On properties of the n-dimensional norm residue symbol in higher local class field theory

Abstract

Let k be a finite extension of Q p which contains the roots of unity μ. Here ¦μ¦ = q = p u, p ≠ 2 . We consider an n-dimensional local field given explicitly as a power series in n − 1 variables by X n = k t 1… t n−1 . The norm residue symbol has been generalized by Vostokov for mixed characteristic local fields X = t 1 t n−1 of dimension n. It is a non-degenerate pairing given by K n(X) (K n(X)) q × X ∗ (X ∗) q → μ where K n ( X) is the nth Milnor K-group of X = X n and X ∗ is the multiplicative group of X. It is shown here that the Vostokov pairing on the n-dimensional local field X n = k t 1 t n−1 commutes with the Vostokov pairing on the n − 1 dimensional local field X n − 1 = k t 1 t n − 2 . We achieve this by constructing a map M which projects the roots of unity from the pairing on X = X n onto the roots of unity from the pairing on X n − 1 .