Twisted sheaves and the period-index problem

• Published in: Compositio mathematica Vol. 144; no. 1; pp. 1 - 31
• Main Author: Lieblich, Max
• Format: Journal Article
• Language: English
• Published: London, UK London Mathematical Society 01.01.2008; Cambridge University Press
• Subjects: 14D20; gerbe; twisted sheaf; 16K50; period-index problem; Brauer group
• ISSN: 0010-437X; 1570-5846
• DOI: 10.1112/S0010437X07003144

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.