BRAUER EQUIVALENCE IN A HOMOGENEOUS SPACE WITH CONNECTED STABILIZER

MIKHAIL BOROVOI AND BORIS KUNYAVSKIĬ

Abstract. Let G be a simply connected algebraic group over a field k of characteristic 0, H a connected k-subgroup of G, X = H\G. When k is a local field or a number field, we compute the set of Brauer equivalence classes in X(k).

0. Introduction

In this note we investigate the Brauer equivalence in a homogeneous space X = H\G, where G is a simply connected algebraic group over a local field or a number field, and H is a connected subgroup of G.

In more detail, let k be a field of characteristic 0, and let k̄ be a fixed algebraic closure of k. For a smooth algebraic variety Y over k, set \overline{Y} = Yk̄ = Y ×_{k} k̄. Let Br Y denote the cohomological Brauer group of Y, Br Y = H_{ét}^{2}(Y, 𝔾_{m}). Set Br_{1} Y = ker[Br Y → Br \overline{Y}]. There is a canonical pairing

Y (k) × Br_{1} Y → Br k,  (y, b) ↦ b(y)  (0.1)

called the Manin pairing. We define the Brauer equivalence on Y (k) as follows: y_{1} ∼ y_{2} if (y_{1}, b) = (y_{2}, b) for all b ∈ Br_{1} Y. We denote the set of classes of Brauer equivalence in Y (k) by Y (k)/Br . Note that we define the Brauer equivalence in terms of Br_{1} Y, not in terms of Br_{1} Y^{c} or Br Y^{c}, where Y^{c} is a smooth compactification of Y.

The notion of B-equivalence for a subgroup B of the Brauer group Br Y was introduced by Manin [Ma1], [Ma2]. Colliot-Thélène and Sansuc [CT/Sa1] investigated the Brauer equivalence in algebraic tori (they defined the Brauer equivalence in terms of the Brauer group of a smooth

This research was supported by the Israel Science Foundation founded by the Israel Academy of Sciences and Humanities — Center of Excellence Program.

The first named author was partially supported by the Hermann Minkowski Center for Geometry.

The second named author was partially supported by the Ministry of Absorption (Israel) and the Minerva Foundation through the Emmy Noether Research Institute of Mathematics.

2 MIKHAIL BOROVOI AND BORIS KUNYAVSKIĬ

compactification). The Brauer equivalence in reductive groups was studied in [Th].

Let G be a simply connected semisimple algebraic group over k. Let H be a connected subgroup of G. We denote by H^{tor} the biggest toric quotient group of H. We are interested in the Brauer equivalence in the set X(k) where X = H\G.

We compute X(k)/Br when k is a local field. Namely, we prove that there is a bijection

X(k)/Br ⟶^{∼}im [ker[H^{1}(k, H) → H^{1}(k, G)] → H^{1}(k, H^{tor})]

(Theorem 2.1). Moreover, when k is a nonarchimedean local field, we prove that there is a bijection X(k)/Br → H^{1}(k, H^{tor}) (Theorem 2.2).

We also compute X(k)/Br when k is a number field. We prove that there is a bijection

X(k)/Br ⟶^{∼}im [ker[H^{1}(k, H) → H^{1}(k, G)] → ⨁_{v} H^{1}(k_{v}, H^{tor})]

(Theorem 3.1), where v runs over the set of places of k. Moreover when k is a totally imaginary number field, we prove that there is a bijection

X(k)/Br ⟶^{∼}H^{1}(k, H^{tor})/� ^{1}(k, H^{tor})

(Theorem 3.6), where � ^{1} denotes the Shafarevich–Tate kernel.

In Example 3.11 we compute X(k)/Br when X is a symmetric space of a simply connected almost simple group over a totally imaginary number field.

Note that all our results remain true when G is any simply connected algebraic group (an extension of a simply connected semisimple group by a unipotent group), not necessarily a simply connected semisimple group.

Acknowledgements. The paper was finished while the authors were visiting Sonderforschungsbereich 343 “Diskrete Strukturen in der Mathematik” (University of Bielefeld) whose hospitality and support are gratefully appreciated. We thank D. Harari and A. N. Skorobogatov for help in proving Theorem 1.5.

