Motivic Homology of Semiabelian Varieties

Rin Sugiyama

Received: April 7, 2014 Revised: August 9, 2014

Communicated by Takeshi Saito

Abstract. We generalize some classical results on the Chow group of an abelian variety to semiabelian varieties and to motivic (co)homology, using a result of Ancona–Enright-Ward–Huber [1] on a decomposition of the motive of a semiabelian variety in Voevodsky’s category of motives.

2010 Mathematics Subject Classification: 19E15,14F42

Keywords and Phrases: semiabelian variety, motivic homology, algebraic cycles

1 Introduction

Let k be a field. Let G be a semiabelian variety over k. By definition, G is an extension of an abelian variety A of dimension g by a torus T of rank r, i.e. there is the following exact sequence of smooth group schemes over k:

1 → T → G → A → 1.  (1.1)

We say that G is of rank r if the rank of T is r. A semiabelian variety of rank zero means an abelian variety. In this paper, we study structure of motivic homology H_{p}(G, ℚ(q)) (cf. Notation) and Bloch’s higher Chow group CH^{q}(G, s; ℚ) ([6]). For an abelian variety A, Beauville [2] proved the following theorem (cf. [9, 18]):

Theorem 1.1 (Beauville [2]). Let A be an abelian variety of dimension g over a field k. Let q be an integer with 0 ≤ q ≤ g. For an integer i, let CH^{q}(G; ℚ)^{(i)} denote the subspace of the Chow group CH^{q}(A; ℚ) :

CH^{q}(A; ℚ)^{(i)} := {α ∈ CH^{q}(A; ℚ) | n_{A}^{∗}α = n^{i}α for all n}.

Here n_{A}^{∗} is the pull-back along the map A ⟶^{×n} A of multiplication by an integer n. Then CH^{q}(A; ℚ)^{(i)} is zero for i ∉ [q, g + q] and the Chow group CH^{q}(A; ℚ) is decomposed as follows: for 0 ≤ q ≤ g,

CH^{q}(A; ℚ) = ⨁_{i=q}^{g+q} CH^{q}(A; ℚ)^{(i)}.

Note that the index is different from the one of Beauville [2] (his notation CH_{s}^{p}(A) corresponds to our CH^{p}(A; ℚ)^{(2p−s)}). Bloch [5] studied an iterated Pontryagin product I_{A}^{∗r} of the kernel I_{A} of the degree map CH_{0}(A) → ℤ for an abelian variety A over an algebraically closed field, and he proved the following theorem:

Theorem 1.2 (Bloch [5]). Let A be an abelian variety of dimension g over an algebraically closed field. Then I_{A}^{∗i} = 0 for i > g.

For a smooth k-scheme X, motivic cohomology H^{p}(X, R(q)) (cf. Notation) is isomorphic to its higher Chow groups for any coefficients R ([26]): for any p, q ∈ ℤ,

H^{p}(X, R(q)) ≃ CH^{q}(X, 2q − p; R).  (1.2)

By Poincaré duality (cf. [11]), for a smooth proper scheme X of pure dimension d, motivic homology agrees with motivic cohomology:

H_{p}(X, R(q)) ≃ H^{2d−p}(X, R(d − q)).  (1.3)

In this paper, we generalize Theorem 1.1 and Theorem 1.2 to semiabelian varieties and to motivic homology and higher Chow groups. Let G be a semiabelian variety over k. For an integer n, let n_{G} denote the map G ⟶^{×n} G of multiplication by n. For an integer i, we define the subgroup CH^{q}(G, s; ℚ)^{(i)} of the higher Chow group CH^{q}(G, s; ℚ) as follows (cf. Notation):

CH^{q}(G, s; ℚ)^{(i)} := {α ∈ CH^{q}(G, s; ℚ) | n_{G}^{∗}α = n^{i}α for all n}.

Here n_{G}^{∗} is the pull-back along n_{G}. Similarly, for an integer i, we define the subgroup H_{p}(G, ℚ(q))_{(i)} of the motivic homology H_{p}(G, ℚ(q)) as follows:

H_{p}(G, ℚ(q))_{(i)} := {α ∈ H_{p}(G, ℚ(q)) | n_{G}_{∗}α = n^{i}α for all n}.

Here n_{G}_{∗} denotes the push-forward map along n_{G}.

Theorem 1.3 (Corollary 3.5). Let G be a semiabelian variety over a perfect field k, which is an extension of an abelian variety of dimension g by a torus of rank r.

(1) We have the following natural decomposition

CH^{q}(G, s; ℚ) = ⨁_{i=j_{0}}^{j_{1}} CH^{q}(G, s; ℚ)^{(i)}

where j_{0} = max{0, q − s} and j_{1} = min{2g + r, g + q}.

In the special case where s = 0 and q ≥ 0, the decomposition is

CH^{q}(G; ℚ) = ⨁_{i=q}^{g+q} CH^{q}(G; ℚ)^{(i)}.

(2) We have the following natural decomposition

H_{p}(G, ℚ(q)) = ⨁_{i=i_{0}}^{i_{1}} H_{p}(G, ℚ(q))_{(i)}

where i_{0} = max{0, q} and i_{1} = min{2g + r, g + r + p − q}.

In the special case where q = 0 and p ≤ g, the decomposition is

H_{p}(G, ℚ) = ⨁_{i=0}^{g+r+p} H_{p}(G, ℚ)_{(i)}

Remark 1.4. (a) For an abelian variety, Theorem 1.3 (1) and (2) are the same by the isomorphism (1.3). The special case in Theorem 1.3 (1) for abelian varieties is Beauville’s result (Theorem 1.1).

(b) Let notation be as in Theorem 1.3. Let q ∈ [0, g] be an integer. Beauville [2] has conjectured that for i > 2q,

CH^{q}(A; ℚ)^{(i)} = 0.

Beauville’s conjecture implies that H_{2q−r}(G, ℚ(q))_{(i)} = 0 for i < 2q − r and that CH^{q}(G; ℚ)^{(i)} = 0 for i > 2q (see Theorem 3.1 (4)).

We denote the 0-th Suslin homology of G with ℤ-coefficient by H_{0}^{S}(G, ℤ). Using Theorem 1.3 (2) in case that p = q = 0 and results on Rojtman’s theorem (see Theorem 4.3) by Spiess–Szamuely [22] and Geisser [13], we obtain Bloch’s result for semiabelian varieties and 0-th Suslin homology:

Corollary 1.5 (Corollary 4.1). Let G be a semiabelian variety over k, which is an extension of an abelian variety of dimension g by a torus of rank r. Let I_{G} be the kernel of the degree map deg : H_{0}^{S}(G, ℤ) → ℤ. Then an iterated Pontryagin product I_{G}^{∗i} is torsion for i > g + r. In particular, if k is an algebraically closed field, then I_{G}^{∗i} = 0 for i > g + r.

Remark 1.6. The subgroups H_{0}(G, ℚ)_{(i)} have a description in terms of Kgroups attached to the semiabelian variety G (see Proposition 4.8).

Remark 1.7. One can remove the assumption that a base field k is perfect in the above theorems, by a result of Cisinski-Déglise on a comparison of Beilinson motives DM_{𝔟} (see [7, §14]) with Voevodsky motives: If k is not perfect, then we take the perfect closure k′ of k. By [7, Proposition 2.1.9, Theorem 14.3.3], the pull-back functor

DM_{𝔟}(k) → DM_{𝔟}(k′)

is an equivalence. By [7, Theorem 16.1.4], the above categories of Beilinson motives are equivalent to DM(k, ℚ) and DM(k′, ℚ) respectively.

