Galois co-descent for étale wild kernels and capitulation
type: Item
instance of: scholarly article
title: Galois co-descent for étale wild kernels and capitulation
zbMATH Open document ID: 0951.11029
publication date: 2000-03-01T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/75419
review text: For a number field \(K\) and a prime \(p\), let us denote by \(S\) a finite set of primes containing the primes above \(p\) and the archimedian primes. For \(i\geq 2\), the higher étale wild kernels \[ WK_{2i-2}^{\text{ét}} (F):= \operatorname {Ker} (H_{\text{ét}}^2 ({\mathcal O}_F^S, \mathbb{Z}_p(i))\to \bigoplus_{v\in S} H^2(F_v, \mathbb{Z}_p(i))) \] are natural generalizations of the \(p\)-part of the classical wild kernel (inside \(K_2\)). They play a similar role in étale cohomology, étale \(K\)-theory and Iwasawa theory as the \(p\)-primary part \(A_F'\) of the \((p)\)-class group of \(F\). In this paper, given a cyclic extension \(L/F\) of degree \(p\) and \(G= \operatorname {Gal} (L/F)\), the authors study Galois co-descent for \(WK_{2i-2}^{\text{ét}} (L)\). The analogous problem for the étale \(K\)-groups \(K_{2i-2}^{\text{ét}} ({\mathcal O}_L^S)\) had been settled previously by \textit{J. Assim} [Manuscr. Math. 86, 499-518 (1995; Zbl 0835.11043)] in terms of primitive ramification. But the difficulty here lies in the cokernel \(K_{2i-2}^{\text{ét}} ({\mathcal O}_L^S)/ WK_{2i-2}^{\text{ét}} (L)\), and it is overcome by interpreting this quotient in terms of a Brauer group. The authors show that the transfer map \(WK_{2i-2}^{\text{ét}} (L)_G\to WK_{2i-2}^{\text{ét}} (F)\) is onto except in a very special situation, and they give a formula for the order of the cokernel in the style of genus theory. As an application, they determine all Galois \(p\)-extensions \(F/\mathbb{Q}\) such that \(WK_2^{\text{ét}} (F)= 0\). A related problem concerns the étale capitulation kernels \(\operatorname {Cap}_{i-1} (F_\infty)\) \((i\geq 2)\), where \(F_\infty= \bigcup_n F_n\) is the cyclotomic \(\mathbb{Z}_p\)-extension of \(F\). These capitulation kernels are defined as \[ \varprojlim_n \operatorname {Ker} (K_{2i-2}^{\text{ét}} ({\mathcal O}_n^S)\to K_{2i-2}^{\text{ét}} ({\mathcal O}_\infty^S)) \] (with obvious notations) and are higher analogs of \[ \operatorname {Cap}_0 (F…
Mathematics Subject Classification ID: 11R23
Mathematics Subject Classification ID: 11R34
Mathematics Subject Classification ID: 11R70
zbMATH DE Number: 1409063
zbMATH Keywords: wild kernel
zbMATH Keywords: codescent
zbMATH Keywords: étale capitulation
zbMATH Keywords: étale cohomology
zbMATH Keywords: étale \(K\)-theory
zbMATH Keywords: Iwasawa theory
zbMATH Keywords: Greenberg's conjecture
author: Manfred Kolster
author: A. Movahhedi
reviewed by: Thong Nguyen Quang Do
MaRDI profile type: Publication
OpenAlex ID: W2330445913
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DOI: 10.5802/AIF.1746
published in: Annales de l’institut Fourier
description: scientific article; zbMATH DE number 1409063