Gelfand-Kirillov dimension in some crossed products
type: Item
instance of: scholarly article
title: Gelfand-Kirillov dimension in some crossed products
zbMATH Open document ID: 1014.16022
author: T. Guédénon
published in: Beiträge zur Algebra und Geometrie
publication date: 2003-02-10T00:00:00Z
full work available at URL: https://eudml.org/doc/122697
full work available at URL: 5.html
review text: Let \(k\) be a field. The author considers a PBW-extension \(A=R[\theta_1,\dots,\theta_n]\) of an algebra \(R\) over \(k\). This is a ring extension \(A\supset R\) generated by elements \(\{\theta_1,\dots,\theta_n\}\) such that (i) \([\theta_i,r]\in R\) for each \(i=1,\dots,n\); (ii) \([\theta_i,\theta_j]\in R+R\theta_1+\cdots+R\theta_n\); (iii) \(A\) is free as (left and right) \(R\)-module with basis \(\theta_1^{i_1}\cdots\theta_n^{i_n}\), where \(i_j\) are positive integers. Let \(\delta_i\) be the respective derivation of \(R\), i.e. \(\delta_i(r)=[\theta_i,r]\) for all \(r\in R\). Theorem. Let \(R\) be locally finite-dimensional over \(\Delta=\{\delta_1,\dots,\delta_n\}\). Then the Gelfand-Kirillov dimensions are connected as \(\text{GKdim}(A)=\text{GKdim}(R)+n\). Earlier, the same fact was known for a finitely generated ring \(R\) over \(k\), now this condition is replaced by the local finite dimensionality over \(\Delta\). From this the author deduces some results concerning incomparability, prime length and Tauvel's height formula for prime ideals in the crossed products \(R*G\), where \(G\) is a finite-dimensional Lie algebra acting by derivations on \(R\). Similar results are obtained for \(R*G\), where \(G\) is a free Abelian group of finite rank.
Mathematics Subject Classification ID: 16P90
Mathematics Subject Classification ID: 16S35
Mathematics Subject Classification ID: 16P40
Mathematics Subject Classification ID: 16S30
Mathematics Subject Classification ID: 16D25
zbMATH DE Number: 1865996
zbMATH Keywords: Gelfand-Kirillov dimension
zbMATH Keywords: growth
zbMATH Keywords: crossed products
zbMATH Keywords: Noetherian algebras
zbMATH Keywords: prime ideals
zbMATH Keywords: locally finite-dimensional algebras
zbMATH Keywords: Poincaré-Birkhoff-Witt extensions
zbMATH Keywords: incomparability
reviewed by: Victor Petrogradsky
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1865996