Theorem 1.3 is deduced form a result on vanishing of motivic (co)homology (Theorem 3.1) and a description of the motive M(G) in DM_{−,Nis}^{eff}(k, ℚ) in terms of symmetric products:

φ_{G} : M(G) ⟶^{≃} Sym(M_{1}(G))

which is due to Ancona–Enright-Ward–Huber [1] (see Theorem 2.5, Corollary 2.7). Here M_{1}(G) denotes the complex consisting of the homotopy invariant Nisnevich sheaf G̃ := Hom_{Sch/k}(−, G)⊗ℚ with transfers concentrated in degree zero (see Definition 2.1). There are two key ingredients in the proof of vanishing of motivic (co)homology. One is an exact triangle which relates G to a semiabelian variety of rank rk(G)− 1 (Lemma 3.9, Lemma 3.10). The exact triangle allows us to use induction on rank of G. Another is to interpret the Weil-Barsotti formula for an abelian variety in terms of motives in DM_{−,Nis}^{eff}. Using this interpretation, one can generalize Beauville’s result to higher Chow groups, i.e. Theorem 1.3 for an abelian variety.

This paper is organized as follows: In Section 2, we recall the result (Theorem 2.5) on the description of the motive M(G) of a semiabelian variety G in DM_{−,Nis}^{eff}(k, ℚ). We give a consequence of the description which is a decomposition of motivic (co)homology of semiabelian varieties. In Section 3, we state and prove our main result on the vanishing of motivic (co)homology (Theorem 3.1) and a consequence of the main result (Theorem 1.3). We first prove the main result in case of a torus and an abelian variety. Lastly, we prove the general case by induction, using key triangles (Lemma 3.9, Lemma 3.10) in DM_{−,Nis}^{eff}. Lemma 3.10 is proved by applying a result on a filtration on symmetric products ([1, Proposition C.3.4]) to the category ShT_{Nis}(k, ℚ) of Nisnevich sheaves of ℚ-modules with transfers. To apply the result, we need the exactness of the tensor product in ShT_{Nis}(k, ℚ). The exactness is proved in Appendix A of this paper (Proposition A.1). In Section 4, we consider 0-th Suslin homology and prove Corollary 1.5 (Corollary 4.1). We also give a description of rational 0-th Suslin homology of a semiabelian variety G in terms of K-groups attached to G (Proposition 4.8, Remark 4.9). In Appendix A, we make some remarks about the tensor product in the category ShT_{Nis}(k, ℚ). The main part is to show the exactness of the tensor product (Proposition A.1).

Acknowledgements The author expresses his gratitude to Professors Thomas Geisser and Marc Levine for helpful suggestions and comments. He also thanks Giuseppe Ancona for a lot of valuable discussions and comments for the preliminary version of this paper and for Remark 1.7. He thanks Professors Annette Huber and Bruno Kahn for a valuable discussion about Appendix A. Thanks are also due to Federico Binda, Jin Cao, Utsav Choudhury and Bradely Drew for helpful comments. He thanks the referee for his/her careful reading and helpful suggestions. In particular, the suggestions allow him to remove an assumption on the characteristic of k in Corollary 1.5.

Notation

1. Let k be a perfect field. For a commutative ring R, let DM_{−,Nis}^{eff}(k, R) denote Voevodsky’s tensor triangulated category of effective motives over k with R-coefficients (cf. [25]). In case R = ℚ, we simply write DM_{−,Nis}^{eff} for DM_{−,Nis}^{eff}(k, ℚ).

2. Let X be a smooth scheme of finite type over k. Let R be a commutative ring. We denote the motive of X in DM_{−,Nis}^{eff}(k, R) by M(X). For any integers p and q, we denote the motivic (co)homology of X with R-coefficients by

H_{p}(X, R(q)) := Hom_{DM_{−,Nis}^{eff}(k,R)}(R(q)[p], M(X));
H^{p}(X, R(q)) := Hom_{DM_{−,Nis}^{eff}(k,R)}(M(X), R(q)[p]).

3. For an integer i ≥ 0, we define the subgroup H_{p}(G, ℚ(q))^{(i)} (resp. H^{p}(G, ℚ(q))^{(i)}) of H_{p}(G, ℚ(q)) (resp. H^{p}(G, ℚ(q)))) as follows:

H_{p}(G, ℚ(q))_{(i)} := {α ∈ H_{p}(G, ℚ(q)) | n_{G∗}α = n^{i}α for all n},
H^{p}(G, ℚ(q))^{(i)} := {α ∈ H_{p}(G, ℚ(q)) | n_{G}^{∗}α = n^{i}α for all n}.

Here n_{G} denotes the map G ⟶^{×n} G of multiplication by an integer n.

4. For a positive integer n, let Σ_{n} denote the group of permutations on n letters. Let C be a idempotent complete ℚ-linear symmetric tensor category, and let M be an object of C. Then we have a representation Σ_{n} → End(M^{⊗n}), and we let σ_{M} denote the endomorphism of M corresponding to σ ∈ Σ_{n}. We define s_{M}^{n} as follows:

s_{M}^{n} := \begin{cases}id_{�} & if n = 0, \\ \frac{1}{n!} ∑_{σ∈Σ_{n}} σ_{M} & if n > 0. \end{cases}

Here � denotes the unit object of C. Then one can easily see that s_{M}^{n} is idempotent. We define the n-th symmetric product Sym^{n}M of M as follows:

Sym^{n}M := Im(s_{M}^{n}).

This image exists since C is idempotent complete. We denote the canonical projection M^{⊗n} → Sym^{n}M by s_{M}^{n}, and the canonical embedding Sym^{n}M → M^{⊗n} by ι_{M}^{n}.

5. Let F ↪^{f} G be a subobject in C. We define a subobject Fil_{i}^{F} Sym^{n}(G) of Sym^{n}(G) as follows: for an integer i with 0 ≤ i ≤ n,

Fil_{i}^{F} Sym^{n}(G) :=

Im(s_{G}^{n} ∘ (f^{⊗i} ⊗ id_{G}^{⊗(n−i)}) : F^{⊗i} ⊗ G^{⊗(n−i)} → G^{⊗n} → Sym^{n}(G)).

For i > n, put Fil_{i}^{F} Sym^{n}(G) = 0. By definition, there is an inclusion map Fil_{i+1}^{F}Sym^{n}(G) → Fil_{i}^{F} Sym^{n}(G).

2 The motive of a semiabelian variety

2.1 The isomorphism φ_{G}

We recall here a recent result on the decomposition of the motive of semiabelian varieties in DM_{−,Nis}^{eff}(k, ℚ), which is due to Enright-Ward [10]/Ancona– Enright-Ward–Huber [1].

Definition 2.1. (1) Let G be a semiabelian variety over k. We denote by \underline{G} the étale sheaf Hom_{Sch/k}(−, G) associated to G. Then the étale sheaf \underline{G} has a canonical structure of an étale sheaf with transfers, and furthermore \underline{G} is homotopy invariant ([22, proof of Lemma 3.2] [3, Lemma 1.3.2]).

We denote by G̃ the étale sheaf of ℚ-modulus with transfer attached to G, i.e., G̃(S) = \underline{G}(S) ⊗_{ℤ} ℚ = Hom_{Sch/k}(S, G) ⊗_{ℤ} ℚ for any smooth scheme S.

