ALGEBRAIC TORI AS NISNEVICH SHEAVES WITH TRANSFERS

BRUNO KAHN

Abstract. We relate R-equivalence on tori with Voevodsky’s theory of homotopy invariant Nisnevich sheaves with transfers and effective motivic complexes.

Contents

1. Main results 1 2. Proofs of Theorem 1 and Corollary 2 5 3. Stable birationality 6 4. Some open questions 12 References 14

1. Main results

Let k be a field and let T be a k-torus. The R-equivalence classes on T have been extensively studied by several authors, notably by Colliot-Thélène and Sansuc in a series of papers including [3] and [4]: they play a central rôle in many rationality issues. In this note, we show that Voevodsky’s triangulated category of motives sheds a new light on this question: see Corollaries 1, 3 and 4 below.

More generally, let G be a semi-abelian variety over k, which is an extension of an abelian variety A by a torus T. Denote by HI the category of homotopy invariant Nisnevich sheaves with transfers over k in the sense of Voevodsky [19]. Then G has a natural structure of an object of HI ([17, proof of Lemma 3.2], [1, Lemma 1.3.2]). Let L be the group of cocharacters of T.

Proposition 1. There is a natural isomorphism G_{−1} ⟶^{∼} L in HI.

Here _{−1} is the contraction operation of [18, p. 96], whose definition is recalled in the proof below.

Date: March 9, 2012.

2010 Mathematics Subject Classification. 14L10, 14E08, 14G27, 14F42.

Proof. Recall that if F is a presheaf [with transfers] on smooth kschemes, the presheaf [with transfers] F_{−1}^{p} is defined by

U ↦ Coker(F(U × A^{1}) → F(U × 𝔾_{m})).

If F is homotopy invariant, we may replace U × A^{1} by U and the rational point 1 ∈ 𝔾_{m} realises F_{−1}^{p}(U) as a functorial direct summand of F(U × 𝔾_{m}).

If F is a Nisnevich sheaf [with transfers], F_{−1} is defined as the sheaf associated to F_{−1}^{p}.

Now A(U × A^{1}) ⟶^{∼} A(U × 𝔾_{m}) since A is an abelian variety, hence A_{−1}^{p} = 0. We therefore have an isomorphism of presheaves T_{−1}^{p} ⟶^{∼} G_{−1}^{p}, and a fortiori an isomorphism of Nisnevich sheaves T_{−1} ⟶^{∼} G_{−1}.

Let p : 𝔾_{m} → Spec k be the structural map. One easily checks that the étale sheaf Coker(T ⟶^{i} p_{∗}p^{∗}T) is canonically isomorphic to L. Since i is split, its cokernel is still L if we view it as a morphism of presheaves, hence of Nisnevich sheaves. □

From now on, we assume k perfect. Let DM_{−}^{eff} be the triangulated category of effective motivic complexes introduced in [19]: it has a tstructure with heart HI. It also has a tensor structure and a (partially defined) internal Hom. We then have an isomorphism

L[0] = G_{−1}[0] ≃ \underline{Hom}_{DM}{eff \atop −}(𝔾_{m}[0], G[0])

[10, Rk. 4.4], hence by adjunction a morphism in DM_{−}^{eff}

(1)  L[0] ⊗ 𝔾_{m}[0] → G.

Let ν_{≤0}G[0] denote the cone of (1): by [11, Lemma 6.3] or [8, §2], ν_{≤0}G[0] is the birational motivic complex associated to G. We want to compute its homology sheaves.

For this, consider a coflasque resolution

(2)  0 → Q → L_{0} → L → 0

of L in the sense of [3, p. 179]. Taking a coflasque resolution of Q and iterating, we get a resolution of L by invertible lattices^{1}:

(3)  ⋯ → L_{n} → ⋯ → L_{0} → L → 0.

We set

Q_{n} = \begin{cases}Q & for n = 1 \\ Ker(L_{n−1} → L_{n−2}) & for n > 1. \end{cases}

1Recall that a lattice is a free finitely generated Galois module; a lattice is invertible if it is a direct summand of a permutation lattice.

