On the transcendental Brauer group

Transcription

1 On the transcendenta Brauer group Jean-Louis Coiot-Théène and Aexei N. Skorobogatov 30th June, 2011 Abstract For a smooth and projective variety X over a fied k of characteristic zero we prove the finiteness of the cokerne of the natura map from the Brauer group of X to the Gaois-invariant subgroup of the Brauer group of the same variety over an agebraic cosure of k. Under further conditions on k, e.g. over number fieds, we give estimates for the order of this cokerne. We emphasise the rôe payed by the exponent of the discriminant groups of the intersection pairing between the groups of divisors and curves moduo numerica equivaence. Résumé Soit X une variété projective et isse sur un corps k de caractéristique zéro. Le groupe de Brauer de X s envoie dans es invariants, sous e groupe de Gaois absou de k, du groupe de Brauer de a même variété considérée sur une côture agébrique de k. Nous montrons que e quotient est fini. Sous des hypothèses suppémentaires, par exempe sur un corps de nombres, nous donnons des estimations sur ordre de ce quotient. L accoupement d intersection entre es groupes de diviseurs et de 1-cyces moduo équivaence numérique joue ici un rôe important. Introduction Let X be a smooth, projective and geometricay integra variety over a fied k of characteristic zero. Let k be an agebraic cosure of k. Let Γ = Ga(k/k) and X = X k k. The kerne of the natura map of Brauer groups Br(X) Br(X) is denoted by Br 1 (X) and is caed the agebraic Brauer group of X. The image of this map is caed the transcendenta Brauer group of 1

2 X; it is a subgroup of the group of invariants Br(X) Γ. We thus have the incusion of groups Br(X)/Br 1 (X) Br(X) Γ. One woud ike to compute these groups, for exampe, in connection with appications to the Brauer Manin obstruction. The foowing question was recenty discussed in [20] and [2]. If k is finitey generated over Q, are these two groups finite? In this paper we show that this doube question reduces to a singe one. For an arbitrary ground fied k of characteristic 0 we prove that the cokerne of the natura map α : Br(X) Br(X) Γ is finite. Under further conditions on the ground fied, for instance over number fieds, we give estimates for the exponent and the order of this finite group. The main too of the paper is a natura compex (see Subsection 1.3) Br(X) α Br(X) Γ β H 2 (k, Pic(X)) which for X with a k-point or for k a number fied is an exact sequence. We have two approaches to the cacuation of the (finite) image of β which give cosey reated though not identica estimates. The first method is presented in Section 2. The main idea is to use the functoriaity of the above compex with respect to morphisms of k-varieties and the triviaity of the Brauer group of curves over an agebraicay cosed fied (Tsen s theorem). It foows that the image of β has trivia restriction over any cosed curve in X. This eventuay eads to Theorems 2.1 and 2.2. Our second approach expoits a genera remark about differentias in the spectra sequence of composed functors. Let Br 0 (X) be the maxima divisibe subgroup of Br(X), and et NS(X) be the Néron Severi group. We interpret the composed map Br 0 (X) Γ Br(X) Γ β H 2 (k, Pic(X)) H 2 (k, NS(X)/ tors ) as the connecting homomorphism attached to a certain natura 2-extension of Γ-modues provided by the Kummer sequence, see Coroary 3.4. A theorem of Lieberman, recaed in Subsection 1.1, states that numerica and homoogica equivaences coincide on 1-dimensiona agebraic cyces. (For surfaces Lieberman s resut reduces to a cassica theorem of Matsusaka.) Using this we show in Proposition 4.1 that the image of the above composite map is 2

3 annihiated by the exponent of either of the two discriminant groups defined by the intersection pairing between the groups of divisors and curves on X moduo numerica equivaence. The estimates for the cokerne of α obtained by this method are proved in Theorems 4.2 and 4.3. In Section 5 we discuss K3-surfaces and products of two curves. For such a surface with a k-point our resut is particuary easy to state: the cokerne of α is annihiated by the exponent of the discriminant group defined by the intersection pairing on NS(X)/ tors, see Propositions 5.1 and 5.2. T. Szamuey asked whether the finiteness resut in Theorem 2.1 aso hods for smooth, quasiprojective varieties. In Section 6 we give a positive answer in the case when the ground fied k is finitey generated over Q. The authors are gratefu to the organisers of the conference Arithmetic of surfaces at the Lorentz Centre in Leiden in October, 2010, where the work on this paper began. We woud ike to thank L. Iusie, B. Kahn, J. Riou and T. Szamuey for usefu discussions and their hep with references. 1 Preiminaries Let k be a fied of characteristic 0 with an agebraic cosure k and the absoute Gaois group Γ = Ga(k/k). Let X be a smooth, projective and geometricay integra variety over k. Let X = X k k. Let d = dim(x). If A is an abeian group we denote by A[n] the set of eements a A such that na = 0. For a prime number we denote by A{} the set of eements a A such that m a = 0 for some m Agebraic cyces Let CH i (X), 0 i d, be the Chow group of codimension i cyces on X, i.e., the group of inear combinations of irreducibe subvarieties of codimension i with coefficients in Z moduo rationa equivaence. We have CH 1 (X) = Pic(X). Let NS(X) be the Néron Severi group of X, defined as the quotient of Pic(X) by its divisibe subgroup Pic 0 (X). Since X is projective, the intersection index defines the Γ-equivariant biinear form CH i (X) CH d i (X) Z. (1) Let N i = Num i (X) be the group of codimension i cyces on X moduo numerica equivaence, defined as the quotient of CH i (X) by the (eft) kerne 3

4 of the pairing (1). Write N i = N d i. We obtain a Γ-equivariant biinear form N i N i Z (2) with trivia eft and right kernes. For every i 0 the abeian group N i is free and finitey generated. This foows from the existence of a Wei cohomoogy theory with coefficients in a fied of characteristic zero, equipped with cyce maps transforming intersections of cyces into cup-product in cohomoogy [9, Thm. 3.5, p. 379]. The pairing (2) for i = 1 gives rise to the exact sequence of Γ-modues 0 N 1 Hom(N 1, Z) D 0, (3) which is the definition of the finite Γ-modue D. This group is one of the two discriminant groups associated to the pairing N 1 N 1 Z. For a i 0 we have the cyce cass maps CH i (X) H 2i ét(x, Z (i)), see [12, Section VI.9] and [19, Cyce]. These maps transform cup-product in -adic cohomoogy into intersection of agebraic cyces, see [12, Prop. VI.9.5]. Let us define the Γ-modues N i = N i Z, N i, = N i Z, H 2i = H 2i ét(x, Z (i))/ tors. For any cohomoogica theory with cyce maps for which intersection on cyces is compatibe with cup-product on cohomoogy, homoogica equivaence impies numerica equivaence. It is a part of the standard conjectures that for any good cohomoogy theory, homoogica and numerica equivaences coincide. In the -adic étae set-up the -adic cyce map shoud factorise as CH i (X) Z N i H 2i. (4) This is true for i = 1 by a cassica theorem of T. Matsusaka [11], who proved that N 1 = NS(X)/ tors. In the case of Betti cohomoogy, with cyce maps CH i (X) H 2i Betti(X(C), Q(i)), where Z(i) = Z(2π 1) i, for i = d 1, this was proved by D. Lieberman [10, Cor. 1]. A more agebraic version is given by Keiman [9, Remark 3.10]. These references prove more resuts, some of which rey on the Hodge index theorem. The case i = d 1 is simper, as we now expain. 4

