F-structures and Bredon-Galois cohomology

TL;DR

This paper develops a cohomological framework called Bredon-Galois cohomology to analyze group actions and field extensions, providing new tools for understanding Brauer groups and Galois cohomology.

Contribution

It introduces Bredon-Galois cohomology, establishes an analog of Hilbert's Theorem 90, and relates second Bredon-Galois cohomology to relative Brauer groups.

Findings

Second Bredon-Galois cohomology equals an intersection of relative Brauer groups.

Realizes the relative Brauer group as a second Bredon cohomology group.

Provides methods to find nonzero elements in Br(L/K).

Abstract

Let F be an arbitrary family of subgroups of a group G and let Orb be the associated orbit category. We investigate interpretations of low dimensional F-Bredon cohomology of G in terms of abelian extensions of Orb. Specializing to fixed point functors as coefficients, we derive several group theoretic applications and introduce Bredon-Galois cohomology. We prove an analog of Hilbert's Theorem 90 and show that the second Bredon-Galois cohomology is a certain intersection of relative Brauer groups. As applications, we realize the relative Brauer group Br(L/K) of a finite separable non-normal extension of fields L/K as a second Bredon cohomology group and show that this approach is quite suitable for finding nonzero elements in Br(L/K).