Geodesic
Indentification of the Hilbert function and Poincar\'e series, and the dimension of modules over filtered rings
Izvestiya. Mathematics , Tome 44 (1995) no. 2, pp. 225-246

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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. 
@article{IM2_1995_44_2_a1,
     author = {V. V. Bavula},
     title = {Indentification of the {Hilbert} function and {Poincar\'e} series, and the dimension of modules over filtered rings},
     journal = {Izvestiya. Mathematics },
     pages = {225--246},
     publisher = {mathdoc},
     volume = {44},
     number = {2},
     year = {1995},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_1995_44_2_a1/}
}
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V. V. Bavula. Indentification of the Hilbert function and Poincar\'e series, and the dimension of modules over filtered rings. Izvestiya. Mathematics , Tome 44 (1995) no. 2, pp. 225-246. http://geodesic.mathdoc.fr/item/IM2_1995_44_2_a1/