Equivariant Witt groups of finite groups of odd order
type: Item
instance of: scholarly article
title: Equivariant Witt groups of finite groups of odd order
zbMATH Open document ID: 0707.20004
DOI: 10.1016/0021-8693(90)90078-3
author: Masahiko Miyamoto
published in: Journal of Algebra
publication date: 1990-01-01T00:00:00Z
review text: In a sense, this paper is a continuation of researches initiated by \textit{J. P. Alexander}, \textit{P. E. Conner} and \textit{G. C. Hamrick} in the mid- seventies [Odd order group actions and Witt classification of inner products (Lect. Notes Math. 625) (1977; Zbl 0363.57025)] and based on work of \textit{A. W. M. Dress} [see, for example, Ann. Math., II. Ser. 102, 291-325 (1975; Zbl 0315.20007)]. It would be helpful to be informed also about some other previous work by the author [see J. Algebra 91, 32-35 (1984; Zbl 0547.20005)]. The important result is a (positive) answer to a question of Alexander, Conner and Hamrick, namely that the Dress exact sequence of equivariant Witt groups \[ 0\quad \to \quad W_ 0({\mathbb{Z}},{\mathbb{Z}}G)\quad \to \quad W_ 0({\mathbb{Q}},{\mathbb{Q}}G)\to^{\partial}W_ 0({\mathbb{Q}}/{\mathbb{Z}},{\mathbb{Z}}G) \] has \(\partial\) an epimorphism when G is a finite group of odd order. Another result is a generalization of a theorem of the above mentioned authors implying that \(W_ 0({\mathbb{Z}},{\mathbb{Z}}G)\) is ``sufficiently large'' to study \(W_ 0({\mathbb{Z}}G)\). Thus, the following is proven. Theorem. Let G be a finite p-group (p odd prime), M a projective \({\mathbb{Z}}G\)-module, b a metabolic, non-singular, \({\mathbb{Z}}\)-valued, G- invariant bilinear form on M. Then (M,b) has a projective metabolizer.
Mathematics Subject Classification ID: 20C15
Mathematics Subject Classification ID: 19G12
Mathematics Subject Classification ID: 20C05
Mathematics Subject Classification ID: 19G24
Mathematics Subject Classification ID: 20D10
Mathematics Subject Classification ID: 16S34
Mathematics Subject Classification ID: 16E20
Mathematics Subject Classification ID: 57R85
zbMATH DE Number: 4160991
zbMATH Keywords: Dress exact sequence
zbMATH Keywords: equivariant Witt groups
zbMATH Keywords: finite group of odd order
zbMATH Keywords: finite p-group
zbMATH Keywords: projective \({\mathbb{Z}}G\)-module
zbMATH Keywords: bilinear form
reviewed by: Héctor A. Merklen
MaRDI profile type: Publication
full work available at URL: 0021-8693(90)90078-3
OpenAlex ID: W1993249434
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description: scientific article; zbMATH DE number 4160991