Singular Homology and Class Field Theory of Varieties over Finite Fields

By Alexander Schmidt and Michael Spieß

Dedicated to the memory of Jürgen Neukirch

The aim of global class field theory is the description of abelian extensions of a finitely generated field k in terms of arithmetic invariants attached to k. 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 [KS2], 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 X̄ of k), then class field theory has a nice geometric description using the Chow group CH_{0}(X̄) of zero-cycles modulo rational equivalence (see [KS1], [Sa]).

The objective of this paper is to give a description of a similar geometric flavor of the tamely ramified abelian extensions of k (w.r.t. a finite set Y_{1}, …, Y_{n} of prime divisors of X̄). We do this in the case of positive characteristic, where the role of CH_{0}(X̄) is taken over by the 0th singular homology group (introduced by A. Suslin) h_{0}(X), where X = X̄ − {Y_{1} ∪ ⋯ ∪ Y_{n}}. The arithmetic case, i.e. if k is a finitely generated extension of ℚ, will be considered in a separate paper.

Let X̄ be a smooth, projective and geometrically connected variety over a finite field and let X be an open subvariety of X̄. Let π^{t}(X) be the tame étale fundamental group of X (cf. [GM]), i.e. π^{t}(X) classifies étale coverings of X which are tamely ramified over all prime divisors of X̄ not lying on X. Our main result is the existence of a natural map (reciprocity homomorphism)

rec : h_{0}(X) ⟶ π_{1}^{t}(X)^{ab},

which is injective with dense image and which induces an isomorphism on the profinite completions. Passing to small open subschemes, this completely describes the tame part of the abelianized absolute Galois group of the function field of X.

Let us formulate the result in more detail. Sending a closed point x of X to its Frobenius in π_{1}^{t}(X)^{ab} defines a homomorphism

Z_{0}(X) ⟶ π_{1}^{t}(X)^{ab},

where Z_{0}(X) is the group of 0-cycles on X. This map is known to have a dense image (see [L2] or [Ra], lemma 1.7), and we will see below that it factors through h_{0}(X), which is a quotient of Z_{0}(X) by definition.

The structural map X → Spec(k) induces natural surjective homomorphisms

π_{1}^{t}(X)^{ab} ↠ π_{1}(Spec(k))^{ab} ≅ ℤ̂ and h_{0}(X) ↠^{deg} h_{0}(Spec(k)) ≅ ℤ

and we denote the kernels of these maps by (π_{1}^{t}(X)^{ab})^{0} and h_{0}(X)^{0}, respectively.

Theorem 0.1 The reciprocity homomorphism rec : h_{0}(X) → π_{1}^{t}(X)^{ab} induces an isomorphism

rec_{0} : h_{0}(X)^{0} ⟶^{∼} (π_{1}^{t}(X)^{ab})^{0}

of finite abelian groups which fits into the commutative diagram with exact rows

0 ⟶ h_{0}(X)^{0} ⟶ h_{0}(X) ⟶ ℤ ⟶ 0
↓^{rec}^{0}  ↓^{rec}  ^{∩}^{can}
0 ⟶ (π_{1}^{t}(X)^{ab})^{0} ⟶ π_{1}^{t}(X)^{ab} ⟶ ℤ̂ ⟶ 0.

In particular, rec induces an isomorphism between the profinite completion of h_{0}(X) and π_{1}^{t}(X)^{ab}.

Theorem 0.1 is well known in the following cases:

1. If dim X = 1, then (see [SV1], th.3.1) h_{0}(X) is naturally isomorphic to the relative Picard group Pic(X̄, X̄ − X). A straightforward computation identifies this relative Picard group with the ray class group of the function field k(X) of X with modulus 𝔪_{X}, where 𝔪_{X} is the product of all primes of X̄ − X. In this case, rec is the reciprocity homomorphism of the classical (one-dimensional) class field theory for global fields of positive characteristic.

2. If X is projective, then h_{0}(X) coincides with the Chow group CH_{0}(X) of zero-dimensional cycles on X modulo rational equivalence. In this case, theorem 0.1 is a reformulation of Kato’s and Saito’s unramified class field theory for X, see [KS1], [Sa] (and for two-dimensional X we will eventually deduce theorem 0.1 from their result in the projective case).

Let us make a remark on the role of the smooth compactification X̄. First of all, π_{1}^{t}(X)^{ab} does not depend on X̄. Indeed, by the purity of the branch locus ([SGA1]), Exp.X, Th.3.1), we have a decomposition

π_{1}^{t}(X)^{ab} ≅ π_{1}(X)^{ab}(prime-to-p) ⊕ π_{1}(X̄)^{ab}(p)

and π_{1}(X̄) is a birational invariant ([SGA1], Exp.X, Cor.3.4). We need X̄ just for technical reasons. Its existence is known in dimension ≤ 2 and any kind of desingularization theorem in positive characteristic would imply that theorem 0.1 holds for an arbitrary smooth, quasiprojective and geometrically connected variety over k.

In the arithmetic case, i.e. when X is a regular flat ℤ-scheme of finite type, one should expect a result similar to theorem 0.1. The first problem one encounters is that the singular homology groups are defined only for varieties over fields. However, one can give an absolute version of singular homology groups for any scheme X of finite type over ℤ (see [Sch] for their definition and basic properties). The relation between the 0th absolute singular homology and the abelianized tame fundamental group will be investigated in a future work.

We want to thank D. Huybrechts for helpful e-mail conversation concerning the Bertini-type theorem 5.1.

1 Notation and frequently used facts

In this section, we collect some facts and notation from the papers [SV1], [SV2], [SV3], [V1], [V2] which will be frequently used in the subsequent sections. Recall Suslin’s definition of the singular homology of varieties [SV1]. Let k be a field which will always assumed to be perfect throughout this paper. Let Δ^{•} be the standard cosimplicial object in the category of smooth schemes over k, i.e. Δ^{n} is the standard n-dimensional simplex given as a subscheme in 𝔸_{k}^{n+1} = Spec(k[T_{0}, …, T_{n}]) by the equation Σ T_{i} = 1 and the simplicial structure is given by the obvious face and degeneracy morphisms.

Let X be a scheme of finite type over k. Denote by C_{n}(X) the free abelian group generated by closed integral subschemes Z ⊂ X × Δ^{n} such that the projection Z → Δ^{n} is finite and surjective. One verifies immediately that if Z is as above, then for each face map δ^{i} : Δ^{n−1} → Δ^{n} each component of (δ^{i})^{−1}(Z) ⊂ X × Δ^{n−1} is finite and surjective over Δ^{n−1} and hence has the ”correct” dimension. So the cycle theoretic inverse image (δ^{i})^{∗}(Z) is well-defined and lies in C_{n−1}(X). This gives us face operators

∂_{i} = (δ^{i})^{∗} : C_{n}(X) ⟶ C_{n−1}(X).

The homology groups of the complex

(C_{•}(X), d), d = ∑(−1)^{i}δ_{i}

will be denoted by h_{i}(X) and are called the (integral) singular homology groups of X. For a positive integer n, the singular homology groups of X with ℤ/nℤcoefficients are defined as the homology groups of the complex C_{•}(X) ⊗ ℤ/nℤ. An essential feature of these singular homology groups is that they naturally occur as homomorphism groups in Voevodsky’s ”triangulated category of motivic complexes” DM_{−}^{eff}(k). As in [V1], we consider the following categories:

Sm(k)  - the category of smooth schemes over Spec(k)

with scheme morphisms,

SmCor(k)  - the category of smooth schemes over Spec(k)

with finite correspondences as morphisms,

Shv_{Nis}(SmCor(k)) - the category of Nisnevich sheaves of

abelian groups on SmCor(k).

We refer to presheaves on SmCor(k) also as presheaves with transfer on Sm(k). A presheaf F on Sm(k) is called homotopy invariant if for every U ∈ Sm(k) the natural projection p : U ×𝔸_{k}^{1} → U induces an isomorphism F (U) ⟶^{∼} F (U ×𝔸_{k}^{1}). If F is a homotopy invariant presheaf with transfers, then also the sheafification F_{Nis} is homotopy invariant (see [V1], th.3.1.12). We use the following notation.