(2) For a homotopy invariant Nisnevich sheaf with transfers F, we denote by F [0] the complex consisting of F concentrated in degree one.

Following [1, 10], for a semiabelian variety G, we define

M_{1}(G) := G̃[0] in DM_{−,Nis}^{eff}.

Construction of φ_{G}: According to Spiess–Szamuely [22], we have morphism ℤ_{tr}(G) → \underline{G} of étale sheaves with transfers, and therefore there is a morphism in DM_{−,Nis}^{eff}:

a_{G} : M(G) → M_{1}(G).

Using this map, we define φ_{G}^{i} : M(G) → Sym^{i}(M_{1}(G)) to be the composite morphism in DM_{−,Nis}^{eff}:

\xymatrix{M(G) \ar[r]^{M(Δ_{G}^{i})} & M(G)^{⊗i} \ar[r]^{a_{G}^{⊗i}} & M_{1}(G)^{⊗i} \ar[r]^{s_{M_{1}(G)}^{i}} & Sym^{i}(M_{1}(G)).}

Proposition 2.2 ([1, Proposition 5.1.1],[10, Lemma 5.7.4]). Let G be a semiabelian variety over k, which is an extension of an abelian variety of dimension g by a torus of rank r. Then Sym^{i}(M_{1}(G)) is zero in DM_{−,Nis}^{eff} for i > 2g + r. In particular, Sym^{sg+r}(M_{1}(G)) ≃ Λ(g+r)[2g+r] with a tensor-invertible Artin motive Λ. If the torus part of G is split, then Λ = ℚ (the unit motive).

Proposition 2.2 for a torus is proved by Biglari [4, Proposition 2.3]. For the general case, by Lemma 3.10, we may assume that G is an abelian variety A. The statement for A is essentially deduced from the oddly finite-dimensionality for the Chow motive h^{1}(A)([17], [18]).

Let Sym(M_{1}(G)) denote the direct sum ⨁_{i≥0} Sym^{i}(M_{1}(G)). Here the sum is

finite from Proposition 2.2. Then we define

φ_{G} := ⨁ φ_{G}^{i} : M(G) → Sym(M_{1}(G)).  (2.1)

The motive M(G) has a canonical Hopf algebra structure defined by morphisms of schemes:

• the multiplication by the group law m_{G} : G × G → G;

• the comultiplication by the diagonal map Δ_{G} : G → G × G;

• the antipodal map by the inverse on G;

• the unity by the neutral element;

• the counit by the structure map G → Spec(k).

Definition 2.3. We define a Hopf algebra structure on Sym(M_{1}(G)) as follows:

(multiplication) For any i, j ≥ 0,

\frac{(i + j)!}{i!j!}s_{M_{1}(G)}^{i+j} ∘ (ι_{M_{1}(G)}^{i} ⊗ ι_{M_{1}(G)}^{j}) :

Sym^{i}(M_{1}(G)) ⊗ Sym^{j}(M_{1}(G)) → Sym^{i+j}(M_{1}(G)); (comultiplication) For any i, j ≥ 0,

(s_{M_{1}(G)}^{i} ⊗ s_{M_{1}(G)}^{j}) ∘ ι_{M_{1}(G)}^{i+j} :
Sym^{i+j}(M_{1}(G)) → Sym^{i}(M_{1}(G)) ⊗ Sym^{j}(M_{1}(G)).

The antipodal map, the unity and the counit are induced by the inverse map, the unit map and the structure map of M_{1}(G).

Remark 2.4. The definition of the bialgebra structure on Sym in the above definition is not the standard one. The coefficient \frac{(i+j)!}{i!j!} is usually used for the comultiplication. The reasons why we use the above definition are to fit into the classical result on motivic decomposition of Chow motive of abelian varieties (cf. [18]) and to make the map (2.1) to be an isomorphism of Hopf algebra objects in DM_{−,Nis}^{eff}. For details see [1].

Theorem 2.5 (Ancona–Enright-Ward–Huber [1, Theorem 7.1.1]). Let G be a semiabelian variety over a perfect field k and let M(G) be the motive of G in DM_{−,Nis}^{eff}(k, ℚ). Then there exists a natural isomorphism of Hopf algebras in DM_{−,Nis}^{eff}(k, ℚ):

φ_{G} : M(G) ⟶^{≃} Sym(M_{1}(G)).

In particular, the following diagrams commute:

\xymatrix{M(G) ⊗ M(G) \ar[d]_{φ_{G}⊗φ_{G}}^{≃} \ar[r]^{M(m_{G})} & M(G) \ar[d]_{≃}^{φ_{G}} \\ Sym(M_{1}(G)) ⊗ Sym(M_{1}(G)) \ar[r] & Sym(M_{1}(G));}  (2.2)

\xymatrix{M(G) \ar[d]_{≃}^{φ_{G}} \ar[r]^{M(Δ_{G})} & M(G) ⊗ M(G) \ar[d]_{φ_{G}⊗φ_{G}}^{≃} \\ Sym(M_{1}(G)) \ar[r] & Sym(M_{1}(G)) ⊗ Sym(M_{1}(G)).}  (2.3)

2.2 Decomposition of motivic (co)homology of a semiabelian variety

We give a decomposition of motivic (co)homology and a relationship between the decomposition and product structure, which is a consequence of Theorem 2.5. We first introduce product on motivic (co)homology.

Definition 2.6. (1) Let a and b be elements in H_{p}(G, ℚ(q)) and H_{p′}(G, ℚ(q′)) respectively. Then we define the Pontryagin product a ∗ b of a and b to be the image of (a, b) under the morphism

H_{p}(G, ℚ(q)) × H_{p′}(G, ℚ(q′)) →H_{p+p′}(G × G, ℚ(q + q′))
→^{m_{G∗}} H_{p+p′}(G, ℚ(q + q′)).  (2.4)

Here the first map is induced by the tensor structure in DM_{−,Nis}^{eff}. For any subgroups F ⊂ H_{p}(G, ℚ(q)) and G ⊂ H_{p′}(G, ℚ(q′)), we write F ∗ G for the subgroup generated by the image of F × G under the morphism (2.4).

(2) For a ∈ H^{p}(G, ℚ(q)) and b ∈ H^{p′}(G, ℚ(q′)), we define the cup product a ∪ b to be the image of (a, b) under the morphism

H^{p}(G, ℚ(q)) × H^{p′}(G, ℚ(q′)) →H^{p+p′}(G × G, ℚ(q + q′))
→^{Δ_{G}^{∗}} H^{p+p′}(G, ℚ(q + q′)).  (2.5)

For any subgroups F ⊂ H^{p}(G, ℚ(q)) and G ⊂ H^{p′}(G, ℚ(q′)), we write F ∪ G for the subgroup generated by the image of F × G under the morphism (2.5).

Corollary 2.7. Let G be a semiabelian variety over k, which is an extension of an abelian variety of dimension g by a torus of rank r. Let p, q be integers. Then we have the following natural decomposition of motivic (co)homology of G:

H_{p}(G, ℚ(q)) = ⨁_{i=0}^{2g+r} H_{p}(G, ℚ(q))_{(i)}, H^{p}(G, ℚ(q)) = ⨁_{i=0}^{2g+r} H^{p}(G, ℚ(q))^{(i)}

which satisfies

H_{p}(G, ℚ(q))_{(i)} ∗ H_{p′}(G, ℚ(q′))_{(i′)} ⊂ H_{p+p′}(G, ℚ(q + q′))_{(i+i′)},
H^{p}(G, ℚ(q))^{(i)} ∪ H^{p′}(G, ℚ(q′))^{(i′)} ⊂ H^{p+p′}(G, ℚ(q + q′))^{(i+i′)}.