5 Proposition 1.1 Let X be a smooth, connected, projective variety over C of dimension d. Define the homoogica equivaence on cyces via Betti cohomoogy with rationa coefficients. Then the natura map is an isomorphism. CH 1 (X)/hom CH 1 (X)/num Proof. We may assume d = dim(x) 3. Let L CH 1 (X) be the cass of a hyperpane section. Mutipication by L d 2 CH d 2 (X) induces a commutative diagram of Q-vector spaces (where A Q := A Z Q): CH d 1 (X) Q /num CH d 1 (X) Q /hom Hdg d 2 (X, Q) H 2d 2 (X, Q(d 1)) CH 1 (X) Q /num CH 1 (X) Q /hom Hdg 2 (X, Q) H 2 (X, Q(1)) The eftward horizonta arrows are onto. A the rightward horizonta arrows are by definition injective. The fourth vertica map is an isomorphism by the hard Lefschetz theorem. The third vertica map, on Hodge casses, is then an isomorphism, because of the Hodge decomposition of the groups H i (X, C), and the fact that (p, q)-type casses go to (p + d 2, q + d 2)-type casses for p = 0, 1, 2. The map CH 1 (X) Q /hom Hdg 2 (X, Q) is an isomorphism by the Lefschetz (1,1)-theorem. A this impies that the second vertica arrow is aso an isomorphism. The eft vertica arrow is thus surjective. By definition, the two finite dimensiona vector spaces CH 1 (X) Q /num and CH d 1 (X) Q /num have the same dimension. The eft vertica arrow is thus an isomorphism. The eft bottom arrow is an isomorphism by Matsusaka s theorem. We concude that the map CH d 1 (X) Q /hom CH d 1 (X) Q /num is an isomorphism. This impies that the map CH d 1 (X)/hom CH d 1 (X)/num is an isomorphism of finitey generated free abeian groups. QED Let us now reca how severa comparison theorems impy (4) for i = d 1, where X is a smooth, projective and geometricay integra variety over a fied k of characteristic zero. Reca that for an agebraicay cosed fied L containing k the Néron Severi group of X L = X k L does not depend on the fied L, because it is the group of connected components of the Picard scheme Pic XL /L. So we can use the notation N 1 without the risk of confusion. Let C be a 1-cyce on X which is numericay equivaent to zero. There exist a subfied K k finitey generated over Q, a variety X over K, and a 5

6 1-cyce C on X such that X = X K k and C = C K k. Without oss of generaity we can assume that N 1 is generated by the casses of reduced, absoutey irreducibe, effective divisors D 1,..., D r defined over K. We choose an embedding K C. Let K be the agebraic cosure of K in C, and et X K = X K K, XC = X K C. The cyce C goes to zero in N 1 = Num 1 (X) if and ony if C intersects triviay with D 1,..., D r. But then C goes to zero in Num 1 ( X C ). By Prop. 1.1 C goes to zero in the Betti cohomoogy group H 2d 2 ( X C, Q(d 1)). The comparison theorem for étae and Betti cohomoogy ([18] XI, XVI, see aso [12, Thm. III.3.12]) gives natura isomorphisms H 2i ét( X C, Q (i)) = H 2i Betti( X C (C), Q(i)) Q Q. The cyce cass maps transform cup-product in cohomoogy into intersection pairing of Chow groups. Starting from this, one can prove that the Betti cyce map and the -adic cyce map with rationa coefficients are compatibe with these isomorphisms. A proof of this is sketched in [3, p. 21]. J. Riou showed us how a forma proof can be deduced from the uniqueness statement for cyce maps in [17, Prop. 1.2]. Since the natura map H 2d 2 ét ( X K, Q (d 1)) H 2d 2 ét ( X C, Q (d 1)) is an isomorphism of vector spaces over Q (cf. [12, Cor. VI.4.3]), the cyce cass map sends C to zero in H 2d 2 ét ( X K, Q (d 1)). Extending the ground fied from K to k we get (4) for i = d 1. As recaed above, the pairing (1) is compatibe with the cup-product pairing H 2i ét(x, Z (i)) H 2d 2i ét (X, Z (d i)) Z via the cyce cass map. Hence we obtain the commutative diagram of pairings of Γ-modues N 1 N 1 Z (5) H 2 H 2d 2 Z with injective vertica maps. We sha use the foowing statement: the bottom pairing in (5) is perfect, i.e., it induces isomorphisms H 2 = Hom Z (H 2d 2, Z ), H 2d 2 = Hom Z (H 2, Z ). 6

7 L. Iusie tes us that this Z -version can be proved using Deigne s Z -adic formaism [4, 1.1]. Poincaré duaity for the compex RΓ(X, Z/ n ) (see [18, XVIII]) gives rise to a perfect duaity for the perfect compex RΓ(X, Z ). Then one appies a universa coefficient theorem argument. The exact sequence (3) gives rise to the exact sequence where the second arrow factors as 0 N 1 Hom Z (N 1,, Z ) D{} 0, (6) N 1 H 2 Hom Z (H 2d 2, Z ) Hom Z (N 1,, Z ). 1.2 The Brauer group Let us reca Grothendieck s description of the Brauer group Br(X) from [5, III.8, p ]. Let ρ = dim Q (NS(X) Q) be the Picard number of X, and et b 2 be the second Betti number of X. Let us denote the maxima divisibe subgroup of Br(X) by Br 0 (X). There is an isomorphism of abeian groups Br 0 (X) = (Q/Z) b 2 ρ. The quotient Br(X)/Br 0 (X) is finite, more precisey there is an exact sequence of Γ-modues 0 Br 0 (X) Br(X) H 3 ét(x, Z (1)) tors 0, (7) where runs through a prime numbers. Let B be the -adic Tate modue of Br(X), defined as the inverse imit of Br(X)[ m ] over m. Note that B is free as a Z -modue. The Gaois modue B ony contros the maxima divisibe subgroup Br 0 (X) Br(X), in the sense that B is aso isomorphic to the Tate modue of Br 0 (X), and there is a canonica isomorphism of Γ-modues, cf. [5, II.8.1, p. 144]: The Kummer sequence Br 0 (X) = (B Z Q /Z ). gives rise to exact sequences of Γ-modues 1 μ n G m x x n G m 1 (8) 0 Pic(X)/ m H 2 ét(x, μ m) Br(X)[ m ] 0. 7

