Criterion for surjectivity of localization in Galois cohomology of a reductive group over a number field

Let $G$ be a connected reductive group over a number field $F$, and let $S$ be a set (finite or infinite) of places of $F$. We give a necessary and sufficient condition for the surjectivity of the localization map from $H^1(F,G)$ to the"direct sum"of the sets $H^1(F_v,G)$ where $v$ runs over $S$. In the appendices, we give a new construction of the abelian Galois cohomology of a reductive group over a field of arbitrary characteristic.

1. Introduction 1.1.Let G be a (connected) reductive group over a number field F (we follow the convention of SGA3, where reductive groups are assumed to be connected).Let F be a fixed algebraic closure of F .We denote by V (F ) the set of places of F .For v ∈ V (F ), we denote by F v the completion of F at v. We refer to Serre's book [20] for the definition of the first Galois cohomology set H 1 (F, G).
In general, H 1 (F, G) is just a pointed set and has no natural groups structure.Let H 1 ab (F, G) denote the abelian Galois cohomology group of G introduced in [4, Section 2]; see also Labesse [14,Section 1.3].This is an abelian group depending functorially on G and F .There is a canonical abelianization map ab : H 1 (F, G) → H 1 ab (F, G).We give a new, better construction of H 1 ab (F, G) in Appendix A. Let S ⊆ V (F ) be a subset (finite or infinite).We consider the localization map In fact this map takes values in the subgroup v∈S H Similarly, consider the localization map In fact it takes values in the subset v∈S H 1 (F v , G) consisting of the families (ξ v ) v∈S with ξ v ∈ H 1 (F v , G) and such that ξ v = 1 for all v except maybe finitely many of them.This well-known fact follows, for instance, from the corresponding assertion for (1.We wish to find conditions under which the localization maps (1.1.2) and (1.1.3)are surjective.
1.2.We denote by M = π 1 (G) the algebraic fundamental group of G (also known as the Borovoi fundamental group of G) introduced in [4, Section 1], and also introduced by Merkurjev [16, Section 10.1] and Colliot-Thélène [8, Proposition-Definition 6.1].See Subsection 2.3 for our definition of π 1 (G).This is a finitely generated abelian group, on which the absolute Galois group Gal(F /F ) naturally acts.Let E/F be a finite Galois extension in F such that Gal(F /E) acts on M trivially and that E has no real places.Then the Galois group Γ := Gal(E/F ) naturally acts on M and on the set of places V (E) of the field E.
After explaining our notation in Section 2, we compute in Section 3 the finite abelian group Q 1 S (F, G) in terms of the action of Γ on M and on V (E) = V f (E) ∪ V C (E); see Corollary 3.8.See Subsection 2.4 for the notations V f and V C .
Concerning the map loc S of (1.1.3),in Section 3 we compute the image of this map; see Main Theorem 3.7.Using this result, we give a criterion (necessary and sufficient condition) for the map loc S to be surjective; see Corollary 3.9.This is also a criterion for the vanishing of Q 1 S (F, G).Again, our criterion is given in terms of the action of Γ on M and on Using this criterion, we give a simple proof of the result of Borel and Harder [2, Theorem 1.7] (see also Prasad and Rapinchuk [19,Proposition 1]) on the surjectivity of the map loc S when G is semisimple and there exists a finite place v 0 of F outside S; see Proposition 3.14 below.
Let Γ be a finite group.In Section 4, we construct an exact sequence arising from a short exact sequence of Γ-modules.In Section 5, using this exact sequence and Main Theorem 3.7, we generalize a result of Prasad and Rapinchuk giving a sufficient condition for the surjectivity of the localization map loc S when G is reductive, in terms of the radical (largest central torus) of G; see Theorem 5.1.As a particular case, we obtain the following corollary.
Corollary 1.4 (of Theorem 5.1).Let G be a reductive group over a number field F , and let C denote the radical of G (the identity component of the center of G).Let S ⊂ V (F ) be a set of places of F .Assume that the F -torus C splits over a finite Galois extension of F of prime degree p and that there exists a finite place v 0 in the complement S ∁ := V (F ) S of S such that C does not split over F v 0 .Then the localization map loc S of (1.1.3)is surjective.1.5.Let G be a reductive group over a field F of characteristic 0. In [4], the author defined the abelian group H 1 ab (F, G) as a set in a canonical way as the Galois hypercohomology of a certain crossed module.However, the definition of the structure of abelian group on H 1 ab (F, G) in [4] was complicated.In Appendix A, we define H 1 ab (F, G) (in arbitrary characteristic) following the letter of Breen to the author [7] and the article by Noohi [17] (written at the author's request), as the Galois hypercohomology H 1 (F, G ab ) of a certain stable crossed module, that is, a crossed module endowed with a symmetric braiding.The structure of abelian group comes from the symmetric braiding.Note that our specific crossed module and specific symmetric braiding were constructed by Deligne [11].
In Appendix B, Zev Rosengarten shows that certain equivalences of crossed modules of algebraic groups over a field F of arbitrary characteristic induce equivalences on F s -points where F s is a separable closure of F .This permits us to use in Appendix A the Galois hypercohomology of these crossed modules rather than fppf hypercohomology.

