Tame Class Field Theory for Singular Varieties

over Algebraically Closed Fields

Thomas Geisser^{1} and Alexander Schmidt^{2}

Received: May 25, 2015 Revised: November 25, 2015

Communicated by Stephen Lichtenbaum

Abstract. Let X be a separated scheme of finite type over an algebraically closed field k and let m be a natural number. By an explicit geometric construction using torsors we construct a pairing between the first mod m Suslin homology and the first mod m tame étale cohomology of X. We show that the induced homomorphism from the mod m Suslin homology to the abelianized tame fundamental group of X mod m is surjective. It is an isomorphism of finite abelian groups if (m, char(k)) = 1, and for general m if resolution of singularities holds over k.

2010 Mathematics Subject Classification: 14F35, 14F43, 14C25 Keywords and Phrases: Suslin homology, higher dimensional class field theory, tame fundamental group

1 Introduction

Let X be a (possibly singular) separated scheme of finite type over an algebraically closed field k of characteristic p ≥ 0 and let m be a natural number. We construct a pairing between the first mod m algebraic singular homology H_{1}^{S}(X, ℤ/mℤ) and the first mod m tame étale cohomology group H_{t}^{1}(X, ℤ/mℤ). For π_{1}^{t,ab}(X) = H_{t}^{1}(X, ℚ/ℤ)^{∨} we prove the following analogue of Hurewicz’s theorem in algebraic topology:

1Supported by JSPS Grant-in-Aid (B) 23340004 2Supported by DFG-Forschergruppe FOR 1920

Documenta Mathematica 21 (2016) 91–123

Theorem 1.1. The induced homomorphism

rec_{X} : H_{1}^{S}(X, ℤ/mℤ) ⟶ π_{1}^{t,ab}(X)/m

is surjective. It is an isomorphism of finite abelian groups if (m, p) = 1, and for general m if resolution of singularities holds for schemes of dimension ≤ dim X + 1 over k.

For p ∤ m, the groups H_{1}^{S}(X, ℤ/mℤ) and π_{1}^{t,ab}(X)/m are known to be isomorphic by the work of Suslin and Voevodsky [SV1]. Theorem 1.1 above provides an explicit isomorphism which extends to the case p | m (under resolution of singularities). Moreover, in the last section we show that for p ∤ m our isomorphism coincides with the one constructed in [SV1].

The motivation for constructing our pairing between the groups H_{1}^{S}(X, ℤ/mℤ) and H_{t}^{1}(X, ℤ/mℤ) comes from topology: For a locally contractible Hausdorff space X and a natural number m, the canonical duality pairing

⟨·, ·⟩ : H_{1}^{sing}(X, ℤ/mℤ) × H^{1}(X, ℤ/mℤ) ⟶ ℤ/mℤ,

between singular homology and sheaf cohomology with mod m coefficients can be given explicitly in the following way: represent b ∈ H^{1}(X, ℤ/mℤ) by a ℤ/mℤtorsor T → X and a ∈ H_{1}^{sing}(X, ℤ/mℤ) by a 1-cycle α in the singular complex of X. Then

⟨a, b⟩ = Φ_{par}^{−1} ∘ Φ_{taut} ∈ ℤ/mℤ, where Φ_{taut}, Φ_{par} : α^{∗}(T )|_{0} →^{∼} α^{∗}(T )|_{1}

are the isomorphisms between the fibres over 0 and 1 of the pull-back torsor α^{∗}(T ) → Δ^{1} = [0, 1] given tautologically (0^{∗}α = 1^{∗}α) and by parallel transport (every ℤ/mℤtorsor on [0, 1] is trivial). For a variety X, the pairing between H_{1}^{S}(X, ℤ/mℤ) and H_{t}^{1}(X, ℤ/mℤ) inducing the homomorphism rec_{X} of our Main Theorem 1.1 will be constructed in the same way. However, 1-cycles in the algebraic singular complex are not linear combinations of morphisms but finite correspondences from Δ^{1} to X. In order to mimic the above construction, we thus have to define the pull-back of a torsor along a finite correspondence, which requires the construction of the push-forward torsor along a finite surjective morphism. To prove Theorem 1.1, we first consider the case of a smooth curve C. If A is the Albanese variety of C, then we have isomorphisms

H_{1}^{S}(C, ℤ/mℤ) ⟶_{∼}^{δ} _{m}H_{0}^{S}(C, ℤ) ≅ _{m}A(k).  (1)

The first isomorphism follows from the coefficient sequence together with the divisibility of H_{1}^{S}(C, ℤ), and the second from the Abel-Jacobi theorem. On the other hand,

Hom(_{m}A(k), ℤ/mℤ) ⟶_{∼}^{τ} H_{t}^{1}(C, ℤ/mℤ).  (2)

This follows because the maximal étale subcovering Ã → A of the m-multiplication map A →^{m} A is the quotient of A by the connected component of the finite group scheme _{m}A, and the maximal abelian tame étale covering of C with Galois group annihilated by m is C̃ := C ×_{A} Ã. The heart of the proof of Theorem 1.1 for smooth curves is to show that under the above identifications, our pairing agrees with the evaluation map. We then show surjectivity of rec_{X} for general X by reducing to the case of smooth curves. Finally, we use duality theorems to show that both sides of rec_{X} have the same order: For the p-primary part, we use resolution of singularities to reduce to the smooth projective case considered in [Ge3]. For (m, char(k)) = 1, Suslin and Voevodsky [SV1] construct an isomorphism

α_{X} : H_{et}^{1}(X, ℤ/mℤ) →^{∼} H_{S}^{1}(X, ℤ/mℤ).

Hence the source and the target of rec_{X} have the same order and therefore rec_{X} is an isomorphism. In Section 7 we show that rec_{X} is dual to the map α_{X}. Thus, for char(k) ∤ m, our construction gives an explicit description of the Suslin-Voevodsky isomorphism α_{X}, which zig-zags through Ext-groups in various categories and is difficult to understand.

The authors thank Takeshi Saito and Changlong Zhong for discussions during the early stages of the project. It is a pleasure to thank Johannes Anschütz whose comments on an earlier version of this paper led to a substantial simplification of the proof of Theorem 4.1. Finally, we want to thank the referee for his helpful comments.

2 Torsors and finite correspondences

All occurring schemes in this section are separated schemes of finite type over a field k. For any abelian group A and a finite surjective morphism π : Z → X with Z integral and X normal, connected, we have transfer maps

π_{∗} : H_{et}^{i}(Z, A) → H_{et}^{i}(X, A)

for all i ≥ 0 (see [MVW], 6.11, 6.21). The group H_{et}^{1}(Z, A) classifies isomorphism classes of étale A-torsors (i.e., principal homogeneous spaces) over the scheme Z. We are going to construct a functor

π_{∗} : PHS(Z, A) ⟶ PHS(X, A)

from the category of étale A-torsors on Z to the category of étale A-torsors on X, which induces the transfer map π_{∗} : H_{et}^{1}(Z, A) → H_{et}^{1}(X, A) above on isomorphism classes. We recall how to add and subtract torsors. For an abelian group A and A-torsors T_{1}, T_{2} on a scheme Y, define

T_{1} + T_{2}

to be the quotient scheme of T_{1} ×_{Y} T_{2} by the action of A given by (t_{1}, t_{2}) + a = (t_{1} + a, t_{2} − a). It carries the structure of an A-torsor by setting

\overline{(t_{1}, t_{2})} + a := \overline{(t_{1} + a, t_{2})} (= \overline{(t_{1}, t_{2} + a)}).

The functor

+ : PHS(Y, A) × PHS(Y, A) ⟶ PHS(Y, A), (T_{1}, T_{2}) ⟼ T_{1} + T_{2},

lifts the addition in H_{et}^{1}(Y, A) to torsors (cf. [Mi], III, Rem. 4.8 (b)). Note that “+” is associative and commutative up to natural functor isomorphisms. In particular, we can multiply a torsor by any natural number m, putting m · T = T + ⋯ + T (m times). If mA = 0, then we have a natural isomorphism of torsors

m · T ⟶^{∼} Y × A, \overline{(t_{1}, …, t_{m})} ↦ (t_{2} − t_{1}) + ⋯ + (t_{m} − t_{1}) ∈ A, (3)

where Y × A is the trivial A-torsor on Y representing the constant sheaf \underline{A} over Y. Here t_{i} − t_{j} denotes the unique element a ∈ A with t_{i} = t_{j} + a.

Furthermore, given a torsor T , define (−T ) to be the torsor which is isomorphic to T as a scheme and on which a ∈ A acts as −a. This yields a functor

(−1) : PHS(Y, A) ⟶ PHS(Y, A), T ⟼ (−T ),

which lifts multiplication by (−1) from H_{et}^{1}(Y, A) to an endofunctor of PHS(Y, A). We have a natural isomorphism of torsors

T + (−T ) ⟶^{∼} Y × A, \overline{(t_{1}, t_{2})} ↦ t_{1} − t_{2} ∈ A.  (4)

Now let π : Z → X be finite and surjective, Z integral, X normal, connected, and let T be an A-torsor on Z. For every point x ∈ X, the base change Z×_{X} X_{x}^{sh} is a product of strictly henselian local schemes. Therefore we find an étale cover (U_{i} → X)_{i∈I} of X such that T trivializes over the pull-back étale cover (π^{−1}(U_{i}) → Z)_{i∈I} of Z. Next choose a pseudo-Galois covering π̃ : Z̃ → X dominating Z → X. Recall that this means that k(Z̃)|k(X) is a normal field extension and that the natural map Aut_{X}(Z̃) → Aut_{k(X)}(k(Z̃)) is bijective (cf. [SV1], Lemma 5.6). Let π_{in} : X_{in} → X be the quotient scheme Z̃/G, where G = Aut_{X}(Z̃). Then X_{in} is the normalization of X in the maximal purely inseparable subextension k(X)^{in}/k(X) of k(Z̃)/k(X). Consider the object

T̃ := ∑_{φ∈Mor_{X}(Z̃,Z)} φ^{∗}(T ) ∈ PHS(Z̃, A),

which is defined up to unique isomorphism. Starting from any trivialization of T over (π^{−1}(U_{i}) → Z)_{i∈I}, we obtain a trivialization of the restriction of T̃ to (π^{−1}(U_{i}) → Z̃)_{i∈I} of the form

T̃|_{π^{−1}(U_{i})} ≅ π^{−1}(U_{i}) × A,

where G = Aut_{X}(Z̃) acts on the right hand side in the canonical way on π^{−1}(U_{i}) and trivially on A. Therefore the quotient scheme T̃/G is an A-torsor on Z̃/G = X_{in} in a natural way. Since X_{in} → X is a topological isomorphism, T̃/G comes by base change from a unique A-torsor T′ on X.

