On Two Higher Chow Groups of Schemes

over a Finite Field

Satoshi Kondo^{1} and Seidai Yasuda

Received: May 16, 2009 Revised: May 21, 2011

Communicated by Stephen Lichtenbaum

Abstract. Given a separated scheme X of finite type over a finite field, its higher Chow groups CH_{−1}(X, 1) and CH_{−2}(X, 3) are computed explicitly.

2010 Mathematics Subject Classification: Primary 14F42; Secondary 14F35 Keywords and Phrases: higher Chow group, motivic cohomology.

1. Introduction

Let 𝔽_{q} be the field of q elements of characteristic p. For a separated scheme X which is essentially of finite type over Spec 𝔽_{q}, we define the Borel-Moore motivic homology group H_{i}^{BM}(X, ℤ(j)) as the homology group H_{i−2j}(z_{j}(X, •)) = CH_{j}(X, i − 2j) of Bloch’s cycle complex z_{j}(X, •) ([Bl, Introduction, p. 267] see also [Ge-Le2, 2.5, p. 60] to remove the condition that X is quasi-projective; see [Le1] for the labeling using dimension and not codimension). If j > i or j > dim X, then H_{i}^{BM}(X, ℤ(j)) = 0 for trivial reason. When X is essentially smooth over Spec 𝔽_{q}, it coincides with the motivic cohomology group defined in [Le1, Part I, Chapter I, 2.2.7, p. 21] or [Vo-Su-Fr] (cf. [Le2, Theorem 1.2, p. 300], [Vo, Corollary 2, p. 351]). For an abelian group M, we set H_{i}^{BM}(X, M(j)) = H_{i−2j}(z_{j}(X, •) ⊗_{ℤ} M). For a scheme X, we let O(X) = H^{0}(X, O_{X}). The aim of this paper is to prove the following theorem.

1Faculty of Mathematics, National Research University Higher School of Economics, Russia; Institute for the Physics and Mathematics of the Universe, University of Tokyo (WPI), Japan

Theorem 1.1. Let X be a connected scheme which is separated and of finite type over Spec 𝔽_{q}. Then for j = −1, −2, the pushforward map

α_{X} : H_{−1}^{BM}(X, ℤ(j)) → H_{−1}^{BM}(Spec O(X), ℤ(j))

is an isomorphism if X is proper, and the group H_{−1}^{BM}(X, ℤ(j)) is zero if X is not proper.

Theorem 1.1 is a generalization of a theorem of Akhtar [Ak, Theorem 3.1, p. 285] where the claim is proved for j = −1 and X smooth projective over Spec 𝔽_{q}. Our proof of Theorem 1.1 is independent of [Ak], and we do not require a Bertini-type theorem. If we assume Parshin’s conjecture, then the statement in the theorem holds for any j ≤ −1. Moreover we also obtain H_{i}^{BM}(X, ℤ(j)) = 0 for any i ≤ −2 and j ≤ −1. The method is explained in Section 4 We define the étale Borel-Moore (not motivic) homology with ℤ_{ℓ}-coefficients, where ℓ is a prime different from p, in Remark 4.3. Then we compute it explicitly, and deduce that H_{i}^{BM}(X, ℤ(j)) ⊗_{ℤ} ℤ_{ℓ} ≅ H_{i}^{BM,et}(X, ℤ_{ℓ}(j)) in the range i ≤ −1 and j ≤ −1 (using Parshin’s conjecture where it is needed for the computation of the Borel-Moore motivic homology groups). The original version of this paper was written without using the Bloch-Kato-Milnor conjecture. We use it as a theorem of Rost and Voevodsky. It is used via theorems of Geisser and Levine (e.g., [Ge-Le2, Corollary 1.2, p.56]). Acknowledgment The first author thanks Shuji Saito and Thomas Geisser for valuable comments in the course of revision. The second author would like to thank Akio Tamagawa for helpful suggestions on the proof of a lemma in the earlier version which no longer exists. The authors thank Thomas Geisser and Tohru Korita for pointing out a mistake in a former version of the manuscript. We thank the referee for numerous suggestions which shortened this paper and clarified many things at many points. During this research, the first author was supported as a Twenty-First Century COE Kyoto Mathematics Fellow and was partially supported by JSPS Grantin-Aid for Scientific Research 17740016, 21654002, 25800005 and by World Premier Research Center Initiative (WPI Initiative), MEXT, Japan. The second author was partially supported by JSPS Grant-in-Aid for Scientific Research 16244120, 21540013, and 24540018.

2. Higher Chow groups of smooth curves over a finite field

A curve will mean a scheme of pure dimension one, separated and of finite type over a field. The aim of this section is to compute the higher Chow groups CH^{i}(X, j) for a smooth curve X over a finite field in the range i, j ≥ 0.

Lemma 2.1. Let X be a connected smooth curve over a finite field. Then

CH^{i}(X, 0) ≅ \{ \begin{matrix}ℤ, & i = 0, \\ Pic(X), & i = 1, \\ 0, & i ≥ 2. \end{matrix}

Proof. These are the classical Chow groups and the computation is known. For i ≥ 2, it vanishes by dimension reason. See also [Bl, THEOREM (6.1), p.287]. □

Lemma 2.2. Let j ≥ 2. Let X be a smooth curve over a finite field 𝔽_{q} of characteristic p. Then we have CH^{i}(X, j) = 0 for i > j, and for i ≤ j, the cycle map in [Ge-Le2, (1.2), p.56] gives an isomorphism

CH^{i}(X, j) ≅ ⨁_{ℓ≠p} H_{et}^{2i−j−1}(X, ℚ_{ℓ}/ℤ_{ℓ}(i)).

The right hand side is zero unless 2i − j = 1, 2, 3. If moreover X is affine, the right hand side is zero for 2i − j = 3.

