Gelfand-Kirillov dimension and associated graded modules.

*(English)*Zbl 0688.16030Let 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.

Reviewer: T.H.Lenagan

##### MSC:

16P60 | Chain conditions on annihilators and summands: Goldie-type conditions |

16W50 | Graded rings and modules (associative rings and algebras) |

16W60 | Valuations, completions, formal power series and related constructions (associative rings and algebras) |

17B35 | Universal enveloping (super)algebras |

##### Keywords:

filtered k-algebra; graded ring; graded module; Gelfand-Kirillov dimensions; finitely generated module; enveloping algebras of Lie algebras; ring of differential operators
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\textit{J. C. McConnell} and \textit{J. T. Stafford}, J. Algebra 125, No. 1, 197--214 (1989; Zbl 0688.16030)

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