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In this paper, we show that the etale index of a torsion cohomological Brauerclass is divisible by the period of the class. The tool used to make thiscomputation is the Cech approximation of the title. To create theapproximation, we use the folklore theorem that the homotopy limit andPostnikov spectral sequences for a cosimplicial space agree beginning with theE2-page. As far we know, this folklore theorem has no proof in the literature,so we include a proof.

Cech approximation to the Brown-Gersten spectral sequence