On the injectivity and non-injectivity of the $l$-adic cycle class maps

Bruno Kahn

On the injectivity and non-injectivity of the $l$-adic cycle class maps

Algebraic Geometry 2024-04-11 v2 Number Theory

Authors: Bruno Kahn

Abstract

We study the injectivity of the cycle class map with values in Jannsen's continuous \'etale cohomology, by using refinements that go through \'etale 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

cycle class map continuous etale cohomology etale motivic cohomology tate conjecture beilinson conjecture injectivity positive characteristic

Cite

@article{arxiv.2307.13281,
  title  = {On the injectivity and non-injectivity of the $l$-adic cycle class maps},
  author = {Bruno Kahn},
  journal= {arXiv preprint arXiv:2307.13281},
  year   = {2024}
}

Comments

Slight changes; to appear in the proceedings of the BIRS-CMO Oaxaca workshop "Moduli, Motives and Bundles" (September 2022), Cambridge LMS Lecture Notes Series

On the injectivity and non-injectivity of the $l$-adic cycle class maps

Abstract

Keywords

Cite

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