New Approach to Arakelov Geometry

By N. Durov. 2007

104 citations recorded by the source corpus.

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.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/ce7d950d7770c1f89683ec44c53240c3ff872d53

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Geometry and Topology

1. Martin Brandenburg
2. Noah Giansiracusa
3. Jeffrey Giansiracusa
4. Oliver Lorscheid
5. Javier L'opez Pena
6. Jaiung Jun
7. A. Gibney

In Mathematical Physics

1. Martin Brandenburg
2. Noah Giansiracusa
3. Jeffrey Giansiracusa
4. Oliver Lorscheid
5. Javier L'opez Pena

Citations in the corpus

• Equations of tropical varieties (2013, background): “quasi-coherent ideal sheaves onX. Over a semiring, the equivalence between the above two characterizations of a closed immersion breaks down; see Remark 3.3.2 below. The prevailing attitude (e.g., in [Dur07]), and the choice that we follow here, is to adopt the second perspective: for a scheme X ove”
• F1-schemes and toric varieties (2006, unknown): “ rigorous. In [10] the authors formalize the transition from rings to schemes on a categorial level and apply this machinery to the category of sets to obtain the category of F1-schemes as in [1]. In [3] and [5] the authors extend the definition of rings in order to capture a structure that deserves ”
• Tensor categorical foundations of algebraic geometry (2014, methodology): “The special case C = Set has already been considered by Durov in his thesis [Dur07] who defined generalized commutative rings as finitary symmetric monoidal monads on Set.”
• Mapping F_1-land:An overview of geometries over the field with one element (2009, background): “In [16], Durov introduces a generalization of classical algebraic geometry, into that Arakelov geometry fits nat urally.”
• Fun with $\F_1$ (2008, unknown): “Recovering the analytic endomotive 28 References 28”