′_{i}} ⨁_{x′∈Ỹ_{0}} K_{i}(k(x′)) ⟶^{N} ⨁_{x∈Y_{0}} K_{i}(k(x)) ⟶ ⨁_{x∈X_{0}} K_{i}(k(x))

where Y is the closure of {y} in X, Ỹ is the normalization of Y and the map ∂′_{i} is defined as

∂′_{i} = \{ \begin{matrix}∑_{x′∈Ỹ_{0}} ord_{x′} for i = 0, \\ ∑_{x′∈Ỹ_{0}} d_{x′} for i = 1, \end{matrix}

where d_{x′} is defined for t = (f, g) ∈ K_{2}(k(y)) by

d_{x′}(t) = (−1)^{ord_{x′} (f)ord_{x′} (g)}f^{ord_{x′} (g)}g^{−ord_{x′} (f)}.

Finally, the map N is defined as follows: For x′ ∈ Ỹ_{0} and x ∈ Y_{0}, the (x′, x)- component of N: K_{i}(k(x′)) → K_{i}(k(x)) is the zero map if x′ does not lie over x ; otherwise it is multiplication by [k(x′) : k(x)]: ℤ → ℤ if i = 0, and the usual norm homomorphism k(x′)^{×} → k(x)^{×} if i = 1.

Assume that X is a variety over a field k and that k (and hence also every finite extension of k) carries a natural topology. Then we give SK_{0}(X) the discrete topology and we endow SK_{1}(X) with the finest topology such that for every x ∈ X_{0} the natural homomorphism

k(x)^{×} ⟶ SK_{1}(X)

is continuous.

A 1. The reciprocity homomorphism for an arithmetic scheme.

Let X be a connected scheme of finite type over Spec(ℤ). For each closed point x ∈ X_{0}, k(x) is a finite field, so there is an isomorphism

ℤ̂ ⟶^{∼} π_{1}^{ab}(x) = Gal(k(x)^{sep}|k(k)),

which sends 1 ∈ ℤ̂ to the Frobenius f_{x} over k(x). We define the Frobenius element F_{x} of x in π_{1}^{ab}(X) to be the image of f_{x} under the natural homomorphism π_{1}^{ab}(x) → π_{1}^{ab}(X). The assignment 1 ↦ F_{x}, x ∈ X_{0}, defines a homomorphism

(2)  ⨁_{x∈X_{0}} ℤ ⟶ π_{1}^{ab}(X).

By the results of Lang [La], we have the

Lemma A.1 If X is irreducible and the reduced subscheme X_{red} of X is normal, then the map (2) has a dense image.

Now we consider the abelianized modified fundamental group π̃_{1}^{ab}(X) which is the unique quotient of π_{1}^{ab}(X) classifying finite abelian étale coverings of X in which every real-valued point of X splits completely.

Lemma A.2 ([Sa], lemma 2.4) If X is proper over Spec(ℤ), then the composite of the map (2) and the natural surjection π_{1}^{ab}(X) → π̃_{1}^{ab}(X) annihilates the image of

∂ : ⨁_{x∈X_{1}} k(x)^{×} ⟶ ⨁_{x∈X_{0}} ℤ,

which is the map (1) for i = 0.

Consequently, we obtain a natural map

SK_{0}(X) ⟶ π̃_{1}^{ab}(X),

which is called the reciprocity map for X.

A 2. Class field theory for varieties over ℝ and ℂ.

We follow [Sa], §4. Let k^{∗} be ℝ or ℂ and let X^{∗} be a connected, proper and smooth scheme over k^{∗}. For x ∈ X_{0}^{∗}, k(x) ≅ ℝ or ℂ and we have a canonical surjection

k(x)^{×} ⟶ π_{1}^{ab}(x) = Gal(k(x)^{sep}|k(x)).

