Witt groups of smooth projective quadrics
type: Item
instance of: scholarly article
title: Witt groups of smooth projective quadrics
zbMATH Open document ID: 1426.19005
author: Xianqiang Yang
published in: Advances in Mathematics
publication date: 2019-03-13T00:00:00Z
full work available at URL: http://hdl.handle.net/21.11116/0000-0004-84FD-A
review text: One way of studying quadratic forms over a field \(F\) of characteristic different from \(2\) is to consider quadratic forms over the function field \(K(Q_d)\) of the projective quadric \(Q_d\) given by a certain homogeneous quadratic equation. Let \(W(F)\) denotes the Witt group of quadratic forms over \(F\). Then a natural problem arises, that is: What is the kernel of the homomorphism \(W(F) \rightarrow W(K(Q_d))\). There have been solutions to the problem in some cases. The author of the paper under review proves that the sequence \( W(C_0(q)_{\sigma}) \rightarrow W(F) \rightarrow W(K(Q_d))\) is exact, where the map \( W(C_0(q)_{\sigma}) \rightarrow W(F)\) is the trace map (see Proposition 8.1 for the precise statement and a proof). The author also investigates the case when the quardric \(Q_d\) is defined by an isotropic quadratic form \(q\) (see Proposition 8.5 for the details). As an application of the exact sequences, the author derive the computation of the Witt group of quadrics over commutative semi-local ring (see subsection 8.3 for more details) and Witt group of quadrics defined by \(q_{m,n}=m \left\langle -1 \right\rangle \perp n \left\langle 1 \right\rangle \), assuming that \(-1\) is not a sum of two squares (see subsection 8.4 for more details).
reviewed by: Mohamed Elhamdadi
Mathematics Subject Classification ID: 19G12
zbMATH DE Number: 7035895
zbMATH Keywords: quadratic forms
zbMATH Keywords: Witt group
MaRDI profile type: Publication
OpenAlex ID: W2914579758
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Wikidata QID: Q128479860
DOI: 10.1016/J.AIM.2019.01.038
description: scientific article; zbMATH DE number 7035895