On Suslin’s Singular Homology and Cohomology

Dedicated to A. A. Suslin on his 60th birthday

Thomas Geisser^{1}

Received: December 10, 2009 Revised: April 4, 2010

Abstract. We study properties of Suslin homology and cohomology over non-algebraically closed base fields, and their p-part in characteristic p. In the second half we focus on finite fields, and consider finite generation questions and connections to tamely ramified class field theory.

2010 Mathematics Subject Classification: 19E15, 14F42, 14G15 Keywords and Phrases: Suslin homology, motivic homology, algebraic cycles, albanese map

1 Introduction

Suslin and Voevodsky defined Suslin homology (also called singular homology) H_{i}^{S}(X, A) of a scheme of finite type over a field k with coefficients in an abelian group A as Tor_{i}(Cor_{k}(Δ^{∗}, X), A). Here Cor_{k}(Δ^{i}, X) is the free abelian group generated by integral subschemes Z of Δ^{i} × X which are finite and surjective over Δ^{i}, and the differentials are given by alternating sums of pull-back maps along face maps. Suslin cohomology H_{S}^{i}(X, A) is defined to be Ext_{Ab}^{i}(Cor_{k}(Δ^{∗}, X), A). Suslin and Voevodsky showed in [22] that over a separably closed field in which m is invertible, one has

H_{S}^{i}(X, ℤ/m) ≅ H_{et}^{i}(X, ℤ/m)  (1)

(see [2] for the case of fields of characteristic p). In the first half of this paper, we study both the situation that m is a power of the characteristic of k, and that k is not algebraically closed. In the second half, we focus on finite base fields and discuss a modified version of Suslin homology, which is closely related to etale cohomology on the one hand, but is also expected to be finitely generated. Moreover, its zeroth homology is ℤ^{π_{0}(X)}, and its first homology is expected to be an integral model of the abelianized tame fundamental group. We start by discussing the p-part of Suslin homology over an algebraically closed field of characteristic p. We show that, assuming resolution of singularities, the groups H_{i}^{S}(X, ℤ/p^{r}) are finite abelian groups, and vanish outside the range 0 ≤ i ≤ dim X. Thus Suslin cohomology with finite coefficients is etale cohomology away from the characteristic, but better behaved than etale cohomology at the characteristic (for example, H_{et}^{1}(𝔸^{1}, ℤ/p) is not finite). Moreover, Suslin homology is a birational invariant in the following strong sense: If X has a resolution of singularities p : X′ → X which is an isomorphism outside of the open subset U, then H_{i}^{S}(U, ℤ/p^{r}) ≅ H_{i}^{S}(X, ℤ/p^{r}). It was pointed out to us by N.Otsubo that this can be applied to generalize a theorem of Spiess-Szamuely [20] to include p-torsion:

1Supported in part by NSF grant No.0901021

Documenta Mathematica · Extra Volume Suslin (2010) 223–249

Theorem 1.1 Let X be a smooth, connected, quasi-projective variety over an algebraically closed field and assume resolution of singularities. Then the albanese map

alb_{X} : H_{0}^{S}(X, ℤ)^{0} → Alb_{X}(k)

from the degree-0-part of Suslin homology to the k-valued points of the Albanese variety induces an isomorphism on torsion groups.

Next we examine the situation over non-algebraically closed fields. We redefine Suslin homology and cohomology by imposing Galois descent. Concretely, if G_{k} is the absolute Galois group of k, then we define Galois-Suslin homology to be

H_{i}^{GS}(X, A) = H^{−i}RΓ(G_{k}, Cork̄(Δ∗ ; k̄, X̄) × A),

and Galois-Suslin cohomology to be

H_{GS}^{i}(X, A) = Ext_{G_{k}}^{i}(Cork̄(Δ∗ ; k̄, X̄), A).

Ideally one would like to define Galois-Suslin homology via Galois homology, but we are not aware of such a theory. With rational coefficients, the newly defined groups agree with the original groups. On the other hand, with finite coefficients prime to the characteristic, the proof of (1) in [22] carries over to show that H_{GS}^{i}(X, ℤ/m) ≅ H_{et}^{i}(X, ℤ/m). As a corollary, we obtain an isomorphism between H_{0}^{GS}(X, ℤ/m) and the abelianized fundamental group π_{1}^{ab}(X)/m for any separated X of finite type over a finite field and m invertible. The second half of the paper focuses on the case of a finite base field. We work under the assumption of resolution of singularities in order to see the picture of the properties which can expected. The critical reader can view our statements as theorems for schemes of dimension at most three, and conjectures in general. A theorem of Jannsen-Saito [11] can be generalized to show that Suslin homology and cohomology with finite coefficients for any X over a finite field is finite. Rationally, H_{0}^{S}(X, ℚ) ≅ H_{S}^{0}(X, ℚ) ≅ ℚ^{π_{0}(X)}. Most other properties are equivalent to the following Conjecture P_{0} considered in [7]: For X smooth and proper over a finite field, CH_{0}(X, i) is torsion for i ≠ 0. This is a particular case of Parshin’s conjecture that K_{i}(X) is torsion for i ≠ 0. Conjecture P_{0} is equivalent to the vanishing of H_{i}^{S}(X, ℚ) for i ≠ 0 and all smooth X. For arbitrary X of dimension d, Conjecture P_{0} implies the vanishing of H_{i}^{S}(X, ℚ) outside of the range 0 ≤ i ≤ d and its finite dimensionality in this range. Combining the torsion and rational case, we show that H_{i}^{S}(X, ℤ) and H_{S}^{i}(X, ℤ) are finitely generated for all X if and only if Conjecture P_{0} holds. Over a finite field and with integral coefficients, it is more natural to impose descent by the Weil group G generated by the Frobenius endomorphism φ instead of the Galois group [14, 3, 4, 7]. We define arithmetic homology

H_{i}^{ar}(X, A) = Tor_{i}^{G}(Cork̄(Δ∗ ; k̄, X̄), A)

and arithmetic cohomology

H_{ar}^{i}(X, ℤ) = Ext_{G}^{i}(Cork̄(Δ∗ ; k̄, X̄), ℤ).

We show that H_{0}^{ar}(X, ℤ) ≅ H_{ar}^{0}(X, ℤ) ≅ ℤ^{π_{0}(X)} and that arithmetic homology and cohomology lie in long exact sequences with Galois-Suslin homology and cohomology, respectively. They are finitely generated abelian groups if and only if Conjecture P_{0} holds. The difference between arithmetic and Suslin homology is measured by a third theory, which we call Kato-Suslin homology, and which is defined as H_{i}^{KS}(X, A) = H_{i}((Cork̄(Δ∗ ; k̄, X̄) ⊗ A)_{G}). By definition there is a long exact sequence

⋯ → H_{i}^{S}(X, A) → H_{i+1}^{ar}(X, A) → H_{i+1}^{KS}(X, A) → H_{i−1}^{S}(X, A) → · · · .

It follows that H_{0}^{KS}(X, ℤ) = ℤ^{π_{0}(X)} for any X. As a generalization of the integral version [7] of Kato’s conjecture [12], we propose

Conjecture 1.2 The groups H_{i}^{KS}(X, ℤ) vanish for all smooth X and i > 0.

Equivalently, there are short exact sequences

0 → H_{i+1}^{S}(X̄, ℤ)_{G} → H_{i}^{S}(X, ℤ) → H_{i}^{S}(X̄, ℤ)^{G} → 0

for all i ≥ 0 and all smooth X. We show that this conjecture, too, is equivalent to Conjecture P_{0}. This leads us to a conjecture on abelian tamely ramified class field theory:

Conjecture 1.3 For every X separated and of finite type over 𝔽_{q}, there is a canonical injection

H_{1}^{ar}(X, ℤ) → π_{1}^{t}(X)^{ab}

with dense image.

It might even be true that the relative group H_{1}^{ar}(X, ℤ)^{∘} := ker(H_{1}^{ar}(X, ℤ) → ℤ^{π_{0}(X)}) is isomorphic to the geometric part of the abelianized fundamental group defined in SGA 3X§6. To support our conjecture, we note that the generalized Kato conjecture above implies H_{0}^{S}(X, ℤ) ≅ H_{1}^{ar}(X, ℤ) for smooth X, so that in this case our conjecture becomes a theorem of Schmidt-Spiess [19]. In addition, we show (independently of any conjectures)

Proposition 1.4 If 1/l ∈ 𝔽_{q}, then H_{1}^{ar}(X, ℤ)^{∧l} ≅ π_{1}^{t}(X)^{ab}(l) for arbitrary X.

In particular, the conjectured finite generation of H_{1}^{ar}(X, ℤ) implies the conjecture away from the characteristic. We also give a conditional result at the characteristic. Notation: In this paper, scheme over a field k means separated scheme of finite type over k. The separable algebraic closure of k is denoted by k̄, and if X is a scheme over k, we sometimes write X̄ or Xk̄ for X ×_{k} k̄. We thank Uwe Jannsen for interesting discussions related to the subject of this paper, and Shuji Saito and Takeshi Saito for helpful comments during a series of lectures I gave on the topic of this paper at Tokyo University.