8 Since the divisibe group Pic 0 (X) goes to zero in H 2 ét (X, μ m) we obtain the exact sequences 0 NS(X)/ m H 2 ét(x, μ m) Br(X)[ m ] 0. Passing to the inverse imit over m gives the exact sequence (8.7) of [5, III.8.2]: 0 NS(X) Z H 2 ét(x, Z (1)) B 0. (9) The second arrow in (9) induces an isomorphism on torsion subgroups (NS(X) Z ) tors = H 2 ét(x, Z (1)) tors. Thus we deduce the exact sequence of finitey generated Z -free Γ-modues 0 N 1 H 2 B 0. (10) As a sequence of Z -modues, it is spit. In particuar, for any prime number the Z -submodue N 1 H 2 is primitive, in the sense that the quotient is torsion-free. On tensoring (10) with Q /Z and taking the direct sum over a primes we obtain an exact sequence of Γ-modues 0 N 1 Q/Z (H 2 Z Q /Z ) Br 0 (X) 0, which gives rise to a 2-extension of Γ-modues 0 N 1 N 1 Q (H 2 Z Q /Z ) Br 0 (X) 0. (11) The foowing easy emma wi be used ater on. Lemma 1.2 Let F be a finite -primary torsion group, and et n 0. Let A be a finite subquotient of (Q /Z ) n F. If the exponent of A is m, then the order of A divides the product of mn and the order of F [ m ]. Proof. The group A is a quotient of (Q /Z ) r F (Q /Z ) n F, where F is a finite group. Then A is a quotient of F / m. The order of F / m equas the order of F [ m ], which is a subgroup of (Z/ m ) n F [ m ]. QED 8

9 1.3 Basic exact sequence Proposition 1.3 Let X be a (not necessariy smooth or projective) scheme over a fied k of characteristic zero. (i) There is a compex that is functoria in X and k: Br(X) α Br(X) Γ β H 2 (k, Pic(X)). (ii) Assume that H 0 ét (X, G m) = k. Assume, moreover, that the natura map H 3 ét (k, k ) H 3 ét (X, G m) is injective, which is the case when X has a k-point or when k is a number fied. Then the above compex is an exact sequence, and we have Im(α) = Ker(β) and Coker(α) = Im(β). Proof. This foows from the Leray spectra sequence E pq 2 = H p (k, H q ét (X, G m)) H p+q ét (X, G m ). (12) Note that a k-point on X defines a section of the map H 3 ét (k, k ) H 3 ét (X, G m); if k is a number fied, then H 3 ét (k, k ) = 0. QED 1.4 Restriction and corestriction The foowing emma is certainy we known, and is proved here for the sake of competeness. Lemma 1.4 Let X be a scheme over a fied k of characteristic zero, and et L k be a finite extension of k of degree n. There are restriction and corestriction homomorphisms res L/k : Br(X) Br(X L ), cores L/k : Br(X L ) Br(X) such that cores L/k (res L/k (x)) = nx. The foowing diagram commutes: Br(X) res L/k Br(X L ) cores L/k Br(X) α α L α Br(X) Γ Br(X) Γ L σ Br(X) Here Γ L = Ga(k/L), and σ(x) = σ i (x), where σ i Γ are coset representatives of Γ/Γ L. 9

10 Proof Let us reca the definition of res L/k and cores L/k. Let f : Y X be a finite fat morphism of connected smooth k-schemes, of degree n. Then we have morphisms of étae sheaves G m,x f G m,y G m,x defined on staks by a natura injection and the norm map, respectivey. The composition of the two maps coincides with raising to the power n. The functor f from the category of étae sheaves on Y to the category of étae sheaves on X is exact [12, Cor. II.3.6]. Thus the Leray spectra sequence gives an isomorphism H ṕ et (X, f G m,y ) H ṕ et (Y, G m,y ). Hence we obtain the desired maps H ṕ et (X, G m,x) res H ṕ et (Y, G m,x) cores H ṕ et (X, G m,x) whose composition is mutipication by n. Now et X be a scheme over the fied k. Let L k be a fied such that [L : K] = n, and et Y = X L = X k L. Ceary, we have the isomorphism L k k k n, where the components correspond to the n distinct embeddings of L into k. By changing the base from X to X we get a commutative diagram H ṕ et (X, G m) res L/k H ṕ et (X L, G m ) cores L/k H ṕ et (X, G m) H ṕ et (X, G m) H ṕ et (X, G m) n H ṕ et (X, G m) where the maps in the bottom row are the diagona embedding and the product. The representation of the Gaois group Γ in H ṕ et (X, G m) n is induced from the natura representation of Γ L in H ṕ et (X, G m). Thus passing to Γ- invariant subgroups, and taking p = 2, we obtain the statement of the emma. QED 2 Approach via Brauer groups of curves 2.1 Finiteness Theorem 2.1 Let X be a smooth, projective and geometricay integra variety over a fied k of characteristic zero. Then the cokerne of the natura map α : Br(X) Br(X) Γ is finite. Proof. By Grothendieck s computation of Br(X) recaed in Subsection 1.2, for any prime power n and any subquotient B of Br(X) the subgroup B[ n ] is finite. Therefore, it is enough to show that Coker(α) has finite exponent. 10

11 For this we can repace k by any finite extension. Indeed, if k L k, [L : k] = n, then it foows from Lemma 1.4 that we have natura maps Coker(α) Coker(α L ) Coker(α), whose composition is the mutipication by n. It is therefore enough to prove that Coker(α L ) has finite exponent. In particuar, we may assume that X has a k-point. By Proposition 1.3(ii) we have Coker(α) = Im(β). Let us show that Im(β) has finite exponent. If f : C X is a morphism, where C is a smooth, projective and geometricay integra curve over k, then the maps f : Pic(X) Pic(C) and f : Br(X) Br(C) fit into the commutative diagram Br(X) Γ Br(C) Γ β X β C H 2 (k, Pic(X)) H 2 (k, Pic(C)) By a theorem of Tsen and Grothendieck ([5], Cor. 1.3, p. 90) we have Br(C) = 0. Hence ( ) For any f : C X, the group Im(β X ) is contained in the kerne of the right vertica map in the diagram. The degree map Pic(C) NS(C) = Z defines the exact sequence of Gaois modues 0 Pic 0 (C) Pic(C) NS(C) 0, so that we have a commutative diagram with exact rows H 2 (k, Pic 0 (X)) H 2 (k, Pic(X)) H 2 (k, NS(X)) 0 H 2 (k, Pic 0 (C)) H 2 (k, Pic(C)) H 2 (k, NS(C)) (13) The zero in the bottom row is due to the fact that H 1 (k, Z) = 0. Over the infinite fied k, the Bertini theorem [7] for hyperpane sections of smooth projective varieties ensures that there exists a inear curve section C X, defined over k, which is smooth and geometricay connected. A combination of the Bertini theorem and Zariski s connectedness theorem (see [6, Lemme 2.10, p. 210]) then impies that over an agebraic cosure of k the inverse image under C X of any connected finite étae cover of X is 11

