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

TL;DR

This paper investigates the conditions under which the $l$-adic cycle class map is injective or not, using advanced cohomological techniques, and explores implications for major conjectures in algebraic geometry.

Contribution

It introduces refined methods to analyze the injectivity of the $l$-adic cycle class map via étale motivic and Jannsen's cohomology, connecting to key conjectures.

Findings

Tate and Beilinson conjectures imply the kernel is torsion in positive characteristic.

Revisits recent counterexamples to injectivity.

Provides new insights into the structure of the cycle class map.

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.