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

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.

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Benjamin Antieau <antieau@math.uic.edu>