Theorem 1. a) Let T_{n} denote the torus with cocharacter group L_{n}. Then ν_{≤0}G[0] is isomorphic to the complex

⋯ → T_{n} → ⋯ → T_{0} → G → 0.

b) Let S_{n} be the torus with cocharacter group Q_{n}. For any connected smooth k-scheme X with function field K, we have

H_{n}(ν_{≤0}G[0])(X) = \begin{cases}0 & if n < 0 \\ G(K)/R & if n = 0 \\ S_{n}(K)/R & if n > 0. \end{cases}

The proof is given in Section 2.

Corollary 1. The assignment Sm(k) ∋ X ↦ ⨁_{x∈X^{(0)}} G(k(x))/R provides G/R with the structure of a homotopy invariant Nisnevich sheaf with transfers. In particular, any morphism φ : Y → X of smooth connected k-schemes induces a morphism φ^{∗} : G(k(X))/R → G(k(Y))/R. □

This functoriality is essential to formulate Theorem 2 below. For φ a closed immersion of codimension 1, it recovers a specialisation map on R-equivalence classes with respect to a discrete valuation of rank 1 which was obtained (for tori) by completely different methods, e.g. [4, Th. 3.1 and Cor. 4.2] or [7]. (I am indebted to Colliot-Thélène for pointing out these references.)

Corollary 2. a) If k is finitely generated, the n-th homology sheaf of ν_{≤0}G[0] takes values in finitely generated abelian groups, and even in finite groups if n > 0 or G is a torus.

b) If G is a torus, then ν_{≤0}G[0] = 0 if G is split by a Galois extension E/k whose Galois group has cyclic Sylow subgroups. This condition is automatic if k is (quasi-)finite.

The proof is also given in Section 2.

Given two semi-abelian varieties G, G′, we would now like to understand the maps

Hom_{k}(G, G′) → Hom_{DM}{eff \atop −}(ν_{≤0}G[0], ν_{≤0}G′[0]) → Hom_{HI}(G/R, G′/R).

In Section 3, we succeed in elucidating the nature of their composition to a large extent, at least if G is a torus. Our main result, in the spirit of Yoneda’s lemma, is

Theorem 2. Let G, G′ be two semi-abelian varieties, with G a torus. Suppose given, for every function field K/k, a homomorphism f_{K} : G(K)/R → G′(K)/R such that f_{K} is natural with respect to the functoriality of Corollary 1. Then a) There exists an extension G̃ of G by a permutation torus, and a homomorphism f : G̃ → G′ inducing (f_{K}).

b) f_{K} is surjective for all K if and only if there exist extensions G̃, G̃′ of G and G′ by permutation tori such that f_{K} is induced by a split surjective homomorphism G̃ → G̃′.

The proof is given in §3.3. See Proposition 2, Corollary 5, Remark 4 and Proposition 3 for complements.

This relates to questions of stable birationality studied by Colliot-Thélène and Sansuc in [3] and [4], providing alternate proofs and strengthening of some of their results (at least over a perfect field). More precisely:

Corollary 3. a) Let G′ be a semi-abelian k-variety such that G′(K)/R = 0 for any function field K/k. Then G′ is an invertible torus.

b) In Theorem 2 b), assume that f_{K} is bijective for all K/k. Then there exist extensions G̃, G̃′ of G and G′ by invertible tori such that f_{K} is induced by an isomorphism G̃ ⟶^{∼} G̃′.

Proof. a) This is the special case G = 0 of Theorem 2 b).

b) By Theorem 2 b), we may replace G and G′ by extensions by permutation tori such that f_{K} is induced by a split surjection f : G → G′. Let T = Ker f. Then T/R = 0 universally. By a), T is invertible.

□

Corollary 3 a) is a version of [4, Prop. 7.4] (taking [3, p. 199, Th. 2] into account). Theorem 2 was inspired by the desire to understand this result from a different viewpoint.

Corollary 4. Let f : G −−→ G′ be a rational map of semi-abelian varieties, with G a torus. Then the following conditions are equivalent:

(i) f_{∗} : ν_{≤0}G[0] → ν_{≤0}G′[0] is an isomorphism (see Proposition 2). (ii) f_{∗} : G(K)/R → G′(K)/R is bijective for any function field

K/k. (iii) f is an isomorphism, up to extensions of G and G′ by invertible

tori and up to a translation. (See Lemma 6.) □

Acknowledgements. Part of Theorem 1 was obtained in the course of discussions with Takao Yamazaki during his stay at the IMJ in October 2010: I would like to thank him for inspiring exchanges. I also thank Daniel Bertrand for a helpful discussion. Finally, I wish to acknowledge inspiration from the work of Colliot-Thélène and Sansuc, which will be obvious throughout this paper.