Proof. We first note that CH^{i}(X, j) = 0 if i > j + 1 by dimension reason. Henceforth we consider the case i ≤ j + 1. Recall Bloch’s formula ([Bl, THEOREM(9.1), p.296]):

(2.1)  K_{j}(X) ⊗_{ℤ} ℚ ≅ ⨁_{i≥0} CH^{i}(X, j) ⊗_{ℤ} ℚ

where K_{j} is the j-th algebraic K-group. Recall also Harder’s result (the result [Hard, 3.2.5 Korollar, p.175] is not correctly stated; we refer to [Gr, THEOREM 0.5, p.70] and the remark there for the explanation and the corrected statement) which implies that K_{j}(X) ⊗_{ℤ} ℚ = 0 for j ≥ 2. Hence CH^{i}(X, j) is torsion for j ≥ 2. We recall the definition of motivic cohomology given in [Ge-Le2, Section 2.5, p.60]. For a smooth scheme X over a field, define the cohomological cycle complex by Z^{j}(X, i) = z^{j}(X, 2j − i) where z^{∗}(−, ∗) is Bloch’s cycle complex ([Bl, INTRODUCTION, p.267], see also [Ge-Le2, 2.2, p.58]). Then, for an abelian group A, define H_{M}^{j}(X, A(i)) = H^{i}(Z^{j}(X, ∗) ⊗_{ℤ} A). We have H_{M}^{2i−j}(X, ℤ(i)) = CH^{i}(X, j). The exact sequence 0 → ℤ → ℚ → ℚ/ℤ → 0 gives an exact sequence

\begin{matrix}H_{M}^{2i−j−1}(X, ℚ(i)) → H_{M}^{2i−j−1}(X, ℚ/ℤ(i)) ⟶^{(1)} H_{M}^{2i−j}(X, ℤ(i)) \\ → H_{M}^{2i−j}(X, ℚ(i)). \end{matrix}

The first and the last terms are zero as was remarked above. The map (1) is hence an isomorphism. Since we are in the range i ≤ j+1 (equivalently, 2i−j−1 ≤ i), we apply [Ge-Le2, Corollary 1.2, p.56] and the computation ([Ge-Le1, Theorem 8.4, p.491]) by Geisser and Levine of p-torsion motivic cohomology to obtain

H_{M}^{2i−j−1}(X, ℚ/ℤ(i)) ≅ ⨁_{ℓ≠p} H_{et}^{2i−j−1}(X, ℚ_{ℓ}/ℤ_{ℓ}(i)) ⊕ \underrightarrow{lim}_{r} H^{i−j−1}(X_{Zar}, ν_{r}^{i}).

(We refer to [Ge-Le1] for the definition of H^{∗}(X_{Zar}, ν_{r}^{i}).) One can compute the right hand side explicitly. The p-part is zero since we are in the range i ≤ j + 1 and j ≥ 2.

Set a = 2i − j − 1. Let us show that H_{et}^{a}(X, ℚ_{ℓ}/ℤ_{ℓ}(i)) = 0 if a ≥ 3. If a ≥ 4, it follows from the fact that the cohomological dimension of a curve over a finite field is 3 ([SGA4-3, Exposé X, Corollaire 4.3, p.15]). Suppose a = 3. We have an exact sequence

H_{et}^{3}(X, ℚ_{ℓ}(i)) → H_{et}^{3}(X, ℚ_{ℓ}/ℤ_{ℓ}(i)) → H_{et}^{4}(X, ℤ_{ℓ}(i)).

The third term is zero because of the cohomological dimension reason. The Hochschild-Serre spectral sequence reads

E_{2}^{p,q} = H_{et}^{p}(Gal(\overline{𝔽}_{q}/𝔽_{q}), H_{et}^{q}(\overline{X}, ℚ_{ℓ}(i)) ⇒ H_{et}^{p+q}(X, ℚ_{ℓ}(i))

where \overline{X} = X ×_{Spec 𝔽_{q}} Spec \overline{𝔽}_{q}. We have E_{2}^{0,3} = 0 since H_{et}^{3}(\overline{X}, ℚ_{ℓ}(i)) = 0. To show E_{2}^{1,2} = 0, note that the weight of H_{et}^{2}(\overline{X}, ℚ_{ℓ}(i)) is 2 − 2i. Since j ≥ 2 and a = 3, the weight 2 − 2i is nonzero, hence E_{2}^{1,2} = 0. We have E_{2}^{2,1} = 0 because the cohomological dimension of Gal(\overline{𝔽}_{q}/𝔽_{q}) is one. This proves the claim in this case. Suppose a = 2 and X is affine. We have an exact sequence

H_{et}^{2}(X, ℚ_{ℓ}(i)) → H_{et}^{2}(X, ℚ_{ℓ}/ℤ_{ℓ}(i)) → H_{et}^{3}(X, ℤ_{ℓ}(i)).

The third term is zero since the cohomological dimension of an affine curve over a finite field is 2 ([SGA4-3, Exposé XIV, Théorème 3.1, p.15]). We use the Hochshild-Serre spectral sequence as above. We have E_{2}^{0,2} = 0 using the same cohomological dimension reason. Note that the (possible) weights of H_{et}^{1}(\overline{X}, ℚ_{ℓ}(i)) are 1 − 2i and 2 − 2i. Since neither of them is zero, we have E_{2}^{1,1} = 0. These imply that H_{et}^{2}(X, ℚ_{ℓ}(i)) = 0 hence the claim in this case.

□

Lemma 2.3. Let X be a smooth curve over a finite field. We have

CH^{i}(X, 1) = \{ \begin{matrix}0, & i = 0, \\ O(X)^{×}, & i = 1, \\ 0, & i ≥ 3. \end{matrix}

Proof. The case i = 0 is trivial. The case i = 1 is found in Bloch’s paper ([Bl, THEOREM (6.1), p.287]). For i ≥ 3, the claim follows by dimension reason. □

Lemma 2.4. Let U be an affine smooth curve over a finite field. Then CH^{2}(U, 1) = 0.

Proof. The group SK_{1}(U) sits in the following exact sequence:

0 → SK_{1}(U) → K_{1}(U) → O(U)^{×} → 0.

