Towards a more general notion of Gelfand-Kirillov dimension
type: Item
instance of: scholarly article
title: Towards a more general notion of Gelfand-Kirillov dimension
zbMATH Open document ID: 0865.16018
DOI: 10.1007/BF02761094
author: R. A. Beaulieu
author: Anders Jensen
author: Søren Jøndrup
published in: Israel Journal of Mathematics
publication date: 1997-07-07T00:00:00Z
review text: The classical Gelfand-Kirillov dimension GK(\ ) is an important tool for the study of algebras over a field. The purpose of the paper under review is to adapt this concept to algebras over commutative noetherian rings, replacing the vector space dimension in the original dimension by the reduced rank, denoted by \(\rho\) in what follows. Let \(C\) be a commutative base ring with finitely generated prime radical \(N(C)\) such that \(C/N(C)\) is Goldie, let \(A=C\{a_1,...,a_n\}\) be an affine algebra over \(C\), and let \(M=1\cdot C+a_1C+\dots+a_nC\). The (generalized) Gelfand-Kirillov dimension is then defined as \[ \widehat{\text{GK}}_C(A)=\limsup_{n\to\infty}\log_n(\rho_C(M^n)). \] If \(A\) is not affine over \(C\), then \(\widehat{\text{GK}}_C(A)=\sup\widehat{\text{GK}}_C(A')\), where \(A'\) runs through the set of all affine subalgebras of \(A\). For an affine \(C\)-algebra \(A\) this analogue of the classical definition is related to the latter by the formula \[ \widehat{\text{GK}}_C(A)=\max_i\{\text{GK}_{Q(C/P_i)}(A/AP_i\otimes Q(C/P_i))\}, \] where the maximum is taken over all minimal prime ideals of \(C\) and \(Q(C/P_i)\) denotes the field of fractions of \(C/P_i\). The definition of \(\widehat{\text{GK}}\) can also be extended to modules over \(A\) in the usual way. The authors establish most of the basic results known for the classical Gelfand-Kirillov dimension, many of the proofs being fairly straightforward adaptations to the new set-up. However, \(\widehat{\text{GK}}\) has one serious deficiency, in so far as it is not sensitive to the presence of torsion within the algebra. Thus, \(\widehat{\text{GK}}_C(A)=-\infty\) whenever \(A\) is \({\mathcal C}(N(C))\)-torsion. This defect makes \(\widehat{\text{GK}}\) illsuited for some of the applications one might want to get, so, for example, a generalization of the theorem by \textit{L. W. Small, J. T. Stafford} and \textit{R. B. Warfield} [Math. Proc. Camb. Philos. Soc. 97, 407-414 (1985; Zbl 0561.16005)] th…
reviewed by: Günter Krause
Mathematics Subject Classification ID: 16P90
Mathematics Subject Classification ID: 16P60
Mathematics Subject Classification ID: 16P40
zbMATH DE Number: 878282
zbMATH Keywords: Gelfand-Kirillov dimension
zbMATH Keywords: algebras over commutative Noetherian rings
zbMATH Keywords: reduced rank
zbMATH Keywords: affine algebras
MaRDI profile type: Publication
cites work: Q3967681
cites work: Q3994888
cites work: Q3677877
cites work: Prime ideals in finite extensions of Noetherian rings
cites work: On the Gel′fand-Kirillov dimension of Noetherian P.I.-algebras
cites work: Q3786586
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cites work: Affine algebras of Gelfand-Kirillov dimension one are PI
full work available at URL: https://doi.org/10.1007/bf02761094
OpenAlex ID: W2022130077
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description: scientific article; zbMATH DE number 878282