2.1.
Let A be an abelian group.We denote by A Tors the torsion subgroup of A. We set A t.f.= A/A Tors , which is a torsion-free group.

2.2.
Let Γ be a finite group, and let B be a Γ-module.We denote by B Γ the group of coinvariants of Γ in B, that is, We write B Γ, Tors := (B Γ ) Tors (which is the torsion subgroup of B Γ ), B Γ, t.f.= B Γ /B Γ, Tors (which is a torsion-free group).
which in general is neither injective nor surjective.
For a maximal torus T ⊆ G, we write T sc = ρ −1 (T ) ⊆ G sc and consider the natural homomorphism ρ : T sc → T.
We consider the algebraic fundamental group where X * denotes the cocharacter group.The Galois group Gal(F s /F ) naturally acts on M , and the Gal(F s /F )-module M is well defined (does not depend on the choice of T up to a transitive system of isomorphisms); see [4, Lemma 1.2].

2.4.
From now on (except for the appendices), F is a number field.We denote by , and V C (F ) the sets of all places of F , of finite places, of infinite places, of real places, and of complex places, respectively.
Let E/F be a finite Galois extension of number fields with Galois group Γ = Gal(E/F ); then Γ acts on V (E).If w ∈ V (E), we write Γ w for the stabilizer of w in Γ; then Γ w ∼ = Gal(E w /F v ) where v ∈ V (F ) is the restriction of w to F .

Main theorem
In this section we state and prove Main Theorem 3.7 computing the images of the localization maps (1.1.2) and (1.1.3).We deduce Corollary 3.8 computing the group Q 1 S (F, G), and Corollary 3.9 giving a necessary and sufficient condition for the surjectivity of the localization map (1.1.3).

3.1.
Let G be a reductive group over a number field F , and let v ∈ V f (F ) be a finite place of F .In [4] we computed . Let E/F be a finite Galois extension in F such that Gal(F /E) acts on M trivially and that E has no real places.Write Γ = Gal(E/F ).
where w is a place of E over v, and a canonical bijection Let v be a finite place of F .We have a surjective (even bijective) map We consider two composite maps with the same image where ω v : M Γw, Tors → M Γ, Tors is the homomorphism induced by the inclusion Γ w ֒→ Γ.Since the maps α ab v and α v are surjective (even bijective), and ω v is a homomorphism, we see that the set im be a complex place.We have zero maps Clearly, the set im λ ab v = im λ v is a subgroup of M Γ, Tors , namely, the subgroup {0}.3.4.Let v ∈ V R (F ) be a real place; then Γ w is a group of order 2, Γ w = {1, γ} where γ = γ w induces the nontrivial automorphism of E w over F v .We consider the Tate cohomology group We see immediately that the abelian group There is a canonical surjective map of Kottwitz [13, Theorem 1.2] (see also [4,Theorem 5.4]) and a canonical embedding Thus we obtain composite maps with the same image im α ab v = im α v , which is a subgroup of M Γw, Tors .Consider the composite maps with the same image Since the set im α ab v = im α v is a subgroup of M Γw, Tors , and ω v is a homomorphism, we conclude that the set im ) .Lemma 3.5.Let S ⊆ V (F ) be any subset, finite or infinite.Consider the summation maps Then the sets im Σ ab S and im Σ S are subgroups of M Γ, Tors , and they are equal.
Proof.Indeed, we have Here we write im λ v v∈S for the subgroup of M Γ, Tors generated by the subgroups im λ v for v ∈ S.
Theorem 3.6.The following sequences are exact: where for brevity we write V for V (F ).
Here (3.6.1) is an exact sequence of abelian groups, and (3.6.2) is an exact sequence of pointed sets.Main Theorem 3.7.Let G be a reductive group over a number field F .Let S ⊆ V := V (F ) be a subset.Write S ∁ = V S, the complement of S in V .Then: Proof.By Lemma 3.5, the sets im Σ ab S ∁ and im Σ S ∁ are (equal) subgroups of M Γ, Tors , and therefore it suffices to prove (3.7.1) with −im Σ ab S ∁ instead of im Σ ab S ∁ , and to prove (3.7.2) with −im Σ S ∁ instead of im Σ S ∁ .Now the corresponding assertions follow easily from the exactness of (3.6.1) and (3.6.2),respectively.
For the reader's convenience, we provide an easy proof of (3.7.2) with We conclude that Σ S (ξ S ) ∈ im Σ S ∩ − im Σ S ∁ , as required.
Conversely, let an element v∈V Proof.The homomorphism χ ab S is clearly surjective, and by Theorem 3.7 its kernel is the image im loc ab S of the localization homomorphism loc ab S of (1.1.2).The corollary follows.Corollary 3.9.The localization map loc S of (1.1.3)is surjective if and only if (3.9.1) im Σ S ⊆ im Σ S ∁ .
Proof.Consider the map By Lemma 3.5 the sets im Σ S = im Σ ab S and im Σ S ∩ im Σ S ∁ = im Σ ab S ∩ im Σ ab S ∁ are abelian groups.The morphism of pointed sets χ S is clearly surjective, and by Theorem 3.7 its kernel is im loc S .We see that the following assertions are equivalent: (a) the map loc S is surjective, that is, im loc This completes the proof.
Proof.Indeed, in our case condition (3.11.1) is equivalent to (3.9.1), and we conclude by Corollary 3.9.
Corollary 3.12.For a subset S ⊂ V (F ), let v 0 ∈ S ∁ , and assume that Then the localization map loc S of (1.1.3)is surjective.
Corollary 3.13.Let v 0 ∈ S ∁ , and assume that the map λ v 0 : Then the localization map loc S of (1.1.3)is surjective.
Proof.Indeed, then and we conclude by Corollary 3.9.
Proposition 3.14 (Borel and Harder [2, Theorem 1.7]).Let G be a semisimple group over a number field F , and let S ⊂ V (F ) be a subset such that the complement S ∁ of S contains a finite place v 0 ∈ V f (F ).Then the localization map loc S of (1.1.3)is surjective.
Proof.Since G is semisimple, the Γ-module M is finite, and so are the groups M Γ and M Γw where w is a place of E over v 0 .It follows that M Γw, Tors = M Γw and M Γ, Tors = M Γ .
The natural homomorphism M Γw → M Γ is clearly surjective.Therefore, the homomorphism is surjective.Since v 0 is finite, we have im λ v 0 = im ω v 0 , whence the map λ v 0 is surjective.We conclude by Corollary 3.13.