1. Generalities over an arbitrary field

1.1. We introduce some notation. For a smooth algebraic variety Y over a field k of characteristic 0 let U(Y) = k[Y]^{×}/k^{×}. Let Pic Y denote the Picard group of Y. Let Br Y and Br_{1} Y be as in the Introduction.

Set Br_{a} Y = coker [Br k → Br_{1} Y]. Assume that Y has a k-rational point y, and define

Br_{y}Y = ker[Br_{1} Y ⟶^{y^{∗}} Br k]

where y^{∗} is the specialization map.

We prove that Br_{y}Y ≃ Br_{a} Y. Consider the composed map

Br k → Br_{1} Y ⟶^{y^{∗}} Br k ,

it is the identity of Br k. It follows that the exact sequence

0 → Br_{y}Y → Br_{1} Y ⟶^{y^{∗}} Br k → 0

splits, and we obtain an isomorphism Br_{y}Y ⊕ Br k ≃ Br_{1} Y. Thus we obtain an isomorphism Br_{y}Y → Br_{a} Y and a splitting Br_{a} Y → Br_{1} Y of the exact sequence

0 → Br k → Br_{1} Y → Br_{a} Y → 0.

1.2. We wish to investigate the Brauer equivalence in homogeneous spaces. Let G be a simply connected semisimple algebraic group over a field k of characteristic 0. Let H ⊂ G be a connected k-subgroup. Set X = H\G; then X is a right homogeneous space of G with connected stabilizer. The variety X has a distinguished k-rational point x_{0}, the image in X(k) of the unit element e ∈ G(k).

We construct a map X(k) → H^{1}(k, H^{tor}) taking x_{0} to 1. From an exact sequence in Galois cohomology of nonabelian groups (cf. [Se], I-5.4, Prop. 36) we obtain a canonical bijection

X(k)/G(k) ⟶^{∼} ker[H^{1}(k, H) → H^{1}(k, G)] .

Thus we obtain a map X(k) → H^{1}(k, H). Composing this map with the canonical map H^{1}(k, H) → H^{1}(k, H^{tor}) induced by the homomorphism H → H^{tor}, we obtain a map

α: X(k) → H^{1}(k, H^{tor}) .  (1.1)

By construction this map is constant on the orbits of G(k) in X(k).

Let X(H) denote the group of k-characters of H, i.e. X(H) = Hom_{k}(H, 𝔾_{m}). We have X(H) = X(H^{tor}).

Proposition 1.3. There is a canonical isomorphism β: X(H)⟶^{∼}Pic X.

Proof. Let χ ∈ X(H). Set L(χ) = H\(𝔾_{a} × G), where H acts on 𝔾_{a} × G by h ∗ (u, g) = (χ(h)u, hg) (here h ∈ H, u ∈ 𝔾_{a}, g ∈ G). There is a canonical map L(χ) → H\G = X, and one can easily see that L(χ) is a linear bundle over X. We denote its class in Pic X by β(χ). One can easily see that β: X(H) → Pic X is a homomorphism.

4 MIKHAIL BOROVOI AND BORIS KUNYAVSKIĬ

We wish to prove that β is an isomorphism. We need the following fact: U(G) = 1. Indeed, by Rosenlicht’s theorem [Ro] U(G) = X(G) and clearly X(G) = 1 because G is semisimple. We need also the fact that Pic G = 0, cf. [Sa], 6.9(iv).

We construct a map which is inverse to β. We follow an idea of Mumford [Mu], Ch. I, §2. Let π denote the canonical map G → X = H\G. Let L → X be a line bundle over X. Consider the pullback π^{∗}L, it is a line bundle over G. Since Pic G = 0, the line bundle π^{∗}L is trivial. This means that there exists an isomorphism

μ: π^{∗}L → 𝔾_{a} × G .

There is a canonical action of H on π^{∗}L, and using μ we obtain an action of H on 𝔾_{a} × G extending the standard action on G, h ∗ g = hg. The action of H on 𝔾_{a} × G can be written as

h ∗ (u, g) = (χ_{g}(h)u, hg) where h ∈ H, g ∈ G, u ∈ 𝔾_{a},

and χ: G×H → 𝔾_{m} is a regular map. If we fix h, then χ_{g}(h): G → 𝔾_{m} is a regular map, in other words it is a nowhere zero regular function on G. Since U(G) = 1, we conclude that χ_{g}(h) is constant in g. We may therefore write χ(h) instead of χ_{g}(h).