2 Motivic homology

Suslin homology H_{i}^{S}(X, ℤ) of a scheme X over a field k is defined as the homology of the global sections C_{∗}^{X}(k) of the complex of etale sheaves C_{∗}^{X}(−) = Cor_{k}(− × Δ^{∗}, X). Here Cor_{k}(U, X) is the group of universal relative cycles of U × Y/U [23]. If U is smooth, then Cor_{k}(U, X) is the free abelian group generated by closed irreducible subschemes of U × X which are finite and surjective over a connected component of U. Note that C_{∗}^{X}(−) = C_{∗}^{X^{red}}(−), and we will use that all contructions involving C_{∗}^{X} agree for X and X^{red} without further notice. More generally [1], motivic homology of weight n are the extension groups in Voevodsky’s category of geometrical mixed motives

H_{i}(X, ℤ(n)) = Hom_{DM_{k}^{−}}(ℤ(n)[i], M(X)),

and are isomorphic to

H_{i}(X, ℤ(n)) = \begin{cases}H_{(0)}^{2n−i}(𝔸^{n}, C_{∗}^{X}) & n ≥ 0 \\ H_{i−2n−1}(C_{∗}(\frac{c_{0}(X×(𝔸^{n}−\{0\}))}{c_{0}(X×\{1\})})(k)) & n < 0. \end{cases}

Here cohomology is taken for the Nisnevich topology. There is an obvious version with coefficients. Motivic homology is a covariant functor on the category of schemes of finite type over k, and has the following additional properties, see [1] (the final three properties require resolution of singularities)

a) It is homotopy invariant. b) It satisfies a projective bundle formula

H_{i}(X × ℙ^{1}, ℤ(n)) = H_{i}(X, ℤ(n)) ⊕ H_{i−2}(X, ℤ(n − 1)).

c) There is a Mayer-Vietoris long exact sequence for open covers.

d) Given an abstract blow-up square

Z′ ⟶ X′
↓  ↓
Z ⟶ X

there is a long exact sequence

⋯ → H_{i+1}(X, ℤ(n)) → H_{i}(Z′, ℤ(n)) → H_{i}(X′, ℤ(n)) ⊕ H_{i}(Z, ℤ(n)) → H_{i}(X, ℤ(n)) → ⋯ (2)

e) If X is proper, then motivic homology agrees with higher Chow groups indexed by dimension of cycles, H_{i}(X, ℤ(n)) ≅ CH_{n}(X, i − 2n).

f) If X is smooth of pure dimension d, then motivic homology agrees with motivic cohomology with compact support,

H_{i}(X, ℤ(n)) ≅ H_{c}^{2d−i}(X, ℤ(d − n)).

In particular, if Z is a closed subscheme of a smooth scheme X of pure dimension d, then we have a long exact sequence

⋯ → H_{i}(U, ℤ(n)) → H_{i}(X, ℤ(n)) → H_{c}^{2d−i}(Z, ℤ(d − n)) → · · · . (3)

In order to remove the hypothesis on resolution of singularities, it would be sufficient to find a proof of Theorem 5.5(2) of Friedlander-Voevodsky [1] that does not require resolution of singularities. For all arguments in this paper (except the p-part of the Kato conjecture) the sequences (2) and (3) and the existence of a smooth and proper model for every function field are sufficient.

2.1 Suslin cohomology

Suslin cohomology is by definition the dual of Suslin homology, i.e. for an abelian group A it is defined as

H_{S}^{i}(X, A) := Ext_{Ab}^{i}(C_{∗}^{X}(k), A).

We have H_{S}^{i}(X, ℚ/ℤ) ≅ Hom(H_{i}^{S}(X, ℤ), ℚ/ℤ), and a short exact sequence of abelian groups gives a long exact sequence of cohomology groups, in particular long exact sequences

⋯ → H_{S}^{i}(X, ℤ) → H_{S}^{i}(X, ℤ) → H_{S}^{i}(X, ℤ/m) → H_{S}^{i+1}(X, ℤ) → · · · . (4)

and

⋯ → H_{S}^{i−1}(X, ℚ/ℤ) → H_{S}^{i}(X, ℤ) → H_{S}^{i}(X, ℚ) → H_{S}^{i}(X, ℚ/ℤ) → · · · .

Consequently, H_{S}^{i}(X, ℤ)_{ℚ} ≅ H_{S}^{i}(X, ℚ) if Suslin-homology is finitely generated. If A is a ring, then H_{S}^{i}(X, A) ≅ Ext_{A}^{i}(C_{∗}^{X}(k) ⊗ A, A), and we get a spectral sequence

E_{2}^{s,t} = Ext_{A}^{s}(H_{t}^{S}(X, A), A) ⇒ H_{S}^{s+t}(X, A).  (5)

In particular, there are perfect pairings

H_{i}^{S}(X, ℚ) × H_{S}^{i}(X, ℚ) → ℚ
H_{i}^{S}(X, ℤ/m) × H_{S}^{i}(X, ℤ/m) → ℤ/m.

Lemma 2.1 There are natural pairings

H_{S}^{i}(X, ℤ)/tor × H_{i}^{S}(X, ℤ)/tor → ℤ

and

H_{S}^{i}(X, ℤ)_{tor} × H_{i−1}^{S}(X, ℤ)_{tor} → ℚ/ℤ.

Proof. The spectral sequence (5) gives a short exact sequence

0 → Ext^{1}(H_{i−1}^{S}(X, ℤ), ℤ) → H_{S}^{i}(X, ℤ) → Hom(H_{i}^{S}(X, ℤ), ℤ) → 0. (6)

The resulting map H_{S}^{i}(X, ℤ)/tor ↠ Hom(H_{i}^{S}(X, ℤ), ℤ) induces the first pairing. Since Hom(H_{i}^{S}(X, ℤ), ℤ) is torsion free, we obtain the map

H_{S}^{i}(X, ℤ)_{tor} ↪ Ext^{1}(H_{i−1}^{S}(X, ℤ), ℤ) ↠

Ext^{1}(H_{i−1}^{S}(X, ℤ)_{tor}, ℤ) ←^{∼} Hom(H_{i−1}^{S}(X, ℤ)_{tor}, ℚ/ℤ)

for the second pairing. □

2.2 Comparison to motivic cohomology

Recall that in the category DM_{k}^{−} of bounded above complexes of homotopy invariant Nisnevich sheaves with transfers, the motive M(X) of X is the complex of presheaves with transfers C_{∗}^{X}. Since a field has no higher Nisnevich cohomology, taking global sections over k induces a canonical map

Hom_{DM_{k}^{−}}(M(X), A[i]) → Hom_{DM^{−}(Ab)}(C_{∗}^{X}(k), A[i]),

hence a natural map

H_{M}^{i}(X, A) → H_{S}^{i}(X, A).  (7)

If X is a schene over L ⊇ k, then even though the cohomology groups do not depend on the base field, the map does. For example, if L/k is an extension of degree d, then the diagram of groups isomorphic to ℤ,

H_{M}^{0}(Spec k, ℤ)  H_{S}^{0}(Spec k, ℤ)
∥  ↓^{×d}
H_{M}^{0}(Spec L, ℤ) ⟶ H_{S}^{0}(Spec L, ℤ)

shows that the lower horizontal map is multiplication by d. We will see below that conjecturally (7) is a map between finitely generated groups which is rationally an isomorphism, and one might ask if its Euler characteristic has any interpretation.

3 The mod p Suslin homology in characteristic p

We examine the p-part of Suslin homology in characteristic p. We assume that k is perfect and resolution of singularities exists over k in order to obtain stronger results. We first give an auxiliary result on motivic cohomology with compact support:

Proposition 3.1 Let d = dim X.

a) We have H_{c}^{i}(X, ℤ/p^{r}(n)) = 0 for n > d.

b) If k is algebraically closed, then H_{c}^{i}(X, ℤ/p^{r}(d)) is finite, H_{c}^{i}(X, ℚ_{p}/ℤ_{p}(d)) is of cofinite type, and the groups vanish unless d ≤ i ≤ 2d.

Proof. By induction on the dimension and the localization sequence, the statement for X and a dense open subset of X are equivalent. Hence replacing X by a smooth subscheme and then by a smooth and proper model, we can assume that X is smooth and proper. Then a) follows from [8]. If k is algebraically closed, then

H^{i}(X, ℤ/p(d)) ≅ H^{i−d}(X_{Nis}, ν^{d}) ≅ H^{i−d}(X_{et}, ν^{d}),

by [8] and [13]. That the latter group is finite and of cofinite type, respectively, can be derived from [16, Thm.1.11], and it vanishes outside of the given range by reasons of cohomological dimension. □

Theorem 3.2 Let X be separated and of finite type over k.

a) The groups H_{i}(X, ℤ/p^{r}(n)) vanish for all n < 0.

b) If k is algebraically closed, then the groups H_{i}^{S}(X, ℤ/p^{r}) are finite, the groups H_{i}^{S}(X, ℚ_{p}/ℤ_{p}) are of cofinite type, and both vanish unless 0 ≤ i ≤ d.

