In general, for a fixed field k, it is an important but difficult problem to give bounds on the index of Brauer classes of k based only on their periods. In this thesis, a new invariant of Brauer classes, the spectral index, is defined, and bounds are proven for the spectral index based on the period. For certain fields, it is conjectured that the index and spectral index coincide.

More specifically, let U be a geometrically connected quasi-separated scheme, and let α be a class in H²( U_ét, [special characters omitted]). 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 m². The spectral index of α, denoted spi(α), is defined to be the least positive integer such that all of the associated obstructions vanish. Let per(α) be the order of α in H ²(U_ét, [special characters omitted]). Methods from stable homotopy theory give an upper bound on the spectral index that depends on the period of α, the étale cohomological dimension of U, the exponents of the stable homotopy groups of spheres, and the exponents of the stable homotopy groups of B ([special characters omitted]/(per(α))). As a corollary, if U is the spectrum of a field of finite cohomological dimension d = 2c or d = 2c + 1, then spi(α)|per(α) ^c whenever per(α) is not divided by any primes that are small relative to d. This result is to be contrasted with the period-index conjecture of Colliot-Thélène.