Since h_{1} ∗ (h_{2} ∗ (u, g)) = (h_{1}h_{2}) ∗ (u, g), we obtain that χ(h_{1})χ(h_{2}) = χ(h_{1}h_{2}). Hence χ: H → 𝔾_{m} is a regular homomorphism, χ ∈ X(H).

If we start from another isomorphism

μ′: π^{∗}L → 𝔾_{a} × G ,

then μ′ differs from μ by a nowhere zero regular function f on G, and this f must be a constant because U(G) = 1. Then the homomorphism χ is replaced by χ′ where χ′_{g}(h) = χ_{g}(h)f(hg)f(g)^{−1}, i.e. χ′ = χ. Thus χ is defined uniquely by L. One can easily check that we have constructed a homomorphism δ: Pic X → X(H) and that this homomorphism is inverse to β. Thus β is an isomorphism.

1.3.1. Remark. We will use Proposition 1.3 only in the case when k is algebraically closed. In this case Proposition 1.3 was proved in [Po], Cor. to Thm. 4.

1.4. We have seen in the proof of Proposition 1.3 that U(G) = 1. Hence U(\overline{G}) = 1. It follows that U(\overline{X}) = 1.

Since X(k) ≠ ∅ and U(\overline{X}) = 1, we have by [Sa], 6.3(iii)

Br_{a} X = H^{1}(k, Pic \overline{X}) .

We have Br_{x_{0}} X ≃ Br_{a} X. By Proposition 1.3, Pic \overline{X} = X(\overline{H}). We obtain

Br_{x_{0}} X = H^{1}(k, X(\overline{H})) = H^{1}(k, X(\overline{H}^{tor})) .  (1.2)

There is a canonical cup product pairing

H^{1}(k, H^{tor}) × H^{1}(k, X(\overline{H}^{tor})) → Br k .  (1.3)

The pairing (1.3) together with the map (1.1) X(k) → H^{1}(k, H^{tor}) and the isomorphism (1.2) defines a pairing

X(k) × Br_{x_{0}} X → Br k .  (1.4)

Theorem 1.5. The pairing (1.4) up to sign coincides with the restriction of the Manin pairing (0.1) to X(k) × Br_{x_{0}}X ⊂ X(k) × Br_{1} X.

This theorem will be proved in the Appendix.

2. Brauer equivalence over a local field

Theorem 2.1. Let G, H, X be as in 1.2. Assume that k is a local field (archimedean or not). Then the map (1.1) α: X(k) → H^{1}(k, H^{tor}) induces a bijection

X(k)/Br ⟶^{∼}im [ker[H^{1}(k, H) → H^{1}(k, G)] → H^{1}(k, H^{tor})] .

Proof. It follows from Theorem 1.5 that two points x_{1}, x_{2} ∈ X(k) are Brauer-equivalent if and only if (α(x_{1}), η) = (α(x_{2}), η) for every η ∈ H^{1}(k, X(\overline{H}^{tor})). Since k is a local field, the cup product pairing (1.3) is perfect (Tate–Nakayama duality, cf. [Mi], Cor. I.2.4), and it follows that x_{1} and x_{2} are Brauer-equivalent if and only if α(x_{1}) = α(x_{2}). Thus the set of classes of Brauer equivalence is in a bijective correspondence with im α. We see that we must only describe the image of X(k) in H^{1}(k, H^{tor}). But the image of X(k) in H^{1}(k, H) is the same as the image of X(k)/G(k) and it equals ker[H^{1}(k, H) → H^{1}(k, G)]. Hence the image of X(k) in H^{1}(k, H^{tor}) is

im [ker[H^{1}(k, H) → H^{1}(k, G)] → H^{1}(k, H^{tor})] ,

and the assertion of the theorem follows.

Theorem 2.2. Let G, H, X be as in 1.2, and assume that k is a nonarchimedean local field. Then the map (1.1) α induces a bijection

X(k)/Br → H^{1}(k, H^{tor}).

