Adams operations on higher arithmetic K-theory

TL;DR

This paper constructs Adams operations on higher arithmetic K-groups of proper arithmetic varieties, ensuring compatibility with algebraic K-theory and regulator maps, advancing the understanding of arithmetic K-theory structures.

Contribution

It introduces a new construction of Adams operations on rational higher arithmetic K-groups, compatible with existing algebraic K-theory operations and regulator maps.

Findings

Adams operations are defined on higher arithmetic K-groups.

The operations are compatible with algebraic K-theory Adams operations.

A modified chain morphism commutes with the Beilinson regulator.

Abstract

We construct Adams operations on the rational higher arithmetic K-groups of a proper arithmetic variety. The definition applies to the higher arithmetic K-groups given by Takeda as well as to the groups suggested by Deligne and Soule, by means of the homotopy groups of the homotopy fiber of the regulator map. They are compatible with the Adams operations on algebraic K-theory. The definition relies on the chain morphism representing Adams operations in higher algebraic K-theory given previously by the author. In this paper it is shown that a slight modification of this chain morphism commutes strictly with the representative of the Beilinson regulator given by Burgos and Wang.