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arXiv · 0704.2030

New Approach to Arakelov Geometry

Nikolai Durov
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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.

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BibTeXRIS

Nikolai Durov. 2007-04-16. New Approach to Arakelov Geometry. https://arxiv.org/abs/0704.2030

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