Proof. Since G is a simply connected group, by Kneser’s theorem (see [Pl/Ra], 6.1, Thm. 4) H^{1}(k, G) = 1. We see now from Theorem 2.1 that X(k)/Br is in a bijective correspondence with im [H^{1}(k, H) → H^{1}(k, H^{tor})]. Let H^{ssu} denote ker[H → H^{tor}], it is an extension of a semisimple group by a unipotent group. Since k is local nonarchimedean and (H^{ssu})^{tor} = 1, the map H^{1}(k, H) → H^{1}(k, H^{tor}) is surjective, cf. [Bo], Cor. 6.4. This proves the theorem.

6 MIKHAIL BOROVOI AND BORIS KUNYAVSKIĬ

3. Brauer equivalence over a number field

Theorem 3.1. Let k be a number field, and let G, H, X be as in 1.2. Then the map

X(k) → X(k)/G(k)⟶^{∼} ker[H^{1}(k, H) → H^{1}(k, G)] → ⨁_{v} H^{1}(k_{v}, H^{tor})

induces a bijection

X(k)/Br ⟶^{∼}im [ker[H^{1}(k, H) → H^{1}(k, G)] → ⨁_{v} H^{1}(k_{v}, H^{tor})]

where v runs over the set of places of k.

3.2. Before proving Theorem 3.1 we note that Br_{1} G = Br k (cf. [Sa], 6.9(iv)), hence every orbit of G(k) in X(k) is contained in one class of Brauer equivalence. It follows that the map X(k) → X(k)/Br factors through X(k)/G(k):

X(k) → X(k)/G(k) → X(k)/Br

and these maps are surjective.

For a place v of k consider the map

X(k) → X(k)/Br → X(k_{v})/Br → H^{1}(k_{v}, H^{tor})  (3.1)

where the last arrow is defined by Theorem 2.1.

Lemma 3.3. The map (3.1) X(k) → H^{1}(k_{v}, H^{tor}) factors

X(k) → X(k)/G(k)⟶^{∼} ker[H^{1}(k, H) → H^{1}(k, G)] → H^{1}(k_{v}, H^{tor})

with the obvious arrows.

Proof. The statement of the lemma shows how to define a map from X(k)/G(k) to H^{1}(k, H^{tor}). We can define a map from X(k_{v})/G(k_{v}) to H^{1}(k_{v}, H^{tor}) in a similar way. Consider the diagram

[figure: X(k_{v})/G(k_{v}); X(k)/G(k); X(k_{v})/Br; X(k)/Br; H^{1}(k_{v}, H^{tor}); H^{1}(k, H^{tor})]

It is commutative (the lower triangle is commutative by Theorem 2.1). It follows that the diagram

[figure: X(k_{v})/G(k_{v}); X(k)/G(k); X(k)/Br; H^{1}(k_{v}, H^{tor}); X(k_{v})/Br]

is commutative, which proves the lemma.

Lemma 3.4. (cf. [Ma/Ts], 4.5). Let Y be a variety over a global field k. Then the map Y (k)/Br → ∏_{v} Y (k_{v})/Br is injective.

Proof. Let y_{1}, y_{2} ∈ Y (k), and assume that y_{1} and y_{2} are Brauerequivalent in Y (k_{v}) for all places v of k. This means that for every b_{v} ∈ Br_{1} Y_{k_{v}}, (y_{1}, b_{v}) = (y_{2}, b_{v}). Let now b ∈ Br_{1} Y. We wish to compare (y_{1}, b) and (y_{2}, b). Consider loc_{v}(y_{i}, b) ∈ Br k_{v}, i = 1, 2, where loc means localization. We have loc_{v}(y_{i}, b) = (y_{i}, loc_{v}b), where loc_{v}b ∈ Br_{1} Y_{k_{v}}. By assumption we have (y_{1}, loc_{v}b) = (y_{2}, loc_{v}b). We see that loc_{v}(y_{1}, b) = loc_{v}(y_{2}, b) for all v. It follows that (y_{1}, b) = (y_{2}, b), because the map loc: Br k → ∏_{v} Br k_{v} is injective. Thus y_{1} and y_{2} are Brauer-equivalent in Y (k).