2. Proofs of Theorem 1 and Corollary 2

Lemma 1. The exact sequence

0 → T (k) → G(k) → A(k)

induces an exact sequence

0 → T (k)/R ⟶^{i} G(k)/R → A(k).

Proof. Let f : P^{1} −−→ G be a k-rational map defined at 0 and 1. Its composition with the projection G → A is constant: thus the image of f lies in a T -coset of G defined by a rational point. This implies the injectivity of i, and the rest is clear. □

Let NST denote the category of Nisnevich sheaves with transfers. Recall that DM_{−}^{eff} may be viewed as a localisation of D^{−}(NST), and that its tensor structure is a descent of the tensor structure on the latter category [19, Prop. 3.2.3].

Lemma 2. If G is an invertible torus, there is a canonical isomorphism in D^{−}(NST)

L[0] ⊗ 𝔾_{m} ⟶^{∼} G[0].

In particular, ν_{≤0}G[0] = 0.

Proof. We reduce to the case T = R_{E/k}𝔾_{m}, where E is a finite extension of k. Let us write more precisely NST(k) and NST(E). There is a pair of adjoint functors

NST(k) ⟶^{f^{∗}} NST(E), NST(E) ⟶^{f_{∗}} HI(k)

where f : Spec E → Spec k is the projection. Clearly,

f_{∗}Z = Z_{tr}(Spec E), f_{∗}𝔾_{m} = T

where Z_{tr}(Spec E) is the Nisnevich sheaf with transfers represented by Spec E. Since Z_{tr}(Spec E) = L, this proves the claim. □

Proof of Theorem 1. a) Recall that L_{0} is an invertible lattice chosen so that L_{0}(E) → L(E) is surjective for any extension E/k. In particular,

(2) and (3) are exact as sequences of Nisnevich sheaves; hence L[0] is isomorphic in D^{−}(NST) to the complex

L_{·} = ⋯ → L_{n} → ⋯ → L_{0} → 0.

(We may view (3) as a version of Voevodsky’s “canonical resolutions” as in [19, §3.2 p. 206].)

By Lemma 2, L_{n}[0] ⊗ 𝔾_{m}[0] ≃ T_{n}[0] is homologically concentrated in degree 0 for all n. It follows that the complex

T_{·} = ⋯ → T_{n} → ⋯ → T_{0} → 0

is isomorphic to L[0] ⊗ 𝔾_{m}[0] in D^{−}(NST), hence a fortiori in DM_{−}^{eff}. b) For any nonempty open subscheme U ⊆ X we have isomorphisms

(4) H_{n}(ν_{≤0}G[0])(X) ⟶^{∼} H_{n}(ν_{≤0}G[0])(U) ⟶^{∼} H_{n}(ν_{≤0}G[0])(K)

(e.g. [8, p. 912]). By a), the right hand term is the n-th homology group of the complex

⋯ → T_{n}(K) → ⋯ → T_{0}(K) → G(K) → 0

with G(K) in degree 0. By [3, p. 199, Th. 2], the sequences

0 → S_{1}(K) → T_{0}(K) → T (K) → T (K)/R → 0
0 → S_{n+1}(K) → T_{n}(K) → S_{n}(K) → S_{n}(K)/R → 0

are all exact. Using Lemma 1 for H_{0}, the conclusion follows from an easy diagram chase. □

Remark 1. As a corollary to Theorem 1, S_{n}(K)/R only depends on G. This can be seen without mentioning DM_{−}^{eff}: in view of the reasoning just above, it suffices to construct a homotopy equivalence between two resolutions of the form (3), which easily follows from the definition of coflasque modules.