D^{−}(Shv_{Nis}(SmCor(k))) - the derived category of bounded above

complexes in Shv_{Nis}(SmCor(k)),

DM_{−}^{eff}(k)  - the full subcategory of D^{−}(Shv_{Nis}(SmCor(k)))

consisting of complexes with homotopy invariant cohomology sheaves.

DM_{−}^{eff}(k) is a triangulated category in a natural way. For a presheaf F on Sm(k), we consider the complex \underline{C}_{•}(F) of presheaves on Sm(k) with \underline{C}_{n}(F)(X) = F (Δ^{n}×X) and with differentials given by the alternating sum of the morphisms which correspond to the boundary morphisms of Δ^{•}. We denote the homology presheaves of \underline{C}_{•}(F) by \underline{h}_{i}(F). We set C_{•}(F) = \underline{C}_{•}(F)(Spec(k)) and h_{i}(F) = \underline{h}_{i}(F)(Spec(k)). Further, \underline{h}_{i}^{Nis}(F) denotes the homology sheaves of \underline{C}_{•}(F)_{Nis} and we write h_{i}^{Nis}(F) for \underline{h}_{i}^{Nis}(F)(Spec(k)). Note that \underline{C}_{•}(F) is a complex of sheaves if F is a sheaf, but in general \underline{C}_{•}(F_{Nis}) and \underline{C}_{•}(F)_{Nis} differ. If F is a presheaf on SmCor(k), then the sheaves \underline{h}_{i}^{Nis}(F) are homotopy invariant (see [V1], th.3.1.12). This yields a functor

\underline{C}_{•} : Shv_{Nis}(SmCor(k)) → DM_{−}^{eff}(k).

Suppose we are given a scheme X of finite type over k. We consider the presheaves z_{0}(X) and c_{0}(X) on SmCor(k) which are given for U ∈ Sm(k) by

z_{0}(X)(U) - free abelian group over closed integral subschemes Z ⊂ X × U,

quasifinite and dominant over an irreducible component of U;

c_{0}(X)(U) - free abelian group over closed integral subschemes Z ⊂ X × U,

finite and surjective over an irreducible component of U.

These presheaves are Nisnevich sheaves (see [SV2], th.4.2.9). If X is smooth, then c_{0}(X)(U) = Hom_{SmCor(k)}(U, X). The complex \underline{C}_{•}(c_{0}(X)) defines an object M(X) in DM_{−}^{eff}(k) which is called the motivic complex attached to X. For an integer n, we use the notation M(X, ℤ/nℤ) for the object in DM_{−}^{eff}(k) which is given by the complex \underline{C}_{•}(c_{0}(X) ⊗ ℤ/nℤ). M(X) is covariant functorial with respect to scheme morphisms and for smooth schemes with respect to finite correspondences. We write ℤ for M(Spec(k)).

By ([V1], prop.3.2.3), \underline{C}_{•} naturally extends to a functor

R\underline{C} : D^{−}(Shv_{Nis}(SmCor(k))) ⟶ DM_{−}^{eff}(k),

and (loc. cit.) one can give an alternative description of DM_{−}^{eff}(k) as the localization of D^{−}(Shv_{Nis}(SmCor(k))) with respect to the localizing subcategory

generated by complexes of the form c_{0}(X × 𝔸_{k}^{1}) ⟶^{c_{0}(pr)} c_{0}(X) for smooth schemes X over k.

For a smooth scheme X, we have (see [V1], th.3.2.6) a natural isomorphism in DM_{−}^{eff}(k)

\underline{C}_{•}(c_{0}(X)) ⟶^{∼} R\underline{C}(c_{0}(X)).

If X and Y are smooth, then (see [V1], cor.3.2.7)

Hom_{DM_{−}^{eff}(k)}(M(X), M(Y)[n]) = H_{Nis}^{n}(X, \underline{C}_{•}(c_{0}(Y))).

In particular, for smooth X

Hom_{DM_{−}^{eff}(k)}(ℤ[n], M(X)) = h_{n}^{Nis}(c_{0}(X)) = h_{n}(c_{0}(X)) = h_{n}(X).

Using left resolutions by presheaves with transfers of the form c_{0}(X) with X smooth, and using the characterization of DM_{−}^{eff}(k) as a localization of the category D^{−}(Shv_{Nis}(SmCor(k))), one constructs a tensor product on DM_{−}^{eff}(k) (see [V1], §3).

Recall the definition of the Tate object ℤ(1) in DM_{−}^{eff}(k) (cf. [SV3], §3). Let 𝔾_{m} denote the multiplicative group scheme 𝔸_{k}^{1} − {0}. Let, for i = 1, ⋯, n, D_{i}(𝔾_{m}^{×(n−1)}) be the direct summand in c_{0}(𝔾_{m}^{×n}) which is given by the homomorphism

c_{0}(𝔾_{m}^{×(n−1)}) ⟶ c_{0}(𝔾_{m}^{×n}), (x_{1}, …, x_{n−1}) ↦ (x_{1}, …, 1_{i}, …, x_{n−1})

We consider the sheaf c_{0}(𝔾_{m}^{∧n}) which is defined as the direct summand of c_{0}(𝔾_{m}^{×n}) complementary to the direct summand ∑_{i=1}^{n} D_{i}(𝔾_{m}^{×(n−1)}). The Tate object ℤ(n) is defined as C_{•}(c_{0}(𝔾_{m}^{∧n}))[−n]. There are natural quasiisomorphisms ℤ(n)⊗ℤ(m) ⟶^{∼} ℤ(n+m) for n, m ≥ 0. For an arbitrary M ∈ DM_{−}^{eff}(k), one defines

M(n) = M ⊗ ℤ(n).

By ([V1], prop.3.5.1), we have a natural decomposition

M(X × ℙ_{k}^{n}) ≅ ⨁_{i=0}^{n} M(X)(i)[2i].

The complex ℤ(n) is acyclic in (cohomological) degrees > n, thus for an étale k-scheme Z, we have

Hom_{DM_{−}^{eff}(k)}(ℤ, M(Z)(q)[p]) = 0 for p > q.

Furthermore, by ([SV3], th.3.4), we have a natural isomorphism

Hom_{DM_{−}^{eff}(k)}(ℤ, M(Z)(2)[2]) ≅ K_{2}(Z).

Since K_{2} of finite fields vanishes, we obtain the

Proposition 1.1 If k is finite and if Z is an étale k-scheme, then

Hom_{DM_{−}^{eff}(k)}(ℤ, M(Z)(2)[i]) = 0 for i ≥ 2

and the same is true with M(Z) replaced by M(Z, ℤ/nℤ) for an arbitrary positive integer n.

Next we recall some well-known facts on the higher Chow groups defined by Bloch [B1]. Let X be an equidimensional scheme of finite type over k. Let z^{r}(X, n) be the free abelian group generated by codimension r subvarieties V ⊂ X × Δ^{n} which intersect all faces X × Δ^{m}, m < n properly (i.e. every irreducible component of the intersection has the right dimension). Using the alternating sum of the homomorphisms associated to the face morphisms, one obtains a complex z^{r}(X, •) in the usual way, and the higher Chow groups (respectively the higher Chow groups with ℤ/nℤ-coefficients) are defined by

CH^{r}(X, i) := H^{−i}(z^{r}(X, •)) and
CH^{r}(X, i, ℤ/nℤ) := H^{−i}(z^{r}(X, •, ℤ/nℤ)),

respectively (where z^{r}(X, •, ℤ/nℤ) = z^{r}(X, •) ⊗ ℤ/nℤ). Immediately from the definition, we have

CH^{r}(X, i) = 0 for r > dim X + i.

