Essential dimension: A functorial point of view (after A. Merkurjev)
type: Item
instance of: scholarly article
title: Essential dimension: A functorial point of view (after A. Merkurjev)
zbMATH Open document ID: 1101.14324
author: Grégory Berhuy
author: Giordano Favi
published in: Documenta Mathematica
publication date: 2004-01-19T00:00:00Z
full work available at URL: https://eudml.org/doc/126738
full work available at URL: 11.html
review text: Summary: We develop a systematic study of the essential dimension of functors. This approach is due to A. Merkurjev and can be found in his unpublished notes. The notion of essential dimension was earlier introduced for finite groups by \textit{J. Buhler} and \textit{Z. Reichstein} [Compos. Math. 106, 159--179 (1997; Zbl 0905.12003)] and for an arbitrary algebraic group over an algebraically closed field by \textit{Z. Reichstein} [Transform. Groups 5, No. 3, 265-304 (2000; Zbl 0981.20033)]. This is a numerical invariant depending on the group \(G\) and the field \(k\). This number is denoted by \(\text{ed}_k(G)\). In this paper we insist on the behaviour of the essential dimension under field extension \(k'/k\) and try to compute \(\text{ed}_k(G)\) for any \(k\). This will be done in particular for the group \(\mathbb{Z}/n\) when \(n\leq5\) and for the circle group. Along the way we define the essential dimension of functor with versal pairs and prove that all the different notions of essential dimension agree in the case of algebraic groups. Applications to finite groups are given. Finally we give a proof of the so-called homotopy invariance, that is \(\text{ed}_k(G)=\text{ed}_{k(t)}(G)\), for an algebraic group \(G\) defined over an infinite field \(k\).
Mathematics Subject Classification ID: 14L30
Mathematics Subject Classification ID: 11E72
Mathematics Subject Classification ID: 12G05
zbMATH DE Number: 2028917
zbMATH Keywords: essential dimension
zbMATH Keywords: algebraic groups
zbMATH Keywords: Galois cohomology
zbMATH Keywords: cohomological invariants
zbMATH Keywords: group scheme actions
zbMATH Keywords: torsors
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 2028917