Specialization of quadratic and symmetric bilinear forms. Transl. by Thomas Unger
type: Item
instance of: scholarly article
title: Specialization of quadratic and symmetric bilinear forms. Transl. by Thomas Unger
zbMATH Open document ID: 1203.11033
DOI: 10.1007/978-1-84882-242-9
author: Manfred Knebusch
published in: Algebra and Applications
publication date: 2010-08-11T00:00:00Z
review text: In a series of seminal papers in the 1970s [Acta Arith. 24, 279--299 (1973; Zbl 0287.15010)], [Proc. Lond. Math. Soc. (3) 33, 65--93 (1976; Zbl 0351.15016)], Proc. Lond. Math. Soc., III. Ser. 34, 1--31 (1977; Zbl 0359.15013)], \textit{M. Knebusch} initiated a systematic theory of specialization of quadratic forms and their generic splitting theory, in particular the study of quadratic forms over function fields of quadrics, a theory that has blossomed ever since, in particular from the 1990s on in the work of Izhboldin, Karpenko, Merkurjev, Laghribi, Vishik and the reviewer to name but a few, and that has yielded some spectacular breakthroughs, partly based on the incorporation of algebro-geometric techniques introduced by Rost, Voevodsky and others that also lead to Voevodsky's proof of the Milnor conjecture. The present book represents the state of the art in the theory of specialization of quadratic forms, containing a substantial amount of previously unpublished new results. It is a welcome addition to the literature as this is the first monograph in which this original and important approach to the algebraic theory of quadratic forms has been given such a comprehensive treatment. In the sequel, we give an overview of the contents of the book. In Chapter 1, the author introduces the basic idea of specialization: Let \(\lambda\,:\,K\to L\cup\infty\) be a place with valuation ring \({\mathfrak o}\subset K\) and maximal ideal \({\mathfrak m}\). Let \(\varphi\) be a symmetric bilinear form over \(K\). We say that \(\varphi\) has good reduction if \(\varphi\) is isometric to \((a_{ij})\) with \(a_{ij}\in {\mathfrak o}\) and \(\det (a_{ij})\in {\mathfrak o}^*\). This gives rise to a well-defined (up to isometry) form \(\lambda_*(\varphi)=(\lambda (a_{ij}))\) provided \(\text{char}(L)\neq 2\). If \(\text{char}(L)=2\), then at least the anisotropic part of \(\lambda_*(\varphi)\) is well defined up to isometry. Thus, in any case, \(\lambda\) induces an additive map betwe…
Mathematics Subject Classification ID: 11E04
Mathematics Subject Classification ID: 11-02
Mathematics Subject Classification ID: 11E08
Mathematics Subject Classification ID: 11E39
Mathematics Subject Classification ID: 11E81
Mathematics Subject Classification ID: 13A18
zbMATH DE Number: 5769117
zbMATH Keywords: quadratic form
zbMATH Keywords: symmetric bilinear form
zbMATH Keywords: place
zbMATH Keywords: quadratic place
zbMATH Keywords: valuation
zbMATH Keywords: specialization
zbMATH Keywords: good reduction
zbMATH Keywords: fair reduction
zbMATH Keywords: generic splitting
zbMATH Keywords: generic splitting tower
zbMATH Keywords: Pfister form
zbMATH Keywords: multiplicative form
zbMATH Keywords: height
zbMATH Keywords: degree
reviewed by: Detlev Hoffmann
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/978-1-84882-242-9
OpenAlex ID: W613399094
description: scientific article; zbMATH DE number 5769117