FORMULAS FOR THE UNRAMIFIED BRAUER

GROUP OF A PRINCIPAL HOMOGENEOUS

SPACE OF A LINEAR ALGEBRAIC GROUP

Mikhail Borovoi and Boris Kunyavskiĭ

Abstract. For a smooth compactification V of a principal homogeneous space E under a connected linear algebraic group G defined over a field k of characteristic zero, we present two formulas expressing Br V/Br k in terms of G.

Introduction

Let G be a connected linear algebraic group over a field k of characteristic zero, E a principal homogeneous space (torsor) under G, and V a smooth complete variety over k containing E as an open subset. The Brauer group Br V = H_{ét}^{2}(V, 𝔾_{m}) does not depend on the choice of V; this birational invariant of E is denoted Br_{nr}(k(E)/k) and is called the unramified Brauer group of E. In this paper we give two formulas for Br V/Br k in terms of G.

Formulas for Br V/Br k were first given by Voskresenskiĭ [V] and Sansuc [S] in the case when k is a number field. A generalization to an arbitrary ground field was presented in [CT/K]. In all these three papers, G is of some special type: either a torus, or a semisimple group, or a group admitting a finite cover of the type G_{0} × S where G_{0} is a semisimple simply connected group and S is a quasi-trivial torus.

In this paper we compute Br V/Br k for any connected k-group G. The paper is based on results of [CT/K]. We use the method of z-extensions developed by Kottwitz [K2], [K3].

We now briefly describe our results. Let G^{u} denote the unipotent radical of G. Set G^{red} = G/G^{u}; it is a reductive group. Let G^{ss} denote the derived group of G^{red}; it is semisimple. Let G^{sc} denote the universal covering of G^{ss}; it is simply connected. Consider the composite map

ρ: G^{sc} → G^{ss} → G^{red}.

Let T ⊂ G^{red} be a maximal torus. Set T^{sc} = ρ^{−1}(T) ⊂ G^{sc}. Denote L^{−1} = X^{∗}(T), L^{0} = X^{∗}(T^{sc}), where X^{∗}(·) = Homk̄(·, 𝔾_{m}) stands for the character group, and k̄ denotes an algebraic closure of k. Consider the complex

L^{•} = (0 → L^{−1} → L^{0} → 0) = (0 → X^{∗}(T^{sc}) → X^{∗}(T) → 0).

Typeset by AMS-TEX

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

Research of the second named author partially supported by the Ministry of Absorption (Israel)

and the Israel Science Foundation founded by the Israel Academy of Sciences and Humanities.

The Galois group 𝔤 = Gal(k̄/k) acts on L^{•}. Let ℍ^{i}(𝔤, L^{•}) denote the hypercohomology group of 𝔤 with coefficients in the complex L^{•}. Set

� _{ω}^{i}(𝔤, L^{•}) = ker[ℍ^{i}(𝔤, L^{•}) → ∏_{γ} ℍ^{i}(γ, L^{•})]

where γ runs over all closed procyclic subgroups of 𝔤. Let \overline{V} = V ×_{k} k̄, and denote by Pic \overline{V} the Picard group; it is a 𝔤-module. Our main results are the following theorem and corollary.

Theorem A. With the above assumptions and notation,

H^{1}(k, Pic \overline{V}) = � _{ω}^{1}(𝔤, L^{•}).

Corollary B. There is an injection

Br V/Br k↪� _{ω}^{1}(𝔤, L^{•})

which is an isomorphism provided either V (k) ≠ ∅, or H^{3}(k, k̄^{×}) = 0.

Let now Z(Ĝ) denote the center of a connected Langlands dual group for a connected reductive group G, cf. [K2], 1.5. It is a C-group of multiplicative type. It turns out that Z(Ĝ) = ker[L^{−1} ⊗ C^{×} → L^{0} ⊗ C^{×}].

Proposition C.

� _{ω}^{1}(𝔤, L^{•}) = � _{ω}^{1}(𝔤, Z(Ĝ)).

Thus

H^{1}(k, Pic \overline{V}) = � _{ω}^{1}(𝔤, Z(Ĝ^{red})).

We obtain a new case of the following Kottwitz principle [K2]: an invariant of reductive groups which is trivial for semisimple simply connected groups can be computed from the Galois module Z(Ĝ).

The structure of the paper is as follows. In Section 1 we collect required information on linear algebraic groups, Brauer groups, Galois cohomology and hypercohomology and prove Proposition C (Proposition 1.3.2). In Section 2 we state and prove our main results (Theorem A = Theorems 2.1 and 2.4, and Corollary B = Corollary 2.2). In Section 3 we present some comments and remarks relating our results to previously known ones.

Acknowledgements. We thank J.-L. Colliot-Thélène for very valuable remarks. The second named author thanks the Max-Planck-Institut für Mathematik (Bonn) for hospitality and support.