3.5. Proof of Theorem 3.1. By Lemma 3.3 the obvious map

ker[H^{1}(k, H) → H^{1}(k, G)] → ∏_{v} H^{1}(k_{v}, H^{tor})  (3.2)

has the same image as the map

X(k)/Br → ∏_{v} X(k_{v})/Br → ∏_{v} H^{1}(k_{v}, H^{tor}),  (3.3)

where the last arrow is described in Theorem 2.1. But the image of map (3.2) is contained in ⨁_{v} H^{1}(k_{v}, H^{tor}) (because the image of H^{1}(k, H^{tor}) in ∏_{v} H^{1}(k_{v}, H^{tor}) is contained in ⨁_{v} H^{1}(k_{v}, H^{tor}), cf. e.g. [Vo], 11.3, Cor. 1 of Prop. 1). Thus the image of map (3.3) is contained in ⨁_{v} H^{1}(k_{v}, H^{tor}) and is equal to the image of ker[H^{1}(k, H) → H^{1}(k, G)].

The first arrow in (3.3) is injective by Lemma 3.4, and the second arrow is injective by Theorem 2.1, and so the composition (3.3) is injective. Since (3.2) and (3.3) have the same images, we obtain a bijection

X(k)/Br ⟶^{∼}im [ker[H^{1}(k, H) → H^{1}(k, G)] → ⨁_{v} H^{1}(k_{v}, H^{tor})] .

This proves Theorem 3.1.

Theorem 3.6. In Theorem 3.1 assume that k is a totally imaginary number field. Then the bijection of Theorem 3.1 induces a bijection

X(k)/Br → H^{1}(k, H^{tor})/� ^{1}(k, H^{tor}).

To prove Theorem 3.6 we need a proposition and two corollaries.

Proposition 3.7. Let k be a totally imaginary number field, and L = (\overline{F} , κ) a k-kernel (k-lien) (see [Sp], [Bo], [F/S/S] for a definition), where \overline{F} is a connected linear k̄-group such that \overline{F}^{tor} = 1. Then every element of H^{2}(k, L) is neutral.

Proof. The proposition follows from [Bo], Thm. 6.8(iii) and Thm. 6.3(ii). Note that in the case when \overline{F} is semisimple, the proposition was proved by Douai ([Do], Cor. 5.1), see also [Bo], Cor. 6.9. The proposition follows also from Douai’s result and [Bo], Prop. 4.1.

Corollary 3.8. Let k be a totally imaginary number field and let

1 → G_{1} → G_{2} → G_{3} → 1

be an exact sequence of linear k-groups. If G_{1} is connected and G_{1}^{tor} = 1, then the map H^{1}(k, G_{2}) → H^{1}(k, G_{3}) is surjective.

Proof. We argue as in [Bo], the proof of Cor. 6.4. Let ξ ∈ H^{1}(k, G_{3}), and let ψ ∈ Z^{1}(k, G_{3}) be a cocycle from the class ξ. According to Springer ([Sp], 1.20), one can associate to ψ a k-kernel L_{ψ} = (G_{1k̄}, κ_{ψ}) and a cohomology class δ(ψ) ∈ H^{2}(k, L_{ψ}) which is the obstruction to lifting ξ to H^{1}(k, G_{2}). Since G_{1k̄}^{tor} = 1, by Prop. 3.7 the class δ(ψ) is neutral, and hence ξ comes from H^{1}(k, G_{2}).

Corollary 3.9. Let F be a connected linear group over a totally imaginary number field k. Then the map H^{1}(k, F) → H^{1}(k, F^{tor}) is surjective.

Proof. We have an exact sequence

1 → F^{ssu} → F → F^{tor} → 1

where (F^{ssu})^{tor} = 1. Now the corollary follows from Corollary 3.8.

3.10. Proof of Theorem 3.6. Since G is simply connected and k is a totally imaginary number field, H^{1}(k, G) = 1 (Kneser–Harder–Chernousov, see [Pl/Ra] §6.1, Thm. 6). Thus ker[H^{1}(k, H) → H^{1}(k, G)] = H^{1}(k, H). By Theorem 3.1 X(k)/Br is in a bijective correspondence with

im [H^{1}(k, H) → H^{1}(k, H^{tor}) → ⨁_{v} H^{1}(k_{v}, H^{tor})] .