Proof. If X is smooth, then H_{i}(X, ℤ/p^{r}(n)) ≅ H_{c}^{2d−i}(X, ℤ/p^{r}(d − n)) and we conclude by the Proposition. In general, we can assume by (2) and induction on the number of irreducible components that X is integral. Proceeding by induction on the dimension, we choose a resolution of singularities X′ of X, let Z be the closed subscheme of X where the map X′ → X is not an isomorphism, and let Z′ = Z ×_{X} X′. Then we conclude by the sequence (2). □

Example. If X′ is the blow up of a smooth scheme X in a smooth subscheme Z, then the strict transform Z′ = X′ ×_{X} Z is a projective bundle over Z, hence by the projective bundle formula H_{i}^{S}(Z, ℤ/p^{r}) ≅ H_{i}^{S}(Z′, ℤ/p^{r}) and H_{i}^{S}(X, ℤ/p^{r}) ≅ H_{i}^{S}(X′, ℤ/p^{r}). More generally, we have

Proposition 3.3 Assume X has a desingularization p : X′ → X which is an isomorphism outside of the dense open subset U. Then H_{i}^{S}(U, ℤ/p^{r}) ≅ H_{i}^{S}(X, ℤ/p^{r}). In particular, the p-part of Suslin homology is a birational invariant.

The hypothesis is satisfied if X is smooth, or if U contains all singular points of X and a resolution of singularities exists which is an isomorphism outside of the singular points.

Proof. If X is smooth, then this follows from Proposition 3.1a) and the localization sequence (3). In general, let Z be the set of points where p is not an isomorphism, and consider the cartesian diagram

Z′ ⟶ U′ ⟶ X′
↓  ↓  ↓
Z ⟶ U ⟶ X.

Comparing long exact sequence (2) of the left and outer squares,

→ H_{i}^{S}(Z′, ℤ/p^{r}) ⟶ H_{i}^{S}(U′, ℤ/p^{r}) ⊕ H_{i}^{S}(Z, ℤ/p^{r}) ⟶ H_{i}^{S}(U, ℤ/p^{r}) →
∥  ∥  ↓
→ H_{i}^{S}(Z′, ℤ/p^{r}) ⟶ H_{i}^{S}(X′, ℤ/p^{r}) ⊕ H_{i}^{S}(Z, ℤ/p^{r}) ⟶ H_{i}^{S}(X, ℤ/p^{r}) →

we see that H_{i}^{S}(U′, ℤ/p^{r}) ≅ H_{i}^{S}(X′, ℤ/p^{r}) implies H_{i}^{S}(U, ℤ/p^{r}) ≅ H_{i}^{S}(X, ℤ/p^{r}). □

Example. If X is a node, then the blow-up sequence gives H_{i}^{S}(X, ℤ/p^{r}) ≅ H_{i−1}^{S}(k, ℤ/p^{r})⊕H_{i}^{S}(k, ℤ/p^{r}), which is ℤ/p^{r} for i = 0, 1 and vanishes otherwise. Reid constructed a normal surface with a singular point whose blow-up is a node, showing that the p-part of Suslin homology is not a birational invariant for normal schemes.

Corollary 3.4 The higher Chow groups CH_{0}(X, i, ℤ/p^{r}) and the logarithmic de Rham-Witt cohomology groups H^{i}(X_{et}, ν_{r}^{d}), for d = dim X, are birational invariants.

Proof. Suslin homology agrees with higher Chow groups for proper X, and with motivic cohomology for smooth and proper X. □

Note that integrally CH_{0}(X) is a birational invariant, but the higher Chow groups CH_{0}(X, i) are generally not. Suslin and Voevodsky [22, Thm.3.1] show that for a smooth compactification X̄ of the smooth curve X, H_{0}^{S}(X, ℤ) is isomorphic to the relative Picard group Pic(X̄, Y) and that all higher Suslin homology groups vanish. Proposition 3.3 implies that the kernel and cokernel of Pic(X̄, Y) → Pic(X̄) are uniquely pdivisible. Given U with compactification j : U → X, the normalization X^{∼} of X in U is the affine bundle defined by the integral closure of O_{X} in j_{∗}O_{U}. We call X normal in U if X^{∼} → X is an isomorphism.

Proposition 3.5 If X is normal in the curve U, then H_{i}^{S}(U, ℤ/p) ≅ H_{i}^{S}(X, ℤ/p).

Proof. This follows by applying the argument of Proposition 3.3 to X′ the normalization of X, Z the closed subset where X′ → X is not an isomorphism, Z′ = X′ ×_{X} Z and U′ = X′ ×_{X} U. Since X is normal in U, we have Z ⊆ U and Z′ ⊆ U′. □

3.1 The albanese map

The following application was pointed out to us by N.Otsubo. Let X be a smooth connected quasi-projective variety over an algeraically closed field k of characteristic p. Then Spiess and Szamuely defined in [20] an albanese map

alb_{X} : H_{0}^{S}(X, ℤ)^{0} → Alb_{X}(k)

from the degree-0-part of Suslin homology to the k-valued points of the Albanese variety in the sense of Serre. They proved that if X is a dense open subscheme in a smooth projective scheme over k, then alb_{X} induces an isomorphism of the prime-to-p-torsion subgroups. We can remove the last hypothesis:

Theorem 3.6 Assuming resolution of singularities, the map alb_{X} induces an isomorphism on torsion groups for any smooth, connected, quasi-projective variety over an algebraically closed field.

Proof. In view of the result of Spiess and Szamuely, it suffices to consider the p-primary groups. Let T be a smooth and projective model of X. Since both sides are covariantly functorial and alb_{X} is functorial by construction, we obtain a commutative diagram

H_{0}^{S}(X, ℤ)^{0} ⟶^{alb_{X}} Alb_{X}(k)
↓  ↓
H_{0}^{S}(T, ℤ)^{0} ⟶^{alb_{T}} Alb_{T} (k)

The lower map is an isomorphism on torsion subgroups by Milne [15]. To show that the left vertical map is an isomorphism, consider the map of coefficient sequences

H_{1}^{S}(X, ℤ) ⊗ ℚ_{p}/ℤ_{p} ⟶ H_{1}^{S}(X, ℚ_{p}/ℤ_{p}) ⟶ _{p}H_{0}^{S}(X, ℤ) ⟶ 0
↓  ↓  ↓
H_{1}^{S}(T, ℤ) ⊗ ℚ_{p}/ℤ_{p} ⟶ H_{1}^{S}(T, ℚ_{p}/ℤ_{p}) ⟶ _{p}H_{0}^{S}(T, ℤ) ⟶ 0

The right vertical map is an isomorphism because the middle map vertical map is an isomorphism by Proposition 3.3, and because H_{1}^{S}(T, ℤ) ⊗ ℚ_{p}/ℤ_{p} ≅ CH_{0}(T, 1) ⊗ ℚ_{p}/ℤ_{p} vanishes by [6, Thm.6.1]. Finally, the map Alb_{X}(k) → Alb_{T} (k) is an isomorphism on p-torsion groups because by Serre’s description [18], the two Albanese varieties differ by a torus, which does not have any p-torsion k-rational points in characteristic p, □

4 Galois properties

Suslin homology is covariant, i.e. a separated map f : X → Y of finite type induces a map f_{∗} : Cor_{k}(T, X) → Cor_{k}(T, Y) by sending a closed irreducible subscheme Z of T × X, finite over T, to the subscheme [k(Z) : k(f(Z))] · f(Z) (note that f(Z) is closed in T × Y and finite over T). On the other hand, Suslin homology is contravariant for finite flat maps f : X → Y, because f induces a map f^{∗} : Cor_{k}(T, Y) → Cor_{k}(T, X) by composition with the graph of f in Cor_{k}(Y, X) (note that the graph is a universal relative cycle in the sense of [23]). We examine the properties of Suslin homology under change of base-fields.

Lemma 4.1 Let L/k be a finite extension of fields, X a scheme over k and Y a scheme over L. Then Cor_{L}(Y, X_{L}) = Cor_{k}(Y, X) and if X is smooth, then Cor_{L}(X_{L}, Y) = Cor_{k}(X, Y). In particular, Suslin homology does not depend on the base field.

Proof. The first statement follows because Y ×_{L} X_{L} ≅ Y ×_{k} X. The second statement follows because the map X_{L} → X is finite and separated, hence a closed subscheme of X_{L} ×_{L} Y ≅ X ×_{k} Y is finite and surjective over X_{L} if and only if it is finite and surjective over X. □

Given a scheme over k, the graph of the projection X_{L} → X in X_{L} × X gives elements Γ_{X} ∈ Cor_{k}(X, X_{L}) and Γ_{X}^{t} ∈ Cor_{k}(X_{L}, X).

