A generalization of a theorem of Swan with applications to Iwasawa theory

TL;DR

This paper extends Swan's theorem to Iwasawa theory, providing new insights into the structure of Iwasawa modules over local and global fields, and determining their structure up to pseudo-isomorphism.

Contribution

It generalizes Swan's theorem to an Iwasawa-theoretic context, offering a novel approach to understanding Iwasawa modules in number theory.

Findings

Established an Iwasawa-theoretic analogue of Swan's theorem.

Determined the structure of natural Iwasawa modules up to pseudo-isomorphism.

Applied the results to local and global field Iwasawa theory.

Abstract

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 $P'$ are isomorphic if and only if $\mathbb Q_p \otimes_{\mathbb Z_p} P$ and $\mathbb Q_p \otimes_{\mathbb Z_p} P'$ 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.