The higher Chow groups are covariant functorial for proper maps and contravariant for flat map (for arbitrary maps, if X is smooth and affine). The higher Chow sheaves

CH^{d}(X, i)

are defined as the sheaves on X_{Zar} associated to U ↦ CH^{d}(U, i). By ([B1], th.3.2), there exists a local to global spectral sequence

E_{2}^{pq} = H_{Zar}^{p}(X, CH^{r}(X, −q)) ⟹ CH^{r}(X, −p − q).

Let X be a smooth variety and denote by X^{j} the set of points of codimension j on X. There are flasque resolutions of Zariski sheaves for all r (Gersten resolutions; see [B1], th.10.1)

CH^{r}(X, q) ↪ ⨁_{x∈X^{0}}i_{x∗}CH^{r}(Spec(k(x)), q)) → ⨁_{x∈X^{1}}i_{x∗}CH^{r−1}(Spec(k(x)), q − 1))

→ ⋯ → ⨁_{x∈X^{q}}i_{x∗}CH^{r−q}(Spec(k(x)), 0)) → 0,

and a similar result holds for ℤ/nℤ-coefficients as well.

Furthermore, we have the following connection to Milnor K-theory (see [NS], th.4.9 or [To]): For any field k

CH^{r}(Spec(k), r) ≅ K_{r}^{M}(k).

From the partial degeneration of the local to global spectral sequence, one obtains the

Proposition 1.2 For X smooth and quasiprojective, we have isomorphisms

CH^{p}(X, i) ≅ H_{Zar}^{p−i}(X, CH^{p}(X, p)) for 0 ≤ i ≤ 2.

and the same holds for ℤ/nℤ-coefficients as well.

The theorem of Suslin/Nesterenko/Totaro together with the Gersten resolution for ℤ/nℤ-coefficients imply the

Proposition 1.3 If X is smooth, quasiprojective of dimension d, then there exists a flasque resolution

0 → CH^{d}(X, d, ℤ/nℤ) → ⨁_{x∈X^{0}} i_{x∗}(K_{d}^{M}(k(x))/n) → ⨁_{x∈X^{1}} i_{x∗}(K_{d−1}^{M}(k(x))/n)

→ ⋯ → ⨁_{x∈X^{d}} i_{x∗}(K_{0}^{M}(k(x))/n)) → 0.

2 Comparison results

Many of the deeper theorems on algebraic cycle complexes depend on the fact that if F is a presheaf on Sm(k) satisfying the property (∗) below, then \underline{h}_{i}^{Nis}(F) = 0 for all i. This was proved in ([FV], th.5.5.2) under the assumption that the field k admits resolution of singularities (ROS, for short) in the sense of ([FV], definition 3.4). Following the arguments in [V2], [FV], we prove a weaker result, which, however, only depends on ROS for schemes of dimension ≤ 2 (which is known).

For a presheaf F on Sm(k), we consider the following property (∗):

”For any smooth scheme X over k and any x ∈ F (X) there exists a proper birational morphism p : X′ → X such that p^{∗}(x) = 0.”

By (∗)^{≤2} we will refer to the property that the statement of (∗) holds for all smooth schemes over k of dimension ≤ 2. For a smooth scheme X, we denote by ℤ(X) the presheaf of abelian groups on Sm(k) freely generated by the presheaf of sets represented by X. We need the following results.

Proposition 2.1 Let k be a perfect field, let X be a smooth scheme over k and let Z ⊂ X be a smooth closed subscheme. Denote by p : X_{Z} → X the blow-up of X with center in Z. Then, for every homotopy invariant G ∈ Shv_{Nis}(SmCor(k)) and any i, one has

Ext_{Shv_{Nis}(Sm(k))}^{i}(coker(ℤ(X_{Z}) →^{ℤ(p)} ℤ(X))_{Nis}, G) = 0.

Proof: This follows from ([V2], prop.5.21) and ([V1], prop.3.1.11). □

Lemma 2.2 Let F be a presheaf on Sm(k) and let G ∈ Shv_{Nis}(SmCor(k)) be homotopy invariant. Then for all i

Hom_{D^{−}(Shv_{Nis}(Sm(k)))}(\underline{C}_{•}(F)_{Nis}, G[i]) ≅ Ext_{Shv_{Nis}(Sm(k))}^{i}(F, G).

Proof: See ([FV], lemma 5.6) resp. the proof of ([V2], th.5.9). □

Corollary 2.3 Let p : X_{Z} → X be as in proposition 2.1. Then

\underline{h}_{i}^{Nis}(coker(ℤ(X_{Z}) →^{ℤ(p)} ℤ(X))) = 0 for all i.

Proof: (cf. the last part of the proof of [FV], th.5.5). Put F = coker(ℤ(X_{Z}) → ℤ(X)). Suppose \underline{h}_{n}^{Nis}(F) ≠ 0 for some n. We may assume that \underline{h}_{i}^{Nis}(F) = 0 for all i < n. Then there is a nontrivial morphism in D^{−}(Shv_{Nis}(Sm(k))) of the form \underline{C}_{•}(F)_{Nis} → \underline{h}_{n}^{Nis}(F)[n]. Therefore the statement follows from 2.2 and 2.1 □

Now we are prepared to prove the

Theorem 2.4 Let F be a presheaf on Sm(k) satisfying the property (∗)^{≤2}. Then

h_{i}^{Nis}(F) = 0 for i ≤ 2.

Proof: A proper birational morphism of smooth schemes of dimension ≤ 1 is an isomorphism, thus F (U) = 0 for any smooth U of dimension ≤ 1. In particular,

\underline{C}_{i}(F)_{Nis}(Spec(k)) = 0 for i = 0, 1,

which shows the assertion for i = 0, 1.

In order to deal with the case i = 2, we first consider the special case that F is of the form

coker(ℤ(X′) ⟶^{ℤ(p)} ℤ(X)),

where p : X′ ⟶ X is a birational morphism of smooth k-schemes of dimension 2 which is a composition of n blow-ups with smooth centers (presheaves of this type satisfy property (∗)). We proceed by induction on n as in the proof of ([FV], lemma 5.4). Let p = p_{0} ∘ p_{1}, where p_{0} is a blow-up with a smooth center and p_{1} is a composition of n − 1 blow-ups with smooth centers. By induction on n and corollary 2.3, we may assume that h_{i}^{Nis}(coker(ℤ(p_{ϵ}))) = 0 for i ≤ 2 and ϵ = 0, 1. We have an exact sequence of presheaves on Sm(k)

0 → Ψ → coker(ℤ(p_{1})) → coker(ℤ(p)) → coker(ℤ(p_{0})) → 0.

We rewrite this as two short exact sequences

0 → Ψ → coker(ℤ(p_{1})) → Ψ′ → 0,
0 → Ψ′ → coker(ℤ(p)) → coker(ℤ(p_{0})) → 0.

As a subpresheaf of coker(ℤ(p_{1})) (resp. of coker(ℤ(p))), the presheaf Ψ (resp. Ψ′) satisfies property (∗). Thus h_{i}^{Nis}(Ψ) = 0 = h_{i}^{Nis}(Ψ′) for i = 0, 1 and the long exact h_{i}^{Nis}-sequences associated with the two short exact sequences above imply that

(1) h_{i}^{Nis}(coker(ℤ(p))) = 0 for i ≤ 2.

