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 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.

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

References in corpus (2)

• Period and index in the Brauer group of an arithmetic surface (with an appendix by Daniel Krashen)
• Rational families of vector bundles on curves, I

Cited by in corpus (42)

• The topological period-index problem over 6-complexes
• Open Problems on Central Simple Algebras
• Fibrations in complete intersections of quadrics, Clifford algebras, derived categories, and rationality problems
• The period-index problem for twisted topological K-theory
• Period and index in the Brauer group of an arithmetic surface (with an appendix by Daniel Krashen)
• Twistor spaces for supersingular K3 surfaces
• Automorphisms of pointless surfaces
• Boundedness of families of canonically polarized manifolds: A higher dimensional analogue of Shafarevich's conjecture
• The Brauer group of the moduli stack of elliptic curves
• Autoequivalences of twisted K3 surfaces
• Cohomological obstruction theory for Brauer classes and the period-index problem
• Variations on a theme of Grothendieck
• On the behaviour of Brauer $p$-dimensions under finitely-generated field extensions
• The period-index problem for real surfaces
• Models of Brauer-Severi surface bundles
• Twisted sheaves and $\mathrm{SU}(r)/{\mathbb{Z}}_r$ Vafa-Witten theory
• Experiments on the Brauer map in High Codimension
• On Brauer $p$-dimensions and index-exponent relations over finitely-generated field extensions
• Extending torsors on the punctured Spec(A_inf)
• An embedding problem for finite local torsors over twisted curves
• On the construction of moduli stack of projective Higgs bundles over surfaces
• On the essential (p)-dimension of parabolic bundles on a curve
• Rational curves in the moduli of supersingular K3 surfaces
• On the relative dual of an $S^1$-gerbe over an orbifold
• Notes on De Jong's period=index theorem for central simple algebras over fields of transcendence degree two
• On the Rost divisibility of henselian discrete valuation fields of cohomological dimension 3
• Colliot-Thélène's conjecture and finiteness of u-invariants
• The topological period-index conjecture
• The derived Brauer map via twisted sheaves
• The Étale Homotopy Type and Obstructions to the Local-Global Principle
• Index reduction for Brauer classes via stable sheaves (with an appendix by Bhargav Bhatt)
• GAGA problems for the Brauer group via derived geometry
• The Brauer group of desingularization of moduli spaces of vector bundles over a curve
• Essential dimension of simple algebras in positive characteristic
• Etale Fundamental group of moduli of torsors under Bruhat-Tits group scheme over a curve
• Sur la catégorie dérivée des faisceaux tordus
• The bigger Brauer group and twisted sheaves
• Lower bounds and infinity criterion for Brauer $p$-dimensions of finitely-generated field extensions
• Unramified division algebras do not always contain Azumaya maximal orders
• Period-index bounds for arithmetic threefolds
• Maximal orders in unramified central simple algebras
• Arithmetic aspects of moduli of sheaves on curves