The Brauer group and the Brauer-Manin set of products of varieties
aa r X i v : . [ m a t h . AG ] D ec THE BRAUER GROUP AND THE BRAUER–MANIN SET OFPRODUCTS OF VARIETIES
ALEXEI N. SKOROBOGATOV AND YURI G. ZARHIN
Abstract.
Let X and Y be smooth and projective varieties over a field k finitely generated over Q , and let X and Y be the varieties over an algebraicclosure of k obtained from X and Y , respectively, by extension of the groundfield. We show that the Galois invariant subgroup of Br( X ) ⊕ Br( Y ) has finiteindex in the Galois invariant subgroup of Br( X × Y ). This implies that thecokernel of the natural map Br( X ) ⊕ Br( Y ) → Br( X × Y ) is finite when k is anumber field. In this case we prove that the Brauer–Manin set of the productof varieties is the product of their Brauer–Manin sets. Let k be a field with a separable closure ¯ k , Γ = Aut (¯ k/k ). For an algebraicvariety X over k we write X for the variety over ¯ k obtained from X by extendingthe ground field. Let Br( X ) be the cohomological Brauer–Grothendieck groupH ( X, G m ), see [4]. The group Br( X ) is naturally a Galois module. The imageof the natural homomorphism Br( X ) → Br( X ) lies in Br( X ) Γ ; the kernel of thishomomorphism is denoted by Br ( X ), so that Br( X ) / Br ( X ) is a subgroup ofBr( X ) Γ . Recall that Br( X ) and Br( X ) are torsion abelian groups whenever X issmooth, see [4, II, Prop. 1.4]. Theorem A
Let k be a field finitely generated over Q . Let X and Y be smooth,projective and geometrically integral varieties over k . Then the cokernel of thenatural injective map Br( X ) Γ ⊕ Br( Y ) Γ → Br( X × Y ) Γ is finite. See Theorem 3.1 for an analogue in characteristic p = 2. The proof uses the resultsof Faltings and the second named author on Tate’s conjecture for abelian varieties.Let k be a field finitely generated over its prime subfield. We proved in ourprevious paper [14] that Br( X ) Γ is finite when X is an abelian variety and char( k ) =2, or X is a K3 surface and char( k ) = 0. As a corollary we obtain that if Z is asmooth and projective variety over k such that Z is birationally equivalent to aproduct of curves, abelian varieties and K3 surfaces, then the groups Br( Z ) Γ andBr( Z ) / Br ( Z ) are finite.The following result easily follows from Theorem A, see Section 4. Mathematics Subject Classification.
Primary 14F22; Secondary 14G25.
Theorem B
Let k be a field finitely generated over Q . Let X and Y be smooth,projective and geometrically integral varieties over k . Assume that ( X × Y )( k ) = ∅ or H ( k, ¯ k ∗ ) = 0 . Then the cokernel of the natural map Br( X ) ⊕ Br( Y ) → Br( X × Y ) is finite. Now let k be a number field. In this case H ( k, ¯ k ∗ ) = 0, see [9, Cor. I.4.21],so by Theorem B the Brauer group Br( X × Y ) is generated, modulo the image ofBr( X ) ⊕ Br( Y ), by finitely many elements. The following result shows that theseelements do not give any new Brauer–Manin conditions on the adelic points of X × Y besides those already given by the elements of Br( X ) ⊕ Br( Y ). For thedefinition of the Brauer–Manin set X ( A k ) Br we refer to [13, Section 5.2]. Theorem C
Let X and Y be smooth, projective, geometrically integral varietiesover a number field k . Then we have ( X × Y )( A k ) Br = X ( A k ) Br × Y ( A k ) Br . The key topological fact behind our proof of Theorem C is this: for any path-connected non-empty CW-complexes X and Y , and any commutative ring R with1 there is a canonical isomorphismH ( X × Y, R ) = H ( X, R ) ⊕ H ( Y, R ) ⊕ (cid:0) H ( X, R ) ⊗ R H ( Y, R ) (cid:1) . See Proposition 2.2 for this exercise in algebraic topology. (This formula does notgeneralise to the third cohomology group, see Remark 2.3.) The proof of TheoremC uses Theorem 2.6 that gives a similar result for the ´etale cohomology of connectedvarieties over ¯ k .T. Schlank and Y. Harpaz, using ´etale homotopy of Artin and Mazur, recentlyproved a statement similar to our Theorem C where the Brauer–Manin set is re-placed by the ´etale Brauer–Manin set. In their result the varieties X and Y do notneed to be proper, see [11, Cor. 1.3].The first named author thanks the University of Tel Aviv, where a part of thispaper was written, for hospitality. He is grateful to J.-L. Colliot-Th´el`ene, B. Kahnand A. P´al for useful discussions.1. Preliminaries . In this paper ‘almost all’ means ‘all but finitelymany’. If B is an abelian group, we write B tors for the torsion subgroup of B . Let B/ tors := B/B tors . If ℓ is a prime, then B ( ℓ ) is the subgroup of B tors consisting ofthe elements whose order is a power of ℓ , and B (non − ℓ ) is the subgroup of B torsHE BRAUER GROUP AND THE BRAUER–MANIN SET OF PRODUCTS OF VARIETIES 3 consisting of the elements whose order is not divisible by ℓ . If m is a positive integer,then B m is the kernel of the multiplication by m in B . . Let us recall some useful elementary statements that are dueto Tate [16, 18]. Let B be an abelian group. The projective limit of the groups B ℓ n (where the transition maps are the multiplications by ℓ ) is called the ℓ -adic Tatemodule of B , and is denoted by T ℓ ( B ). This limit carries a natural structure of a Z ℓ -module; there is a natural injective map T ℓ ( B ) /ℓ ֒ → B ℓ . One may easily checkthat T ℓ ( B ) ℓ = 0, and hence T ℓ ( B ) is torsion-free.Let us assume that B ℓ is finite. Then all the B ℓ n are obviously finite, and T ℓ ( B )is finitely generated by Nakayama’s lemma. Therefore, T ℓ ( B ) is isomorphic to Z rℓ for some non-negative integer r ≤ dim F ℓ ( B ℓ ). Moreover, T ℓ ( B ) = 0 if and onlyif B ( ℓ ) is finite. We denote by V ℓ ( B ) the Q ℓ -vector space T ℓ ( B ) ⊗ Z ℓ Q ℓ . Clearly, V ℓ ( B ) = 0 if and only if B ( ℓ ) is finite.If A is an abelian variety over a field k , and ℓ is a prime different from char( k ), n = ℓ i , then we write A n for the kernel of multiplication by n in A (¯ k ). The group A n is a free Z /n -module of rank 2dim ( A ) equipped with the natural structure of aΓ-module [10]. We write T ℓ ( A ) for T ℓ ( A (¯ k )), and V ℓ ( A ) for V ℓ ( A (¯ k )), respectively.The Q ℓ -vector space V ℓ ( A ) has dimension 2dim ( A ) and carries the natural structureof a Γ-module. . Let X be a smooth,projective and geometrically integral variety over k . For a positive integer n coprimeto char( k ) we have the Kummer exact sequence of sheaves of abelian groups in ´etaletopology: 0 → µ n → G m → G m → . Recall that H ( X, G m ) = Pic ( X ). Thus the Kummer sequence gives rise to anisomorphism of Γ-modules(1) H ( X, µ n ) = Pic ( X ) n . Let Pic ( X ) be the Γ-submodule of Pic ( X ) consisting of the classes of divisors al-gebraically equivalent to 0. By definition, the N´eron–Severi group of X is the quo-tient Γ-module NS ( X ) = Pic ( X ) / Pic ( X ). The abelian group NS ( X ) is finitelygenerated by a theorem of N´eron and Severi.Let A be the Picard variety of X , see [7]. Then A is an abelian variety over k such that the Γ-module A (¯ k ) is identified with Pic ( X ). Since the multiplicationby n is a surjective endomorphism of A , we have an exact sequence of Γ-modules(2) 0 → A n → Pic ( X ) n → NS ( X ) n → . ALEXEI N. SKOROBOGATOV AND YURI G. ZARHIN
Setting n = ℓ m , where ℓ = char( k ) is a prime, we deduce from (1) and (2) acanonical isomorphism of Γ-modules(3) H ( X, Z ℓ (1)) = T ℓ ( A ) = T ℓ (Pic ( X )) . In particular, this is a free Z ℓ -module of finite rank.Again, by surjectivity of the multiplication by ℓ m on A (¯ k ) = Pic ( X ) we obtainfrom the Kummer sequence the following exact sequence of Γ-modules(4) 0 → NS ( X ) /ℓ m → H ( X, µ ℓ m ) → Br( X ) ℓ m → . Passing to the projective limit in m gives rise to the well known exact sequence(5) 0 → NS ( X ) ⊗ Z ℓ → H ( X, Z ℓ (1)) → T ℓ (Br( X )) → . It shows that the torsion subgroup of H ( X, Z ℓ (1)) coincides with NS ( X )( ℓ ), cf.[14, Sect. 2.2]. In particular, if ℓ does not divide the order of the torsion subgroupof NS ( X ), then H ( X, Z ℓ (1)) is a free Z ℓ -module of finite rank. . Let X and Y be smooth, projective and geometricallyintegral varieties over k . We have the natural projection maps π X : X × Y → X, π Y : X × Y → Y. We denote by the same symbols the projections X × Y → X and X × Y → Y .Fixing ¯ k -points x ∈ X (¯ k ) and y ∈ Y (¯ k ), we define closed embeddings q y : X = X × y ֒ → X × Y , q x : Y = x × Y ֒ → X × Y .
Then π X q y = id X and π Y q x = id Y . On the other hand, π Y q y ( X ) = y ⊂ Y , π X q x ( Y ) = x ⊂ X. Let F be an ´etale sheaf defined by a commutative k -group scheme, see [8, Cor.II.1.7]. For example, F can be the sheaf defined by the multiplicative group G m ,or by the finite k -groups Z /n or µ n , where n is not divisible by the characteristicof k . The induced map π ∗ X : H i ´et ( X, F ) → H i ´et ( X × Y , F ) is a homomorphism of Γ-modules, whereas q ∗ x : H i ´et ( X × Y , F ) → H i ´et ( Y , F ) is a priori only a homomorphismof abelian groups. If x ∈ X ( k ), then q ∗ x is also a homomorphism of Γ-modules.The next proposition easily follows from definitions and the above considerations. Proposition 1.5.
For any i ≥ we have the following statements. (i) The induced maps π ∗ X : H i ´et ( X, F ) → H i ´et ( X × Y , F ) , π ∗ Y : H i ´et ( Y , F ) → H i ´et ( X × Y , F ) are injective homomorphisms of Γ -modules. (ii) The induced maps q ∗ y and q ∗ x define isomorphisms of abelian groups q ∗ y : π ∗ X (H i ´et ( X, F )) → H i ´et ( X, F ) , q ∗ x : π ∗ Y (H i ´et ( Y , F )) → H i ´et ( Y , F ) . HE BRAUER GROUP AND THE BRAUER–MANIN SET OF PRODUCTS OF VARIETIES 5 (iii)
The subgroup π ∗ X (H i ´et ( X, F )) lies in the kernel of q ∗ x : H i ´et ( X × Y , F ) → H i ´et ( Y , F ) , and, similarly, π ∗ Y (H i ( Y , F )) lies in the kernel of q ∗ y : H i ´et ( X × Y , F ) → H i ´et ( X, F ) . Hence we have π ∗ X (H i ´et ( X, F )) ∩ π ∗ Y (H i ´et ( Y , F )) = 0 . (iv) The map ( a, b ) π ∗ X ( a ) + π ∗ Y ( b ) defines an injective homomorphism of Γ -modules H i ´et ( X, F ) ⊕ H i ´et ( Y , F ) → H i ´et ( X × Y , F ) . We identify Pic ( X ) ⊕ Pic ( Y )with its image in Pic ( X × Y ). It is well known thatPic ( X × Y ) = Pic ( X ) ⊕ Pic ( Y ) . Let A be the Picard variety of X , and let B be the Picard variety of Y . The dualabelian variety A t of A is the Albanese variety of X . When X ( k ) = ∅ , the choiceof a point x ∈ X ( k ) defines a morphism Alb x : X → A t that sends x to 0. Thepair ( A t , Alb x ) can be characterized by the universal property that any morphismfrom X to an abelian variety A ′ that sends x to 0 is the composition of Alb x anda morphism of abelian varieties A t → A ′ . See [10], [7] for more details. It is clearthat the Albanese variety of X is A t . Proposition 1.7.
We have a commutative diagram of Γ -modules with exact rowsand columns, where the exact sequence in the bottom row is split: ↓ ↓ A (¯ k ) ⊕ B (¯ k ) = A (¯ k ) ⊕ B (¯ k ) ↓ ↓ → Pic ( X ) ⊕ Pic ( Y ) → Pic ( X × Y ) → Hom( B t , A ) → ↓ ↓ || → NS ( X ) ⊕ NS ( Y ) → NS ( X × Y ) → Hom( B t , A ) → ↓ ↓ If ( X × Y )( k ) = ∅ , then the exact sequence in the middle row is also split. Proof.
