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A Generalization of a Theorem of Swan with Applications to Iwasawa Theory
- Part of
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- Journal:
- Canadian Journal of Mathematics / Volume 72 / Issue 3 / June 2020
- Published online by Cambridge University Press:
- 16 November 2018, pp. 656-675
- Print publication:
- June 2020
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- Article
- Export citation
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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.
