## 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 H^{2}(U _{ét,}double-struck G sign_{m})_{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 H^{2}(U_{ét},double-struck G sign_{m})_{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 language | English (US) |
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Pages (from-to) | 419-435 |

Number of pages | 17 |

Journal | Journal of K-Theory |

Volume | 8 |

Issue number | 3 |

DOIs | |

State | Published - Dec 2011 |

Externally published | Yes |

## Keywords

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

## ASJC Scopus subject areas

- Algebra and Number Theory
- Geometry and Topology