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

TL;DR

This paper develops a cohomological obstruction theory for Brauer classes on schemes, providing bounds on the etale index related to the period and cohomological dimension, with implications for the period-index problem.

Contribution

It introduces a novel cohomological obstruction framework using K-theory for Brauer classes and derives new bounds on the etale index based on stable homotopy theory.

Findings

The etale index divides the period raised to the power of half the cohomological dimension.

Stable homotopy theory provides explicit bounds on the etale index.

For fields of finite cohomological dimension, the etale index divides the period to the power of [d/2].

Abstract

Let U be a connected scheme of finite cohomological dimension in which every finite set of points is contained in an affine open subscheme. Suppose that alpha is a class in H^2(U_et,Gm)_{tors}. For each positive integer m, the K-theory of alpha-twisted sheaves is used to identify obstructions to alpha being representable by an Azumaya algebra of rank m^2. The etale index of alpha, denoted eti(alpha), is the least positive integer such that all the obstructions vanish. Let per(alpha) be the order of alpha in H^2(U_{et},Gm)_{tors}. Methods from stable homotopy theory give an upper bound on the etale index that depends on the period of alpha and the etale 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(alpha)). As a corollary, if U is the spectrum of a field of finite cohomological dimension d, then eti(alpha) divides per(alpha)^[d/2], where [d/2] is the integer part of d/2, whenever per(alpha) is divided neither by the characteristic of k nor by any primes that are small relative to d.