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.
| Identifer | oai:union.ndltd.org:bl.uk/oai:ethos.bl.uk:575360 |
| Date | January 2012 |
| Creators | Braeunling, Oliver |
| Publisher | University of Nottingham |
| Source Sets | Ethos UK |
| Detected Language | English |
| Type | Electronic Thesis or Dissertation |
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