Indentification of the Hilbert function and Poincaré series, and the dimension of modules over filtered rings

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 $A_n$, the Krull dimension is computed, and Bernstein's inequality is proved and strengthened.