Notation and conventions

Throughout the paper, k denotes a field of characteristic zero, k̄ is a fixed algebraic closure of k, 𝔤 = Gal(k̄/k) is the absolute Galois group of k, k^{×} denotes the multiplicative group of k. An algebraic k-torus T is called quasi-trivial if it is a direct product of tori of the form R_{K/k}𝔾_{m} where K/k is a finite extension and R_{K/k} stands for Weil’s restriction of scalars. We denote by X^{∗}(G) the group of characters of a linear algebraic group G and by X_{∗}(T) = Homk̄(𝔾_{m}, T) the group of cocharacters of a torus T; one can view X^{∗}(G) and X_{∗}(T) as 𝔤-modules. For a torus T, X^{∗}(T) is a Z-free 𝔤-module of finite rank; if T is quasi-trivial, X^{∗}(T) is a permutation module (i.e. it has a Z-basis permuted by 𝔤). If M is a Galois module, we denote by H^{i}(k, M) (or by H^{i}(𝔤, M)) the i-th Galois cohomology group. For a smooth projective k-variety X we denote \overline{X} = X ×_{k} k̄. Set Pic X = H_{ét}^{1}(X, 𝔾_{m}), Br X = H_{ét}^{2}(X, 𝔾_{m}), these are the Picard group and the Brauer group of X, respectively. Other notation is explained in the introduction.

1. Preliminaries

1.1. Linear algebraic groups.

1.1.1. Let G be a connected linear algebraic group over a field k of characteristic zero. By Chevalley’s theorem [C] the k-variety G is rational, i.e. k̄-birationally equivalent to an affine space.

Every unipotent k-group is k-rational, i.e. k-birationally equivalent to an affine space.

Every quasi-trivial k-torus Z is k-rational, and by Hilbert 90 H^{1}(K, Z) = 1 for every extension K of k.

1.1.2. Levi decomposition. For any connected linear algebraic group G over a field k of characteristic zero, there is an isomorphism G^{u} ⋊ G^{red}⟶^{∼}G (Levi decomposition) which gives rise to a k-biregular morphism of varieties

G^{u} × G^{red}⟶^{∼}G.

1.1.3. z-extensions. A z-extension of a connected reductive k-group G is an epimorphism of reductive groups α: H → G with kernel Z, such that H^{ss} is simply connected and Z is central and is a quasi-trivial k-torus. The notion of z-extension was introduced by Langlands [L]. We say that a z-extension α_{1}: H_{1} → G dominates a z-extension α_{2}: H_{2} → G if there exists a homomorphism ϕ: H_{1} → H_{2} such that α_{2} = ϕ ∘ α_{1}.

Lemma 1.1.4. (Kottwitz) (1) For every connected reductive k-group G and a cohomology class ξ ∈ H^{1}(k, G) there exists a z-extension α: H → G such that ξ ∈ im [α_{∗}: H^{1}(k, H) → H^{1}(k, G)].

(2) For every two z-extensions α_{1}: H_{1} → G and α_{2}: H_{2} → G of G there exists a z-extension α_{3}: H_{3} → G that dominates both α_{1} and α_{2}.

(3) Let G_{1} → G_{2} be a homomorphism, and let H_{i} → G_{i} (i = 1, 2) be z-extensions. Then there exists a commutative diagram

H_{1} ⟵ H_{3} ⟶ H_{2}
↓  ↓  ↓
G_{1} ⟵^{id} G_{1} ⟶ G_{2}

in which H_{3} → G_{1} is a z-extension.

Proof. (1) For a proof of existence of some z-extension of G see [M/S], Prop. 3.1. The existence of a z-extension such that ξ lifts to H^{1}(k, H) is proved in [K3] in the proof of Theorem 1.2, p. 369.

(2) See [K1], Lemma 1.1(2).

(3) See [K2], Lemma 2.4.4. □

1.2. Birational invariants.

1.2.1. Permutation modules. A permutation 𝔤-module P can be written as a direct sum of induced modules Z[𝔤/𝔥], where 𝔥 is a closed subgroup of finite index in 𝔤. By Shapiro’s lemma, H^{1}(𝔤, Z[𝔤/𝔥]) = H^{1}(𝔥, Z) = 0, hence H^{1}(𝔤, P) = 0. Moreover, H^{1}(γ, P) = 0 for any closed subgroup γ ⊂ 𝔤.

We also have � _{ω}^{2}(𝔤, P) = 0 (cf. [S], (1.9.1), for the case where k is a number field). Indeed, it suffices to prove this for an induced module M = Z[𝔤/𝔥]. We have H^{2}(𝔤, M) = H^{2}(𝔥, Z) = Hom(𝔥, Q/Z). Since any continuous homomorphism 𝔥 → Q/Z factors through a finite quotient of 𝔤, we may assume that 𝔥 and 𝔤 are finite. Since any non-trivial homomorphism 𝔥 → Q/Z is non-trivial on some cyclic subgroup of 𝔥, we conclude that � _{ω}^{2}(𝔤, M) = 0.