Proof. The map n_{G} on Sym^{i}(M_{1}(G)) is n^{i} · id_{Sym^{i}(M_{1}(G))}. Thus we have

H_{p}(G, ℚ(q))_{(i)} = Hom_{DM}{eff \atop −,Nis}(ℚ(q)[p], Sym^{i}(M_{1}(G))),
H^{p}(G, ℚ(q))^{(i)} = Hom_{DM}{eff \atop −,Nis}(Sym^{i}(M_{1}(G)), ℚ(q)[p]).

The assertion follows from this description and Theorem 2.5.

3 Main result

In this section, we state the main result on the vanishing of motivic (co)homology of a semiabelian variety and its consequences.

Theorem 3.1. Let G be a semiabelian variety over k, which is an extension of an abelian variety of dimension g by a torus of rank r. Let p, q be integers. Then

(1) H_{p}(G, ℚ(q)) vanishes in the following cases

(a) p < q; (b) g + r < q;

(c) p − 2q + r < 0;

(d) q = g + r and p − 2q + r ≥ 1;

(e) q = g + r − 1 and p − 2q + r ≥ 2.

(2) Let i_{0} = max{0, q} and i_{1} = min{2g+r, g+r+p−q}. Then H_{p}(G, ℚ(q))_{(i)} vanishes for i ∉ [i_{0}, i_{1}].

(3) Let s ≥ 0 be an integer. Let j_{0} = max{0, q −s} and j_{1} = min{2g +r, g +q}. Then CH^{q}(G, s; ℚ)^{(i)} vanishes for i ∉ [j_{0}, j_{1}].

(4) Beauville’s conjecture implies that H_{2q−r}(G, ℚ(q))_{(i)} = 0 for i < 2q −r and that CH^{q}(G; ℚ)^{(i)} = 0 for i > 2q.

Remark 3.2. By the definition of higher Chow groups and a computation of codimension one cycles (see Bloch [6]), we know the following result on vanishing of higher Chow groups:

Theorem 3.3. For any smooth scheme Xof dimension d over a field k and for any abelian group R, we have CH^{i}(X, s; R) = 0 in the cases:

(1) s < 0; (2) i ∉ [0, d + s]; (3) i = 0 and s ≥ 1; (4) i = 1 and s ≥ 2.

Remark 3.4. For an abelian variety A of dimension g, by Poincaré duality ([11, 25]), we have

M(A) ≃ M(A)^{∗}(g)[2g] in DM_{−,Nis}^{eff}.

Here ∗ denotes the dual in DM_{−,Nis}^{eff}. Thus we have an isomorphism (cf. (1.3))

H_{p}(A, ℚ(q)) ≃ H^{2g−p}(A, ℚ(g − q)) ≃ CH^{g−q}(A, p − 2q; ℚ).

The last isomorphism is the isomorphism (1.2) in the introduction. Furthermore, for 0 ≤ i ≤ 2g, the isomorphism induces an isomorphism

H_{p}(A, ℚ(q))_{(i)} ≃ H^{2g−p}(A, ℚ(g − q))^{(2g−i)} ≃ CH^{g−q}(A, p − 2q; ℚ)^{(2g−i)}.

Therefore Theorem 3.1 (1) is the same as Theorem 3.3, and Theorem 3.1(2) and (3) are the same.

The following corollary immediately follows from Corollary 2.7 and Theorem 3.1.

Corollary 3.5 (Theorem 1.3). Let notation be as in Theorem 3.1. Then the decompositions in Corollary 2.7 are

H_{p}(G, ℚ(q)) = ⨁_{i=i_{0}}^{i_{1}} H_{p}(G, ℚ(q))_{(i)}, CH^{q}(G, s; ℚ) = ⨁_{i=j_{0}}^{j_{1}} CH^{q}(G, s; ℚ)^{(i)}.

3.1 Proof of main result

Let G be a semiabelian variety over k, which is an extension of an abelian variety of dimension g by a torus of rank r.

3.1.1 Reduction to the case of an algebraically closed base field

Let L/k be a finite extension. Let G_{L} denote the scalar extension G ⊗_{k} L and let f_{L/k} : G_{L} → G be the projection. Then the composition of the following homomorphisms is the multiplication by the degree [L : k]:

H_{p}(G, ℚ(q)) ⟶^{f_{L/k}^{∗}} H_{p}(G_{L}, ℚ(q)) ⟶^{f_{L/k}_{∗}} H_{p}(G, ℚ(q)).

Hence we obtain that the push- forward map f_{L/k}^{∗} is injective, and that there is a natural injection

H_{p}(G, ℚ(q)) ↪ colim_{L/k} H_{p}(G_{L}, ℚ(q)).

By a result of Ivorra [14, Proposition 4.16], we have a bijection

colim_{L/k} H_{p}(G_{L}, ℚ(q)) ≃ H_{p}(Gk̄, ℚ(q)).

Here k̄ denotes an algebraic closure of k. Thus we have an injection

H_{p}(G, ℚ(q)) ↪ H_{p}(Gk̄, ℚ(q)).

Note that a similar assertion holds for H_{p}(G, ℚ(q))_{(i)}, CH^{q}(G, s; ℚ) and CH^{q}(G, s; ℚ)^{(i)}. Therefore we may assume that k is algebraic closed.

3.1.2 The case of a torus

Let G = T be a torus of rank r. By the above argument, we may assume that T = 𝔾_{m}^{r}. In this case, we know that

M(𝔾_{m}^{r}) ≃ ⨁_{i=0}^{r} (ℚ(i)[i])^{⊕c_{i}},

where c_{i} = \binom{r}{i} denotes the binomial coefficient. Thus, by isomorphism 2.1, we obtain that for 0 ≤ i ≤ r,

H_{p}(T, ℚ(q))_{(i)} ≃ CH^{i−q}(k, i + p − 2q; ℚ)^{⊕c_{i}}.  (3.1)

From this equation and Theorem 3.3, one can easily obtain the following proposition which induces Theorem 3.1 for a torus.

Proposition 3.6. Let T be the torus 𝔾_{m}^{r} of rank r. Let p, q be integers. Let i be an integer with 0 ≤ i ≤ r. Then H_{p}(T, ℚ(q))_{(i)} vanishes in the following cases: (1) q > p; (2) q > i; (3) i < 2q − p; (4) q = i and p − 2q + i ≥ 1;

(5) q = i − 1 and p − 2q + i ≥ 2.

3.1.3 The case of abelian varieties

Let G be an abelian variety A of dimension g over an algebraically closed field k. By Remark 3.4, it enough to prove Theorem 3.1 (2).

Lemma 3.7. Let Â be the dual abelian variety of A. There is an isomorphism in DM_{−,Nis}^{eff}

M_{1}(Â) ≃ (M_{1}(A))^{∗}(1)[2].

Here ∗ denotes the dual in DM_{−,Nis}^{eff}.

Proof. By the Weil-Barsotti formula, we have an isomorphism

Â ≃ \underline{Ext}^{1}(A, 𝔾_{m}).

The assertion follows from a result of Barbieri-Viale–Kahn ([3, §4]) on a comparison between Cartier dual of 1-motives and motivic dual of corresponding motives.