Exact sequence
In this section we construct an exact sequence that we shall use in Section 5.
Theorem 4.1.A finite group Γ and a short exact sequence of Γ-modules give rises to an exact sequence depending functorially on Γ and on the sequence (4.1.1).

4.2.
We specify the homomorphism δ.Let x 3 ∈ B 3 be such that the image (x 3 ) Γ of x 3 in (B 3 ) Γ is contained in (B 3 ) Γ, Tors .This means that there exist n ∈ Z >0 and y 3,γ ∈ B 3 such that We lift x 3 to some x 2 ∈ B , we lift each y 3,γ to some y 2,γ ∈ B 2 , and we consider the element Then j(z 2 ) = 0 ∈ B 3 , whence z 2 = i(z 1 ) for some z 1 ∈ B 1 .We consider the image (z 1 ) Γ, t.f. of z 1 ∈ B 1 in (B 1 ) Γ, t.f., and we put where we write 1 n for the image in Q/Z of 1 n ∈ Q. Below we give the proof of Theorem 4.1 suggested by Vladimir Hinich (private communication).For another proof, due to Alexander Petrov, see [18].

Proof Theorem 4.1 due to Vladimir Hinich. The functor from the category Γ-modules to the category of abelian groups
is the group ring of Γ. From the short exact sequence of Γ-modules (4.1.1),we obtain a long exact sequence depending functorially on Γ and on (4.1.1);see Weibel [21].Now Theorem 4.1 follows from the next proposition.
Proposition 4.4.For a finite group Γ and a Γ-module B, there is a canonical and functorial isomorphism Proof.Consider the short exact sequence regarded as a short exact sequence of Γ-modules with trivial action of Γ. Tensoring with B, we obtain a long exact sequence (4.4.1) By Lemma 4.5 below, we have Tor Λ 1 (Q, B) = 0, and the proposition follows from (4.4.1).
Lemma 4.5.For a finite group Γ and any Γ-module B, we have be a Λ-free resolution of the trivial Γ-module Z, for example, the standard complex; see Atiyah and Wall [1, Section 2].Tensoring with Q over Z, we obtain a flat resolution of Q Tensoring with B over Λ = Z[Γ], we obtain the complex (Q ⊗ Z P • ) ⊗ Λ B : (4.5.1) By definition, Tor Λ 1 (Q, B) is the first homology group of this complex.However, we can obtain the complex (4.5.1) from P • by tensoring first with B over Λ, and after that with Q over Z: Since Q is a flat Z-module, we obtain canonical isomorphisms Alternatively, one can check directly that the map δ constructed in Subsection 4.2 is well-defined (does not depend on the choices made) and that the sequence (4.1.2) is exact.