We use the result [Gr, THEOREM 0.5, p.70], which says that SK_{1}(U)⊗_{ℤ}ℚ = 0 for an affine smooth curve U. Using Lemma 2.3, it follows from counting the dimension of both sides of Bloch’s formula (2.1) that dim_{ℚ} H_{M}^{3}(U, ℚ(2)) = 0. For the rest of the proof, one proceeds as in Lemma 2.2. □

Lemma 2.5. Let Z be a scheme which is finite over Spec 𝔽_{q}. Then we have isomorphisms

H_{−1}^{BM}(Z, ℤ(j)) ≅^{(1)} H_{−1}^{BM}(Z_{red}, ℤ(j))

≅^{(2)} H_{0}^{BM}(Z_{red}, ℚ/ℤ(j)) ≅^{(3)} ⨁_{ℓ≠p} H_{et}^{0}(Z_{red}, ℚ_{ℓ}/ℤ_{ℓ}(−j))

for j ≤ −1, which are functorial with respect to pushforwards. Here Z_{red} denotes the reduced scheme associated to Z.

Proof. For any scheme W of finite type over 𝔽_{q} and an abelian group A, we have H_{i}^{BM}(W, A(j)) ≅ H_{i}^{BM}(W_{red}, A(j)) for any i, j, since the cycle complexes are canonically isomorphic by definition. This gives the isomorphism (1). For (2), we use the long exact sequence of the universal coefficient theorem for higher Chow groups:

H_{0}^{BM}(Z_{red}, ℚ(j)) → H_{0}^{BM}(Z_{red}, ℚ/ℤ(j))
→ H_{−1}^{BM}(Z_{red}, ℤ(j)) → H_{−1}^{BM}(Z_{red}, ℚ(j)).

We know that the higher K-groups of a finite field are torsion from [Qu, THE-OREM 8, p.583]. Then using a formula of Bloch (2.1), we see that the groups in the sequence above with ℚ-coefficient are zero. Since 𝔽_{q} is perfect, Spec Z_{red} is smooth over 𝔽_{q}. The map (3) is the cycle map in [Ge-Le2] (which is defined for smooth schemes over a field). The fact that the cycle map is an isomorphism follows from [Ge-Le2, Corollary 1.2, p. 56] and [Me-Su, (11.5), THEOREM, p. 328], and [Ge-Le1, Theorem 1.1, p406]. It is clear that the isomorphisms (1) and (2) are functorial with respect to pushforwards. Let Z′ be another scheme which is finite over Spec 𝔽_{q} and let f : Z′ → Z be a morphism over Spec 𝔽_{q}. We prove that the isomorphisms (3) for Z and Z′ are compatible with the pushforward maps with respect to f. We are easily reduced to the case when both Z and Z′ are spectra of finite extensions of 𝔽_{q}. Let Z′′ = Z′ ×_{Z} Z′ and let pr_{1}, pr_{2} : Z′′ → Z′ denote the projections to the first and the second factor, respectively. Then the diagram

H_{0}^{BM}(Z′, ℚ/ℤ(j)) ⟶^{f_{∗}} H_{0}^{BM}(Z, ℚ/ℤ(j))
pr_{2}^{∗}↓  ↓^{f^{∗}}
H_{0}^{BM}(Z′′, ℚ/ℤ(j)) ⟶^{(pr_{1})_{∗}} H_{0}^{BM}(Z′, ℚ/ℤ(j))

is commutative, and a similar commutativity holds for the corresponding étale cohomology groups. Since the pullback map f^{∗} : H_{et}^{0}(Z, ℚ_{ℓ}/ℤ_{ℓ}(−j)) → H_{et}^{0}(Z′, ℚ_{ℓ}/ℤ_{ℓ}(−j)) is injective, it suffices to prove that the isomorphisms (3) for Z′′ and Z′ are compatible with the pushforward maps with respect to pr_{1}. Since Z′′ is isomorphic to the disjoin union of a finite number of copies of Z′, the last claim can be checked easily. The lemma is proved. □

The statement is better understood using étale Borel-Moore homology groups. See Remark 4.3.

Remark 2.6. Suppose X is a connected scheme which is proper over Spec 𝔽_{q}. Then Theorem 1.1 says that the group H_{−1}^{BM}(X, ℤ(j)) is isomorphic to H_{−1}^{BM}(Spec O(X), ℤ(j)). We can then use Lemma 2.5 and compute this group explicitly. The computation of étale cohomology group shows that this group is a cyclic group whose order is

|O(X_{red})|^{−j} − 1

for each j = −1, −2.

Lemma 2.7. Let 𝔽′_{1}, 𝔽′_{2} be two finite extensions of 𝔽_{q} with 𝔽′_{1} ⊂ 𝔽′_{2}. Then for j ≤ −1, the pushforward map H_{−1}^{BM}(Spec 𝔽′_{2}, ℤ(j)) → H_{−1}^{BM}(Spec 𝔽′_{1}, ℤ(j)) is surjective.

Proof. By Lemma 2.5, the cycle class map gives an isomorphism α : H_{−1}^{BM}(Spec 𝔽′_{k}, ℤ(j)) ≅ ⨁_{ℓ≠p} H_{et}^{0}(Spec 𝔽′_{k}, ℚ_{ℓ}/ℤ_{ℓ}(−j)) for k = 1, 2. The cycle class map is compatible with the pushforward by a finite morphism ([Ge-Le2, Lemma 3.5(2), p.69]). Thus the claim follows from the corresponding statement for the étale cohomology groups. (See [So, Lemme 6 iii), p.269] and [So, IV.1.7, p.283].) □

3. Proof of Theorem 1.1

Lemma 3.1. Let X be an integral scheme which is of finite type over Spec 𝔽_{q}. Let 𝔽 be the algebraic closure of 𝔽_{q} in O(X). Then the degree [𝔽 : 𝔽_{q}] divides the degree [κ(x) : 𝔽_{q}] for all closed points x ∈ X_{0}. If moreover X is normal, we have the equality [𝔽 : 𝔽_{q}] = gcd_{x∈X_{0}}[κ(x) : 𝔽_{q}].