Proof of Corollary 2. a) This follows via Theorem 1 and Lemma 1 from [3, p. 200, Cor. 2] and the Mordell-Weil-Néron theorem. b) We may choose the L_{n}, hence the S_{n} split by E/k. The conclusion now follows from Theorem 1 and [3, p. 200, Cor. 3]. The last claim is clear. □

Remark 2. In characteristic p > 0, all finitely generated perfect fields are finite. To give some contents to Corollary 2 a) in this characteristic, one may pass to the perfect [one should say radicial] closure k of a finitely generated field k_{0}. If G is a semi-abelian k-variety, it is defined over some finite extension k_{1} of k_{0}. If k_{2}/k_{1} is a finite (purely inseparable) subextension of k/k_{1}, then the composition

G(k_{2}) ⟶^{N_{k_{2}/k_{1}}} G(k_{1}) → G(k_{2})

equals multiplication by [k_{2} : k_{1}]. Hence Corollary 2 a) remains true at least after inverting p.

3. Stable birationality

If X is a smooth variety over a field k, we write Alb(X) for its generalised Albanese variety in the sense of Serre [16]: it is a semiabelian variety, and a rational point x_{0} ∈ X determines a morphism X → Alb(X) which is universal for morphisms from X to semi-abelian varieties sending x_{0} to 0.

We also write NS(X) for the group of cycles of codimension 1 on X modulo algebraic equivalence. This group is finitely generated if k is algebraically closed [9, Th. 3].

3.1. Well-known lemmas. I include proofs for lack of reference.

Lemma 3. a) Let G, G′ be two semi-abelian k-varieties. Then any k-morphism f : G → G′ can be written uniquely f = f(0) + f′, where f′ is a homomorphism.

b) For any semi-abelian k-variety G, the canonical map G → Alb(G) sending 0 to 0 is an isomorphism.

Proof. a) amounts to showing that if f(0) = 0, then f is a homomorphism. By an adjunction game, this is equivalent to b). Let us give two proofs: one of a) and one of b).

Proof of a). We may assume k to be a universal domain. The staement is classical for abelian varieties [15, p. 41, Cor. 1] and an easy computation for tori. In the general case, let T, T′ be the toric parts of G and G′ and A, A′ be their abelian parts. Let g ∈ G(k). As any morphism from T to A′ is constant, the k-morphism

φ_{g} : T ∋ t ↦ f(g + t) − f(g) ∈ G′

(which sends 0 to 0) lands in T′, hence is a homomorphism. Therefore it only depends on the image of g in A(k). This defines a morphism φ : A → \underline{Hom}(T, T′), which must be constant with value φ_{0} = f. It follows that

(g, h) ↦ f(g + h) − f(g) − f(h)

induces a morphism A × A → T′. Such a morphism is constant, of value 0.

Proof of b). This is true if G is abelian, by rigidity and the equivalence between a) and b). In general, any morphism from G to an abelian variety is trivial on T. This shows that the abelian part of Alb(G) is A. Let T′ = Ker(Alb(G) → A). We also have the counit morphism Alb(G) → G, and the composition G → Alb(G) → G is the identity. Thus T is a direct summand of T′. It suffices to show that dim T′ = dim T. Going to the algebraic closure, we may reduce to T = 𝔾_{m}.

Then consider the line bundle completion Ḡ → A of the 𝔾_{m}-bundle G → A. It is sufficient to show that the kernel of

Alb(G) → Alb(Ḡ) = A

is 1-dimensional. This follows for example from [1, Cor. 10.5.1]. □

Lemma 4. Suppose k algebraically closed, and let G be a semi-abelian k-variety. Let A be the abelian quotient of G. Then the map

(5)  NS(A) → NS(G)

is an isomorphism.

Proof. Let T = Ker(G → A) and X(T) be its character group. Choosing a basis (e_{i}) of X(T), we may complete the 𝔾_{m}^{n}-torsor G into a product of line bundles Ḡ → A. The surjection

Pic(A) ⟶^{∼} Pic(Ḡ) → → Pic(G)

show the surjectivity of (5). Its kernel is generated by the classes of the irreducible components D_{i} of the divisor with normal crossings Ḡ − G. These components correspond to the basis elements e_{i}. Since the corresponding 𝔾_{m}-bundle is a group extension of A by 𝔾_{m}, the class of the 0 section of its line bundle completion lies in Pic^{0}(A), hence goes to 0 in NS(Ḡ). □

Lemma 5. Let X be a smooth k-variety, and let U ⊆ X be a dense open subset. Then there is an exact sequence of semi-abelian varieties

0 → T → Alb(U) → Alb(X) → 0

with T a torus. If NS(Ū) = 0 (this happens if U is small enough), there is an exact sequence of character groups

0 → X(T) → ⨁_{x∈X^{(1)}−U^{(1)}} Z → NS(X̄) → 0.

