On the Brauer groups of fibrations Yanshuai Qin December 3, 2020 Contents 1 Introduction 1 1.1 TateconjectureandBrauergroup . .. 2 1.2 Brauergroupsforfibrations . 3 1.3 The finiteness of the Galois invariant part of geometric Brauer group . 4 1.4 Sketch of the proof of Theorem 1.4 ...................... 4 1.5 NotationandTerminology ........................... 5 2 Geometric Brauer groups and Tate-Shafarevich groups 8 2.1 Preliminary on Tate’s conjecture and Brauer groups . .... 8 2.2 The Galois invariant part of geometric Brauer group . ... 8 2.3 Tate-Shafarevichgroup. .. .. 12 2.4 Cofiniteness of Brauer groups and Tate-Shafarevich groups . ........ 14 3 Proof of the main theorem 15 arXiv:2012.01324v1 [math.AG] 2 Dec 2020 3.1 The left exactness in Theorem 1.4 ....................... 15 3.2 Thepull-backtrick ............................... 18 3.3 Proof of Theorem 1.4 .............................. 20 4 Applications 21 4.1 Proof of Theorem 1.8 .............................. 21 4.2 Proof of Theorem 1.2 .............................. 22 4.3 Brauer groups of integral models of abelian varieties and K3 surfaces . 22 1 Introduction Let C be the spectrum of the ring of integers in a number field or a smooth projective geometrically connected curve over a finite field with function field K. Let X be a 2- dimensional regular scheme and π : X −→ C be a proper flat morphism such that 1 the generic fiber X of π is smooth and geometrically connected over K. Artin and Grothendieck (cf.[Gro3, § 4] or [Ulm, Prop. 5.3]) proved that there is an isomorphism X 0 ∼ (PicX/K ) = Br(X ) X 0 up to finite groups. Assuming the finiteness of (PicX/K ) or Br(X ), Milne [Mil4], Gonzales-Aviles [Goa] and Geisser [Gei1] gave a precise relation between their orders. In this paper, we generalize the original result of Artin and Grothendieck to fibrations of high relative dimensions over arbitrary finitely generated fields. As a consequence, we reduce the Tate conjecture for divisors on smooth projective varieties over a finitely generated field K to the Tate conjecture for divisors on smooth projective varieties over the prime subfield of K. For any noetherian scheme X, the cohomological Brauer group 2 Br(X) := H (X, Gm)tor 2 is defined to be the torsion part of the etale cohomology group H (X, Gm). 1.1 Tate conjecture and Brauer group Let us first recall the Tate conjecture for divisors over finitely generated fields. Conjecture 1.1. ( Conjecture T 1(X,ℓ)) Let X be a projective and smooth variety over a finitely generated field k of characteristic p ≥ 0, and ℓ =6 p be a prime number. Then the cycle class map 2 Gk Pic(X) ⊗Z Qℓ −→ H (Xks , Qℓ(1)) is surjective. The conjecture is proved for abelian varieties by Tate [Tat3] (over finite fields), Zarhin [Zar1, Zar2] (over positive characteristics) and Faltings [Fal1, Fal2] (over characteristic zero), for K3 surfaces over characteristic zero by Andr´e[And] and Tankeev [Tan4, Tan5], for K3 surfaces over positive characteristics by Nygaard [Nyg], Nygaard-Ogus [NO], Artin- Swinnerton-Dyer[ASD], Maulik [Mau], Charles [Cha] and Madapusi-Pera [MP]. By a well-known result (see Proposition 2.1 ), T 1(X,ℓ) is equivalent to the finiteness Gk of Br(Xks ) (ℓ). Theorem 1.2. Let k be a prime field and K be a finitely generated field over k. Let ℓ be a prime different from char(k). Assuming that T 1(X ,ℓ) is true for all smooth projective varieties X over k, then T 1(X,ℓ) holds for all smooth projective varieties X over K. Remark 1.3. T 1(X,ℓ) for smooth projective varieties over finite fields has been reduced 1 1 to T (X,ℓ) for smooth projective surfaces X over finite fields with H (X, OX ) = 0 by the work of de Jong [deJ2], Morrow[Mor, Thm. 4.3] and Yuan [Yua1, Thm. 1.6]. The above reduction theorem was proved by Ambrosi[Amb]. The idea of our proof is to work on the obstructions of the Tate conjecture, i.e. Brauer groups directly. We obtain the following relation between Brauer groups for fibrations. 