Minimal resolutions of Iwasawa modules

Abstract

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