Proof. This follows for example from [1, Cor. 10.5.1]. □

Lemma 6. Let f : G −−→ G′ be a rational map between semi-abelian k-varieties, with G a torus. Then there exists an extension G̃ of G by a permutation torus and a homomorphism f̃: G̃ → G′ which extends f up to translation in the following sense: there exists a rational section s : G −−→ G̃ of the projection π : G̃ → G and a rational point g′ ∈ G′(k) such that f = f̃s + g′. If f is defined at 0_{G} and sends it to 0_{G′}, then g′ = 0.

Proof. Let U be an open subset of G where f is defined. We define G̃ = Alb(U). Applying Lemmas 5 and 3 b) and using NS(Ḡ) = 0, we get an extension

0 → P → G̃ → G → 0

where P is a permutation torus, as well as a morphism f̃ = Alb(f) : G̃ → G′.

Let us first assume k infinite. Then U(k) ≠ ∅ because G is unirational. A rational point g ∈ U defines an Albanese map s : U → G̃ sending g to 0G̃. Since P is a permutation torus, g ∈ G(k) lifts to g̃ ∈ G̃(k) (Hilbert 90) and we may replace s by a morphism sending g to g̃. Then s is a rational section of π. Moreover, f = f̃s + g′ with g′ = f(g) − f̃(g̃). The last assertion follows.

If k is finite, then U has at least a zero-cycle g of degree 1, which is enough to define the Albanese map s. We then proceed as above (lift every closed point involved in g to a closed point of G̃ with the same residue field). □

Lemma 7. Let G be a finite group, and let A be a finitely generated G-module. Then

a) There exists a short exact sequence of G-modules 0 → P → F → A → 0, with F torsion-free and flasque, and P permutation.

b) Let B be another finitely generated G-module, and let 0 → P′ → E → B → 0 be an exact sequence with P′ an invertible module. Then any G-morphism f : A → B lifts to f̃: F → E.

Proof. a) is the contents of [4, Lemma 0.6, (0.6.2)]. b) The obstruction to lifting f lies in Ext_{G}^{1}(F, P′) = 0 [3, p. 182, Lemme 9]. □

3.2. Functoriality of ν_{≤0}G. We now assume k perfect.

Lemma 8. Let

(6)  0 → P → G → H → 0

be an exact sequence of semi-abelian varieties, with P an invertible torus. Then ν_{≤0}G[0] ⟶^{∼} ν_{≤0}H[0].

Proof. As P is invertible, (6) is exact in NST hence defines an exact triangle

P [0] → G[0] → H[0] ⟶^{+1}

in DM_{−}^{eff}. The conclusion then follows from Lemma 2. □

Proposition 2. Let G, G′ be two semi-abelian k-varieties, with G a torus. Then a rational map f : G −−→ G′ induces a morphism f_{∗} : ν_{≤0}G[0] → ν_{≤0}G′[0], hence a homomorphism f_{∗} : G(K)/R → G′(K)/R for any extension K/k. If K is infinite, f_{∗} agrees up to translation with the morphism induced by f via the isomorphism U(K)/R ⟶^{∼} G(K)/R from [3, p. 196 Prop. 11], where U is an open subset of definition of f.

Proof. By Lemma 6, f induces a homomorphism G̃ → G′ where G̃ is an extension of G by a permutation torus. By Lemma 8, the induced morphism

ν_{≤0}G̃[0] → ν_{≤0}G′[0]

factors through a morphism f_{∗} : ν_{≤0}G[0] → ν_{≤0}G′[0].

The claims about R-equivalence classes follow from Theorem 1 b) and Lemma 6. □

Remark 3. The proof shows that f′_{∗} = f_{∗} if f′ differs from f by a translation by an element of G(k) or G′(k).

Corollary 5. If T and T′ are birationally equivalent k-tori, then ν_{≤0}T [0] ≃ ν_{≤0}T′[0]. In particular, the groups T (k)/R and T′(k)/R are isomorphic.