4.1 Covariance

Lemma 4.2 a) If X and Y are separated schemes of finite type over k, then the two maps

Cor_{L}(X_{L}, Y_{L}) → Cor_{k}(X, Y)

induced by composition and precomposition, respectively, with Γ_{Y}^{t} and Γ_{X} agree. Both maps send a generator Z ⊆ X_{L} ×_{k} Y ≅ X ×_{k} Y_{L} to its image in X × Y with multiplicity [k(Z) : k(f(Z))], a divisor of [L : k].

b) If F/k is an infinite algebraic extension, then lim_{L/k} Cor_{L}(X_{L}, Y_{L}) = 0.

Proof. The first part is easy. If Z is of finite type over k, then k(Z) is a finitely generated field extension of k. For every component Z_{i} of Z_{F}, we obtain a map F → F ⊗_{k} k(Z) → k(Z_{i}), and since F is not finitely generated over k, neither is k(Z_{i}). Hence going up the tower of finite extensions L/k in F, the degree of [k(W_{L}) : k(Z)], for W_{L} the component of Z_{L} corresponding to Z_{i}, goes to infinity. □

4.2 Contravariance

Lemma 4.3 a) If X and Y are schemes over k, then the two maps

Cor_{k}(X, Y) → Cor_{L}(X_{L}, Y_{L})

induced by composition and precomposition, respectively with Γ_{Y} and Γ_{X}^{t} agree. Both maps send a generator Z ⊆ X × Y to the cycle associated to Z_{L} ⊆ X×_{k}Y_{L} ≅ X_{L}×_{k}Y. If L/k is separable, this is a sum of the integral subschemes lying over Z with multiplicity one. If L/k is Galois with group G, then the maps induce an isomorphism

Cor_{k}(X, Y) ≅ Cor_{L}(X_{L}, Y_{L})^{G}.

b) Varying L, Cor_{L}(X_{L}, Y_{L}) forms an etale sheaf on Spec k with stalk M = colim_{L} Cor_{L}(X_{L}, Y_{L}) ≅ Cork̄(Xk̄, Yk̄), where L runs through the finite extensions of k in a separable algebraic closure k̄ of k. In particular, Cor_{L}(X_{L}, Y_{L}) ≅ M^{Gal(k̄/L)}.

Proof. Again, the first part is easy. If L/k is separable, Z_{L} is finite and etale over Z, hence Z_{L} ≅ ∑_{i} Z_{i}, a finite sum of the integral cycles lying over Z with multiplicity one each. If L/k is moreover Galois, then Cor_{k}(X, Y) ≅ Cor_{L}(X_{L}, Y_{L})^{G} and Cork̄(Xk̄, Yk̄) ≅ colim_{L/k} Cor_{L}(X_{L}, Y_{L}) by EGA IV Thm. 8.10.5. □

The proposition suggests to work with the complex C_{∗}^{X} of etale sheaves on Spec k given by

C_{∗}^{X}(L) := Cor_{L}(Δ_{L}^{∗}, X_{L}) ≅ Cor_{k}(Δ_{L}^{∗}, X).

Corollary 4.4 If k̄ is a separable algebraic closure of k, then H_{i}^{S}(Xk̄, A) ≅ colim_{L/k} H_{i}^{S}(X_{L}, A), and there is a spectral sequence

E_{2}^{s,t} = lim_{L/k}^{s}H_{S}^{t}(X_{L}, A) ⇒ H_{S}^{s+t}(Xk̄, A).

The direct and inverse system run through finite separable extensions L/k, and the maps in the systems are induced by contravariant functoriality of Suslin homology for finite flat maps.

Proof. This follows from the quasi-isomorphisms

R Hom_{Ab}(C_{∗}^{X}(k̄), ℤ) ≅ R Hom_{Ab}(colim_{L} C_{∗}^{X}(L), ℤ) ≅ R lim_{L} R Hom_{Ab}(C_{∗}^{X}(L), ℤ).

□

4.3 Coinvariants

If G_{k} is the absolute Galois group of k, then Cork̄(X̄, Ȳ)_{G_{k}} can be identified with Cor_{k}(X, Y) by associating orbits of points of X̄ ×k̄ Ȳ with their image in X ×_{k} Y. However, this identification is neither compatible with covariant nor with contravariant functoriality, and in particular not with the differentials in the complex C_{∗}^{X}(k). But the obstruction is torsion, and we can remedy this problem by tensoring with ℚ: Define an isomorphism

τ : (Cork̄(X̄, Ȳ)_{ℚ})_{G_{k}} → Cor_{k}(X, Y)_{ℚ}.

as follows. A generator 1Z̄ corresponding to the closed irreducible subscheme Z̄ ⊆ X̄ × Ȳ is sent to \frac{1}{g_{Z}} 1_{Z}, where Z is the image of Z̄ in X × Y and g the number of irreducible components of Z ×_{k} k̄, i.e. g_{Z} is the size of the Galois orbit of Z̄.

Lemma 4.5 The isomorphism τ is functorial in both variables, hence it induces an isomorphism of complexes

(C_{∗}^{X}(k̄)_{ℚ})_{G_{k}} ≅ C_{∗}^{X}(k)_{ℚ}.

Proof. This can be proved by explicit calculation. We give an alternate proof. Consider the composition

Cor_{k}(X, Y) → Cork̄(X̄, Ȳ)^{G_{k}} → Cork̄(X̄, Ȳ)_{G_{k}} ⟶^{τ} Cor_{k}(X, Y)_{ℚ}.

The middle map is induced by the identity, and is multiplication by g_{Z} on the component corresponding to Z. All maps are isomorphisms upon tensoring with ℚ. The first map, the second map, and the composition are functorial, hence so is τ. □

5 Galois descent

Let k̄ be the algebraic closure of k with Galois group G_{k}, and let A be a continuous G_{k}-module. Then C_{∗}^{X}(k̄) ⊗ A is a complex of continuous G_{k}-modules, and if k has finite cohomological dimension we define Galois-Suslin homology to be

H_{i}^{GS}(X, A) = H^{−i}RΓ(G_{k}, C_{∗}^{X}(k̄) ⊗ A).

By construction, there is a spectral sequence

E_{s,t}^{2} = H^{−s}(G_{k}, H_{t}^{S}(X̄, A)) ⇒ H_{s+t}^{GS}(X, A).

The case X = Spec k shows that Suslin homology does not agree with Galois-Suslin homology, i.e. Suslin homology does not have Galois descent. We define Galois-Suslin cohomology to be

H_{GS}^{i}(X, A) = Ext_{G_{k}}^{i}(C_{∗}^{X}(k̄), A).  (8)

This agrees with the old definition if k is algebraically closed. Let τ_{∗} be the functor from G_{k}-modules to continuous G_{k}-modules which sends M to colim_{L} M^{G_{L}}, where L runs through the finite extensions of k. It is easy to see that R^{i}τ_{∗}M = colim_{H} H^{i}(H, M), with H running through the finite quotients of G_{k}.

Lemma 5.1 We have H_{GS}^{i}(X, A) = H^{i}RΓG_{k}Rτ_{∗} Hom_{Ab}(C_{∗}^{X}(k̄), A). In particular, there is a spectral sequence

E_{2}^{s,t} = H^{s}(G_{k}, R^{t}τ_{∗} Hom_{Ab}(C_{∗}^{X}(k̄), A)) ⇒ H_{GS}^{s+t}(X, A).  (9)

Proof. This is [17, Ex. 0.8]. Since C_{∗}^{X}(k̄) is a complex of free ℤ-modules, Hom_{Ab}(C_{∗}^{X}(k̄), −) is exact and preserves injectives. Hence the derived functor of τ_{∗} Hom_{Ab}(C_{∗}^{X}(k̄), −) is R^{t}τ_{∗} applied to Hom_{Ab}(C_{∗}^{X}(k̄), −). □

Lemma 5.2 For any abelian group A, the natural inclusion C_{∗}^{X}(k) ⊗ A → (C_{∗}^{X}(k̄) ⊗ A)^{G_{k}} is an isomorphism.

Proof. Let Z be a cycle corresponding to a generator of C_{∗}(k). If Z ⊗_{k} k̄ is the union of g irreducible components, then the corresponding summand of C_{∗}(k̄) is a free abelian group of rank g on which the Galois group permutes the summands transitively. The claim is now easy to verify. □

Proposition 5.3 We have

H_{i}^{GS}(X, ℚ) ≅ H_{i}^{S}(X, ℚ)
H_{GS}^{i}(X, ℚ) ≅ H_{S}^{i}(X, ℚ).

Proof. By the Lemma, H_{i}^{S}(X, ℚ) = H_{i}(C_{∗}^{X}(k) ⊗ ℚ) ≅ H_{i}((C_{∗}^{X}(k̄) ⊗ ℚ)^{G_{k}}). But the latter agrees with H_{i}^{GS}(X, ℚ) because higher Galois cohomology is torsion. Similarly, we have R^{t}τ_{∗} Hom(C_{i}^{X}(k̄), ℚ) = 0 for t > 0, and H^{s}(G_{k}, τ_{∗} Hom(C_{∗}^{X}(k̄), ℚ)) = 0 for s > 0. Hence H_{GS}^{i}(X, ℚ) is isomorphic to the ith cohomology of

Hom_{G_{k}}(C_{∗}^{X}(k̄), ℚ) ≅ Hom_{Ab}(C_{∗}^{X}(k̄)_{G_{k}}, ℚ) ≅ Hom_{Ab}(C_{∗}^{X}(k), ℚ).

