ANNALES
DE LA FACULTÉ
DES SCIENCES
DE TOULOUSE
Mathématiques

BRUNO KAHN

Algebraic tori as Nisnevich sheaves with transfers

Tome XXIII, n° 3 (2014), p. 699-715.

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# Algebraic tori as Nisnevich sheaves with transfers

BRUNO KAHN(1)

**ABSTRACT.** — We relate *R*-equivalence on tori with Voevodsky's theory of homotopy invariant Nisnevich sheaves with transfers and effective motivic complexes.

**RÉSUMÉ.** — On relie la *R*-équivalence sur les tores aux faisceaux Nisnevich avec transferts invariants par homotopie et aux complexes motiviques effectifs, étudiés par Voevodsky.

## Contents

<table><tr><td>1</td><td>Main results</td><td>699</td></tr><tr><td>2</td><td>Review of terminology for tori</td><td>703</td></tr><tr><td>3</td><td>Proofs of Theorem 1.2 and Corollay 1.4</td><td>704</td></tr><tr><td>4</td><td>Stable birationality</td><td>706</td></tr><tr><td>5</td><td>Some opren questions</td><td>712</td></tr><tr><td></td><td>References</td><td>715</td></tr></table>

### 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 [4] and [5]: 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, 1.7 and 1.8 below.

<footnote>(1) IMJ-PRG, UMR 7586, Case 247, 4 place Jussieu, 75252 Paris Cedex 05, France
bruno.kahn@imj-prg.fr</footnote>
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 1.1.** — There is a natural isomorphism $G_{-1} \xrightarrow{\sim} L$ in HI.

Here $\mathbb{F}_{-1}$ is the contraction operation of [18, p. 96], whose definition is recalled in the proof below.

*Proof.* — Recall that if $\mathcal{F}$ is a presheaf [with transfers] on smooth $k$-schemes, the presheaf [with transfers] $\mathcal{F}_{-1}^p$ is defined by

$$U \mapsto \text{Coker}(\mathcal{F}(U \times \mathbf{A}^1) \to \mathcal{F}(U \times \mathbb{G}_m)).$$

If $\mathcal{F}$ is homotopy invariant, we may replace $U \times \mathbf{A}^1$ by $U$ and the rational point $1 \in \mathbb{G}_m$ realises $\mathcal{F}_{-1}^p(U)$ as a functorial direct summand of $\mathcal{F}(U \times \mathbb{G}_m)$.

If $\mathcal{F}$ is a Nisnevich sheaf [with transfers], $\mathcal{F}_{-1}$ is defined as the sheaf associated to $\mathcal{F}_{-1}^p$.

Now $A(U \times \mathbf{A}^1) \xrightarrow{\sim} A(U \times \mathbb{G}_m)$ since $A$ is an abelian variety, hence $A_{-1}^p = 0$. We therefore have an isomorphism of presheaves $T_{-1}^p \xrightarrow{\sim} G_{-1}^p$, and a *fortiori* an isomorphism of Nisnevich sheaves $T_{-1} \xrightarrow{\sim} G_{-1}$.

Let $p : \mathbb{G}_m \to \text{Speck}$ be the structural map. One easily checks that the *étale* sheaf Coker($T \xrightarrow{i} p_*p^*T$) 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. $\square$

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

$$L[0] = G_{-1}[0] \simeq \text{Hom}_{\text{DM}^{\text{eff}}} (\mathbb{G}_m[0], G[0])$$

[10, Rk. 4.4], hence by adjunction a morphism in $\text{DM}^{\text{eff}}$

$$L[0] \otimes \mathbb{G}_m[0] \to G. \qquad (1.1)$$

Let $\nu_{\le 0}G[0]$ denote the cone of (1.1): by [11, Lemma 6.3] or [8, §2], $\nu_{\le 0}G[0]$ is the birational motivic complex associated to $G$. We want to compute its homology sheaves.
For this, consider a coflasque resolution¹

$$0 \to Q \to L_0 \to L \to 0 \qquad (1.2)$$

of $L$ in the sense of [4, p. 179]. Taking a coflasque resolution of $Q$ and iterating, we get a resolution of $L$ by invertible lattices:

$$\dots \to L_n \to \dots \to L_0 \to L \to 0. \qquad (1.3)$$

We set

$$Q_n = \begin{cases} Q & \text{for } n = 1 \\ \text{Ker}(L_{n-1} \to L_{n-2}) & \text{for } n > 1. \end{cases}$$

**THEOREM 1.2.** — a) Let $T_n$ denote the torus with cocharacter group $L_n$. Then $\nu_{\le 0}G[0]$ is isomorphic to the complex

$$\dots \to T_n \to \dots \to T_0 \to G \to 0.$$

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

$$H_n(\nu_{\le 0}G[0])(X) = \begin{cases} 0 & \text{if } n < 0 \\ G(K)/R & \text{if } n = 0 \\ S_n(K)/R & \text{if } n > 0. \end{cases}$$

The proof is given in Section 3.

