Hilbert function and numerical polynomial

Question

Let $R=\bigoplus_{ n\in\mathbb N}R_{ n}$ be a Noetherian standard ring defined over an Artinian local ring. Let $M=\bigoplus_{ n\in\mathbb N}M_{ n}$ be an $\mathbb N$-graded $R$-module (not necessarily finite) with $\lambda_{R_{ 0}}(M_{ n})<\infty.$

Question: Does there exist a numerical polynomial $P$ such that $P( n)=\lambda_{R_{ 0}}(M_{ n})$ for all large $n$? (The module is not finite.)

I am not sure whether this question is suitable for Mathoverflow or not. But please at least give some hint or link.

Answer 1

This is obviously false. For example, let $R = k[x]$ (where $x$ has degree $1$). Write $N(d)_m := N_{m+d}$. Then consider $M=\bigoplus_{n \geq 0} R(-n)^{d_n}$ (for some $d_0, d_1, \ldots \in \mathbb Z_{\geq 0}$). We have $$M_m = \bigoplus_{n \geq 0} (R_{m-n})^{d_n} \cong \bigoplus_{n = 0}^m k^{d_n},$$ which is finite-dimensional of dimension $\sum_{n=0}^m d_n$ (and $0$ if $m < 0$, which you require). This function is generally not (eventually) a polynomial; in fact we can get it to be any monotone increasing function $\mathbb Z_{\geq 0} \to \mathbb Z_{\geq 0}$.