On the Injectivity and Non-injectivity of the l-Adic Cycle Class Maps

doi:10.1017/9781009497206.008

7 - On the Injectivity and Non-injectivity of the l-Adic Cycle Class Maps

Published online by Cambridge University Press: 31 October 2025

Bruno Kahn

Edited by

Pedro L. del Ángel R. ,

Frank Neumann and

Alexander H. W. Schmitt

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Pedro L. del Ángel R.: Affiliation:
Centro de Investigación en Matemáticas
Frank Neumann: Affiliation:
Università di Pavia
Alexander H. W. Schmitt: Affiliation:
Freie Universität Berlin

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Summary

We study the injectivity of the cycle class map with values in Jannsen’s continuous étale cohomology, by using refinements that go through étale motivic cohomology and the “tame” version of Jannsen’s cohomology. In particular, we use this to show that the Tate and the Beilinson conjectures imply that its kernel is torsion in positive characteristic, and to revisit recent counterexamples to injectivity.

Keywords

l-adic cohomology cycle class map motivic cohomology

Information

Type: Chapter
Information: Moduli, Motives and Bundles
New Trends in Algebraic Geometry
, pp. 201 - 232

DOI: https://doi.org/10.1017/9781009497206.008 [Opens in a new window]

Publisher: Cambridge University Press

Print publication year: 2025

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