Regulators and cycle maps in higher-dimensional differential algebraic K-theory

TL;DR

This paper develops a new differential algebraic K-theory framework for regular arithmetic schemes, introducing a spectrum-level Beilinson regulator and a cycle map linking geometric bundles to K-theory classes.

Contribution

It presents a novel construction of a functorial Beilinson regulator at the spectrum level and a cycle map representing classes via geometric vector bundles.

Findings

Constructed a spectrum-level Beilinson regulator using differential forms.

Developed a cycle map linking geometric bundles to algebraic K-theory.

Derived Lott's relation connecting short exact sequences and higher analytic torsion.

Abstract

We develop differential algebraic K-theory of regular arithmetic schemes. Our approach is based on a new construction of a functorial, spectrum level Beilinson regulator using differential forms. We construct a cycle map which represents differential algebraic K-theory classes by geometric vector bundles. As an application we derive Lott's relation between short exact sequences of geometric bundles with a higher analytic torsion form.