Quadratic algebras, Clifford algebras, and arithmetic Witt groups
type: Item
instance of: scholarly article
title: Quadratic algebras, Clifford algebras, and arithmetic Witt groups
zbMATH Open document ID: 0791.11021
author: Alexander J. Hahn
published in: Universitext
publication date: 1994-01-18T00:00:00Z
review text: ``The goal of this volume is to introduce the reader in an elemental and accessible way to the large and dynamic area of algebras and forms over commutative rings. Quadratic algebras and their analysis give the volume its direction. Indeed, its defining moment is the fact that quadratic algebras lie at the heart of the theory of quadratic forms and Clifford algebras over commutative rings.'' The book is written in the style of lecture notes. Up to Chapter 12 it is nearly self-contained with full proofs. It can serve very well as an introductory text to more standard but also more abstract books like \textit{M.-A. Knus} [Quadratic and Hermitian forms over rings, Grundlehren Math. Wiss. 294 (1991; Zbl 0756.11008)]. One of the positive, if not the best, features of Hahn's book is the large number of exercises of varying degree of difficulty, and the hints for solving them. In the following I mention the headlines and main topics treated in the fifteen chapters of the book, occasionally with some additionally explanation. Ch. 1 studies free quadratic algebras \(R[X]/(X^ 2 - aX - b)\), their classification, involutions and gradings \(\pmod 2\). Ch. 2 is on separable \(R\)-algebras \(A\). \(A\) is called separable if the natural \(A^ e\)-module sequence \[ 0 \to \text{ker }\phi \to A^ e @>\phi>> A \to 0 \] splits. Here \(A^ e = A \otimes A^{\text{opp}}\) is the enveloping algebra of \(A\), \(\phi\) is given by \(\phi(a \otimes b^ 0) = ab\). This concept generalizes the usual separability of a field extension. The above free quadratic algebra is separable iff \(a^ 2 + 4b \in R^*\). Ch. 3 investigates the group structure of the set of isomorphism classes \([a,b]\) of separable free quadratic \(R\)-algebras, their discriminants and a related graded version. Ch. 4 introduces some standard facts on bilinear and quadratic forms over \(R\) via localization. An important topic is the group \(\text{Dis}(R)\) of ``discriminant modules'' \((P,f)\). Here \(P\) is a finitely generated…
Mathematics Subject Classification ID: 11E88
Mathematics Subject Classification ID: 15A66
Mathematics Subject Classification ID: 11-02
Mathematics Subject Classification ID: 13-02
Mathematics Subject Classification ID: 15-02
Mathematics Subject Classification ID: 11E04
Mathematics Subject Classification ID: 13A99
Mathematics Subject Classification ID: 11E81
Mathematics Subject Classification ID: 15A63
Mathematics Subject Classification ID: 15A75
Mathematics Subject Classification ID: 13F05
zbMATH DE Number: 493075
zbMATH Keywords: discriminant modules
zbMATH Keywords: standard modules
zbMATH Keywords: Arf algebra
zbMATH Keywords: Clifford modules
zbMATH Keywords: arithmetic Dedekind ring
zbMATH Keywords: Hasse domain
zbMATH Keywords: Clifford index
zbMATH Keywords: quadratic algebras
zbMATH Keywords: quadratic forms
zbMATH Keywords: commutative rings
zbMATH Keywords: large number of exercises
zbMATH Keywords: free quadratic algebras
zbMATH Keywords: Clifford algebras
zbMATH Keywords: exterior algebras
zbMATH Keywords: quadratic module
zbMATH Keywords: special elements
zbMATH Keywords: separable quadratic algebras
zbMATH Keywords: Picard group
zbMATH Keywords: Witt groups
reviewed by: Albrecht Pfister
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 493075