For a general presheaf F on Sm(k) satisfying property (∗)^{≤2}, we consider the homomorphism

⨁ ϕ_{α} : ⨁ ℤ(X_{α}) ⟶ F,

where the sum is taken over all pairs of the form (X_{α}, ϕ_{α}), where X_{α} is a smooth scheme of dimension 2 over k and ϕ_{α} ∈ F (X_{α}). It follows from property (∗)^{≤2} and from the resolution of singularities for schemes of dimension 2 [Li] that the above homomorphism may by factored through a homomorphism of the form

⨁ coker(ℤ(X′) →^{ℤ(p)} ℤ(X)) ⟶ F, (2)

where each of the morphisms p is a birational morphism between smooth kschemes of dimension 2 which a composition of finitely many blow-ups with smooth centers and such that the cokernel Ψ_{2} of (2) has the property Ψ_{2}(U) = 0 for every smooth k-scheme U of dimension ≤ 2. Denote by Ψ_{0} and Ψ_{1} the kernel and the image of the homomorphism (2). Thus we have short exact sequences of presheaves on Sm(k)

0 → Ψ_{0} → ⨁ coker(ℤ(X′) →^{ℤ(p)} ℤ(X)) → Ψ_{1} → 0, (3)

0 ⟶ Ψ_{1} ⟶ F ⟶ Ψ_{2} ⟶ 0. (4)

Each of the presheaves coker(ℤ(X′) →^{ℤ(p)} ℤ(X)) satisfies property (∗), and therefore also Ψ_{0} has this property. Furthermore, as a subpresheaf of F, also Ψ_{1} satisfies property (∗)^{≤2}. Thus

h_{i}^{Nis}(Ψ_{0}) = h_{i}^{Nis}(Ψ_{1}) = 0 for i = 0, 1.

Since Ψ_{2}(X) = 0 for every smooth scheme of dimension ≤ 2, we obtain

h_{i}^{Nis}(Ψ_{2}) = 0 for i ≤ 2.

The long exact sequences associated to (3) and (4) together with (1) therefore yield the assertion. □

Let us recall the following results.

Theorem 2.5 (Suslin, [S]) Let X be an affine, equidimensional scheme of finite type over k of dimension d. Then the natural inclusion of complexes of abelian groups

C_{•}(z_{0}(X)) ⟶ z^{d}(X, •)

is a quasiisomorphism.

Theorem 2.6 (Suslin-Voevodsky, [SV2] th.4.3.1) Let X be a scheme of finite type over k, let Y be a closed subscheme of X and let j : U → X be the open complement of Y in X. Then the sequence of presheaves on SmCor(k)

0 ⟶ z_{0}(Y) ⟶ z_{0}(X) ⟶^{j^{∗}} z_{0}(U)

is exact and coker(j^{∗}) satisfies property (∗).

Now we are in the position to prove the

Theorem 2.7 Let X be a smooth, projective scheme of dimension d ≤ 2 over a perfect field k. Then the canonical map

h_{i}(X) ⟶ CH^{d}(X, i)

is an isomorphism for i = 0, 1. The same result holds for ℤ/nℤ-coefficients.

Proof: The case d = 0 is trivial. We proceed by induction on d. Assume that d ≥ 1 and that the theorem is true in dimension < d.

Choose a smooth hyperplane section H ⊂ X and let Y = X − H (observe that Y is affine). By 2.6, we have an exact sequence of presheaves on SmCor(k)

0 ⟶ c_{0}(H) ⟶ c_{0}(X) ⟶^{j^{∗}} z_{0}(Y) (5)

and F := coker(j^{∗}) satisfies property (∗). We split (5) into two short exact sequences

0 ⟶ c_{0}(H) ⟶ c_{0}(X) ⟶ G ⟶ 0,
0 ⟶ G ⟶ z_{0}(Y) ⟶ F ⟶ 0.

Consider the following diagram with exact columns (the right column is the localization sequence for higher Chow groups, [B2])

⋮  ⋮
↓  ↓
h_{i}(H) ⟶^{α_{i}} CH^{d−1}(H, i)
↓  ↓
h_{i}(X) ⟶^{β_{i}}  CH^{d}(X, i)
↓  ↓
h_{i}^{Nis}(G) ⟶^{γ_{i}}  CH^{d}(Y, i)
↓  ↓
h_{i−1}(H) ⟶^{α_{i−1}} CH^{d−1}(H, i − 1)
↓  ↓
⋮  ⋮  .

The map γ_{i} is the composition

h_{i}^{Nis}(G) ⟶ h_{i}^{Nis}(z_{0}(Y)) = h_{i}(z_{0}(Y)) ⟶^{∼} CH^{d}(Y, i).

The right hand isomorphism follows from 2.5, since Y is affine. By 2.4, γ_{i} is an isomorphism for i = 0, 1 and surjective for i = 2. Therefore the induction step follows from the five-lemma.

If the field k is finite, then the existence of a smooth hyperplane section H is only granted over a finite extension field of k whose degree, however, can be chosen prime to an arbitrary given integer. Considering two hyperplane sections H_{1}, H_{2} defined over finite extensions k_{1}, k_{2} of k with coprime degree, we conclude the induction step by the usual restriction-corestriction argument.

□

Corollary 2.8 Let C be a smooth, projective curve over k. Then we have a natural isomorphism

Hom_{DM_{−}^{eff}(k)}(ℤ, M(C)(1)[1]) ≅ CH^{2}(C, 1).

If the field k is finite, then both groups are finite of order prime to char(k). Furthermore,

Hom_{DM_{−}^{eff}(k)}(ℤ, M(C)(1)[2]) = 0.

The same result holds for ℤ/nℤ-coefficients.

Proof: For the surface C × ℙ^{1}, we have the following decompositions for the Suslin homology ([V1], prop.3.5.1) and for the higher Chow groups ([B1], th.7.1)

h_{1}(C × ℙ^{1})  ≅  h_{1}(C)  ⊕ Hom_{DM_{−}^{eff}(k)}(ℤ, M(C)(1)[1]),
CH^{2}(C × ℙ^{1}, 1) ≅ CH^{1}(C, 1) ⊕  CH^{2}(C, 1).

By construction, the natural homomorphism h_{1}(C × ℙ^{1}) → CH^{2}(C × ℙ^{1}, 1) respects these decompositions, hence the first isomorphism of the corollary follows from 2.7. By 1.2, we have an isomorphism

CH^{2}(C, 1) ≅ H_{Zar}^{1}(C, CH^{2}(C, 2)).

The Gersten resolution for CH^{2}(C, 2) implies that

CH^{2}(C, 1) ≅ coker(K_{2}(k(C)) ⟶ ⨁_{x∈C^{1}} k(x)^{∗}) (6)

If k is finite, a theorem of C. Moore ([Mo], th.7.4) shows that this cokernel is isomorphic to the product of the multiplicative groups of the fields of constants of the function fields of the connected components of C. In particular, it is finite of order prime to p. In the same manner as above, we obtain natural decompositions

h_{0}(C × ℙ^{1})  ≅  h_{0}(C)  ⊕ Hom_{DM_{−}^{eff}(k)}(ℤ, M(C)(1)[2]),
CH^{2}(C × ℙ^{1}, 0) ≅ CH^{1}(C, 0) ⊕  CH^{2}(C, 0),

showing the remaining statement. □

3 Construction of the reciprocity map

In this section we will construct natural homomorphisms

c_{i} = c_{i,n} : h_{i}(X, ℤ/nℤ) ⟶ H_{c, et}^{2d−i}(X, ℤ/nℤ(d))

for any equidimensional scheme X ∈ Sm(k) of dimension d and any natural number n prime to p = char(k). Having done this, we can pass to the inverse limit over all ℓ^{r}, r ≥ 0 (ℓ a prime number ≠ p), and we obtain homomorphisms

h_{0}(X) → h_{0}(X) ⊗ ℤ_{ℓ} → H_{c,et}^{2d}(X, ℤ_{ℓ}(d)).

If the field k is finite, Poincaré duality identifies the right hand group with the pro-ℓ-part π_{1}(X)^{ab}(ℓ) of the abelianized étale fundamental group of X. The composition of these maps defines the prime-to-p-part of the reciprocity homomorphism rec of the introduction. We define the p-part of rec as the composition map

h_{0}(X) → h_{0}(X̄) ≅ CH_{0}(X̄) →^{α} π_{1}(X̄)^{ab}(p) ≅ π_{1}^{t}(X)^{ab}(p),

where α is the p-part of the reciprocity map CH_{0}(X̄)→π_{1}(X̄)^{ab} defined by Kato and Saito [KS1].

This yields a natural homomorphism

rec : h_{0}(X) ⟶ π_{1}^{t}(X)^{ab}.

