Presenting powers of augmentation ideals and Pfister forms
Let \(G\) be a group, \({\mathbb Z}[G]\) the integral group ring, and let \(J(G)\) be the augmentation ideal in \({\mathbb Z}[G]\). Its \(n\)-th power \(J^n(G):=(J(G))^n\) is generated as an abelian group by elements of type \([g_1,\cdots ,g_n]=\prod_{i=1}^n(1-g_i)\) with \(g_i\in G\setminus\{ 1\}\). The main result of the paper gives a complete set of relations for the generators \([g_1,\cdots ,g_n]\) of \(J^n(G)\) for a group \(G\) of exponent \(2\). A first step in the proof is to derive a complete set of relations for such symbols in the case of the \(n\)-th power of the augmentation ideal of the group ring \({\mathbb F}_2[G]\). If \(F\) is a field of characteristic \(\neq 2\), then there is a natural surjective ring homomorphism from \({\mathbb Z}[F^*/F^{*2}]\) onto the Witt ring \(WF\) sending \(aF^*\) to the class of the quadratic form \(\langle a\rangle\), and thus inducing a homomorphism of additive groups \(J^n(F^*/F^{*2})\to I^nF\) by mapping a symbol \([a_1F^{*2},\dots ,a_nF^{*2}]\) onto the class of an \(n\)-fold Pfister form \(\langle\langle a_1,\cdots ,a_n\rangle\rangle\). The authors show that for \(n=1,2\), one obtains a complete set of additive relations for classes of \(n\)-fold Pfister forms by taking the relations obtained from those in \(J^n(F^*/F^{*2})\) plus certain obvious ones, and they ask whether, for \(n\geq 3\), the relations valid in \(J^n(F^*/F^{*2})\) plus one further explicitly given relation suffice to get all relations for \(n\)-fold Pfister forms in \(I^nF\). They point out that a positive answer would yield a proof of Milnor's conjecture stating that there is an isomorphism \(I^nF/I^{n+1}F\to K_2F/2K_2F\). It should be noted that a proof of this conjecture is announced by \textit{D. Orlov, A. Vishik} and \textit{V. Voevodsky} [An exact sequence for Milnor's \(K\)-theory with applications to quadratic forms, \(K\)-theory Preprint Archives 454 (2000)]. Applying these results, \textit{R. Elman} and \textit{J. Kr. Arason} [J. Algebra 239, 150-160 (2001; Zbl 0990.11021)] have obtained a complete set of relations for \(n\)-fold Pfister forms in \(I^nF\). Using the so-called chain-\(p\)-equivalence for Pfister forms, it is not difficult to show that Elman and Arason's set of relations is equivalent to the one given by the authors (in fact, relation \((T)\) in the paper under review is redundant for Pfister forms as it follows from the other relations). The present paper and the one by Arason-Elman give no reference to each other.
scientific article; zbMATH DE number 1563903