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Let $p$ be a prime and let $G$ be a finite group. By a celebrated theorem of Swan, two finitely generated projective $\mathbb{Z}_{p}[G]$-modules $P$ and $P^{\prime }$ are isomorphic if and only if $\mathbb{Q}_{p}\otimes _{\mathbb{Z}_{p}}P$ and $\mathbb{Q}_{p}\otimes _{\mathbb{Z}_{p}}P^{\prime }$ are isomorphic as $\mathbb{Q}_{p}[G]$-modules. We prove an Iwasawa-theoretic analogue of this result and apply this to the Iwasawa theory of local and global fields. We thereby determine the structure of natural Iwasawa modules up to (pseudo-)isomorphism.

Let $H$ be a torsion-free compact $p$-adic analytic group whose Lie algebra is split semisimple. We show that the quotient skewfield of fractions of the Iwasawa algebra $\varLambda_H$ of $H$ has trivial centre and use this result to classify the prime $c$-ideals in the Iwasawa algebra $\varLambda_G$ of $G:=H\times\mathbb{Z}_p$. We also show that a finitely generated torsion $\varLambda_G$-module having no non-zero pseudo-null submodule is completely faithful if and only if it is has no central torsion. This has an application to the study of Selmer groups of elliptic curves.

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A Generalization of a Theorem of Swan with Applications to Iwasawa Theory

CENTRES OF SKEWFIELDS AND COMPLETELY FAITHFUL IWASAWA MODULES

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A Generalization of a Theorem of Swan with Applications to Iwasawa Theory

CENTRES OF SKEWFIELDS AND COMPLETELY FAITHFUL IWASAWA MODULES