The maps π_{1}^{ab}(x) → π_{1}^{ab}(X^{∗}) for x ∈ X_{0}^{∗} induce a well-defined continuous homomorphism

(3)  τ : SK_{1}(X^{∗}) ⟶ π_{1}^{ab}(X^{∗}),

which is the zero map if k^{∗} = ℂ or X^{∗}(ℝ) = ∅. Let k ⊂ k^{∗} be a subfield and suppose that the following conditions (a)–(c) are satisfied

(a) k is dense in k^{∗} for the usual topology of k^{∗}.

(b) If k^{∗} = ℂ, k is algebraically closed. If k^{∗} = ℝ, the algebraic closure k̄ of k is a quadratic extension of k.

(c) There is a proper, smooth scheme X over k such that X ⊗_{k} k^{∗} ≅ X^{∗}.

By the proper base change theorem, the natural map π_{1}^{ab}(X^{∗}) → π_{1}^{ab}(X) is an isomorphism and the image of the natural homomorphism

(4)  SK_{1}(X) ⟶ π_{1}^{ab}(X)

coincides with that of the map (3). Slightly more general, let V ⊂ X be a nonempty open subscheme and V^{∗} = V ⊗_{k} k^{∗}. Using the smooth base change theorem instead (we are in characteristic zero), we have again the isomorphism π_{1}^{ab}(V^{∗}) ⟶^{∼} π_{1}^{ab}(V) and, sending a point x ∈ V (ℝ) = V^{∗}(ℝ) to the image of the unique nontrivial element of π_{1}^{ab}(x) ≅ Gal(ℂ|ℝ) in π_{1}^{ab}(V), we obtain a natural map

i : V (ℝ) ⟶ π_{1}^{ab}(V).

The map i is locally constant on V (ℝ) and has finite image (see [Sa], lemma 4.8 for the case V = X, the proof in the general case is the same). If χ ∈ H_{et}^{1}(V, ℚ/ℤ) corresponds to a cyclic étale covering V′ → V, then a point x ∈ V (ℝ) splits completely in V′ if and only if χ is trivial on i(x). In particular, the subset of points in V (ℝ) which split completely in V′ is (norm) open and closed in V (ℝ).

A 3. Class field theory of schemes over henselian discrete valuation fields.

We follow [Sa], §3. Let O_{k} be a henselian discrete valuation ring with finite residue field F and quotient field k. Let X be a proper smooth scheme over k. For x ∈ X_{0}, k(x) is a finite extension of k, so that the local class field theory for k(x) gives us a canonical homomorphism

ρ_{x} : k(x)^{×} ⟶ π_{1}^{ab}(x) ⟶ π_{1}^{ab}(X).

The sum of these ρ_{x} is trivial on the diagonal image of ⨁_{x∈X_{1}} K_{2}(k(X)) and we obtain a canonical morphism

(5)  τ : SK_{1}(X) ⟶ π_{1}^{ab}(X).

Giving each k(x), x ∈ X_{0} the usual adic topology, we obtain a natural topology on SK_{1}(X) (see the beginning of this section). The map τ is continuous with respect to this topology and the natural profinite topology on π_{1}^{ab}(X). For every natural number n which is prime to char(k), the subgroup n · SK_{1}(X) is open in SK_{1}(X). In particular, if char(k)=0, then every homomorphism SK_{1}(X) → ℚ/ℤ with finite image is continuous.

Now assume that X has a proper regular model 𝔛 over O_{k} and let Y be its special fibre. By [Sa], lemma 3.11, we have a natural continuous and surjective homomorphism ∂ : SK_{1}(X) → SK_{0}(Y) and a commutative diagram

SK_{1}(X) ⟶^{τ} π_{1}^{ab}(X)
↓^{∂}  ↓^{sp}
SK_{0}(Y) ⟶^{ϕ} π_{1}^{ab}(Y),

where sp is the specialization homomorphism (cf. [SGA1], X) and ϕ is the reciprocity map for Y as defined in paragraph 1 above. Important for us is the

Proposition A.3 ([Sa], prop. 3.3) Let χ ∈ H_{et}^{1}(X, ℚ/ℤ) and let χ̃ : SK_{1}(X) → ℚ/ℤ be the induced homomorphism via (5). Then χ comes from the subgroup H_{et}^{1}(Y, ℚ/ℤ) ≅ H_{et}^{1}(𝔛, ℚ/ℤ) ⊂ H_{et}^{1}(X, ℚ/ℤ) if and only if χ̃ factors though the map ∂.

