The transcendental part of higher Brauer groups in weight 2 [thesis]

Thomas Götzer
2017
The classical cohomological Brauer group Br(X) = H 2 et (X, G m ) of a smooth scheme X is related to many deep questions in algebraic geometry. For example, it yields an obstruction to the Tate conjecture for X in codimension 1 and it is related to the behaviour of the zeta function of X at 1. Furthermore, in some cases, the failure of the local-global Hasse principle can be explained in terms of the Brauer group. Therefore it becomes an interesting question to attempt to actually compute
more » ... Colliot-Thélène and Skorobogatov showed that for a smooth, projective and geometrically integral variety X over a field k of characteristic 0 the cokernel of the natural map Br(X) → Br(X) G k is finite. The cycle complexes constructed by Bloch define complexes Z X (r) of étale sheaves on X which allow one to define 'higher' Brauer groups Br r (X) := H 2r+1 et (X, Z X (r)). Since Br 1 (X) and Br(X) are isomorphic, these groups Br r (X) can be seen as a natural generalisation of the classical Brauer groups. In this dissertation we generalise the result of Colliot-Thélène and Skorobogatov to Br 2 (X) under some additional assumptions, i.e. we show: Let X be a smooth, projective and geometrically irreducible variety of dimension at most 4 over a field k of characteristic 0 and cohomological dimension at most 2. If the third Betti number of X is 0 and H 4 et (X, Z ℓ (2)) is torsion free for every prime ℓ (equivalently H 4 B (X C , Z(2)) tors = 0), then the cokernel of the natural map Br 2 (X) → Br 2 (X) G k is finite.
doi:10.5282/edoc.20456 fatcat:ehpyodpmvfbnhdke34l3q4alwq