Cohomological obstruction theory for Brauer classes and the period-index problem

Benjamin Antieau*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

4 Scopus citations

Abstract

Let U be a connected noetherian scheme of finite étale cohomological dimension in which every finite set of points is contained in an affine open subscheme. Suppose that α is a class in H2(U ét,double-struck G signm)tors. For each positive integer m, the K-theory of α-twisted sheaves is used to identify obstructions to α being representable by an Azumaya algebra of rank m 2. The étale index of α, denoted eti(α), is the least positive integer such that all the obstructions vanish. Let per(α) be the order of α in H2(Uét,double-struck G signm)tors. Methods from stable homotopy theory give an upper bound on the étale index that depends on the period of α and the étale cohomological dimension of U; this bound is expressed in terms of the exponents of the stable homotopy groups of spheres and the exponents of the stable homotopy groups of B(ℤ/(per(α))). As a corollary, if U is the spectrum of a field of finite cohomological dimension d, then eti(α)|per(α)⌊d/2⌋, where ⌊d/2⌋ is the integer part of d/2, whenever per(α) is divided neither by the characteristic of k nor by any primes that are small relative to d.

Original languageEnglish (US)
Pages (from-to)419-435
Number of pages17
JournalJournal of K-Theory
Volume8
Issue number3
DOIs
StatePublished - Dec 2011
Externally publishedYes

Keywords

  • Brauer groups
  • higher algebraic K-theory
  • homotopy theory
  • stable
  • twisted sheaves

ASJC Scopus subject areas

  • Algebra and Number Theory
  • Geometry and Topology

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