The crucial step in the proof of the last proposition is the following lemma, which we also need in the proof of theorem 6.2.

Lemma A.4 ([Sa], lemma 3.15) Let A be a henselian regular local ring of dimension ≥ 2 with perfect residue field F and quotient field K. If char(K) = 0 and char(F) = p > 0 assume that there exists exactly one height-one prime ideal 𝔭 ⊂ A dividing p and let T ∈ A be an element with 𝔭 = (T). In the remaining cases let T ≠ 0 be any non-unit of A. Put U = Spec(A[1/T]). Then, if χ ∈ H_{et}^{1}(U, ℚ/ℤ) induces an unramified character χ_{u} ∈ H_{et}^{1}(u, ℚ/ℤ) for each u ∈ U_{0}, χ comes from H_{et}^{1}(Spec(A), ℚ/ℤ).

A 4. Unramified class field theory of arithmetic schemes.

Let 𝔛 be a connected regular, proper and flat scheme over Spec(ℤ), and suppose that X_{η} = 𝔛 ⊗_{ℤ} ℚ is projective over Spec(ℚ). Let k be the algebraic closure of ℚ in the function field of X and put S = Spec(O_{k}). The structural morphism 𝔛 → Spec(ℤ) factors through S and X_{η} is geometrically irreducible as a variety over k. Let S_{f} be the set of closed points of S and let S_{∞} be the set of archimedean places of the number field k. For v ∈ S_{f} let Y_{v} = 𝔛 ⊗_{S} v be the special fibre of 𝔛 over v. For v ∈ S_{∞} ∪ S_{f}, let k_{v} be the algebraic closure of k in the completion of k at v and X_{v} = X_{η} ⊗_{k} k_{v}.

For v ∈ S_{f} we recall the continuous and surjective homomorphism (5) from paragraph 3:

τ_{v} : SK_{1}(X_{v}) ⟶ SK_{0}(Y_{v}) = CH_{0}(Y_{v}).

For a nonempty open subscheme U ⊂ S we consider the topological group

I(𝔛/U) = ( ∏_{v∈P_{U}} SK_{1}(X_{v})) × ( ⨁_{v∈U_{0}} CH_{0}(Y_{v})) ,

where P_{U} denote the set of places of k (including the archimedean ones) which are not in U_{0}. There exists a natural homomorphism ([Sa], 5.3)

I(𝔛/U) ⟶ π_{1}(𝔛_{U})^{ab},

which annihilates the image of the diagonal map SK_{1}(X_{η}) ⟶ I(𝔛/U). It therefore induces a continuous homomorphism ([Sa], 5.5)

τ : C(𝔛/U) := coker(SK_{1}(X_{η}) → I(𝔛)) ⟶ π_{1}(𝔛)^{ab}.

By [Sa], 5.6., every subgroup of finite index in C(𝔛/U) is open and, for every positive integer n, τ induces an isomorphism of finite abelian groups

C(𝔛/U)/n ⟶^{∼} π_{1}(𝔛_{U})^{ab}/n.

References

[BMS] H. Bass, J. W. Milnor, J.-P. Serre Solution of the congruence subgroup problem for SL_{n}(n ≥ 3) and Sp_{2n}(n ≥ 2) Publ. Math., Inst. Hautes Étud. Sci. 33 (1967), 59–137

[B-T] H. Bass, J. Tate The Milnor ring of a global field. Algebr. K-Theory II, Proc. Conf. Battelle Inst. 1972, Springer LNM 342, 349–446 (1973)

[B-O] S. Bloch, A. Ogus Gersten’s conjecture and the homology of schemes. Ann. Sci. Éc. Norm. Supér., IV. Sér. 7 (1974), 181–201