1.2.2. Smooth compactifications. By Hironaka [H], any smooth affine k-variety X can be embedded into a smooth complete k-variety V (X) containing X as an open subset. Indeed, one has to map X biregularly onto a closed subscheme of an affine space, embed it into the projective space, take the projective closure, and resolve its singularities. We call V (X) a smooth k-compactification of X. If V_{1} and V_{2} are two smooth k-compactifications of X, then there exists an isomorphism of 𝔤modules Pic \overline{V}_{1} ⊕P_{1} ≅ Pic \overline{V}_{2} ⊕P_{2} where P_{1} and P_{2} are permutation 𝔤-modules (cf. [V], Th. 1). By 1.2.1, this gives an isomorphism H^{1}(k, Pic \overline{V}_{1}) → H^{1}(k, Pic \overline{V}_{2}), and the construction in [V] shows that this isomorphism is canonical. This also shows that H^{1}(k, Pic \overline{V} (X)) is a birational invariant of X.

Moreover, the group H^{1}(k, Pic \overline{V} (X)) is functorial in X. Indeed, let f: X_{1} → X_{2} be a k-morphism of smooth integral k-varieties. We wish to extend f to a k-morphism f′: V_{1} → V_{2} where V_{i} is a suitable smooth compactification of X_{i}, i = 1, 2. We are very grateful to J.-L. Colliot-Thélène for communicating us the following construction. Let U denote the graph of f in X_{1} ×_{k} X_{2}. Choose smooth compactifications W_{i} of X_{i}, i = 1, 2. Let W be the closure of U in W_{1} ×_{k} W_{2}. Then U is a smooth open subvariety of W. By Hironaka [H] there exists a proper morphism (desingularization) π: V_{1} → W such that V_{1} is smooth and the restriction π^{−1}(U) → U is an isomorphism. Clearly V_{1} is a smooth compactification of X_{1}. Set V_{2} = W_{2} and define f′: V_{1} → V_{2} to be the composite map V_{1} → W → V_{2} where the second arrow is the restriction of the canonical projection W_{1} ×W_{2} → W_{2} = V_{2}. The map f′ induces a homomorphism H^{1}(k, Pic \overline{V}_{1}) → H^{1}(k, Pic \overline{V}_{2}), as required.

We prove the following property of the functor H^{1}(k, Pic \overline{V} (X)): if Z is a krational variety, then H^{1}(k, Pic \overline{V} (X ×_{k} Z)) ≅ H^{1}(k, Pic \overline{V} (X)). Indeed, let V_{X}, V_{Z} be smooth compactifications of X, Z, respectively. One can then take V_{X} × V_{Z} as a smooth compactification of X × Z. The variety \overline{V}_{Z} is rational, hence by [CT/S1], Lemme 11, p. 188, the canonical homomorphism Pic \overline{V}_{X} ⊕ Pic \overline{V}_{Z} → Pic (\overline{V}_{X} × \overline{V}_{Z}) is an isomorphism. Since Z is k-rational, H^{1}(k, Pic \overline{V}_{Z}) = 0. Thus H^{1}(k, Pic \overline{V} (X × Z)) ≅ H^{1}(k, Pic \overline{V} (X)), as required. This isomorphism is induced by the canonical projection pr_{X}: X × Z → X.

1.2.3. Brauer group. For a geometrically integral smooth projective k-variety X we have an exact sequence

Br k → ker[Br X → Br \overline{X}] → H^{1}(k, Pic \overline{X}) → H^{3}(k, k̄^{×});

if X has a k-point, we have an exact sequence

0 → Br k → ker[Br X → Br \overline{X}] → H^{1}(k, Pic \overline{X}) → 0

(cf. [CT/S2], 1.5.0). If X is k-rational, this gives an isomorphism Br k⟶^{∼}Br X; if X is a smooth k-compactification of a G-torsor E with X(k) ≠ ∅, this gives an isomorphism

Br X/Br k ≅ H^{1}(k, Pic \overline{X}),

because by 1.1.1 \overline{X} is rational, and since \overline{X} is projective and rational, Br \overline{X} = 0.

1.3. Hypercohomology.

1.3.1. Let M^{•} = (0 → M^{−1} → M^{0} → 0) be a short complex of 𝔤-modules. We often shorten notation to (M^{−1} → M^{0}). We define the hypercohomology ℍ^{i}(𝔤, M^{•}) as the cohomology H^{i} of the ordinary chain complex corresponding to the double complex

0 ⟶ M^{0} ⟶ C^{1}(𝔤, M^{0}) ⟶ C^{2}(𝔤, M^{0}) ⟶ …
↑  ↑  ↑
0 ⟶ M^{−1} ⟶ C^{1}(𝔤, M^{−1}) ⟶ C^{2}(𝔤, M^{−1}) ⟶ …

where C^{i} is the usual group of non-homogeneous continuous i-cochains and the bidegree of M^{−1} is (−1, 0).

For a subgroup γ ⊂ 𝔤 one can define the restriction map ℍ^{i}(𝔤, M^{•}) → ℍ^{i}(γ, M^{•}) and define

� _{ω}^{i}(𝔤, M^{•}) = ker [ℍ^{i}(𝔤, M^{•}) → ∏_{γ} ℍ^{i}(γ, M^{•})]

where γ runs over all closed procyclic subgroups of 𝔤.

1.3.2. Let T be a maximal k-torus in a connected reductive k-group G, and let Z(Ĝ) denote the center of a connected Langlands dual group for G (see Introduction). Denote L^{−1} = X^{∗}(T), L^{0} = X^{∗}(T^{sc}), L^{•} = (L^{−1} → L^{0}).

