  In this paper, we show that the etale index of a torsion cohomological Brauer
class is divisible by the period of the class. The tool used to make this
computation is the Cech approximation of the title. To create the
approximation, we use the folklore theorem that the homotopy limit and
Postnikov spectral sequences for a cosimplicial space agree beginning with the
E2-page. As far we know, this folklore theorem has no proof in the literature,
so we include a proof.