The latter equality follows with Lemma 4.5. □

Theorem 5.4 If m is invertible in k and A is a finitely generated m-torsion G_{k}-module, then

H_{GS}^{i}(X, A) ≅ H_{et}^{i}(X, A).

Proof. This follows with the argument of Suslin-Voevodsky [22]. Indeed, let f : (Sch/k)_{h} → Et_{k} be the canonical map from the large site with the htopology of k to the small etale site of k. Clearly f_{∗}f^{∗}F ≅ F, and the proof of Thm.4.5 in loc.cit. shows that the cokernel of the injection f^{∗}f_{∗}F → F is uniquely m-divisible, for any homotopy invariant presheaf with transfers (like, for example, C_{i}^{X} : U ↦ Cor_{k}(U × Δ^{i}, X)). Hence

Ext_{h}^{i}(F_{h}^{∼}, f^{∗}A) ≅ Ext_{h}^{i}(f^{∗}f_{∗}F_{h}^{∼}, f^{∗}A) ≅ Ext_{Et_{k}}^{i}(f_{∗}F_{h}^{∼}, A) ≅ Ext_{G_{k}}^{i}(F(k̄), A).

Then the argument of section 7 in loc.cit. together with Theorem 6.7 can be descended from the algebraic closure of k to k. □

Duality results for the Galois cohomology of a field k lead via theorem 5.4 to duality results between Galois-Suslin homology and cohomology over k.

Theorem 5.5 Let k be a finite field, A a finite G_{k}-module, and A^{∗} = Hom(A, ℚ/ℤ). Then there is a perfect pairing of finite groups

H_{i−1}^{GS}(X, A) × H_{GS}^{i}(X, A^{∗}) → ℚ/ℤ.

Proof. According to [17, Example 1.10] we have

Ext_{G_{k}}^{r}(M, ℚ/ℤ) ≅ Ext_{G_{k}}^{r+1}(M, ℤ) ≅ H^{1−r}(G_{k}, M)^{∗}

for every finite G_{k}-module M, and the same holds for any torsion module by writing it as a colimit of finite modules. Hence

Ext_{G_{k}}^{r}(C_{∗}^{X}(k̄), Hom(A, ℚ/ℤ)) ≅ Ext_{G_{k}}^{r}(C_{∗}^{X}(k̄) ⊗ A, ℚ/ℤ) ≅

H^{1−r}(G_{k}, C_{∗}^{X}(k̄) ⊗ A)^{∗} = H_{r−1}^{GS}(X, A)^{∗}.

□

The case of non-torsion sheaves is discussed below.

Theorem 5.6 Let k be a local field with finite residue field and separable closure k^{s}. For a finite G_{k}-module A let A^{D} = Hom(A, (k^{s})^{×}). Then we have isomorphisms

H_{GS}^{i}(X, A^{D}) ≅ Hom(H_{i−2}^{GS}(X, A), ℚ/ℤ).

Proof. According to [17, Thm.2.1] we have

Ext_{G_{k}}^{r}(M, (k^{s})^{×}) ≅ H^{2−r}(G_{k}, M)^{∗}

for every finite G_{k}-module M. This implies the same statement for torsion modules, and the rest of the proof is the same as above. □

6 Finite base fields

From now on we fix a finite field 𝔽_{q} with algebraic closure 𝔽̄_{q}. To obtain the following results, we assume resolution of singularities. This is needed to use the sequences (2) and (3) to reduce to the smooth and projective case on the one hand, and the proof of Jannsen-Saito [11] of the Kato conjecture on the other hand (however, Kerz and Saito announced a proof of the prime to p-part of the Kato conjecture which does not require resolution of singularities). The critical reader is invited to view the following results as conjectures which are theorems in dimension at most 3. We first present results on finite generation in the spirit of [11] and [7].

Theorem 6.1 For any X/𝔽_{q} and any integer m, the groups H_{i}^{S}(X, ℤ/m) and H_{S}^{i}(X, ℤ/m) are finitely generated.

Proof. It suffices to consider the case of homology. If X is smooth and proper of dimension d, then H_{i}^{S}(X, ℤ/m) ≅ CH_{0}(X, i, ℤ/m) ≅ H_{c}^{2d−i}(X, ℤ/m(d)), and the result follows from work of Jannsen-Saito [11]. The usual devisage then shows that H_{c}^{j}(X, ℤ/m(d)) is finite for all X and d ≥ dim X, hence H_{i}^{S}(X, ℤ/m) is finite for smooth X. Finally, one proceeds by induction on the dimension of X with the blow-up long-exact sequence to reduce to the case X smooth. □

6.1 Rational Suslin-homology

We have the following unconditional result:

Theorem 6.2 For every connected X, the map H_{0}^{S}(X, ℚ) → H_{0}^{S}(𝔽_{q}, ℚ) ≅ ℚ is an isomorphism.

Proof. By induction on the number of irreducible components and (2) we can first assume that X is irreducible and then reduce to the situation where X is smooth. In this case, we use (3) and the following Proposition to reduce to the smooth and proper case, where H_{0}^{S}(X, ℚ) = CH_{0}(X)_{ℚ} ≅ CH_{0}(𝔽_{q})_{ℚ}. □

Proposition 6.3 If n > dim X, then H_{c}^{i}(X, ℚ(n)) = 0 for i ≥ n + dim X.

Proof. By induction on the dimension and the localization sequence for motivic cohomology with compact support one sees that the statement for X and a dense open subscheme of X are equivalent. Hence we can assume that X is smooth and proper of dimension d. Comparing to higher Chow groups, one sees that this vanishes for i > d + n for dimension (of cycles) reasons. For i = d + n, we obtain from the niveau spectral sequence a surjection

⨁_{X_{(0)}} H_{M}^{n−d}(k(x), ℚ(n − d)) ↠ H_{M}^{d+n}(X, ℚ(n)).

But the summands vanish for n > d because higher Milnor K-theory of finite fields is torsion. □

By definition, the groups H_{i}(X, ℚ(n)) vanish for i < n. We will consider the following conjecture P_{n} of [5]:

Conjecture P_{n}: For all smooth and projective schemes X over the finite field 𝔽_{q}, the groups H_{i}(X, ℚ(n)) vanish for i ≠ 2n.

This is a special case of Parshin’s conjecture: If X is smooth and projective of dimension d, then

H_{i}(X, ℚ(n)) ≅ H_{M}^{2d−i}(X, ℚ(d − n)) ≅ K_{i−2n}(X)^{(d−n)}

and, according to Parshin’s conjecture, the latter group vanishes for i ≠ 2n. By the projective bundle formula, P_{n} implies P_{n−1}.

Proposition 6.4 a) Let U be a curve. Then H_{i}^{S}(U, ℚ) ≅ H_{i}^{S}(X, ℚ) for any X normal in U.

b) Assume conjecture P_{−1}. Then H_{i}(X, ℚ(n)) = 0 for all X and n < 0, and if X has a desingularization p : X′ → X which is an isomorphism outside of the dense open subset U, then H_{i}^{S}(U, ℚ) ≅ H_{i}^{S}(X, ℚ). In particular, Suslin homology and higher Chow groups of weight 0 are birational invariant.

c) Under conjecture P_{0}, the groups H_{i}^{S}(X, ℚ) are finite dimensional and vanish unless 0 ≤ i ≤ d.

d) Conjecture P_{0} is equivalent to the vanishing of H_{i}^{S}(X, ℚ) for all i ≠ 0 and all smooth X.

Proof. The argument is the same as in Theorem 3.2. To prove b), we have to show that H_{c}^{i}(X, ℚ(n)) = 0 for n > d = dim X under P_{−1}, and for c) we have to show that H_{c}^{i}(X, ℚ(d)) is finite dimensional and vanishes unless d ≤ i ≤ 2d under P_{0}. By induction on the dimension and the localization sequence we can assume that X is smooth and projective. In this case, the statement is Conjecture P_{−1} and P_{0}, respectively, plus the fact that H_{0}^{S}(X, ℚ) ≅ CH_{0}(X)_{ℚ} is a finite dimensional vector space. The final statement follows from the exact sequence (3) and the vanishing of H_{c}^{i}(X, ℚ(n)) = 0 for n > d = dim X under P_{−1}. □

Proposition 6.5 Conjecture P_{0} holds if and only if the map H_{M}^{i}(X, ℚ) → H_{S}^{i}(X, ℚ) of (7) is an isomorphism for all X/𝔽_{q} and i.

