Extendability of quadratic modules with sufficient Witt index
type: Item
instance of: scholarly article
title: Extendability of quadratic modules with sufficient Witt index
zbMATH Open document ID: 0537.10013
DOI: 10.1016/0021-8693(84)90062-0
published in: Journal of Algebra
publication date: 1984-01-01T00:00:00Z
review text: Let K be a field of characteristic different from two. A well-known theorem of Harder states that any quadratic space over the polynomial ring K[X] in one variable is extended from K and hence is decomposable. It is also known that isotropic quadratic spaces over \(K[X_ 1,...,X_ m]\) are extended from K for any m and hence these are again decomposable. On the other hand, if K admits a quaternion division algebra, then there do exist indecomposable anisotropic quadratic spaces of ranks 3 and 4 over K[X,Y] (Parimala's counter-examples to the quadratic analogue of Serre's conjecture). Suslin and Kopeiko considered the corresponding question for a general commutative ring R with characteristic different from two. They proved: Any quadratic space q over \(R[T_ 1,...,T_ m]\) with Witt-index \(Witt(\bar q)\geq \dim R+1\) which is stably extended from R is actually extended from R. Here the bar denotes \(''modulo\quad(T_ 1,...,T_ m)''.\) In the paper under review some improvements of this result for a regular ring R are given: (1) If R is an equicharacteristic complete regular local ring, then every quadratic space q over \(R[T_ 1,...,T_ m]\) with \(Witt(\bar q)\geq 1\) is extended from R. (2) If \(\dim(R)=d\), \(d=2\) or 3, and q is a quadratic \(R[T_ 1,...,T_ m]-space\) with \(Witt(\bar q)\geq d,\) then q is extended. (3) If R is a regular ring of essentially finite type over an infinite field K of dimension \(d\geq 1\), and q is a quadratic \(R[T_ 1,...,T_ m]-space\) with \(Witt(\bar q)\geq d,\) then q is extended. Here a ring R is of essentially finite type over K if \(R=S^{- 1} C\), where S is a multiplicatively closed subset of a finitely generated K-algebra C.
Mathematics Subject Classification ID: 11E08
Mathematics Subject Classification ID: 13H05
Mathematics Subject Classification ID: 13C10
Mathematics Subject Classification ID: 13C15
zbMATH DE Number: 3853184
zbMATH Keywords: extendability
zbMATH Keywords: quadratic analogue of Serre conjecture
zbMATH Keywords: quadratic spaces over polynomial rings
zbMATH Keywords: Witt-index
zbMATH Keywords: regular local ring
author: Ravi A. Rao
reviewed by: Hans-Jochen Bartels
MaRDI profile type: Publication
full work available at URL: 0021-8693(84)90062-0
OpenAlex ID: W4213042615
cites work: The Brauer Group of a Commutative Ring
cites work: Clifford Algebras and Spinor Norms Over a Commutative Ring
cites work: Locally polynomial algebras are symmetric algebras
cites work: Q5648449
cites work: Q5602064
cites work: Q4165471
cites work: Serre's conjecture
cites work: On the Bass-Quillen conjecture concerning projective modules over polynomial rings
cites work: Projective A [X ]-Modules
cites work: Efficient generation of ideals in polynomial rings
cites work: Q4179114
cites work: Q3757970
cites work: A splitting theorem for quadratic forms
cites work: Projective modules over polynomial rings
cites work: Quadratic Forms Over Polynomial Rings Over Dedekind Domains
cites work: Cancellation of quadratic forms over principal ideal domains
cites work: Quadratic spaces over polynomial extensions of regular rings of dimension 2
cites work: Cancellation of quadrate forms over commutative rings
cites work: Application of patching diagrams to some questions about projective modules
cites work: ON THE STRUCTURE OF THE SPECIAL LINEAR GROUP OVER POLYNOMIAL RINGS
cites work: Q3205434
cites work: Projective Modules over Laurent Polynomial Rings
cites work: The general linear group of polynomial rings over regular rings
cites work: Q5611913
cites work: Q5732722
description: scientific article; zbMATH DE number 3853184