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The associated graded ring of an integral group ring. (English) Zbl 0358.16006
Let \(G\) be an abelian group, \(S(G)\) the symmetric algebra of \(G\) and \(gr ZG\) the graded ring associated to the filtration of the ring \(ZG\) by powers of the augmentation ideal \(A_G\). There is a natural homogeneous surjection \(\theta:S(G) \to gr ZG\), with \(n\)-th component given by \[ \theta_n(x_1 \overset \wedge\otimes \dots \overset \wedge\otimes x_n) = \prod_{i=1}^n (x_i-1) + A^{n+1}_G. \]
Several results on the kernel of \(\theta\) are obtained; the main result is a determination (in the case of finite \(p\)-groups) of when \(\theta_n\), is an isomorphism. The discussion is highly technical but interesting.

MSC:
16S34 Group rings
20C05 Group rings of finite groups and their modules (group-theoretic aspects)
15A69 Multilinear algebra, tensor calculus
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