Infinite CW-complexes, Brauer groups and phantom cohomology

TL;DR

This paper extends Serre's result on the Brauer group to arbitrary compact spaces, explores cases where equality fails, and relates phantom cohomology to homology using advanced homotopy theory techniques.

Contribution

It generalizes the equivalence of Brauer and cohomological Brauer groups to all compact spaces and provides new proofs and formulas involving phantom cohomology.

Findings

Brauer group equals cohomological Brauer group for all compact spaces

Counterexamples for Eilenberg--MacLane spaces of type K(Z/nZ,2)

Formula relating phantom cohomology to homology

Abstract

Expanding a result of Serre on finite CW-complexes, we show that the Brauer group coincides with the cohomological Brauer group for arbitrary compact spaces. Using results from the homotopy theory of classifying spaces for Lie groups, we give another proof of the result of Antieau and Williams that equality does not hold for Eilenberg--MacLane spaces of type K(Z/nZ,2). Employing a result of Dwyer and Zabrodsky, we show the same for the classifying spaces BG where G is an infinite-dimensional F_p-vector space. In this context, we also give a formula expressing phantom cohomology in terms of homology.