In this paper we show that the period equals the index for elements of Brauer groups of (function fields of) surfaces. A key idea of the proof is that any Azumaya algebra over a surface can be transformed into an Azumaya algebra that is unobstructed.
A. J. de Jong. "The period-index problem for the Brauer group of an algebraic surface." Duke Math. J. 123 (1) 71 - 94, 15 May 2004. https://doi.org/10.1215/S0012-7094-04-12313-9