Choose a ¯ k -point ( x , y ) in X × Y , and let P x ,y be the kernel of thegroup homomorphismPic ( X × Y ) → Pic ( X ) ⊕ Pic ( Y ) , L ( q ∗ y L, q ∗ x L ) . Let N x ,y be the image of P x ,y in NS ( X × Y ). By Proposition 1.5 the intersec-tion of P x ,y with Pic ( X ) ⊕ Pic ( Y ) inside Pic ( X × Y ) is zero, hence the naturalsurjective map P x ,y → N x ,y is an isomorphism of abelian groups. ALEXEI N. SKOROBOGATOV AND YURI G. ZARHIN
For any L ∈ P x ,y we have q ∗ y L = 0, hence q ∗ y L ∈ Pic ( X ) for any y ∈ Y (¯ k ).Thus N x ,y is the kernel of the group homomorphismNS ( X × Y ) → NS ( X × y ) ⊕ NS ( x × Y )for any x ∈ X (¯ k ) and y ∈ Y (¯ k ). In particular, N x ,y does not depend on the choiceof ( x , y ), so we can drop x and y , and write N = N x ,y . It follows that N is aGalois submodule of NS ( X × Y ), so that we have a decomposition of Γ-modulesNS ( X × Y ) = NS ( X ) ⊕ NS ( Y ) ⊕ N. It remains to show that Γ-modules N and Hom( B t , A ) are canonically isomorphic.The Poincar´e sheaf P X on A t × A is a certain canonical invertible sheaf thatrestricts trivially to both { } × A and A t × { } , see [10], [7]. Every morphism ofabelian varieties u : B t → A gives rise to the invertible sheaf (Alb x , u Alb y ) ∗ ( P X )on X × Y , whose isomorphism class is in P x ,y . It is well known that this definesa group isomorphism(6) Hom( B t , A ) ˜ −→ P x ,y . The Poincar´e sheaf is defined over k so from (6) we deduce a canonical isomorphismof Γ-modules Hom( B t , A ) ˜ −→ N . The last statement of the proposition is clear: itis enough to choose ( x , y ) ∈ ( X × Y )( k ). QEDThe following corollary is well known. See Proposition 2.2 below for a topologicalanalogue. Corollary 1.8.
Let n be a positive integer not divisible by char( k ) . Then we havea canonical decomposition of Γ -modules (7) H ( X × Y , µ n ) = H ( X, µ n ) ⊕ H ( Y , µ n ) . Proof . The middle row of the diagram of Proposition 1.7 gives an isomorphism ofΓ-modules Pic ( X ) n ⊕ Pic ( Y ) n and Pic ( X × Y ) n . It remains to use the canonicalisomorphism (1). QED Remark 1.9.
The abelian group Hom( B t , A ) is finitely generated and torsion-free,hence H ( k, Hom( B t , A )) is finite. It follows that the cokernel of the natural mapH ( k, Pic ( X )) ⊕ H ( k, Pic ( Y )) −→ H ( k, Pic ( X × Y )) , and the kernel of the natural mapH ( k, Pic ( X )) ⊕ H ( k, Pic ( Y )) −→ H ( k, Pic ( X × Y ))are both finite. HE BRAUER GROUP AND THE BRAUER–MANIN SET OF PRODUCTS OF VARIETIES 7 K¨unneth decompositions . We continue toassume that X and Y are smooth, projective and geometrically integral varietiesover k . Let ℓ = char( k ) be a prime. We have the K¨unneth decomposition ofΓ-modulesH ( X × Y , Q ℓ ) = H ( X, Q ℓ ) ⊕ H ( Y , Q ℓ ) ⊕ (cid:0) H ( Y , Q ℓ ) ⊗ Q ℓ H ( X, Q ℓ ) (cid:1) , see [8, Cor. VI.8.13]. From (3) we have a canonical isomorphismH ( X, Q ℓ (1)) = V ℓ ( A ) . When n is a positive integer coprime to char( k ), the non-degeneracy of the Weilpairing gives rise to a canonical isomorphism of Galois modules B n = Hom( B tn , µ n ),and hence to a canonical isomorphismH ( Y , Q ℓ ) = V ℓ ( B )( −
1) = Hom Q ℓ ( V ℓ ( B t ) , Q ℓ ) . Therefore we have an isomorphism of Γ-modules [17, p. 143]H ( Y , Q ℓ ) ⊗ Q ℓ H ( X, Q ℓ (1)) = Hom Q ℓ ( V ℓ ( B t ) , V ℓ ( A )) , and hence a decomposition of Galois modules(8) H ( X × Y , Q ℓ (1)) = H ( X, Q ℓ (1)) ⊕ H ( Y , Q ℓ (1)) ⊕ Hom Q ℓ ( V ℓ ( B t ) , V ℓ ( A )) . Our next result, Theorem 2.6, is probably well known to experts; we give aproof as we could not find it in the literature. As a motivation and for the sake ofcompleteness we present a similar result for CW-complexes (we shall not need it inthe rest of the paper).
Proposition 2.2.
Let X and Y be non-empty path-connected CW-complexes. Forany commutative ring R with we have canonical isomorphisms of abelian groups H ( X × Y, R ) = H ( X, R ) ⊕ H ( Y, R ) and H ( X × Y, R ) = H ( X, R ) ⊕ H ( Y, R ) ⊕ (cid:0) H ( X, R ) ⊗ R H ( Y, R ) (cid:1) . Proof.
To simplify notation we write H n ( X ) for H n ( X, Z ). The universal coeffi-cients theorem [6, Thm. 3.2] gives the following (split) exact sequence of abeliangroups(9) 0 → Ext(H n − ( X ) , R ) → H n ( X, R ) → Hom(H n ( X ) , R ) → . Since X is non-empty and path-connected we have H ( X ) = Z , see [6, Prop. 2.7].This gives a canonical isomorphism(10) H ( X, R ) = Hom(H ( X ) , R ) . ALEXEI N. SKOROBOGATOV AND YURI G. ZARHIN
The K¨unneth formula for homology is0 → n M i =0 (cid:0) H i ( X ) ⊗ H n − i ( Y ) (cid:1) → H n ( X × Y ) → n − M i =0 Tor(H i ( X ) , H n − − i ( Y )) → , see [6, Thm. 3.B.6]. We deduce from it canonical isomorphisms(11) H ( X × Y ) = H ( X ) ⊕ H ( Y )and(12) H ( X × Y ) = H ( X ) ⊕ H ( Y ) ⊕ (cid:0) H ( X ) ⊗ H ( Y ) (cid:1) . Our first isomorphism follows from (10) and (11).The exact sequence (9) for n = 2 gives rise to the following commutative diagram0 → Ext(H ( X ) ⊕ H ( Y ) , R ) → H ( X, R ) ⊕ H ( Y, R ) → Hom(H ( X ) ⊕ H ( Y ) , R ) → ↑ ↑ ↑ → Ext(H ( X × Y ) , R ) → H ( X × Y, R ) → Hom(H ( X × Y ) , R ) → ( X ) ⊗ H ( Y ) , R ), which by (10) is isomorphic toH ( X, R ) ⊗ R H ( Y, R ). Moreover, H ( X, R ) and H ( Y, R ) are direct factors ofH ( X × Y, R ), so our second isomorphism follows. QED
Remark 2.3.