2 1.2 Brauer groups for fibrations For a torsion abelian group M and a prime number p, let M(non-p) denote the subgroup of elements of order prime to p. We also define M(non-p) := M for p = 0. For two abelian groups M and N, we say that they are almost isomorphic if there exists a subquotient M1/M0 of M(resp. N1/N0 of N) such that M0 (resp. N0) and M/M1 (resp. N/N1) are finite and M1/M0 ∼= N1/N0. Let f : M → N be a homomorphism with a finite cokernel and a kernel almost isomorphic to an abelian group H, in this situation, we say that the sequence 0 → H → M → N → 0 is exact up to finite groups. Theorem 1.4. Let k be a finitely generated field of characteristic p ≥ 0. Let X be a smooth geometrically connected variety over k and C be a smooth projective geometrically connected curve over k with function field K. Let π : X −→ C be a dominant morphism such that the generic fiber X is smooth projective geometrically connected over K. Let K′ s 0 denote Kk and write P for PicX/K,red. Define 1 ′ 1 X ′ sh K (P ) := Ker(H (K ,P ) −→ Y H (Kv ,P )). v∈Cks Then there is an exact sequence up to finite groups Gk Gk GK 0 → XK′ (P ) (non-p) → Br(Xks ) (non-p) → Br(XKs ) (non-p) → 0. In the case that k is a finite field and X is projective over k, the following result generalizes Artin-Grothendieck’s theorem to high relative dimensions. Corollary 1.5. Let π : X −→ C be a proper flat morphism, where C is a smooth projective geometrically connected curve over a finite field k of characteristic p with function field K. Assuming that X is smooth projective over k and the generic fiber X of π is smooth projective geometrically connected over K, then there is an exact sequence up to finite groups X 0 GK 0 → (PicX/K,red)(non-p) → Br(X )(non-p) → Br(XKs ) (non-p) → 0. Gk In Theorem 1.4, when k is finite, the canonical maps X(P )(non-p) → XK′ (P ) (non-p) Gk and Br(X ) → Br(Xks ) have finite kernels and cokernels (cf. Proposition 2.8 and [Yua2, Cor. 1.4]). Thus, Corollary 1.5 follows from Theorem 1.4. It is well-known that the BSD conjecture for an abelian variety over a global function field is equivalent to the finiteness of the Tate-Shafarevich group (cf. [Mil5] and [Sch]). Thus, Corollary 1.5 implies the following result: Corollary 1.6. Notations as in Corollary 1.5, then T 1(X ,ℓ) is equivalent to the BSD 0 1 conjecture for PicX/K,red and T (X,ℓ). Remark 1.7. Geisser[Gei2] first studied fibrations of high relative dimensions over finite fields by using etale motivic cohomology theory. He proved Corollary 1.6 and gave a 3 precise relation between the orders of the Tate-Shafarevich group of the Albanese variety of the generic fiber and the Brauer group of X under the assumption of the finiteness of Br(X ). For fibrations over the spectrum of the ring of integers in a number field of relative dimension > 1, the above question was first studied by Tankeev (cf.[Tan1, Tan2, Tan3]). In our forthcoming work, we will prove a version of Corollary 1.5 over the spectrum of the ring of integers in a number field. 1.3 The finiteness of the Galois invariant part of geometric Brauer group Gk In [SZ1], Skorobogatov and Zarkhin conjectured that Br(Xks ) is finite for any smooth projective variety X over a finitely generated field k. By the following result, assuming Gk Tate conjecture for divisors, we have that Br(Xks ) (non-p) is finite if k is a finitely generated field of characteristic p> 0. Theorem 1.8. Let k be a prime field and K be a field finitely generated over k. Then the following statements are true. Gk (1) If k = Q and Br(Xks ) is finite for any smooth projective variety X over k, then GK Br(XKs ) is finite for any smooth projective variety X over K. (2) If k = Fp and the Tate conjecture for divisors holds for any smooth projective variety GK over k, then Br(XKs ) (non-p) is finite for any smooth projective variety X over K. Gk Remark 1.9. In [SZ1, SZ2], Skorobogatov and Zarkhin proved the finiteness of Br(Xks ) (non-p) for abelian varieties and K3 surfaces(char(k) =6 2). 1.4 Sketch of the proof of Theorem 1.4 For the proof of Theorem 1.4, the left exactness will follow from Grothendieck’s argument in [Gro3, § 4]. The tricky part is to show that the natural map Gk GK Br(Xks ) (non-p) −→ Br(XKs ) (non-p) has a finite cokernel. The key idea is to use a pull-back trick developed by Colliot-Th´el`ene and Skorobogatov (cf.[CTS] or [Yua2] ) to reduce it to cases of relative dimension 1. Choose an open dense subset U ⊆ C such that π−1(U) −→ U is smooth and proper. Write V for π−1(U). By purity for Brauer groups, the natural map Gk Gk Br(Xks ) (non-p) −→ Br(Vks ) (non-p) is injective and has a finite cokernel. It suffices to show that GK Br(V)(non-p) −→ Br(XKs ) (non-p) 4 has a finite cokernel.