Definition 2.1. The push-forward A-torsor π_{∗}(T ) on X is defined by

π_{∗}(T ) = [k(Z) : k(X)]_{in} · T′.

The assignment T ↦ π_{∗}(T ) defines a functor

π_{∗} : PHS(Z, A) ⟶ PHS(X, A).

The functor π_{∗} is additive in the sense that it commutes with the functors “+” and “(−1)” up to a natural functor isomorphism.

Let T ∈ PHS(Z, A) and assume that there exists a section s : Z → T to the projection T → Z (so T is trivial and s gives a trivialization). Let again π : Z → X be finite and surjective, Z integral, X normal, connected. Then

T̃ := ∑_{φ∈Mor_{X}(Z̃,Z)} φ^{∗}(T ) ∈ PHS(Z̃, A)

has the canonical section ∑_{φ∈Mor_{X}(Z̃,Z)} φ^{∗}(s) over Z̃. It descends to a section of

T /G over Z̃/G = X_{in}. Descending to X and multiplying by [k(X_{in}) : k(X)], we obtain a section

π_{∗}(s) : X → π_{∗}(T ).

In other words, we obtain a map

π_{∗} : Γ(Z, T ) ⟶ Γ(X, π_{∗}(T ));

hence every trivialization of T gives a trivialization of π_{∗}(T ) in a natural way.

In order to see that π_{∗} induces the transfer map π_{∗} : H_{et}^{1}(Z, A) → H_{et}^{1}(X, A) after passing to isomorphism classes, we formulate the construction of π_{∗} on the level of Čech 1-cocycles. As explained above, we find an étale cover (U_{i} → X)_{i∈I} such that T trivializes over the étale cover (π^{−1}(U_{i}) → Z)_{i∈I} of Z. We fix a trivialization and obtain a Čech 1-cocycle

a = (a_{ij} ∈ Γ(π^{−1}(U_{i} ×_{X} U_{j}), A))

over (π^{−1}(U_{i}) → Z)_{i∈I} which defines T . As before choose a pseudo-Galois covering π̃ : Z̃ → X dominating Z → X. Now for all i, j consider the element

∑_{φ∈Mor_{X}(Z̃,Z)} φ^{∗}(a_{ij}) ∈ Γ(π^{−1}(U_{i} ×_{X} U_{j}), A)

which, by Galois invariance, lies in

Γ(π_{in}^{−1}(U_{i} ×_{X} U_{j}), A) = Γ(U_{i} ×_{X} U_{j}, A).

The Čech 1-cocycle given by

[k(Z) : k(X)]_{in} · ( ∑_{φ∈Mor_{X}(Z̃,Z)} φ^{∗}(a_{ij})) ∈ Γ(U_{i} ×_{X} U_{j}, A).

now defines a trivialization of π_{∗}(T ) over (U_{i} → X)_{i∈I}. Since the transfer map on étale cohomology is defined on Čech cocycles in exactly this way (see [MVW], 6.11, 6.21), we obtain

Lemma 2.2. Passing to isomorphism classes, the functor π_{∗} : PHS(Z, A) → PHS(X, A) constructed above induces the transfer homomorphism

π_{∗} : H_{et}^{1}(Z, A) → H_{et}^{1}(X, A).

If any finite subset of closed points of X is contained in an affine open, then symmetric powers exist, and another description of the push-forward for torsors is the following: Associated with the finite morphism π : Z → X of degree d, there is a section s_{π} : X → Sym^{d}(Z/X) to the natural projection Sym^{d}(Z/X) → X (see ([SV1], p. 81). We denote the composite of s_{π} with pr : Sym^{d}(Z/X) → Sym^{d}(Z) by S_{π}. Defining f : Z̃ → Sym^{d}(Z/X) by repeating each element in Mor_{X}(Z̃, Z) exactly [k(Z) : k(X)]_{in}-times, the diagram

\begin{aligned}Z̃ & Sym^{d}(Z/X) & Sym^{d}(Z) \\ & X \end{aligned} f π̃ pr s_{π} S_{π}

commutes. For an A-torsor T → Z, the d-fold self-product T ×_{k} ⋯ ×_{k} T is an A^{d}-torsor over the d-fold self-product of Z in a natural way. Taking the quotient by the A^{d−1}-action

(a_{1}, …, a_{d−1})(t_{1}, …, t_{d}) = (t_{1} + a_{1}, t_{2} − a_{1} + a_{2}, t_{3} − a_{2} + a_{3}, …, t_{d} − a_{d−1}),

we obtain an A-torsor over Z^{d}. Dividing out the by the action of the symmetric group S_{d}, we obtain an A-torsor over Sym^{d}(Z) and denote it by Sym_{A}^{d}(T ). We obtain natural isomorphisms in PHS(Z̃, A):

[k(Z) : k(X)]_{in} · ∑_{φ∈Mor_{X}(Z̃,Z)} φ^{∗}(T ) ≅ (pr ∘ f)^{∗} Sym_{A}^{d}(T )
≅ π^{∗} ∘ (pr ∘ s_{π})^{∗} Sym_{A}^{d}(T ).

By our construction of π_{∗}(T ) we obtain

Lemma 2.3. We have a natural isomorphism in PHS(X, A):

π_{∗}(T ) ≅ S_{π}^{∗}(Sym^{d}(T )),

where S_{π} = pr ∘ s_{π} : X → Sym^{d}(Z).

Assume now that X is regular and Y arbitrary. The group of finite correspondences Cor(X, Y) is defined as the free abelian group on the set of integral subschemes Z ⊂ X×Y which project finitely and surjectively to a connected component of X. For such a Z, we define p_{[Z→X] ∗} : PHS(Z, A) → PHS(X, A) by extending (if X is not connected) the push-forward torsor defined above in a trivial way to those connected components of X which are not dominated by Z. We consider the functor

[Z]^{∗} = p_{[Z→X] ∗} ∘ p_{[Z→Y]}^{∗} : PHS(Y, A) ⟶ PHS(X, A).

Using the operations “+” and “(−1)” we extend this construction to arbitrary finite correspondences.

Definition 2.4. Let X be regular, Y arbitrary and α = ∑ n_{i}Z_{i} ∈ Cor(X, Y) a finite correspondence. Then

α^{∗} : PHS(Y, A) ⟶ PHS(X, A)

is defined by setting

α^{∗}(T ) := ∑ n_{i}[Z_{i}]^{∗}(T ).

Using the isomorphism (4) above, we immediately obtain

Lemma 2.5. For α_{1}, α_{2} ∈ Cor(X, Y) and T_{1}, T_{2} ∈ PHS(Y, A), n_{1}, n_{2} ∈ ℤ, we have a natural isomorphism

(α_{1} + α_{2})^{∗}(n_{1}T_{1} + n_{2}T_{2}) ≅ n_{1}α_{1}^{∗}(T_{1}) + n_{1}α_{2}^{∗}(T_{1}) + n_{2}α_{1}^{∗}(T_{2}) + n_{2}α_{2}^{∗}(T_{2}).

If X and Y are regular and Z is arbitrary, we have a natural composition law

Cor(X, Y) × Cor(Y, Z) ⟶ Cor(X, Z), (α, β) ↦ β ∘ α,

(see [MVW], Lecture 1). A straightforward but lengthy computation unfolding the definitions shows Proposition 2.6. Let X and Y be regular and Z arbitrary. Let α ∈ Cor(X, Y) and β ∈ Cor(Y, Z). Then, for any T ∈ PHS(Z, A), we have a canonical isomorphism

α^{∗}(β^{∗}(T )) ≅ (β ∘ α)^{∗}(T ).

Finally, assume that mA = 0 for some natural number m. Then (using the isomorphism (3) above), we have for any α, β ∈ Cor(X, Y), T ∈ PHS(Y, A), a natural isomorphism

(α + mβ)^{∗}(T ) ≅ α^{∗}(T ).

Therefore, we have an A-torsor

ᾱ^{∗}(T ) ∈ PHS(X, A)

given up to unique isomorphism for any ᾱ ∈ Cor(X, Y) ⊗ ℤ/mℤ. In other words, we obtain the

Lemma 2.7. Assume that mA = 0, and let α, β ∈ Cor(X, Y) have the same image in Cor(X, Y) ⊗ ℤ/mℤ. Then there is a natural isomorphism of functors

α^{∗} ≅ β^{∗} : PHS(Y, A) → PHS(X, A).

For a regular connected curve C we consider the subgroup H_{t}^{1}(C, A) ⊆ H_{et}^{1}(C, A) of tame cohomology classes (corresponding to those continuous homomorphisms π_{1}^{et}(C) → A which factor through the tame fundamental group π_{1}^{t}(C̄, C̄ − C), where C̄ is the unique regular compactification of C). For a general scheme X over k we call a cohomology class in a ∈ H_{et}^{1}(X, A) curvetame (or just tame) if for any morphism f : C → X with C a regular curve, we have f^{∗}(a) ∈ H_{t}^{1}(C, A). The tame cohomology classes form a subgroup

H_{t}^{1}(X, A) ⊆ H_{et}^{1}(X, A).

The groups coincide if X is proper or if p = 0 or if p > 0 and A is p-torsion free, where p is the characteristic of the base field k.

Definition 2.8. We call an étale A-torsor T on X tame if its isomorphism class lies in H_{t}^{1}(X, A) ⊆ H_{et}^{1}(X, A).

Lemma 2.9. Let Z be integral, X normal, connected, π : Z → X finite, surjective and f : Z → Y any morphism. Let T be a tame torsor on Y. Then π_{∗}(f^{∗}(T )) is a tame torsor on X.

Proof. By definition, f^{∗} preserves curve-tameness. So we may assume Z = Y, f = id. Again by the definition of curve-tameness and using Proposition 2.6, we may reduce to the case that X is a regular curve. Since étale cohomology commutes with direct limits of coefficients, we may assume that A is a finitely generated abelian group. Furthermore, we may assume that char(k) = p > 0 and A = ℤ/p^{r}ℤ, r ≥ 1. Let Z̄ be the canonical compactification of Z, i.e., the unique proper curve over k which contains Z as a dense open subscheme and such that all points of Z̄ ∖ Z are regular points of Z̄. By the definition of tame coverings of curves, T extends to a ℤ/p^{r}ℤ-torsor on Z̄. Hence also π_{∗}(T ) extends to the canonical compactification X̄ of X and so is tame.