Proof. For each x ∈ X_{0}, the composite 𝔽 ↪ O(X) → κ(x), where the second map is induced from the pullback map by the closed immersion, is injective since 𝔽 is a field. Hence [𝔽 : 𝔽_{q}] divides [κ(x) : 𝔽_{q}]. Suppose that X is normal of dimension d. Then X is geometrically irreducible as a scheme over Spec 𝔽. Let d_{0} = (gcd_{x∈X_{0}}[κ(x) : 𝔽_{q}])/[𝔽 : 𝔽_{q}]. Let 𝔽 ⊂ 𝔽_{0} denote an extension of degree d_{0}. The canonical morphism f : X ×_{Spec 𝔽} Spec 𝔽_{0} → X is a finite étale cover in which every closed point of X splits completely. It follows from a Chebotarev density type theorem ([La]; we refer to [Ra, Lemma 1.7, p.98] for the statement which is ready for our use) that f is an isomorphism. Hence d_{0} = 1. This completes the proof. □

Lemma 3.2. Let d ≥ 0 be an integer. Suppose that Theorem 1.1 holds for all connected normal schemes over Spec 𝔽_{q} of dimension d which are not proper over Spec 𝔽_{q}. Then Theorem 1.1 holds for all connected normal schemes of dimension d which are proper over Spec 𝔽_{q}.

Proof. Let X be a connected normal scheme of dimension d which is proper over Spec 𝔽_{q}. Let j ∈ {−1, −2}. Let x ∈ X_{0} be a closed point. The pushforward map α_{X} in the statement of Theorem 1.1 is surjective since its composite with the pushforward map ι_{x∗} : H_{−1}^{BM}(Spec κ(x), ℤ(j)) → H_{−1}^{BM}(X, ℤ(j)) is surjective by Lemma 2.7. By assumption, the group H_{−1}^{BM}(X \ {x}, ℤ(j)) is zero. Hence the localization sequence shows that the pushforward map ι_{x∗} is surjective. This implies that H_{−1}^{BM}(X, ℤ(j)) is of order dividing gcd_{x∈X_{0}}(|κ(x)|^{−j} − 1) = q^{(−j)·gcd_{x∈X_{0}} [κ(x):𝔽_{q}]} − 1. This equals q^{[𝔽:𝔽_{q}]} by Lemma 3.1 where 𝔽 is the algebraic closure of 𝔽_{q} in O(X). We know the order of the target of α_{X} is also equal to this value from Lemma 2.5. Hence the bijectivity of α_{X} follows. □

Lemma 3.3. Let S_{1} ⟵^{α_{1}} S_{3} ⟶^{α_{2}} S_{2} be a diagram of sets and let R be an integral domain. For i = 1, 2, 3, let Map(S_{i}, R) denote the R-module of Rvalued functions on S_{i}. Then the cokernel of the homomorphism

β : Map(S_{1}, R) ⊕ Map(S_{2}, R) → Map(S_{3}, R)

which sends (f_{1}, f_{2}) to f_{1} ∘ α_{1} − f_{2} ∘ α_{2} is R-torsion free.

Proof. Let e : Map(S_{3}, R) → Coker β denote the quotient map. Let f ∈ Map(S_{3}, R) and suppose that e(f) is an R-torsion element in Coker β. We prove that e(f) = 0. Since e(f) is an R-torsion element, there exist a non-zero element a ∈ R and an element (f_{1}, f_{2}) ∈ Map(S_{1}, ℤ) ⊕ Map(S_{2}, ℤ) satisfying af = f_{1} ∘ α_{1} − f_{2} ∘ α_{2}. Let us take a complete set T ⊂ R of representatives of R/aR. For i = 1, 2, let \overline{f}_{i} denote the unique T -valued function on S_{i} satisfying \overline{f}_{i}(x) ≡ f_{i}(x) mod aR for every x ∈ S_{i}. We then have

\overline{f}_{1} ∘ α_{1} ≡ f_{1} ∘ α_{1} ≡ f_{2} ∘ α_{2} ≡ \overline{f}_{2} ∘ α_{2}

modulo aMap(S_{3}, R). Since both \overline{f}_{1} ∘ α_{1} and \overline{f}_{2} ∘ α_{2} are T -valued functions, we have \overline{f}_{1} ∘ α_{1} = \overline{f}_{2} ∘ α_{2}. For i = 1, 2, let g_{i} denote the unique R-valued function on S_{i} satisfying f_{i} = \overline{f}_{i} + ag_{i}. Then

af = (\overline{f}_{1} + ag_{1}) ∘ α_{1} − (\overline{f}_{2} + ag_{2}) ∘ α_{2} = a(g_{1} ∘ α_{1} − g_{2} ∘ α_{2}).

Since Map(S_{3}, R) is R-torsion free, we have f = g_{1} ∘ α_{1} − g_{2} ∘ α_{2}. This shows that e(f) = 0, which proves the claim. □

Lemma 3.4. Let d ≥ 0 be an integer. Suppose that Theorem 1.1 holds for all connected schemes of dimension smaller than d which are proper over Spec 𝔽_{q} and for all connected normal schemes over Spec 𝔽_{q} of dimension d. Then Theorem 1.1 holds for all connected schemes of dimension d which are proper over Spec 𝔽_{q}.

Proof. Let X be a connected scheme of dimension d which is proper over Spec 𝔽_{q}. Without loss of generality we may assume that X is reduced. Suppose that X is not normal. Let π : X′ → X denote the normalization of X. The scheme X′ is proper over Spec 𝔽_{q} since π is finite by [EGAII, Remarque 6.3.10, p. 120]. Take a reduced closed subscheme Y ⊂ X of dimension less than that of X such that X \ Y is normal and set Y′ = (Y ×_{X} X′)_{red}. By assumption, Theorem 1.1 holds for each connected component of X \ Y, X′, Y and Y′.

Let j ∈ {−1, −2}. Let us consider the commutative diagram

H_{−1}^{BM}(Y, ℤ(j))  ⟶^{β}  H_{−1}^{BM}(X, ℤ(j))
α_{Y} ↓≅  ^{α_{X}} ↓
H_{−1}^{BM}(Spec O(Y), ℤ(j)) ⟶^{γ} H_{−1}^{BM}(Spec O(X), ℤ(j))

where all the homomorphisms are pushforwards. Since α_{Y} is an isomorphism and we know that γ is surjective using Lemma 2.5 and Lemma 2.7, the homomorphism α_{X} is surjective. Since H_{−1}^{BM}(X \ Y, ℤ(j)) is zero, the localization sequence shows that the map β is surjective. We use the following notation for short: for a scheme Z, we denote Spec O(Z)_{red} by a(Z).

