Manin obstruction to strong approximation for homogeneous spaces

14 Mikhail Borovoi and Cyril Demarchebe an exact sequence of connected algebraic groups over a field k of characteristic0. Assume that G ′ is linear. Then there is a commutative diagram with exact rowsPic(G)i ∗ Pic(G ′ ) ∆ G/G ′′ Br(G ′′ )ι ′′j ∗ Br(G) m∗ −p ∗ G Br(G ′ × G)iPic(G)∗ Pic(G ′ ) ∆ G/G ′′ Br 1,e (G ′′ , G) Br 1,e (G) Br 1,e (G ′ ) .(10)Here p G : G ′ × G → G is the projection map, the map m : G ′ × G → G is definedby m(g ′ , g) := i(g ′ ).g (where the product denotes the group law in G), ι ′′ and ι arethe inclusion homomorphisms, and the injective homomorphism ν is defined as inTheorem 2.8.If the homomorphism Pic(G) → Pic(G ′ ) is surjective (e.g. when G ′ is a k-<strong>to</strong>rus,or when G ′ ss is simply connected, or when all the three groups G ′ , G and G ′′ arelinear), then Br 1,e (G ′′ , G) = Br 1,e (G ′′ ), and we have a commutative diagram withexact rowsj ∗ιi ∗νPic(G)i ∗ Pic(G ′ ) ∆ G/G ′′ Br(G ′′ )j ∗ Br(G) m∗ −p ∗ G Br(G ′ × G)(11)ι ′′ινPic(G)i ∗ Pic(G ′ ) ∆ G/G ′′ Br 1,e (G ′′ )j ∗Br 1,e (G)i ∗Br 1,e (G ′ ) .Proof of the corollary. The short exact sequence of algebraic groups defines astructure of (left) G ′′ -<strong>to</strong>rsor under G ′ on G (G ′ acts on G by left translations).Now from the diagram with exact rows (8) we obtain diagram (10), which differsfrom diagram (11) by the middle term in the bot<strong>to</strong>m row.From diagram (8) we obtain an exact sequencePic(G) −→ i∗Pic(G ′ ) ∆ G/G−−−−→ ′′Br(G ′′ ) −→ j∗Br(G) . (12)If the homomorphism i ∗ : Pic(G) → Pic(G ′ ) is surjective, then the homomorphismj ∗ : Br(G ′′ ) → Br(G) is injective, hence Br 1,e (G ′′ , G) = Br 1,e (G ′′ ), and we obtaindiagram (11) from diagram (10).If G ′ is a k-<strong>to</strong>rus or if G ′ ss is simply connected, then Pic(G ′ ) = 0, and thehomomorphism Pic(G) → Pic(G ′ ) is clearly surjective. If all the three groupsG ′ , G and G ′′ are linear, then again the homomorphism Pic(G) → Pic(G ′ ) issurjective, see [38], proof of Corollary 6.11, p. 44.For the proof of Theorem 2.8 we need a crucial lemma.Lemma 2.13. Let k be a field of characteristic zero. Let H a connected lineark-group, X a smooth k-variety and π : Y H −→ X a (left) <strong>to</strong>rsor under H. Let