Furthermore, the class [x] of a closed point x of X in h_{0}(X) is sent to the Frobenius of x in the abelianized tame fundamental group. For the p-part, this is true by construction. For the prime-to-p part, this follows by considering the case of dimension zero, the morphism {x} → X and the fact that the maps c_{i} constructed below are natural with respect to scheme morphisms. The commutative diagram

h_{0}(X) ⟶^{deg} h_{0}(Spec(k)) ≅ ℤ
↓^{rec}  ^{∩}^{can}
π_{1}^{t}(X)^{ab} ⟶  G_{k} ≅ ℤ̂

implies the existence of

rec_{0} : h_{0}(X)^{0} ⟶ (π_{1}^{t}(X)^{ab})^{0}.

Now we come to the construction of the maps c_{i}. Recall that by de Jong’s theorem on alterations ([dJ], th.4.1) a scheme of finite type over any field is locally smooth in the h-topology. Therefore, if k is perfect, the rigidity theorem ([SV1], th.4.5) together with the arguments of section 7 of [SV1] implies natural isomorphisms

H_{et}^{i}(X, ℤ/nℤ) ≅ Ext_{Shv_{ét}(Spec(k))}^{i}(\underline{C}_{•}(c_{0}(X)), ℤ/nℤ),

for all i. Here we consider \underline{C}_{•}(c_{0}(X)) as a sheaf on the small étale site on Spec(k). If X is smooth and k is finite, then the Yoneda pairing

R^{i}Γ(Spec(k)_{et}, \underline{C}_{•}(c_{0}(X))) × Ext_{Shv_{ét}(Spec(k)}^{j}(\underline{C}_{•}(c_{0}(X)), ℤ/nℤ)

⟶ R^{i+j}Γ(Spec(k)_{et}, ℤ/nℤ),

together with H^{1}(Spec(k)_{et}, ℤ/nℤ) ≅ ℤ/nℤ induces homomorphisms

h_{i}(X, ℤ/nℤ) ⟶ R^{−i}Γ(Spec(k)_{et}, \underline{C}_{•}(c_{0}(X)) ⊗ ℤ/nℤ)

⟶ Hom(H_{et}^{1+i}(X, ℤ/nℤ), ℤ/nℤ) ⟶^{∼} H_{c,et}^{2d−i}(X, ℤ/nℤ(d)),

where the last isomorphism is induced by Poincaré duality. We define c_{i,n} as the composition of these homomorphism.

The assignment

K ⟼ z^{r}(X_{K}, •, ℤ/nℤ)

defines a complex of sheaves \underline{z}^{r}(X, •, ℤ/nℤ) in Shv_{ét, ℤ/nℤ}(Spec(k)), and if X is quasiprojective of dimension d, then Suslin [S] (see again [Ge] for the necessary modifications in positive characteristic) shows that the natural inclusion

\underline{C}_{•}(z_{0}(X) ⊗ ℤ/nℤ) ⟶ \underline{z}^{d}(X, •, ℤ/nℤ)

induces an isomorphism in D^{−}(Shv_{ét, ℤ/nℤ}(Spec(k))). Thus, if X is projective, we obtain a commutative diagram

\underline{C}_{•}(c_{0}(X) ⊗ ℤ/nℤ))(Spec(k)) ⟶ \underline{z}^{d}(X, •, ℤ/nℤ)(Spec(k))
↓ ↓
RΓ(k, \underline{C}_{•}(c_{0}(X) ⊗ ℤ/nℤ)) ⟶^{∼}  RΓ(k, \underline{z}^{d}(X, •, ℤ/nℤ))

Therefore, if X is smooth and projective, the map

c_{i,n} : h_{i}(X, ℤ/nℤ) ⟶ H_{et}^{2d−i}(X, ℤ/nℤ(d))

factors through CH^{d}(X, i, ℤ/nℤ). Summarizing, we obtain the

Proposition 3.1 Let k be a finite field and let X be a smooth, geometrically connected variety of dimension d over k. Let n be an integer prime to the characteristic of k. Then we have natural homomorphisms for all i

c_{i,n} : h_{i}(X, ℤ/nℤ) ⟶ H_{c, et}^{2d−i}(X, ℤ/nℤ(d)).

If X is projective, then c_{i,n} factors through the canonical homomorphism

h_{i}(X, ℤ/nℤ) ⟶^{can} CH^{d}(X, i, ℤ/nℤ) ⟶ H_{et}^{2d−i}(X, ℤ/nℤ(d)).

Remarks. (a) The assumption that k is finite is not necessary, prop.3.1 holds over any perfect field k. The argument is essentially the same, but one has to use the relative version of Poincaré duality, instead of its absolute form over finite fields.

(b) Applying the result of prop.3.1 to X × ℙ_{k}^{i} and using the natural decomposition of M(X × ℙ_{k}^{i}), we obtain natural maps

Hom_{DM_{−}^{eff}(k)}(ℤ, M(X, ℤ/nℤ)(i)[j]) ⟶ H_{et,c}^{2d+j}(X, ℤ/nℤ(d + i))

for X as in prop.3.1 and integers i, j with i ≥ 0.

(c) The morphisms c_{i} are functorial with respect to Gysin sequences. More precisely, let Z a smooth closed subscheme of X everywhere of codimension c. Denote by Y the complement of Z in X. Then we have a commutative diagram with exact columns

Hom_{DM_{−}^{eff}(k)}(ℤ, M(Z, ℤ/nℤ)(c)[2c − i − 1]) ⟶ H_{et,c}^{2d−i−1}(Z, ℤ/nℤ(d))
↓  ↓
h_{i}(Y, ℤ/nℤ)  ⟶ H_{et,c}^{2d−i}(Y, ℤ/nℤ(d))
↓  ↓
h_{i}(X, ℤ/nℤ)  ⟶ H_{et,c}^{2d−i}(X, ℤ/nℤ(d))
↓  ↓
Hom_{DM_{−}^{eff}(k)}(ℤ, M(Z, ℤ/nℤ)(c)[2c − i]) ⟶ H_{et,c}^{2d−i}(Z, ℤ/nℤ(d))

where the first column is induced by the Gysin triangle of Z ⊆ X in DM_{−}^{eff}(k) ([V1], prop.3.5.4) and the horizontal maps are the c_{i}.

4 Proof of the main theorem for dim X ≤ 2

In this section we will prove theorem 0.1 for dim X ≤ 2. The general case will follow from this by a hypersurface section argument.

Theorem 4.1 Let k be a finite field and let X be smooth, projective and geometrically connected scheme of dimension d ≤ 2 over k. Let n be an integer prime to p =char(k). Then the natural homomorphism

c_{i,n} : h_{i}(X, ℤ/nℤ) ⟶ H_{et}^{2d−i}(X, ℤ/nℤ(d))

is an isomorphism for i = 0, 1.

Proof: By 2.7 and 3.1, it suffices to show that

CH^{d}(X, i, ℤ/nℤ) ⟶ H_{et}^{2d−i}(X, ℤ/nℤ(d))

is an isomorphism for i = 0, 1. For i = 0, this follows from ([KS1], th.1). For d = 1 = i, it follows from ([B1], th.6.1), together with the Kummer sequence. Let d = 2, i = 1. Consider the Zariski sheaf on X

H^{2}(X, ℤ/nℤ(2)),

which is associated to the presheaf U ↦ H_{et}^{2}(U, ℤ/nℤ(2)). By ([BO], th.4.2.2), we have a flasque resolution

0 → H^{2}(X, ℤ/nℤ(2)) → ⨁_{x∈X^{0}} i_{x∗} (H^{2}(k(x), ℤ/nℤ(2))) →

⨁_{x∈X^{1}} i_{x∗} (H^{1}(k(x), ℤ/nℤ(1))) → ⨁_{x∈X^{2}} i_{x∗} (H^{0}(k(x), ℤ/nℤ)) → 0.

