TY - JOUR

T1 - Refined unramified cohomology of schemes

AU - Schreieder, Stefan

N1 - Funding Information: I am grateful for conversations with and comments from Giuseppe Ancona, Theodosis Alexandrou, Samet Balkan, Jean-Louis Colliot-Thélène, Matthias Paulsen, Anand Sawant, Domenico Valloni, Claire Voisin, and Lin Zhou. Thanks also go to Hélène Esnault and the referees for helpful comments on the presentation and to Bhargav Bhatt for checking (a previous version of) § . This project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme under grant agreement No 948066 (ERC-StG RationAlgic).

PY - 2023/7

Y1 - 2023/7

N2 - We introduce the notion of refined unramified cohomology of algebraic schemes and prove comparison theorems that identify some of these groups with cycle groups. This generalizes to cycles of arbitrary codimension previous results of Bloch-Ogus, Colliot-Thélène-Voisin, Kahn, Voisin, and Ma. We combine our approach with the Bloch-Kato conjecture, proven by Voevodsky, to show that on a smooth complex projective variety, any homologically trivial torsion cycle with trivial Abel-Jacobi invariant has coniveau. This establishes a torsion version of a conjecture of Jannsen originally formulated. We further show that the group of homologically trivial torsion cycles modulo algebraic equivalence has a finite filtration (by coniveau) such that the graded quotients are determined by higher Abel-Jacobi invariants that we construct. This may be seen as a variant for torsion cycles modulo algebraic equivalence of a conjecture of Green. We also prove -adic analogues of these results over any field which contains all -power roots of unity.

AB - We introduce the notion of refined unramified cohomology of algebraic schemes and prove comparison theorems that identify some of these groups with cycle groups. This generalizes to cycles of arbitrary codimension previous results of Bloch-Ogus, Colliot-Thélène-Voisin, Kahn, Voisin, and Ma. We combine our approach with the Bloch-Kato conjecture, proven by Voevodsky, to show that on a smooth complex projective variety, any homologically trivial torsion cycle with trivial Abel-Jacobi invariant has coniveau. This establishes a torsion version of a conjecture of Jannsen originally formulated. We further show that the group of homologically trivial torsion cycles modulo algebraic equivalence has a finite filtration (by coniveau) such that the graded quotients are determined by higher Abel-Jacobi invariants that we construct. This may be seen as a variant for torsion cycles modulo algebraic equivalence of a conjecture of Green. We also prove -adic analogues of these results over any field which contains all -power roots of unity.

KW - Abel-Jacobi maps

KW - algebraic cycles

KW - integral Hodge conjecture

KW - unramified cohomology

UR - http://www.scopus.com/inward/record.url?scp=85164372942&partnerID=8YFLogxK

U2 - 10.48550/arXiv.2010.05814

DO - 10.48550/arXiv.2010.05814

M3 - Article

AN - SCOPUS:85164372942

VL - 159

SP - 1466

EP - 1530

JO - Compositio mathematica

JF - Compositio mathematica

SN - 0010-437X

IS - 7

ER -