Remark 3.8. Lemma 3.7 also follows from the Weil-Barsotti formula in Chow motives and Voevodsky’s functor ψ from the category of non-effective Chow motives to the category DM_{gm} of geometric motives deduced from [25, Proposition 2.1.4]. We used here the fact that the functor commutes with duality as a tensor functor, and Proposition 4.3.3 in [1] that ψ sends the Chow motive h^{1}(A) to M_{1}(A).

Proof of Theorem 3.1 (2). By Lemma 3.7 and replacing (A, Â) by (Â, A), for an integer i ≥ 0, we have an isomorphism

Sym^{i}(M_{1}(A)) ≃ (Sym^{i}(M_{1}(Â)))^{∗}(i)[2i].

¿From this isomorphism, we have

H_{p}(A, ℚ(q))_{(i)} = Hom_{DM}{eff \atop −,Nis}(ℚ(q)[p], Sym^{i}(M_{1}(A)))
= Hom_{DM}{eff \atop −,Nis}(ℚ(q)[p], (Sym^{i}(M_{1}(Â)))^{∗}(i)[2i])
= Hom_{DM}{eff \atop −,Nis}(Sym^{i}(M_{1}(A)), ℚ(i − q)[2i − p])
= CH^{i−q}(Â, p − 2q; ℚ)^{(i)}
⊂ CH^{i−q}(Â, p − 2q; ℚ).

¿From Theorem 3.3, CH^{i−q}(Â, p − 2q; ℚ) = 0 for i − q ∉ [0, g + p − 2q]. Thus H_{p}(A, ℚ(q))_{(i)} = 0 for i ∉ [i_{0}, i_{1}], since Sym^{i}(M_{1}(A)) = 0 for i ∉ [0, 2g] by Proposition 2.2.

3.1.4 The general case

We prove Theorem 3.1 by induction on rank of a semiabelian variety G. Following [10], we first give some triangles in DM_{−,Nis}^{eff} which allow us to use induction on the rank of a semiabelian variety. We first give an exact sequence of smooth group schemes. Let G be a semiabelian variety of rank r over an algebraically closed field k. Then there is an exact sequence of smooth group schemes over k of the following form:

1 → 𝔾_{m}^{r} →^{i} G →^{f} A → 1.

Then, following [10, Section 5.2], we consider the cokernel H of the composite map

𝔾_{m} → 𝔾_{m}^{r} →^{i} G,

where the first map is the inclusion to the first factor of 𝔾_{m}^{r}. Then H is a semiabelian variety of rank r − 1 over k which fits into

1 → 𝔾_{m} → G → H → 1,  (3.2)

1 → 𝔾_{m}^{r−1} →^{i} H →^{f̄} A → 1.

Lemma 3.9 ([1, 10]). Let notation as above. Then we have the following exact triangles in DM_{−,Nis}^{eff}:

M(H)(1)[1] → M(G) → M(H) → M(H)(1)[2].

Proof. From (3.2), we may regard G as a 𝔾_{m}-torsor on H. Let E be the line bundle over H associated to the 𝔾_{m}-torsor G. Let s : H → E be the zero section. By the Gysin triangle attached to E and s(H), we have an exact triangle in DM_{−,Nis}^{eff}

M(E − s(H)) → M(E) → M(s(H))(1)[2] → M(E − s(H))[1]

Since E − s(H) is isomorphic to G and M(E) ≃ M(H) ≃ M(s(H)), we get the desired triangle after shifting.

Lemma 3.10 ([1, 10]). Let notation as above. Then we have the following exact triangles in DM_{−,Nis}^{eff}:

Sym^{n−1}(M_{1}(H))(1)[1] → Sym^{n}(M_{1}(G))
→ Sym^{n}(M_{1}(H)) → Sym^{n−1}(M_{1}(H))(1)[2].

Proof. By Corollary A.3, there is an isomorphism in DM_{−,Nis}^{eff}

Sym^{n}(M_{1}(G)) ≃ Sym^{n}(G̃)[0].  (3.3)

For the definition of G̃ and a complex F [0], see Definition 2.1. By the isomorphism (3.3), we may work in the category ShT_{Nis}(k, ℚ) of Nisnevich sheaves of ℚ-modules with transfers. We have an exact sequence in ShT_{Nis}(k, ℚ):

1 → 𝔾̃_{m} → G̃ → H̃ → 1.

Proposition A.1 allows us to apply [1, Proposition C.3.4] to the above exact sequence in ShT_{Nis}(k, ℚ). Then we have exact triangles for i ≥ 0,

Fil𝔾̃_{m} ; i+1Sym^{n}(G̃)[0] → Fil𝔾̃_{m} ; iSym^{n}(G̃)[0]

→ Sym^{i}(M_{1}(𝔾_{m})) ⊗ Sym^{n−i}(M_{1}(H)) → Fil𝔾̃_{m} ; i+1Sym^{n}(G̃)[1].

For the definition of Fil𝔾̃_{m} ; iSym^{n}(G̃), see Notation. We know that M_{1}(𝔾_{m}) ≃ 𝔾̃_{m}[0] ≃ ℚ(1)[1] in DM_{−,Nis}^{eff}. Therefore, by [4, Proposition 2.3], Sym^{n}(M_{1}(𝔾_{m})) ≃ Alt^{n}(ℚ)(n)[n] = 0 for n ≥ 2 in DM_{−,Nis}^{eff}. Thus we have

Fil𝔾̃_{m} ; iSym^{n}(G̃)[0] ≃ \begin{cases}Sym^{n}(M_{1}(G)) & if i = 0, \\ Sym^{n−1}(M_{1}(H))(1)[1] & if i = 1, \\ 0 & otherwise. \end{cases}

Hence we have the desired triangle.

Proof of Theorem 3.1. By Lemma 3.9, we have an exact sequence

⋯ → H_{p−1}(H, ℚ(q − 1)) → H_{p}(G, ℚ(q)) → H_{p}(H, ℚ(q)) → · · · .

By Lemma 3.10, we also have exact sequences

⋯ → H_{p−1}(H, ℚ(q − 1))_{(i−1)} → H_{p}(G, ℚ(q))_{(i)} → H_{p}(H, ℚ(q))_{(i)} → ⋯,

⋯ ← CH^{q−1}(H, 2q − p − 1; ℚ)^{(i−1)} ← CH^{q}(G, 2q − p; ℚ)^{(i)}

← CH^{q}(H, 2q − p; ℚ)^{(i)} ← · · · .

By these exact sequences, the assertion for a semiabelian variety G follows from the induction hypothesis and the assertion for abelian varieties.

4 0-th Suslin homology

We consider here 0-th Suslin homology H_{0}^{S}(G, ℤ) of a semiabelian variety G, and prove Corollary 1.5 (Corollary 4.1). We also give a description of rational 0-th Suslin homology H_{0}^{S}(G, ℚ) in terms of K-groups (Proposition 4.8), using a result of Kahn–Yamazaki [16] (see Theorem 4.5).

4.1 Bloch’s result for a semiabelian variety

Corollary 4.1 (Corollary 1.5). Let G be a semiabelian variety over k, which is an extension of an abelian variety of dimension g by a torus of rank r. Let I_{G} be the kernel of the degree map H_{0}^{S}(G, ℤ) → ℤ. Then an iterated Pontryagin product I_{G}^{∗i} is torsion for i > g + r. In particular, if k is an algebraically closed field, then I_{G}^{∗i} = 0 for i > g + r.

Proof. ¿From Theorem 3.5, we have

I_{G} ⊗ ℚ = ⨁_{n=1}^{g+r} H_{0}(G, ℚ)_{(n)}.

