Minimal resolutions of Iwasawa modules

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 Zₚ-extension K_/K, we consider X₊_, S= Gal (M₊_, S/K_) where S is a finite set of places of k containing all ramifying places in K_ and archimedean places, and M₊_, S is the maximal abelian pro-p-extension of K_ unramified outside S. We give lower and upper bounds of the minimal numbers of generators and of relations of X₊_, S as a Zₚ[ Gal (K_/k) ]-module, using the p-rank of Gal (K/k). This result explains the complexity of X₊_, S as a Zₚ[ Gal (K_/k) ]-module when the p-rank of 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₊_, 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₊_, S.