12 connected. In particuar, the map of abeian varieties Pic 0 X/k Pic 0 C/k has trivia kerne. By the Poincaré reducibiity theorem [15, 19, Thm. 1] there exists an abeian subvariety A Pic 0 C/k such that the natura map Pic 0 X/k A Pic 0 C/k is an isogeny of abeian varieties over k. Since H 2 (k, Pic 0 (C)) H 2 (k, Pic(C)) is injective, this impies: ( ) The kerne of the composite map has finite exponent. H 2 (k, Pic 0 (X)) H 2 (k, Pic(X)) H 2 (k, Pic(C)) Since N 1 = NS(X)/ tors is a finitey generated free abeian group, we can choose finitey many, say m, curves in X such that the intersection pairing with the casses of these curves defines an injective group homomorphism ι : N 1 Z m. By taking normaisation we obtain m morphisms from smooth projective curves defined over k to X. For the purpose of the proof we can repace k by a finite extension over which a of these curves are defined. We now have morphisms f i : C i X, i = 1,..., m, from smooth, projective and geometricay irreducibe curves to X defined over the ground fied k. The maps induce a map of Γ-modues NS(X) m i=1 NS(C i ) = Z m. In view of (13), ( ) and ( ), to compete the proof it is enough to show that the kerne of the induced map H 2 (k, NS(X)) H 2 (k, Z m ) has finite exponent. This map is the composite of two maps: H 2 (k, NS(X)) H 2 (k, N 1 ) H 2 (k, Z m ). It is enough to show that the kerne of each of these maps is of finite exponent. From the cohomoogy sequence attached to the exact sequence of Γ- modues 0 NS(X) tors NS(X) N 1 0 we deduce that the map H 2 (k, NS(X)) H 2 (k, N 1 ) 12

13 has its kerne annihiated by mutipication by the exponent of NS(X) tors. There exists a homomorphism Z m N 1 such that the composition of homomorphisms of abeian groups with trivia Gaois action N 1 ι Z m N 1 is mutipication by a positive integer. The kerne of H 2 (k, N 1 ) H 2 (k, Z m ) is annihiated by mutipication by this integer. QED 2.2 Upper bounds, I Let δ 0 be the exponent of the finite group D defined in (3), and et ν 0 be the exponent of the finite group NS(X) tors. Let α be the natura map Br(X) Br(X) Γ. Theorem 2.2 Let X be a smooth, projective and geometricay integra variety over a fied k of characteristic zero. Let L/k be a finite fied extension such that the finitey generated abeian group N 1 = Num 1 (X) is generated by the casses of integra curves on X defined over L. Let λ = [L : k]. (i) If H 1 (X, O X ) = 0 and the map H 3 (k, G m ) H 3 (X, G m ) is injective, then the exponent of Coker(α) divides λδ 0 ν 0. (ii) If k is a number fied, the exponent of Coker(α) divides 2λδ 0 ν 0, and it divides λδ 0 ν 0 if k is totay imaginary. Proof. We foow the proof of Theorem 2.1 making necessary cacuations aong the way. We first appy Proposition 1.3. For any number fied k we have H 3 (k, k ) = 0, thus in this case we aways have Im(α) = Ker(β), hence Coker(α) = Im(β). This aso hods under the assumption that the map H 3 (k, G m ) H 3 (X, G m ) is injective. We choose finitey many, say m, integra curves C 1,..., C m in X whose casses generate N 1, and then repace k by a finite extension L over which a of these curves are defined. The restriction-corestriction argument at the beginning of the proof of Theorem 2.1 shows that when we repace k by the finite extension L, of degree λ, the exponent of Coker(α) divides the product of the exponent of Coker(α L ) and λ. To prove the theorem, we may now assume k = L, that is λ = 1. Let Pic(X) Z m be the map given by restriction to the curves C i, foowed by the degree map on each curve. The proof of Theorem 2.1 shows that the 13

14 image of β is contained in the kerne of the induced map H 2 (k, Pic(X)) H 2 (k, Z m ). The map Pic(X) Z m factorises as foows Pic(X) NS(X) N 1 Hom(N 1, Z) Z m. We sha bound the exponent of each induced map on H 2 (k, ). Sending each curve C i to its cass in N 1 yieds an exact sequence of trivia Γ-modues 0 Z r Z m N 1 0. Duaising we obtain a spit exact sequence of trivia Γ-modues The map 0 Hom(N 1, Z) Z m Z r 0. H 2 (k, Hom(N 1, Z)) H 2 (k, Z m ) is therefore injective. From the exact sequence (3) we concude that the kerne of H 2 (k, N 1 ) H 2 (k, Hom(N 1, Z)) is annihiated by the exponent of H 1 (k, D), hence by δ 0, the exponent of D. As we have seen in the proof of the previous theorem, the kerne of the map H 2 (k, NS(X)) H 2 (k, N 1 ) is kied by the exponent ν 0 of NS(X) tors. Finay we have the exact sequence 0 Pic 0 X/k(k) Pic(X) NS(X) 0. If H 1 (X, O X ) = 0, then Pic 0 X/k = 0. When k is a number fied and A is an abeian variety, we have H 2 (k, A) = v H 2 (k v, A), where v runs through the rea competions of k [13, Thm (c), p. 92]. Hence the exponent of H 2 (k, A) is at most 2. This competes the proof of the theorem. QED Remark. An expicit bound for the order of Coker(α) immediatey foows from Theorem 2.2 via Lemma

