Compositio Mathematica

Research Article

Twisted sheaves and the period-index problem

Max Lieblicha1

a1 Department of Mathematics, Fine Hall, Washington Road, Princeton NJ 08544-1000, USA (email: lieblich@math.princeton.edu)

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.

(Received November 15 2005)

(Accepted May 19 2007)

(Online publication January 23 2008)

Keywords

  • twisted sheaf;
  • gerbe;
  • period-index problem;
  • Brauer group

2000 Mathematics subject classification

  • 14D20;
  • 16K50

Footnotes

The author was partially supported by a Clay Liftoff Fellowship and an NSF Postdoctoral Fellowship during the preparation of this paper.