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