15 3 Differentias 3.1 A genera remark about differentias in spectra sequences Let us reca the standard set-up of the spectra sequence of composed functors. Let A, B, C be abeian categories such that A and B have enough injectives. Let G : A B and F : B C be eft exact additive functors such that G sends injective objects into F -acycic. Then for every object B Ob(A) we have the spectra sequence Let E pq 2 = (RF p )(R q G)B R p+q (F G)B. (14) p,q : (R p F )(R q G)B (R p+2 F )(R q 1 G)B be the canonica maps in this spectra sequence. Suppose we have an exact sequence in A: 0 A B C 0. (15) Appying the right derived functors of G we get a ong exact sequence in B. Truncating it we obtain for any q 1 the exact sequence 0 B 1 (R q 1 G)C (R q G)A B 2 0, (16) together with the surjective map s : (R q 1 G)B B 1 and the injective map i : B 2 (R q G)B. Let : (R p F )B 2 (R p+2 F )B 1 be the connecting homomorphism defined by (16). Let s = (R p+2 F )(s) : (R p+2 F )(R q 1 G)B (R p+2 F )B 1 be the map induced by s, and simiary et i = (R p F )(i) : (R p F )B 2 (R p F )(R q G)B be the map induced by i. Lemma 3.1 We have = s p,q i. 15

16 Proof. There exists a short exact sequence 0 A B C 0 of injective resoutions of A, B, C, respectivey. Let a n : A n A n+1 be the differentias in A, and simiary for B and C. We have the commutative diagram A q 1 /Im(a q 2 ) B q 1 /Im(b q 2 ) C q 1 /Im(c q 2 ) 0 0 Ker(a q ) Ker(b q ) Ker(c q ) Appying the snake emma we obtain the exact sequence (R q 1 G)A (R q 1 G)B (R q 1 G)C (R q G)A (R q G)B (R q G)C. Truncating it we get (16). Chasing the diagram one checks that (16) is equivaent to the 2-extension 0 (R q 1 G)B B q 1 /b q 2 (B q 2 ) Ker(b q ) (R q G)B 0 (17) pued back via i : B 2 (R q G)B and pushed out via s : (R q 1 G)B B 1. By definition, the canonica map p,q is the connecting homomorphism (R p F )(R q G)B (R p+2 F )(R q 1 G)B defined by (17), hence s p,q i =. QED 3.2 Appications to the Brauer group The Kummer sequence (8) gives rise to the 2-extension of Γ-modues 0 Pic(X)/Pic(X)[n] Pic(X) H 2 ét(x, μ n ) Br(X)[n] 0, (18) where the second arrow is defined by mutipication by n on Pic(X). Proposition 3.2 The foowing diagram commutes: Br(X)[n] Γ Br(X) Γ β H 2 (k, Pic(X)/Pic(X)[n]) H 2 (k, Pic(X)) Here is the connecting homomorphism defined by (18), and the vertica arrows are the obvious natura maps. 16

17 Proof. In the set-up of (14) and Lemma 3.1, et now A be the category of étae sheaves on X, et B be the category of continuous discrete Γ-modues, and et C be the category of abeian groups. Let G = π, where π : X Spec(k) is the structure morphism. Let F (M) = M Γ. Let A = μ n,x, B = C = G m,x, and et (15) be the Kummer sequence (8). Take p = 0 and q = 2. The associated sequence (16) is precisey the sequence (18). It remains to appy Lemma 3.1: the eft vertica map in the diagram is i, the bottom horizonta map is β = 0,2, and the right vertica map is s. QED It is easy to check that the exact sequence gives rise to the exact sequence 0 Pic 0 (X) Pic(X) NS(X) 0 0 Pic 0 (X)/Pic 0 (X)[n] Pic(X)/Pic(X)[n] NS(X)/NS(X)[n] 0. The divisibe subgroup Pic 0 (X) Pic(X) is contained in the kerne of Pic(X) H 2 ét (X, μ n), hence (18) gives rise to the 2-extension of Γ-modues 0 NS(X)/NS(X)[n] NS(X) H 2 ét(x, μ n ) Br(X)[n] 0, (19) where the second arrow is induced by mutipication by n on NS(X). Coroary 3.3 The foowing diagram commutes: Br(X)[n] Γ Br(X) Γ β H 2 (k, NS(X)/NS(X)[n]) H 2 (k, Pic(X)) Here is the connecting homomorphism defined by (19), and the vertica arrows are the obvious natura maps. Proof. This immediatey foows from Proposition 3.2. QED Coroary 3.4 The foowing diagram commutes: Br 0 (X) Γ Br(X) Γ H 2 (k, N 1 ) β H 2 (k, Pic(X)) Here is the connecting homomorphism defined by (11), and the vertica arrows are the obvious natura maps. 17

18 Proof. Let n be a positive integer divisibe by ν 0, the exponent of NS(X) tors. For such an n the exact sequence (19) takes the form 0 N 1 NS(X) H 2 ét(x, μ n ) Br(X)[n] 0, the map N 1 NS(X) being induced by mutipication by n on NS(X). Let us write n = n, where n is a power of the prime. Let P = NS(X){}, and et Im(P ) be the image of P under the composite map NS(X) H 2 ét(x, Z (1)) H 2 ét(x, μ n ). We have the foowing commutative diagram of Γ-modues, with exact rows: 0 N 1 N 1 Q (H 2 Z Q /Z ) Br 0 (X) 0 0 N 1 n N 1 H 2/n Br 0 (X)[n] 0 0 N 1 n N 1 H 2 ét (X, μ n )/Im(P ) Br(X)[n] 0 0 N 1 n NS(X) H 2 ét (X, μ n) Br(X)[n] 0. The exact sequence in the first row is obtained by tensoring (10) with Q /Z and then taking the direct sum over a primes. To obtain the exact sequence in the second row, tensor (10) with Z/n and then take the direct sum over a primes. The vertica map N 1 N 1 Q sends x to x 1. A other n vertica arrows are natura maps. Using this diagram, we deduce from Coroary 3.3 that the restriction of the composite map Br(X) Γ β H 2 (k, Pic(X)) H 2 (k, N 1 ) to Br 0 (X) Γ Br(X) Γ, is the connecting homomorphism defined by (11), the top 2-extension in the diagram. QED 4 Approach via transcendenta cyces In this section X is a smooth, projective and geometricay integra variety of dimension d over a fied k of characteristic zero. 18

