Minimal resolutions of Iwasawa modules

Abstract

In this paper, we study the module-theoretic structure of classical Iwasawa modules. More precisely, for a finite abelian $p$-extension $K/k$ of totally real fields and the cyclotomic $\mathbb{Z}_p$-extension $K_{\infty}/K$, we consider $X_{K_{\infty},S}={\rm Gal}(M_{K_{\infty},S}/K_{\infty})$ 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 $S$. We give lower and upper bounds of the minimal numbers of generators and of relations of $X_{K_{\infty},S}$ as a $\mathbb{Z}_p[[{\rm Gal}(K_{\infty}/k)]]$-module, using the $p$-rank of ${\rm Gal}(K/k)$. This result explains the complexity of $X_{K_{\infty},S}$ as a $\mathbb{Z}_p[[{\rm Gal}(K_{\infty}/k)]]$-module when the $p$-rank of ${\rm Gal}(K/k)$ is large. Moreover, we prove an analogous theorem in the setting that $K/k$ is non-abelian. We 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 theorems, we systematically study the minimal resolutions of $X_{K_{\infty},S}$.