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Title: Higher-dimensional adèles and their applications
Author: Braeunling, Oliver
ISNI:       0000 0004 2742 4946
Awarding Body: University of Nottingham
Current Institution: University of Nottingham
Date of Award: 2012
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THIS thesis studies higher-dimensional adeles, It can be divided into two larger parts (Chapters 2 & 3). The first part constructs and studies a certain notion of idele sheaf on smooth surfaces over a field. These idele sheaves form a flasque resolution of the Zariski sheaves arising from Rost cycle modules. The second part studies a generaliza- tion of Tate's construction of the 'l-dimensional local residue to arbitrary dimensions. This generalization is due to Beilinson, but was only explained in a 2 page article with- out proofs. We rework this approach and give full details for the construction of the higher local residue along this path. Parshin posed the problem to give an explicit for- mula for Beilinson's rather abstract approach. We give such a formula. This formula also gives a natural generalization of Tate's central extension class in Lie cohomology for multiloop Lie algebras. To the best of the author's knowledge no such explicit for- mula has appeared in the literature for dimension n > 1. This thesis develops ideas in the theory of multidimensional adeles and should be seen as a part of 1. B. Fesenko's program to investigate arithmo-geometric questions through adele /Idele theories. In Chapter 2 we use certain finiteness/integrality conditions on adeles which can be seen as multiplicative and K-delic analogues of the rank 2 integral structure of adeles as introduced by Fesenko. Chapter 3 is related to the multiple loop space nature of the additive group of higher local fields. As a residue over C is usually expressed as an integral over a loop, the multidimensional residue should always arise as a special case of integration. While we have not established such a link, this question has certainly been an inspiration and links to Fesenko's theory of higher integration and potential connections to physics.
Supervisor: Not available Sponsor: Not available
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available