On Higher Arithmetic Intersection Theory
The results of this thesis contribute to the program of developing a higher arithmetic intersection theory. These results constitute chapters 3 and 5. Chapters 2 and 4 consist of the preliminary results needed for chapters 3 and 5, in the area of homotopy theory of simplicial sheaves and algebraic K...
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| Tipo de recurso: | tesis doctoral |
| Estado: | Versión publicada |
| Fecha de publicación: | 2007 |
| País: | España |
| Institución: | CBUC, CESCA |
| Repositorio: | TDR. Tesis Doctorales en Red |
| OAI Identifier: | oai:www.tdx.cat:10803/658 |
| Acceso en línea: | http://www.tdx.cat/TDX-1220107-112706 http://hdl.handle.net/10803/658 |
| Access Level: | acceso abierto |
| Palabra clave: | Teoria K Geometria d'Arakelov Teoria homotòpica de feixos. Ciències Experimentals i Matemàtiques 512 |
| Sumario: | The results of this thesis contribute to the program of developing a higher arithmetic intersection theory. These results constitute chapters 3 and 5. Chapters 2 and 4 consist of the preliminary results needed for chapters 3 and 5, in the area of homotopy theory of simplicial sheaves and algebraic K-theory.<br/><br/>In chapter 3, we develop a higher intersection theory on arithmetic varieties, "à la Bloch". We construct a representative of the Beilinson regulator using the Deligne complex of differential forms. Next, we develop a theory of higher arithmetic Chow groups, for any arithmetic variety X over a field. We prove that the construction is functorial and that there is a commutative and associative product structure, compatible with the algebraic intersection product. Therefore, we provide an arithmetic intersection product for arithmetic varieties over a field.<br/><br/>Chapters 4 and 5 are devoted to the definition of Adams operations on higher arithmetic K-theory. By the nature of the definition of the higher arithmetic K-groups, it is apparently necessary to have a description of the Adams operations in algebraic K-theory in terms of a chain morphism, compatible with the representative of the Beilinson regulator "ch".<br/><br/>In chapter 4, we obtain a chain morphism inducing Adams operations on higher algebraic K-theory over the field of rational numbers. This definition is of combinatory nature. This chain morphism is designed to commute with the Beilinson regulator "ch" given by Burgos and Wang.<br/><br/>In chapter 5 it is shown that this chain morphism indeed commutes with the representative of the Beilinson regulator "ch" and we use this fact to define Adams operations on the rational higher arithmetic K-groups.<br/><br/>The development of this study required tools to compare morphisms from algebraic K-groups to a suitable cohomology theory or to the K-groups themselves. In chapter 2, we study these comparisons at a general level, providing theorems giving sufficient conditions for two morphisms to agree. The theory underlying the proofs is the homotopy theory of simplicial sheaves. As an application, we prove that the Adams operations defined by Grayson agree for any regular noetherian scheme of finite Krull dimension with the Adams operations defined by Gillet and Soulé by means of homotopy theory of sheaves. In particular, this implies that the Adams operations defined by Grayson's work. |
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