Towards a more general notion of Gelfand-Kirillov dimension
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)] that algebras with Gelfand-Kirillov dimension 1 are PI is beyond reach. However, in a subsequent paper [Commun. Algebra 24, No. 2, 605-622 (1996; Zbl 0847.16014)], the authors present a modification of \(\widehat{\text{GK}}\) which allows to obtain the above theorem, although it has other drawbacks.
algebras over commutative Noetherian rings
scientific article; zbMATH DE number 878282
scientific article