By Corollary 3.9 the map H^{1}(k, H) → H^{1}(k, H^{tor}) is surjective. We see that X(k)/Br is in a bijective correspondence with

im [H^{1}(k, H^{tor}) → ⨁_{v} H^{1}(k_{v}, H^{tor})] = H^{1}(k, H^{tor})/� ^{1}(k, H^{tor}).

Example 3.11. Let G be a simply connected almost simple group over a number field k, H ⊂ G a connected k-subgroup, X = H\G. Assume that X is a symmetric space, i.e. H is the group of invariants of an involution of G. From the classification of involutions of simple Lie algebras (see e.g. [He], X-5, p. 514) it follows that dim H^{tor} ≤ 1.

If H^{tor} = 1 or H^{tor} is a one-dimensional split torus, then H^{1}(k_{v}, H^{tor}) = 1 for all v, and by Theorem 3.1 X(k)/Br consists of one element.

If H^{tor} is a one-dimensional nonsplit torus, then H^{tor} splits over a quadratic extension K of k. Assume in addition that k is totally imaginary. Then by Theorem 3.6 X(k)/Br = H^{1}(k, H^{tor})/� ^{1}(k, H^{tor}). Since K/k is cyclic, we have � ^{1}(k, H^{tor}) = 1 ([Vo], 11.6, Cor.3), and we see that

X(k)/Br = H^{1}(k, H^{tor}) = k^{×}/N_{K/k}K^{×}

where N_{K/k} denotes the norm map.

10 MIKHAIL BOROVOI AND BORIS KUNYAVSKIĬ

4. Appendix

In this appendix, we prove Theorem 1.5.

We use the description of the Manin pairing with the help of torsors given in [CT/Sa2], §2. We first recall some generalities.

4.1. Torsors. Let X be a smooth geometrically integral k-variety such that U(\overline{X}) = 1 and Pic \overline{X} is a finitely generated 𝔤-module, where 𝔤 = Gal(k̄/k). Let S be a k-torus. Let (Y, p_{Y} : Y → X) be a right X-torsor under S.

The set of isomorphism classes of X-torsors under S is the first étale cohomology group H^{1}(X, S). We have a canonical map

χ: H^{1}(X, S) → Hom_{𝔤}(X(\overline{S}), Pic \overline{X})

defined as follows. To any X-torsor Y under S we associate a homomorphism χ_{Y} sending a character λ ∈ X(\overline{S}), λ: \overline{S} → 𝔾_{m,k̄} to the push-forward λ_{∗}(Y) which is an X-torsor under 𝔾_{m,k̄}; since Pic \overline{X} = H^{1}(\overline{X}, 𝔾_{m}), we identify the isomorphism class of λ_{∗}(Y) with an element of Pic \overline{X}. We call χ_{Y} the type of Y. If S is the Néron–Severi torus of X (i.e. X(\overline{S}) = Pic \overline{X}) and χ_{Y} is the identity map, the torsor Y is called universal.

4.2. Evaluation map. Let Y be an X-torsor under S. We then have a natural evaluation map θ_{Y} : X(k) → H^{1}(k, S) which associates to every rational point x ∈ X(k) the isomorphism class of the fibre Y_{x} of Y at x. The map θ_{Y} allows us to identify the Manin pairing with a cupproduct. We use the following construction. To x ∈ X(k) we can associate a homomorphism of 𝔤-modules σ: k̄(X)^{×} → k̄^{×}, σ(f) = f(x) for f ∈ k̄(X)^{×}. There is a canonical isomorphism H^{1}(k, Pic \overline{X})⟶^{∼}Br_{a} X, cf. [Sa], 6.3(iii). Together with the isomorphism Br_{a} X⟶^{∼}Br_{x}X (see 1.1) this yields an embedding t_{x}: H^{1}(k, Pic \overline{X}) → Br_{1} X (in [CT/Sa2], p. 449, this embedding is denoted by t_{σ}).

Let Y^{x} denote an X-torsor under S of type χ_{Y} : X(\overline{S}) → Pic \overline{X} which is trivial at x (such a torsor exists and is unique up to isomorphism, cf. [CT/Sa2], p. 449).