**COROLLARY 1.3.** — The assignment $Sm(k) \ni X \mapsto \bigoplus_{x \in X^{(0)}} G(k(x))/R$ provides $G/R$ with the structure of a homotopy invariant Nisnevich sheaf with transfers. In particular, any morphism $\varphi : Y \to X$ of smooth connected $k$-schemes induces a morphism $\varphi^* : G(k(X))/R \to G(k(Y))/R$.

This functoriality is essential to formulate Theorem 1.5 below. For $\varphi$ 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. [5, Th. 3.1 and Cor. 4.2] or [7]. (I am indebted to Colliot-Thélène for pointing out these references.)

**COROLLARY 1.4.** — a) If $k$ is finitely generated, the $n$-th homology sheaf of $\nu_{\le 0}G[0]$ 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 $\nu_{\le 0}G[0] = 0$ if $G$ is split by a Galois extension $E/k$ whose Galois group has cyclic $\text{Sylow}$ subgroups. This condition is automatic if $k$ is (quasi-)finite.

<footnote>(1) See Section 2 for this and further terminology.</footnote>
The proof is also given in Section 3.

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

$$
\operatorname{Hom}_k(G, G') \to \operatorname{Hom}_{\operatorname{DM}^{\text{eff}}}(\nu_{\le 0} G[0], \nu_{\le 0} G'[0]) \to \operatorname{Hom}_{\operatorname{HI}}(G/R, G'/R).
$$

In Section 4, 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 1.5. — Let $G, G'$ be two semi-abelian varieties, with $G$ a torus.
Suppose given, for every function field $K/k$, a homomorphism $f_K :$
$G(K)/R \to G'(K)/R$ such that $f_K$ is natural with respect to the functo-riality of Corollary 1.3. Then

a) There exists an extension $\tilde{G}$ of $G$ by a permutation torus, and a
homomorphism $f : \tilde{G} \to G'$ inducing $(f_K)$.

b) $f_K$ is surjective for all $K$ if and only if there exist extensions $\tilde{G}, \tilde{G}'$ of $G$ and $G'$ by permutation tori such that $f_K$ is induced by a split surjective homomorphism $\tilde{G} \to \tilde{G}'$.

The proof is given in §4.3. See Proposition 4.7, Corollary 4.9, Remark 4.10 and Proposition 4.11 for complements.

This relates to questions of stable birationality studied by Colliot-Thélène
and Sansuc in [4] and [5], providing alternate proofs and strengthening of
some of their results (at least over a perfect field). More precisely, let us
introduce the following terminology:

*Definition 1.6.* — a) *A torus is quasi-invertible if it is a quotient of a invertible torus by a permutation torus.*

b) An extension $0 \to T' \to T \to T'' \to 0$ of tori is Nisnevich-exact if $T(K) \to T''(K)$ is surjective for any function field $K/k$.

(a) was suggested by Xun Jiang; see also [2]. See §2 for “permutation torus” and “invertible torus”.)

Thanks to [18, Cor. 4.18], Nisnevich-exact sequences of tori are exact
in the Nisnevich topology and even in the Zariski topology. It is easy to
see that an extension as in b) is Nisnevich-exact if $T'$ is invertible, but
not necessarily if $T'$ is only quasi-invertible. Using [4, Th. 2], one sees that
quasi-invertible tori are universally $R$-trivial. Conversely:
COROLLARY 1.7. — a) Let $G'$ be a semi-abelian k-variety such that $G'(K)/R = 0$ for any function field $K/k$. Then $G'$ is a quasi-invertible torus.

b) In Theorem 1.5 b), assume that $f_K$ is bijective for all $K/k$. Then there exists an extension $\tilde{G}$ of $G$ by a permutation torus and a Nisnevich-exact extension $\tilde{G}'$ of $G'$ by a quasi-invertible torus such that $f_K$ is induced by an isomorphism $\tilde{G} \stackrel{\sim}{\to} \tilde{G}'$.

