Classes in Zakharevich K-groups constructed from Quillen K-theory

TL;DR

This paper demonstrates that certain algebraic K-groups related to integers and CM fields are embedded as summands in Zakharevich's K-theory space, leading to new classes of infinite order in higher K-groups.

Contribution

It establishes the presence of classical K-groups as summands in Zakharevich's K-theory and constructs the first infinite order classes in higher Zakharevich K-groups.

Findings

K-groups for integers and CM fields are summands in Zakharevich's K-theory.

Rationalisation and connective cover make these summands retracts of spaces.

First construction of infinite order classes in higher Zakharevich K-groups.

Abstract

We show that the K-groups K_{n}(O) for O the integers or an order in a CM field and n>0 appear as direct summands of the homotopy groups of various localisations of Zakharevich's K-theory space. After rationalisation and going to the 1-connective cover, this even becomes a retract of spaces. As an application, we provide the first construction of classes of infinite order in the higher Zakharevich K-groups.