Minimal resolutions of Iwasawa modules (Q6573194)

In this paper, the authors study the module-theoretic structure of Iwasawa modules under the classical case. For a finite abelian \(p\)-extension \(K/k\) of totally real fields and the cyclotomic \(\mathbb{Z}_P\)-extension \(K_\infty/K\), they consider \(X_{{K_\infty}, S}=\mathrm{Gal}(M_{K_\infty,S}/K_\infty)\) where \(S\) is a finite set of places of \(k\) containing all ramifying places in \(K_{\infty}\) and archimedean places, and \(M_{K_\infty, S}\) is the maximal abelian pro-\(p\)-extension of \(K_\infty\) unramified outside \(S\). They give lower and upper bounds of the minimal numbers of generators and of relations of \(X_{K_\infty,S}\) as a \(\mathbb{Z}_p[[\mathrm{Gal}(K_\infty/k)]]\)-module, using the \(p\)-rank of \(\mathrm{Gal}(K/k)\). This result explains the complexity of \(X_{K_\infty,S}\) as a \(\mathbb{Z}_p[[\mathrm{Gal}(K_\infty/k)]]\)-module when the \(p\)-rank of \(\mathrm{Gal}(K/k)\) is large. Moreover, they provide an noncommutative analogous theorem whenever \(K/k\) is non-abelian. The authors also study the Iwasawa adjoint of \(X_{K_\infty,S}\), and the minus part of the unramified Iwasawa module for a CM-extension. In order to prove these results, they systematically investigate the minimal resolutions of \(X_{K_\infty,S}\).\N\NA key component to the proof of their theorems is the existence of certain exact sequences, which is the so-called Tate sequences. It should be remarked here that \textit{C. Greither} also used a different kind of Tate sequence in [Acta Arith. 160, No. 1, 55--66 (2013; Zbl 1284.11150)] to get information on the minimal numbers of generators of class groups of number fields. The current method using the Tate sequence in this paper under review is totally different from Greither's.\NThis is a well-organized paper with clear expository style which gives a new insight towards understanding the structure of classical Iwasawa modules.