Details
| Originalsprache | Englisch |
|---|---|
| Seiten (von - bis) | 1466-1530 |
| Seitenumfang | 65 |
| Fachzeitschrift | Compositio Mathematica |
| Jahrgang | 159 |
| Ausgabenummer | 7 |
| Frühes Online-Datum | 15 Juni 2023 |
| Publikationsstatus | Veröffentlicht - Juli 2023 |
Abstract
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.
ASJC Scopus Sachgebiete
- Mathematik (insg.)
- Algebra und Zahlentheorie
Zitieren
- Standard
- Harvard
- Apa
- Vancouver
- BibTex
- RIS
in: Compositio Mathematica, Jahrgang 159, Nr. 7, 07.2023, S. 1466-1530.
Publikation: Beitrag in Fachzeitschrift › Artikel › Forschung › Peer-Review
}
TY - JOUR
T1 - Refined unramified cohomology of schemes
AU - Schreieder, Stefan
N1 - Publisher Copyright: © 2023 The Author(s). The publishing rights in this article are licensed to Foundation Compositio Mathematica under an exclusive licence.
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 - 10.1112/S0010437X23007236
DO - 10.1112/S0010437X23007236
M3 - Article
VL - 159
SP - 1466
EP - 1530
JO - Compositio Mathematica
JF - Compositio Mathematica
SN - 0010-437X
IS - 7
ER -