Hermitian forms over ordered *-fields
type: Item
instance of: scholarly article
title: Hermitian forms over ordered *-fields
zbMATH Open document ID: 0933.11020
author: Thomas Craven
author: Tara L. Smith
published in: Journal of Algebra
publication date: 1999-10-05T00:00:00Z
review text: Let \((D,{}^*)\) be a skew field with an involution *. Let \(Y_D\) and \(X_D\) denote space of Baer orderings and its subspace of *-orderings on \((D,{}^*).\) The main object of interest of the paper is the subring \(WS(D,{}^*)\) of the ring \({\mathcal C}(Y_D,Z)\) of continuous functions from \(Y_D\) to the ring of integers generated by the image of the Witt group \(W(D,{}^*)\) under the total semisignature mapping. As the authors show \(WS(D,{}^*)\) is isomorphic to \(WR_{\text{red}}(D,{}^*) \oplus C\), where \(WR_{\text{red}}(D,{}^*)\) is the ring defined by the first author in [J. Algebra 147, 96-127 (1992; Zbl 0752.11017)] and \(C\) is the kernel of the natural ring homomorphism \(WS(D,{}^*) \longrightarrow WR_{\text{red}}(D,{}^*)\) obtained by the restriction mapping \({\mathcal C}(Y_D, Z) \rightarrow {\mathcal C}(X_D, Z).\) When \((D,{}^*)\) is equipped with a smooth *-valuation \(v\) one can consider the quotient ring \(W_v(D)\) of \(WS(D,{}^*)\) obtained by restricting all functions on \(Y_D\) to the subset of all Baer orderings of \(D\) which are compatible with \(v\). The authors describe the ring \(W_v(D)\) in terms of a group ring over a tensor product of some number of copies of \(WS(D_v,{}^*)\). The paper sums up with a discussion on the relation between \(W(D,{}^*), WS(D,{}^*)\) and the Witt rings associated with fields such as the center of \(D\) or the field of central symmetric elements.
Mathematics Subject Classification ID: 11E39
Mathematics Subject Classification ID: 11E81
Mathematics Subject Classification ID: 12E15
Mathematics Subject Classification ID: 16K40
zbMATH DE Number: 1309251
zbMATH Keywords: hermitian form
zbMATH Keywords: Baer ordering
zbMATH Keywords: Witt group
zbMATH Keywords: valuation
zbMATH Keywords: space of orderings
zbMATH Keywords: signature
zbMATH Keywords: Witt rings
reviewed by: Andrzej Sładek
MaRDI profile type: Publication
full work available at URL: jabr.1998.7774
OpenAlex ID: W2010423296
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DOI: 10.1006/JABR.1998.7774
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description: scientific article; zbMATH DE number 1309251