Proposition 2.10. Let X̄ be a proper and regular scheme over k and let X ⊂ X̄ be a dense open subscheme. Let p = char(k) > 0. Then for any r ≥ 1 the natural inclusion

H_{et}^{1}(X̄, ℤ/p^{r}ℤ) ↪ H_{et}^{1}(X, ℤ/p^{r}ℤ)

induces an isomorphism

H_{et}^{1}(X̄, ℤ/p^{r}ℤ) = H_{t}^{1}(X̄, ℤ/p^{r}ℤ) ⟶^{∼} H_{t}^{1}(X, ℤ/p^{r}ℤ) ⊆ H_{et}^{1}(X, ℤ/p^{r}ℤ).

Proof. Let T_{0} be any connected component of a tame ℤ/p^{r}ℤ-torsor T on X. Then the morphism T_{0} → X is curve-tame in the sense of [KS], §4, and T_{0} is the normalization of X in the abelian field extension of p-power degree k(T_{0})/k(X). By [KS], Thm. 5.4. (b), T_{0} → X is numerically tamely ramified along X̄ ∖ X. This means that the inertia groups in Gal(k(T_{0})/k(X)) of all points x̄ ∈ X̄ ∖ X are of order prime to p, hence trivial. Therefore T_{0}, and thus T extends to X̄.

Corollary 2.11. Let Δ^{n} = Spec(k[T_{0}, …, T_{n}]/ ∑ T_{i} = 1) be the n-dimensional standard simplex over k and let A be an abelian group. Then

H_{t}^{1}(Δ^{n}, A) ≅ H_{et}^{1}(k, A).

In particular, H_{t}^{1}(Δ^{n}, A) = 0 if k is separably closed.

Proof. Since tame cohomology commutes with direct limits of coefficients, and since H_{et}^{1}(Δ^{n}, ℤ) = 0, we may assume that A ≅ ℤ/mℤ for some m ≥ 1. If p ∤ m, we obtain:

H_{t}^{1}(Δ^{n}, ℤ/mℤ) ≅ H_{et}^{1}(𝔸^{n}, ℤ/mℤ) ≅ H_{et}^{1}(k, ℤ/mℤ).

If p = char(k) > 0 and m = p^{r}, r ≥ 1, Proposition 2.10 yields

H_{t}^{1}(Δ^{n}, ℤ/p^{r}ℤ) ≅ H_{t}^{1}(𝔸^{n}, ℤ/p^{r}ℤ) ←^{∼} H_{t}^{1}(ℙ^{n}, ℤ/p^{r}ℤ) = H_{et}^{1}(ℙ^{n}, ℤ/p^{r}ℤ).

Finally note that H_{et}^{1}(ℙ^{n}, ℤ/p^{r}ℤ) ≅ H_{et}^{1}(k, ℤ/p^{r}ℤ).

In the following, let k be an algebraically closed field of characteristic p ≥ 0 and let X be a separated scheme of finite type over k. Let H_{i}^{S}(X, ℤ/mℤ) denote the mod-m Suslin homology, i.e., the i-th homology group of the complex

Cor(Δ^{•}, X) ⊗ ℤ/mℤ.

Let A be an abelian group with mA = 0. We are going to construct a pairing

H_{1}^{S}(X, ℤ/mℤ) × H_{t}^{1}(X, A) ⟶ A

as follows: let T → X be a tame A-torsor representing a class in H_{t}^{1}(X, A) and let α ∈ Cor(Δ^{1}, X) be a finite correspondence representing a 1-cocycle in the mod-m Suslin complex. Then

α^{∗}(T )

is a torsor over Δ^{1}. Since α is a cocycle modulo m, (0^{∗} − 1^{∗})(α) is of the form m · z for some z ∈ Cor(Δ^{0}, X) = ℤ^{(X(k))}. We therefore obtain a canonical identification

Φ_{taut} : 0^{∗}(α^{∗}(T )) ⟶^{∼} 1^{∗}(α^{∗}(T ))

of A-torsors over Δ^{0} = Spec(k). Furthermore, by Corollary 2.11, the tame torsor α^{∗}(T ) on Δ^{1} is trivial, hence a disjoint union of copies of Δ^{1}. By parallel transport, we obtain another identification

Φ_{par} : 0^{∗}(α^{∗}(T )) ⟶^{∼} 1^{∗}(α^{∗}(T )).

Hence there is a unique γ(α, T ) ∈ A such that

Φ_{par} = (translation by γ(α, T )) ∘ Φ_{taut}.

Proposition 2.12. The element γ(α, T ) ∈ A only depends on the class of T in H_{t}^{1}(X, A) and on the class of α in H_{1}^{S}(X, ℤ/mℤ). We obtain a bilinear pairing

⟨·, ·⟩ : H_{1}^{S}(X, ℤ/mℤ) × H_{t}^{1}(X, A) ⟶ A.

Proof. Replacing T by another torsor isomorphic to T does not change anything. The nontrivial statement is that ⟨α, T ⟩ only depends on the class of α in H_{1}^{S}(X, ℤ/mℤ). For β ∈ Cor(Δ^{1}, X), we have

⟨α + mβ, T ⟩ = ⟨α, T ⟩ + m⟨β, T ⟩ = ⟨α, T ⟩.

It therefore remains to show that

⟨∂^{∗}(Φ), T ⟩ = 0,

for all Φ ∈ Cor(Δ^{2}, X), where ∂_{i} : Δ^{1} → Δ^{2}, i = 0, 1, 2, are the face maps and ∂^{∗}(Φ) = Φ ∘ ∂_{0} − Φ ∘ ∂_{1} + Φ ∘ ∂_{2}. Considering ∂ = ∂_{0} − ∂_{1} + ∂_{2} as a finite correspondence from Δ^{1} to Δ^{2}, it represents a cocycle in the singular complex Cor(Δ^{•}, Δ^{2}). Proposition 2.6 implies that

⟨∂^{∗}(Φ), T ⟩ = ⟨Φ ∘ ∂, T ⟩ = ⟨∂, Φ^{∗}(T )⟩.

By Corollary 2.11, the tame torsor Φ^{∗}(T ) is trivial on Δ^{2}. Hence ⟨∂, Φ^{∗}(T )⟩ = 0.

In the following, we use the notation π_{1}^{t,ab}(X) := H_{t}^{1}(X, ℚ/ℤ)^{∗}. If X is connected, then π_{1}^{t,ab}(X) is the abelianized (curve-)tame fundamental group of X, see [KS], §4.

Definition 2.13. For m ≥ 1 we define

rec_{X} : H_{1}^{S}(X, ℤ/mℤ) ⟶ π_{1}^{t,ab}(X)/m

as the homomorphism induced by the pairing of Proposition 2.12 for A = ℤ/mℤ combined with the isomorphism H_{t}^{1}(X, ℤ/mℤ)^{∗} ≅ π_{1}^{t,ab}(X)/m.

The statement of the next lemma immediately follows from the definition of rec.

Lemma 2.14. Let f : X′ → X be a morphism of separated schemes of finite type over k. Then the induced diagram

\begin{matrix}H_{1}^{S}(X′, ℤ/mℤ) & π_{1}^{t,ab}(X′)/m \\ H_{1}^{S}(X, ℤ/mℤ) & π_{1}^{t,ab}(X)/m \end{matrix} rec_{X′} f_{∗} f_{∗} rec_{X}

commutes.

3 Rigid Čech complexes

We consider étale sheaves F on the category Sch/k of separated schemes of finite type over a field k. By a result of M. Artin, Čech cohomology Ȟ^{•}(X, F) and sheaf cohomology H_{et}^{•}(X, F) coincide in degree ≤ 1 and in arbitrary degree if X is quasiprojective (cf. [Mi], III Thm. 2.17). Comparing the Čech complex for a covering U and that for a finer covering V, the refinement homomorphism

Č^{•}(U, F) ⟶ Č^{•}(V, F)

is canonical only up to chain homotopy and hence only the induced map Ȟ^{•}(U, F) → Ȟ^{•}(V, F) is well-defined. We can remedy this problem in the spirit of Friedlander [Fr], chap.4, by using rigid coverings: We fix an algebraic closure k̄/k. A rigid étale covering U of X is a family of pointed separated étale morphisms

(U_{x}, u_{x}) ⟶ (X, x), x ∈ X(k̄),

with U_{x} connected and u_{x} ∈ U_{x}(k̄) mapping to x. For an étale sheaf F the rigid Čech complex is defined by

Č^{•}(U, F) : Č^{n}(U, F) = ∏_{(x_{0},...,x_{n})∈X(k̄)^{n+1}} Γ(U_{x_{0}} ×_{X} ⋯ ×_{X} U_{x_{n}}, F)

with the usual differentials. It is clear what it means for a rigid covering V to be a refinement of U. Because the marked points map to each other, there is exactly one refinement morphism, hence we obtain a canonical refinement morphism on the level of complexes

Č^{•}(U, F) → Č^{•}(V, F).

The set of rigid coverings is cofiltered (form the fibre product for each x ∈ X(k̄) and restrict to the connected components of the marked points). Therefore we can define the rigid Čech complex of X with values in F as the filtered direct limit

Č^{•}(X, F) := \underrightarrow{lim}_{U} Č^{•}(U, F),

where U runs through all rigid coverings of X. Note that the rigid Čech complex depends on the structure morphism X → k and not merely on the scheme X. Forgetting the marking, we can view a rigid covering as a usual covering. Every covering can be refined by a covering which arises by forgetting the marking of a rigid covering. Hence the cohomology of the rigid Čech complex coincides with the usual Čech cohomology of X with values in F.

For a morphism f : Y → X and a rigid Čech covering U/X, we obtain a rigid Čech covering f^{∗}U/Y by taking base extension to Y and restricting to the connected components of the marked points, and in the limit we obtain a homomorphism

f^{∗} : Č^{•}(X, F) ⟶ Č^{•}(Y, F).

Lemma 3.1. If π : Y → X is quasi-finite, then the rigid coverings of the form π^{∗}U are cofinal among the rigid coverings of Y.

Proof. This is an immediate consequence of the fact that a quasi-finite and separated scheme Y over the spectrum X of a henselian ring is of the form Y = Y_{0}⊔Y_{1}⊔…⊔Y_{r} with Y_{0} → X not surjective and Y_{i} → X finite surjective with Y_{i} the spectrum of a henselian ring, i = 1, …, r, cf. [Mi], I, Thm. 4.2.

Lemma 3.2. If F is qfh-sheaf on Sch/k, then for any n ≥ 0 the presheaf \underline{Č}^{n}(−, F) given by

X ⟼ Č^{n}(X, F)

is a qfh-sheaf. The obvious sequence

0 → F → \underline{Č}^{0}(−, F) → \underline{Č}^{1}(−, F) → \underline{Č}^{2}(−, F) → ⋯

is exact as a sequence of étale (and hence also of qfh) sheaves.