Lemma 3.5. The diagram

H_{et}^{0}(a(Y′), ℚ_{ℓ}/ℤ_{ℓ}(−j)) ⟶ H_{et}^{0}(a(X′), ℚ_{ℓ}/ℤ_{ℓ}(−j))
↓  ↓
H_{et}^{0}(a(Y), ℚ_{ℓ}/ℤ_{ℓ}(−j)) ⟶ H_{et}^{0}(a(X), ℚ_{ℓ}/ℤ_{ℓ}(−j)),

where the arrows are the pushforward homomorphisms, is cocartesian for every prime number ℓ ≠ p.

Proof. Let \overline{X} = X ×_{Spec 𝔽_{q}} Spec \overline{𝔽}_{q} and define \overline{Y}, \overline{X}′ and \overline{Y}′ in a similar manner. By [EGAIII-I, (1.4.16.1), p.94], we have a(\overline{X}) = a(X)×_{Spec 𝔽_{q}} Spec \overline{𝔽}_{q} and similarly for Y, X′, and Y′. Since j ≠ 0, the weight argument shows that the Gal(\overline{𝔽}_{q}/𝔽_{q})-coinvariants of any quotient Gal(\overline{𝔽}_{q}/𝔽_{q})-module and of any divisible Gal(\overline{𝔽}_{q}/𝔽_{q})-submodule of H_{et}^{0}(a(\overline{Y}′), ℚ_{ℓ}/ℤ_{ℓ}(−j)) vanish. Hence it suffices to show that the diagram

H_{et}^{0}(a(\overline{Y}′), ℚ_{ℓ}/ℤ_{ℓ}(−j)) ⟶ H_{et}^{0}(a(\overline{X}′), ℚ_{ℓ}/ℤ_{ℓ}(−j))
↓  ↓
H_{et}^{0}(a(\overline{Y}), ℚ_{ℓ}/ℤ_{ℓ}(−j)) ⟶ H_{et}^{0}(a(\overline{X}), ℚ_{ℓ}/ℤ_{ℓ}(−j))

is cocartesian in the category of Gal(\overline{𝔽}_{q}/𝔽_{q})-modules and that the kernel of the homomorphism H_{et}^{0}(a(\overline{Y}′), ℚ_{ℓ}/ℤ_{ℓ}(−j)) → H_{et}^{0}(a(\overline{X}′), ℚ_{ℓ}/ℤ_{ℓ}(−j)) ⊕ H_{et}^{0}(a(\overline{Y}), ℚ_{ℓ}/ℤ_{ℓ}(−j)) is divisible. By taking the Pontryagin dual, we prove that the diagram

H_{et}^{0}(a(\overline{X}), ℤ_{ℓ}(j)) ⟶ H_{et}^{0}(a(\overline{X}′), ℤ_{ℓ}(j))
↓  ↓
H_{et}^{0}(a(\overline{Y}), ℤ_{ℓ}(j)) ⟶ H_{et}^{0}(a(\overline{Y}′), ℤ_{ℓ}(j)),

where the arrows are the pullback homomorphisms, is cartesian in the category of Gal(\overline{𝔽}_{q}/𝔽_{q})-modules and that the cokernel of the homomorphism

(3.1)  H_{et}^{0}(a(\overline{X}′), ℤ_{ℓ}(j)) ⊕ H_{et}^{0}(a(\overline{Y}), ℤ_{ℓ}(j)) → H_{et}^{0}(a(\overline{Y}′), ℤ_{ℓ}(j))

is torsion free. Let Z be a scheme which is of finite type over 𝔽_{q}. Let us write \overline{Z} = Z ×_{Spec 𝔽_{q}} Spec \overline{𝔽}_{q}. Then we have an isomorphism

(3.2)  H_{et}^{0}(a(\overline{Z}), ℤ_{ℓ}(j)) ≅ Map(π_{0}(\overline{Z}), ℤ_{ℓ}) ⊗_{ℤ_{ℓ}} ℤ_{ℓ}(j)

of Gal(\overline{𝔽}_{q}/𝔽_{q})-modules, which is functorial with respect to pullbacks. It then follows from Lemma 3.3 that the homomorphism (3.1) has a torsion free cokernel. Hence it suffices to show that the diagram

(3.3)  \begin{matrix}π_{0}(\overline{X}) & ⟵ & π_{0}(\overline{X}′) \\ ↑ & & ↑^{φ} \\ π_{0}(\overline{Y}) & ⟵^{ψ} & π_{0}(\overline{Y}′) \end{matrix}

is cocartesian in the category of sets. As X′ → X is a normalization, it is surjective. As surjectivity is preserved under base change, the map \overline{Y}′ → \overline{Y} is surjective, hence ψ is surjective. This implies that the pushout of the diagram

π_{0}(\overline{X}′) ⟵^{φ} π_{0}(\overline{Y}′) ⟶^{ψ} π_{0}(\overline{Y})

is isomorphic to the quotient of π_{0}(\overline{X}′) by the following equivalence relation. We define a binary relation ∼ on π_{0}(\overline{X}′) as follows. We say x′_{1} ∼ x′_{2} if there exist y′_{1}, y′_{2} ∈ π_{0}(\overline{Y}′) such that x′_{1} = φ(y′_{1}), x′_{2} = φ(y′_{2}), and ψ(y′_{1}) = ψ(y′_{2}). We also write ∼ for the equivalence relation on π_{0}(\overline{X}′) generated by the binary relation above. Let us show that the map ϕ : π_{0}(\overline{X}′)/ ∼→ π_{0}(\overline{X}) obtained from the diagram (3.3) is an isomorphism. As the étale base change of a normalization, \overline{X}′ → \overline{X} is a normalization. Hence π_{0}(\overline{X}′) coincides with the set of irreducible components of \overline{X}. As a normalization is a surjective morphism, the map ϕ is surjective. Let C_{1}, C_{2} be two distinct irreducible components of \overline{X}. We claim that if C_{1} ∩ C_{2} ≠ ∅ then the classes of C_{1} and C_{2} in π_{0}(\overline{X}′)/ ∼ coincide. Let y ∈ C_{1} ∩ C_{2}. Then the local ring O_{\overline{X},y} is not an integral domain. Since we chose Y

