Twisted sheaves and the periodindex 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 reprove 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 periodindex 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 periodindex problems for Brauer groups of function fields over algebraically closed fields to characteristic 0, 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 localtoglobal conjecture for geometrically rational varieties over the function field of the projective plane.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 November 2005
 arXiv:
 arXiv:math/0511244
 Bibcode:
 2005math.....11244L
 Keywords:

 Mathematics  Algebraic Geometry;
 Mathematics  Number Theory;
 14D22;
 14G05;
 14G15
 EPrint:
 31 pages. Major changes to section 4.2.3