Proof. We show that each \underline{Č}^{n}(−, F) is a qfh-sheaf. For this, let π : Y → X be a qfh-covering, i.e., a quasi-finite universal topological epimorphism. We denote the projection by Π : Y ×_{X} Y → X. By Lemma 3.1, we have to show that the sequence

\underrightarrow{lim}_{U} Č^{n}(U, F) → \underrightarrow{lim}_{U} Č^{n}(π^{∗}U, F) ⇉ \underrightarrow{lim}_{U} Č^{n}(Π^{∗}U, F)

is an equalizer, where U runs through the rigid coverings of X. Since filtered colimits commute with finite limits, it suffices to show the exactness for a single, sufficiently small U. This, however, follows from the assumption that F is a qfh-sheaf. Finally, the exactness of 0 → F → \underline{Č}^{0}(−, F) → \underline{Č}^{1}(−, F) → ⋯ as a sequence of étale sheaves follows by considering stalks.

Being qfh-sheaves, the sheaves F and \underline{Č}^{n}(−, F) admit transfer maps, see [SV1], §5. For later use, we make the relation between the transfers of F and of \underline{Č}^{n}(−, F) explicit: Let Z be integral, X regular and π : Z → X finite and surjective. Let F be a qfh-sheaf on Sch/k. For x ∈ X(k̄) we have

X_{x}^{sh} ×_{X} Z = ∐_{z∈π^{−1}(x)} Z_{z}^{sh},

where π^{−1}(x) denotes the set of morphisms z : Spec(k̄) → Z with π ∘ z = x. For sufficiently small étale (U_{x}, u_{x}) → (X, x), the set of connected components of U_{x} ×_{X} Z is in 1-1-correspondence with the set π^{−1}(x), and to each family of étale morphisms

(V_{z}, v_{z}) ⟶ (Z, z), z ∈ π^{−1}(x),

there is (after possibly making U_{x} smaller) a unique morphism

U_{x} ×_{X} Z ⟶ ∐_{z∈π^{−1}(x)} V_{z},

over Z, which sends the connected component associated with z of U_{x} ×_{X} Z to V_{z}, and the point (u_{x}, z) to v_{z}.

In this way we obtain, for finitely many points (x_{0}, …, x_{n}), n ≥ 0, and for every family

(V_{z_{i},v_{z_{i}}}) ⟶ (Z, z_{i}), z_{i} ∈ π^{−1}(x_{i}), and sufficiently small chosen

(U_{x_{i}}, u_{x_{i}}) ⟶ (X, x_{i}), i = 0, …, n,

a homomorphism

∏_{\begin{matrix}(z_{0},...,z_{n}) \\ z_{i}∈π^{−1}(x_{i}) \end{matrix}} Γ(V_{z_{0}} ×_{Z} ⋯ ×_{Z} V_{z_{n}}, F ) ⟶ Γ(U_{x_{0}} ×_{X} ⋯ ×_{X} U_{x_{n}} ×_{X} Z, F ).

Since F is a qfh-sheaf, we can compose this with the transfer map associated with the finite morphism

U_{x_{0}} ×_{X} ⋯ ×_{X} U_{x_{n}} ×_{X} Z → U_{x_{0}} ×_{X} ⋯ ×_{X} U_{x_{n}}.

Forming for fixed n the product over all (x_{0}, …, x_{n}) ∈ X(k̄)^{n+1} and passing to the limit over all rigid coverings, we obtain the transfer homomorphism

π_{∗} : Č^{•}(Z, F) ⟶ Č^{•}(X, F).

Passing to cohomology, we obtain the usual transfer on étale cohomology in degree 0 and 1, and in any degree if the schemes are quasi-projective.

Next we give the pairing

⟨·, ·⟩ : H_{1}^{S}(X, ℤ/mℤ) × H_{t}^{1}(X, A) ⟶ A.

constructed in Proposition 2.12for k algebraically closed and an abelian group A with mA = 0 the following interpretation in terms of the rigid Čech complex:

Let a ∈ H_{1}^{S}(X, ℤ/mℤ) and b ∈ H_{t}^{1}(X, A) be given, and let α ∈ Cor_{k}(Δ^{1}, X) and β ∈ ker(Č^{1}(X, A) →^{d} Č^{2}(X, A)) be representing elements. Note that (0^{∗} −1^{∗})(α) ∈ m Cor(Δ^{0}, X) by assumption. Consider the diagram

\begin{aligned}& Č^{0}(X, A) & Č^{1}(X, A) & Č^{2}(X, A) \\ & Č^{0}(Δ^{1}, A) & Č^{1}(Δ^{1}, A) & Č^{2}(Δ^{1}, A) \\ A & Č^{0}(Δ^{0}, A) & Č^{1}(Δ^{0}, A) & Č^{2}(Δ^{0}, A) \end{aligned} d d d d d d α^{∗} α^{∗} α^{∗} 0^{∗} − 1^{∗} 0^{∗} − 1^{∗} 0^{∗} − 1^{∗}

Since β represents a tame torsor T on X, α^{∗}(β) represents the torsor α^{∗}(T ), which is tame by Lemma 2.9. By Corollary 2.11, there exists γ ∈ Č^{0}(Δ^{1}, A) with dγ = α^{∗}(β). Since

d(0^{∗} − 1^{∗})(γ) = (0^{∗} − 1^{∗})α^{∗}(β) = 0, we conclude that (0^{∗} − 1^{∗})(γ) lies in

A = H^{0}(Δ^{0}, A) = ker(Č^{0}(Δ^{0}, A) →^{d} Č^{1}(Δ^{0}, A)).

It is easy to verify that the assignment

⟨·, ·⟩ : (a, b) ⟼ (0^{∗} − 1^{∗})(γ) ∈ A

does not depend on the choices made. By the explicit geometric relation between Čech 1-cocycles and torsors, and since our construction of finite push-forwards of torsors is compatible with the construction of transfers for qfh-sheaves given in [SV1], §5, we see that the pairing constructed above coincides with the one constructed in Proposition 2.12. Finally, let

A ↪ I^{0} → I^{1} → I^{2}  (5)

be a (partial) injective resolution of the constant sheaf A in the category of ℤ/mℤmodule sheaves on (Sch/k)_{qfh}. Let ϕ : (Sch/k)_{qfh} → (Sch/k)_{et} denote the natural map of sites. Since ϕ^{∗} is exact, ϕ_{∗} sends injective sheaves to injective sheaves. By [SV1], Thm. 10.2, we have R^{0}ϕ_{∗}(A) = A and R^{i}ϕ_{∗}(A) = 0 for i ≥ 1. Hence (5) is also a partial resolution of A by injective, étale sheaves of ℤ/mℤ-modules. We choose a quasi-isomorphism

[0 → \underline{Č}^{0}(−, A) → \underline{Č}^{1}(−, A) → \underline{Č}^{2}(−, A)] ⟶ [0 → I^{0} → I^{1} → I^{2}]

of truncated complexes of qfh-sheaves. Since Čech- and étale cohomology agree in dimension ≤ 1, the induced map on global sections is a quasi-isomorphism of truncated complexes of abelian groups. Hence the pairing of Proposition 2.12 can also be obtained by the same procedure as above but using the diagram

\begin{aligned}& I^{0}(X) & I^{1}(X) & I^{2}(X) \\ & I^{0}(Δ^{1}) & I^{1}(Δ^{1}) & I^{2}(Δ^{1}) \\ A & I^{0}(Δ^{0}) & I^{1}(Δ^{0}) & I^{2}(Δ^{0}). \end{aligned} d d d d d d α^{∗} α^{∗} α^{∗} 0^{∗} − 1^{∗} 0^{∗} − 1^{∗} 0^{∗} − 1^{∗}

By [SV1], Theorem 10.7, the same argument applies with a partial injective resolution of the constant sheaf A in the category of ℤ/mℤ-module sheaves on (Sch/k)_{h}.

4 The case of smooth curves

In this section we prove Theorem 1.1 in the case that X = C is a smooth curve.

Let k be an algebraically closed field of characteristic p ≥ 0, and let C be a smooth, but not necessarily projective, curve over k. Let the semi-abelian variety A be the generalized Jacobian of C with respect to the modulus given by the sum of the points on the boundary of the regular compactification C̄ of C (cf. [Se], Ch. 5). The group A(k) is the subgroup of degree zero elements of the relative Picard group Pic(C̄, C̄ ∖ C). By [SV1], Thm. 3.1 (see [Li], for the case C = C̄), there is an isomorphism

H_{0}^{S}(C, ℤ)^{0} := ker(H_{0}^{S}(C, ℤ) ⟶^{deg} ℤ) ≅ A(k),

in particular, A(k) is a quotient of the group of zero cycles of degree zero on C. From the coefficient sequence together with the divisibility of H_{1}^{S}(C, ℤ) (which is isomorphic to k^{×} if C is proper and zero otherwise), we obtain an isomorphism

H_{1}^{S}(C, ℤ/mℤ) ⟶_{∼}^{δ} _{m}H_{0}^{S}(C, ℤ) ≅ _{m}A(k).  (6)

After fixing a closed point P_{0} of C, the morphism C → A, P ↦ P − P_{0}, is universal for morphisms of C to semi-abelian varieties, i.e., A is the generalized Albanese variety of C ([Se], V, Th. 2). Consider the m-multiplication map A →^{m} A. Its maximal étale subcovering Ã → A is the quotient of A by the connected component of the finite group scheme _{m}A (if (p, m) = 1, the connected component is trivial). The projection A → Ã induces an isomorphism A(k) →^{∼} Ã(k) on rational points, and we identify A(k) and Ã(k) via this isomorphism. With respect to this identification, the projection Ã(k) → A(k) is the m-multiplication map on A(k). By [Se], Ch. IV, C̃ := C ×_{A} Ã is the maximal abelian tame étale covering of C with Galois group annihilated by m. Because Aut_{A}(Ã) ≅ _{m}A(k), we obtain an isomorphism

Hom(_{m}A(k), A) ⟶_{∼}^{τ} H_{t}^{1}(C, A)  (7)

for any finite abelian group A with mA = 0.

Theorem 4.1. For any finite abelian group A with mA = 0, the diagram

\begin{matrix}H_{1}^{S}(C, ℤ/mℤ) & × & H_{t}^{1}(C, A) & A \\ _{m}A(k) & × & Hom(_{m}A(k), A) & A \end{matrix} ⟨ , ⟩ ∼_{δ} ∼_{τ} eval

where ⟨ , ⟩ is the pairing from Proposition 2.12 and eval is the evaluation map, commutes. In particular, the upper pairing is perfect and the induced homomorphism H_{1}^{S}(C, ℤ/mℤ) → π_{1}^{t,ab}(C)/m is an isomorphism.

