Geometric presentations of Milnor $K$-groups of certain Artin algebras and Bass-Tate-Kato norms

Abstract

For an arbitrary field $k$, and an arbitrary regular henselian local $k$-scheme $X$ of dimension $1$ with the residue field $k$, we introduce two subcomplexes of the higher Chow complexes of $X$ using certain extended face intersection conditions. We define suitable equivalence relations on them, and prove that their Milnor range cycle class groups offer geometric presentations of the improved (Gabber-Kerz) Milnor $K$-groups of Artin local $k$-algebras of the embedding dimension $1$, and their relative groups, generalizing the theorem of Nesterenko-Suslin and Totaro. Using these, we prove the existence of the norm and trace maps for the Milnor $K$-groups of the Artin local algebras associated to arbitrary finite extensions of fields, generalizing the Bass-Tate and Kato norms on the Milnor $K$-theory of fields.