Continuous group actions on profinite spaces

TL;DR

This paper develops a model structure for profinite spaces with continuous group actions, enabling spectral sequence computations related to Galois actions and Grothendieck's section conjecture.

Contribution

It introduces a new model structure on profinite spaces and spectra with continuous actions, facilitating homotopy fixed point analysis in algebraic geometry.

Findings

Constructed a model structure on profinite spaces with continuous group actions.

Derived descent spectral sequences for homotopy fixed points and orbit spaces.

Applied framework to Galois actions on étale topological types of varieties.

Abstract

For a profinite group, we construct a model structure on profinite spaces and profinite spectra with a continuous action. This yields descent spectral sequences for the homotopy groups of homotopy fixed point space and for stable homotopy groups of homotopy orbit spaces. Our main example is the Galois action on profinite \'etale topological types of varieties over a field. One motivation is to understand Grothendieck's section conjecture in terms of homotopy fixed points.