Proof. We have to show that ϕ(δ(ζ)) = ⟨ζ, τ(ϕ)⟩ for any ζ ∈ H_{1}^{S}(C, ℤ/mℤ) and any ϕ ∈ Hom(_{m}A(k), A). By functoriality, it suffices to consider the universal case A = _{m}A(k), ϕ = id. In this case τ(id) is the torsor π̃ : C̃ → C. Let C′ be the regular compactification of C. By [SV1], Thm. 3.1, δ(ζ) ∈ mH_{0}^{S}(C, ℤ) = _{m}A(k) is the class [z] of some z ∈ Z_{0}(C) (the group of zero-cycles on C) such that

mz = γ^{∗}(0) − γ^{∗}(1)

for some finite morphism γ : C′ → ℙ^{1} with C′ ∖ C ⊂ γ^{−1}(∞). The diagram

\begin{matrix}C′ ∖ γ^{−1}(∞) & C & C′ \\ Δ^{1} & & ℙ^{1} \end{matrix} γ|_{C′∖γ−1(∞)} γ

shows that γ induces a finite correspondence, say g, from Δ^{1} to C. The class of g in H_{1}^{S}(C, ℤ/mℤ) is a pre-image of δ(ζ) under H_{1}^{S}(C, ℤ/mℤ) →^{∼} _{m}H_{0}^{S}(C, ℤ), i.e., ζ is represented by g. It therefore suffices to show that

[z] = ⟨g, C̃⟩.

Let d be the degree of γ and γ^{∗}(0) = ∑_{i=1}^{d} P_{i}, γ^{∗}(1) = ∑_{i=1}^{d} Q_{i}. Each point in γ^{∗}(0) and γ^{∗}(1) occurs with multiplicity divisible by m, in particular d = mr for some integer r. After reindexing, we may assume that P_{i} = P_{j} and Q_{i} = Q_{j} for i ≡ j mod r, hence

z = ∑_{i=1}^{r} P_{i} − ∑_{i=1}^{r} Q_{i}.

On the level of closed points, C̃ = C ×_{A} Ã can be identified with the set of a ∈ Ã(k) = A(k) such that ma = P − P_{0} for some point P ∈ C (a projects to P in C, i.e., π̃(a) = P). The _{m}A(k)-principal homogeneous space 0^{∗}g^{∗}C̃ can be identified with the quotient of the set

∏_{i=1}^{d} π^{−1}(P_{i})

by the action of_{m}A(k)^{d−1} given by

(β_{1}, …, β_{d−1})(a_{1}, …, a_{d}) = (a_{1} + β_{1}, a_{2} − β_{1} + β_{2}, …, a_{d} − β_{d−1}).

We fix points a_{1}, …, a_{d} ∈ C̃ over P_{1}, …, P_{d} subject to the condition a_{i} = a_{j} for P_{i} = P_{j}. Then 0^{∗}g^{∗}C̃ is identified with the quotient of the set

(a_{1} +_{m}A(k)) × ⋯ × (a_{d} +_{m}A(k))

by the action of_{m}A(k)^{d−1}. Since each a_{i} occurs with multiplicity divisible by m, the trivialization 0^{∗}g^{∗}(C̃) →^{∼} _{m}A(k) given by

\overline{(a_{1} + α_{1}, …, a_{d} + α_{d})} ⟼ α_{1} + ⋯ + α_{d} ∈ _{m}A(k)

Documenta Mathematica 21 (2016) 91–123

does not depend on the choice of the a_{i}. We do the same with 1^{∗}g^{∗}(C̃) by choosing b_{i} ∈ C̃ over Q_{i}. Then we see that the tautological identification Φ_{taut} : 0^{∗}g^{∗}(C̃) →^{∼} 1^{∗}g^{∗}(C̃) is given by

\overline{(a_{1} + α_{1}, …, a_{d} + α_{d})} ⟼ \overline{(b_{1} + α_{1}, …, b_{d} + α_{d})}.

Now consider the morphism

Σ : Sym^{d}(C) ⟶ A, (x_{1}, …, x_{d}) ⟼ [∑(x_{i} − P_{0})].

Associated with the _{m}A(k)-torsor C̃ over C, we have the _{m}A(k)-torsor Sym_{_{m}A(k)}^{d}(C̃) over Sym^{d}(C) (cf. the paragraph preceding Lemma 2.3). The commutative diagram

\begin{matrix}Sym_{_{m}A(k)}^{d}(C̃) & Ã \\ Sym^{d}(C) & A \end{matrix} Σ Σ

induces a map (hence an isomorphism) of _{m}A(k)-torsors Sym_{_{m}A(k)}^{d}(C̃) →^{∼} Ã ×_{A} Sym^{d}(C). Consider the morphism S_{g} : Δ_{k}^{1} → Sym^{d}(C) associated with the finite correspondence g. Since the generalized Jacobian of Δ_{k}^{1} ≅ 𝔸_{k}^{1} is Spec(k), the composite

Δ_{k}^{1} ⟶^{S_{g}} Sym^{d}(C) ⟶^{Σ} A

is constant with value a := [∑_{i=1}^{d}(P_{i} − P_{0})] = [∑_{i=1}^{d}(Q_{i} − P_{0})] ∈ A(k). By Lemma 2.3, we obtain an isomorphism

g^{∗}(C̃) = S_{g}^{∗}(Sym^{d}(C̃)) = Σ^{∗}S_{g}^{∗}Ã = Δ_{k}^{1} × π^{−1}(a)

(giving a trivialization after choosing a point in π^{−1}(a)). On the fibre over 0 it is given by

\overline{(a_{1} + α_{1}, …, a_{d} + α_{d})} ⟼ ∑_{i=1}^{d}(a_{i} + α_{i}) ∈ π^{−1}(a) ⊂ Ã

and similarly on the fibre over 1. We conclude that Φ_{par} ∘ Φ_{taut}^{−1} is translation by

∑_{i=1}^{d}(a_{i} − b_{i}) = ∑_{i=1}^{r} m(a_{i} − b_{i}) = ∑_{i=1}^{r}[P_{i} − Q_{i}] = [z].

This concludes the proof.

5 The blow-up sequences

All schemes in this section are separated schemes of finite type over the spectrum of a perfect field k. A curve on a scheme X is a closed one-dimensional subscheme. The normalization of a curve C is denoted by C̃.

Now let

\begin{matrix}Z′ & X′ \\ Z & X \end{matrix} π i

be an abstract blow-up square, i.e., a cartesian diagram of schemes such that π : X′ → X is proper, i : Z → X is a closed embedding and π induces an isomorphism

(X′ ∖ Z′)_{red} →^{∼} (X ∖ Z)_{red}.

Proposition 5.1. Given an abstract blow-up square and an abelian group A, assume that π is finite or A is torsion. Then there is a natural exact sequence

0 → H_{et}^{0}(X, A) → H_{et}^{0}(X′, A) ⊕ H_{et}^{0}(Z, A) → H_{et}^{0}(Z′, A)

→^{δ} H_{t}^{1}(X, A) → H_{t}^{1}(X′, A) ⊕ H_{t}^{1}(Z, A) → H_{t}^{1}(Z′, A).

Proof. We call an abstract blow-up square trivial, if i is surjective (i.e., s_{red} is an isomorphism) or if π_{red} : X′_{red} → X_{red} has a section. Every abstract blow-up square with X a connected regular curve is trivial. Now let an arbitrary abstract blow-up square be given. If A is torsion, the proper base change theorem implies (cf. [Ge2], 3.2 and 3.6) that we have a long exact sequence

⋯ → H_{et}^{i}(X, A) → H_{et}^{i}(X′, A) ⊕ H_{et}^{i}(Z, A) → H_{et}^{i}(Z′, A) → H_{et}^{i+1}(X, A) → ⋯

If π is finite, the same is true for arbitrary A since π_{∗} is exact. If the blow-up square is trivial, this long exact sequence splits into short exact sequences 0 → H_{et}^{i}(X, A) → H_{et}^{i}(X′, A) ⊕ H_{et}^{i}(Z, A) → H_{et}^{i}(Z′, A) → 0 for all i.

Next we show the exact sequence of the proposition. We omit the coefficients A and put H_{t}^{0}(X) = H_{et}^{0}(X). We first show, that the image of the boundary map δ : H_{et}^{0}(Z′) → H_{et}^{1}(X) has image in H_{t}^{1}(X), thus showing the existence of H_{t}^{0}(Z′) → H_{t}^{1}(X) and, at the same time, the exactness of the sequence at H_{t}^{1}(X). Let C̃ → X be the normalization of a curve in X. The base change

\begin{matrix}Z′ ; C̃ & X′ ; C̃ \\ ZC̃ & C̃ \end{matrix} π̃ ĩ

of our abstract blow-up square to C̃ is a trivial abstract blow-up square. Therefore, for any α ∈ H_{et}^{0}(Z′), the pull-back of α to H_{et}^{0}(Z′ ; C̃) lies in the image of H_{et}^{0}(X′ ; C̃) ⊕ H_{et}^{0}(ZC̃) → H_{et}^{0}(Z′ ; C̃) and has therefore trivial image under δ : H_{et}^{0}(Z′ ; C̃) → H_{et}^{1}(C̃).

Therefore, δ(α) ∈ H_{et}^{1}(X) has trivial image in H_{et}^{1}(C̃) for every curve C ⊂ X, in particular, it lies in H_{t}^{1}(X). It remains to show exactness at H_{t}^{1}(X′) ⊕ H_{t}^{1}(Z). Let α be in this group with trivial image in H_{t}^{1}(Z′). Then there exists β ∈ H_{et}^{1}(X) mapping to α and it remains to show that β lies in the subgroup H_{t}^{1}(X). But this is clear, because for every curve C ⊂ X

we have H_{t}^{1}(C̃) = ker(H_{t}^{1}(X′ ; C̃) ⊕ H_{t}^{1}(ZC̃) → H_{t}^{1}(Z′ ; C̃)).

Proposition 5.2. Given an abstract blow-up square

\begin{matrix}Z′ & X′ \\ Z & X \end{matrix} π i

and an abelian group A, there is a natural exact sequence of Suslin homology groups

H_{1}^{S}(Z′, A) → H_{1}^{S}(X′, A) ⊕ H_{1}^{S}(Z, A) → H_{1}^{S}(X, A)

→^{δ} H_{0}^{S}(Z′, A) → H_{0}^{S}(X′, A) ⊕ H_{0}^{S}(Z, A) → H_{0}^{S}(X, A) → 0.