Theorem 4.3. ([CT/Sa2], Prop. 2.7.10) Let

ρ: H^{1}(k, X(\overline{S})) → H^{1}(k, Pic \overline{X}) ⟶^{t_{x}} Br_{1} X

be the composed map where the left arrow is induced by χ_{Y}. Then the following diagram is commutative up to sign:

\begin{matrix}X(k) & × & Br_{1} X & ⟶ & Br k \\ ^{θ_{Y^{x}}} ↓ & ↑^{ρ} & ∥ \\ H^{1}(k, S) & × & H^{1}(k, X(\overline{S})) & ⟶ & Br k \end{matrix}  (4.1)

Here the top row is the Manin pairing and the bottom row is the cupproduct.

4.4. Abelianization of torsors. Let us now suppose that X is as in 1.2, i.e. X = H\G. We have a canonical map G → X. Define a right action of H on G by g ∗ h = h^{−1}g, where g ∈ G, h ∈ H. With this action the map G → H is a right (non-abelian) X-torsor under H. Set S = H^{tor}. We consider the natural homomorphism ψ: H → S. Let H^{ssu} denote the kernel of ψ. Let ψ_{∗}: H^{1}(k, H) → H^{1}(k, S) and ψ_{∗}^{X}: H^{1}(X, H) → H^{1}(X, S) be the induced push-forward maps, they send non-abelian torsors under H to abelian torsors under S. Explicitly, to any right Xtorsor Z under H we associate an X-torsor ψ_{∗}^{X}(Z) = Z/H^{ssu} under S (abusing notation, we use the same symbol ψ_{∗} for maps of torsors and their cohomology classes). Denote by Y the torsor under S obtained by applying ψ_{∗}^{X} to G → X, i.e. Y = H^{ssu}\G. Note that by Proposition 1.3, we have an isomorphism X(\overline{S})⟶^{∼}Pic \overline{X} so that Y is a universal torsor. With the notation of Theorem 4.3, Y = Y^{x_{0}} where x_{0} ∈ X(k) is the image of the unit element of G.

Lemma 4.5. The map θ_{Y} : X(k) → H^{1}(k, H^{tor}) coincides with the map α defined by (1.1).

Proof. Indeed, α is the composition

ψ_{∗} ∘ δ: X(k) ⟶^{δ}H^{1}(k, H) ⟶^{ψ_{∗}} H^{1}(k, H^{tor})

where δ is the connecting map. Note that the image of x ∈ X(k) under δ coincides with the (isomorphism class of the) fibre of G → X at x (cf. [Se], I.5.4, Prop. 36, [Ha/Sk], 4.1). Furthermore, push-forward commutes with specialization, i.e. for any X-torsor Z under H and any x ∈ X(k) the fibre [ψ_{∗}^{X}(Z)]_{x} coincides with ψ_{∗}(Z_{x}). Hence ψ_{∗}(δ(x)) coincides with the fibre of the abelian X-torsor Y under H^{tor} at x, which is equal to θ_{Y} (x) (by the definition of θ_{Y}). Thus α = θ_{Y}.

4.6. To finish the proof of Theorem 1.5, it only remains to apply diagram (4.1). Indeed, denote by i the isomorphism Br_{x_{0}}X⟶^{∼}H^{1}(k, X(\overline{S})). Then the pairing (1.4) X(k) × Br_{x_{0}}X → Br k is given by

(x, b) ↦ α(x) ∪ i(b),  (4.2)

whereas the Manin pairing (0.1) X(k) × Br_{1} X → Br k restricted to X(k) × Br_{x_{0}}X is given by

(x, b) ↦ b(x).  (4.3)

Any b ∈ Br_{x_{0}}X can be written as b = ρ(z) with z ∈ H^{1}(k, X(\overline{S})), where in the definition of ρ (in Theorem 4.3) we use x_{0} instead of x. On applying diagram (4.1) to the Manin pairing (4.3), we get an equality (up to sign)

(x, ρ(z)) = θ_{Y} (x) ∪ z,

where Y = Y^{x_{0}} in θ_{Y}. On the other hand, taking into account that α = θ_{Y} (Lemma 4.5) and i(ρ(z)) = z, formula (4.2) gives the same value θ_{Y} (x) ∪ z for (x, ρ(z)). Thus formulas (4.2) and (4.3) coincide (up to sign), and the Manin pairing restricted to Br_{x_{0}}X coincides (up to sign) with pairing (1.4). The theorem is proved.