19 4.1 Lattices of agebraic and transcendenta cyces Let be a prime. If M is a Z -modue, we write M = Hom Z (M, Z ). Reca from Subsection 1.1 the foowing commutative diagram of Γ-equivariant pairings N 1 N 1, Z H 2 H 2d 2 Z with injective vertica maps (by Matsusaka s and Lieberman s theorems). Moreover, the bottom pairing induces isomorphisms H 2 = (H 2d 2 ) and H 2d 2 = (H 2). By the exact sequence (6) and the remark after it we see that the composition N 1 H 2 (H 2d 2 ) N1, is an injective map with cokerne D{}, where D is the finite abeian group defined in (3). In particuar, this is an isomorphism if does not divide δ = D, the order of D. As we have seen in Subsection 1.2 the subgroup N 1 H 2 is primitive. However, Koár (see [22], Thm. 14) showed that N 1, is not necessariy a primitive subgroup of H 2d 2. This can be remedied as foows. First of a, if does not divide δ, the natura map (H 2d 2 ) N1, is surjective, hence N 1, is primitive in H 2d 2. For every we define the Γ-modue M as the saturation of N 1, in H 2d 2, in other words, M = H 2d 2 N 1, Z Q H 2d 2 Z Q. If does not divide δ, then M = N 1,. Now we define the Γ-modue M as the subgroup of N 1 Q consisting of the eements that map to M under the natura map N 1 Q N 1 Q = M Z Q, for every prime. Thus N 1 M N 1 Q. We aso obtain a Γ-equivariant biinear form N 1 M Q. By tensoring with Z for every prime we see that this is actuay an integra biinear form N 1 M Z, which extends the intersection pairing form on N 1 N 1. It gives the exact sequence of Γ-modues 0 N 1 Hom(M, Z) E 0, (20) 19

20 which is the definition of the finite Γ-modue E, and for each it gives the exact sequence 0 N 1 M E{} 0. (21) It is cear that E is a Γ-submodue of D, and D/E = Hom(M/N 1, Q/Z), hence D/E = M/N 1. Note that if d = 2, i.e., X is a surface, then N 1, = N 1 H2 is primitive, hence M = N 1 and D = E. Let S H 2 be the orthogona compement to N 1, (or to M ). We et T H 2d 2 be the orthogona compement to N 1 with respect to the cupproduct pairing. Duaising the exact sequence of finitey generated, Z free, Γ-modues 0 T H 2d 2 (N 1 ) 0 we obtain the exact sequence of finitey generated, Z free, Γ-modues 0 N 1 H 2 T 0. (22) It gives a canonica identification T = B, where B is the Tate modue of the Brauer group defined in Subsection 1.2. Since M H 2d 2 is a primitive subgroup, cup-product gives the foowing exact sequence: 0 S H 2 M 0. (23) The composite map N 1 H 2 (H2d 2 ) M (N 1,) is injective. Thus S N 1 = 0. Using (22) and (23) we see that for every prime we have canonica isomorphisms of Γ-modues E{} = M /N 1 = H 2 /(N 1 S ) = T /S. We thus have a natura exact sequence 0 M /N 1 (S Z Q /Z ) (N 1 Z Q /Z ) H 2 Z Q /Z 0. (24) The foowing commutative diagram of Γ-modues with exact rows and coumns wi be usefu to us: M /N 1 S Z Q /Z Br 0 (X){} 0 0 N 1 Z Q /Z H 2 Z Q /Z Br 0 (X){} 0 (25) M Z Q /Z = M Z Q /Z

21 The midde row, respectivey, coumn, here is (22), respectivey, (23), tensored with Q /Z. The rest of the diagram foows from (24). Taking the direct sum over a primes we obtain from (25) the equivaence of 2-extensions 0 N 1 Hom(M, Z) (S Z Q /Z ) Br 0 (X) 0 0 N 1 N 1 Q (H 2 Z Q /Z ) Br 0 (X) 0 The top extension is the Yoneda product of 1-extensions of Γ-modues (20) and 0 E (S Z Q /Z ) Br 0 (X) 0 (26) Let us denote by 1 : Br 0 (X) Γ H 1 (k, E) and 2 : H 1 (k, E) H 2 (k, N 1 ) the differentias defined by these 1-extensions. Proposition 4.1 The composed map Br 0 (X) Γ Br(X) Γ coincides, up to sign, with the composed map β H 2 (k, Pic(X)) H 2 (k, N 1 ) Br 0 (X) Γ 1 H 1 (k, E) 2 H 2 (k, N 1 ). In particuar, the image of β(br 0 (X) Γ ) in H 2 (k, N 1 ) is annihiated by the exponent of E. Proof. We have seen that (11) is equivaent to the Yoneda product of (20) and (26), so the proposition foows from Coroary 3.4. QED 4.2 Upper bounds, II Let us denote the order of the finite group H 3 ét (X, Z (1)) tors by γ, and its exponent by γ 0. Let ε 0 be the exponent of the finite group E defined in (20). The integer ε 0 divides δ 0, which is the exponent of the finite group D defined in (3). Reca that ν is the order of the finite group NS(X) tors, and ν 0 is its exponent. If d = 2, i.e., X is a surface, then H 2 ét (X, Z (1)) tors is dua to H 3 ét (X, Z (1)) tors, hence γ = ν and γ 0 = ν 0. Here N 1 = N 1, and we have the symmetric biinear pairing N 1 N 1 Z, 21

22 with trivia kerne. The integer δ = D is the absoute vaue of the determinant of this pairing. The groups D and E coincide, hence δ = ε and δ 0 = ε 0. Here is our main resut over a genera fied of characteristic zero. Theorem 4.2 Let X be a smooth, projective and geometricay integra variety over a fied k of characteristic zero such that H 1 (X, O X ) = 0. Assume that the canonica map H 3 (k, k ) H 3 ét (X, G m) is injective (for exampe, X has a k-point). Then we have the foowing statements. (i) The exponent of the cokerne of α : Br(X) Br(X) Γ divides γ 0 ε 0 ν 0. The order of Coker(α) divides γ(ε 0 ν 0 ) b 2 ρ. (ii) If X is a surface, the exponent of Coker(α) divides δ 0 ν 2 0; the order of Coker(α) divides ν(δ 0 ν 0 ) b 2 ρ. Proof. Under our hypotheses Coker(α) = Im(β) by Proposition 1.3, so we ony need to estimate the size of β(br(x) Γ ). From (7) we deduce the exact sequence 0 Br 0 (X) Γ Br(X) Γ H 3 ét(x, Z (1)) Γ tors. This impies that β(br(x) Γ ) divides γ β(br 0 (X) Γ ), and the exponent of β(br(x) Γ ) divides the product of γ 0 and the exponent of β(br 0 (X) Γ ). By Proposition 4.1, the group ε 0.β(Br 0 (X) Γ ) is a subgroup of We have the short exact sequence and the short exact sequence Ker[H 2 (k, Pic(X)) H 2 (k, N 1 )]. H 2 (k, Pic 0 (X)) H 2 (k, Pic(X)) H 2 (k, NS(X)) H 2 (k, NS(X) tors ) H 2 (k, NS(X)) H 2 (k, N 1 ). Our assumption H 1 (X, O X ) = 0 impies Pic 0 (X) = 0, hence the exponent of ε 0.β(Br 0 (X) Γ ) divides ν 0. Thus the exponent of β(br 0 (X) Γ ) divides ε 0.ν 0. The group β(br 0 (X) Γ ) is a subquotient of (Q/Z) b 2 ρ. Lemma 1.2 then gives a bound for the order of β(br 0 (X) Γ ), from which then foows the caimed 22

