Noetherian \({\mathbb{Z}}_ pT\)-modules, adjoints, and Iwasawa theory
The results of this paper are going out an unpublished course notes of \textit{K. Iwasawa} [Seminar on the Riemann-Hurwitz formula for number fields, Princeton, New Jersey, Spring 1980] concerning the category of Noetherian torsion \(\Lambda\)-modules, where \(\Lambda ={\mathbb{Z}}_ pT\), p a prime. Instead of the ordinary class groups the author is working with R-generalized S-class groups. Here, for a number field k, S and R are disjoint finite sets of places of k with S containing all the non- Archimedian places and R containing no primes above p. Let \(k_ n\) be the n th layer of the basic \({\mathbb{Z}}_ p\)-extension \(k_{\infty}\) of k, \(\vartheta_ n\) the ring of S-integers of \(k_ n\), \(I_ n\) the invertible \(\vartheta_ n\)-submodules of \(k_ n\) which are prime to R (the ''S-ideals'') and \(P_ n=\{\alpha \vartheta_ n\), \(\alpha\) \(\equiv 1 mod {\mathfrak p}\) \(\forall {\mathfrak p}\in R\}\). The group \(I_ n/P_ n\) is called the R-generalized S-class group. Let \(A_ n\) be the p-part of \(I_ n/P_ n\), \(H=Hom_{{\mathbb{Z}}_ p}(\lim_{\to}A_ n, {\mathbb{Q}}_ p/{\mathbb{Z}}_ p)\) (the limit is with respect to the natural maps induced by extension of S-ideals), \(A=\lim_{\to}A_ n\) (the limit is with respect to the norm maps). The notions of H and A are studied in this article, and in particular it is shown that if \({\mathbb{Z}}_ p[G(k_{\infty}/k)]\)-modules and \(\Lambda\)-modules are identified by requiring that \((T+1)\) act as a topological generator \(\gamma\) of \(G(k_{\infty}/k)\), then H is a Noetherian \(\Lambda\)-module, A is pseudoisomorphic to H and there is a canonical \(\Lambda\)-module isomorphism \(A\overset \sim \rightarrow G(L_{\infty},k_{\infty})\), where \(L_ n\) denotes the maximal abelian p-extension of \(k_ n\) which is unramified outside R and in which every prime above S is completely decomposed.
Noetherian torsion \(\Lambda \) -modules
canonical \(\Lambda \) -module isomorphism
scientific article; zbMATH DE number 3959567