Proof. Consider the exact sequences

C_{•}(Z′, A) ↪ C_{•}(X′, A) ⊕ C_{•}(Z, A) → C_{•}(X, A) ↠ K_{•}^{A}

and

C_{•}(Z′) ↪ C_{•}(X′) ⊕ C_{•}(Z) → C_{•}(X) ↠ K_{•},

where K_{•}^{A} and K_{•} are defined to make the sequences exact. Since the complexes C_{•}(−) consist of free abelian groups, in order the show the statement of the proposition, it suffices to show that H_{i}(K_{•}) = 0 for i ≤ 2. Let Sm/k be the full subcategory of Sch/k consisting of smooth schemes. For Y ∈ Sch/k we consider the presheaf c(Y) on Sm/k given by c(Y)(U) = Cor(U, Y). Then, by [SV2], Thm. 5.2, 4.7 and its proof, the sequence

0 → c(Z′) → c(X′) ⊕ c(Z) →^{(π_{∗},i_{∗})} c(X)

is exact and F := coker(π_{∗}, i_{∗}) has the property that, for any U ∈ Sm/k of dimension ≤ 2 and any x ∈ F (U), there exists a proper birational morphism ϕ : V → U with V smooth such that ϕ^{∗}(x) = 0. Let F_{•} be the complex of presheaves given by F_{n}(U) = F (U × Δ^{n}) with the obvious differentials and let (F_{•})_{Nis} be the associated complex of sheaves on (Sm/k)_{Nis}. Then by [SS], Thm. 2.4, the Nisnevich sheaves

H_{i}((F_{•})_{Nis})

vanish for i ≤ 2. Evaluating at U = Spec(k) yields the result.

Now assume that k is algebraically closed. Let

rec_{1,X} : H_{1}^{S}(X, ℤ/mℤ) → H_{t}^{1}(X, ℤ/mℤ)^{∗}

be the reciprocity map constructed in Section 2 and let

rec_{0,X} : H_{0}^{S}(X, ℤ/mℤ) → H_{et}^{0}(X, ℤ/mℤ)^{∗}

be the homomorphism induced by the pairing

⟨·, ·⟩ : H_{0}^{S}(X, ℤ/mℤ) × H_{et}^{0}(X, ℤ/mℤ) ⟶ ℤ/mℤ

defined as follows: Given a ∈ H_{0}^{S}(X, ℤ/mℤ) and b ∈ H_{et}^{0}(X, ℤ/mℤ), we represent a by a correspondence α ∈ Cor(Δ^{0}, X) and put ⟨a, b⟩ = α^{∗}(b) ∈ H_{et}^{0}(Δ^{0}, ℤ/mℤ) ≅ ℤ/mℤ. This is well-defined since the homomorphisms 0^{∗}, 1^{∗} : H_{et}^{0}(Δ^{1}, ℤ/mℤ) → H_{et}^{0}(Δ^{0}, ℤ/mℤ) agree.

Lemma 5.3. For any m, rec_{0,X} is an isomorphism.

Proof. For connected X, we have the commutative diagram

\begin{matrix}H_{0}^{S}(X, ℤ/mℤ) & H_{et}^{0}(X, ℤ/mℤ)^{∗} \\ H_{0}^{S}(k, ℤ/mℤ) & H_{et}^{0}(k, ℤ/mℤ)^{∗}. \end{matrix} deg rec_{0,X} ≀ rec_{0,k} ∼

Hence, for connected X, it suffices by functoriality to consider the mod m degree map. In particular, rec_{0,X} is surjective for arbitrary X and is an isomorphism if dim X = 0. If X is a smooth connected curve, then H_{0}^{S}(X, ℤ) = Pic(X̄, X̄ ∖ X), where X̄ is the smooth compactification of X (cf. [SV1], Thm. 3.1). The subgroup Pic^{0}(X̄, X̄ ∖ X) of degree zero elements is the group of k-rational points of the Albanese of X, and hence divisible. Therefore, rec_{0,X} is an isomorphism for connected, and hence for all smooth curves. Considering the normalization morphism of an arbitrary scheme of dimension 1 and the exact sequences of Propositions 5.1 and 5.2, the five-lemma shows that rec_{0,X} is a isomorphism for dim X ≤ 1. It remains to show that rec_{0,X} is injective for arbitrary X. We may assume X to be connected. Let a ∈ ker(rec_{0,X}) and let α ∈ Z_{0}(X) be a representing 0-cycle. Since supp(α) is finite, we can find a connected 1-dimensional closed subscheme Z ⊂ X containing supp(α) (use, e.g., [Mu], II §6 Lemma). Since rec_{0,Z} is injective and a is in the image of H_{0}^{S}(Z, ℤ/mℤ) → H_{0}^{S}(X, ℤ/mℤ), we conclude that a = 0.

Corollary 5.4. Let k be an algebraically closed field and let X ∈ Sch/k be connected. Then the kernel of the degree map

deg : H_{0}^{S}(X, ℤ) ⟶ H_{0}^{S}(k, ℤ) ≅ ℤ is divisible.

Proposition 5.5. Let k be algebraically closed and let

\begin{matrix}Z′ & X′ \\ Z & X \end{matrix} i′ π′ π i

be an abstract blow-up square. Then for any integer m ≥ 1 the diagram

\begin{matrix}H_{1}^{S}(X, ℤ/mℤ) & H_{0}^{S}(Z′, ℤ/mℤ) \\ H_{t}^{1}(X, ℤ/mℤ)^{∗} & H_{et}^{0}(Z′, ℤ/mℤ)^{∗}, \end{matrix} δ rec_{1,X} δ^{∗} rec_{0,X}

commutes. Here δ is the boundary map of Proposition 5.2 and δ^{∗} is the dual of the boundary map of Proposition 5.1.

Proof. We have to show that the diagram

\begin{matrix}H_{1}^{S}(X, ℤ/mℤ) & × & H_{t}^{1}(X, ℤ/mℤ) & ℤ/mℤ \\ H_{0}^{S}(Z′, ℤ/mℤ) & × & H_{et}^{0}(Z′, ℤ/mℤ) & ℤ/mℤ \end{matrix} ⟨ , ⟩ ⟨ , ⟩ δ δ

commutes. Given a ∈ H_{1}^{S}(X, ℤ/mℤ) and b ∈ H_{et}^{0}(Z′, ℤ/mℤ), we choose a representing correspondence α ∈ C_{1}(X, ℤ/mℤ) = Cor(Δ^{1}, X) ⊗ ℤ/mℤ in such a way that it has a pre-image α̂ ∈ C_{1}(X′, ℤ/mℤ) ⊕ C_{1}(Z, ℤ/mℤ) (see the proof of Proposition 5.2). By definition, δa ∈ H_{0}^{S}(Z′, ℤ/mℤ) is represented by a correspondence γ ∈ C_{0}(Z′, ℤ/mℤ) such that the diagram

\begin{matrix}Δ^{0} & Δ^{1} \\ Z′ & X′ ⨿ Z \end{matrix} 0 − 1 γ i′ − π′ α̂

of correspondences commutes modulo m. Next choose an injective resolution ℤ/mℤ → I^{•} of ℤ/mℤ in the category of sheaves of ℤ/mℤ-modules on (Sch/k)_{h} in order to compute the pairings (cf. the end of section 3). Consider the following diagram

I^{0}(X′) ⊕ I^{0}(Z) I^{0}(Z′)

I^{1}(X)  I^{1}(X′) ⊕ I^{1}(Z) I^{1}(Z′)

I^{0}(Δ^{1})  I^{1}(Δ^{1})  I^{1}(Δ^{1})  I^{2}(Δ^{1})

I^{0}(Δ^{0})  I^{1}(Δ^{0})  I^{1}(Δ^{0})  I^{2}(Δ^{0})

α^{∗} d i^{′∗}−π^{′∗} d (π^{∗},i^{∗}) α^{∗} i^{′∗}−π^{′∗} α^{∗} d 0^{∗}−1^{∗} 0^{∗}−1^{∗} 0^{∗}−1^{∗} d 0^{∗}−1^{∗} d d

By the argument of [MVW] Lemma 12.7, the sequence

0 → F (X) → F (X′) ⊕ F (Z) → F (Z′)

is exact for every h-sheaf F. Therefore the second line in the diagram is exact. The proper base change theorem implies (cf. [Ge2], 3.2 and 3.6) that

I^{•}(X) ⟶ I^{•}(X′) ⊕ I^{•}(Z) ⟶ I^{•}(Z′) ⟶^{[1]}

is an exact triangle in D(Ab). For the exact sequence of complexes

0 → I^{•}(X) → I^{•}(X′) ⊕ I^{•}(Z) → I^{•}(Z′) → coker^{•} → 0,

this implies that the complex coker^{•} is exact. Therefore, b ∈ ker(I^{0}(Z′) → I^{1}(Z′)) has a pre-image β̂ ∈ I^{0}(X′) ⊕ I^{0}(Z). Then

dβ̂ ∈ ker(I^{1}(X′) ⊕ I^{1}(Z) → I^{1}(Z′)),

and there exists a unique ε ∈ I^{1}(X) with (π^{∗}, i^{∗})(ε) = dβ̂ representing δb ∈ H_{t}^{1}(X). We see that α^{∗}(dβ̂) = α^{∗}(ε). It follows that

d(α^{∗}(β̂)) = α^{∗}(dβ̂) = α^{∗}(ε) ∈ ker(I^{1}(Δ^{1}) ⟶^{0^{∗}−1^{∗}} I^{1}(Δ^{0})).

By definition of ⟨ , ⟩, we obtain

⟨a, δ(b)⟩ = (0^{∗} − 1^{∗})α^{∗}β̂ ∈ ker(I^{0}(Δ^{0}) → I^{1}(Δ^{0})) = ℤ/mℤ.

On the other hand, ⟨δa, b⟩ = γ^{∗}(b) ∈ H_{et}^{0}(Δ^{0}) is represented by γ^{∗}β ∈ I^{0}(Δ^{0}) and the commutative diagram of correspondences above implies

γ^{∗}β = γ^{∗}(i^{′∗} − π^{′∗})(β̂) = (0^{∗} − 1^{∗})α^{∗}β̂.

This finishes the proof.

Proposition 5.6. Let X be a normal, generically smooth, connected scheme of finite type over a field k and let M ⊆ H_{et}^{1}(X, ℤ/mℤ) be a finite subgroup. Then there exists a regular curve C over k and a finite morphism ϕ : C → X such that M has trivial intersection with the kernel of ϕ^{∗} : H_{et}^{1}(X, ℤ/mℤ) → H_{et}^{1}(C, ℤ/mℤ).