Comparing this resolution with the Gersten resolution 1.3, the theorem of Merkur’ev-Suslin induces a natural isomorphism

CH^{2}(X, 2, ℤ/nℤ) ⟶^{∼} H^{2}(X, ℤ/nℤ(2)).

Since CH^{2}(X, 1, ℤ/nℤ) ≅ H_{Zar}^{1}(X, CH^{2}(X, 2, ℤ/nℤ)) by 1.2, in order to prove the theorem, it remains to show that

H_{Zar}^{1}(X, H^{2}(X, ℤ/nℤ(2)) ≅ H_{et}^{3}(X, ℤ/nℤ(2)).

This follows from the exact sequence

H_{Zar}^{1}(X, H^{2}(X, ℤ/nℤ(2))) ↪ H_{et}^{3}(X, ℤ/nℤ(2)) → H_{Zar}^{0}(X, H^{3}(X, ℤ/nℤ(2))),

since the group on the right hand vanishes by ([CSS], §2.4, Rem.2). □

Corollary 4.2 Let C be a smooth, projective curve over k. Then there is a canonical isomorphism

Hom_{DM_{−}^{eff}(k)}(ℤ, M(C, ℤ/nℤ)(1)[1]) ⟶^{∼} H_{et}^{3}(C, ℤ/nℤ(2)).

Proof: This follows from 4.1 by considering the surface C ×ℙ_{k}^{1} and the decompositions of h_{1}(C ×ℙ_{k}^{1}, ℤ/nℤ) and of H_{et}^{3}(C ×ℙ_{k}^{1}, ℤ/nℤ(2)) via correspondences in a similar way as in the proof of 2.8. □

Proof of theorem 0.1 for dimension 2: There exists open immersions X ⊂ X′ ⊂ X̄ such that C := X′ − X is a smooth curve (not necessarily connected) and Y := X̄ − X′ is an étale k-scheme. Let C′ be a compactification of C over k, i.e. C′/k is a smooth, projective curve and C is an open dense subscheme of C′. We denote by Z the complement of C in C′. By ([V1], prop.3.5.4), we get three exact triangles in DM_{−}^{eff}(k)

M(X) ⟶ M(X′) ⟶ M(C)(1)[2] ⟶ M(X)[1], (7)

M(C) ⟶ M(C′) ⟶ M(Z)(1)[2] ⟶ M(C)[1], (8)

M(X′) ⟶ M(X̄) ⟶ M(Y)(2)[4] ⟶ M(X′)[1]. (9)

By 1.1, the triangle (9) induces an isomorphism

h_{0}(X′) ⟶^{∼} h_{0}(X̄).

The triangle (8) together with 1.1 induces isomorphisms for i ≥ 1

Hom_{DM_{−}^{eff}(k)}(ℤ, M(C)(1)[i]) ⟶^{∼} Hom_{DM_{−}^{eff}(k)}(ℤ, M(C′)(1)[i])

By 2.8,

Hom_{DM_{−}^{eff}(k)}(ℤ, M(C′)(1)[1]) ≅ CH^{2}(C′, 1)

is finite of order prime to p and

Hom_{DM_{−}^{eff}(k)}(ℤ, M(C′)(1)[2]) = 0.

Therefore, (7) induces an exact sequence

Hom_{DM_{−}^{eff}(k)}(ℤ, M(C)(1)[1]) ⟶ h_{0}(X) ⟶ h_{0}(X′) ⟶ 0.

Thus the above sequences show that h_{0}(X)^{0} is finite provided that this holds for h_{0}(X̄)^{0}. But this follows from ([KS1], th.1), since h_{0}(X̄) = CH_{0}(X̄). Furthermore, we obtain an isomorphism of the p-torsion subgroups

h_{0}(X){p} ⟶^{∼} h_{0}(X̄){p}. (10)

By ([KS1], th.1), we have h_{0}(X̄){p} ≅ (π_{1}(X̄)^{ab})^{0}{p} and, as explained in the introduction, the latter group coincides with (π_{1}^{t}(X)^{ab})^{0}{p}. Therefore, in order to conclude the proof of theorem 0.1 for dimension 2, it suffices to show that for all n ≥ 1 and every prime number ℓ ≠ p, the homomorphism

c_{0,ℓ^{n}} : h_{0}(X, ℤ/ℓ^{n}ℤ) ⟶ H_{et,c}^{4}(X, ℤ/ℓ^{n}ℤ(2)) (11)

is an isomorphism. Indeed, the finiteness of h_{0}(X)^{0} and (11) imply that

h_{0}(X)^{0}{ℓ} ≅ h_{0}(X)^{0} ⊗ ℤ_{ℓ}
= ker(h_{0}(X) ⊗ ℤ_{ℓ} ⟶^{deg} ℤ_{ℓ})
= ker(lim_{⟵}_{n} h_{0}(X, ℤ/ℓ^{n}ℤ) ⟶ ℤ_{ℓ})
≅ ker(lim_{⟵}_{n} H_{et,c}^{4}(X, ℤ/ℓ^{n}ℤ(2)) ⟶ ℤ_{ℓ})
= ker(H_{et,c}^{2d}(X, ℤ_{ℓ}(d)) → H_{c}^{2d}(Xk̄, ℤ_{ℓ}(d)) ≅ ℤ_{ℓ})
≅ (π_{1}^{t}(X)^{ab})^{0} ⊗ ℤ_{ℓ}.

Consider the commutative diagrams with exact columns

Hom_{DM_{−}^{eff}(k)}(ℤ, M(C, ℤ/ℓ^{n}ℤ)(1)[1 − i]) ⟶ H_{et,c}^{3−i}(C, ℤ/ℓ^{n}(2))
↓  ↓
h_{i}(X, ℤ/ℓ^{n}ℤ)  ⟶ H_{et,c}^{4−i}(X, ℤ/ℓ^{n}ℤ(2))
↓ ↓
h_{i}(X′, ℤ/ℓ^{n}ℤ)  ⟶ H_{et,c}^{4−i}(X′, ℤ/ℓ^{n}ℤ(2))
↓ ↓
Hom_{DM_{−}^{eff}(k)}(ℤ, M(C, ℤ/ℓ^{n}ℤ)(1)[2 − i]) ⟶ H_{et,c}^{4−i}(C, ℤ/ℓ^{n}ℤ(2)),

(12)

Hom_{DM_{−}^{eff}(k)}(ℤ, M(Z, ℤ/ℓ^{n}ℤ)(2)[3 − i]) ⟶ H_{et}^{3−i}(Z, ℤ/ℓ^{n}ℤ(2))
↓  ↓
Hom_{DM_{−}^{eff}(k)}(ℤ, M(C, ℤ/ℓ^{n}ℤ)(1)[2 − i]) ⟶ H_{et,c}^{4−i}(C, ℤ/ℓ^{n}ℤ(2))
↓  ↓
Hom_{DM_{−}^{eff}(k)}(ℤ, M(C′, ℤ/ℓ^{n}ℤ)(1)[2 − i]) ⟶ H_{et}^{4−i}(C′, ℤ/ℓ^{n}ℤ(2))
↓  ↓
Hom_{DM_{−}^{eff}(k)}(ℤ, M(Z, ℤ/ℓ^{n}ℤ)(2)[4 − i]) ⟶ H_{et}^{4−i}(Z, ℤ/ℓ^{n}ℤ(2)),

(13)

Hom_{DM_{−}^{eff}(k)}(ℤ, M(Y, ℤ/ℓ^{n}ℤ)(2)[3 − i]) ⟶ H_{et}^{3−i}(Y, ℤ/ℓ^{n}ℤ(2))
↓  ↓
h_{i}(X′, ℤ/ℓ^{n}ℤ)  ⟶ H_{et,c}^{4−i}(X′, ℤ/ℓ^{n}ℤ(2))
↓ ↓
h_{i}(X̄, ℤ/ℓ^{n}ℤ)  ⟶ H_{et}^{4−i}(X̄, ℤ/ℓ^{n}ℤ(2))
↓ ↓
Hom_{DM_{−}^{eff}(k)}(ℤ, M(Y, ℤ/ℓ^{n}ℤ)(2)[4 − i]) ⟶ H_{et}^{4−i}(Y, ℤ/ℓ^{n}ℤ(2)).