Proposition. With the above notation

� _{ω}^{1}(𝔤, L^{•}) = � _{ω}^{1}(𝔤, Z(Ĝ)).

1.3.3. We first prove the following lemma.

Lemma. There is a short exact sequence of C-groups

1 → Z(Ĝ) → L^{−1} ⊗ C^{×} → L^{0} ⊗ C^{×} → 1.

Proof. Set π_{1}(G) = X_{∗}(T)/ρ(X_{∗}(T^{sc})), where X_{∗}(·) = Homk̄(𝔾_{m}, ·) is the cocharacter group. By [B], 1.2, the Galois module π_{1}(G) does not depend on the choice of T ⊂ G. We call π_{1}(G) the algebraic fundamental group of G. By [B], 1.10, we have π_{1}(G) = Hom(Z(Ĝ), C^{×}). Hence Z(Ĝ) = Hom(π_{1}(G), C^{×}).

By the definition of π_{1}(G), there is an exact sequence

0 → X_{∗}(T^{sc}) → X_{∗}(T) → π_{1}(G) → 0.

We thus obtain an exact sequence

1 → Hom(π_{1}(G), C^{×}) → Hom(X_{∗}(T), C^{×}) → Hom(X_{∗}(T^{sc}), C^{×}) → 1,

or

1 → Z(Ĝ) → L^{−1} ⊗ C^{×} → L^{0} ⊗ C^{×} → 1. □

1.3.4. Proof of Proposition 1.3.2. From Lemma 1.3.3 we obtain a quasi-isomor-phism of complexes (Z(Ĝ) → 1) → L^{•} ⊗ C^{×}. Hence

ℍ^{0}(𝔤, L^{•} ⊗ C^{×}) = ℍ^{0}(𝔤, Z(Ĝ) → 1) = H^{1}(𝔤, Z(Ĝ)).  (1.3.4.1)

On the other hand, the short exact sequence

0 → Z → C → C^{×} → 1

induces a short exact sequence of complexes

0 → L^{•} → L^{•} ⊗ C → L^{•} ⊗ C^{×} → 1

and a hypercohomology exact sequence

ℍ^{0}(𝔤, L^{•} ⊗ C) → ℍ^{0}(𝔤, L^{•} ⊗ C^{×}) → ℍ^{1}(𝔤, L^{•}) → ℍ^{1}(𝔤, L^{•} ⊗ C).  (1.3.4.2)

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We prove that ℍ^{0}(𝔤, L^{•}⊗C) = 0 and ℍ^{1}(𝔤, L^{•}⊗C) = 0. Let T^{ss} = T ∩G^{ss}. Then X^{∗}(T^{ss}) is a subgroup of finite index of X^{∗}(T^{sc}), and so X^{∗}(T^{sc})⊗C = X^{∗}(T^{ss})⊗C. We see that

L^{•} ⊗ C = (X^{∗}(T) ⊗ C → X^{∗}(T^{ss}) ⊗ C).

It follows that L^{•} ⊗ C is quasi-isomorphic to (X^{∗}(G^{tor}) ⊗ C → 0). Hence

ℍ^{0}(𝔤, L^{•} ⊗ C) = ℍ^{0}(𝔤, (X^{∗}(G^{tor}) ⊗ C → 0)) = H^{1}(𝔤, X^{∗}(G^{tor}) ⊗ C).

But H^{1}(𝔤, X^{∗}(G^{tor}) ⊗ C) = 0 because X^{∗}(G^{tor}) ⊗ C is a uniquely divisible group. Thus ℍ^{0}(𝔤, L^{•}⊗C) = 0. Similarly ℍ^{1}(𝔤, L^{•}⊗C) = 0. From exact sequence (1.3.4.2) we then obtain

ℍ^{0}(𝔤, L^{•} ⊗ C^{×}) = ℍ^{1}(𝔤, L^{•}).

We see from (1.3.4.1) that

H^{1}(𝔤, Z(Ĝ)) = ℍ^{1}(𝔤, L^{•}).

Similarly, H^{1}(γ, Z(Ĝ)) = ℍ^{1}(γ, L^{•}) for every closed subgroup γ ⊂ 𝔤. We conclude that

� _{ω}^{1}(𝔤, L^{•}) = � _{ω}^{1}(𝔤, Z(Ĝ)). □

1.3.5. Remark. Proposition 1.3.2 shows that � _{ω}^{1}(𝔤, L^{•}) does not depend on the choice of T. Indeed, it only depends on the algebraic fundamental group π_{1}(G) which does not depend on T ([B], 1.2).

2. Main results

Theorem 2.1. Let k be a field of characteristic zero, 𝔤 = Gal(k̄/k), G a connected linear algebraic k-group, V a smooth k-compactification of G, T ⊂ G a maximal k-torus, L^{•} = (X^{∗}(T) → X^{∗}(T^{sc})). Then the group H^{1}(k, Pic \overline{V}) is canonically isomorphic to � _{ω}^{1}(𝔤, L^{•}), and this isomorphism is functorial in G.