By compatibility of this decomposition with Pontryagin product (Corollary 2.7),

I_{G}^{∗i} ⊗ ℚ ⊂ ⨁_{n=i}^{g+r} H_{0}(G, ℚ)_{(n)}.

Thus by Theorem 3.1 (2), we obtain that I_{G}^{∗i} ⊗ ℚ = 0 for i > g + r. For second assertion, we assume that k is algebraically closed. Then I_{G} is generated by cycles of the form [a] − [0_{G}], where a, 0_{G} ∈ G(k) and 0_{G} is the identity element of G. Thus I_{G}^{∗2} is generated by cycles of the form

[a + b] − [a] − [b] − [0_{G}]

for a, b ∈ G(k). Let alb_{G} be the albanese map from I_{G} → G(k). Then it is easily seen that

alb_{G}(I_{G}^{∗2}) = 0_{G}.

Since I_{G}^{∗i} ⊂ I_{G}^{∗2} for i ≥ 2, we have

alb_{G}(I_{G}^{∗i}) = 0 for i ≥ 2.  (4.1)

We claim that G has a smooth compactification. The following argument is attributed by J.-L. Colliot-Thélène and M. Brion (see also [15, p. 13]): Let 0 → T → G → A → 0 be the canonical decomposition of G. Then we may assume T = 𝔾_{m}^{r} since k is algebraically closed. Let Y := (ℙ^{1})^{r} be a smooth compactification of T. Then the bundle G ×^{T} Y associated to the T -torsor G → A exists, which is a smooth compactification of G. Now the second assertion follows from the first assertion, (4.1) and Theorem 4.3 below.

Remark 4.2. A smooth comapctification of a semiabelian variety over any field exists, since the case of a torus is shown in [8] and the argument in the proof of Corollary 4.1 works.

Theorem 4.3. Let X be a smooth, quasi-projective scheme of finite type over an algebraically closed field k of characteristic p ≥ 0. Assume that X has a smooth compactification. 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 Alb_{X} of X induces an isomorphism on torsion groups.

Remark 4.4. For an open variety of a smooth projective variety, the theorem is proved by Spieß–Szamuely [22, Theorem 1.1] (prime-to-p-part) and Geisser [13, Theorem 1.1] (p-part).

4.2 A description of rational Suslin homology

We first briefly recall the definition of K-groups attached to semiabelian varieties. Let F_{1}, …, F_{r} be homotopy invariant Nisnevich sheaves with transfers. We then define the K-group K(k; F_{1} …, F_{r}) to be the quotient group

( ⨁_{L/k:finite} F_{1}(L) ⊗ ⋯ ⊗ F_{r}(L))/R  (4.2)

where R is a subgroup whose elements corresponds to Projection formula and Weil reciprocity. For example, if F_{i} = \underline{𝔾_{m}} (cf. Definition 2.1 (1)) for i = 1, …, r, then we have the r-th Milnor K-group

K(k; \underline{𝔾_{m}}, …, \underline{𝔾_{m}}) = K_{r}^{M}(k).

For the precise definition, see [21, 16].

Theorem 4.5 (Kahn–Yamazaki [16]). Let F_{1} …, F_{r} be homotopy invariant Nisnevich sheaves with transfers . Then there is an isomorphism

K(k; F_{1} …, F_{r}) ≃ Hom_{DM_{−,Nis}^{eff}(k,ℤ)}(ℤ, F_{1}[0] ⊗ ⋯ ⊗ F_{r}[0]).

Definition 4.6. (1) For a semiabelian variety G over a perfect field k, let K_{i}(k; G)_{ℚ} denotes the K-group K(k; \underline{G}, …, \underline{G}) ⊗ ℚ attached to i copies of the homotopy invariant Nisnevich sheaf with transfers \underline{G} attached to G (cf. Definition 2.1 (1)).

(2) For a_{i} ∈ \underline{G}(L), let \{a_{1}, …, a_{i}\}_{L/k} denote the element of K_{i}(k; G)_{ℚ} represented by a_{1} ⊗ ⋯ ⊗ a_{i}. We define an action of the permutation group Σ_{i} on K_{i}(k; G) as follows: for σ ∈ Σ_{i} and a_{i} ∈ \underline{G}(L),

σ({a_{1}, …, a_{i}}_{L/k}) = {a_{σ(1)}, …, a_{σ(i)}}_{L/k}.

Then we define S_{i}(k; G)_{ℚ} to be the image of an idempotent map s^{i} := \frac{1}{i!} ∑_{σ∈Σ_{i}} σ:

S_{i}(k; G)_{ℚ} := Im(s^{i} : K_{i}(k; G)_{ℚ} → K_{i}(k; G)_{ℚ}).

Example 4.7. For G = 𝔾_{m} and an integer i ≥ 0, we have

K_{i}(k; G) ≃ K_{i}^{M}(k).

Here K_{i}^{M}(k) is the i-th Milnor K-group of k.

Proposition 4.8 (Gazaki [12](for an abelian variety)). Let G be a semiabelian variety over k, which is an extension of an abelian variety by a torus of rank r. Let H_{0}^{S}(G, ℚ)_{(i)} denote H_{0}(G, ℚ(0))_{(i)}. Then for any 0 ≤ i ≤ r, we have

H_{0}^{S}(G, ℚ)_{(i)} ≃ S_{i}(k; G)_{ℚ}.

In particular,

H_{0}^{S}(G, ℚ) ≃ ⨁_{i=0}^{g+r} S_{i}(k; G)_{ℚ}

Proof. ¿From Theorem 4.5, we have the following commutative diagram

[figure: K_{i}(k; G)_{ℚ}; s^{i}; Hom_{DM}{eff \atop −,Nis}(ℚ, M_{1}(G)^{⊗i}); s_{M_{1}(G)}^{i}; ((G̃)^{⊗_{HI}^{i}})(k); si ; G̃; K_{i}(k; G)_{ℚ}; Hom_{DM}{eff \atop −,Nis}(ℚ, M_{1}(G)^{⊗i}); ((G̃)^{⊗_{HI}^{i}})(k).]

Here G̃ denotes the Nisnevich sheaf of ℚ-modules with transfers attached to G (Definition 2.1 (1)). Since H_{0}^{S}(G, ℚ)^{(i)} = Hom_{DM}{eff \atop −,Nis}(ℚ, Sym^{i}(M_{1}(G))), the assertion follows from this diagram.

Remark 4.9. Let notation be as in Propostion 4.8. Let F^{n}H_{0}^{S}(G) denote ⨁_{i=n}^{g+r} S_{i}(k; G)_{ℚ}. Then the filtration F^{•}H_{0}^{S}(G) on H_{0}^{S}(G, ℚ) satisfies the following:

(a) F^{1}H_{0}^{S}(G) = Ker(deg : H_{0}^{S}(G) → ℚ) = I_{G,ℚ};

(b) F^{2}H_{0}^{S}(G) = Ker(alb_{G/k} : F^{1}H_{0}^{S}(G) → G(k) ⊗ ℚ). This map is induced by the albanese map G → Alb_{G};

(c) F^{n}H_{0}^{S}(G) ∗ F^{m}H_{0}^{S}(G) ⊂ F^{n+m}H_{0}^{S}(G). Here ∗ denote the Pontryagin product;

(d) (F^{1}H_{0}^{S}(G))^{∗n} = 0 for n > g + r (see Corollary 4.1).