so that X \ Y is normal, y belongs to \overline{Y}. One can take y_{1}, y_{2} ∈ \overline{X}′ lying over y such that y_{i} lies in the same connected component as C_{i} for each i = 1, 2. Note that y_{1}, y_{2} ∈ \overline{Y}′ since they both lie over y ∈ \overline{Y}. Then using the definition of the equivalence relation above for y_{1} and y_{2}, we see that C_{1} ∼ C_{2}. Let C′_{1} and C′_{2} be two irreducible components of \overline{X}. It follows from the discussion above that if they belong to the same connected component, then C′_{1} ∼ C′_{2}. This implies the injectivity of ϕ. This proves the claim. □

We return to the proof of Lemma 3.4. It follows from Lemma 2.5 and Lemma 3.5 that the diagram

H_{−1}^{BM}(Spec O(Y′), ℤ(j)) ⟶ H_{−1}^{BM}(Spec O(X′), ℤ(j))
↓  ↓
H_{−1}^{BM}(Spec O(Y), ℤ(j)) ⟶ H_{−1}^{BM}(Spec O(X), ℤ(j))

is cocartesian. We saw that γ is surjective. Taking a lift and composing with β∘(α_{Y})^{−1} we obtain a map H_{−1}^{BM}(Spec O(X), ℤ(j)) → H_{−1}^{BM}(X, ℤ(j)). The fact that the diagram above is cocartesian and some diagram chasing imply that this map does not depend on the choice of a lift and this map is a homomorphism. We then see that the homomorphism β factors through the homomorphism

H_{−1}^{BM}(Y, ℤ(j)) ⟶^{α_{Y}} H_{−1}^{BM}(Spec O(Y), ℤ(j))

⟶^{γ} H_{−1}^{BM}(Spec O(X), ℤ(j)).

This proves that the order of H_{−1}^{BM}(X, ℤ(j)) divides the order of

H_{−1}^{BM}(Spec O(X), ℤ(j)).

Hence α_{X} is an isomorphism. This completes the proof. □

Lemma 3.6. Let U be a nonempty open subscheme of a separated connected scheme V over Spec 𝔽_{q} such that V \ U ≠ ∅. Then U is not proper over Spec 𝔽_{q}.

Proof. As V is separated, the diagonal Δ ⊂ V ×_{Spec 𝔽_{q}} V is closed, hence the restriction Δ∩(U ×_{Spec 𝔽_{q}} V) ⊂ U ×_{Spec 𝔽_{q}} V is closed. The image of this closed set under the second projection U ×_{Spec 𝔽_{q}} V → V is U, hence it is not closed in V since V is connected. This shows the structure map U → Spec 𝔽_{q} is not universally closed, hence it is not proper. □

Proof of Theorem 1.1. First suppose d = 1. The claim for X normal and nonproper follows from Lemmas 2.4 and 2.2. Then the claim for X proper follows from Lemmas 3.2 and 3.4. Let us prove the claim for non-proper X. We use induction on the number of irreducible components n of X. Suppose n = 1. We may without loss of generality assume X is reduced so that X is integral. Take an open immersion from X to a connected scheme X′ of dimension one which is proper over Spec 𝔽_{q} such that the complement X′ \ X is zero dimensional. Let j ∈ {−1, −2}. We have proved that the pushforward map H_{−1}^{BM}(X′, ℤ(j)) → H_{−1}^{BM}(Spec O(X′), ℤ(j)) is an isomorphism. This implies, using Lemma 2.7, that the pushforward map H_{−1}^{BM}(X′ \ X, ℤ(j)) → H_{−1}^{BM}(X′, ℤ(j)) is surjective. Hence, by the localization sequence, we have H_{−1}^{BM}(X, ℤ(j)) = 0. Suppose n ≥ 2. We take a (non-empty) zero dimensional closed subscheme Y ⊂ X such that X \ Y = X_{1} ∐ ⋯ ∐ X_{r} (disjoint union of schemes) with the following properties:

(1) X_{i} is a connected one dimensional open subscheme of X, (2) the number of irreducible components of X_{i} is less than n,

(3) the closure \overline{X}_{i} of X_{i} in X equals X_{i} ∪ Y, for each 1 ≤ i ≤ r. We can for example take as Y the following scheme. Let Y_{0} be a zero dimensional subscheme of X such that the complement X \ Y_{0} is not connected. We order the set of such Y_{0}’s by inclusion, and let Y be a minimal one with respect to this ordering. Let us check the properties (1)(2)(3). Let {X_{i}}_{1≤i≤r} be the set of connected components of X \ Y then (1) holds true. We have r ≥ 2 by construction. Since the number of irreducible components of X equals the sum of the number of irreducible components of the X_{i}’s, the property (2) holds true. The closure \overline{X}_{i} of X_{i} in X is contained in X_{i} ∪ Y by construction. Suppose \overline{X}_{i} ≠ X_{i} ∪ Y for some 1 ≤ i ≤ r. Let y ∈ (X_{i} ∪ Y) \ \overline{X}_{i}. Then the minimality condition on the construction of Y implies that X \ (Y \ {y}) = (X_{1} ∐ ⋯ ∐ X_{r}) ∪ {y} is connected. This implies in particular that y ∈ \overline{X}_{i}, which is a contradiction, so (3) holds true. Taking U = X_{i} and V = \overline{X}_{i} in Lemma 3.6, we see that X_{i} is not proper. By the non-properness of X (and changing the indexing) we may suppose that \overline{X}_{1} is not proper. The localization sequence gives the exact sequence

H_{0}^{BM}(X_{1}, ℤ(j)) ⟶^{φ} H_{−1}^{BM}(Y, ℤ(j)) → H_{−1}^{BM}(\overline{X}_{1}, ℤ(j)).