Proof. The second statement implies the first, because if the map is an isomorphism, then H_{S}^{i}(X, ℚ) = 0 for i ≠ 0 and X smooth and proper, and hence so is the dual H_{i}^{S}(X, ℚ). To show that P_{0} implies the second statement, first note that because the map is compatible with long exact blow-up sequences, we can by induction on the dimension assume that X is smooth of dimension d. In this case, motivic cohomology vanishes above degree 0, and the same is true for Suslin cohomology in view of Proposition 6.4d). To show that for connected X the map (7) is an isomorphism of ℚ in degree zero, we consider the commutative diagram induced by the structure map

H_{M}^{0}(𝔽_{q}, ℚ) ⟶ H_{S}^{0}(𝔽_{q}, ℚ)
↓  ↓
H_{M}^{0}(X, ℚ) ⟶ H_{S}^{0}(X, ℚ)

This reduces the problem to the case X = Spec 𝔽_{q}, where it can be directly verified. □

6.2 Integral coefficients

Combining the torsion results [11] with the rational results, we obtain the following

Proposition 6.6 Conjecture P_{0} is equivalent to the finite generation of H_{i}^{S}(X, ℤ) for all X/𝔽_{q}.

Proof. If X is smooth and proper, then according to the main theorem of Jannsen-Saito [11], the groups H_{i}^{S}(X, ℚ/ℤ) = CH_{0}(X, i, ℚ/ℤ) are isomorphic to etale homology, and hence finite for i > 0 by the Weil-conjectures. Hence finite generation of H_{i}^{S}(X, ℤ) implies that H_{i}^{S}(X, ℚ) = 0 for i > 0. Conversely, we can by induction on the dimension assume that X is smooth and has a smooth and proper model. Expressing Suslin homology of smooth schemes in terms of motivic cohomology with compact support and again using induction, it suffices to show that H_{M}^{i}(X, ℤ(n)) is finitely generated for smooth and proper X and n ≥ dim X. Using the projective bundle formula we can assume that n = dim X, and then the statement follows because H_{M}^{i}(X, ℤ(n)) ≅ CH_{0}(X, 2n − i) is finitely generated according to [7, Thm 1.1]. □ Recall the pairings of Lemma 2.1. We call them perfect if they identify one group with the dual of the other group. In the torsion case, this implies that the groups are finite, but in the free case this is not true: For example, ⊕_{I}ℤ and ∏_{I} ℤ are in perfect duality.

Proposition 6.7 Let X be a separated scheme of finite type over a finite field. Then the following statements are equivalent:

a) The groups H_{i}^{S}(X, ℤ) are finitely generated for all i.

b) The groups H_{S}^{i}(X, ℤ) are finitely generated for all i.

c) The groups H_{S}^{i}(X, ℤ) are countable for all i.

d) The pairings of Lemma 2.1 are perfect for all i.

Proof. a) ⇒ b) ⇒ c) are clear, and c) ⇒ a) follows from [9, Prop.3F.12], which states that if A is not finitely generated, then either Hom(A, ℤ) or Ext(A, ℤ) is uncountable. Going through the proof of Lemma 2.1 it is easy to see that a) implies d). Conversely, if the pairing is perfect, then _{tor}H_{i}^{S}(X, ℤ) is finite. Let A = H_{S}^{i}(X, ℤ)/tor and fix a prime l. Then A/l is a quotient of H_{S}^{i}(X, ℤ)/l ⊆ H_{S}^{i}(X, ℤ/l), and which is finite by Theorem 6.1. Choose lifts b_{i} ∈ A of a basis of A/l and let B be the finitely generated free abelian subgroup of A generated by the b_{i}. By construction, A/B is l-divisible, hence H_{i}^{S}(X, ℤ)/tor = Hom(A, ℤ) ⊆ Hom(B, ℤ) is finitely generated. □

6.3 The algebraically closure of a finite field

Suslin homology has properties similar to a Weil-cohomology theory. Let X_{1} be separated and of finite type over 𝔽_{q}, X_{n} = X ×_{𝔽_{q}} 𝔽_{q^{n}} and X = X_{1} ×_{𝔽_{q}} 𝔽̄_{q}. From Corollary 4.4, we obtain a short exact sequence

0 → lim^{1}H_{S}^{t+1}(X_{n}, ℤ) → H_{S}^{t}(X, ℤ) → lim H_{S}^{t}(X_{n}, ℤ) → 0.

The outer terms can be calculated with the 6-term lim-lim^{1}-sequence associated to (6). The theorem of Suslin and Voevodsky implies that

lim H_{S}^{i}(X, ℤ/l^{r}) ≅ H_{et}^{i}(X, ℤ_{l})

for l ≠ p = char 𝔽_{q}. For X is proper and l = p, we get the same result from [6]

H_{S}^{i}(X, ℤ/p^{r}) ≅ Hom(CH_{0}(X, i, Z/p^{r}), ℤ/p^{r}) ≅ H_{et}^{i}(X, ℤ/p^{r}).

We show that this is true integrally:

Proposition 6.8 Let X be a smooth and proper curve over the algebraic closure of a finite field k of characteristic p. Then the non-vanishing cohomology groups are

H_{S}^{i}(X, ℤ) ≅ \begin{cases}ℤ & i = 0 \\ lim_{r} Hom_{GS}(μ_{p^{r}}, Pic X) × ∏_{l≠p} T_{l} Pic X(−1) & i = 1 \\ ∏_{l≠p} ℤ_{l}(−1) & i = 2. \end{cases}

Here Hom_{GS} denotes homomorphisms of group schemes.

Proof. By properness and smoothness we have

H_{i}^{S}(X, ℤ) ≅ H_{M}^{2−i}(X, ℤ(1)) ≅ \begin{cases}Pic X & i = 0; \\ k^{×} & i = 1; \\ 0 & i ≠ 0, 1. \end{cases}

Now

Ext^{1}(k^{×}, ℤ) = Hom(colim_{p|̸m} μ_{m}, ℚ/ℤ) ≅ ∏_{l≠p} ℤ_{l}(−1)

and since Pic X is finitely generated by torsion,

Ext^{1}(Pic X, ℤ) ≅ Hom(colim_{m}_{m} Pic X, ℚ/ℤ) ≅

lim Hom_{GS}(_{m} Pic X, ℤ/m) ≅ lim_{m} Hom_{GS}(μ_{m},_{m} Pic X)

by the Weil-pairing. □

Proposition 6.9 Let X be smooth, projective and connected over the algebraic closure of a finite field. Assuming conjecture P_{0}, we have

H_{S}^{i}(X, ℤ) ≅ \begin{cases}ℤ & i = 0 \\ ∏_{l} H_{et}^{i}(X, ℤ_{l}) & i ≥ 1. \end{cases}

In particular, the l-adic completion of H_{S}^{i}(X, ℤ) is l-adic cohomology H_{et}^{i}(X, ℤ_{l}) for all l.

Proof. Let d = dim X. By properness and smoothness we have

H_{i}^{S}(X, ℤ) ≅ H_{M}^{2d−i}(X, ℤ(d)).

Under hypothesis P_{0}, the groups H_{i}^{S}(X, ℤ) are torsion for i > 0, and H_{0}^{S}(X, ℤ) = CH_{0}(X) is the product of a finitely generated group and a torsion group. Hence for i ≥ 1 we get by (6) that

H_{S}^{i}(X, ℤ) ≅ Ext^{1}(H_{i−1}^{S}(X, ℤ), ℤ) ≅ Hom(H_{i−1}^{S}(X, ℤ)_{tor}, ℚ/ℤ)

≅ Hom(H_{M}^{2d−i+1}(X, ℤ(d))_{tor}, ℚ/ℤ) ≅ Hom(H_{et}^{2d−i}(X, ℚ/ℤ(d)), ℚ/ℤ)

≅ Hom(colim_{m} H_{et}^{2d−i}(X, ℤ/m(d)), ℚ/ℤ) ≅ lim_{m} Hom(H_{et}^{2d−i}(X, ℤ/m(d)), ℤ/m).

By Poincare-duality, the latter agrees with lim H_{et}^{i}(X, ℤ/m) ≅ ∏_{l} H_{et}^{i}(X, ℤ_{l}). □

7 Arithmetic homology and cohomology

We recall some definitions and results from [3]. Let X be separated and of finite type over a finite field 𝔽_{q}, X̄ = X ×_{𝔽_{q}} 𝔽̄_{q} and G be the Weil-group of 𝔽_{q}. Let γ : T_{G} → TĜ be the functor from the category of G-modules to the category of continuous Ĝ = Gal(𝔽_{q})-modules which associated to M the module γ_{∗} = colim_{m} M^{mG}, where the index set is ordered by divisibility. It is easy to see that the forgetful functor is a left adjoint of γ_{∗}, hence γ_{∗} is left exact and preserves limits. The derived functors γ_{∗}^{i} vanish for i > 1, and γ_{∗}^{1}M = R^{1}γ_{∗}M = colim M_{mG}, where the transition maps are given by M_{mG} → M_{mnG}, x ↦ ∑_{s∈mG/mnG} sx. Consequently, a complex M^{·} of Gmodules gives rise to an exact triangle of continuous Ĝ-modules

γ_{∗}M^{·} → Rγ_{∗}M^{·} → γ_{∗}^{1}M^{·}[−1].  (10)

If M = γ^{∗}N is the restriction of a continuous Ĝ-module, then γ_{∗}M = N and γ_{∗}^{1}M = N ⊗ ℚ. In particular, Weil-etale cohomology and etale cohomology of torsion sheaves agree. Note that the derived functors γ_{∗} restricted to the category of Ĝ-modules does not agree with the derived functors of τ_{∗} considered in Lemma 5.1. Indeed, R^{i}τ_{∗}M = colim_{L} H^{i}(G_{L}, M) is the colimit of Galois cohomology groups, whereas R^{i}γ_{∗}M = colim_{m} H^{i}(mG, M) is the colimit of cohomology groups of the discrete group ℤ.

