### Cohomological Obstruction Theory for Brauer Classes and the Period-Index Problem, by Benjamin Antieau

[Apr 27, 2010: The preprint now resides at http://arxiv.org/abs/0909.2352.]

Let U be a noetherian, quasi-compact, and connected scheme. Let [a] be a class
in Br(U). For each positive integer m, we use the K-theory of [a]-twisted
sheaves to identify obstructions to [a] being representable by an Azumaya
algebra of rank m^2. We define the spectral index of [a], denoted spi([a]), to
be the least positive integer such that all of the associated obstructions
vanish. Let per([a]) be the order of [a] in Br(U). We give an upper bound on
the spectral index that depends on the etale cohomological dimension of U, the
exponents of the stable homotopy groups of spheres, and the exponents of the
stable homotopy groups of B(\mu_{per([a])}). As a corollary, we prove that when
U is the spectrum of a field of finite cohomological dimension d=2c or d=2c+1,
then spi([a])|per([a])^c whenever per([a]) is not divided by any primes that
are small relative to d.

Benjamin Antieau <antieau@math.uic.edu>