By the inductive hypothesis, we have H_{−1}^{BM}(\overline{X}_{1}, ℤ(j)) = 0, hence φ is a surjection. Now use the following localization sequence

⨁_{i=1}^{r} H_{0}^{BM}(X_{i}, ℤ(j)) ⟶^{ψ} H_{−1}^{BM}(Y, ℤ(j)) → H_{−1}^{BM}(X, ℤ(j))

→ ⨁_{i=1}^{r} H_{−1}^{BM}(X_{i}, ℤ(j)).

Since φ is surjective, ψ is surjective. By the inductive hypothesis, ⨁_{i=1}^{r} H_{−1}^{BM}(X_{i}, ℤ(j)) = 0. It follows that H_{−1}^{BM}(X, ℤ(j)) = 0. The claim is proved in the case d = 1. Next suppose that d ≥ 2 and X is affine. Let j ∈ {−1, −2}. The localization sequence gives an exact sequence

\underrightarrow{lim}_{Y} H_{−1}^{BM}(Y, ℤ(j)) → H_{−1}^{BM}(X, ℤ(j))
→ \underrightarrow{lim}_{Y} H_{−1}^{BM}(X \ Y, ℤ(j)),

where Y runs over the reduced closed subschemes of X of pure codimension one. For dimension reasons, we have \underrightarrow{lim}_{Y} H_{−1}^{BM}(X \ Y, ℤ(j)) = 0. Hence by induction on d, we have H_{−1}^{BM}(X, ℤ(j)) = 0. Next suppose that d ≥ 2 and X is not proper. Using a similar argument as above, we are reduced, by induction on the number of irreducible components of X, to the case where X is integral. Take an open immersion from X to a connected scheme X′ of dimension d, which is proper over Spec 𝔽_{q}, such that X is dense in X′. Take a non-empty affine open subscheme U ⊂ X and set Y = X′ \ U. Let us take an algebraic closure \overline{𝔽}_{q} of 𝔽_{q}. By [Hart, Chapter II, Proposition 3.1, p. 66] and [Hart, Chapter II, Proposition, p. 67] (originally due to [Go]), for each irreducible component X′′ of X′ ×_{Spec 𝔽_{q}} Spec \overline{𝔽}_{q}, we know that X′′ \ U ×_{Spec 𝔽_{q}} Spec \overline{𝔽}_{q} is connected and is of pure codimension one in X′. This shows that Y is of pure codimension one in X′. Let us show that Y is connected. Write f: X′ ×_{Spec 𝔽_{q}} Spec \overline{𝔽}_{q} → X′ for the canonical projection. We note that the map f is surjective, and, as the canonical morphism Spec \overline{𝔽}_{q} → Spec 𝔽_{q} is universally closed by [EGAII, Proposition (6.1.10)], the map f is a closed map. Let ξ ∈ X denote the generic point of X. As X is dense in X′, the closure of ξ in X′ equals X′. Take ξ′ ∈ f^{−1}(ξ) and let X′′ be an irreducible component of X′ ×_{Spec 𝔽_{q}} Spec \overline{𝔽}_{q} that contains ξ′. Using that an irreducible component is closed, we see that X′′ contains the closure in X′ ×_{Spec 𝔽_{q}} Spec \overline{𝔽}_{q} of ξ′. Then as f is a closed map, the morphism f|_{X′′} : X′′ → X′ is surjective. Using the fact above by Goodman, we have that X′′ \U ×_{Spec 𝔽_{q}} Spec \overline{𝔽}_{q} is connected. Then as X′′ \(U ×_{Spec 𝔽_{q}} Spec \overline{𝔽}_{q}) surjects onto X′ \ U = Y, we have that Y is connected as the continuous image of a connected space. Write X ∩ Y = Z_{1} ∐ ⋯ ∐ Z_{r} so that each Z_{i} is connected. We claim that each Z_{i} is not proper. As X ⊂ X′ is an open subset, X ∩ Y ⊂ Y is an open subset of Y, hence each Z_{i} ⊂ Y is an open subset of Y. As Y is connected, \overline{Z}_{i}^{Y} ≠ Z_{i} where \overline{Z}_{i}^{Y} denotes the closure of Z_{i} in Y. This implies that Z_{i} is the complement of a non-empty closed set, namely \overline{Z}_{i}^{Y} \ Z_{i}, of a connected proper scheme \overline{Z}_{i}^{Y}. It follows from Lemma 3.6 that Z_{i} is not proper. Let j ∈ {−1, −2}. Since U is affine, from the localization sequence

H_{−1}^{BM}(Y ∩ X, ℤ(j)) → H_{−1}^{BM}(X, ℤ(j)) → H_{−1}^{BM}(U, ℤ(j))

it follows by induction on d that H_{−1}^{BM}(X, ℤ(j)) is zero (to remove the hypothesis that the schemes in the localization sequence are quasi-projective, we refer to [Le2, Theorem 1.7, p. 301] and [Ge-Le2, 2.6, p. 60]). This proves the claim for X not proper. The claim for X proper follows from Lemmas 3.2 and 3.4. This completes the proof. □

4. Under Parshin’s conjecture

We assume Parshin’s conjecture in this section and draw some consequences. Parshin’s conjecture states that for any projective smooth scheme Z over a finite field, H_{M}^{a}(Z, ℚ(b)) = 0 unless a = 2b. We note that it is a theorem of Harder for curves (we refer to [Gr, THEOREM 0.5, p.70] for the correct implication of Harder’s result).

Proposition 4.1. Assume that Parshin’s conjecture holds. Then the statement in Theorem 1.1 holds true for any j ≤ −1. We also have H_{i}^{BM}(X, ℤ(j)) = 0 for i ≤ −2 and j ≤ −1.

We begin with a lemma.

Lemma 4.2. (1) Let U be a connected scheme of pure dimension d ≥ 1 over 𝔽_{q}. Then \underrightarrow{lim}_{V} H_{i}^{BM}(V, ℤ(j)) = 0, where V runs over the (non-empty) open subschemes of U, for i ≤ −1 and j ≤ −1 assuming Parshin’s conjecture.

