An elementary proof of the fundamental theorem of projective geometry
type: Item
instance of: scholarly article
title: An elementary proof of the fundamental theorem of projective geometry
zbMATH Open document ID: 0996.51001
DOI: 10.1023/A:1014933313332
author: C. Faure
published in: Geometriae Dedicata
publication date: 2002-05-30T00:00:00Z
review text: The author proves the following generalization of the classical result of projective geometry which states that every isomorphism between Desarguesian projective spaces is induced by some semilinear map between the associated vector spaces. Theorem: Let \(V\), \(W\) denote vector spaces over division rings \(K\) and \(L\), respectively. Let \(g:{\mathcal P}(V) \to {\mathcal P}(W)\) be a morphism between the associated projective spaces. If the image of \(g\) is not contained in some line of \({\mathcal P}(W)\), then there exists a semilinear map \(f:V\to W\) such that \(g\) is induced by \(f\), i.e. \(g(\langle x\rangle) = \langle f(x)\rangle\) for any admissible \(x\). Moreover, as in the classical case, the map \(f\) is unique up to scalar multiplication. As an application, the author gives a short proof of Wigner's theorem.
Mathematics Subject Classification ID: 51A05
Mathematics Subject Classification ID: 51A10
Mathematics Subject Classification ID: 81P10
Mathematics Subject Classification ID: 46C05
zbMATH DE Number: 1747827
zbMATH Keywords: projective geometry
zbMATH Keywords: fundamental theorem
zbMATH Keywords: Wigner's theorem
zbMATH Keywords: semilinear map
MaRDI profile type: Publication
full work available at URL: a:1014933313332
OpenAlex ID: W99664993
description: scientific article; zbMATH DE number 1747827