Proof. The proof of Proposition 2 shows that f ↦ f_{∗} is functorial for composable rational maps between tori. Let f : T −−→ T′ be a birational isomorphism, and let g : T′ −−→ T be the inverse birational isomorphism. Then we have g_{∗}f_{∗} = 1_{ν_{≤0}T [0]} and f_{∗}g_{∗} = 1_{ν_{≤0}T′[0]}. The last claim follows from Theorem 1. □

Remark 4. It is proven in [3] that a birational isomorphism of tori f : T −−→ T′ induces a set-theoretic bijection f_{∗} : T (k)/R ⟶^{∼} T′(k)/R (p. 197, Cor. to Prop. 11) and that the group T (k)/R is abstractly a birational invariant of T (p. 200, Cor. 4). The proof above shows that f_{∗} is an isomorphism of groups if f respects the origins of T and T′. This solves the question raised in [3, mid. p. 397]. The proofs of Lemma 6 and Proposition 2 may be seen as dual to the proof of [3, p. 189, Prop. 5], and are directly inspired from it.

3.3. Faithfulness and fullness.

Proposition 3. Let f : G −−→ G′ be a rational map between semiabelian varieties, with G a torus. Assume that the map f_{∗} : G(K)/R → G′(K)/R from Proposition 2 is identically 0 when K runs through the finitely generated extensions of k. Then there exists a permutation torus P and a factorisation of f as

G −−→^{f̃} P ⟶^{g} G′

where f̃ is a rational map and g is a homomorphism. If f is a morphism, we may choose f̃ as a homomorphism. Conversely, if there is such a factorisation, then f_{∗} : ν_{≤0}G[0] → ν_{≤0}G′[0] is the 0 morphism.

Proof. By Lemma 6, we may reduce to the case where f is a morphism. Let K = k(G). By hypothesis, the image of the generic point η_{G} ∈ G(K) is R-equivalent to 0 on G′(K). By a lemma of Gille [6, Lemme II.1.1 b)], it is directly R-equivalent to 0: in other words, there exists a rational map h : G × A^{1} −−→ G′, defined in the neighbourhood of 0 and 1, such that h_{|G×{0}} = 0 and h_{|G×{1}} = f.

Let U ⊆ G × A^{1} be an open set of definition of h. The 0 and 1-sections of G × A^{1} → G induce sections

s_{0}, s_{1} : G → Alb(U)

of the projection π : Alb(U) → Alb(G × A^{1}) = G such that Alb(h) ∘ s_{0} = 0 and Alb(h) ∘ s_{1} = f. If P = Ker π, then s_{0} − s_{1} induces a homomorphism f̃: G → P such that the composition

G ⟶^{f̃} P → Alb(U) ⟶^{Alb(h)} G′

equals f. Finally, P is a permutation torus by Lemma 5.

The last claim follows from Lemma 2. □

Proof of Theorem 2. a) Take K = k(G). The image of the generic point η_{G} by f_{K} lifts to a (non unique) rational map f : G −−→ G′. Using Lemma 6, we may extend f to a homomorphism

f̃: G̃ → G′

where G̃ is an extension of G by a permutation torus P. Since G̃(K)/R ⟶^{∼} G(K)/R, we reduce to G̃ = G and f̃ = f.

Let L/k be a fonction field, and let g ∈ G(L). Then g arises from a morphism g : X → G for a suitable smooth model X of L. By assumption on K ↦ f_{K}, the diagram

G(K)/R ⟶^{f_{K}} G′(K)/R
g^{∗}↓  ^{g^{∗}}↓
G(L)/R ⟶^{f_{L}} G′(L)/R

commutes. Applying this to η_{K} ∈ G(K), we find that f_{L}([g]) = [g ∘ f], which means that f_{L} is the map induced by f.

b) The hypothesis implies that G′(E)/R = 0 for any algebraically closed extension E/k, which in turn implies that G′ is also a torus. Applying a), we may, and do, convert f into a true homomorphism by replacing G by a suitable extension by a permutation torus. Applying Lemma 7 a) to the cocharacter group of G, we get a resolution 0 → P_{1} → Q → G → 0 with Q coflasque and P_{1} permutation. Hence we may (and do) further assume G coflasque.