(2) Let V be a zero dimensional scheme over 𝔽_{q}. Then H_{i}^{BM}(V, ℤ(j)) = 0 for i ≤ −2 and j ≤ −1 assuming Parshin’s conjecture.

Proof. Let K denote the function field of U. If d > i − j, then H_{i}^{BM}(V, ℤ(j)) = CH_{j}(V, i−2j) = CH^{d−j}(V, i−2j). So the limit is CH^{d−j}(Spec K, i−2j), which equals zero by dimension reason. Suppose d ≤ i − j. Let V be an open smooth subscheme of U. We proceed as in the proof of Lemma 2.2. We have

H_{i}^{BM}(V, ℤ(j)) = H_{M}^{2d−i}(V, ℤ(d − j)) = H_{M}^{2d−i−1}(V, ℚ/ℤ(d − j)),

where the second equality follows from [Ge, Theorem 4.7 ii), p.312] (this uses Parshin’s conjecture). Since we are in the range d ≤ i − j, we can use [Ge-Le2, Corollary 1.2, p.56] and [Ge-Le1, Theorem 8.4, p.491] to see that the quantity above is isomorphic to ⨁_{ℓ≠p} H_{et}^{2d−i−1}(V, ℚ_{ℓ}/ℤ_{ℓ}(d − j)) ⊕ \underrightarrow{lim}_{r} H^{d+j−i−1}(X_{Zar}, ν_{r}^{d−j}). The p-part is zero since we are in the range d ≤ i − j. We may assume that V is affine. Then H_{et}^{2d−i−1}(V, ℚ_{ℓ}/ℤ_{ℓ}(d − j)) = 0 for d − 3 ≥ i since the cohomological dimension of V is d + 1 ([SGA4-3, Exposé XIV, Théorème 3.1, p.15]). Suppose d ≥ 2. Then the claim follows from this immediately since i ≤ −1. Suppose d = 1. The vanishing H_{et}^{2}(V, ℚ_{ℓ}/ℤ_{ℓ}(1 − j)) = 0 can be shown using the same method as in the proof of Lemma 2.2. This proves (1). Let V be as in (2). The remaining case is i = −2. We proceed as in the proof of Lemma 2.2. We have an exact sequence

H_{et}^{1}(V, ℚ_{ℓ}(−j)) → H_{et}^{1}(V, ℚ_{ℓ}/ℤ_{ℓ}(−j)) → H_{et}^{2}(V, ℤ_{ℓ}(−j)).

The third term is zero since V is zero dimensional. We use the Hochschild-Serre spectral sequence as before. We have E_{2}^{0,1} = E_{2}^{1,0} = 0 using the weight argument. The claim then follows. This completes the proof. □

Proof of Proposition 4.1. Suppose i = −1. For the proof of Theorem 1.1 to work for general i, we need as an input the vanishing of \underrightarrow{lim}_{Y} H_{−1}^{BM}(X \ Y, ℤ(j)) (the notation as in the proof of Theorem 1.1), and this is all that we need. As we have seen in Lemma 4.2 above, the vanishing holds under Parshin’s conjecture, hence the claim follows if i = −1. Suppose i ≤ −2. We show by induction on the dimension d that H_{i}^{BM}(X, ℤ(j)) = 0. The case d = 0 is Lemma 4.2(2). Consider the exact sequence

\underrightarrow{lim}_{Y} H_{i}^{BM}(Y, ℤ(j)) → H_{i}^{BM}(X, ℤ(j)) → \underrightarrow{lim}_{Y} H_{i}^{BM}(X \ Y, ℤ(j))

where Y runs over closed subschemes of X. By the inductive hypothesis, the first term is zero, and the third term is zero by Lemma 4.2(1). □

For fixed (i, j), we only need to assume Parshin’s conjecture for projective smooth schemes of dimension (less than or equal to) i − j. Theorem 1.1 treats the cases (i, j) = (−1, −1) and (−1, −2). Hence we can use Harder’s result and need not assume Parshin’s conjecture.

Remark 4.3. For this remark, we do not use Parshin’s conjecture. Let us define and compute the étale Borel-Moore (not motivic) homology groups for a scheme X separated and of finite type over 𝔽_{q} in the same range as that of Proposition 4.1. We will see that the étale Borel-Moore homology groups and the Borel-Moore motivic homology groups are isomorphic in this range. Let ℓ be a prime number prime to p. We define the étale Borel-Moore homology group to be

H_{i}^{BM,et}(X, ℤ/ℓ^{n}(j)) = H_{et}^{−i}(X, Rf^{!}(ℤ/ℓ^{n}(−j)))

for i, j ∈ ℤ and n ≥ 1. Since this is isomorphic to

Hom_{ℤ/ℓ^{n}}(H_{et,c}^{i+1}(X, ℤ/ℓ^{n}(−j)), ℤ/ℓ^{n}),

we set

H_{i}^{BM,et}(X, ℤ_{ℓ}(j)) = Hom_{ℚ_{ℓ}/ℤ_{ℓ}}(H_{et,c}^{i+1}(X, ℚ_{ℓ}/ℤ_{ℓ}(−j)), ℚ_{ℓ}/ℤ_{ℓ}).

Then it is easy to see that a statement similar to the one in Proposition 4.1 holds for étale Borel-Moore (not motivic) homology groups with ℤ_{ℓ}-coefficient. Namely, we have H_{i}^{BM,et}(X, ℤ_{ℓ}(j)) = 0 for i ≤ −2, and, for X connected and for i = −1, the pushforward by the structure morphism is an isomorphism if X is proper, and H_{−1}^{BM,et}(X, ℤ_{ℓ}(j)) = 0 otherwise.

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Satoshi Kondo Faculty of Mathematics National Research University Higher School of Economics 7 Vavilova Str. Moscow 117312 Russia skondo@hse.ru Institute for the Physics and Mathematics of the University of Tokyo (WPI) Japan Seidai Yasuda Department of Mathematics Graduate School of Science Osaka University Toyonaka, Osaka 560-8502 JAPAN s-yasuda@math.sci.osaka-u.ac.jp