Milnor $K$-groups modulo $p^n$ of a complete discrete valuation field

TL;DR

This paper determines the structure of Milnor K-groups modulo p^n for certain local fields, linking their graded quotients to differential forms of the residue field, advancing understanding in algebraic K-theory.

Contribution

It explicitly describes the graded quotients of Milnor K-groups modulo p^n for complete discrete valuation fields containing p^n-th roots of unity, in terms of differential forms.

Findings

Graded quotients of K-groups are characterized in terms of differential forms.

Results apply to mixed characteristic complete discrete valuation fields.

Provides explicit descriptions linking K-theory and differential forms.

Abstract

For a mixed characteristic complete discrete valuation field $K$ which contains a $p^n$-th root of unity, we determine the graded quotients of the filtration on the Milnor $K$-groups $K_q^M(K)$ modulo $p^n$ in terms of differential forms of the residue field of $K$.