2007/04/16 by Nikolai Durov, Durov, Nikolai · 3 citations
Mathematics · #08A40 (Secondary) #14A20 #14G40 (Primary) #18G55 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Number Theory (math.NT) #math.AG #math.NT #msc:08A40 #msc:14A20 #msc:14G40 #msc:18G55
paper · pdf · doi:10.48550/arxiv.0704.2030
568 pages, with hyperlinks
arxiv created 2007/04/16 · openalex publication_date 2007/04/16 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 F1 ("field with one element"), Z_∞ ("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.