Bloch [5] studied a filtration on the Chow group CH_{0}(A) of an abelian variety over an algebraically closed field, which is defined by iterated Pontryagin product I_{A}^{∗r} of the kernel I_{A} of the degree map CH_{0}(A) → ℤ. A similar filtration on CH_{0}(A) for an abelian variety over a field is studied by Gazaki [12], using K-groups attached to A.

A Remarks on tensor product on ShT_{Nis}(k, ℚ)

We make here two remarks about the tensor product ⊗_{ShT} on ShT_{Nis}(k, ℚ) One is that the tensor product is exact (Proposition A.1). A key of a proof of the exactness are results of Suslin–Voevodsky [23, 24] and Cisinski–Déglise [7]. Another is to give an example of étale sheaves with transfers for which étale sheafification of presheaf tensor product is not isomorphic to the tensor product ⊗_{ShT}. Let us fix notation:

• Sch_{k} : the category of separated schemes of finite type over k;

• Sm_{k} : the subcategory of Sch_{k} of smooth k-schemes;

• Sch_{k}^{cor} : the category of separated schemes of finite type over k with morphism finite correspondences (which are given by universally integral relative cycles ([24]));

• Sm_{k}^{cor} : the subcategory of Sch_{k}^{cor} of smooth k-schemes;

• PST(k, ℚ) : the category of presheaves of ℚ-modules on Sm_{k}^{cor};

• ShT_{Nis}(k, ℚ) : the category of Nisnevich sheaves of ℚ-modules on Sm_{k}^{cor};

• \underline{Sh}_{Nis}(k, ℚ) : the category of Nisnevich sheaves of ℚ-modules on Sch_{k};

• \underline{ShT}_{Nis}(k, ℚ) : the category of Nisnevich sheaves of ℚ-modules on Sch_{k}^{cor};

• \underline{Sh}_{qfh}(k, ℚ) : the category of qfh-sheaves of ℚ-modules on Sch_{k}.

We call an object of PST(k, ℚ) (resp. ShT_{Nis}(k, ℚ)) a presheaf (resp. Nisnevich) with transfers. We call an object of \underline{ShT}_{Nis}(k, ℚ) a generalized Nisnevich sheaf with transfers.

A.1 Exactness of the tensor product on ShT_{Nis}(k, ℚ)

We recall the tensor product on PST: For X ∈ Sm_{k}, L(X) denotes the representable presheaf with transfers. We first define the tensor product L(X) ⊗_{PST} L(Y) as

L(X) ⊗_{PST} L(Y) := L(X × Y).

Let F be presheaf with transfers. We have a canonical projective resolution L(F) → F of F of the following form

⋯ → ⨁_{j} L(Y_{j}) → ⨁_{i} L(X_{i}) → F → 0

Then for presheaves with transfers F and G, the tensor product F ⊗_{PST} G of F and G is defined to be

H_{0}(Tot(L(F) ⊗_{PST} L(G))).

For Nisnevich sheaves with transfers F and G, the tensor product F ⊗_{ShT} G is given by

F ⊗_{ShT} G = (F ⊗_{PST} G)_{Nis}.

Proposition A.1. The bifunctor ⊗_{ShT} on ShT_{Nis}(k, ℚ) is exact.

Remark A.2. (1) The tensor product on ShT_{Nis}(k, ℤ) is right exact by construction.

(2) If F is a presheaf of ℚ-modules with transfers, then F_{Nis} = F_{ét} ([19, Corollary 14.22] [20, Ex. 1.21]). Therefore, Proposition A.1 holds for ShT_{ét}(k, ℚ).

¿From Proposition A.1 and the definition of tensor product on the derived category D^{−}(ShT_{Nis}(k, ℚ)) of ShT_{Nis}(k, ℚ), we obtain the following:

Corollary A.3. Let F and G be Nisnevich sheaves of ℚ-modules with transfers. Let F [0] denote the complex consisting of F concentrated in degree zero. Then

(1) F [0] ⊗ G[0] is isomorphic to (F ⊗_{ShT} G)[0] in D^{−}(ShT_{Nis}(k, ℚ)).

(2) Sym^{i}(F [0]) is isomorphic to Sym^{i}(F)[0] in D^{−}(ShT_{Nis}(k, ℚ)).

We recall results of Suslin–Voevodsky [23, Corollary 6.6, Theorem 6.7][24, Theorem 4.2.12] (cf. Cisinski–Déglise [7, Theorem 10.5.5]) to prove Proposition A.1.

Theorem A.4. Let X ∈ Sch_{k} be a separated scheme of finite type over k. Let \underline{L}(X)_{ℚ} ∈ \underline{ShT}_{Nis}(k, ℚ) denote the generalized Nisnevich sheaf with transfers represented by X. Then \underline{L}(X)_{ℚ} is a qfh-sheaf. Furthermore, let ℚ_{qfh}(X) ∈ \underline{Sh}_{qfh}(k, ℚ) denote the qfh-sheaf represented by X. Then \underline{L}(X)_{ℚ} is isomorphic to the qfh-sheaf ℚ_{qfh}(X).

By this theorem, for any qfh-sheaf F ∈ \underline{Sh}_{qfh}(k, ℚ), Cisinski–Déglise [7, §10] defined a generalized Nisnevich sheaf with transfers ρ(F) as follows: for any X ∈ Sch_{k},

ρ(F)(X) := Hom_{Sh_{qfh}(k,ℚ)}(\underline{L}(X)_{ℚ}, F).

Thus we have a functor

ρ : \underline{Sh}_{qfh}(k, ℚ) ⟶ \underline{ShT}_{Nis}(k, ℚ).

Now we have a functor

ψ : \underline{Sh}_{qfh}(k, ℚ) →^{ρ} \underline{ShT}_{Nis}(k, ℚ) →^{ι^{∗}} ShT_{Nis}(k, ℚ).

where the second functor is the pull-back with respect to ι : Sm_{k}^{cor} → Sch_{k}^{cor}. By [7, Lemma 10.4.6], the functor ι^{∗} admits a left adjoint ι_{!} such that ι_{!}(L(X)) = \underline{L}(X) for any X ∈ Sm_{k}. Thus we also have a functor

ϕ : ShT_{Nis}(k, ℚ) →^{ι_{!}} \underline{ShT}_{Nis}(k, ℚ) → \underline{Sh}_{Nis}(k, ℚ) ⟶ \underline{Sh}_{qfh}(k, ℚ)

where the second one is the forgetful functor and the third one is the qfhsheafification. Cisinski-Déglise showed the following properties of the functors ψ, ϕ:

Proposition A.5 ([7, Proposition 10.5.14]). The following holds:

(1) For any smooth k- scheme X, ψ(ℚ_{qfh}(X)) ≃ L(X)_{ℚ};

(2) The functor ψ admits the left adjoint ϕ;

(3) For any smooth k-scheme X, ϕ(L(X)_{ℚ}) ≃ ℚ_{qfh}(X);

(4) The functor ψ is exact and preserves colimits;

(5) The functor ϕ is exact, fully faithful and preserves colimits.

