Witt-Burnside ring and Burnside ring over a special \(\lambda \)-ring
Let \(G\) be a profinite group. The Witt-Burnside ring of \(G\) was introduced by A.~Dress and C.~Siebeneicher in 1988 as a group-theoretical generalization of the classical ring of \(p\)-typical Witt vectors of Teichmüller and Witt and the genealized Witt vectors of Cartier. More precisely, this is a covariant functor \(\mathbb{W}_G\) from the category of commutative rings \(A\) with unity into itself, having certain properties. This was further generalized by \textit{J. J.~Graham} [Adv. Math. 99, No.2, 248--263 (1993; Zbl 0783.13022)] by constructing a covariant functor \(F_G\) for any group \(G\), and showed that when \(G\) is profinite, there is a functorial isomorphism between \(\mathbb{W}_G(A)\) and \(F_G(A)/I_G(A)\), where \(I_G(A)\) is an ideal of \(F_G(A)\) determined by a suitable condition. In a previous paper [Adv. Math. 190, No. 1, 1--46 (2005); corrigendum ibid. 192, No. 1, 226--227 (2005; Zbl 1105.19001)], the author constructed, for each group \(G\), a covariant functor \(Nr_G\) from the category of special \(\lambda\)-rings to the category of commutative rings with unity. In the paper under review, the author constructs a functor \(\widehat{F}_G\) from the category of special \(\lambda\)-rings to that of commutative rings with unity, and when \(G\) is a strongly complete profinite group, he proves that for a special \(\lambda\)-ring \(R\), \(Nr_G(R)\) is functorially isomorphic to \(\widehat{F}_G(R)/J_G(R)\) for some ideal \(J_G(R)\) of \(\widehat{F}_G(R)\).
scientific article; zbMATH DE number 6099538
scientific article