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 Z_p-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 Z_p[[Gal(K_∞/k)]]-module, using the p-rank of Gal(K/k). This result explains the complexity of X_K∞,S as a Z_p[[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.