K-theoretic descent and a motivic Atiyah-Segal theorem

TL;DR

This paper proves a p-adic derived completion conjecture relating Galois representation spectra to algebraic K-theory, utilizing properties of absolute Galois groups of geometric fields.

Contribution

It establishes a new interpretation of p-adic derived completions as K-theory of pro-schemes, advancing the understanding of Galois groups and algebraic K-theory connections.

Findings

Proved the p-adic derived completion conjecture for geometric Galois groups.

Identified the role of total torsion freeness in Galois representation theory.

Connected derived completions to algebraic K-theory of pro-schemes.

Abstract

This paper concerns our earlier conjecture about the equivalence of a derived completion construction applied to the representation spectrum of the absolute Galois group of a geometric field is equivalent to the algebraic K-theory of the field. We prove that the p-adic version of the derived completion construction can be interpreted as the K-theory of a certain pro-scheme over an algebraically closed field contained within the field. In order to make this construction, we use a special property of absolute Galois groups of geometric field, namely that of total torsion freeness, and the paper also contains some poperties of the representation theory of such groups.