Hermitian forms in additive categories: Finiteness results
type: Item
instance of: scholarly article
title: Hermitian forms in additive categories: Finiteness results
zbMATH Open document ID: 0689.10031
DOI: 10.1016/0021-8693(89)90050-1
published in: Journal of Algebra
publication date: 1989-01-01T00:00:00Z
review text: In a ring E with involution \({}^*\) the set of units fixed by the involution is denoted by \(S(^*, E^{\times})\). Two elements \(u,v\in S(^*, E^{\times})\) are similar if there is some \(w\in E^{\times}\) such that \(u=w^*vw\). The set of similarity classes of \(S(^*, E^{\times})\) is denoted by \(H(^*, E^{\times})\). These similarity classes are related to isometry classes of certain Hermitian forms [cf. the authors, J. Reine Angew. Math. 375/376, 167-183 (1987; Zbl 0607.57013), section 5]. If E is Q-finite, i.e., the torsion group of E is finite and \(E\otimes_{{\mathbb{Z}}} {\mathbb{Q}}\) is finite dimensional over \({\mathbb{Q}}\), then \(E_ p\) denotes the p-adic completion of E at the rational prime p. There is a canonical localization map \(H(^*, E^{\times})\to \prod_{p}H(^*, E_ p^{\times})\). It is shown that the fibres of this map are finite. The paper also contains a number of related finiteness results as well as applications to forms with group actions, to systems of forms and to knot theory.
reviewed by: Niels Schwartz
Mathematics Subject Classification ID: 11E16
Mathematics Subject Classification ID: 57Q45
Mathematics Subject Classification ID: 11R52
zbMATH DE Number: 4128905
zbMATH Keywords: order
zbMATH Keywords: stable knot
zbMATH Keywords: isometry classes of Hermitian forms
zbMATH Keywords: similarity classes
zbMATH Keywords: p-adic completion
zbMATH Keywords: localization
author: Eva Bayer-Fluckiger
author: C. Kearton
author: Stephen M. J. Wilson
MaRDI profile type: Publication
full work available at URL: 0021-8693(89)90050-1
OpenAlex ID: W2054479894
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description: scientific article; zbMATH DE number 4128905