ABSTRACT
Let A = ⊕n∈ℕ An be a noetherian graded ring of finite Krull dimension where A 0 is an artinian ring and M = ⊕ n∈ℕ Mn be a graded A-module of finite type with Krull dimension d. The Hilbert function H(M, −) of M is defined by https://www.w3.org/1998/Math/MathML"> H ( M , n ) = l A 0 ( M n ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429222641/bd60306a-e05b-46de-85c8-902aafc6d56a/content/eq1109.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> . The Hilbert-Theorem asserts that H(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 https://www.w3.org/1998/Math/MathML"> e A ( M ) = lim n → ∞ d ! n d H * ( M , n ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429222641/bd60306a-e05b-46de-85c8-902aafc6d56a/content/eq1110.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> as in the homogeneous case. Another point of view is to consider the Hilbert series https://www.w3.org/1998/Math/MathML"> S H M ( T ) = ∑ n ∈ ℕ H ( M , n ) T n https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429222641/bd60306a-e05b-46de-85c8-902aafc6d56a/content/eq1111.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> 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 filtrations. 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 f = (In ) a noetherian filtration on A such that https://www.w3.org/1998/Math/MathML"> I 1 = r ( A ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429222641/bd60306a-e05b-46de-85c8-902aafc6d56a/content/eq1112.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> the Jacobson ideal of A, then the function n ↦ ℓA (M/InM) is a uniform quasi-polynomial function of degree d.
