Gelfand-Kirillov dimension and associated graded modules
type: Item
instance of: scholarly article
title: Gelfand-Kirillov dimension and associated graded modules
zbMATH Open document ID: 0688.16030
DOI: 10.1016/0021-8693(89)90301-3
author: J. C. McConnell
author: J. T. Stafford
published in: Journal of Algebra
publication date: 1989-01-01T00:00:00Z
review text: Let S be a filtered k-algebra with a filtration \(\{S_ n\}\), \(n\geq 0\), and write gr(S) for the associated graded ring \(gr(S)=\oplus S_ n/S_{n-1}\). In a similar manner, let M be a finitely generated S-module with associated graded module \(gr(M)=\oplus M_ n/M_{n-1}\). The authors are interested in relating the Gelfand-Kirillov dimensions of M and gr(M) in the case that gr(M) is a finitely generated module and gr(S) is a finitely generated k-algebra. One always has GK dim(gr(M))\(\leq GK \dim (M)\), and strict inequality is possible. However, equality holds in many natural cases and if the graded ring has a simpler structure this is a considerable aid to computation. If the \(S_ n\) are finite dimensional equality is not too difficult to establish and this is a very useful result, for example, in the study of enveloping algebras of Lie algebras. Here, the authors consider the case that gr(S) is a finitely generated commutative k-algebra and prove that GK dim(gr(M))\(=GK \dim (M)\). This is an imprtant case: for example, it arises in many cases when S is a ring of differential operators filtered by the degree of differential operators. The proof is too technical to summarize here, but the authors have made a good job of presenting the necessarily complicated computations.
Mathematics Subject Classification ID: 16P60
Mathematics Subject Classification ID: 16W50
Mathematics Subject Classification ID: 16W60
Mathematics Subject Classification ID: 17B35
zbMATH DE Number: 4127447
zbMATH Keywords: filtered k-algebra
zbMATH Keywords: graded ring
zbMATH Keywords: graded module
zbMATH Keywords: Gelfand-Kirillov dimensions
zbMATH Keywords: finitely generated module
zbMATH Keywords: enveloping algebras of Lie algebras
zbMATH Keywords: ring of differential operators
reviewed by: T. H. Lenagan
MaRDI profile type: Publication
full work available at URL: 0021-8693(89)90301-3
OpenAlex ID: W2092725080
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description: scientific article; zbMATH DE number 4127447