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

Jinhyun Park

TL;DR

The paper constructs geometric presentations for improved Milnor K-groups of Artin local k-algebras of embedding dimension 1 by defining two subcomplexes of higher Chow cycles, the d- and v-cycles, on a regular henselian local 1-dimensional k-scheme X. It introduces mod I^{m+1}-equivalence and extended face conditions to ensure well-behaved cycle operations, then proves isomorphisms between the Milnor K-groups (including the Gabber–Kerz improvements) and the corresponding Chow groups in both the Milnor range and relative settings. Central to the approach are graph maps from Milnor symbols to graph cycles, regulator maps arising from Kerz’s Gersten framework, and a moving/triangular reduction that expresses cycle classes via graph cycles, enabling the construction of norm and trace maps for arbitrary finite field extensions. These results generalize the Bass–Tate–Kato norms to Artin local algebras and illuminate a geometric, motivic perspective on relative Milnor K-theory, with implications for de Rham–Witt connections and motivic cohomology in the Milnor range. Overall, the work provides a robust toolkit linking cycle-theoretic presentations to Milnor K-theory for Artin algebras and establishes transitive norm/traces consistent with classical field theories.

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.

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

TL;DR

The paper constructs geometric presentations for improved Milnor K-groups of Artin local k-algebras of embedding dimension 1 by defining two subcomplexes of higher Chow cycles, the d- and v-cycles, on a regular henselian local 1-dimensional k-scheme X. It introduces mod I^{m+1}-equivalence and extended face conditions to ensure well-behaved cycle operations, then proves isomorphisms between the Milnor K-groups (including the Gabber–Kerz improvements) and the corresponding Chow groups in both the Milnor range and relative settings. Central to the approach are graph maps from Milnor symbols to graph cycles, regulator maps arising from Kerz’s Gersten framework, and a moving/triangular reduction that expresses cycle classes via graph cycles, enabling the construction of norm and trace maps for arbitrary finite field extensions. These results generalize the Bass–Tate–Kato norms to Artin local algebras and illuminate a geometric, motivic perspective on relative Milnor K-theory, with implications for de Rham–Witt connections and motivic cohomology in the Milnor range. Overall, the work provides a robust toolkit linking cycle-theoretic presentations to Milnor K-theory for Artin algebras and establishes transitive norm/traces consistent with classical field theories.

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.

Paper Structure

This paper contains 33 sections, 66 theorems, 188 equations.

Table of Contents

  1. Introduction
  2. Motivation and the results
  3. Sketch of the main ideas
  4. Recollection of certain definitions and results
  5. Recollection on Milnor $K$-theory
  6. The Milnor $K$-groups
  7. The improved Milnor $K$-groups of Gabber-Kerz
  8. Higher Chow cycles
  9. ${\rm d}$-cycles, ${\rm v}$-cycles and mod $I^{m+1}$-equivalence
  10. Dominance / flatness conditions $(DO)$ / $(DF)$
  11. Special fiber condition $(SF)$
  12. Extended conditions $(GP)_*$ and $(SF)_*$
  13. The ${\rm d}$-cycles over $X$
  14. The ${\rm v}$-cycles over $X$
  15. The ${\rm v}$-cycles of higher vanishing order
  16. ...and 18 more sections

Key Result

Theorem 1.1.1

Let $k$ be an arbitrary field and let $m, n \geq 1$ be integers. Let $X$ be any regular henselian local $k$-scheme of dimension $1$ with the residue field $k$.

Theorems & Definitions (174)

  • Theorem 1.1.1
  • Theorem 1.1.2
  • Lemma 1.2.1
  • Corollary 1.2.2
  • Theorem 1.2.3
  • Theorem 2.1.1: Kerz finite
  • Lemma 2.1.2
  • proof
  • Lemma 2.1.3
  • proof
  • ...and 164 more