Galois descent for higher Brauer groups
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© Springer-Verlag GmbH Germany, part of Springer Nature 2019
H. Anthony Diaz
Galois descent for higher Brauer groups Received: 9 October 2018 / Accepted: 17 November 2019 Abstract. For X a smooth projective variety over a field k, we consider the problemof Galois descentfor higher Brauer groups. More precisely, we extend a finiteness result of Colliot-Thélène and Skorobogatov (J Reine Angew Math 682:141–165, 2013) to higher Brauer groups.
For X a smooth projective variety over a field k, the Brauer group Br (X ) = He´2t (X, Gm ) is a fundamental invariant in arithmetic geometry. Of particular interest is its role in the Tate conjecture in codimension 1. Recall that the Tate conjecture in codimension m, which we denote by T C m (X )Q , states that when k is a finitely generated field, k is its separable closure and X = X ×k k, the cycle class map C H m (X ) ⊗ Q → He´2m t (X , Q (m))
(1)
surjects onto the subspace of Tate classes: U He´2m t (X , Q (m)) U
(where U ranges over all open subgroups of Gal(k/k)). When k is a finite field, Tate showed in [18] that the Tate conjecture T C 1 (X )Q holds ⇔ the -primary torsion in Br (X ) is finite (for = char k). For arbitrary fields, the Tate conjecture for divisors is equivalent to the finiteness of the -primary torsion in Br (X )Gal(k/k) (see, for instance, [3] Prop. 2.1.1). In higher codimension, the m th higher Brauer groups Br m (X ) are defined by HL2m+1 (X, Z(m)), where HL∗ (X, Z(m)) denote the étale motivic cohomology groups. These latter are (étale) hyper-cohomology groups of the étale sheafification of Bloch’s cycle complexes [2], denoted by Z(m). When m = 1, the complex Z(1) is quasi-isomorphic to Gm [−1], which recovers the usual Brauer group. As motivation for the utility of higher Brauer groups, we begin with the following observation: Proposition 0.1. Let k be a finitely generated field of characteristic 0. Then, T C m (X )Q holds ⇔ the -primary torsion subgroup Br m (X )[∞ ]Gal(k/k) is finite. H. Anthony Diaz (B): Department of Mathematics, Washington University, St. Louis, MO 63130, USA. e-mail: [email protected] Mathematics Subject Classification: Primary 14F20; Secondary 14F22 · 19E15
https://doi.org/10.1007/s00229-019-01170-5
H. Anthony Diaz
The finiteness of Br m (X )Gal(k/k) in particular would imply that the cokernel of the restriction map: Br m (X )[∞ ] → Br m (X )Gal(k/k) [∞ ] is at worst finite. Thus, if one believes in the truth of the Tate conjecture, one should expect that the failure of Galois descent for (higher) Brauer classes is at worst finite. An unconditional result in this direction was proved by Colliot-Thélène and Skorobogatov for fields of characteristic 0 when m = 1; i.e., for the usual Brauer group. Our main result is to extend this Galois descent property to higher Brauer groups: Theorem 0.2. Let X be a smooth projective variety over a finitely generated field k and suppose that one of the following holds: (a) k has characteristic 0; (b) X satisfies the standard conjectures (i.e., conjectures B, C
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