Negligibility of projective linear automorphisms
type: Item
instance of: scholarly article
title: Negligibility of projective linear automorphisms
zbMATH Open document ID: 0894.12001
author: Hamza Ahmad
author: Mowaffaq Hajja
author: Ming-Chang Kang
published in: Journal of Algebra
publication date: 1998-09-08T00:00:00Z
review text: Let \(k\) be a field, \(s=(a_{ij})_{1\leq i,j\leq n+1}\in GL_{n+1}(k)\), and \(K=k(x_1/x_{n+1},\dots,x_n/x_{n+1},y_1/y_{n+1},\dots,\) \(y_n/y_{n+1})\) be a field of transcendence degree \(2n\) over \(k\). The diagonal action of \(s\) on \(K\) is defined as \[ s(x_j/x_{n+1})=\sum^{n+1}_{i=1} a_{ij} x_i/\sum^{n+1}_{i=1} a_{i,n+1} x_i, \] \[ s(y_j/y_{n+1})=\sum^{n+1}_{i=1} a_{ij} y_i/\sum^{n+1}_{i=1} a_{i,n+1} y_i. \] The authors show that there exist \(z_1,\dots,z_n\in K\) such that \(K=k(x_1/x_{n+1},\dots,x_n/x_{n+1},z_1,\dots,\) \(z_n)\) and \(s(z_i)=z_i\) for \(i=1,\dots,n\). In particular, if \(k(x_1/x_{n+1},\dots,x_n/x_{n+1})^{\langle s\rangle}\) is pure transcendental over \(k\), then \(K^{\langle s\rangle}\) is also pure transcendental over \(k\).
Mathematics Subject Classification ID: 12F20
zbMATH DE Number: 1115997
zbMATH Keywords: projective linear automorphisms
zbMATH Keywords: transcendental field extensions
zbMATH Keywords: pure transcendental fields
zbMATH Keywords: diagonal action
reviewed by: John N. Mordeson
MaRDI profile type: Publication
full work available at URL: jabr.1997.7199
OpenAlex ID: W1988225703
cites work: Rationality of some projective linear actions
cites work: Q3963168
cites work: A note on affine automorphisms
cites work: A Description of an algebraic automorphism in terms of the radical of its minimal polynomial
cites work: Some actions of symmetric groups
cites work: A note on algebraic automorphisms
cites work: Efficient generation of the ring of invariants
cites work: Q5732722
DOI: 10.1006/JABR.1997.7199
Recommended article: Relative negligibility of linear automorphisms
Recommended article: Projective subschemes with many linear automorphisms
Recommended article: Automorphisms of certain affine complements in projective space
Recommended article: Q4463314
Recommended article: Q3128009
Recommended article: Automorphic equivalence of linear algebras
Recommended article: Q3353145
Recommended article: Componentwise linearity of projective varieties with almost maximal degree
Recommended article: LINEAR AND PROJECTIVE BOUNDARY OF NILPOTENT GROUPS
Recommended article: MINIMAL AUTOMORPHIC POSETS AND THE PROJECTION PROPERTY
description: scientific article; zbMATH DE number 1115997