On the asymptotic behaviour of Hilbert functions of graded rings and modules: applications to the Hilbert–Samuel functions of well filtered modules, multiplicity and analytic spread of nice filtrations

Abstract

This paper is devoted to the study of the asymptotic behaviour of Hilbert functions on finitely generated graded modules and particularly to some investigations on the asymptotic behaviour of Hilbert–Samuel functions on finitely generated well filtered modules. Consider a graded noetherian ring $A=\bigoplus _{n\in \mathbb {N}}A_{n}$ which may be assumed to be of the form $A=A_{0}[x_{1},\ldots ,x_{s}],$ where each element $x_{i}$ is homogeneous and deg $x_{i}=$$k_{i}\ge 1$ for all i. Suppose that the ring $A_{0}$ is artinian. Let M$=\bigoplus _{n\in \mathbb {Z}}M_{n}$ be a finitely generated positively $\mathbb {Z}$-graded A-module. Then the length$\ l_{A_{0}}(M_{n})$ of the $A_{0}$-module $M_{n}$ is finite for all $n\in \mathbb {Z}$. The Hilbert function $H(M,-)$ of M is defined as$\ H(M,n)=$$\ l_{A_{0}}(M_{n})$ for all $n\in \mathbb {Z}$. By a well known Hilbert Theorem, if $k_{i}=1$ for all i,  then there exists a unique polynomial $P(X)\in \mathbb {Q}[X]$ with $\deg$$P(X)=d-1$ such that $H(M,n)=$P(n) for all large integers n,  where d denotes the Krull dimension of the A-module M. But if at least one of the homogeneous generators $x_{i}$ has degree $k_{i}$$\ge 2,$ then such a polynomial need not exist. Dichi and Sangaré (J Pure Appl Algebra 138:205–213, 1999) had shown in that case that the Hilbert function $H(M,-)$ of M has a good asymptotic behaviour. More precisely, they had shown that $H(M,-)$ is a quasi-polynomial function. Then this result is applied, with some minor but necessary adaptation, to the graded $G_{f}(A)$-module $G_{\Phi }(M)=\bigoplus _{n\text { }\ge \text { }0}\frac{M_{n}}{M_{n+1}}$, where $f=(I_{n})_{n\text { }\in \text { }\mathbb {Z}}$ is a nice filtration of the ring A and where M is a finitely generated A-module which is no longer graded but which is endowed with a filtration $\Phi =(M_{n})_{n\text { }\in \text { }\mathbb {Z}}$ compatible with the filtration f. Here $G_{f}(A)=\bigoplus _{n\text { }\ge \text { }0}\frac{I_{n}}{I_{n+1}}$ is the graded ring associated with f. This leads in particular to the definition of a concept of multiplicity $e_{f}(M)$ of M with respect to the filtration f which is a rational number. If the filtration f is non I-adic, then its multiplicity $e_{f}(M)$ need not be an integer unlikely for the multiplicity of an ideal. So it does not have any geometrical interpretation. However, it is shown in Dichi and Sangaré (1999) that, for a given noetherian filtration $f=(I_{n})_{n\text { }\in \text { }\mathbb {Z}}$ on the noetherian finite dimensional ring A, if $l_{A}(\frac{M}{I_{1}M})<+\infty ,$ then the multiplicity function $M\longmapsto e_{f}$ (M) behaves well on short exact sequences of finitely generated A-modules. As a second application of the Hilbert Theorem, we introduce a satisfactory concept of analytic spread for filtrations and we show that this analytic spread is of “asymptotic nature” for nice filtrations.