7.1 Homology

We define arithmetic homology with coefficients in the G-module A to be

H_{i}^{ar}(X, A) := Tor_{i}^{G}(C_{∗}^{X}(k̄), A).

A concrete representative is the double complex

C_{∗}^{X}(k̄) ⊗ A ⟶^{1−φ} C_{∗}^{X}(k̄) ⊗ A,

with the left and right term in homological degrees one and zero, respectively, and with the Frobenius endomorphism φ acting diagonally. We obtain short exact sequences

0 → H_{i}^{S}(X̄, A)_{G} → H_{i}^{ar}(X, A) → H_{i−1}^{S}(X̄, A)^{G} → 0.  (11)

Lemma 7.1 The groups H_{i}^{ar}(X, ℤ/m) are finite. In particular, H_{i}^{ar}(X, ℤ)/m and _{m}H_{i}^{ar}(X, ℤ) are finite.

Proof. The first statement follows from the short exact sequence (11). Indeed, if m is prime to the characteristic, then we apply (1) together with finite generation of etale cohomology, and if m is a power of the characteristic, we apply Theorem 3.2 to obtain finiteness of the outer terms of (11). The final statements follows from the long exact sequence

⋯ → H_{i}^{ar}(X, ℤ) ⟶^{×m} H_{i}^{ar}(X, ℤ) → H_{i}^{ar}(X, ℤ/m) → ⋯

□

If A is the restriction of a Ĝ-module, then (10), applied to the complex of continuous Ĝ-modules C_{∗}^{X}(k̄)⊗A, gives upon taking Galois cohomology a long exact sequence

⋯ → H_{i}^{GS}(X, A) → H_{i+1}^{ar}(X, A) → H_{i+1}^{GS}(X, A_{ℚ}) → H_{i−1}^{GS}(X, A) → ⋯

With rational coefficients this sequence breaks up into isomorphisms

H_{i}^{ar}(X, ℚ) ≅ H_{i}^{S}(X, ℚ) ⊕ H_{i−1}^{S}(X, ℚ).  (12)

7.2 Cohomology

In analogy to (8), we define arithmetic cohomology with coefficients in the G-module A to be

H_{ar}^{i}(X, A) = Ext_{G}^{i}(C_{∗}^{X}(k̄), A).  (13)

Note the difference to the definition in [14], which does not give well-behaved (i.e. finitely generated) groups for schemes which are not smooth and proper. A concrete representative is the double complex

Hom(C_{∗}^{X}(k̄), A) ⟶^{1−φ} Hom(C_{∗}^{X}(k̄), A),

where the left and right hand term are in cohomological degrees zero and one, respectively. There are short exact sequences

0 → H_{S}^{i−1}(X̄, A)_{G} → H_{ar}^{i}(X, A) → H_{S}^{i}(X̄, A)^{G} → 0.  (14)

The proof of Lemma 7.1 also shows

Lemma 7.2 The groups H_{ar}^{i}(X, ℤ/m) are finite. In particular, _{m}H_{ar}^{i}(X, ℤ) and H_{ar}^{i}(X, ℤ)/m are finite.

Lemma 7.3 For every G-module A, we have an isomorphism

H_{ar}^{i}(X, A) ≅ H_{GS}^{i}(X, Rγ_{∗}γ^{∗}A).

Proof. Since M^{G} = (γ_{∗}M)Ĝ, arithmetic cohomology is the Galois cohomology of the derived functor of γ_{∗} Hom_{Ab}(C_{∗}^{X}(k̄), −) on the category of G-modules. By Lemma 5.1, it suffices to show that this derived functor agrees with the derived functor of τ_{∗} Hom_{Ab}(C_{∗}^{X}(k̄), γ_{∗}−) on the category of G-modules. But given a continuous Ĝ-modules M and a G-module N, the inclusion

τ_{∗} Hom_{Ab}(M, γ_{∗}N) ⊆ γ_{∗} Hom_{Ab}(γ^{∗}M, N)

induced by the inclusion γ_{∗}N ⊆ N is an isomorphism. Indeed, if f : M → N is H-invariant and m ∈ M is fixed by H′, then f(m) is fixed by H ∩ H′, hence f factors through γ_{∗}N. □

Corollary 7.4 If A is a continuous Ĝ-module, then there is a long exact sequence

⋯ → H_{GS}^{i}(X, A) → H_{ar}^{i}(X, A) → H_{GS}^{i−1}(X, A_{ℚ}) → H_{GS}^{i+1}(X, A) → · · · .

Proof. This follows from the Lemma by applying the long exact Ext∗ ; Ĝ(C_{∗}^{X}(k̄), −)-sequence to (10). □

7.3 Finite generation and duality

Lemma 7.5 There are natural pairings

H_{ar}^{i}(X, ℤ)/tor × H_{i}^{ar}(X, ℤ)/tor → ℤ

and

H_{ar}^{i}(X, ℤ)_{tor} × H_{i−1}^{ar}(X, ℤ)_{tor} → ℚ/ℤ.

Proof. From the adjunction Hom_{G}(M, ℤ) ≅ Hom_{Ab}(M_{G}, ℤ) and the fact that L(−)_{G} = R(−)^{G}[−1], we obtain by deriving a quasi-isomorphism

R Hom_{G}(C_{∗}^{X}(k̄), ℤ) ≅ R Hom_{Ab}(C_{∗}^{X}(k̄) ⊗_{G}^{L} ℤ, ℤ).

Now we obtain the pairing as in Lemma 2.1 using the resulting spectral sequence

Ext_{Ab}^{s}(H_{t}^{ar}(X, ℤ), ℤ) ⇒ H_{ar}^{s+t}(X, ℤ).

□

Proposition 7.6 For a given separated scheme X of finite type over 𝔽_{q}, the following statements are equivalent:

a) The groups H_{i}^{ar}(X, ℤ) are finitely generated.

b) The groups H_{ar}^{i}(X, ℤ) are finitely generated. c) The groups H_{ar}^{i}(X, ℤ) are countable.

d) The pairings of Lemma 7.5 are perfect.

Proof. This is proved exactly as Proposition 6.7, with Theorem 6.1 replaced by Lemma 7.1. □

We need a Weil-version of motivic cohomology with compact support. We define H_{c}^{i}(X_{W} , ℤ(n)) to be the ith cohomology of RΓ(G, RΓ_{c}(X̄, ℤ(n))), where the inner term is a complex defining motivic cohomology with compact support of X̄. We use this notation to distinguish it from arithmetic homology with compact support considered in [4], which is the cohomology of RΓ(G, RΓ_{c}(X̄_{et}, ℤ(n))). However, if n ≥ dim X, which is the case of most importance for us, both theories agree. Similar to (3) we obtain for a closed subscheme Z of a smooth scheme X of pure dimension d with open complement U a long exact sequence

⋯ → H_{i}^{ar}(U, ℤ) → H_{i}^{ar}(X, ℤ) → H_{c}^{2d+1−i}(Z_{W} , ℤ(d)) → · · · .  (15)

The shift by 1 in degrees occurs because arithmetic homology is defined using homology of G, whereas cohomology with compact support is defined using cohomology of G.

Proposition 7.7 The following statements are equivalent:

a) Conjecture P_{0}.

b) The groups H_{i}^{ar}(X, ℤ) are finitely generated for all X.

Proof. a) ⇒ b): By induction on the dimension of X and the blow-up square, we can assume that X is smooth of dimension d, where

H_{i}^{ar}(X, ℤ) ≅ H_{c}^{2d+1−i}(X_{W} , ℤ(d)).