Corollary 2.2. With the notation of Theorem 2.1, there is an injection

Br V/Br k↪� _{ω}^{1}(𝔤, L^{•})

which is an isomorphism provided either V (k) ≠ ∅, or H^{3}(k, k̄^{×}) = 0.

Proof. By 1.1.1 G is rational, hence V is projective and rational, and Br \overline{V} = 0, see 1.2.3. The corollary now follows from 1.2.3 and Theorem 2.1. □

2.3. Proof of Theorem 2.1. We first assume that G is reductive and G^{ss} is simply connected. We use Voskresenskiĭ’s exact sequence

0 → X^{∗}(G) → P → Pic \overline{V} → Pic \overline{G} → 0  (2.3.1)

which is valid for any connected k-group G [V] (see also [S], (9.0.0)). Here P is a permutation 𝔤-module. The exact sequence of k̄-groups

1 → \overline{G}^{ss} → \overline{G} → \overline{G}^{tor} → 1

induces the exact sequence ([S], (6.11.4))

0 → X^{∗}(G^{tor}) → X^{∗}(G) → X^{∗}(G^{ss}) → Pic \overline{G}^{tor} → Pic \overline{G} → Pic \overline{G}^{ss} → 0.

Since \overline{G}^{ss} is simply connected, we have Pic \overline{G}^{ss} = 0; since G^{ss} is semisimple, we have X^{∗}(G^{ss}) = 0; since G^{tor} is a torus, we have Pic \overline{G}^{tor} = 0, cf. [S], 6.9. We conclude that Pic \overline{G} = 0 and X^{∗}(G) = X^{∗}(G^{tor}). Exact sequence (2.3.1) is thus reduced to

0 → X^{∗}(G^{tor}) → P → Pic \overline{V} → 0.  (2.3.2)

We can now use the following fundamental property of the Picard group Pic \overline{V} proved in [CT/K], Prop. 3.2: H^{1}(γ, Pic \overline{V}) = 0 for all closed procyclic subgroups γ ⊂ 𝔤. Since P is a permutation 𝔤-module, we obtain

H^{1}(𝔤, Pic \overline{V}) = � _{ω}^{2}(𝔤, X^{∗}(G^{tor}))  (2.3.3)

by [CT/S2], Prop. 9.5(ii).

We now prove that � _{ω}^{2}(𝔤, X^{∗}(G^{tor})) ≅ � _{ω}^{1}(𝔤, L^{•}). Since G^{ss} is simply connected, one has an exact sequence of tori

1 → T^{sc} → T → G^{tor} → 1

and the dual exact sequence of character groups

0 → X^{∗}(G^{tor}) → X^{∗}(T) → X^{∗}(T^{sc}) → 0.

It induces a morphism of complexes

(X^{∗}(G^{tor}) → 0) → L^{•}

which is a quasi-isomorphism. Thus

H^{i+1}(γ, X^{∗}(G^{tor})) = ℍ^{i}(γ, L^{•})

for every natural i and every closed subgroup γ ⊆ 𝔤. We conclude that

� _{ω}^{2}(𝔤, X^{∗}(G^{tor})) ≅ � _{ω}^{1}(𝔤, L^{•}).

Thus H^{1}(𝔤, Pic \overline{V}) ≅ � _{ω}^{1}(𝔤, L^{•}), and this isomorphism is functorial in G. The theorem is proved for reductive G with G^{ss} simply connected.

Let now G be an arbitrary connected reductive k-group. Let H →^{α} G be a zextension with kernel Z. Let V_{G} be a a smooth compactification of G and let V_{H} be a smooth compactification of H. We have a homomorphism

α_{∗}: H^{1}(k, Pic \overline{V}_{H}) → H^{1}(k, Pic \overline{V}_{G}).

We prove that α_{∗} is an isomorphism, hence

H^{1}(k, Pic \overline{V}_{H}) ≅ H^{1}(k, Pic \overline{V}_{G}).  (2.3.4)

Since Z is a quasi-trivial torus, the map α admits a rational k-section s: G → H. Indeed, the obstruction to the existence of such a section lies in H^{1}(k(G), Z) = 0. The rational section s gives rise to a biregular k-isomorphism i: U_{H} → U_{G} × Z, where U_{H} ⊂ H and U_{G} ⊂ G are open k-subvarieties, U_{G} is an open subvariety on which s is defined, and U_{H} = s(U_{G}) · Z. The projections are defined as follows: pr_{U_{G}}(h) = α(h), pr_{Z}(h) = h · s(α(h))^{−1}, where h ∈ U_{H}. Since Z is a quasi-trivial torus, it is k-rational, and by 1.2.2 we obtain a canonical isomorphism H^{1}(k, Pic \overline{V} (U_{H}))⟶^{∼}H^{1}(k, Pic \overline{V} (U_{G})) induced by the projection pr_{U_{G}}: U_{H} → U_{G}. This gives us (2.3.4), and we see that (2.3.4) is the canonical isomorphism induced by α: H → G. It does not depend on s.

