Extension of the Hilbert-Samuel Theorem
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HENRI DICHI, Laboratoire de Mathematigues pures, Universite Blaise Pascal, Les Cezeaux, 63177 Aubiere Cedex, France, e-mail: [email protected]
INTRODUCTION Let A = 0n€N An be a noetherian graded ring of finite Krull dimension where AQ is an artinian ring and M = ®n€p,jMn be a graded yl-module of finite type with Krull dimension d. The Hilbert function H(M, -) of M is defined by H(M,n) = £Ao(Mn). The Hilbert-Theorem asserts that ff(M,-) is a polynomial function of degree d — 1 when A is a homogeneous graded ring. Here we are concerned with the case of a not necessarily homogeneous graded ring A. We prove that the Hilbert function H(M,—) and the cumulative Hilbert function H*(M, —) are quasi-polynomial functions and in addition that H*(M,—} is a uniform quasi-polynomial function. Then it is possible to define the multiplicity of a graded module of finite type by the asymptotic formula e^(M) — linin-^oo ^?H*(M,n) as in the homogeneous case. Another point of view is to consider the Hilbert series SHM(T] = ]T}neN #(M,n)Tn of M. In the second section some well-known results concerning SHM(T} and some arithmetic and geometric examples are given. In the last part, we give an extension of the Hilbert-Samuel Theorem to good nitrations. In particular we prove that if A is a noetherian semi local ring, M an A-module of finite type with Krull dimension d and / = (/n) a noetherian filtration on A such that \fl[ = r(A) the Jacobson ideal of A, then the function n ^ ^(M//nM) is a uniform quasi-polynomial function of degree d.