Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Two Cycle Class Maps on Torsion Cycles

Theodosis Alexandrou

Abstract

We compare two cycle class maps on torsion cycles and show that they agree up to a minus sign. The first one goes back to Bloch (1979), with recent generalizations to non-closed fields. The second is the étale motivic cycle class map $α^{i}_{X}\colon \text{CH}^{i}(X)_{\mathbb{Z}_{\ell}}\to H^{2i}_{L}(X,\mathbb{Z}_{\ell}(i))$ restricted to torsion cycles.

Two Cycle Class Maps on Torsion Cycles

Abstract

We compare two cycle class maps on torsion cycles and show that they agree up to a minus sign. The first one goes back to Bloch (1979), with recent generalizations to non-closed fields. The second is the étale motivic cycle class map restricted to torsion cycles.
Paper Structure (7 sections, 19 theorems, 70 equations)

This paper contains 7 sections, 19 theorems, 70 equations.

Table of Contents

  1. Introduction
  2. Notations
  3. Twisted Borel-Moore Étale Motivic Cohomology
  4. Borel-Moore motivic étale cohomology
  5. Twisted Borel-Moore axioms
  6. Comparison to pro-étale Borel-Moore Cohomology
  7. Reformulating the étale motivic cycle class map

Key Result

Theorem 1.1

Let $X$ be a smooth variety over a field $k$ and let $\ell$ be a prime invertible in $k$. Let $\lambda^{i}_{X}$ and $\alpha^{i}_{X,\operatorname{tors}}$ be the cycle class maps on $\ell$-power torsion cycles from def:blochs-map and def:etale-motivic-torsion-map, respectively. Then $I^{2i-1}(X)=M^{2i

Theorems & Definitions (38)

  • Theorem 1.1
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Proposition 2.3
  • proof
  • Lemma 2.4
  • proof
  • Remark 2.5
  • ...and 28 more