Galois descent for completed Algebraic K-theory

TL;DR

This paper establishes Galois descent for completed algebraic K-theory of fields using rigidity and a derived Atiyah-Segal completion theorem, primarily for pro-l Galois groups and K-theory completed at prime l.

Contribution

It provides a proof of Galois descent for equivariant algebraic K-theory under specific conditions using new rigidity results and a derived completion theorem.

Findings

Galois descent holds for pro-l Galois groups with K-theory completed at prime l.

Rigidity results for Borel-style equivariant cohomology are established.

A derived Atiyah-Segal completion theorem is applied to prove the main result.

Abstract

In this paper we consider the problem of Galois descent for suitably completed algebraic K-theory of fields. One of the main results is a suitable form of rigidity for Borel-style generalized equivariant cohomology with respect to certain spectra. In order to apply this to the problem at hand, we need to invoke a derived Atiyah-Segal completion theorem for pro-groups. In the present paper, the authors apply such a derived completion theorem proven by the first author elsewhere. These two results provide a proof of the Galois descent problem for equivariant algebraic K-theory as formulated by the first author, at least when restricted to the case where the absolute Galois groups are pro-$l$ groups for some prime $l$ different from the characteristic of the base field and the K-theory spectrum is completed at the same prime $l$. Work in progress hopes to remove these restrictions.