Geodesic
On Bloch’s map for torsion cycles over non-closed fields
Forum of Mathematics, Sigma, Tome 11 (2023)

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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 $\ell $-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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     author = {Theodosis Alexandrou and Stefan Schreieder},
     title = {On {Bloch{\textquoteright}s} map for torsion cycles over non-closed fields},
     journal = {Forum of Mathematics, Sigma},
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     year = {2023},
     doi = {10.1017/fms.2023.51},
     language = {en},
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Theodosis Alexandrou; Stefan Schreieder. On Bloch’s map for torsion cycles over non-closed fields. Forum of Mathematics, Sigma, Tome 11 (2023). doi: 10.1017/fms.2023.51

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