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Motivic Cohomology and K-groups of varieties over higher local fields

Rahul Gupta, Amalendu Krishna, Jitendra Rathore

Abstract

For quasi-projective varieties over a higher local field $k_N$, we prove that its $K$-groups, above a suitable degree, are divisible-by-finite. We also prove the finiteness of the prime-to-$p$ torsion subgroup of certain higher Chow groups for smooth projective varieties over such fields, where $p$ denotes the final residue characteristic of $k_N$. As an application, we show that the kernel of the tame reciprocity map is uniquely $p'$-divisible. A key ingredient in achieving these results is the finiteness of étale cohomology groups over such fields.

Motivic Cohomology and K-groups of varieties over higher local fields

Abstract

For quasi-projective varieties over a higher local field , we prove that its -groups, above a suitable degree, are divisible-by-finite. We also prove the finiteness of the prime-to- torsion subgroup of certain higher Chow groups for smooth projective varieties over such fields, where denotes the final residue characteristic of . As an application, we show that the kernel of the tame reciprocity map is uniquely -divisible. A key ingredient in achieving these results is the finiteness of étale cohomology groups over such fields.
Paper Structure (23 sections, 62 theorems, 63 equations)

This paper contains 23 sections, 62 theorems, 63 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Algebraic $K$-groups for quasi-projective varieties over $N$-local fields
  4. Finiteness of étale cohomology groups over $N$-local fields
  5. Bloch's Higher Chow groups
  6. Application to Class field theory
  7. Outline of proofs
  8. Notations
  9. Finiteness in étale cohomology over finite and local fields
  10. Finiteness in étale cohomology over higher local fields
  11. Finiteness for quasi-projective varieties over $N$-local fields
  12. Finiteness for curves
  13. Finiteness in higher dimensions
  14. Étale $K$-groups over higher local fields
  15. Algebraic $K$-groups over higher local fields
  16. ...and 8 more sections

Key Result

Theorem 1.1

Let $X$ be a connected quasi-projective variety of dimension $d\geq 0$ over $k_{N}$. Then for any integer $m \geq d+N+1$, we have where $F$ is a finite group and $D$ is a uniquely $p'$-divisible group. In particular, $K_{m}(X)_{{\rm tor}}$ (except $p$-torsion) is finite. Moreover, if ${\rm char}(k_N) = {\rm char}(k_0)$, then $D$ is a uniquely divisible group and $K_{m}(X)_{{\rm tor}}$ is finite.

Theorems & Definitions (116)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Conjecture 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Theorem 1.9
  • Theorem 2.1
  • ...and 106 more