Presenting powers of augmentation ideals and Pfister forms
type: Item
instance of: scholarly article
title: Presenting powers of augmentation ideals and Pfister forms
zbMATH Open document ID: 0985.11016
DOI: 10.1023/A:1026736413398
author: Anthony Bak
author: Nikolai Vavilov
published in: \(K\)-Theory
publication date: 2002-05-19T00:00:00Z
review text: 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. Algebr…
Mathematics Subject Classification ID: 11E81
Mathematics Subject Classification ID: 19D45
Mathematics Subject Classification ID: 11E04
Mathematics Subject Classification ID: 19G99
Mathematics Subject Classification ID: 16S34
zbMATH DE Number: 1563903
zbMATH Keywords: quadratic forms
zbMATH Keywords: Pfister form
zbMATH Keywords: Milnor \(K\)-theory
zbMATH Keywords: Milnor conjecture
zbMATH Keywords: integral group ring
zbMATH Keywords: augmentation ideal
zbMATH Keywords: \(2\)-cocycle
Wikidata QID: Q122395037
reviewed by: Detlev Hoffmann
MaRDI profile type: Publication
full work available at URL: a:1026736413398
OpenAlex ID: W2077622851
description: scientific article; zbMATH DE number 1563903