Let T_{G} ⊂ G be a maximal torus and set L_{G}^{•} = (X^{∗}(T_{G}) → X^{∗}(T_{G}^{sc})). Set T_{H} = α^{−1}(T) ⊂ H, L_{H}^{•} = (X^{∗}(T_{H}) → X^{∗}(T_{H}^{sc})). We prove that

� _{ω}^{1}(k, L_{G}^{•}) ≅ � _{ω}^{1}(k, L_{H}^{•}).  (2.3.5)

We note that the z-extension H → G induces an exact sequence of complexes

1 → L_{G}^{•} → L_{H}^{•} → (X^{∗}(Z) → 1) → 1

which leads to the following commutative diagram:

0  0  0
↓  ↓  ∥
0  � _{ω}^{1}(𝔤, X^{∗}(Z)) ⟶ � _{ω}^{1}(𝔤, L_{G}^{•}) ⟶ � _{ω}^{1}(𝔤, L_{H}^{•}) ⟶ � _{ω}^{2}(𝔤, X^{∗}(Z))
↓  ↓  ↓  ↓
0  H^{1}(𝔤, X^{∗}(Z)) ⟶ ℍ^{1}(𝔤, L_{G}^{•}) ⟶ ℍ^{1}(𝔤, L_{H}^{•}) ⟶ H^{2}(𝔤, X^{∗}(Z))
↓  ↓  ↓  ↓
0  ∏_{γ} H^{1}(γ, X^{∗}(Z)) ⟶ ∏_{γ} ℍ^{1}(γ, L_{G}^{•}) ⟶ ∏_{γ} ℍ^{1}(γ, L_{H}^{•}) ⟶ ∏_{γ} H^{2}(γ, X^{∗}(Z)).

In this diagram, the middle and the bottom rows are exact, and the terms in the left column are all zero since X^{∗}(Z) is a permutation 𝔤-module. The group � _{ω}^{2}(𝔤, X^{∗}(Z)) is zero by the same reason (see 1.2.1). By diagram chasing one can easily prove that the top row is exact. We have proved isomorphism (2.3.5).

We have already proved that H^{1}(k, Pic \overline{V}_{H}) ≅ � _{ω}^{1}(k, L_{H}^{•}), because H^{ss} is simply connected. Together with isomorphisms (2.3.4) and (2.3.5) we obtain an isomorphism

H^{1}(k, Pic \overline{V}_{G}) ≅ � _{ω}^{1}(k, L_{G}^{•}).  (2.3.6)

Using Lemma 1.1.4(2), one can easily prove that isomorphism (2.3.6) does not depend on the choice of a z-extension H → G. Using Lemma 1.1.4(3), one can easily check that isomorphism (2.3.6) is functorial in G. This establishes the theorem for reductive k-groups.

Let now G be an arbitrary connected k-group. Let V_{G} be a smooth compactification of G and let V_{G^{red}} be a smooth compactification of G^{red}. By 1.1.2, there is an isomorphism of k-varieties G ≅ G^{red} × G^{u} where G^{u} is k-rational; by 1.2.2, we then obtain

H^{1}(𝔤, Pic \overline{V}_{G}) ≅ H^{1}(𝔤, Pic \overline{V}_{G^{red}}).  (2.3.7)

Let the complex L^{•} be defined as in the introduction, i.e. in terms of G^{red}. Since G^{red} is reductive, we have already proved that H^{1}(𝔤, Pic \overline{V}_{G^{red}}) ≅ � _{ω}^{1}(𝔤, L^{•}). Together with (2.3.7) we obtain

H^{1}(𝔤, Pic \overline{V}_{G}) ≅ � _{ω}^{1}(𝔤, L^{•}).

This isomorphism is functorial in G. □

Theorem 2.4. Let k be a field of characteristic zero, 𝔤 = Gal(k̄/k), G a connected linear algebraic k-group, E a G-torsor, V a smooth compactification of E, T ⊂ G a maximal k-torus, L^{•} = (X^{∗}(T) → X^{∗}(T^{sc})). Then the group H^{1}(k, Pic \overline{V}) is isomorphic to � _{ω}^{1}(𝔤, L^{•}).

2.5. Proof of Theorem 2.4. First assume that G is reductive and G^{ss} is simply connected. If E = G, Theorem 2.4 coincides with Theorem 2.1. If E has no rational points, one just has to make use of the device of passage to the generic point (see [CT/S3], App. 2B, p. 462–463; cf. [CT/K], proof of Th. 4.1). We reproduce here this argument adapted to our setting. Denote K = k(E), L = k̄(E), and let M be an algebraic closure of K containing L. We have Gal(L/K) = Gal(k̄/k) = 𝔤. Let 𝔤_{1} = Gal(M/K), 𝔥 = Gal(M/L): 𝔥 is a normal subgroup of 𝔤_{1}, and 𝔤 = 𝔤_{1}/𝔥. Since \overline{V} is a proper smooth rational variety, the natural inclusions of free abelian groups of finite rank

Pic (V ×_{k} k̄)↪Pic (V ×_{k} L)↪Pic (V ×_{k} M)

are in fact equalities. Denote this abelian group by Pic \overline{V}. The group 𝔥 acts trivially on Pic (V ×_{k} M). We write down the restriction-inflation exact sequence for the extensions M/L/K:

0 → H^{1}(𝔤, Pic \overline{V}) →^{inf} H^{1}(𝔤_{1}, Pic \overline{V}) →^{res} H^{1}(𝔥, Pic \overline{V}) = 0.