(14)

Since the groups in the top and bottom row of diagram (14) vanish for i = 0, 1 we deduce from 4.1 that

c_{i,ℓ^{n}} : h_{i}(X′, ℤ/ℓ^{n}ℤ) ⟶ H_{et,c}^{4−i}(X′, ℤ/ℓ^{n}ℤ(2))

is an isomorphism for i = 0, 1. Moreover, by 4.2 and 2.8, the map

Hom_{DM_{−}^{eff}(k)}(ℤ, M(C′, ℤ/ℓ^{n}ℤ)(1)[2 − i]) ⟶ H_{et}^{4−i}(C′, ℤ/ℓ^{n}(2))

is an isomorphism for i = 0, 1. Since Hom_{DM_{−}^{eff}(k)}(ℤ, M(Z, ℤ/ℓ^{n}ℤ)(2)[i + 1]) = 0 = H_{et}^{3−i}(Z, ℤ/ℓ^{n}ℤ(2)) for i = 0, 1, diagram (13), 2.8 and (4.2) show that

Hom_{DM_{−}^{eff}(k)}(ℤ, M(C, ℤ/ℓ^{n}ℤ)(1)[i]) ⟶ H_{et}^{4−i}(C, ℤ/ℓ^{n}ℤ(2))

is an isomorphism for i = 0, 1, as well. Finally, diagram (12) and the five-lemma imply that (11) is bijective. □

5 Proof in the general case

In order to extend the result from dimension ≤ 2 to arbitrary dimensions, we are going to use a hypersurface section argument similar to that proposed by Colliot-Thélène in the projective case (see [KS1]). Unfortunately, the following Bertini-type theorem does not follow directly from ([AK], th.1 and th.7); one has modify the arguments given there.

Theorem 5.1 Let X ⊂ ℙ = ℙ(W) be a smooth, projective, geometrically connected variety of dimension ≥ 3 over an infinite field k. Let Z_{1}, …, Z_{s}, Z_{s+1}, …, Z_{t} be geometrically irreducible subvarieties in X, where Z_{1}, …, Z_{s} are of codimension 1 in X. Furthermore, let P_{1}, …, P_{r} be closed points in X − (∪_{i=1}^{t}Z_{i}). Let I be the ideal defining {P_{1}, …, P_{r}} in ℙ and let n be chosen such I(n) is generated by global sections. Let H ⊂ ℙ be a generic hypersurface of degree n + 1 containing P_{1}, …, P_{r}. Then H ∩ X is smooth and geometrically connected, H does not contain Z_{i}, i = 1, …, t and H ∩ Z_{i} is geometrically irreducible of codimension 1 in Z_{i} for i = 1, …, s.

Sketch of proof following [AK]: Let Y = X −{P_{1}, …, P_{r}}. The O_{Y} -module I(n)|Y is invertible and generated by global sections, so we obtain a morphism

Y ⟶ ℙ(Γ(ℙ, I(n))).

Combining it with inclusion Y ⟶ ℙ(W) and using the Segre embedding, we obtain a composite embedding

i : Y ⟶ ℙ(Γ(ℙ, I(n))) × ℙ(W) ⟶ ℙ(Γ(ℙ, I(n)) ⊗ W).

It is defined by the natural surjection

(Γ(ℙ, I(n)) ⊗ W)_{Y} ⟶ I(n + 1)|Y,

which factors through the natural map

(Γ(ℙ, I(n + 1)))_{Y} ⟶ I(n + 1)|Y.

Therefore, also the latter map is surjective and we obtain a commutative diagram

Y ⟶^{i} ℙ(Γ(ℙ, I(n)) ⊗ W)
∥ ↑
Y ⟶  U
∥  ^{∩}
Y ⟶^{j} ℙ(Γ(ℙ, I(n + 1))),

where U ⊂ ℙ(Γ(ℙ, I(n + 1))) is the open subset on which the morphism ℙ(Γ(ℙ, I(n + 1))) → ℙ(Γ(ℙ, I(n)) ⊗ W) is defined. Since i is an embedding, j is an embedding. Let s be a general element of Γ(ℙ, I(n + 1)). It defines via j a section Y_{1} of Y by a general hyperplane of ℙ(Γ(ℙ, I(n + 1))). Clearly, Y_{1} does not contain Z_{1}, …, Z_{t} and, using the classical Bertini theorems (see [L1], chap.VII, th.7, prop.13) as in the proof of ([AK], th.1), we see that Y_{1} is smooth and geometrically connected and Y_{1} ∩ Z_{i} is geometrically irreducible for i = 1, …, s. Viewing s as a section of O_{ℙ}(n + 1), we obtain a general hypersurface section X_{1} = H ∩ X which contains P_{1}, …, P_{r} and with

Y_{1} = X_{1} ∩ Y.

We conclude that X_{1} is geometrically irreducible and smooth off {P_{1}, …, P_{r}}. Furthermore, X_{1} ∩ Z_{i} = Y_{1} ∩ Z_{i} for i = 1, … t. It remains to show that X_{1} is smooth at P_{1}, …, P_{r}. This follows from ([AK], th.7). □

Corollary 5.2 Let X̄ ⊂ ℙ_{k}^{N} be a smooth, projective, geometrically connected variety of dimension ≥ 3 over an infinite perfect field k and let X ⊂ X̄ be an open subscheme. Let finitely many closed points P_{1}, …, P_{r} ∈ X be given. Let Z_{1}, …, Z_{s} be the irreducible components of (X̄ − X)k̄ which are of codimension 1 in X̄k̄ (k̄ denotes an algebraic closure of k). Then there exists a closed subset T ⊂ X̄ − X of codimension ≥ 2 in X̄ and a hypersurface H ⊆ ℙ_{k}^{N} containing P_{1}, …, P_{r} such that

(i) H ∩ X̄ is smooth and geometrically connected,

(ii) Y_{i} := Z_{i} − (Tk̄ ∩ Z_{i}) is smooth and connected for i = 1, …, s,

(iii) Hk̄ ∩ Y_{i} is smooth and connected for i = 1, …, s,

(iv) H ∩ T is of codimension ≥ 2 in H ∩ X̄,

(v) T contains every irreducible component of X̄ − X which is of codimension ≥ 2 in X̄.

Proof: By 5.1, a generic hypersurface section of sufficiently high degree in ℙN ; k̄ containing {P_{1}, …, P_{r}}k̄ has an irreducible intersection with Z_{i}, i = 1, …, s, does not contain any irreducible component of the singular locus of one of the Z_{i} and does not contain any irreducible component of (X̄ − X)k̄. Since k is infinite, we may choose H defined over k and again by 5.1, H ∩ X̄ is smooth. Now choose T such that Tk̄ contains all singularities of Z_{i}, all singularities of Hk̄ ∩ Z_{i}, i = 1, …, s and every irreducible component of (X̄ − X)k̄ which is of codimension ≥ 2 in X̄k̄. □

We use the above result in order to show the

Theorem 5.3 Let X̄ ⊂ ℙ_{k}^{N} be a smooth, projective, geometrically connected variety of dimension ≥ 3 over a finite field k and let X ⊂ X̄ be an open subscheme. Let finitely many closed points P_{1}, …, P_{r} ∈ X be given and let M be a positive integer. Then, after replacing k by a finite extension of degree prime to M, there exists a hypersurface H ⊂ ℙ_{k}^{N} containing P_{1}, …, P_{r} such that X̄ ∩ H is smooth and geometrically connected and such that the natural homomorphism

π_{1}^{t}(X ∩ H)^{ab} ⟶ π_{1}^{t}(X)^{ab}

is an isomorphism.