Proof of Proposition A.1. Since the tensor product on ShT_{Nis}(k, ℚ) is right exact (cf. Remark A.2), it suffices to show that the tensor product is left exact. Thus, we need to show that for any injection F_{1} → F_{2} in ShT_{Nis}(k, ℚ) and any objects G, H of ShT_{Nis}(k, ℚ), the map

Hom_{ShT_{Nis}}(H, F_{1} ⊗_{ShT} G) → Hom_{ShT_{Nis}}(H, F_{2} ⊗_{ShT} G)

is injective. Since the functor ϕ is fully faithful by Proposition A.5 (5), we have the following commutative diagram

[figure: Hom_{ShT_{Nis}(k,ℚ)}(H, F_{1} ⊗_{ShT} G); Hom_{ShT_{Nis}(k,ℚ)}(H, F_{2} ⊗_{ShT} G); Hom_{\underline{Sh}_{qfh}(k,ℚ)}(ϕ(H), ϕ(F_{1} ⊗_{ShT} G)); Hom_{\underline{Sh}_{qfh}(k,ℚ)}(ϕ(H), ϕ(F_{2} ⊗_{ShT} G))]

Let us construct a natural isomorphism

ϕ(F ⊗_{ShT} G) = ϕ(F) ⊗ ϕ(G).  (∗)

Granting this, the bottom horizontal map is

Hom_{\underline{Sh}_{qfh}(k,ℚ)}(ϕ(H), ϕ(F_{1}) ⊗ ϕ(G)) → Hom_{\underline{Sh}_{qfh}(k,ℚ)}(ϕ(H), ϕ(F_{2}) ⊗ ϕ(G)).

But, this map is injective because the functor ϕ is exact by Proposition A.5 (5) and the tensor on \underline{Sh}_{qfh}(k, ℚ) is exact. Now our task is to construct the natural isomorphism (∗). Let F, G ∈ ShT_{Nis}(k, ℚ) be Nisnevich sheaves with transfers. In case where F = L(X)_{ℚ} and G = L(Y)_{ℚ}, we have

ϕ(L(X)_{ℚ} ⊗_{ShT} L(Y)_{ℚ}) = ϕ(L(X × Y)_{ℚ})
≃ ℚ_{qfh}(X × Y)

= ℚ_{qfh}(X) ⊗ ℚ_{qfh}(Y) ≃ ϕ(L(X)_{ℚ}) ⊗ ϕ(L(Y)_{ℚ}).

In general case, let L_{1}(F) (resp. L_{1}(G)) be the presentation of F (resp. G) truncated above degree one. Then by the right exactness of ⊗_{ShT}, the truncation of Tot(L_{1}(F) ⊗_{ShT} L_{1}(G)) is a presentation of F ⊗_{ShT} G. Thus from the the above special case and the exactness of ϕ, we obtain that

ϕ(F ⊗_{ShT} G) = ϕ(H_{0}(Tot(L_{1}(F) ⊗_{ShT} L_{1}(G))))
≃ H_{0}(Tot(ϕ(L_{1}(F)) ⊗ ϕ(L_{1}(G))))
= ϕ(F) ⊗ ϕ(G).

A.2 An example

We prove that in general, the étale sheafification of a presheaf tensor L(X) ⊗ L(Y) can not be isomorphic to L(X ×_{k} Y), even after tensoring with ℚ. We have a map

ϕ : L(X) ⊗ L(Y) → L(X ×_{k} Y).

and consider the étale stalks at a strictly Henselian local scheme S , which is

ϕ_{S} : Cor(S, X) ⊗ Cor(S, Y) → Cor(S, X ×_{k} Y), Z ⊗ W ↦ [Z ×_{S} W].

The claim in the beginning of this section follows from Proposition A.6 below for a strictly Henselian discrete valuation ring R.

Proposition A.6. Let X = Y = 𝔸_{k}^{1}. Let R be a discrete valuation ring over k and let S := Spec(R). Then ϕ_{S} is not surjective, even after tensoring with ℚ.

To show Proposition A.6, we consider some concrete cycles: Write X = Spec(k[x]) and Y = Spec(k[y]). Let t be a uniformizer of R. We define f(x) ∈ R[x] as

f(x) = \begin{cases}x^{2} − t & if char(k) ≠ 2, \\ x^{3} − t & if char(k) = 2. \end{cases}

and define cycles to be

Z := Spec(R[x]/(f(x))) ∈ Cor(S, X) and W := Spec(R[y]/(f(y))) ∈ Cor(S, Y).

Then we have

T := Z ×_{S} W = Spec(R[x, y]/(f(x), f(y))).

And we define T_{1} and T_{2} as follows: in case char(k) ≠ 2,

T_{1} := Spec(R[x, y]/(f(x), x − y)), T_{2} := Spec(R[x, y]/(f(x), x + y));

in case char(k) = 2,

T_{1} := Spec(R[x, y]/(f(x), x − y)), T_{2} := Spec(R[x, y]/(f(x), x^{2} + xy + y^{2})).

Then, one easily sees that

ϕ_{S}(Z ⊗ W) = [T] = T_{1} + T_{2} ∈ Cor(S, X ×_{k} Y).

Lemma A.7. Let notation be as in above. The only pair (α, β) of integral cycles α ∈ Cor(S, X) and β ∈ Cor(S, Y) such that ϕ_{S}(α ⊗ β) contains T_{1}(or T_{2}) is (Z, W).

Proof. Integral cycles α and β are given by monic irreducible polynomials g(x) ∈ R[x] and h(y) ∈ R[y] respectively, i.e.

α = Spec(R[x]/(g(x))) ⊂ S × X, β = Spec(R[y]/(h(y))) ⊂ S × Y.

Put

γ := α ×_{S} β = Spec(R[x, y]/(g(x), h(y))) ⊂ S × X × Y.

Suppose that our cycle T_{1} ⊂ γ is a irreducible component. Let p, q denote the projections from S ×X ×Y to S ×X and S ×Y respectively. Then,

p(T_{1}) = Z, q(T_{1}) = W.

Hence the images of γ along p and q contain Z and W respectively, i.e.

Z ⊂ Spec(R[x]/(g(x))) and W ⊂ Spec(R[y]/(h(y))).

These inclusions imply that

g(x) = a(x)f(x) and h(y) = b(y)f(y).

Since f, g and h are monic and irreducible, we have

g(x) = f(x), h(y) = f(y).

Hence α = Z, β = W. The above argument works also for T_{2}.

Proof of Proposition A.6. Assume that ϕ_{S} is surjective. Then we have an element x := ∑ n_{ij}(α_{i} ⊗ β_{j}) ∈ Cor(S, X) ⊗ Cor(S, Y) such that ϕ_{S}(x) = T_{1}. For some component α_{i} ⊗ β_{j} of x, ϕ_{S}(α_{i} ⊗ β_{j}) contains T_{1}. By Lemma A.7, such α_{i} ⊗ β_{j} is Z ⊗ W only. Therefore we have

x = m(Z ⊗ W) + ∑ n_{ij}(α_{i} ⊗ β_{j}),

where m ≥ 1 and the sum is taken over all i, j such that (α_{i}, β_{j}) ≠ (Z, W). Then

T_{1} = ϕ_{S}(x) = m(T_{1} + T_{2}) + ∑ n_{ij}ϕ_{S}(α_{i} ⊗ β_{j}).

Hence

(1 − m)T_{1} − mT_{2} = − ∑ n_{ij}ϕ_{S}(α_{i} ⊗ β_{j}).

But this equality cannot be happen by Lemma A.7. Thus ϕ_{S} is not surjective. Since the above argument works even if the coefficients m, n_{ij} are rational numbers, ϕ_{S} is not surjective even after tensoring with ℚ.

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Rin Sugiyama

Fakultät Mathematik

Universität Duisburg-Essen Thea-Leymann-Straße 9 45127 Essen, Germany rin.sugiyama@uni-due.de