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 \(X_{K_{\infty },S}={{\,\textrm{Gal}\,}}(M_{K_{\infty },S}/K_{\infty })\) 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 \(\mathbb {Z}_p[[{{\,\textrm{Gal}\,}}(K_{\infty }/k)]]\) -module, using the p-rank of \({{\,\textrm{Gal}\,}}(K/k)\) . This result explains the complexity of \(X_{K_{\infty },S}\) as a \(\mathbb {Z}_p[[{{\,\textrm{Gal}\,}}(K_{\infty }/k)]]\) -module when the p-rank of \({{\,\textrm{Gal}\,}}(K/k)\) 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 \(X_{K_{\infty },S}\) .
