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

Part of: Algebraic number theory: global fields Algebraic number theory: local and $p$-adic fields

Published online by Cambridge University Press:  16 November 2018

Andreas Nickel
Andreas Nickel*
Affiliation:
Universität Duisburg-Essen, Fakultät für Mathematik, Thea-Leymann-Str. 9, 45127 Essen, Germany Email: andreas.nickel@uni-due.dehttps://www.uni-due.de/∼hm0251/english.html
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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.

Keywords

Swan’s theoremprojective latticeIwasawa algebraIwasawa theoryGalois module structure

MSC classification

Primary: 11R23: Iwasawa theory
Secondary: 11R34: Galois cohomology 11S23: Integral representations 11S25: Galois cohomology
Type
Article
Information
Canadian Journal of Mathematics , Volume 72 , Issue 3 , June 2020 , pp. 656 - 675
Copyright
© Canadian Mathematical Society 2018

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Footnotes

The author acknowledges financial support provided by the Deutsche Forschungsgemeinschaft (DFG) within the Heisenberg programme (No. NI 1230/3-1).

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