Proof. For any normal scheme Z and dense open subscheme Z′ ⊂ Z, the induced map H_{et}^{1}(Z, ℤ/mℤ) → H_{et}^{1}(Z′, ℤ/mℤ) is injective. Hence we may replace X by an open subscheme and assume that X is smooth. Let Y → X be the finite abelian étale covering corresponding to the kernel of π_{1}^{ab}(X) ↠ M^{∗}. We have to find a regular curve C and a finite morphism C → X such that C ×_{X} Y is connected. Choose a separating transcendence basis t_{1}, …, t_{d} of k(X) over k. This yields a rational map X → ℙ_{k}^{d}. Let t be another indeterminate and let X_{t} (resp. Y_{t}) be the base change of X (resp. Y) to the rational function field k(t). Consider the composition ϕ : Y_{t} → X_{t} → ℙ_{k(t)}^{d}. Since k(t) is Hilbertian [FJ], Thm. 12.10, we can find a rational point P ∈ ℙ_{k(t)}^{d} over which ϕ is defined and such that P has exactly one pre-image y_{t} in Y_{t}. The image x_{t} ∈ X_{t} of y_{t} has exactly one pre-image in Y_{t}. Let x be the image of x_{t} in X. If trdeg_{k}k(x) = 1 put x′ = x, if trdeg_{k}k(x) = 0 (i.e., x is a closed point in X) choose any x′ ∈ X with trdeg_{k}k(x′) = 1 such that x is a regular point of the closure of x′. In both cases the normalization C of the closure of x′ in X is a regular curve with the desired property.

6 Proof of the main theorem

In this section we prove our main result. We say that “resolution of singularities holds for schemes of dimension ≤ d over k” if the following two conditions are satisfied.

(1) For any integral separated scheme of finite type X of dimension ≤ d over k, there exists a projective birational morphism Y → X with Y smooth over k which is an isomorphism over the regular locus of X.

(2) For any integral smooth scheme X of dimension ≤ d over k and any birational proper morphism Y → X there exists a tower of morphisms X_{n} → X_{n−1} → ⋯ → X_{0} = X, such that X_{n} → X_{n−1} is a blow-up with a smooth center for i = 1, …, n, and such that the composite morphism X_{n} → X factors through Y → X.

Theorem 6.1 (=Theorem 1.1). Let k be an algebraically closed field of characteristic p ≥ 0, X a separated scheme of finite type over k and m a natural number. Then

rec_{X} : H_{1}^{S}(X, ℤ/mℤ) ⟶ π_{1}^{t,ab}(X)/m

is surjective. It is an isomorphism of finite abelian groups if (m, p) = 1, and for general m if resolution of singularities holds for schemes of dimension ≤ dim X + 1 over k.

The proof will occupy the rest of this section. Following the notation of Section 5, we write H_{t}^{0} = H_{et}^{0} and consider the maps

rec_{i,X} : H_{i}^{S}(X, ℤ/mℤ) → H_{t}^{i}(X, ℤ/mℤ)^{∗}

for i = 0, 1 (i.e., rec_{X} = rec_{1,X}). Given a morphism X′ → X, we have a commutative diagram of pairings defining rec_{i} for i = 0, 1.

\begin{matrix}H_{i}^{S}(X′, ℤ/mℤ) & × & H_{t}^{i}(X′, ℤ/mℤ) & ℤ/mℤ \\ H_{i}^{S}(X, ℤ/mℤ) & × & H_{t}^{i}(X, ℤ/mℤ) & ℤ/mℤ. \end{matrix} ⟨ , ⟩ ⟨ , ⟩ ϕ_{∗} ϕ^{∗}

Step 1: rec_{1,X} is surjective for arbitrary X.

We may assume that X is reduced and proceed by induction on d = dim X. The case dim X = 0 is trivial. Consider the normalization morphism X′ → X, which is an isomorphism outside a closed subscheme Z ⊂ X of dimension ≤ d − 1. Using the exact sequences of Propositions 5.1 and 5.2, which are compatible by Proposition 5.5 and the fact that rec_{0,X} is an isomorphism by Lemma 5.3, a diagram chase shows that it suffices to show surjectivity of rec_{1,X} for normal schemes. Let X be normal. Since H_{t}^{1}(X, ℤ/mℤ) is finite, it suffices to show that the pairing defining rec_{1,X} has a trivial right kernel. We may assume that X is connected. Let b ∈ H_{t}^{1}(X, ℤ/mℤ) be arbitrary but non-zero. By Proposition 5.6, we find a morphism ϕ : C → X with C a smooth curve such that ϕ^{∗}(b) ∈ H_{et}^{1}(C, ℤ/mℤ) is non-zero. Since the pairing for C is perfect by Theorem 4.1, the pairing for X has a trivial right kernel.

Step 2: Theorem 6.1 holds if (m, p) = 1.

If (m, p) = 1, H_{1}^{S}(X, ℤ/mℤ) and H_{et}^{1}(X, ℤ/mℤ)^{∗} are isomorphic finite abelian groups by [SV1]. In particular, they have the same order. Hence the surjective homomorphism rec_{1,X} is an isomorphism.

Step 3: Theorem 6.1 holds for arbitrary X if m = p^{r} and resolution of singularities holds for schemes of dimension ≤ dim X + 1 over k.

We may assume that X is reduced. Using resolution of singularities and Chow’s Lemma, we obtain a morphism X′ → X with X′ smooth and quasi-projective, which is an isomorphism over a dense open subscheme of X. Using the exact sequences of Propositions 5.1 and 5.2, Lemma 5.3, Step 1, induction on the dimension and the five-lemma, it suffices to show the result for smooth, quasi-projective schemes. Let X be smooth, quasi-projective and let X̄ be a smooth, projective variety containing X as a dense open subscheme. Then, by [Ge3, §5], we have an isomorphism

H_{et}^{1}(X̄, ℤ/p^{r}ℤ)^{∗} ≅ CH_{0}(X̄, 1, ℤ/p^{r}ℤ).

Furthermore, by [SS, Thm. 2.7] (which makes the assumption dim ≤ 2 but does not use it in its proof), we have an isomorphism

CH_{0}(X̄, 1, ℤ/p^{r}ℤ) ≅ H_{1}^{S}(X̄, ℤ/p^{r}ℤ).

By Proposition 6.2 below, the natural homomorphism

H_{1}^{S}(X, ℤ/p^{r}ℤ) → H_{1}^{S}(X̄, ℤ/p^{r}ℤ)

is an isomorphism of finite abelian groups and by Proposition 2.10, we have an isomorphism

H_{et}^{1}(X̄, ℤ/p^{r}ℤ) →^{∼} H_{t}^{1}(X, ℤ/p^{r}ℤ).

Hence the finite abelian groups H_{t}^{1}(X, ℤ/p^{r}ℤ)^{∗} and H_{1}^{S}(X, ℤ/p^{r}ℤ) are isomorphic, in particular, they have the same order. Since rec_{1,X} is surjective, it is an isomorphism.

In order to conclude the proof of Theorem 6.1 it remains to show

Proposition 6.2. Let k be a perfect field, X ∈ Sch/k smooth, U ⊂ X a dense open subscheme and n ≥ 0 an integer. Assume that resolution of singularities holds for schemes of dimension ≤ dim X + n over k. Then for any r ≥ 1 the natural map

H_{i}^{S}(U, ℤ/p^{r}ℤ) → H_{i}^{S}(X, ℤ/p^{r}ℤ)

is an isomorphism of finite abelian groups for i = 0, …, n.

Remark 6.3. A proof of Proposition 6.2 for n = 1 and k algebraically closed independent of the assumption on resolution of singularities would relax the condition in Theorem 6.1 to: There exists a smooth, projective scheme X̄′ ∈ Sch/k, dense open subschemes U′ ⊂ X′ ⊂ X̄′, U ⊂ X, and a surjective, proper morphism X′ → X which induces an isomorphism U′_{red} → U_{red} . In particular, Theorem 6.1 would hold for dim X ≤ 3 without any assumption on resolution of singularities [CV].

Proof of Proposition 6.2. We set R = ℤ/p^{r}ℤ. By [MVW], Lecture 14, we have

H_{i}^{S}(X, R) = Hom_{DM_{Nis}^{eff,−}(k,R)}(R[i], M(X, R)).

Let d = dim X. Choose a series of open subschemes U = X_{d} ⊂ ⋯ X_{1} ⊂ X_{0} = X such that Z_{j} := X_{j} ∖ X_{j+1} is smooth of dimension j for j = 0, …, d − 1. Using the exact Gysin triangles [MVW, 15.15]

M(X_{j+1}, R) → M(X_{j}, R) → M(Z_{j}, R)(d − j)[2d − 2j] →^{[1]} M(X_{j+1}, R)[1]

and induction, it suffices to show that

Hom_{DM_{Nis}^{eff,−}(k,R)}(R[i], M(Z_{j}, R)(s)[2s]) = 0

for j = 0, …, d − 1, i = 0, …, n + 1 and s ≥ 1. Using smooth compactifications of the Z_{j} and induction again, it suffices to show

Hom_{DM_{Nis}^{eff,−}(k,R)}(R[i], M(Z, R)(s)[2s]) = 0

for Z connected, smooth, projective, i = 0, …, d − d_{Z} + n and s ≥ 1. By the comparison of higher Chow groups and motivic cohomology [V] and by [GL], Thm. 8.5, the restriction of R(s) to the small Nisnevich site of a smooth scheme Y is isomorphic to ν_{r}^{s}[−s], where ν_{r}^{s} is the logarithmic de Rham Witt sheaf of Milne and Illusie. In particular, R(s)|_{Y} is trivial for s > dim Y. For an étale k-scheme Z we obtain

Hom_{DM_{Nis}^{eff,−}(k,R)}(R[i], M(Z, R)(s)[2s]) = H_{Nis}^{2s−i}(Z, R(s)) = 0

for s ≥ 1 and all i ≥ 0. Now assume dim Z ≥ 1. Using resolution of singularities for schemes of dimension ≤ d + n, the same method as in the proof of [SS], Thm. 2.7 yields isomorphisms

Hom_{DM_{Nis}^{eff,−}(k,R)}(R[i], M(Z, R)) ≅ CH^{d_{Z}} (Z, i, R)

for i = 0, …, d − 1 + n. Applying this to Z × ℙ^{s} and using the decompositions given by the projective bundle theorem on both sides implies isomorphisms

Hom_{DM_{Nis}^{eff,−}(k,R)}(R[i], M(Z, R)(s)[2s]) ≅ CH^{d_{Z}+s}(Z, i, R)

for i = 0, …, d − 1 + n. By [V], the latter group is isomorphic to

Hom_{DM_{Nis}^{eff,−}(k,R)}(M(Z, R)[2d_{Z} +2s−i], R(d_{Z} +s)) ≅ H_{Nis}^{2d_{Z}+2s−i}(Z, R(d_{Z} +s)),

which vanishes for s ≥ 1. This finishes the proof.

Remark 6.4. The assertion of Proposition 6.2 remains true for non-smooth X if U contains the singular locus of X (see [Ge4], Prop. 3.3).

