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On duality and Iwasawa descent. (Sur la dualité et la descente d’Iwasawa.) (French. English summary) Zbl 1254.11098

Summary: 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 \[ R\underset \leftarrow \lim A_{n}\rightarrow \text{RHom} _{\Lambda}(\text{RHom}_{\mathbb Z_p}(\underset \rightarrow \lim A_n,\mathbb Z_p),\Lambda ) \] 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.

MSC:

11R23 Iwasawa theory
11R34 Galois cohomology
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References:

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