*Proof*. — a) This is the special case $G = 0$ of Theorem 1.5 b).

b) By Theorem 1.5 b), we may replace $G$ and $G'$ by extensions by permutation tori such that $f_K$ is induced by a split surjection $f: G \to G'$. Let $T = \text{Ker}f$. Then $T/R = 0$ universally. By a), $T$ is quasi-invertible. Replacing $G'$ by $G' \times T$, we get the desired statement. $\square$

Corollary 1.7 a) is a version of [5, Prop. 7.4] (taking [4, p. 199, Th. 2] into account). Theorem 1.5 was inspired by the desire to understand this result from a different viewpoint. Another characterisation of quasi-invertible tori in loc. cit. is that they are the retract-rational tori.

COROLLARY 1.8. — Let $f: G \to G'$ be a rational map of semi-abelian varieties, with $G$ a torus. Then the following conditions are equivalent:

(i) $f_*: \nu_{\le 0} G[0] \to \nu_{\le 0} G'[0]$ is an isomorphism (see Proposition 4.7).

(ii) $f_*: G(K)/R \to G'(K)/R$ is bijective for any function field $K/k$.

(iii) $f$ is an isomorphism, up to Nisnevich-exact extensions of $G$ and $G'$ by quasi-invertible tori and up to a translation. (See Lemma 4.4.)

**Acknowledgements.** — Part of Theorem 1.2 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, Xun Jiang for pointing out some errors and the referee for suggesting expository improvements. Finally, I wish to acknowledge inspiration from the work of Colliot-Thélène and Sansuc, which will be obvious throughout this paper.

## 2. Review of terminology for tori

We take this terminology from [4] and [5].

*Definition 2.1 Let $G$ be a profinite group.*

a) A lattice is a $G$-module which is finitely generated and free over $\mathbb{Z}$.
b) A lattice L is

• permutation if it affords a $G$-invariant $\mathbb{Z}$-basis.

• invertible if it is isomorphic to a direct summand of a permutation lattice.

• coflasque if $H^1(H, L) = 0$ for any open (hence closed) subgroup $H \subseteq G$.

• flasque if the dual lattice $L^*$ is coflasque.

c) A coflasque resolution of a lattice L is a short exact sequence of lattices

$$0 \to Q \to P \to L \to 0$$

where P is permutation and Q is coflasque. Dually, we have flasque [co]resolution

$$0 \to L \to P \to F \to 0$$

with P permutation and F flasque.

**PROPOSITION 2.2 ([4, p. 181, LEMME 3]).** — Any lattice has a flasque and a coflasque resolution.

In [5, Lemma 0.6], the first statement of c) is extended to $G$-modules which are finitely generated over $\mathbb{Z}$ but not necessarily free.

Let $k_s$ be a separable closure of the field $k$ and take $G = \text{Gal}(k_s/k)$. Let $T$ be a $k$-torus: we shall say that it is *permutation*, *invertible*, *flasque*, *coflasque*, if its character group is (Colliot-Thélène and Sansuc use *quasi-trivial* for “permutation”). Any permutation torus is of the form $R_{E/k}\mathbb{G}_m$ (Weil restriction of scalars) for some *étale* $k$-algebra $E$.

### 3. Proofs of Theorem 1.2 and Corollary 1.4

**LEMMA 3.1.** — The exact sequence

$$0 \to T(k) \to G(k) \to A(k)$$

induces an exact sequence

$$0 \to T(k)/R \xrightarrow{i} G(k)/R \to A(k).$$

*Proof.* — Let $f: \mathbb{P}^1 \longrightarrow G$ be a $k$-rational map defined at 0 and 1. Its composition with the projection $G \to A$ 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. $\square$
Let NST denote the category of Nisnevich sheaves with transfers. Recall that $DM_{-}^{\text{eff}}$ may be viewed as a localisation of $D^{-}(\text{NST})$, and that its tensor structure is a descent of the tensor structure on the latter category [19, Prop. 3.2.3].

**LEMMA 3.2.** — If $G$ is an invertible torus, there is a canonical isomorphism in $D^{-}(\text{NST})$

$$L[0] \otimes \mathbb{G}_m \xrightarrow{\sim} G[0].$$

In particular, $\nu_{\le 0}G[0] = 0$.

