Capitulation for even \(K\)-groups in the cyclotomic \(\mathbb Z_p\)-extension
Let \(F\) be a number field and let \(p\) be a prime (when \(p=2\) assume that \(F\) contains \(\sqrt{-1}\)). Let \(F_\infty/F\) be the cyclotomic \(\mathbb{Z}_p\)-extension of \(F\). For a finite layer \(F_n/F\) of \(F_\infty/F\), let \(A'_{F_n}\) denote the \(p\)-part of the class group of the ring of \(p\)-integers \(o'_{F_n}\) of \(F_n\). While studying the behaviour of \(A'_{F_n}\) for large \(n\) in the framework of classical Iwasawa theory, one is naturally led to consider the direct limit \(A'_{F_\infty}=\lim\limits_{\longrightarrow}A'_{F_n}\) (resp. the projective limit \(X'_{F_\infty}=\lim\limits_{\longleftarrow}A'_{F_n}\)) over the extension (or capitulation) maps \(A'_{F_n}\to A'_{F_m}\) (resp. over the norm maps \(A'_{F_m}\to A'_{F_n}\)) for \(m\geq n\). A well-known result of Iwasawa asserts that the kernels \[ \mathrm{Cap}(F_n)=\ker\left(A'_{F_n}\to A'_{F_\infty}\right) \] stabilize (i.e. become isomorphic) for large \(n\). Moreover, asymptotically, \(\mathrm{Cap}(F_n)\) is isomorphic to the maximal finite \(\Lambda\)-submodule \((X'_{F_\infty})^{0}\) of \(X'_{F_\infty}\) (where \(\Lambda=\mathbb{Z}_p[\![\mathrm{Gal}(F_\infty/F)]\!]\) denotes the Iwasawa algebra, as usual). Thanks to a result of \textit{M. Grandet} and \textit{J.-F. Jaulent} [J. Reine Angew. Math. 362, 213--217 (1985; Zbl 0564.12011)], one also knows that, if the torsion subgroup of \(X'_{F_\infty}\) is finite (i.e. the \(\mu\)-invariant of \(X'_{F_\infty}\) is trivial), then \(\mathrm{Cap}(F_n)\) is a direct summand of \(A'_{F_n}\) for large \(n\). The paper under review offers a cohomological generalization of the result of Grandet-Jaulent in the following context. The group \(A'_{F_n}\) has an interesting interpretation in terms of the étale cohomology of \(\mathrm{Spec}(o'_{F_n})\), namely \[ A'_{F_n}\cong t_{\mathbb{Z}_p}(H^2_{\acute{e}t}(o'_{F_n},\mathbb{Z}_p(1))) \] where \(\mathbb{Z}_p(1)\) is the (first) Tate twist of \(\mathbb{Z}_p\) and \(t_{\mathbb{Z}_p}(\--)\) denotes \(\mathbb{Z}_p\)-torsion. Higher twists \(i\geq 1\) give interesting finite groups \(H^2(o'_{F_n},\mathbb{Z}_p(1+i))\) which are deeply related with the \(K\)-theory of \(o'_{F_n}\). One is then naturally led to consider a higher version of the above situation, namely to define and study the higher capitulation kernels \[ \mathrm{Cap}_i(F_n)=\ker\left(H_{\acute{e}t}^2(o'_{F_n},\mathbb{Z}_p(1+i))\to \lim\limits_{\longrightarrow}H_{\acute{e}t}^2(o'_{F_n},\mathbb{Z}_p(1+i))\right). \] Results of Nguyen Quang Do, Kahn and Kolster-Movahhedi show that \(\mathrm{Cap}_i(F_n)\) has the same stabilization property as \(\mathrm{Cap}(F_n)\). In the present paper the author then proves the following generalization of the result of Grandet-Jaulent: if \(n\) is large enough and the \(\mu\)-invariant of \(X'_{F(\mu_{2p})_\infty}\) is trivial, then \(\mathrm{Cap}_i(F_n)\) is a direct summand of \(H_{\acute{e}t}^2(o'_{F_n},\mathbb{Z}_p(1+i))\) (here \(\mu_{2p}\) denotes the group of \(2p\)-th roots of unity). The heart of the proof is a version of the above splitting result for the so-called étale wild kernels \(W\!K^{\acute{e}t}_{2i}(F_n)\), which are subgroups of \(H_{\acute{e}t}^2(o'_{F_n},\mathbb{Z}_p(1+i))\) defined as kernels of the localization maps \[ H_{\acute{e}t}^2(o'_{F_n},\mathbb{Z}_p(1+i))\to\prod_{v\mid p} H_{\acute{e}t}^2((F_n)_v,\mathbb{Z}_p(1+i)). \] The advantage of using étale wild kernels is that they have a useful description (due to \textit{P. Schneider} [Math. Z. 168, 181--205 (1979; Zbl 0421.12024)]) in terms of Iwasawa theory, namely \[ W\!K_{2i}(F_n)\cong \left(X'_{F(\mu_{2p})_\infty}\otimes\mathbb{Z}_p(i)\right)_{\mathrm{Gal}(F(\mu_{2p})_\infty/F_n)}. \] In particular, one has an isomorphism \[ \mathrm{Cap}_i(F_n)\cong \left((X'_{F(\mu_{2p})_\infty})^0\otimes \mathbb{Z}_p(i)\right)_{\mathrm{Gal}(F(\mu_{2p})_\infty/F_n)} \] which gives an easier description of the natural maps \(\mathrm{Cap}_i(F_n)\to \mathrm{Cap}_i(F_m)\) for \(m\geq n\). As the author also remarks, the splitting results proved in the paper are particular cases of a more general theorem: if \(X\) is a finitely generated torsion \(\Lambda\)-module whose \(\mu\)-invariant is trivial and such that the coinvariant quotients \(X_{\mathrm{Gal}(F_\infty/F_n)}\) are finite for all \(n>\!\!>0\), then there are split exact sequences of abelian groups \[ 0\to X^0\to X_{\mathrm{Gal}(F_\infty/F_n)}\to \left(\lim\limits_{\longrightarrow} X_{\mathrm{Gal}(F_\infty/F_m)}\right)^{\mathrm{Gal}(F_\infty/F_n)}\to 0 \] for all \(n>\!\!>0\) (for a proof see [\textit{R. Validire}, Capitulation des noyaux sauvages étales. PhD Thesis, Université de Limoges (2008)]).
scientific article; zbMATH DE number 5620661
scientific article