A generalization of a theorem of Swan with applications to 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 $P^{\prime }$ 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.

arXiv comment18 pages; minor changes since v2; title has changed slightly; to appear in Canad. J. Math