*Proof.* — We reduce to the case $T = R_{E/k}\mathbb{G}_m$, where $E$ is a finite extension of $k$. Let us write more precisely $\text{NST}(k)$ and $\text{NST}(E)$. There is a pair of adjoint functors

$$\text{NST}(k) \xrightarrow{f^*} \text{NST}(E), \quad \text{NST}(E) \xrightarrow{f_*} \text{HI}(k)$$

where $f : \text{Spec}E \to \text{Spek}$ is the projection. Clearly,

$$f_*\mathbf{Z} = \mathbf{Z}_{\text{tr}}(\text{Spec}E), \quad f_*\mathbb{G}_m = T$$

where $\mathbf{Z}_{\text{tr}}(\text{Spec}E)$ is the Nisnevich sheaf with transfers represented by $\text{Spec}E$. Since $\mathbf{Z}_{\text{tr}}(\text{Spec}E) = L$, this proves the claim. $\square$

*Proof of Theorem 1.2.* — a) Recall that $L_0$ is an invertible lattice chosen so that $L_0(E) \to L(E)$ is surjective for any extension $E/k$. In particular, (1.2) and (1.3) are exact as sequences of Nisnevich sheaves; hence $L[0]$ is isomorphic in $D^{-}(\text{NST})$ to the complex

$$L. = \dots \to L_n \to \dots \to L_0 \to 0.$$

(We may view (1.3) as a version of Voevodsky’s “canonical resolutions” as in [19, §3.2 p. 206].)

By Lemma 3.2, $L_n[0] \otimes \mathbb{G}_m[0] \simeq T_n[0]$ is homologically concentrated in degree 0 for all $n$. It follows that the complex

$$T. = \dots \to T_n \to \dots \to T_0 \to 0$$

is isomorphic to $L[0] \otimes \mathbb{G}_m[0]$ in $D^{-}(\text{NST})$, hence a fortiori in $DM_{-}^{\text{eff}}$.

b) For any nonempty open subscheme $U \subseteq X$ we have isomorphisms

$$H_n(\nu_{\le 0} G[0])(X) \xrightarrow{\sim} H_n(\nu_{\le 0} G[0])(U) \xrightarrow{\sim} H_n(\nu_{\le 0} G[0])(K) \quad (3.1)$$

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

$$\dots \to T_n(K) \to \dots \to T_0(K) \to G(K) \to 0$$
with $G(K)$ in degree 0. By [4, p. 199, Th. 2], the sequences

$$ 
\begin{array}{l}
0 \to S_1(K) \to T_0(K) \to T(K) \to T(K)/R \to 0 \\
0 \to S_{n+1}(K) \to T_n(K) \to S_n(K) \to S_n(K)/R \to 0
\end{array}
$$

are all exact. Using Lemma 3.1 for $H_0$, the conclusion follows from an easy diagram chase. □

*Remark 3.3.* — As a corollary to Theorem 1.2, $S_n(K)/R$ only depends on $G$. This can be seen without mentioning $\mathrm{DM}^\mathrm{eff}$: in view of the reasoning just above, it suffices to construct a homotopy equivalence between two resolutions of the form (1.3), which easily follows from the definition of coflasque modules.

*Proof of Corollary 1.4.* — a) This follows via Theorem 1.2 and Lemma 3.1 from [4, 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 1.2 and [4, p. 200, Cor. 3]. The last claim is clear. □

*Remark 3.4.* — In characteristic $p > 0$, all finitely generated perfect fields are finite. To give some contents to Corollary 1.4 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

$$ G(k_2)^{N_{k_2/k_1}} \xrightarrow{k_1} G(k_1) \to G(k_2) $$

equals multiplication by $[k_2 : k_1]$. Hence Corollary 1.4 a) remains true at least after inverting $p$.

## 4. Stable birationality

If $X$ is a smooth variety over a field $k$, we write $\mathrm{Alb}(X)$ for its generalised Albanese variety in the sense of Serre [16]: it is a semi-abelian variety, and a rational point $x_0 \in X$ determines a morphism $X \to \mathrm{Alb}(X)$ which is universal for morphisms from $X$ to semi-abelian varieties sending $x_0$ to 0.

We also write $\mathrm{NS}(X)$ 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].
## 4.1. Well-known lemmas

I include proofs for lack of reference.

