Fitting ideals and various notions of equivalence for modules
Abstract: Recently there has been a lot of work on determining the Fitting ideals of arithmetic modules over groups rings, first and foremost of class groups and their Pontryagin duals. In particular, it has turned out that these Fitting ideals are usually non-principal and may be described, up to principal ideals, in terms of group-theoretical information only. The involved principal ideal factors are essentially given by values of equivariant L-functions. The present paper is not concerned with these L-functions but r… Show more
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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 $$\mathbb {Z}_p$$ 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 .
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 $$\mathbb {Z}_p$$ 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 .
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