By localization for H_{c}^{∗}(X_{W} , ℤ(d)) and induction on the dimension we can reduce the question to X smooth and projective. In this case ℤ(d) has etale hypercohomological descent over an algebraically closed field by [6], hence H_{c}^{j}(X_{W} , ℤ(d)) agrees with the Weil-etale cohomology H_{W}^{j} (X, ℤ(d)) considered in [3]. These groups are finitely generated for i > 2d by [3, Thm.7.3,7.5]. By conjecture P_{0}, and the isomorphism H_{W}^{i} (X, ℤ(d))_{ℚ} ≅ CH_{0}(X, 2d − i)_{ℚ} ⊕ CH_{0}(X, 2d−i+1)_{ℚ} of Thm.7.1c) loc.cit., these groups are torsion for i < 2d, so that the finite group H^{i−1}(X_{et}, ℚ/ℤ(d)) surjects onto H_{W}^{i} (X, ℤ(d)). Finally, H_{W}^{2d}(X, ℤ(d)) is an extension of the finitely generated group CH_{0}(X̄)^{G} by the finite group H^{2d−1}(X̄_{et}, ℤ(d))_{G} ≅ H^{2d−2}(X̄_{et}, ℚ/ℤ(d))_{G}.

b) ⇒ a) Consider the special case that X is smooth and projective. Then as above, H_{i}^{ar}(X, ℤ) ≅ H_{W}^{2d+1−i}(X, ℤ(d)). If this group is finitely generated, then we obtain from the coefficient sequence that H_{W}^{2d+1−i}(X, ℤ(d)) ⊗ ℤ_{l} ≅ lim H^{2d+1−i}(X_{et}, ℤ/l^{r}(d)), and the latter group is torsion for i > 1 by the Weil-conjectures. Now use (12). □

Theorem 7.8 For connected X, the map H_{0}^{ar}(X, ℤ) → H_{0}^{ar}(𝔽_{q}, ℤ) ≅ ℤ is an isomorphism. In particular, we have H_{0}^{ar}(X, ℤ) ≅ ℤ^{π_{0}(X)}.

Proof. The proof is similar to the proof of Theorem 6.2. Again we use induction on the dimension and the blow-up sequence to reduce to the situation where X is irreducible and smooth. In this case, we can use (15) and the following Proposition to reduce to the smooth and proper case, where we have H_{0}^{ar}(X, ℤ) = CH_{0}(X̄)_{G} ≅ ℤ. □

Proposition 7.9 If n > dim X, then H_{c}^{i}(X_{W} , ℤ(n)) = 0 for i > n + dim X.

Proof. By induction on the dimension and the localization sequence for motivic cohomology with compact support one sees that the statement for X and a dense open subscheme of X are equivalent. Hence we can assume that X is smooth and proper of dimension d. In this case, H_{c}^{i}(X_{W} , ℤ(n)) is an extension of H_{M}^{i}(X̄, ℤ(n))^{G} by H_{M}^{i−1}(X̄, ℤ(n))_{G}. These groups vanish for i − 1 > d + n for dimension (of cycles) reasons. For i = d + n + 1, we have to show that H_{M}^{d+n}(X̄, ℤ(n))_{G} vanishes. From the niveau spectral sequence for motivic cohomology we obtain a surjection

⨁_{X̄_{(0)}} H_{M}^{n−d}(k(x), ℤ(n − d)) ↠ H_{M}^{d+n}(X̄, ℤ(n)).

The summands are isomorphic to K_{n−d}^{M}(𝔽̄_{q}). If n > d + 1, then they vanish because higher Milnor K-theory of the algebraical closure of a finite field vanishes. If n = d + 1, then the summands are isomorphic to (𝔽̄_{q})^{×}, whose coinvariants vanish. □

8 A Kato type homology

We construct a homology theory measuring the difference between Suslin homology and arithmetic homology. The cohomological theory can be defined analogously. Kato-Suslin-homology H_{i}^{KS}(X, A) with coefficients in the G-module A is defined as the ith homology of the complex of coinvariants (C_{∗}^{X}(k̄) ⊗ A)_{G}. If A is trivial as a G-module, then Lemma 5.2 gives a short exact sequence of double complexes

0 ⟶ C_{∗}^{X}(k) ⊗ A ⟶ C_{∗}^{X}(k̄) ⊗ A ⟶  0
↓  ^{1−φ}↓  ↓
0  ⟶ C_{∗}^{X}(k̄) ⊗ A ⟶ (C_{∗}^{X}(k̄) ⊗ A)_{G} ⟶ 0

and hence a long exact sequence

⋯ → H_{i}^{S}(X, A) → H_{i+1}^{ar}(X, A) → H_{i+1}^{KS}(X, A) → H_{i−1}^{S}(X, A) → · · · .

By Theorem 7.8 we have H_{0}^{KS}(X, ℤ) ≅ H_{0}^{ar}(X, ℤ) ≅ ℤ^{π_{0}(X)}. The following is a generalization of the integral version [7] of Kato’s conjecture [12].

Conjecture 8.1 (Generalized integral Kato-conjecture) If X is smooth, then H_{i}^{KS}(X, ℤ) = 0 for i > 0.

Equivalently, the canonical map H_{i}^{S}(X, ℤ) ≅ H_{i+1}^{ar}(X, ℤ) is an isomorphism for all smooth X and all i ≥ 0, i.e. there are short exact sequences

0 → H_{i+1}^{S}(X̄, ℤ)_{G} → H_{i}^{S}(X, ℤ) → H_{i}^{S}(X̄, ℤ)^{G} → 0.

Theorem 8.2 Conjecture 8.1 is equivalent to conjecture P_{0}.

Proof. If Conjecture 8.1 holds, then

H_{i}^{S}(X, ℚ) ≅ H_{i+1}^{ar}(X, ℚ) ≅ H_{i+1}^{S}(X, ℚ) ⊕ H_{i}^{S}(X, ℚ)

implies the vanishing of H_{i}^{S}(X, ℚ) for i > 0. Conversely, we first claim that for smooth and proper Z, the canonical map H_{c}^{i}(Z, ℤ(n)) → H_{c}^{i}(Z_{W} , ℤ(n)) is an isomorphism for all i if n > dim Z, and for i ≤ 2n if n = dim Z. Indeed, if n ≥ dim Z then the cohomology of ℤ(n) agrees with the etale hypercohomology of ℤ(n), see [6], hence satisfies Galois descent. But according to (the proof of) Proposition 6.4b), these groups are torsion groups, so that the derived funtors RΓG_{k} and RΓG agree. Using localization for cohomology with compact support and induction on the dimension, we get next that H_{c}^{i}(Z, ℤ(n)) ≅ H_{c}^{i}(Z_{W} , ℤ(n)) for all i and all Z with n > dim Z. Now choose a smooth and proper compactification C of X. Comparing the exact sequences (3) and (15), we see with the 5-Lemma that the isomorphism H_{i}^{S}(C, ℤ) ≅ H_{c}^{2d−i}(C, ℤ(d)) → H_{i+1}^{ar}(C, ℤ) ≅ H_{c}^{2d−i}(C_{W} , ℤ(d)) for C implies the same isomorphism for X and i ≥ 0. □

9 Tamely ramified class field theory

We propose the following conjecture relating Weil-Suslin homology to class field theory:

Conjecture 9.1 (Tame reciprocity) For any X separated and of finite type over a finite field, there is a canonical injection to the tame abelianized fundamental group with dense image

H_{1}^{ar}(X, ℤ) → π_{1}^{t}(X)^{ab}.

Note that the group H_{1}^{ar}(X, ℤ) is conjecturally finitely generated. At this point, we do not have an explicit construction (associating elements in the Galois groups to algebraic cycles) of the map. One might even hope that H_{1}^{ar}(X, ℤ)^{∘} := ker(H_{1}^{ar}(X, ℤ) → ℤ^{π_{0}(X)} is finitely generated and isomorphic to the abelianized geometric part of the tame fundamental group defined in SGA 3X§6. Under Conjecture 8.1, H_{0}^{S}(X, ℤ) ≅ H_{1}^{ar}(X, ℤ) for smooth X, and Conjecture 9.1 is a theorem of Schmidt-Spiess [19].

Proposition 9.2 a) We have H_{1}^{ar}(X, ℤ)^{∧l} ≅ π_{1}^{t}(X)^{ab}(l). In particular, the prime to p-part of Conjecture 9.1 holds if H_{1}^{ar}(X, ℤ) is finitely generated.

b) The analog statement holds for the p-part if X has a compactification T which has a desingularization which is an isomorphism outside of X.

Proof. a) By Theorem 7.8, H_{0}^{ar}(X, ℤ) contains no divisible subgroup. Hence if l ≠ p, we have by Theorems 5.4 and 5.5

H_{1}^{ar}(X, ℤ)^{∧l} ≅ lim H_{1}^{ar}(X, ℤ/l^{r}) ≅ lim H_{0}^{GS}(X, ℤ/l^{r})

≅ lim H_{et}^{1}(X, ℤ/l^{r})^{∗} ≅ π_{1}^{t}(X)^{ab}(l).

b) Under the above hypothesis, we can use the duality result of [6] for the proper scheme T to get with Proposition 3.3

H_{1}^{ar}(X, ℤ) ⊗ ℤ_{p} ≅ lim H_{0}^{GS}(X, ℤ/p^{r}) ≅ lim H_{0}^{GS}(T, ℤ/p^{r})

≅ lim H_{et}^{1}(T, ℤ/p^{r})^{∗} ≅ π_{1}(T)^{ab}(p) ≅ π_{1}^{t}(X)^{ab}(p).
□

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Thomas Geisser

USC Department of Math. 3620 South Vermont Ave. Los Angeles, CA 90089 geisser@usc.edu 250

Documenta Mathematica · Extra Volume Suslin (2010)