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New approach to Arakelov geometry


Durov,  Nikolai
Max Planck Institute for Mathematics, Max Planck Society;

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Durov, N. (2007). New approach to Arakelov geometry. PhD Thesis, Rheinische Friedrich-Wilhelms-Universität Bonn, Bonn.

Cite as: https://hdl.handle.net/21.11116/0000-0004-2B9F-A
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.