**LEMMA 4.1.** — a) Let $G, G'$ be two semi-abelian $k$-varieties. Then any $k$-morphism $f: G \to G'$ can be written uniquely $f = f(0) + f'$, where $f'$ is a homomorphism.

b) For any semi-abelian $k$-variety $G$, the canonical map $G \to \text{Alb}(G)$ 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 statement 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 $g \in G(k)$. As any morphism from $T$ to $A'$ is constant, the $k$-morphism

$$ \varphi_g : T \ni t \mapsto f(g+t) - f(g) \in G' $$

(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 $\varphi: A \to \text{Hom}(T, T')$, which must be constant with value $\varphi_0 = f$. It follows that

$$ (g, h) \mapsto f(g + h) - f(g) - f(h) $$

induces a morphism $A \times A \to T'$. 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 $\text{Alb}(G)$ is $A$. Let $T' = \text{Ker}(\text{Alb}(G) \to A)$. We also have the counit morphism $\text{Alb}(G) \to G$, and the composition $G \to \text{Alb}(G) \to G$ 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 $T = \mathbb{G}_m$.

Then consider the line bundle completion $\bar{G} \to A$ of the $\mathbb{G}_m$-bundle $G \to A$. It is sufficient to show that the kernel of

$$ \text{Alb}(G) \to \text{Alb}(\bar{G}) = A $$

is 1-dimensional. This follows for example from [1, Cor. 10.5.1]. □
LEMMA 4.2. — Suppose $k$ algebraically closed, and let $G$ be a semi-abelian $k$-variety. Let $A$ be the abelian quotient of $G$. Then the map

$$ \mathrm{NS}(A) \to \mathrm{NS}(G) \qquad (4.2) $$

is an isomorphism.

*Proof.* — Let $T = \mathrm{Ker}(G \to A)$ and $X(T)$ be its character group. Choosing a basis $(e_i)$ of $X(T)$, we may complete the $\mathbb{G}_m^n$-torsor $G$ into a product of line bundles $\bar{G} \to A$. The surjection

$$ \mathrm{Pic}(A) \xrightarrow{\sim} \mathrm{Pic}(\bar{G}) \twoheadrightarrow \mathrm{Pic}(G) $$

show the surjectivity of (4.2). Its kernel is generated by the classes of the irreducible components $D_i$ of the divisor with normal crossings $\bar{G}-G$. These components correspond to the basis elements $e_i$. Since the corresponding $\mathbb{G}_m$-bundle is a group extension of $A$ by $\mathbb{G}_m$, the class of the 0 section of its line bundle completion lies in $\mathrm{Pic}^0(A)$, hence goes to 0 in $\mathrm{NS}(A)$. $\square$

LEMMA 4.3. — Let $X$ be a smooth $k$-variety, and let $U \subseteq X$ be a dense open subset. Then there is an exact sequence of semi-abelian varieties

$$ 0 \to T \to \mathrm{Alb}(U) \to \mathrm{Alb}(X) \to 0 $$

with $T$ a torus. If $\mathrm{NS}(U \otimes_k \bar{k}) = 0$ (this happens if $U$ is small enough), there is an exact sequence of character groups

$$ 0 \to X(T) \to \bigoplus_{x \in X^{(1)} - U^{(1)}} \mathbb{Z} \to \mathrm{NS}(\bar{X}) \to 0. $$

*Proof.* — This follows for example from [1, Cor. 10.5.1]. $\square$

LEMMA 4.4. — Let $f : G \to G'$ be a rational map between semi-abelian $k$-varieties, with $G$ a torus. Then there exists an extension $\tilde{G}$ of $G$ by a permutation torus and a homomorphism $\tilde{f} : \tilde{G} \to G'$ which extends $f$ up to translation in the following sense: there exists a rational section $s : G \to \tilde{G}$ of the projection $\pi : \tilde{G} \to G$ and a rational point $g' \in G'(k)$ such that $f = \tilde{f}s + g'$. 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 $\tilde{G} = \mathrm{Alb}(U)$. Applying Lemmas 4.3 and 4.1 b) and using $\mathrm{NS}(G \otimes_k \bar{k}) = 0$, we get an extension

$$ 0 \to P \to \tilde{G} \to G \to 0 $$

where $P$ is a permutation torus, as well as a morphism $\tilde{f} = \mathrm{Alb}(f) : \tilde{G} \to G'$.
Let us first assume $k$ infinite. Then $U(k) \neq \emptyset$ because $G$ is unirational. A rational point $g \in U$ defines an Albanese map $s : U \to \tilde{G}$ sending $g$ to $0_{\tilde{G}}$. Since $P$ is a permutation torus, $g \in G(k)$ lifts to $\tilde{g} \in \tilde{G}(k)$ (Hilbert 90) and we may replace $s$ by a morphism sending $g$ to $\tilde{g}$. Then $s$ is a rational section of $\pi$. Moreover, $f = \tilde{f}s + g'$ with $g' = f(g) - \tilde{f}(\tilde{g})$. 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 $\tilde{G}$ with the same residue field). $\square$

