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A spectral sequence for Iwasawa adjoints. (English) Zbl 1316.11099

The aim of this paper is to give a purely algebraic tool for treating the so-called generalized Iwasawa adjoints of some naturally occurring Iwasawa modules for \(p\)-adic Lie group extensions, by relating them to certain continuous Galois cohomology groups via a spectral sequence.
Let \(k\) be a number field, \(p\) be a fixed prime and \(k_\infty\) be some Galois extension of \(k\) such that \(\mathcal{G}=\text{Gal}(k_\infty/k)\) is a \(p\)-adic Lie-group (e.g., \(\mathcal{G}\cong \mathbb{Z}_p^r\) for some \(r\geq 1\)). Let \(S\) be a finite set of primes containing all primes above \(p\) and \(\infty\), and all primes ramified in \(k_\infty/k\), and let \(k_S\) be the maximal \(S\)-ramified extension of \(k\); by assumption, \(k_\infty\subseteq k_S\). Let \(G_S=\text{Gal}(k_S/k)\) and \(G_{\infty, S}=\text{Gal}(k_S/k_\infty)\). Let \(A\) be a discrete \(G_S\)-module which is isomorphic to \((\mathbb{Q}_p/\mathbb{Z}_p)^r\) for some \(r\geq 1\) as an abelian group.
Let \(\lambda=\mathbb{Z}_p[[\mathcal{G}]]\) be the Iwasawa algebra of \(\mathcal{G}\) over \(\mathbb{Z}_p\). For a finitely generated \(\Lambda\)-module \(M\) we put \(E^i(M)=\text{Ext}^i_\lambda(M, \Lambda)\). Hence \(E^0(M)=\text{Hom}_\Lambda(M, \Lambda) =: M^+\) is just the \(\Lambda\)-dual of \(M\). This has a natural structure of a \(\Lambda\)-module, by letting \(\sigma\in \mathcal{G}\) act via \(\sigma f(m)=\sigma f(\sigma^{-1})m\) for \(f\in M^+, m\in M\). The main result of this paper is the following theorem.
There is a spectral sequence of finitely generated \(\Lambda\)-modules \[ E^{p, q}_2=E^p(H^q(G_{\infty, S}, A)^\vee)\Rightarrow \underset{k^\prime, m}{\underleftarrow{\text{lim}}}H^{p+q}(G_S(k^\prime), A[p^m])=\underset{k^\prime}{\underleftarrow{\text{lim}}}H^{p+q}(G_S(k^\prime), T_pA), \] where the limit runs through the natural numbers m and the finite extensions \(kk^\prime /k\) contained in \(k_\infty\) respectively, via the natural maps \(H^n(G_S(k^\prime), A[p^{m+1}])\rightarrow H^n(G_S(k^\prime), A[p^m])\) and the corestrictions. The rightmost group is the continuous cohomology of the Tate module \(T_pA=\underset{m}{\underleftarrow{\text{lim}}}A[p^m]\).
Reviewer: Wei Feng (Beijing)

MSC:

11R23 Iwasawa theory
18G40 Spectral sequences, hypercohomology
22E50 Representations of Lie and linear algebraic groups over local fields
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