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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 \(e\).
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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