References

[Bo] M. Borovoi, Abelianization of the second nonabelian Galois cohomology, Duke Math. J. 72 (1993), 217–239.

[CT/Sa1] J.-L. Colliot-Thélène et J.-J. Sansuc, La R-équivalence sur les tores, Ann.

Sci. École Norm. Sup. (4) 10 (1977), 175–229.

[CT/Sa2] J.-L. Colliot-Thélène et J.-J. Sansuc, Descente sur les variétés ra-

tionnelles, II, Duke Math. J. 54 (1987), 375–492.

[Do] J.-C. Douai, Cohomologie galoisienne des groupes semi-simples définis sur les corps globaux, C.R. Acad. Sci. Paris Sér. A 281 (1975), 1077– 1080.

[F/S/S] Y. Z. Flicker, C. Scheiderer, and R. Sujatha, Grothendieck’s theorem on non-abelian H^{2} and local-global principles, J. Amer. Math. Soc. 11 (1998), 731–750.

[Ha/Sk] D. Harari and A. N. Skorobogatov, Non-abelian cohomology and rational

points, Preprint IHES/M/99/42.

[He] S. Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces, Academic Press, New York, 1978.

[Ma1] Yu. I. Manin, Le groupe de Brauer–Grothendieck en géométrie diophantienne, Actes du Congr. Internat. Math., Nice, 1970, Tome 1, pp. 401–411.

[Ma2] Yu. I. Manin, Cubic Forms: Algebra, Geometry, Arithmetic, Nauka, Moscow, 1972; English transl. of the 2nd edition: North-Holland, Amsterdam, 1986.

[Ma/Ts] Yu. I. Manin and M. A. Tsfasman, Rational varieties: algebra, geometry,

and arithmetic, Russian Math. Surveys 41:2 (1986), 51–116.

[Mi] J. S. Milne, Arithmetic Duality Theorems, Academic Press, Boston, 1986.

[Mu] D. Mumford, Abelian Varieties, Oxford Univ. Press, Oxford, 1970. [Pl/Ra] V. P. Platonov and A. S. Rapinchuk, Algebraic Groups and Number The-

ory, Nauka, Moscow, 1991; English transl. Academic Press, Boston, 1994.

[Po] V. L. Popov, The Picard group of homogeneous spaces of linear algebraic groups and one-dimensional homogeneous vector bundles, Math. USSR Izv. 8 (1975), 301–327.

[Ro] M. Rosenlicht, Toroidal algebraic groups, Proc. Amer. Math. Soc. 12 (1961), 984–988.

[Sa] J.-J. Sansuc, Groupe de Brauer et arithmétique des groupes algébriques linéaires sur un corps de nombres, J. Reine Angew. Math. 327 (1981), 12–80.

[Se] J.-P. Serre, Cohomologie galoisienne, Fifth edition, Lecture Notes in Math. 5, Springer-Verlag, Berlin et al., 1994.

[Sp] T. A. Springer, Non-abelian H^{2} in Galois cohomology, in “Algebraic Groups and Discontinuous Subgroups”, Proc. Symp. Pure Math. 9, Amer. Math. Soc., Providence, Rhode Island, 1966, pp. 164–182.

[Th] Nguyêñ Q. Thǎńg, Weak approximation, Brauer and R-equivalence in algebraic groups over arithmetical fields, J. Math. Kyoto Univ., to appear.

[Vo] V. E. Voskresenskiĭ, Algebraic Groups and Their Birational Invariants, Transl. of Math. Monographs, vol. 179, Amer. Math. Soc., Providence, Rhode Island, 1998.

Borovoi: Raymond and Beverly Sackler School of Mathematical Sciences, Tel Aviv University, 69978 Tel Aviv, Israel

E-mail address: borovoi@math.tau.ac.il

Kunyavskiĭ: Department of Mathematics and Computer Science, Bar-Ilan University, 52900 Ramat Gan, Israel

E-mail address: kunyav@macs.biu.ac.il