Let X = RP . Then H ( X ) = Z / n ( X ) = 0 for n ≥
2. Fromthe universal coefficients theorem (9) we obtain H ( X, Z ) = 0, H ( X, Z ) = Z / n ( X, Z ) = 0 for n ≥
3, cf. [6, Ex. 3.9]. Combining the calculation of homology of X in [6, Ex. 3.B.3] with the universal coefficients theorem we obtainH ( X , Z ) = Z / = M i =0 (cid:0) H i ( X, Z ) ⊗ H − i ( X, Z ) (cid:1) . This shows that Proposition 2.2 does not generalise to the third cohomology group,at least when R = Z . . After this digression into algebraic topology we returnto smooth, projective and geometrically integral varieties X and Y over a field k .We now introduce some notation. Let S X be the finite commutative k -group ofmultiplicative type whose Cartier dual ˆ S X := Hom ¯ k − gr . ( S X , G m ) isˆ S X = H ( X, µ n ) = Pic ( X ) n . Then we have a canonical identification(13) Hom( S X , Z /n ) = H ( X, Z /n ) . For any finite ¯ k -group scheme G of multiplicative type annihilated by n we have acanonical isomorphism, functorial in X and G : τ G : H ( X, G ) ˜ −→ Hom( ˆ G, Pic ( X )) = Hom( ˆ G, ˆ S X ) , HE BRAUER GROUP AND THE BRAUER–MANIN SET OF PRODUCTS OF VARIETIES 9 see [13, Cor. 2.3.9] (this uses the assumption that X is projective and connected,and so has no non-constant invertible regular functions). It can be defined via thenatural pairing(14) H ( X, G ) × ˆ G → H ( X, µ n ) = ˆ S X , see [13, Section 2.3]. If Z/X is a torsor under G , then the associated homomorphism τ G ( Z ) : ˆ G → ˆ S X is called the type of Z/X . If we take G = S X , then there existsa torsor T X /X under S X , unique up to isomorphism, whose type is the identitymap. Thus there is a well defined class [ T X ] ∈ H ( X, S X ). This class can beused to describe τ − G explicitly. For ϕ ∈ Hom( ˆ G, ˆ S X ) let ˆ ϕ ∈ Hom( S X , G ) be thehomomorphism that corresponds to ϕ under the identificationHom( ˆ G, ˆ S X ) = Hom( S X , G ) . The functoriality of τ G in G implies that τ − G ( ϕ ) is the push-forward ˆ ϕ ∗ [ T X ], whichcan also be defined as the class of the X -torsor ( T X × k G ) /S X .If we take G = S X in (14) we obtain a natural pairingH ( X, S X ) × ˆ S X → H ( X, µ n ) = ˆ S X . The definition of T X implies that pairing with the class [ T X ] gives the identity mapon ˆ S X . After a twist we obtain a natural pairingH ( X, S X ) × H ( X, Z /n ) → H ( X, Z /n );moreover, pairing with [ T X ] gives the identity map on H ( X, Z /n ). Remark 2.5.
There is a natural cup-product mapH ( X, S X ) ⊗ H ( Y , S Y ) −→ H ( X × Y , S X ⊗ S Y ) . Let us denote the image of [ T X ] ⊗ [ T Y ] by [ T X ] ∪ [ T Y ]. From (13) we obtain a naturalpairingH ( X × Y , S X ⊗ S Y ) × H ( X, Z /n ) ⊗ H ( Y , Z /n ) −→ H ( X × Y , Z /n ) . Since pairing with [ T X ] induces identity on H ( X, Z /n ), and similarly for Y , wesee that pairing with [ T X ] ∪ [ T Y ] gives the cup-product map ∪ : H ( X, Z /n ) ⊗ H ( Y , Z /n ) → H ( X × Y , Z /n ) . Theorem 2.6.
Let n be a positive integer coprime to char( k ) . Then the homomor-phism of Γ -modules H ( X, Z /n ) ⊕ H ( Y , Z /n ) ⊕ (cid:0) H ( X, Z /n ) ⊗ H ( Y , Z /n ) (cid:1) −→ H ( X × Y , Z /n ) given by π ∗ X on the first factor, by π ∗ Y on the second factor, and by the cup-producton the third factor, is an isomorphism. Proof.
