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Gelfand-Kirillov dimension and associated graded modules. (English) Zbl 0688.16030
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.
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
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