On non-commutative twisting in étale and motivic cohomology

[Sur la torsion non commutative en cohomologie étale et motivique]

Hornbostel, Jens ; Kings, Guido

Annales de l'Institut Fourier, Tome 56 (2006), p. 1257-1279 / Harvested from Numdam

Résumé

Cet article confirme une conséquence de la conjecture principale de la théorie d’Iwasawa non abélienne. On démontre que, sous une condition technique, les groupes de cohomologie étale $H^1(\mathcal{O}_K[1/S],H^i(\overline{X},\mathbb{Q}_p(j)))$, où $X\rightarrow \operatorname{Spec}\mathcal{O}_K[1/S]$ est un schéma projectif lisse, sont engendrés par des unités tordues compatible par rapport aux normes dans une tour de corps de nombres associés à $H^i(\overline{X},\mathbb{Z}_p(j))$. On établit un résultat similaire pour la cohomologie motivique à coefficients finis en utilisant la conjecture de Bloch-Kato.

Abstract

This article confirms a consequence of the non-abelian Iwasawa main conjecture. It is proved that under a technical condition the étale cohomology groups $H^1(\mathcal{O}_K[1/S],H^i(\overline{X},\mathbb{Q}_p(j)))$, where $X\rightarrow \operatorname{Spec}\mathcal{O}_K[1/S]$ is a smooth, projective scheme, are generated by twists of norm compatible units in a tower of number fields associated to $H^i(\overline{X},\mathbb{Z}_p(j))$. Using the “Bloch-Kato-conjecture” a similar result is proven for motivic cohomology with finite coefficients.

Détail

Publié le : 2006-01-01

MR 2266890 | Zbl pre05145722

DOI : https://doi.org/10.5802/aif.2212
Classification:  11R23,  14G40,  14F42,  11R32
Mots clés: cohomologie étale, cohomologie motivique, théorie d’Iwasawa non-commutative

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