It is enough to establish this decomposition at the level of abelian groups.Choose x ∈ X (¯ k ), y ∈ Y (¯ k ). Using the notation of Section 1.4 we defineH ( X × Y , Z /n ) prim = Ker [( q ∗ y , q ∗ x ) : H ( X × Y , Z /n ) → H ( X, Z /n ) ⊕ H ( Y , Z /n )] . The ´etale (or Zariski) sheaf R q π X ∗ ( Z /n ) is the constant sheaf associated with thefinite abelian group H q ´et ( Y , Z /n ) (for example, by the proper base change theorem[8, Cor. VI.2.3]). Thus we have the following Leray spectral sequence(15) E p,q = H p ´et ( X, H q ´et ( Y , Z /n )) ⇒ H p + q ´et ( X × Y , Z /n ) . We have seen in Proposition 1.5 that the maps π ∗ X and π ∗ Y make the abelian groupsH m ´et ( X, Z /n ) and H m ´et ( Y , Z /n ) direct factors of H m ´et ( X × Y , Z /n ), for all m ≥
1. Bythe standard theory of spectral sequences this gives a canonical isomorphism β : H ( X × Y , Z /n ) prim ˜ −→ H ( X, H ( Y , Z /n )) . Taking G = ˆ S Y in (14) we get an isomorphism τ ˆ S Y : H ( X, ˆ S Y ) ˜ −→ Hom( S Y , ˆ S X ).Using (13), after a twist we obtain an isomorphism τ : H ( X, H ( Y , Z /n )) ˜ −→ H ( X, Z /n ) ⊗ H ( Y , Z /n ) . This gives some isomorphism as in the statement of the theorem. To complete theproof we need to check that for any x ∈ H ( X, Z /n ) and any y ∈ H ( Y , Z /n ) wehave x ∪ y = β − τ − ( x ⊗ y ) . We have seen above that τ − ( x ⊗ y ) is the push-forward of [ T X ] by the map S X → Hom( S Y , Z /n ) defined by x ⊗ y ∈ H ( X, Z /n ) ⊗ H ( Y , Z /n ) = Hom( S X ⊗ S Y , Z /n ) . In other words, τ − ( x ⊗ y ) is obtained by pairing [ T X ] with x ⊗ y . In view of Remark2.5 in order to finish the proof, it remains to check that β − can be described viathe pairing H ( X, Hom( S Y , Z /n )) × H ( Y , S Y ) −→ H ( X × Y , Z /n ) , namely, as pairing with [ T Y ]. This calculation is more or less standard, cf. [13,Thm. 4.1.1] or, more recently, [5, Thm. 1.4].Let us write D ( Z ) for the bounded derived category of ´etale sheaves of abeliangroups on a variety Z . Let R π X ∗ : D ( X × Y ) → D ( X ) be the derived functor of π X .Let ρ : Y → Spec(¯ k ) be the structure morphism, and let R ρ ∗ : D ( Y ) → D (Ab) bethe corresponding derived functor to the bounded derived category of the categoryof abelian groups Ab. Each of these derived categories has the canonical truncationfunctors τ ≤ m . We need to recall the definition of the type map of a group G ofmultiplicative type, see [13, Section 2.3]. This is the composed map(16) H ( Y , G ) → Ext ( ˆ G, τ ≤ R ρ ∗ G m ) → Hom( ˆ G, Pic ( Y )) . HE BRAUER GROUP AND THE BRAUER–MANIN SET OF PRODUCTS OF VARIETIES11
The Hom- and Ext-groups without subscript are taken in Ab or D (Ab). The secondmap in (16) is induced by the obvious exact triangle in D (Ab)¯ k ∗ → τ ≤ R ρ ∗ G m → (Pic Y )[ − , where we used the facts that H ( Y , G m ) = ¯ k ∗ , since Y is reduced and connected,and H ( Y , G m ) = Pic ( Y ). To define the first map in (16) consider the local-to-global spectral sequence of Ext-groups E p,q = H p ´et ( Y , E xt qY ( ˆ G, G m )) ⇒ Ext p + qY ( ˆ G, G m ) . It completely degenerates since E xt qY ( ˆ G, G m ) = 0 for q ≥
1, thus giving an iso-morphism H q ´et ( Y , G ) ˜ −→ Ext qY ( ˆ G, G m ) [13, Lemma 2.3.7]. It remains to use theidentities Ext qY ( ˆ G, G m ) = Ext q ( ˆ G, R ρ ∗ G m ) = Ext q ( ˆ G, τ ≤ q R ρ ∗ G m )stemming from the fact that R Hom Y ( ρ ∗ ˆ G, · ) = R Hom( ˆ G, R ρ ∗ ( · )). When G isannihilated by n , the image of the type map lies in Hom( ˆ G, (Pic Y ) n ), and thus τ G can be written as the composition of the mapsH ( Y , G ) → Ext ( ˆ G, τ ≤ R ρ ∗ µ n ) → Hom( ˆ G, (Pic Y ) n ) . We claim that these maps fit into the following commutative diagram of pairings:H ( X, ˆ G ) × H ( Y , G ) → H ( X × Y , µ n ) || ↓ ↑ H ( X, ˆ G ) × Ext ( ˆ G, τ ≤ R ρ ∗ µ n ) → H ( X, τ ≤ R π X ∗ µ n ) || ↓ ↓ H ( X, ˆ G ) × Hom( ˆ G, (Pic Y ) n ) → H ( X, (Pic Y ) n )The first pairing is induced by the obvious pairing π ∗ X ˆ G × H om Y ( π ∗ Y ˆ G, µ n,Y ) −→ µ n,X × Y , by taking cohomology of X , Y , X × Y , respectively. If instead of R ρ ∗ ( H om Y ( · , · ))we consider R ( ρ ∗ H om Y )( · , · ) we get the second pairing. This explains the com-patibility of the two upper pairings. The third pairing is the natural one, so thatthe compatibility of the two lower pairings is clear.Now take G = S Y , so that ˆ G = (Pic Y ) n . By pairing with [ T Y ], after a twist weobtain a map γ : H ( X, H ( Y , Z /n )) → H ( X × Y , Z /n ) , which factors through the injective mapH ( X, τ ≤ R π X ∗ µ n ) → H ( X × Y , Z /n ) . Since ¯ k is separably closed we have H ( y , G ) = H (¯ k, G ) = 0, so that q ∗ y γ = 0.A similar argument gives q ∗ x γ = 0, thus Im ( γ ) ⊂ H ( X × Y , Z /n ) prim . By thestandard theory of spectral sequences the map β is obtained from the right handdownward map in the diagram (after a twist). Since the type of T Y is the identity in Hom((Pic Y ) n , (Pic Y ) n ), the commutativity of the diagram implies that βγ = id.QED Corollary 2.7.
Let n be a positive integer coprime to char( k ) , | NS ( X ) tors | and | NS ( Y ) tors | . Then we have a canonical decomposition of Γ -modules (17) H ( X × Y , µ n ) = H ( X, µ n ) ⊕ H ( Y , µ n ) ⊕ Hom( B tn , A n ) . Proof.
