$\boldsymbol{\mathfrak{F}}$ -Structures and Bredon–Galois Cohomology

Abstract

Let \(\mathfrak{F}\) be an arbitrary family of subgroups of a group G and let \(\mathcal{O}_\mathfrak{F}G\) be the associated orbit category. We investigate interpretations of low dimensional \(\mathfrak{F}\)-Bredon cohomology of G in terms of abelian extensions of \(\mathcal{O}_\mathfrak{F}G\). 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).