Hilbert functions, Hilbert-Samuel quasi-polynomials with respect to f-good filtrations, multiplicities

In this paper we prove a generalization of the Hilbert Theorem in terms of quasi-polynomial functions. As an application we give an extension of the Hilbert-Samuel Theorem to finitely generated A-modules filtered by f-good filtrations, where f is a noetherian filtration on a noetherian ring A. We deduce a multiplicity function on the category of finitely generated A-modules with values in the set Q of rational numbers and we show that this multiplicity behaves like that of ideals. (C) 1999 Elsevier Science B.V. All rights reserved.