**LEMMA 4.5.** — 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 $0 \to P \to F \to A \to 0$, with $F$ torsion-free and flasque, and $P$ permutation.

b) Let $B$ be another finitely generated $G$-module, and let $0 \to P' \to E \to B \to 0$ be an exact sequence with $P'$ an invertible module. Then any $G$-morphism $f : A \to B$ lifts to $\tilde{f} : F \to E$.

*Proof.* — a) is the contents of [5, Lemma 0.6, (0.6.2)]. b) The obstruction to lifting $f$ lies in $\text{Ext}_G^1(F, P') = 0$ [4, p. 182, Lemme 9]. $\square$

## 4.2. Functionality of $\nu_{\le 0} G$

We now assume $k$ perfect.

**LEMMA 4.6.** — Let

$$0 \to P \to G \to H \to 0 \qquad (4.3)$$

be an exact sequence of semi-abelian varieties, with $P$ an invertible torus. Then $\nu_{\le 0} G[0] \xrightarrow{\sim} \nu_{\le 0} H[0]$.

*Proof.* — As $P$ is invertible, (4.3) is exact in $\text{NST}$ hence defines an exact triangle

$$P[0] \to G[0] \to H[0] \xrightarrow{+1}$$

in $\text{DM}^\text{eff}_\le$. The conclusion then follows from Lemma 3.2. $\square$

**PROPOSITION 4.7.** — Let $G, G'$ be two semi-abelian $k$-varieties, with $G$ a torus. Then a rational map $f : G \to G'$ induces a morphism $f_* : \nu_{\le 0} G[0] \to \nu_{\le 0} G'[0]$, hence a homomorphism $f_* : G(K)/R \to G'(K)/R$ for any extension $K/k$. If $K$ is infinite, $f_*$ agrees up to translation with the morphism induced by $f$ via the isomorphism $U(K)/R \to G(K)/R$ from [4, p. 196 Prop. 11], where $U$ is an open subset of definition of $f$.
*Proof.* — By Lemma 4.4, $f$ induces a homomorphism $\tilde{G} \to G'$ where $\tilde{G}$ is an extension of $G$ by a permutation torus. By Lemma 4.6, the induced morphism

$$ \nu_{\le 0} \tilde{G}[0] \to \nu_{\le 0} G'[0] $$

factors through a morphism $f_* : \nu_{\le 0} G[0] \to \nu_{\le 0} G'[0]$.

The claims about $R$-equivalence classes follow from Theorem 1.2 b) and Lemma 4.4. $\square$

*Remark 4.8.* — The proof shows that $f'_* = f_*$ if $f'$ differs from $f$ by a translation by an element of $G(k)$ or $G'(k)$.

**COROLLARY 4.9.** — If $T$ and $T'$ are birationally equivalent $k$-tori, then $\nu_{\le 0} T[0] \simeq \nu_{\le 0} T'[0]$. In particular, the groups $T(k)/R$ and $T'(k)/R$ are isomorphic.

*Proof.* — The proof of Proposition 4.7 shows that $f \mapsto f_*$ is functorial for composable rational maps between tori. Let $f : T \dashrightarrow T'$ be a birational isomorphism, and let $g : T' \dashrightarrow T$ be the inverse birational isomorphism. Then we have $g_*f_* = 1_{\nu_{\le 0} T[0]}$ and $f_*g_* = 1_{\nu_{\le 0} T'[0]}$. The last claim follows from Theorem 1.2. $\square$

*Remark 4.10.* — It is proven in [4] that a birational isomorphism of tori $f : T \dashrightarrow T'$ induces a set-theoretic bijection $f_* : T(k)/R \dashrightarrow T'(k)/R$ (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 [4, mid. p. 397]. The proofs of Lemma 4.4 and Proposition 4.7 may be seen as dual to the proof of [4, p. 189, Prop. 5], and are directly inspired from it.

