Interaction of topology and algebra in arithmetic geometry

Camara, Alberto (2013) Interaction of topology and algebra in arithmetic geometry. PhD thesis, University of Nottingham.

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This thesis studies topological and algebraic aspects of higher dimensional local fields and relations to other neighbouring research areas such as nonarchimedean functional analysis and higher dimensional arithmetic geometry.

We establish how a higher local field can be described as a locally convex space once an embedding of a local field into it has been fixed. We study the resulting spaces from a functional analytic point of view: in particular we introduce and study bounded, c-compact and compactoid submodules of characteristic zero higher local fields. We show how these spaces are isomorphic to their appropriately topologized duals and study the implications of this fact in terms of polarity.

We develop a sequential-topological study of rational points of schemes of finite type over local rings typical in higher dimensional number theory and algebraic geometry. These rings are certain types of multidimensional complete fields and their rings of integers and include higher local fields. Our results extend the constructions of Weil over (one-dimensional) local fields. We establish the existence of an appropriate topology on the set of rational points of schemes of finite type over the rings considered, study the functoriality of this construction and deduce several properties.

Item Type: Thesis (University of Nottingham only) (PhD)
Supervisors: Fesenko, I.B.
Trihan, F.
Subjects: Q Science > QA Mathematics > QA150 Algebra
Faculties/Schools: UK Campuses > Faculty of Science > School of Mathematical Sciences
Item ID: 13247
Depositing User: EP, Services
Date Deposited: 21 Oct 2013 13:01
Last Modified: 15 Dec 2017 05:19

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