For any prime ℓ dividing n we have NS ( X )( ℓ ) = 0 and NS ( Y )( ℓ ) = 0.Thus from the isomorphism (1) and the exact sequence (2) we obtain canonicalisomorphisms H ( X, µ ℓ m ) = A ℓ m , H ( Y , µ ℓ m ) = B ℓ m , for any m ≥
1. From the non-degeneracy of the Weil pairing we deduce a canonicalisomorphism of Γ-modulesH ( Y , Z /ℓ m ) = B ℓ m ( −
1) = Hom( B tℓ m , Z /ℓ m ) . We conclude that the Galois modules H ( Y , Z /ℓ m ) ⊗ H ( X, µ ℓ m ) and Hom( B tℓ m , A ℓ m )are canonically isomorphic. Hence, after a twist by µ n , the isomorphism of Theorem2.6 can be written as (17). QED . Let ℓ be a prime different from char( k ). Tensoring (5)with Q ℓ we obtain the following exact sequence of Γ-modules(18) 0 → NS ( Z ) ⊗ Q ℓ → H ( Z, Q ℓ (1)) → V ℓ (Br( Z )) → , The injective maps from (18) and (4) are both called the first Chern class maps(see, e.g., [8], VI.9): c : NS ( X ) ⊗ Q ℓ ֒ → H ( X, Q ℓ (1)) , ¯ c : NS ( X ) /ℓ m ֒ → H ( X, µ ℓ m ) . Proposition 1.7 gives a natural isomorphism of Galois modulesNS ( X × Y ) = NS ( X ) ⊕ NS ( Y ) ⊕ Hom( B t , A ) . Since the maps c and ¯ c are functorial in X , we see that the map c : NS ( X × Y ) ⊗ Q ℓ ֒ → H ( X × Y , Q ℓ (1))forms obvious commutative diagrams with the maps c : NS ( X ) ⊗ Q ℓ ֒ → H ( X, Q ℓ (1)) , c : NS ( Y ) ⊗ Q ℓ ֒ → H ( Y , Q ℓ (1)) ,c : Hom( B t , A ) ⊗ Q ℓ ֒ → Hom Q ℓ ( V ℓ ( B t ) , V ℓ ( A )) . Similarly, for ℓ coprime to char( k ), | NS ( X ) tors | and | NS ( Y ) tors | , the map¯ c : NS ( X × Y ) /ℓ m ֒ → H ( X × Y , µ ℓ m )forms similar commutative diagrams with the maps¯ c : NS ( X ) /ℓ m ֒ → H ( X, µ ℓ m ) , ¯ c : NS ( Y ) /ℓ m ֒ → H ( Y , µ ℓ m ) , ¯ c : Hom( B t , A ) /ℓ m ֒ → Hom( B tℓ m , A ℓ m ) = H ( Y , Z /ℓ m ) ⊗ H ( X, µ ℓ m ) . HE BRAUER GROUP AND THE BRAUER–MANIN SET OF PRODUCTS OF VARIETIES13 . Let ℓ be a prime different fromchar( k ). Applying (18) to X , Y and X × Y , using (8) and the compatibilities fromSection 2.8, we obtain the following decomposition of Galois modules:(19) V ℓ (Br( X × Y )) = V ℓ (Br( X )) ⊕ V ℓ (Br( Y )) ⊕ (cid:0) Hom Q ℓ ( V ℓ ( B t ) , V ℓ ( A )) / Hom( B t , A ) ⊗ Q ℓ (cid:1) . When ℓ is also coprime to | NS ( X ) tors | and | NS ( Y ) tors | , we apply (4) to X , Y and X × Y , and obtain from Corollary 2.7 and Section 2.8 the decomposition ofΓ-modules(20) Br( X × Y ) ℓ m = Br( X ) ℓ m ⊕ Br( Y ) ℓ m ⊕ Hom( B tℓ m , A ℓ m ) / (cid:0) Hom( B t , A ) /ℓ m (cid:1) . The case when X and Y are elliptic curves was considered in [15, Prop. 3.3].3. Proof of Theorem A
The proof of Theorem A crucially uses the following properties. Let C and D beabelian varieties over a field k finitely generated over its prime subfield, char( k ) = 2.Then(1) the Γ-modules V ℓ ( C ) and V ℓ ( D ) are semisimple, and the natural injective mapHom( C, D ) ⊗ Q ℓ ֒ → Hom Γ ( V ℓ ( C ) , V ℓ ( D )is bijective;(2) for almost all primes ℓ the Γ-modules C ℓ and D ℓ are semisimple, and the naturalinjective map Hom( C, D ) /ℓ ֒ → Hom Γ ( C ℓ , D ℓ )is bijective.Statement (1) was proved by the second named author in characteristic p > Theorem 3.1.
Let k be a field finitely generated over its prime subfield. Let X and Y be smooth, projective and geometrically integral varieties over k . Then wehave the following statements. (i) If char( k ) = 0 , then [Br( X × Y ) / (Br( X ) ⊕ Br( Y ))] Γ is finite. (ii) If char( k ) = p > , then the group [Br( X × Y ) / (Br( X ) ⊕ Br( Y ))] Γ (non − p ) is finite. Proof.
Since Br( X × Y ) is a torsion group, it is enough to prove these statements:(a) If ℓ is a prime, ℓ = char( k ) , then V ℓ (cid:0) (Br( X × Y ) / (Br( X ) ⊕ Br( Y ))) Γ (cid:1) = 0 . (b) For almost all primes ℓ we have (cid:0) Br( X × Y ) ℓ / (Br( X ) ℓ ⊕ Br( Y ) ℓ ) (cid:1) Γ = 0 . Let us prove (a). Using (19) we obtain V ℓ (cid:0) (Br( X × Y ) / (Br( X ) ⊕ Br( Y ))) Γ (cid:1) = V ℓ (cid:0) Br( X × Y ) / (Br( X ) ⊕ Br( Y )) (cid:1) Γ = (cid:0) V ℓ (Br( X × Y )) / ( V ℓ (Br( X )) ⊕ V ℓ (Br( Y ))) (cid:1) Γ = (cid:0) Hom Q ℓ (cid:0) V ℓ ( B t ) , V ℓ ( A ) (cid:1) / Hom( B t , A ) ⊗ Q ℓ (cid:1) Γ . By a theorem of Chevalley [1, p. 88] the semisimplicity of Γ-modules V ℓ ( B t ) and V ℓ ( A ) implies the semisimplicity of the Γ-module Hom Q ℓ ( V ℓ ( B t ) , V ℓ ( A )). From thiswe deduce V ℓ (cid:0) (Br( X × Y ) / (Br( X ) ⊕ Br( Y ))) Γ (cid:1) = Hom Γ ( V ℓ ( B t ) , V ℓ ( A )) / Hom( B t , A ) ⊗ Q ℓ = 0 , thus proving (a).Let us prove (b). By (20) it is enough to show that (cid:0) Hom( B tℓ , A ℓ ) / (Hom( B t , A ) /ℓ ) (cid:1) Γ = 0 . Since Hom( B t , A ) Γ = Hom( B t , A ), the exact sequence0 → Hom( B t , A ) Γ /ℓ → (cid:0) Hom( B t , A ) /ℓ (cid:1) Γ → H ( k, Hom( B t , A ))implies that for all but finitely many primes ℓ we have (cid:0) Hom( B t , A ) /ℓ (cid:1) Γ = Hom( B t , A ) /ℓ. If we further assume that ℓ > A ) + 2dim ( B ) −
2, then, by a theorem of Serre[12], the semisimplicity of the Γ-modules B tℓ and A ℓ implies the semisimplicity ofHom( B tℓ , A ℓ ). Hence we obtain (cid:0) Hom( B tℓ , A ℓ ) / (Hom( B t , A ) /ℓ ) (cid:1) Γ = Hom( B tℓ , A ℓ ) Γ / (Hom( B t , A ) /ℓ ) Γ =Hom Γ ( B tℓ , A ℓ ) / (Hom( B t , A ) /ℓ ) = 0 , thus proving (b). QED Corollary 3.2.