[Bo] N. Bourbaki Commutative Algebra Herman Paris, 1972

[Co] K. R. Coombes Relative algebraic K-theory. Appendix to K. R. Coombes, V. Srinivas: Relative K-theory and vector bundles on certain singular varieties. Invent. Math. 70 (1982), 13–25

[De] P. Deligne (after A. Grothendieck) La class de cohomologie associée à un cycle. In: Séminaire de Géométrie Algébrique du Bois-Marie SGA 4\frac{1}{2}, Springer Lect. Notes in Math. vol. 569 (1977), 129–153

[Fu] K. Fujiwara A proof of the absolute purity conjecture (after Gabber). Algebraic geometry 2000, Azumino, 153–183, Adv. Stud. Pure Math., 36, Math. Soc. Japan, Tokyo, 2002

[Gr] D. Grayson (after D. Quillen) Higher algebraic K-theory. II. Springer Lect. Notes in Math. vol. 551 (1976), 217–240

[G-M] A. Grothendieck, J. P. Murre The tame fundamental group of a formal neighbourhood of a divisor with normal crossings on a scheme. Springer Lect. Notes Math. vol. 208, 1971

[J-S] U. Jannsen, S. Saito Katohomology of arithmetic schemes and higher class field theory over local fields. Doc. math. Extra Vol: Kazuya Kato’s fiftieth birthday 2003, 479–538.

[Ka] K. Kato A Hasse principle for two dimensional global fields. J. reine u. angew. Math. 366 (1986), 13–37

[K-S1] K. Kato, S. Saito Unramified class field theory of arithmetical surfaces. Ann. of Math. 118 (1983), 142–183

[K-S2] K. Kato, S. Saito Two dimensional class field theory. Galois groups and their representations, Proc. Symp., Nagoya/Jap. 1981, Adv. Stud. Pure Math. 2 (1983), 103–152

[K-S3] K. Kato, S. Saito Global class field theory of arithmetic schemes. Applications of Algebraic K-theory to Algebraic Geometry and Number Theory (S.Bloch, R.K.Dennis, E.Friedlander, and M.Stein, ed.) Contemp. Math., vol. 55, Amer. Math. Soc., Providence RI, 1986, 255–331 [La] S. Lang Sur le séries L d’une variété algébrique. Bull. Soc. Math. France

84 (1956), 555–563

[Le] M. Levine The indecomposable K_{3} of fields. Ann. Sci. Éc. Norm. Supér., IV. Sér. 22 (1989), 255-344

[Mo] C. Moore Group extensions of p-adic and adelic linear groups. Publ. Math. I.H.E.S. 35 (1969), 5–74

[Po] B. Poonen Bertini theorems over finite fields. Preprint 2002 www.arxiv.org: math.AG/0204002, to appear in Ann. Math.

[Qu] D. Quillen Higher algebraic K-theory. I. Algebr. K-Theory I, Proc. Conf. Battelle Inst. 1972, Springer Lect. Notes in Math. vol. 341 (1973), 85–147

[Sa] S. Saito Unramified class field theory of arithmetical schemes. Ann. of Math. 121 (1985), 251–281

[S1] A. Schmidt Tame coverings of arithmetic schemes. Math. Annalen 322 (2002), 1–18

[S2] A. Schmidt Singular homology of arithmetic schemes. Preprint 2000 http://www.math.uiuc.edu/K-theory/0418/.

[S-S] A. Schmidt, M. Spieß Singular homology and class field theory of varieties over finite fields. J. reine u. angew. Math. 527 (2000), 13–37

[Se] J.-P. Serre Local fields. Springer Verlag New York, Berlin, 1979

[SGA1] A. Grothendieck Revêtements étales et groupe fondamental. Springer Lect. Notes Math. vol. 224 (1971)

[Sn] V. Snaith On the localization sequence in K-theory. Proc. Amer. Math. Soc. 73 (1981), 359–364

Alexander Schmidt, NWF I - Mathematik, Universität Regensburg, D-93040 Regensburg, Deutschland. email: alexander.schmidt@mathematik.uni-regensburg.de