This gives an isomorphism

H^{1}(Gal(k̄/k), Pic (V ×_{k} k̄)) ≅ H^{1}(Gal(M/K), Pic (V ×_{k} M)).  (2.5.1)

Recall that G is reductive and G^{ss} is simply connected. The K-variety V ×_{k} K is a smooth compactification of the torsor E ×_{k} K which has a K-point and is hence isomorphic to G ×_{k} K. Formula (2.3.3) and isomorphism (2.5.1) then show that there is an isomorphism

H^{1}(Gal(k̄/k), Pic (V ×_{k} k̄)) ≅ � _{ω}^{2}(Gal(M/K), X^{∗}(G^{tor})).  (2.5.2)

Since 𝔥 = Gal(M/L) acts trivially on X^{∗}(G^{tor}), and X^{∗}(G^{tor}) is a torsion-free module, we can use the equality H^{1}(𝔥, X^{∗}(G^{tor})) = 0 and write down the restrictioninflation exact sequence for H^{2}:

0 → H^{2}(𝔤, X^{∗}(G^{tor})) →^{inf} H^{2}(𝔤_{1}, X^{∗}(G^{tor})) →^{res} H^{2}(𝔥, X^{∗}(G^{tor})).

Since � _{ω}^{2}(𝔥, X^{∗}(G^{tor})) = 0, we obtain

� _{ω}^{2}(Gal(k̄/k), X^{∗}(G^{tor})) ≅ � _{ω}^{2}(Gal(M/K), X^{∗}(G^{tor})).  (2.5.3)

Putting together isomorphisms (2.5.2) and (2.5.3), we obtain H^{1}(𝔤, Pic \overline{V}) ≅ � _{ω}^{2}(𝔤, X^{∗}(G^{tor})). Since G^{ss} is simply connected, we have � _{ω}^{2}(𝔤, X^{∗}(G^{tor})) ≅ � _{ω}^{1}(𝔤, L^{•}). Thus H^{1}(𝔤, Pic \overline{V}) ≅ � _{ω}^{1}(𝔤, L^{•}), and we obtain the theorem for G reductive with G^{ss} simply connected.

Let now G be an arbitrary reductive group and E_{G} a principal homogeneous space of G. By Lemma 1.1.4(1), there exists a z-extension α: H → G with kernel Z such that the class Cl(E_{G}) ∈ H^{1}(k, G) is the image of some Cl(E_{H}) ∈ H^{1}(k, H) where E_{H} is a principal homogeneous space of H. The cohomology map α_{∗}: H^{1}(k, H) → H^{1}(k, G) is represented by the map E_{H} ↦ E_{H}/Z. We may therefore assume that E_{G} = E_{H}/Z. The canonical projection E_{H} → E_{G} = E_{H}/Z admits a k-rational section, because Z is a quasi-trivial torus. Indeed, the obstruction to the existence of such a section lies in H^{1}(k(E_{G}), Z), and this cohomology group is zero by Hilbert 90. Therefore we have a birational isomorphism f: E_{G} × Z → E_{H} given by f(x, z) = s(x) · z where x ∈ E_{G} and z ∈ Z. The quasi-trivial torus Z is k-rational. By 1.2.2 the birational isomorphism f gives an isomorphism

H^{1}(Gal(k̄/k), Pic (V_{G} ×_{k} k̄)) ≅ H^{1}(Gal(k̄/k), Pic (V_{H} ×_{k} k̄))

where V_{G} (resp. V_{H}) stands for a smooth compactification of E_{G} (resp. E_{H}). Since H^{ss} is simply connected, by the preceding part of the proof we have

H^{1}(Gal(k̄/k), Pic (V_{H} ×_{k} k̄)) ≅ � _{ω}^{2}(Gal(k̄/k), X^{∗}(H^{tor})).

As shown in the proof of Theorem 2.1, � _{ω}^{2}(Gal(k̄/k), X^{∗}(H^{tor})) is isomorphic to � _{ω}^{1}(𝔤, L^{•}). Thus H^{1}(𝔤, Pic V_{G}) ≅ � _{ω}^{1}(𝔤, L^{•}). This proves the theorem for any reductive group.

Let now G be an arbitrary (not necessarily reductive) connected k-group. The canonical homomorphism r: G → G^{red} induces a bijection of Galois cohomology pointed sets r_{∗}: H^{1}(k, G) → H^{1}(k, G^{red}), cf. [S], 1.13. The map r_{∗} is represented by the map of torsors E ↦ E/G^{u}, where E is a torsor under G and E/G^{u} is a torsor under G/G^{u} = G^{red}. We wish to prove that

H^{1}(k, Pic \overline{V}_{E}) ≅ H^{1}(k, Pic \overline{V}_{E/G^{u}}),

where V_{E} and V_{E/G^{u}} are smooth compactifications of E and E/G^{u}, respectively. Since our functor � _{ω}^{1}(𝔤, L^{•}) is, by definition, the same for G and for G^{red}, this will prove the theorem.