Let k be a field finitely generated over its prime subfield. Let X and Y be smooth, projective and geometrically integral varieties over k . Then wehave the following statements. (i) Assume char( k ) = 0 . The group Br( X × Y ) Γ is finite if and only if the groups Br( X ) Γ and Br( Y ) Γ are finite. (ii) Assume that char( k ) is a prime p > . The group Br( X × Y ) Γ (non − p ) isfinite if and only if the groups Br( X ) Γ (non − p ) and Br( Y ) Γ (non − p ) are finite. HE BRAUER GROUP AND THE BRAUER–MANIN SET OF PRODUCTS OF VARIETIES15 Proof of Theorem B
It is enough to prove the following statements:(a)
The subgroup of
Br( X × Y ) generated by Br ( X × Y ) and the images of Br( X ) and Br( Y ) , has finite index. (b) The cokernel of the natural map Br ( X ) ⊕ Br ( Y ) → Br ( X × Y ) is finite. Each of these statements formally follows from Theorem A, the functoriality ofthe Hochschild–Serre spectral sequence(21) E p,q = H p ( k, H q ´et ( X, G m )) ⇒ H p + q ´et ( X, G m )with respect to X , and the finiteness property stated in Remark 1.9.Let us recall how (21) is usually applied. If X ( k ) = ∅ , then the canonical map E , = H ( k, ¯ k ∗ ) → H ( X, G m )has a section given by a k -point on X , and hence is injective. The same is obviouslytrue if H ( k, ¯ k ∗ ) = 0. The standard theory of spectral sequences now implies thatthe kernel of the canonical map E , = Br( X ) Γ → E , = H ( k, Pic ( X ))is the image of Br( X ) in Br( X ) Γ .Let us prove (a). By functoriality of the spectral sequence (21) we have thefollowing commutative diagram with exact rows:Br( X × Y ) → Br( X × Y ) Γ → H ( k, Pic ( X × Y )) ↑ ↑ ↑ Br( X ) ⊕ Br( Y ) → Br( X ) Γ ⊕ Br( Y ) Γ → H ( k, Pic ( X )) ⊕ H ( k, Pic ( Y ))Note that the middle vertical map here is injective. To prove (a) we must showthat the image of Br( X ) ⊕ Br( Y ) in Br( X × Y ) Γ has finite index in the subgroupof the elements that go to 0 in H ( k, Pic ( X × Y )). This follows from Theorem 3.1(i) and Remark 1.9.To prove (b) we consider another commutative diagram with exact rows, alsoconstructed using the functoriality of the spectral sequence (21):Br( k ) → Br ( X × Y ) → H ( k, Pic ( X × Y )) → ↑ ↑ Br ( X ) ⊕ Br ( Y ) → H ( k, Pic ( X )) ⊕ H ( k, Pic ( Y )) → Proof of Theorem C
The inclusion of the left hand side into the right hand side follows from functori-ality of the Brauer group. Thus we can assume that X ( A k ) Br and Y ( A k ) Br are notempty. Since the Brauer group of a smooth projective variety is a torsion group,to prove the opposite inclusion it is enough to show that for any positive integer n the subgroup Br( X × Y ) n is generated by the images of Br( X ) n and Br( Y ) n ,together with some elements that pair trivially with X ( A k ) Br × Y ( A k ) Br with re-spect to the Brauer–Manin pairing. The Kummer sequence gives a surjective mapH ( X × Y, µ n ) → Br( X × Y ) n , so it suffices to show that, modulo the images ofH ( X, µ n ) and H ( Y, µ n ), the group H ( X × Y, µ n ) is generated by the elementsthat pair trivially with X ( A k ) Br × Y ( A k ) Br .If Z/X is a torsor under a k -group of multiplicative type G annihilated by n ,then the type of Z/X , as recalled in Section 2.4, is the image of the class [
Z/X ]under the composed mapH ( X, G ) → H ( X, G ) Γ → Hom k ( ˆ G, Pic ( X )) = Hom k ( ˆ G, Pic ( X ) n ) . Recall that S X denotes the k -group scheme of multiplicative type that is dual tothe Γ-module Pic ( X ) n . Lemma 5.1. If X ( A k ) Br is not empty, then there exists an X -torsor under S X whose type is the identity map on ˆ S X . Proof . One of the main results of the descent theory of Colliot-Th´el`ene and Sansucsays that if X ( A k ) Br = ∅ , then for any homomorphism of Γ-modules ˆ G → Pic ( X )there exists an X -torsor under G of this type, see [13, Cor. 6.1.3 (1)]. QEDLet us choose one such X -torsor under S X , and call it T X . (It is unique up totwisting by a k -torsor under S X .) Then T X is isomorphic to the X -torsor T X fromSection 2. As in Remark 2.5 we form the class[ T X ] ∪ [ T Y ] ∈ H ( X × Y, S X ⊗ S Y ) . Pairing with it gives a map ε : Hom k ( S X ⊗ S Y , µ n ) = Hom k ( S X , ˆ S Y ) −→ H ( X × Y, µ n ) . For ϕ ∈ Hom k ( S X , ˆ S Y ) we can write ε ( ϕ ) = ϕ ∗ [ T X ] ∪ [ T Y ], where ∪ stands for thecup-product pairingH ( X, ˆ S Y ) × H ( Y, S Y ) → H ( X × Y, µ n ) . Remark 2.5 gives a commutative diagram(22) Hom k ( S X , ˆ S Y ) ε −→ H ( X × Y, µ n ) || ↓ (cid:0) H ( X, Z /n ) ⊗ H ( Y , µ n ) (cid:1) Γ ∪ −→ H ( X × Y , µ n ) Γ It is clear that Theorem C is a consequence of Lemmas 5.2 and 5.3 below.
Lemma 5.2.
We have H ( X × Y, µ n ) = π ∗ X H ( X, µ n ) + π ∗ Y H ( Y, µ n ) + Im ( ε ) . Lemma 5.3.
For any positive integer n we have the inclusion X ( A k ) Br ( X )[ n ] × Y ( A k ) Br ( Y )[ n ] ⊂ ( X × Y )( A k ) Im ( ε ) . HE BRAUER GROUP AND THE BRAUER–MANIN SET OF PRODUCTS OF VARIETIES17
Proof of Lemma 5.2.