Let K = k(G′) and choose some g ∈ G(K) mapping modulo Requivalence to the generic point of G′. Then g defines a rational map g : G′ −−→ G such that fg is R-equivalent to 1_{G′}. It follows that the induced map

(7)  1 − fg : G′/R → G′/R

is identically 0.

Reapplying Lemma 6, we may find an extension G̃′ of G′ by a suitable permutation torus which converts g into a true homomorphism. Since G is coflasque, Lemma 7 b) shows that f : G → G′ lifts to f̃: G → G̃′. Then (7) is still identically 0 when replacing (G′, f) by (G̃′, f̃).

Summarising: we have replaced the initial G and G′ by suitable extensions by permutation tori, such that f lifts to these extensions and there is a homomorphism g : G′ → G such that (7) vanishes identically. Hence 1−fg factors through a permutation torus P thanks to Proposition 3. Write u : G′ → P and v : P → G′ for homomorphisms such that 1 − fg = vu. Let G_{1} = G × P and consider the maps

f_{1} = (f, v) : G_{1} → G′, g_{1} = (\begin{matrix}g \\ u \end{matrix}) : G′ → G_{1}.

Then f_{1}g_{1} = 1 and G′ is a direct summand of G_{1} as requested. □

4. Some open questions

Question 1. Are lemma 6 and Proposition 2 still true when G is not a torus?

This is far from clear in general, starting with the case where G is an abelian variety and G′ a torus. Let me give a positive answer in the case of an elliptic curve.

Proposition 4. The answer to Question 1 is yes if the abelian part A of G is an elliptic curve.

Proof. Arguing as in the proof of Proposition 2, we get for an open subset U ⊆ G of definition for f an exact sequence

0 → 𝔾_{m} → P → Alb(U) → G → 0

where P is a permutation torus. Here we used that NS(Ḡ) ≃ Z, which follows from Lemma 4.

The character group X(P) has as a basis the geometric irreducible components of codimension 1 of G − U. Up to shrinking U, we may assume that G − U contains the inverse image D of 0 ∈ A. As the divisor class of 0 generates NS(Ā), D provides a Galois-equivariant splitting of the map 𝔾_{m} → P. Thus its cokernel is still a permutation torus, and we conclude as before. □

Question 2. Can one formulate a version of Theorem 2 and Corollary 3 providing a description of the groups Hom_{DM}{eff \atop −}(ν_{≤0}G[0], ν_{≤0}G′[0]) and Hom_{HI}(G/R, G′/R) (at least when G and G′ are tori)?

The proof of Theorem 2 suggests the presence of a closed model structure on the category of tori (or lattices), which might provide an answer to this question.

For the last question, let G be a semi-abelian variety. Forgetting its group structure, it has a motive M(G) ∈ DM_{−}^{eff}. Recall the canonical morphism

M(G) → G[0]

induced by the “sum” maps

(8)  c(X, G) ⟶^{σ} G(X)

for smooth varieties X ([17, (6), (7)], [1, §1.3]).

The morphism (8) has a canonical section

(9)  G(X) ⟶^{γ} c(X, G)

given by the graph of a morphism: this section is functorial in X but is not additive.

Consider now a smooth equivariant compactification Ḡ of G. It exists in all characteristics. For tori, this is written up in [2]. The general case reduces to this one by the following elegant argument I learned from M. Brion: if G is an extension of an abelian variety A by a torus T, take a smooth projective equivariant compactification Y of T. Then the bundle G ×^{T} Y associated to the T -torsor G → A also exists: this is the desired compactification.

Then we have a diagram of birational motives

(10)  \begin{matrix}ν_{≤0}M(G) & ⟶^{∼} & ν_{≤0}M(Ḡ) \\ ν^{≤0}σ & ↓ \\ ν_{≤0}G[0]. \end{matrix}

By [11], we have H_{0}(ν_{≤0}M(Ḡ))(X) = CH_{0}(Ḡ_{k(X)}) for any smooth connected X. Hence the above diagram induces a homomorphism

(11)  CH_{0}(Ḡ_{k(X)}) → G(k(X))/R

which is natural in X for the action of finite correspondences (compare Corollary 1). One can probably check that this is the homomorphism of [12, (17) p. 78], reformulating [3, Proposition 12 p. 198]. Similarly, the set-theoretic map

(12)  G(k(X))/R → CH_{0}(Ḡ_{k(X)})

of [3, p. 197] can presumably be recovered as a birational version of (9), using perhaps the homotopy category of schemes of Morel and Voevodsky [14].

