New Approach to Arakelov Geometry

TL;DR

This paper introduces a novel algebraic framework for Arakelov geometry that encompasses non-traditional objects like F_1, Z_infinity, and tropical numbers, enabling a broader and more systematic study of arithmetic varieties.

Contribution

It develops a comprehensive theory of generalized rings and schemes, extending classical concepts to include exotic objects, and applies this to construct Arakelov models over Q.

Findings

Established a theory of generalized rings and schemes including exotic objects.

Constructed algebraic K-theory, intersection theory, and Chern classes within this framework.

Proved the existence of Arakelov models for algebraic varieties over Q.

Abstract

This work is dedicated to a new completely algebraic approach to Arakelov geometry, which doesn't require the variety under consideration to be generically smooth or projective. In order to construct such an approach we develop a theory of generalized rings and schemes, which include classical rings and schemes together with "exotic" objects such as F_1 ("field with one element"), Z_\infty ("real integers"), T (tropical numbers) etc., thus providing a systematic way of studying such objects. This theory of generalized rings and schemes is developed up to construction of algebraic K-theory, intersection theory and Chern classes. Then existence of Arakelov models of algebraic varieties over Q is shown, and our general results are applied to such models.