We use the spectral sequence(23) E p,q = H p ( k, H q ´et ( X, µ n )) ⇒ H p + q ´et ( X, µ n ) . Let us make a few observations in the case when X is a smooth, projective andgeometrically integral variety over a number field k such that X ( A k ) = ∅ . Thecanonical maps E p, = H p ( k, µ n ) → H p ´et ( X, µ n )are injective for p ≥
3. Indeed, for such p the natural mapH p ( k, M ) → M k v ≃ R H p ( k v , M )is a bijection for any finite Galois module M , see [9, Thm. I.4.10 (c)]. Next, thenatural map H p ( k v , M ) → H p ´et ( X × k k v , M ) is injective for any p since any k v -pointof X defines a section of it. It follows that the composite mapH p ( k, M ) → H p ´et ( X, M ) → M k v ≃ R H p ´et ( X × k k v , M )is injective, and this implies our claim. We note that E , = H ( k, µ n ) → H ( X, µ n )is also injective. The argument is similar once we identify H ( k, µ n ) = Br( k ) n usingthe Kummer sequence and Hilbert’s theorem 90, and use the embedding of Br( k )into the direct sum of Br( k v ), for all completions k v of k , provided by global classfield theory, together with the existence of k v -points on X for every place v . Thisimplies the triviality of all the canonical maps in the spectral sequence whose targetis E p, = H p ( k, µ n ) for p ≥ ( X, µ n ) for the quotient of H ( X, µ n ) by the (injective) imageof H ( k, µ n ). Using the above remarks we obtain from (23) the following exactsequence:(24) 0 → H ( k, H ( X, µ n )) → ˜H ( X, µ n ) → H ( X, µ n ) Γ → H ( k, H ( X, µ n )) . There are similar sequences for Y and X × Y linked by the maps π ∗ X and π ∗ Y .Let us define H = π ∗ X H ( X, µ n ) + π ∗ Y H ( Y, µ n ) + Im ( ε ) ⊂ H ( X × Y, µ n ) . It is clear that the (injective) image of H ( k, µ n ) in H ( X × Y, µ n ) is containedin H , so to prove Theorem C it is enough to prove that the natural map H → ˜H ( X × Y, µ n ) is surjective.By (7) the image of H ( k, H ( X × Y , µ n )) → ˜H ( X × Y, µ n ) is contained in H .In view of (24) it remains to show that every element of the kernel of the mapH ( X × Y , µ n ) Γ → H ( k, H ( X × Y , µ n )) comes from H . By Theorem 2.6 and (7) this map can be written asH ( X, µ n ) Γ ⊕ H ( Y , µ n ) Γ ⊕ Hom k ( S X , ˆ S Y ) −→ H ( k, H ( X, µ n )) ⊕ H ( k, H ( Y , µ n )) . By the commutativity of the diagram (22) for any ϕ ∈ Hom k ( S X , ˆ S Y ), the element ε ( ϕ ) ∈ H ( X × Y, µ n ) maps to ϕ ∈ Hom( S X , ˆ S Y ) Γ ⊂ H ( X × Y , µ n ) Γ . This implies that for any a ∈ H ( X × Y, µ n ) there exists an element b ∈ H such thatthe image of a − b in H ( X × Y , µ n ) Γ is π ∗ X ( x ) + π ∗ Y ( y ) for some x ∈ H ( X, µ n ) Γ and y ∈ H ( Y , µ n ) Γ . From the exact sequence (24) for X × Y we see that π ∗ X ( x ) + π ∗ Y ( y ) goes to zero in H ( k, H ( X, µ n )) ⊕ H ( k, H ( Y , µ n )), hence x goes to zeroin H ( k, H ( X, µ n )), and y goes to zero in H ( k, H ( Y , µ n )). By (24) for X wesee that x is the image of some c ∈ H ( X, µ n ). Similarly, y is the image ofsome d ∈ H ( Y, µ n ). This proves that a − ( b + π ∗ X ( c ) + π ∗ Y ( d )) goes to zero inH ( X × Y , µ n ) Γ , and hence belongs to H . Thus a ∈ H . QED Proof of Lemma 5.3.
Let M be a finite Γ-module such that nM = 0. Let M D be the dual module Hom( M, ¯ k ∗ ). If v is a non-archimedean place of k , we writeH ( k v , M ) for the unramified subgroup of H ( k v , M ). By definition, it consistsof the classes that are annihilated by the restriction to the maximal unramifiedextension of k v . We write P ( k, M ) for the restricted product of the abelian groupsH ( k v , M ) relative to the subgroups H ( k v , M ), where v is a non-archimedeanplace of k . By [9, Lemma I.4.8] the image of the diagonal mapH ( k, M ) → Y all v H ( k v , M )is contained in P ( k, M ). Let us denote this image by U ( k, M ).The local pairings H ( k v , M ) × H ( k v , M D ) → H ( k v , µ n ) give rise to the globalPoitou–Tate pairing( , ) : P ( k, M ) × P ( k, M D ) −→ Z /n. It is a perfect duality of locally compact abelian groups, moreover, the subgroups U ( k, M ) and U ( k, M D ) are exact annihilators of each other [9, Thm. I.4.10 (b)].Let ϕ ∈ Hom k ( S X , ˆ S Y ). Let ( P v ) ∈ X ( A k ) be an adelic point that is Brauer–Manin orthogonal to Br ( X )[ n ], and let ( Q v ) ∈ Y ( A k ) be an adelic point orthogonalto Br ( Y )[ n ]. The Brauer–Manin pairing of the adelic point ( P v × Q v ) with theimage of ε ( ϕ ) = ϕ ∗ [ T X ] ∪ [ T Y ] in Br( X × Y ) is given by the Poitou–Tate pairing,so to prove Lemma 5.3 we need to show that(25) ( ϕ ∗ [ T X ]( P v ) , [ T Y ]( Q v )) = 0 , where in the above notation M = ˆ S Y , M D = S Y . We point out that for any a ∈ H ( k, ˆ S Y ) we have a ∪ [ T Y ] ∈ Br ( Y )[ n ], and hence ( a, [ T Y ]( Q v )) = 0. If HE BRAUER GROUP AND THE BRAUER–MANIN SET OF PRODUCTS OF VARIETIES19 an element of P ( k, S Y ) is orthogonal to U ( k, ˆ S Y ), then it belongs to U ( k, S Y ).Therefore, we must have(26) [ T Y ]( Q v ) ∈ U ( k, S Y ) . Similarly, for any b ∈ H ( k, S Y ) we have ϕ ∗ [ T X ] ∪ b ∈ Br ( X )[ n ], and hence( ϕ ∗ [ T X ]( P v ) , b ) = 0. Since every element of P ( k, ˆ S Y ) orthogonal to U ( k, S Y )belongs to U ( k, ˆ S Y ), this implies(27) ϕ ∗ [ T X ]( P v ) ∈ U ( k, ˆ S Y ) . Now (26) and (27) imply (25). QED
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Department of Mathematics, South Kensington Campus, Imperial College London,SW7 2BZ England, U.K.Institute for the Information Transmission Problems, Russian Academy of Sciences,19 Bolshoi Karetnyi, Moscow, 127994 Russia
E-mail address : [email protected] Department of Mathematics, Pennsylvania State University, University Park, PA16802, USAInstitute for Mathematical Problems in Biology, Russian Academy of Sciences, Pushchino,Moscow Region, Russia
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