In [12], Merkurjev shows that (11) is an isomorphism for G a torus of dimension at most 3. This suggests:

Question 3. Is the map ν_{≤0}σ of Diagram (10) an isomorphism when G is a torus of dimension ≤ 3?

In [13], Merkurjev gives examples of tori G for which (12) is not a homomorphism; hence its (additive) left inverse (11) cannot be an isomorphism. Merkurjev’s examples are of the form G = R_{K/k}^{1}𝔾_{m} × R_{L/k}^{1}𝔾_{m}, where K and L are distinct biquadratic extensions of k. This suggests:

Question 4. Can one study Merkurjev’s examples from the above viewpoint? More generally, what is the nature of the map ν_{≤0}σ of Diagram (10)?

We leave all these questions to the interested reader.

References

[1] L. Barbieri-Viale, B. Kahn On the derived category of 1-motives,

arXiv:1009.1900.

[2] J.-L. Colliot-Thélène, D. Harari, A. Skorobogatov Compactification équivariante d’un tore (d’après Brylinski et Künnemann), Expo. Math. 23 (2005), 161–170.

[3] J.-L. Colliot-Thélène, J.-J. Sansuc La R-équivalence sur les tores, Ann. Sci. Éc. Norm. Sup. 10 (1977), 175–230.

[4] J.-L. Colliot-Thélène, J.-J. Sansuc Principal homogeneous spaces under flasque tori; applications, J. Alg. 106 (1987), 148–205.

[5] F. Déglise Motifs génériques, Rend. Sem. Mat. Univ. Padova 119 (2008), 173– 244.

[6] P. Gille La R-équivalence pour les groupes algébriques réductifs définis sur un corps global, Publ. Math. IHÉS 86 (1997), 199-235.

[7] P. Gille Spécialisation de la R-équivalence pour les groupes réductifs, Trans. Amer. Math. Soc. 356 (2004), 4465–4474.

[8] A. Huber and B. Kahn The slice filtration and mixed Tate motives, Compositio Math. 142 (2006), 907–936.

[9] B. Kahn Sur le groupe des classes d’un schéma arithmétique (avec un appendice de Marc Hindry), Bull. Soc. Math. France 134 (2006), 395–415.

[10] B. Kahn, T. Yamazaki Somekawa’s K-groups and Voevodsky’s Hom groups,

arXiv:1108.2764.

[11] B. Kahn, R. Sujatha Birational motives, I (preliminary version), preprint,

2002, http://www.math.uiuc.edu/K-theory/0596/.

[12] A. S. Merkurjev R-equivalence on three-dimensional tori and zero-cycles, Algebra Number Theory 2 (2008), 69–89.

[13] A. S. Merkurjev Zero-cycles on algebraic tori, in The geometry of algebraic cycles, 119–122, Clay Math. Proc., 9, Amer. Math. Soc., Providence, RI, 2010.

[14] F. Morel, V. Voevodsky A^{1}-homotopy theory of schemes, Publ. Math. IHÉS 90 (1999), 45–143.

[15] D. Mumford Abelian varieties (corrected reprint), TIFR – Hindustan Book Agency, 2008.

[16] J.-P. Serre Morphismes universels et variétés d’Albanese, in Exposés de séminaires, 1950–1989, Doc. mathématiques 1, SMF, 2001, 141–160.

[17] M. Spiess, T. Szamuely On the Albanese map for smooth quasi-projective varieties, Math. Ann. 325 (2003), 1–17.

[18] V. Voevodsky Cohomological theory of presheaves with transfers, in E. Friedlander, A. Suslin, V. Voevodsky Cycles, transfers and motivic cohomology theories, Ann. Math. Studies 143, Princeton University Press, 2000, 88–137.

[19] V. Voevodsky Triangulated categories of motives over a field, in E. Friedlander, A. Suslin, V. Voevodsky Cycles, transfers and motivic cohomology theories, Ann. Math. Studies 143, Princeton University Press, 2000, 188–238.

Institut de Mathématiques de Jussieu, UMR 7586, Case 247, 4 place Jussieu, 75252 Paris Cedex 05, France

E-mail address: kahn@math.jussieu.fr