Manin obstruction to strong approximation for homogeneous spaces

18 Mikhail Borovoi and Cyril DemarcheLet p ∈ Pic(H), and let us prove that π ∗ (∆ Y/X (p)) = 0. The element pcorresponds <strong>to</strong> the class of an extension 1 → G m → H 1 → H → 1, and we have∆ Y/X (p) = ∂ H1 ([Y ]). We have a commutative diagramH 1 (X, H)∂ H1H 2 (X, G m )π ∗H 1 (Y, H)π ∗∂ H1 H 2 (Y, G m )so that π ∗ (∆ Y/X (p)) = π ∗ (∂ H1 ([Y ])) = ∂ H1 (π ∗ [Y ]). The <strong>to</strong>rsor π ∗ [Y ] is trivial inthe set H 1 (Y, H), hence π ∗ (∆ Y/X (p)) = 0. Consequently the <strong>to</strong>p row of diagram(8) is a complex.Let us prove that the <strong>to</strong>p row of diagram (8) is exact. For the exactness at theterm Br(Y ), see Proposition 2.4. So it remains <strong>to</strong> prove the exactness of the <strong>to</strong>prow at Pic(H) and at Br(X).Let p ∈ Pic(H) be such that ∆ Y/X (p) = 0. Such a p corresponds <strong>to</strong> the classqof an extension 1 → G m → H 1 −→ H → 1 such that ∂H1 ([Y ]) = 0. From exactsequence (7) we see that there exists an X-<strong>to</strong>rsor Z −−→ H1X under H 1 , with anH 1 -equivariant map Z → Y , making Z → Y in<strong>to</strong> a <strong>to</strong>rsor under G m . Then by theuniqueness part of Lemma 2.13 we know that the class of Z → Y in Pic(Y ) maps<strong>to</strong> the class of H 1 in Pic(H), i.e. ϕ 1 ([Z]) = p, which proves the exactness of the<strong>to</strong>p row of diagram (8) at Pic(H).Let us now prove the exactness of the <strong>to</strong>p row of (8) at Br(X).Assume first that the k-variety X is quasi-projective. Let A ∈ Br(X) such thatπ ∗ A = 0 ∈ Br(Y ). By a theorem of Gabber also proven by de Jong (see [12]), weknow that there exists a positive integer n and an X-<strong>to</strong>rsor Z → X under PGL nsuch that −A is the image of the class of Z in H 1 (X, PGL n ) by the coboundarymap H 1 (X, PGL n ) ∂GLn−−−→ H 2 (X, G m ). Let W denote the product Y × X Z. Fromthe commutative diagram with exact rowsH 1 (X, GL n ) H 1 (X, PGL n ) Br(X)H 1 (Y, GL n ) H 1 (Y, PGL n ) Br(Y )π ∗we see that the assumption π ∗ A = 0 implies that the <strong>to</strong>rsor W −−−−→ PGLnY isdominated by some Y -<strong>to</strong>rsor under GL n , i.e. there exists a <strong>to</strong>rsor V −−−→ GLnYand a morphism of Y -<strong>to</strong>rsors V → W compatible with the quotient morphism