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_K∞,S= Gal(M_K∞,S/K∞) where S is a finite set of places of k containing all ramifying places in K∞ and archimedean places, and M_K∞,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_K∞,S as a Zₚ[[ Gal(K∞/k)]]-module, using the p-rank of Gal(K/k). This result explains the complexity of X_K∞,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_K∞,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∞,S.