Proof: Let Z_{1}, …, Z_{s} be the irreducible components of (X̄ − X)k̄ of codimension 1 in X̄. Over any infinite extension of k, we find a hypersurface H and a closed subscheme T ⊂ X̄ − X of codimension ≥ 2 in X̄ satisfying the conditions of 5.2. Replacing k by a finite extension, which we may choose of degree prime to M, we may assume that H and T are already defined over k. Let X_{1} = X̄ − T. We set Y_{i} = Z_{i} ∩ (X_{1})k̄. Then the following holds.

(i) X̄ − X_{1} is of codimension ≥ 2 in X̄,

(ii) (X_{1} − X)k̄ = Y_{1} ∪ ⋯ ∪ Y_{s},

(iii) X̄ ∩ H is smooth and geometrically connected,

(iv) Y_{i} ∩ Hk̄ is smooth, connected and of codimension 1 in Y_{i} for i = 1, …, s,

(v) X̄ − (H ∩ X̄) is affine,

(vi) H ∩ T is of codimension ≥ 2 in H ∩ X̄.

Recall that the étale cohomological dimension of a scheme U of finite type over a finite field is less or equal to 2 dim(U) + 1 (cf. [Mi], VI, cor.1.4) and is equal to dim(U) + 1 if U is affine ([Mi], VI, th.7.2).

Let n be any integer prime to the characteristic p of k. By (i), we have

H_{et}^{2d−i}(X̄ − X_{1}, ℤ/nℤ(d)) = 0 for i < 3.

Thus the exact sequences

H_{et}^{2d−i−1}(X̄ − X_{1}, ℤ/nℤ(d)) → H_{et,c}^{2d−i}(X_{1}, ℤ/nℤ(d)) →^{α_{i}} H_{et}^{2d−i}(X̄, ℤ/nℤ(d))

→ H_{et}^{2d−i}(X̄ − X_{1}, ℤ/nℤ(d))

show that α_{i} is an isomorphism for i = 0, 1. The same reasoning applied to H ∩ (X̄ − X_{1}) shows (using property (vi) above) that

H_{et,c}^{2d−2−i}(X_{1} ∩ H, ℤ/nℤ(d − 1)) ⟶^{β_{i}} H_{et}^{2d−2−i}(X̄ ∩ H, ℤ/nℤ(d − 1))

is an isomorphism for i = 0, 1. Now consider the following commutative diagram for i = 0, 1

H_{et,c}^{2d−2−i}(X_{1} ∩ H, ℤ/nℤ(d − 1)) ⟶^{β_{i}} H_{et}^{2d−2−i}(X̄ ∩ H, ℤ/nℤ(d − 1))
↓^{ϕ_{i}}  ↓^{ψ_{i}}
H_{et,c}^{2d−i}(X_{1}, ℤ/nℤ(d))  ⟶^{α_{i}}  H_{et}^{2d−i}(X̄, ℤ/nℤ(d)).

The arrow ψ_{i} is induced via Poincaré duality from the natural homomorphism H_{et}^{1}(X̄, ℤ/nℤ) → H_{et}^{1}(X̄ ∩ H, ℤ/nℤ) and the definition of ϕ is similar. Since X̄ −(H ∩X̄) is affine, we have H_{et}^{i}(X̄ −(H ∩X̄), ℤ/nℤ(d)) = 0 for i > d+1. Hence the Gysin sequence ([Mi],VI,5.4) shows that ψ_{i} is an isomorphism for i = 0 and is surjective for i = 1. Therefore, the same is true for ϕ_{i}.

Now put D = X_{1} − X, thus Dk̄ = Y_{1} ∪ ⋯ ∪ Y_{s}. We consider the following commutative diagram with exact columns

H_{et,c}^{2d−3}(X_{1} ∩ H, ℤ/nℤ(d − 1)) ⟶^{ϕ_{1}} H_{et,c}^{2d−1}(X_{1}, ℤ/nℤ(d))
↓  ↓
H_{et,c}^{2d−3}(D ∩ H, ℤ/nℤ(d − 1)) ⟶^{a} H_{et,c}^{2d−1}(D, ℤ/nℤ(d))
↓  ↓
H_{et,c}^{2d−2}(X ∩ H, ℤ/nℤ(d − 1)) ⟶^{b} H_{et,c}^{2d}(X, ℤ/nℤ(d))
↓  ↓
H_{et,c}^{2d−2}(X_{1} ∩ H, ℤ/nℤ(d − 1)) ⟶^{ϕ_{0}} H_{et,c}^{2d}(X_{1}, ℤ/nℤ(d))
↓  ↓
H_{et,c}^{2d−2}(D ∩ H, ℤ/nℤ(d − 1)) ⟶ H_{et,c}^{2d}(D, ℤ/nℤ(d)).

Both objects on the lower line are zero by dimension reasons, ϕ_{0} is an isomorphism and ϕ_{1} is surjective. Thus, in order to show that b is a isomorphism, it suffices to show that a is an isomorphism. Since G_{k} ≅ ℤ̂, the Hochschild-Serre spectral sequence induces isomorphisms

H_{et,c}^{2d−3}(D ∩ H, ℤ/nℤ(d − 1)) ≅ H^{1}(G_{k}, H_{et,c}^{2d−4}((D ∩ H)k̄, ℤ/nℤ(d − 1)))

and

H_{et,c}^{2d−1}(D, ℤ/nℤ(d)) ≅ H^{1}(G_{k}, H_{et,c}^{2d−2}(Dk̄, ℤ/nℤ(d))).

Therefore it suffices to show that a⊗k̄ is an isomorphism. This, however, follows via Poincaré duality using property (iv) above. We have shown that the natural homomorphism

H_{et,c}^{2d−2}(X ∩ H, ℤ/nℤ(d − 1)) ⟶ H_{et,c}^{2d}(X, ℤ/nℤ(d))

is an isomorphism. Another application of Poincaré duality therefore implies that the natural map

π_{1}(X ∩ H)^{ab}/n ⟶ π_{1}(X)^{ab}/n

is an isomorphism. Since n was arbitrary prime to p, this shows the statement of the theorem for the prime-to-p-part. The statement on the p-part follows from the commutative diagram

π_{1}^{t}(X ∩ H)^{ab}(p) ⟶ π_{1}^{t}(X)^{ab}(p)
↓^{≀}  ↓^{≀}
π_{1}(X̄ ∩ H)^{ab}(p) ⟶ π_{1}(X̄)^{ab}(p)

and from the fact that the lower horizontal arrow is an isomorphism by ([SGA2], Exp.XII, Cor.3.5). □

Now we complete the proof of theorem 0.1. We proceed by induction on the dimension of X. For dimension ≤ 2 everything has been proved, so assume that dim X = d ≥ 3 and that the theorem is true in dimensions < d. Choose a hypersurface section H according to 5.3 (with any points P_{1}, …, P_{r}). We obtain a commutative diagram

h_{0}(X ∩ H) ⟶ π_{1}^{t}(X ∩ H)^{ab}
↓  ↓
h_{0}(X)  ⟶  π_{1}^{t}(X)^{ab}

which exists over a finite extension of k whose degree can be chosen prime to any given integer M. Using the usual restriction-corestriction argument, the induction hypothesis and theorem 5.3 imply that

rec_{0} : (h_{0}(X))^{0} → (π_{1}^{t}(X)^{ab})^{0}

is surjective and that (π_{1}^{t}(X)^{ab})^{0} is finite. Now let z be a cycle representing an element in the kernel of rec_{0}. Then, over a finite extension of k whose degree can be chosen prime to any given integer M, there exists a hypersurface section H according to 5.3 containing the support of z. The same reasoning as above shows that the class [z] of z in h_{0}(X) is annihilated by the degree of the finite field extension, hence rec_{0} is injective. □

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Alexander Schmidt Michael Spieß Mathematisches Institut School of Mathematical Sciences der Universität Heidelberg University of Nottingham Im Neuenheimer Feld 288 University Park 69120 Heidelberg NG7 2RD Nottingham Germany United Kingdom

e-mail: schmidt@mathi.uni-heidelberg.de mks@maths.nott.ac.uk