Tori and essential dimension
type: Item
instance of: scholarly article
title: Tori and essential dimension
zbMATH Open document ID: 1141.14028
author: Giordano Favi
author: Mathieu Florence
published in: Journal of Algebra
publication date: 2008-07-21T00:00:00Z
full work available at URL: Favi, Giordano and Florence, Mathieu. (2008) Tori and essential dimension. Journal of algebra, 319 (9). pp. 3885-3900.pdf
review text: This paper establishes three interesting, but unrelated results about essential dimensions. The authors use geometric tools, in particular, those of algebraic tori, to prove these results. The study of essential dimension of a linear algebraic group (= a smooth affine group scheme) \(G\) over a field \(k\), denoted by \(ed_k(G)\), was initiated by Reichstein and his coauthors [see \textit{J. Buhler} and \textit{Z. Reichstein}, Compos. Math. 106, No. 2, 159--179 (1997; Zbl 0905.12003); \textit{Z. Reichstein}, Transform. Groups 5, No. 3, 265--304 (2000; Zbl 0981.20033); \textit{G. Berhuy} and \textit{G. Favi}, Doc. Math., J. DMV 8, 279--330 (2003; Zbl 1101.14324)]. Note that it is permissible that \(G\) is a finite group. The definition may be summarized as follows: \(\text{ed}_k(G)= \min \{ \dim_k(Y) \}\) where \(Y\) runs over irreducible \(k\)-varieties satisfying that there is is a \(k\)-variety \(X\) and a \(k\)-morphism \(f: X \rightarrow Y\) so that \(f: X \rightarrow Y\) is a versal (or classifying) \(G\)-torsor over \(Y\) (see the first section of the present paper for the precise definitions of \(G\)-torsor and versal \(G\)-torsors). The definition of essential dimension of a linear algebraic group was generalized by Merkurjev (1999) to that of a functor \(F: C_k \rightarrow Sets\), denoted by \(\text{ed}_k(F)\), where \(C_k\) is the category of field extensions of a field \(k\) and \(Sets\) is the category of sets (see Definition 1.14 of the present paper). In particular, if \(G\) is a linear algebraic group over a field \(k\), then \(\text{ed}_k(G)=\text{ed}_k(F)\) where \(F\) is the functor defined by \(F(K)=H^1(K, G)\) for any field extension \(K\) of the field \(k\). The first two sections of the paper under review provides a clear and convenient preliminary of torsors, essential dimensions and tori. If \(n\) is an odd integer and \(n \geq 5\), a versal torsor for \(\text{PGL}_n\) over any base field \(k\) is constructed explictly. In particular, the dim…
reviewed by: Ming-Chang Kang
Mathematics Subject Classification ID: 14L30
zbMATH DE Number: 5303137
zbMATH Keywords: essential dimension
zbMATH Keywords: tori
zbMATH Keywords: versal tosors
MaRDI profile type: Publication
OpenAlex ID: W2017990934
cites work: Q5658178
cites work: Essential dimension: A functorial point of view (after A. Merkurjev)
cites work: Q3999894
cites work: Q3972725
cites work: Q4345956
cites work: Q4416651
cites work: Q4787521
cites work: On the Essential Dimension of Some Semi-Direct Products
cites work: Essential dimension of algebraic groups and integral representations of Weyl groups
cites work: FIELDS OF DEFINITION FOR DIVISION ALGEBRAS
cites work: Q3871006
cites work: Q5549011
cites work: Geometric Invariant Theory
cites work: On the notion of essential dimension for algebraic groups
cites work: Essential Dimensions of Algebraic Groups and a Resolution Theorem for G-Varieties
cites work: Q5521947
cites work: Linear algebraic groups.
cites work: Q4210019
DOI: 10.1016/J.JALGEBRA.2007.03.048
Recommended article: Essential dimension: a survey
Recommended article: Essential dimension
Recommended article: Essential dimension of algebraic tori
Recommended article: Q5852631
Recommended article: An upper bound on the essential dimension of a central simple algebra.
Recommended article: Essential dimension of the spin groups in characteristic 2
Recommended article: Essential dimension of algebraic groups, including bad characteristic
Recommended article: The essential dimension of the normalizer of a maximal torus in the projective linear group
Recommended article: Essential dimension of spinor and Clifford groups
Recommended article: On the essential dimension of infinitesimal group schemes
description: scientific article; zbMATH DE number 5303137