Basic theory of n-local fields

Official URL: http://risc01.sabanciuniv.edu/record=b1164863 (Table of Contents)

n-local fields arise naturally in the arithmetic study of algebro-geometric objects. For example, let X be a scheme which is integral and of absolute dimension n. Let F be the field of rational functions on X. Then to any complete flag of irreducible subschemes XQ C XI C C Xn_i C Xn = X, with dim(Xj) = i for i = 0, . . . , n, there corresponds a completion F(X0,..., Xn) of the field F introduced by Parshin, which is an example of an n-local field, in case each Xi is non-singular for i = 0, . . . , n. This n-local field F(X0, , Xn) plays a central role in the class field theory of X, introduced by Parshin and Kato. In this thesis, we develop the basic theory of n-local fields, including a complete elementary proof of Parshin's classification theorem; and for an n-local field K, introduce the sequential topology on K+ and Kx, and study the Kato-Zhukov higher ramification theory, including the Hasse-Arf theorem, for K.