## 4.3. Faithfulness and fullness

**PROPOSITION 4.11.** — Let $f : G \dashrightarrow G'$ be a rational map between semi-abelian varieties, with $G$ a torus. Assume that the map $f_* : G(K)/R \to G'(K)/R$ from Proposition 4.7 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

$$ G \xrightarrow{\tilde{f}} P \xrightarrow{g} G' $$

where $\tilde{f}$ is a rational map and $g$ is a homomorphism. If $f$ is a morphism, we may choose $\tilde{f}$ to be a homomorphism.
Conversely, if there is such a factorisation, then $f_* : \nu_{\le 0} G[0] \to \nu_{\le 0} G'[0]$ is the 0 morphism.

*Proof.* — By Lemma 4.4, we may reduce to the case where $f$ is a homomorphism. Let $K = k(G)$. By hypothesis, the image of the generic point $\eta_G \in G(K)$ 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 $h : G \times \mathbf{A}^1 \dashrightarrow G'$, defined in the neighbourhood of 0 and 1, such that $h|_{G \times \{0\}} = 0$ and $h|_{G \times \{1\}} = f$.

Let $U \subseteq G \times \mathbf{A}^1$ be an open set of definition of $h$. The 0 and 1-sections of $G \times \mathbf{A}^1 \to G$ induce sections

$$s_0, s_1 : G \to \mathrm{Alb}(U)$$

of the projection $\pi : \mathrm{Alb}(U) \to \mathrm{Alb}(G \times \mathbf{A}^1) = G$ such that $\mathrm{Alb}(h) \circ s_0 = 0$ and $\mathrm{Alb}(h) \circ s_1 = f$. If $P = \mathrm{Ker}\pi$, then $s_1 - s_0$ induces a homomorphism $\tilde{f} : G \to P$ such that the composition

$$G \xrightarrow{\tilde{f}} P \to \mathrm{Alb}(U) \xrightarrow{\mathrm{Alb}(h)} G'$$

equals $f$. Finally, $P$ is a permutation torus by Lemma 4.3.

The last claim follows from Lemma 3.2. $\square$

<figure><img src="image_1.png" /></figure>

*Proof of Theorem 1.5.* — a) Take $K = k(G)$. The image of the generic point $\eta_G$ by $f_K$ lifts to a (non unique) rational map $f : G \dashrightarrow G'$. Using Lemma 4.4, we may extend $f$ to a homomorphism

$$\tilde{f} : \tilde{G} \to G'$$

where $\tilde{G}$ is an extension of $G$ by a permutation torus $P$. Since $\tilde{G}(K)/R \xrightarrow{\sim} G(K)/R$, we reduce to $\tilde{G} = G$ and $\tilde{f} = f$.

Let $L/k$ be a function field, and let $g \in G(L)$. Then $g$ arises from a morphism $g : X \to G$ for a suitable smooth model $X$ of $L$. By assumption on $K \mapsto f_K$, the diagram

$$\begin{tikzcd}
G(K)/R \arrow[r, "f_K"] & G'(K)/R \\
g^* \arrow[u, "\downarrow"] & g^* \arrow[u, "\downarrow"] \\
G(L)/R \arrow[r, "f_L"] & G'(L)/R
\end{tikzcd}$$

commutes. Applying this to $\eta_K \in G(K)$, we find that $f_L([g]) = [g \circ f]$, 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 4.5 a) to the cocharacter group of $G$, we then get a resolution $0 \to P_1 \to Q \to G \to 0$ with $Q$ coflasque and $P_1$ permutation. Hence we may (and do) further assume $G$ coflasque.

Let $K = k(G')$ and choose some $g \in G(K)$ mapping modulo $R$-equivalence to the generic point of $G'$. Then $g$ defines a rational map $g: G' \to G$ such that $fg$ is $R$-equivalent to $1_{G'}$. It follows that the induced map

$$1 - fg: G'/R \to G'/R \qquad (4.4)$$

is identically 0.

Reapplying Lemma 4.4, we may find an extension $\tilde{G}'$ of $G'$ by a suitable permutation torus which converts $g$ into a true homomorphism. Since $G$ is coflasque, Lemma 4.5 b) shows that $f: G \to G'$ lifts to $\tilde{f}: G \to \tilde{G}'$. Then (4.4) is still identically 0 when replacing $(G', f)$ by $(\tilde{G}', \tilde{f})$.

