On the powers of the fundamental ideal of a Witt ring
type: Item
instance of: scholarly article
title: On the powers of the fundamental ideal of a Witt ring
zbMATH Open document ID: 1061.19001
author: Fabien Morel
published in: Commentarii Mathematici Helvetici
publication date: 2004-11-05T00:00:00Z
review text: Let \(F\) be a field of characteristic \(\neq 2\). A recent result of \textit{J. K. Arason} and \textit{R. Elman} [``Powers of the fundamental ideal in the Witt ring``, J. Algebra 239, 150--160 (2001; Zbl 0990.11021)] describes the powers \(I^n(F)\) of the fundamental ideal \(I(F)\) of the Witt ring \(W(F)\) of \(F\) by generators and relations. The author gives several reformulations of this result. One reformulation is in terms of the Witt \(K\)-theory \(K_*^W(F)\) of \(F\), defined as follows: \(K_*^W(F)\) is the \(\mathbb Z\)-graded ring generated by symbols \(\{a\}\), \(a \in F^*\), in degree \(1\), and a symbol \(\eta\) in degree \(-1\), subject to the following relations: \[ \{ab\} = \{a\} + \{b\} - \eta.\{a\}.\{b\}, \tag{1} \] \[ \{a\}.\{1-a\} = 0 \quad \text{for } a\neq 1, \tag{2} \] \[ \eta.\{-1\} = 2, \tag{3} \] \[ \eta .\{a\} = \{a\}.\eta. \tag{4} \] Let \(I_*(F)\) denote the \(\mathbb Z\)-graded ring, which has the powers \(I^n(F)\) in degrees \(n \geq 1\) and the Witt ring \(W(F)\) in degrees \(n \leq 0\). Then the canonical homomorphism \[ K_*^W(F) \rightarrow I_*(F), \] defined via \[ \{a\} \mapsto \langle\langle a\rangle\rangle = 1-\langle a\rangle \in I(F),\quad \eta \mapsto 1 \in I_{-1}(F) = W(F) \] is an isomorphism.
Mathematics Subject Classification ID: 19D45
Mathematics Subject Classification ID: 15A63
Mathematics Subject Classification ID: 11E81
zbMATH DE Number: 2113817
zbMATH Keywords: quadratic forms
zbMATH Keywords: Witt ring
zbMATH Keywords: Milnor \(K\)-theory
reviewed by: Manfred Kolster
MaRDI profile type: Publication
full work available at URL: s00014-004-0815-z
OpenAlex ID: W2058459054
DOI: 10.1007/S00014-004-0815-Z
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description: scientific article; zbMATH DE number 2113817