Twisted sheaves and the period-index problem

Abstract

We use twisted sheaves and their moduli spaces to study the Brauer group of a scheme. In particular, we (1) show how twisted methods can be efficiently used to re-prove the basic facts about the Brauer group and cohomological Brauer group (including Gabber's theorem that they coincide for a separated union of two affine schemes), (2) give a new proof of de Jong's period-index theorem for surfaces over algebraically closed fields, and (3) prove an analogous result for surfaces over finite fields. We also include a reduction of all period-index problems for Brauer groups of function fields over algebraically closed fields to characteristic zero, which (among other things) extends de Jong's result to include classes of period divisible by the characteristic of the base field. Finally, we use the theory developed here to give counterexamples to a standard type of local-to-global conjecture for geometrically rational varieties over the function field of the projective plane.

Open problems on central simple algebras

“…If F has transcendence degree 1 over F 0 , then F is global and Br.dimÔF Õ 1. If F has transcendence degree 2 over F 0 , then Br.dimÔF Õ 3 by [Lie08].…”

“…Saltman's results on division algebras over the function field of a p-adic curve, see [Sal97a], [Sal07], [Bru10], [PS98], [PSar]; De Jong's result on the Brauer group of the function field of a complex surface, see [dJ04], [Lie08], and §4;…”

Period and index in the Brauer group of an arithmetic surface

“…Inspired by these results, Artin asked if the period and index are equal for classes over a C 2 -field, and he and Tate proved it for classes of period 2 a 3 b in the Appendix to [3]. The equality of period and index was proved in general for function fields of surfaces over algebraically closed fields by de Jong in [10] (with a slight improvement that may be found in [32] for classes with periods divisible by the characteristic of the base field). It is still unknown whether this equality holds for general C 2 -fields, or even for function fields of curves over C 1 -fields.…”

“…The basic fact underlying the usefulness of twisted sheaves for the period-index problem is the following, which may be found in [32], Proposition 3.1.2.1.…”

Erratum for Boundedness of families of canonically polarized manifolds: A higher dimensional analogue of Shafarevich's conjecture$^\ast$

“…By cohomology and base change, the sheaf (p i ) * L univ | P i ×B is a locally free P i -twisted sheaf (see [Lie08, §3.1.1] for the definition and basic properties of twisted sheaves). A choice of sections σ 0 , .…”