IDENTIFICATION OF THE HILBERT FUNCTION AND POINCARÉ SERIES, AND THE DIMENSION OF MODULES OVER FILTERED RINGS

Abstract

In this paper it is shown how to reconstruct a Poincaré series from a known Hilbert (integral) function of a graded module over a commutative Noetherian graded ring, and vice versa. The dimension and multiplicity of modules over a filtered ring whose associated graded ring is commutative and Noetherian are introduced. For one class of generalized Weyl algebras that includes the Weyl algebras An, the Krull dimension is computed, and Bernstein's inequality is proved and strengthened.