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 $g: G' \to G$ such that (4.4) vanishes identically. Hence $1 - fg$ factors through a permutation torus $P$ thanks to Proposition 4.11. Write $u: G' \to P$ and $v: P \to G'$ for homomorphisms such that $1 - fg = vu$. Let $G_1 = G \times P$ and consider the maps

$$f_1 = (f, v): G_1 \to G', \quad g_1 = (g) u: G' \to G_1.$$

Then $f_1 g_1 = 1$ and $G'$ is a direct summand of $G_1$ as requested. $\square$

## 5. Some open questions

*Question 5.1.* — Are lemma 4.4 and Proposition 4.7 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 5.2.** — The answer to Question 5.1 is yes if the abelian part A of G is an elliptic curve.
*Proof*. — Arguing as in the proof of Proposition 4.7, we get for an open subset $U \subseteq G$ of definition for $f$ an exact sequence

$$0 \to \mathbb{G}_m \to P \to \mathrm{Alb}(U) \to G \to 0$$

where $P$ is a permutation torus. Here we used that $\mathrm{NS}(\bar{G}) \simeq \mathbb{Z}$, which follows from Lemma 4.2.

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 $0 \in A$. As the divisor class of 0 generates $\mathrm{NS}(\bar{A})$, $D$ provides a Galois-equivariant splitting of the map $\mathbb{G}_m \to P$. Thus its cokernel is still a permutation torus, and we conclude as before. $\square$

*Question 5.3.* — Can one formulate a version of Theorem 1.5 and Corollary 1.7 providing a description of the groups $\mathrm{Hom}_{\mathrm{DM}^{\mathrm{eff}}}(\nu_{\le 0}G[0], \nu_{\le 0}G'[0])$ and $\mathrm{Hom}_{\mathrm{HI}}(G/R, G'/R)$ (at least when $G$ and $G'$ are tori)?

The proof of Theorem 1.5 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 $M(G) \in \mathrm{DM}^{\mathrm{eff}}$. Recall the canonical morphism

$$M(G) \to G[0]$$

induced by the “sum” maps

$$c(X, G) \xrightarrow{\sigma} G(X) \qquad (5.1)$$

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

The morphism (5.1) has a canonical section

$$G(X) \xrightarrow{\gamma} c(X, G) \qquad (5.2)$$

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

Consider now a smooth equivariant compactification $\bar{G}$ of $G$. It exists in all characteristics. For tori, this is written up in [3]. 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 $G \times^T Y$ associated to the $T$-torsor $G \to A$ also exists: this is the desired compactification.

Then we have a diagram of birational motives

$$ \begin{array}{ccc} \nu_{\le 0} M(G) & \xrightarrow{\sim} & \nu_{\le 0} M(\bar{G}) \\ \nu_{\le 0} \sigma & \longrightarrow & \\ \nu_{\le 0} G[0]. & \longrightarrow & \end{array} \qquad (5.3) $$

By [11], we have $H_0(\nu_{\le 0} M(\bar{G}))(X) = CH_0(\bar{G}_{k(X)})$ for any smooth connected $X$. Hence the above diagram induces a homomorphism

$$ CH_0(\bar{G}_{k(X)}) \to G(k(X))/R \qquad (5.4) $$

which is natural in $X$ for the action of finite correspondences (compare Corollary 1.3). One can probably check that this is the homomorphism of [12, (17) p. 78], reformulating [4, Proposition 12 p. 198]. Similarly, the set-theoretic map

$$ G(k(X))/R \to CH_0(\bar{G}_{k(X)}) \qquad (5.5) $$

of [4, p. 197] can presumably be recovered as a birational version of (5.2), using perhaps the homotopy category of schemes of Morel and Voevodsky [14].

In [12], Merkurjev shows that (5.4) is an isomorphism for $G$ a torus of dimension at most 3. This suggests:

**Question 5.4.** — Is the map $\nu_{\le 0} \sigma$ of Diagram (5.3) an isomorphism when $G$ is a torus of dimension $\le 3$?

In [13], Merkurjev gives examples of tori $G$ for which (5.5) is not a homomorphism; hence its (additive) left inverse (5.4) cannot be an isomorphism. Merkurjev's examples are of the form $G = R_{K/k}^1 \mathbb{G}_m \times R_{L/k}^1 \mathbb{G}_m$, where $K$ and $L$ are distinct biquadratic extensions of $k$. This suggests:

**Question 5.5.** — Can one study Merkurjev's examples from the above viewpoint? More generally, what is the nature of the map $\nu_{\le 0} \sigma$ of Diagram (5.3)?

We leave all these questions to the interested reader.
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