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Arithmetical properties of idempotents in group algebras. (English. Abridged French version) Zbl 0958.20007
In this nice paper the author studies for a finite group \(G\) and a Dedekind ring \(A\) of characteristic \(0\) the possible denominators for the numbers \(e_g\in A\) which can occur for an idempotent \(e=\sum_{g\in G}e_gg\in AG\). In fact the author proves that given a prime ideal \(\wp\) of \(A\) and an idempotent \(\widetilde e\) in \(A_\wp G\), then there is an idempotent \(e\in A_\wp G\) and a natural number \(n\in\mathbb{N}\) so that \(A_\wp Ge=A_\wp G\widetilde e\) and \(|G|^ne\in AG\). To show the result one first semi-localizes outside the primes dividing the order of \(G\) and different from \(\wp\). Then, the observation is to show that an \(AG\)-lattice is indecomposable if and only if its localization at \(\wp\) is indecomposable. From this one constructs the idempotent \(e\).
MSC:
20C10 Integral representations of finite groups
20C05 Group rings of finite groups and their modules (group-theoretic aspects)
16S34 Group rings
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