Surjectivity for a reductive group with nice radical
In this section we prove the following theorem that gives a sufficient condition for the surjectivity of the localization map (1.1.3)for a reductive F -group G in terms of the radical (largest central torus) of G.
We define Γ = Gal(E/F ) for M as in Subsection 3.1.Let S ⊂ V (F ) be a subset, and assume that S ∁ contains a finite place v 0 such that where w is a place of E over v 0 .Then the localization map loc S of (1.1.3)is surjective.
Proof.It follows from (5.1.1)that (M C ) Γw = (M C ) Γ , whence Using Theorem 4.1, we construct an exact commutative diagram Since G is semisimple, its algebraic fundamental group M is finite, and therefore the homomorphism ω in the diagram above is surjective; see the proof of Proposition 3.14.By a four lemma, the homomorphism is surjective as well.Since v 0 is finite, the map bijective, and therefore the map is surjective.We conclude by Corollary 3.13.Proof of Corollary 1.4.We define E, Γ, and Γ w for M = π 1 (G) as in Subsection 3.1.We have It follows that (5.1.1)holds.We conclude by Theorem 5.1.
is an equivalence (quasi-isomorphism), that is, it induces isomorphisms of F -group schemes Following an idea sketched by Labesse and Lemaire [15], we observe that (A.5.1)induces isomorphisms on groups of F s -points (in arbitrary characteristic); see Theorem B.1 in Appendix B below.It follows that the induced map on Galois gypercohomology ) is an isomorphism of abelian groups; see Noohi [17,Proposition 5.6].This shows that the abelian group structure on the pointed set H 1 F, G sc ρ − − → G, θ defined using the bijection (j T ) * (as in [4, Section 3.8]) coincides with the abelian group structure defined by the symmetric braiding { , }.
Remark A.6.González-Avilés [12] defined the abelian fppf cohomology group H 1 fppf, ab (X, G) and the abelianization map ab : H 1 fppf (X, G) → H 1 fppf, ab (X, G) for a reductive group scheme G over an arbitrary base scheme X, which includes the case of a reductive group over a field F of arbitrary characteristic.However, his definition uses the center Z G of G, and hence it is functorial only with respect to the normal homomorphisms G 1 → G 2 (homomorphisms with normal image, hence sending Z G 1 to Z G 2 )), whereas our definition above (over a field only) is functorial with respect to all homomorphisms.

Zev Rosengarten
In this appendix we prove the following theorem: Theorem B.1.Let F be a field of arbitrary characteristic and let F s be a fixed separable closure of F .Let ρ : G sc ։ [G, G] ֒→ G be as in Subsection 2.3.Let T ⊆ G be a maximal torus.We write T sc = ρ −1 (T ).Then the morphism of crossed modules is an equivalence (quasi-isomorphism).
Proof.We must show that the maps For (B.1.1),the injectivity is obvious.Moreover, any element of ker G sc (F s ) → G(F s ) lies in the preimage T sc of T , hence it is an element of T sc (F s ) and of ker T sc (F s ) → T (F s ) , which gives the surjectivity of i ker .
We prove the surjectivity of (B.

Theorem 3 . 2 ([ 4 ,
Proposition 4.1(i) and Corollary 5.4.1]).With the notation and assumptions of Subsection 3.1, for any finite place v of F there is a canonical isomorphism of abelian groups

Remark 3 . 10 .Corollary 3 . 11 .
Since by Lemma 3.5 we have im Σ ab S = im Σ S and im Σ ab S ∁ = im Σ S ∁ , we see from (d) in the proof above and from Corollary 3.8 that the localization map loc S of (1.1.3)is surjective if and only ifQ 1 S (F, G) = {1}.Let v 0 ∈ V (F ), S = V (F ) {v 0 }.Then the localization map loc S of (1.1.3)is surjective if and only if

Theorem 5 . 1 .
Let G be a reductive group over a number field F , and let C denote the radical of G. Write G = G/C, which is a semisimple group, and consider the short exact sequence of fundamental groups[4, Lemma 1.5]

Corollary 5 . 2 (
Prasad and Rapinchuk [19, Proposition 2(a)]).Let G be a reductive group over a number field F , and let C denote the radical of G. Assume that the F -torus C is split and that S ∁ contains a finite place v 0 .Then the localization map loc S of (1.1.3)is surjective.Proof.We define E, Γ, and Γ w for M = π 1 (G) as in Subsection 3.1.Then im[Γ → Aut M C ] = {1}, and hence (5.1.1)holds.We conclude by Theorem 5.1.
1.2).Let C ⊆ G denote the radical (largest central torus) of G. Then the mapψ : C × G sc → G, (c, s) → c • ρ(s) for c ∈ C, s ∈ G sc is surjective with central kernel Z ∼ = ρ −1 (C ∩ [G, G]) (which might be non-smooth).We have an exact commutative diagram of F -group schemes