Manin obstruction to strong approximation for homogeneous spaces

20 Mikhail Borovoi and Cyril Demarchecontained in the central subgroup G m of L. Therefore, since c(e, l ′ ) = c(l, e) = 1,Rosenlicht’s lemma implies that c(l, l ′ ) = 1 for all (l, l ′ ) ∈ L H × L PGLn . Hence themorphism µ is a group homomorphism, and the following diagram is commutativewith exact rows1 G m × G m L H × L PGLnH × PGL n 1mµ1 G m L H × PGL n 1 ,where m: G m × G m → G m is the group law. Therefore we get a commutativediagram of coboundary mapsH 1 (X, H) × H 1 (X, PGL n ) ∂ L H ×∂ LPGLn H 2 (X, G m ) × H 2 (X, G m )∼ =∼ =H 1 (X, H × PGL n )H 1 (X, H × PGL n )∂ LH ×L PGLn H 2 (X, G m × G m )m∂ L H 2 (X, G m ) .In particular, this diagram implies that∂ L ([W ]) = ∂ LH ([Y ]) + ∂ LPGLn ([Z])in H 2 (X, G m ). Since W −−−−−−→ H×PGLnX is dominated by the X-<strong>to</strong>rsor Vknow that ∂ L ([W ]) = 0, therefore the above formula implies thatL −→ X, we∂ LH ([Y ]) = −∂ LPGLn ([Z]) = A ∈ Br(X) ,i.e. A = ∆ Y/X ([L H ]), with [L H ] ∈ Ext c k(H, G m ) ∼ = Pic(H). Hence the first row ofdiagram (8) is exact under the assumption that X is quasi-projective.Let us now deduce the general case: X is not supposed <strong>to</strong> be quasi-projectiveanymore. By Nagata’s theorem (see [32]) we know that there exists a proper k-variety Z and an open immersion of k-varieties X → Z. By Chow’s lemma (see[15], II.5.6.1 and II.5.6.2, or [39], Chapter VI, §2.1), there exists a projective k-variety Z ′ and a projective, surjective birational morphism Z ′ → Z. Moreover,using Hironaka’s resolution of singularities (see [27], see also [3] and [17]), thereexists a smooth projective k-variety ˜Z and a birational morphism ˜Z → Z ′ . DefineX ′ <strong>to</strong> be the fibred product X ′ := ˜Z × Z X. Then X ′ is a open subvariety of ˜Z,hence X ′ is a smooth quasi-projective k-variety, and the natural map X ′ → Xis a birational morphism. Define Y ′ <strong>to</strong> be the product Y × X X ′ . By the quasi-