New Approach to Arakelov Geometry
New Approach to Arakelov Geometry
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阿拉克洛夫几何的新方法
DOI：
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发表时间：
2007
期刊：
影响因子：
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通讯作者：
N. Durov
中科院分区：
文献类型：
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作者：
N. Durov
这项工作致力于阿拉克洛夫几何的一种新的完全代数方法，它不要求所考虑的种类是一般光滑的或射影的。为了构造这种方法，我们发展了一种广义环和方案的理论，它包括经典的环和方案以及“奇异”对象，如F_1(“一元场”)、Z_\infty(“实整数”)、T(热带数字)等，从而为研究这类对象提供了一种系统的方法。
这种广义环论和模论发展到代数K-理论、交理论和陈类的构造。然后证明了Q上代数簇的Arakelov模型的存在性，并将我们的一般结果应用于此类模型。
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.