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David Vauclair
Sur la dualité et la descente d’Iwasawa
(On duality and Iwasawa descent)
Annales de l'institut Fourier, 59 no. 2 (2009), p. 691-767, doi: 10.5802/aif.2446
Article PDF | Reviews MR 2521434 | Zbl pre05549007
Class. Math.: 11R34, 13C05, 11R23
Keywords: Théorie d’Iwasawa, dualité, contrôle

Résumé - Abstract

Guided by the concrete examples of cyclotomic units and the ideal class group in cyclotomic Iwasawa theory, we develop a general tool for studying descent and codescent, with a special interest in relating the two of them.

Given any « normic system » $A=(A_n)$ (that is a collection of Galois modules plus additional data), attached to a fixed $p$-adic Lie extension with Iwasawa algebra $\Lambda $, we mainly show that there is a natural morphism

$$ R \varprojlim A_n \rightarrow {\rm RHom}_\Lambda ({\rm RHom}_{{\mathbb{Z}}_p} (\varinjlim A_n,{\mathbb{Z}}_p),\Lambda ) $$

which can be given a functorial cone measuring the defect of descent as well as the defect of codescent (for the $A_n$’s). Thanks to a sharpening of the usual Poincaré duality, this results in an enlightening relation between these two.

We show in great detail how known results in the cyclotomic situation fit into this setting, and give a generalization to multiple ${\mathbb{Z}}_p$-extensions.

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