On Gelfand-Kirillov dimension and related topics
Martin Lorenz1988, Journal of Algebra
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Abstract
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This article addresses crucial problems concerning the Gelfand-Kirillov (GK) dimension of algebras and modules, specifically when this dimension is an integer and under what conditions exactness holds in short exact sequences of modules. Significant results are obtained related to the conditions needed for integer GK-dimensions and exactness across various classes of algebras, with a detailed analysis of the behavior of filtrations under intersection with submodules and ideals. The study further explores the rationality of Poincare series for graded modules, advancing the understanding of GK-dimensions in non-commutative settings.
Key takeaways
- Some exceptions, where GK-dimension is known to be an integer or co, are: finitely presented monomial algebras (Govorov [9]), almost commutative algebras (Tauvel [16]), an Noetherian PI-algebras (Lorenz and Small [ 131).
- Part (i) of the following lemma explains the interest of finitely controlled submodules for our purposes.
- (d) If S is strongly finitely presented, then gr,(S) is not automatically finitely presented, for any generating subspace V, but V must be carefully chosen.
- Using the filtrations IV(") on So, r~( IV(")) on RO, and In WC") on I, := In S,, we have an isomorphism of associated graded algebras Since gr(R,) is finitely presented and gr(S,) is affine, gr(&) is finitely generated as ideal of gr(S,).
- subspace W= W( V) c S with VC R[ W] and gr w(n[ W]) (right) Noetherian.
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