On Bloch's map for torsion cycles over non-closed fields

2022/10/06 by Theodosis Alexandrou, Stefan Schreieder, Alexandrou, Theodosis +1
Mathematics · #14C15 #14C25 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2210.03201

openalex publication_date 2022/10/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We generalize Bloch's map on torsion cycles from algebraically closed fields to arbitrary fields. While Bloch's map over algebraically closed fields is injective for zero-cycles and for cycles of codimension at most two, we show that the generalization to arbitrary fields is only injective for cycles of codimension at most two but in general not for zero-cycles. Our result implies that Jannsen's cycle class map in integral ℓ-adic continuous étale cohomology is in general not injective on torsion zero-cycles over finitely generated fields. This answers a question of Scavia and Suzuki.

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