Max Lieblich
Published 2005-11-09, updated 2007-05-25Version 4
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 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 local-to-global conjecture for geometrically rational varieties over the function field of the projective plane.
Comments: 31 pages. Major changes to section 4.2.3; the "question" is now answered in the negative, with resulting applications to local-to-global problems. To appear in Compositio Mathematica
On the period-index problem in light of the section conjecture
Isotriviality is equivalent to potential good reduction for endomorphisms of ${\mathbb P}^N$ over function fields
Invariance of the fundamental group under base change between algebraically closed fields
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