We fix a Levi decomposition G^{u} ⋊G^{red}⟶^{∼}G. It defines a natural homomorphism φ: G^{red} → G and a map φ_{∗}: H^{1}(k, G^{red}) → H^{1}(k, G), inverse to r_{∗}. We want to describe φ_{∗} in terms of torsors.

Let X be a torsor under G^{red}. Set Y = X × G^{u}. We define a right action of G = G^{red} ⋉ G^{u} on Y by

(x, v) · (g, u) = (x · g, v^{g} · u),

where x ∈ X, v, u ∈ G^{u}, g ∈ G^{red}, and v^{g} refers to the right action of G^{red} on G^{u} defined by the Levi decomposition. One can easily check that this is a well-defined action and that the map X ↦ Y represents φ_{∗}.

Since Y = X × G^{u} and G^{u} is k-rational, by 1.2.2 we have H^{1}(k, Pic \overline{V} (Y)) ≅ H^{1}(k, Pic \overline{V} (X)). Since φ_{∗} is inverse to r_{∗}, any torsor E of G is isomorphic to Y for X = E/G^{u}. We obtain H^{1}(k, Pic \overline{V}_{E}) ≅ H^{1}(k, Pic \overline{V}_{E/G^{u}}). This proves the theorem. □

3. Comments and remarks

3.1. If G = T is a torus, we have � _{ω}^{1}(𝔤, L^{•}) = � _{ω}^{2}(𝔤, X^{∗}(T)), and the formula of Theorem 2.1 reduces to

H^{1}(k, Pic \overline{V}) ≅ � _{ω}^{2}(k, X^{∗}(T));

cf. [S], Prop. 9.8, in the number field case, [CT/S2] for E = T over an arbitrary field, and [CT/K] in general.

3.2. If G is a semisimple group, we have � _{ω}^{1}(𝔤, L^{•}) = � _{ω}^{1}(𝔤, X^{∗}(B)) where B = ker[G^{sc} → G] is the fundamental group of G, and the formula of Theorem 2.1 reduces to

H^{1}(k, Pic \overline{V}) ≅ � _{ω}^{1}(k, X^{∗}(B));  (3.2.1)

cf. [S], 9.6, in the number field case and [CT/K] in general.

3.3. Let now G be a reductive group admitting a special covering μ: G_{0} × S → G with kernel B, where G_{0} is a simply connected group, S is a quasi-trivial torus, and B is a finite group. We show that in this case Theorem 2.1 reduces to formula (3.2.1), cf. [S], Prop. 9.8, in the number field case and [CT/K] in general. Recall that T is a maximal torus of G, L^{•} = (X^{∗}(T) → X^{∗}(T^{sc})). We have to prove that

� _{ω}^{1}(k, X^{∗}(B)) = � _{ω}^{1}(𝔤, L^{•}).

Write μ^{−1}(T) = T_{0} × S, where T_{0} is a maximal torus of G_{0}. We have G^{sc} = G_{0}, T^{sc} = T_{0}. Set L_{1}^{•} = (X^{∗}(T) → X^{∗}(T_{0}) × X^{∗}(S)). Consider an exact sequence

1 → B → T_{0} × S → T → 1.

We see that the complexes (B → 1) and (T_{0} × S → T) are quasi-isomorphic. Hence the complexes (0 → X^{∗}(B)) and L_{1}^{•} are also quasi-isomorphic, and therefore

� _{ω}^{1}(k, X^{∗}(B)) = � _{ω}^{1}(𝔤, L_{1}^{•}).

We now consider a short exact sequence of complexes

0 → (0 → X^{∗}(S)) → L_{1}^{•} → L^{•} → 0.

It induces the following commutative diagram:

0  0  0
↓  ↓  ↓
0  ⟶ � _{ω}^{1}(𝔤, L_{1}^{•}) ⟶ � _{ω}^{1}(𝔤, L^{•}) ⟶ � _{ω}^{2}(𝔤, X^{∗}(S))
↓  ↓  ↓  ↓
0 = H^{1}(𝔤, X^{∗}(S)) ⟶ ℍ^{1}(𝔤, L_{1}^{•}) ⟶ ℍ^{1}(𝔤, L^{•}) ⟶ H^{2}(𝔤, X^{∗}(S))
↓  ↓  ↓  ↓
0 = ∏_{γ} H^{1}(γ, X^{∗}(S)) ⟶ ∏_{γ} ℍ^{1}(γ, L_{1}^{•}) ⟶ ∏_{γ} ℍ^{1}(γ, L^{•}) ⟶ ∏_{γ} H^{2}(γ, X^{∗}(S))

The middle and the bottom rows are exact. Since S is a quasi-trivial torus, we have � _{ω}^{2}(𝔤, X^{∗}(S)) = 0. By diagram chasing, one can show that the top row of the diagram is also exact. Thus

� _{ω}^{1}(k, X^{∗}(B)) = � _{ω}^{1}(𝔤, L_{1}^{•}) = � _{ω}^{1}(𝔤, L^{•}),

as required. □

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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