7 Comparison with the isomorphism of Suslin-Voevodsky

Theorem 7.1. Let k be an algebraically closed field, X ∈ Sch/k and m an integer prime to char(k). Then the reciprocity isomorphism

rec_{X} : H_{1}^{S}(X, ℤ/mℤ) ⟶ π_{1}^{ab}(X)/m

is the dual of the isomorphism

α_{X} : H_{et}^{1}(X, ℤ/mℤ) ⟶ H_{S}^{1}(X, ℤ/mℤ)

of [SV1], Cor. 7.8.

The proof will occupy the rest of this section. Let i : ℤ/mℤ ↪ I^{0} be an injection into an injective sheaf in the category of ℤ/mℤ-module sheaves on (Sch/k)_{qfh} and put J^{1} = coker(i). Then (see the end of section 3) the pairing between H_{1}^{S}(X, ℤ/mℤ) and H_{et}^{1}(X, ℤ/mℤ) constructed in Proposition 2.12 can be given as follows: For a ∈ H_{1}^{S}(X, ℤ/mℤ) choose a representing correspondence α ∈ Cor(Δ^{1}, X) and for b ∈ H_{et}^{1}(X, ℤ/mℤ) a pre-image β ∈ J^{1}(X). Consider the diagram

I^{0}(X)  J^{1}(X)

I^{0}(Δ^{1})  J^{1}(Δ^{1})  (8)

ℤ/mℤ  I^{0}(Δ^{0})  J^{0}(Δ^{0}). α^{∗} α^{∗} 0^{∗} − 1^{∗} 0^{∗} − 1^{∗}

Then α^{∗}(β) is the image of some element γ ∈ I^{0}(Δ^{1}) and (0^{∗} − 1^{∗})(γ) ∈ ℤ/mℤ = ker(I^{0}(Δ^{0}) → J^{1}(Δ^{0})) equals ⟨a, b⟩.

For Y ∈ Sch/k let ℤ_{Y}^{qfh} be the free qfh-sheaf generated by Y. We set A = ℤ[1/char(k)] and L_{Y} = ℤ_{Y}^{qfh} ⊗ A. For smooth U the homomorphism

Cor(U, X) ⊗ A → Hom_{qfh}(L_{U}, L_{X})

is an isomorphism by [SV1], Thm. 6.7. We have

H_{et}^{1}(X, ℤ/mℤ) = H_{qfh}^{1}(X, ℤ/mℤ) = Ext_{qfh}^{1}(L_{X}, ℤ/mℤ)

= coker(Hom_{qfh}(L_{X}, I^{0}) → Hom_{qfh}(L_{X}, J^{1})).

The diagram (8) can be rewritten in terms of Hom-groups as follows:

Hom_{qfh}(L_{X}, I^{0})  Hom_{qfh}(L_{X}, J^{1})

Hom_{qfh}(L_{Δ^{1}}, I^{0})  Hom_{qfh}(L_{Δ^{1}}, J^{1})(9)

ℤ/mℤ  Hom_{qfh}(L_{Δ^{0}}, I^{0})  Hom_{qfh}(L_{Δ^{0}}, J^{1}). α^{∗} α^{∗} 0^{∗} − 1^{∗} 0^{∗} − 1∗

We denote the morphism L_{X} → J^{1} corresponding to β ∈ J^{1}(X) ≅ Hom_{qfh}(L_{X}, J^{1}) by the same letter β. Putting E := I^{0} ×_{J_{1},β} L_{X}, the extension

0 ⟶ ℤ/mℤ ⟶ E ⟶ L_{X} ⟶ 0

represents b ∈ Ext_{qfh}^{1}(L_{X}, ℤ/mℤ). Consider the diagram

Hom_{qfh}(L_{X}, E)  Hom_{qfh}(L_{X}, L_{X})

Hom_{qfh}(L_{Δ^{1}}, E)  Hom_{qfh}(L_{Δ^{1}}, L_{X})(10)

ℤ/mℤ  Hom_{qfh}(L_{Δ^{0}}, E)  Hom_{qfh}(L_{Δ^{0}}, L_{X}). α^{∗} α^{∗} 0^{∗} − 1^{∗} 0^{∗} − 1^{∗}

Because diagram (10) maps to diagram (9) via β_{∗} and id ∈ Hom_{qfh}(L_{X}, L_{X}) maps under β_{∗} to β ∈ Hom_{qfh}(L_{X}, J^{1}), we can calculate the pairing using diagram (10) after replacing β by id. Since id maps to α ∈ Hom_{qfh}(L_{Δ^{1}}, L_{X}) under α^{∗}, we see, writing the lower part of diagram (10) in the form

\begin{matrix}ℤ/mℤ & E(Δ^{1}) & L_{X}(Δ^{1}) \\ ℤ/mℤ & E(Δ^{0}) & L_{X}(Δ^{0}), \end{matrix} ; 0^{∗} − 1^{∗} ; 0^{∗} − 1^{∗} ; 0 ; h (11)

that

⟨a, b⟩ = h(α) mod m ∈ ker(E(Δ^{0})/m → L_{X}(Δ^{0})/m) = ℤ/mℤ,

where h is the unique homomorphism making diagram (11) commutative. We consider the complex C_{•}(X) = Cor(Δ^{•}, X) ⊗ A = L_{X}(Δ^{•}) with the obvious differentials. By the above considerations, the homomorphism induced by the pairing of Proposition 2.12

H_{et}^{1}(X, ℤ/mℤ) = H_{qfh}^{1}(X, ℤ/mℤ) ⟶

H_{1}^{S}(X, ℤ/mℤ)^{∗} = Ext_{A}^{1}(C_{•}(X), ℤ/mℤ) = Hom_{D(A)}(C_{•}(X), ℤ/mℤ[1]),

is given by sending an extension class [ℤ/mℤ ↪ E ↠ L_{X}] to the morphism C_{•}(X) → ℤ/mℤ[1] in the derived category of A-modules represented by the morphism

C_{•}(X) → [0 → E(Δ^{0}) → L_{X}(Δ^{0}) → 0]

which is given by id : L_{X}(Δ^{0}) → L_{X}(Δ^{0}) in degree zero and by h : L_{X}(Δ^{1}) → E(Δ^{0}) in degree one. The same construction works for any qfh-sheaf of A-modules F instead of L_{X}, i.e., setting C_{•}(F) = F (Δ^{•}) and starting from an element

[ℤ/mℤ ↪ E ↠ F] ∈ Ext_{qfh}^{1}(F, ℤ/mℤ),

we get a map C_{•}(F) → ℤ/mℤ[1] in the derived category of A-modules. We thus constructed a homomorphism

Ext_{qfh}^{1}(F, ℤ/mℤ) ⟶ Ext^{1}(C_{•}(F), ℤ/mℤ),  (12)

which for F = L_{X} and under the canonical identifications coincides with the map

H_{et}^{1}(X, ℤ/mℤ) ⟶ H_{1}^{S}(X, ℤ/mℤ)^{∗}

induced by the pairing constructed in Proposition 2.12.

Now we compare the map (12) with the map

α_{X} : Ext_{qfh}^{1}(F, ℤ/mℤ) ⟶ Ext_{A}^{1}(C_{•}(F), ℤ/mℤ)  (13)

constructed by Suslin-Voevodsky [SV1] (cf. [Ge1] for the case of positive characteristic). Let F_{•}^{∼} be the complex of qfh-sheaves associated with the complex of presheaves F_{•}(U) = F (U × Δ^{•}). By [SV1], the inclusion F → F_{•}^{∼} induces an isomorphism

Ext_{qfh}^{1}(F_{•}^{∼}, ℤ/mℤ) ⟶^{∼} Ext_{qfh}^{1}(F, ℤ/mℤ),  (14)

and evaluation at Spec(k) induces an isomorphism

Ext_{qfh}^{1}(F_{•}^{∼}, ℤ/mℤ) ⟶^{∼} Ext_{A}^{1}(C_{•}(F), ℤ/mℤ).  (15)

The map (13) of Suslin-Voevodsky is the composite of the inverse of (14) with (15).

We construct the inverse of (14). Let a class [ℤ/mℤ ↪ E ↠ F] ∈ Ext_{qfh}^{1}(F, ℤ/mℤ) be given. As a morphism in the derived category this class is given by the homomorphism

\begin{matrix}0 & 0 & F & 0 \\ 0 & E & F & 0 & . \end{matrix}_{id}

We therefore have to construct a homomorphism F_{1} ⟶ E making the diagram

\begin{matrix}F_{2} & F_{1} & F_{0} & 0 \\ 0 & E & F & 0 \end{matrix}_{id}

commutative. The construction is a sheafified version of what we did before. Let U ∈ Sch/k be arbitrary. Consider the diagram

\begin{matrix}0 & ℤ/mℤ(U) & E(U × Δ^{2}) & F (U × Δ^{2}) & 0 \\ 0 & ℤ/mℤ(U) & E(U × Δ^{1}) & F (U × Δ^{1}) & 0 \\ 0 & ℤ/mℤ(U) & E(U) & F (U) & 0 & . \end{matrix} δ^{0} − δ^{1} + δ^{2} δ^{0} − δ^{1} + δ^{2} id 0 0^{∗} − 1^{∗} 0^{∗} − 1^{∗}

Let α_{1} ∈ F (U × Δ^{1}) be given. By the smooth base change theorem and since H_{et}^{1}(Δ^{1}, ℤ/mℤ) = 0, we can lift α_{1} to E(U × Δ^{1}) after replacing U by a sufficiently fine étale cover. Applying 0^{∗} − 1^{∗} to this lift, we get an element in E(U). This gives the homomorphism F_{1} → E. Now let α_{2} ∈ F (U × Δ^{2}) be arbitrary. After replacing U by a sufficiently fine étale cover, we can lift α_{2} to E(U × Δ^{2}). Since (0^{∗} −1^{∗})(δ^{0} −δ^{1} +δ^{2}) = 0 this shows that (δ^{0} −δ^{1} +δ^{2})(α_{2}) maps to zero in E(U). This describes the inverse isomorphism to (14). Evaluating at U = Spec(k) gives back our original construction, hence (12) and (13) are the same maps. This finishes the proof.

References

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[Ge3] T. Geisser Duality via cycle complexes. Ann. of Math. (2) 172 (2010), no. 2, 1095–1126.

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Thomas Geisser Rikkyo University Department of Mathematics 3-34-1 Nishi-Ikebukuro Toshima-ku Tokyo Japan 171-8501 geisser@rikkyo.ac.jp Alexander Schmidt Universität Heidelberg Mathematisches Institut Im Neuenheimer Feld 288 D-69120 Heidelberg Deutschland schmidt@mathi.uniheidelberg.de 124

Documenta Mathematica 21 (2016)