Localization theorem for higher arithmetic K-theory

TL;DR

This paper extends Quillen's localization theorem to higher arithmetic K-theory for arithmetic schemes with group actions, providing new analytic and equivariant insights and generalizations in Arakelov geometry.

Contribution

It introduces an arithmetic analogue of Quillen's localization theorem for equivariant K-theory, extending Burgos-Wang's regulator to the equivariant setting and generalizing Thomason's concentration theorem.

Findings

Established an equivariant arithmetic localization theorem.

Provided an analytic refinement of the higher equivariant Riemann-Roch theorem.

Proved a higher arithmetic concentration theorem generalizing Thomason's result.

Abstract

Quillen's localization theorem is well known as a fundamental theorem in the study of algebraic K-theory. In this paper, we present its arithmetic analogue for the equivariant K-theory of arithmetic schemes, which are endowed with an action of certain diagonalisable group scheme. This equivariant arithmetic K-theory is defined by means of a natural extension of Burgos-Wang's simplicial description of Beilinson's regulator map to the equivariant case. As a byproduct of this work, we give an analytic refinement of the Riemann-Roch theorem for higher equivariant algebraic K-theory. And as an application, we prove a higher arithmetic concentration theorem which generalizes Thomason's corresponding result in purely algebraic case to the context of Arakelov geometry.