Cech approximation to the Brown-Gersten spectral sequence

TL;DR

This paper introduces a Cech approximation method for the Brown-Gersten spectral sequence to analyze torsion cohomological Brauer classes, establishing divisibility properties of the etale index related to the period.

Contribution

It provides a novel proof of the folklore theorem equating homotopy limit and Postnikov spectral sequences for cosimplicial spaces, and applies this to cohomological Brauer classes.

Findings

The etale index of a torsion cohomological Brauer class is divisible by its period.

A proof of the folklore theorem relating homotopy limit and Postnikov spectral sequences is provided.

The Cech approximation effectively computes the etale index in this context.

Abstract

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.