23 bound for the order of β(br(x) Γ ) = Coker(α). The statement for a surface then foows from the genera facts recaed at the beginning of this subsection. QED When k is a number fied, we have a resut for arbitrary varieties. Theorem 4.3 Let X be a smooth, projective and geometricay integra variety over a number fied k. Then we have the foowing statements. (i) The exponent of the cokerne of α : Br(X) Br(X) Γ divides 2γ 0 ε 0 ν 0, and it divides γ 0 ε 0 ν 0 if k is totay imaginary. The order of Coker(α) divides γ(2ε 0 ν 0 ) b2 ρ, and it divides γ(ε 0 ν 0 ) b2 ρ if k is totay imaginary. (ii) If X is a surface, the exponent of Coker(α) divides 2δ 0 ν0 2 and it divides δ 0 ν0 2 if k is totay imaginary; the order of Coker(α) divides ν(2δ 0 ν 0 ) b2 ρ, and it divides ν(δ 0 ν 0 ) b2 ρ if k is totay imaginary. Proof. In this case H 3 (k, k ) = 0. We foow the proof of Theorem 4.2. It is enough to note that H 2 (k, Pic 0 (X)) is a finite group of exponent 2, and it is zero when k is totay imaginary [13, Thm (c), p. 92]. The statement for a surface then foows as before. QED Remark We have the isomorphism NS(X) tors = H 2 ét(x, Z (1)) tors. The Poincaré duaity impies that the finite abeian groups H 2 ét (X, Z (1)) tors and H 2d 1 ét (X, Z (d 1)) tors are dua to each other. The foowing proposition is a usefu compement to Theorem 4.3. For a number fied k we denote by k v the competion of k at a non-archimedean pace v, and by kv nr the maxima unramified extension of k v. If S is a finite set of primes of k and E a finite Γ-modue, we et H 1 S (k, E) be the subgroup of H 1 (k, E) consisting of the eements unramified outside S, that is, the intersection of kernes of the natura restriction maps H 1 (k, E) H 1 (kv nr, E) for a v / S. Proposition 4.4 Let X be a smooth, projective and geometricay integra variety over a number fied k with good reduction outside a finite set S of 23

24 primes of k. Let E be the finite Γ-modue defined in (20). Let T E be the set of primes of k dividing the order of E. Then 1 (Br 0 (X) Γ ) H 1 S T E (k, E). Proof. Let I v = Ga(k v /k v ) be the inertia subgroup. It is we known that if v is a prime of good reduction, and the residua characteristic of k v is different from, then the natura action of I v on H 2 ét (X, μ m), m 1, and hence on H 2 ét (X, Z (1)), is trivia (by the smooth base change theorem, see [12, Cor. VI.4.2]). This impies, by the Kummer sequence, that I v acts triviay on Br(X){}. But I v aso acts triviay on S H 2, hence the differentia : Br 0 (X){} Iv H 1 (k nr v, E{}) defined by the exact sequence of I v -modues (the top row of (25)) 0 E{} S Z Q /Z Br 0 (X){} 0, is zero. Hence, for v / S T E the image of Br 0 (X) Γ in H 1 (k, E) is in the kerne of the restriction map to H 1 (k nr v, E). QED 5 Appications to surfaces Proposition 5.1 Let X be a K3 surface over a fied k of characteristic zero, such that the map H 3 (k, k ) H 3 ét (X, G m) is injective (for exampe, k is a number fied, or X has a k-point). Let α be the natura map Br(X) Br(X) Γ. Then the exponent of Coker(α) divides δ 0, and the order divides δ b 2 ρ 0. Proof. For a K3 surface, H 1 (X, O X ) = 0 and the Néron Severi group NS(X) is torsion-free, so ν = 1. The statement is a specia case of Theorem 4.2 (ii). QED Exampes 1. Let X P 3 k be a diagona quartic surface over a fied of characteristic zero. It is we known that δ = 64 and δ 0 = 8, see [16]. This aready impies that any eement of odd order in Br(X) Γ comes from Br(X). Furthermore, b 2 = 22 and ρ = 20. If we foow the proof of Theorem 4.2, we see that Coker(α) is a subquotient of (Q 2 /Z 2 ) 2. Since its exponent divides 8, Coker(α) is isomorphic to a subgroup of (Z/8) 2. 24

25 2. Let k be a fied of characteristic zero. Suppose X P g k is a very genera K3 surface, that is, NS(X) = Z is generated by the hyperpane section cass H. We have δ = (H.H) = 2g 2. The exact sequence (3) now becomes 0 Z Z D 0, where E = D = Z/(2g 2) with trivia Γ-action. The proof of Theorem 4.2 gives an injection Coker(α) (Z/(2g 2)) 21. Since H 1 (k, Z) = 0, the map 1 : H 1 (k, D) H 2 (k, Z) = H 2 (k, NS(X)) = H 2 (k, Pic(X)) is injective. If k is a number fied, S is the set of bad reduction primes for X, and T is the set of primes dividing 2g 2, then Proposition 4.4 gives an injective homomorphism Coker(α) H 1 S T (k, Z/(2g 2)). When X is a product of two curves we have a resut simiar to Theorem 4.2 (ii) but in a case when H 1 (X, O X ) 0. Proposition 5.2 Let X = C 1 C 2 be a product of two smooth, projective and geometricay integra curves over a fied k of characteristic zero. Let J 1 and J 2 be the Jacobians of C 1 and C 2, respectivey. Assume X has a k-point. Then we have the foowing statements. (i) The exponent of the cokerne of Br(X) Br(X) Γ divides δ 0. (ii) If Hom k (J 1, J 2 ) = 0, then Br(X) Br(X) Γ is surjective. Proof. The foowing facts are we known, see [14, 1; Cor. 6.2]. The natura map of Γ-modues (p 1, p 2) : Pic(C 1 ) Pic(C 2 ) Pic(X) is a spit injection, a retraction of which is given by the choice of a k-point M = (M 1, M 2 ) X(k) = C 1 (k) C 2 (k). This induces an isomorphism of Γ-modues Pic 0 (C 1 ) Pic 0 (C 2 ) Pic 0 (X). The cokerne of (p 1, p 2) is the finitey generated Z-free Γ-modue Hom k grp (J 1, J 2 ). There is an induced exact sequence of finitey generated Z-free Γ-modues 0 NS(C 1 ) NS(C 2 ) NS(X) Hom k grp (J 1, J 2 ) 0, 25

