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

TL;DR
This paper introduces geometric presentations of Milnor K-groups for certain Artin algebras using higher Chow complexes, extending classical theorems and establishing norm and trace maps for these groups.
Contribution
It provides new geometric descriptions of Milnor K-groups for Artin local algebras and generalizes norm and trace maps to these contexts.
Findings
Geometric presentations of Milnor K-groups for Artin local algebras.
Extension of Bass-Tate and Kato norms to these groups.
Proof of existence of norm and trace maps for arbitrary finite extensions.
Abstract
For an arbitrary field , and an arbitrary regular henselian local -scheme of dimension with the residue field , we introduce two subcomplexes of the higher Chow complexes of 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 -groups of Artin local -algebras of the embedding dimension , 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 -groups of the Artin local algebras associated to arbitrary finite extensions of fields, generalizing the Bass-Tate and Kato norms on the Milnor -theory of fields.
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Taxonomy
TopicsAlgebraic Geometry and Number Theory · Algebraic structures and combinatorial models · Homotopy and Cohomology in Algebraic Topology