26 that is 0 Z Z NS(X) Hom k grp (J 1, J 2 ) 0, with a spitting associated to the point M. The k-point M X(k) thus gives rise to a commutative diagram H 2 (k, Pic 0 (X)) H 2 (k, Pic(X)) H 2 (k, Pic 0 (C 1 ) Pic 0 (C 2 )) H 2 (k, Pic(C 1 ) Pic(C 2 )) (27) The injection in the bottom row is due to the fact that H 1 (k, Z) = 0. We proceed as in the proof of Theorem 2.1. By Proposition 1.3, for i = 1, 2, we have commutative diagrams of exact sequences Br(X) Br(X) Γ β X H 2 (k, Pic(X)) Br(C i ) Br(C i ) Γ β Ci H 2 (k, Pic(C i )) By Tsen s theorem we have Br(C i ) = 0. Thus β(br(x) Γ ) Ker[H 2 (k, Pic(X)) H 2 (k, Pic(C 1 ) Pic(C 2 ))]. Since NS(X) is torsion-free, we see (see the Remark after Theorem 4.3) that H 3 ét (X, Z (1)) is torsion-free for any, which impies Br 0 (X) = Br(X). By Proposition 4.1 the image of Br(X) Γ = Br 0 (X) Γ in H 2 (k, N 1 ) = H 2 (k, NS(X)) is annihiated by δ 0. Thus any eement in δ 0.β(Br(X) Γ ) H 2 (k, Pic(X)) comes from H 2 (k, Pic 0 (X)) and goes to zero in H 2 (k, Pic(C 1 ) Pic(C 2 )). From (27) we concude δ 0.β(Br(X) Γ ) = 0. This proves (i). If Hom k grp (J 1, J 2 ) = 0, then NS(C 1 ) NS(C 2 ) NS(X) and δ = 1. QED Exampes 1. In the case where C 1 = E and C 2 = E are eiptic curves not isogenous over k, compare with [21, Prop. 3.3]. 2. If C 1 = C 2 is an eiptic curve E without compex mutipication over k, then NS(X) is a free abeian group of rank 3, generated by the casses of E {0} and {0} E and the diagona Δ. We have δ = 2, b 2 = 6, ρ = 3, hence the cokerne of α : Br(X) Br(X) Γ is isomorphic to a subgroup of (Z/2) 3. See [21, Prop. 4.3] for an exampe with Coker(α) 0. 26

27 6 Open varieties This section partiay answers a question raised by T. Szamuey. Proposition 6.1 Let k be a fied of finite type over Q. Let U be a smooth, quasiprojective k-variety. Then H 1 ét (U, Q/Z)Γ is a finite group. Proof. This is a reformuation of a specia case of a resut of Katz and Lang [8, Thm. 1, p. 295]. One may aso give a proof aong the foowing more genera ines. One may assume that U/k is geometricay connected. By Hironaka s theorem, there exists a smooth, projective, geometricay integra k-variety X which contains U as a dense open set. Let Z = X \ U, and et F Z be the singuar ocus of Z. Let X 0 = X \ F and Z 0 = Z \ F. Since X 0 and F 0 are smooth, the ocaisation sequences for étae cohomoogy with finite coefficients and purity give rise to the exact sequences of Γ-modues 0 H 1 ét(x 0, Q /Z ) H 1 ét(u, Q /Z ) H 0 (Z 0, Q /Z ( 1)). Since F is of codimension at east 2 in X, the incusion X 0 X induces an isomorphism H 1 ét(x, Q /Z ) = H 1 ét(x 0, Q /Z ). We thus get an exact sequence 0 H 1 ét(x, Q /Z ) Γ H 1 ét(u, Q /Z ) Γ H 0 (Z 0, Q /Z ( 1)) Γ. The smooth k-variety Z 0 decomposes as a disjoint union Z 0 = i Z i of connected smooth k-varieties. Let k i denote the integra cosure of k in the function fied k(z i ). We choose a k-embedding k i k, and et Γ i = Ga(k/k i ). The group H 0 (Z 0, Q /Z ( 1)) Γ is the direct sum i (Q /Z ( 1)) Γ i. Each group (Q /Z ( 1)) Γ i is finite; moreover, it is zero for amost a. Indeed, this statement is immediatey reduced to the foowing statement: if k is a number fied and Γ = Ga(k/k), then (Q /Z ( 1)) Γ is finite, and is zero for amost a. One is then reduced to checking that H 1 ét (X, Q /Z ) Γ is finite, and is zero for amost a. This foows from the proper base change theorem and the Wei conjectures (cf. [1, Thm. 1.5]). QED Theorem 6.2 Let k be a fied of finite type over Q, and et U be a smooth, quasiprojective, geometricay integra k-variety. Then we have the foowing statements. 27

28 (i) The quotient Br(U) Γ /Im(Br(U)) is a finite group. (ii) If U is a surface, and the -adic Tate conjecture for divisors hods for a smooth compactification of U, then Br(U){} Γ is finite. (iii) If the -adic Tate conjecture for divisors hods for a smooth compactification X of U, and if, moreover, the Gaois modue H 2 ét (X, Q (1)) is semisimpe, then Br(U){} Γ is finite. Proof. We foow the beginning of proof of Proposition 6.1 and use the same notation. The ocaisation sequences for étae cohomoogy with finite coefficients and purity give rise to the exact sequence of Γ-modues 0 Br(X 0 ) Br(U) H 1 ét(f 0, Q/Z). Since the codimension of F in X is at east 2, the purity theorem for the Brauer group shows that the restriction map Br(X) Br(X 0 ) is an isomorphism. We thus have an exact sequence of Γ-modues 0 Br(X) Br(U) H 1 ét(f 0, Q/Z). Taking invariants under Γ, we get an exact sequence 0 Br(X) Γ Br(U) Γ H 1 ét(f 0, Q/Z) Γ. By Proposition 6.1, the group H 1 ét (F 0, Q/Z) Γ is finite. Statements (ii) and (iii) are then a consequence of the finiteness of Br(X){} Γ, which hods under the respective hypotheses of (ii) and (iii), see [2, Prop. 4.1]. Functoriaity gives a commutative diagram of exact sequences 0 Br(X) Γ Br(U) Γ H 1 ét (F 0, Q/Z) Γ Br(X) Br(U) By Theorem 2.1, the quotient Br(X) Γ /Im(Br(X)) is finite. By Proposition 6.1, the group H 1 ét (F 0, Q/Z) Γ is finite. This impies that the quotient Br(U) Γ /Im(Br(U)) is finite as we, which is statement (i). QED Remark We do not know whether (i) sti hods over an arbitrary fied k of characteristic zero. 28

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31 Department of Mathematics, South Kensington Campus, Imperia Coege London, SW7 2BZ Engand, U.K. Institute for the Information Transmission Probems, Russian Academy of Sciences, 19 Boshoi Karetnyi, Moscow, Russia 31