On the degree of the Hilbert polynomial of a graded module over the Rees algebra

Question

If $A=\oplus_{n=0}^\infty A_n$ is a Noetherian graded ring of finite dimension such that $A_0$ is local and $A=A_0[A_1]$, and if $M=\oplus_{n=0}^\infty M_n$ is a finitely generated graded $A$-module such that each $M_n$ has finite length as an $A_0$-module, then it is well known that the function $n \mapsto \lambda_{A_0}(M_n)$ is eventually of polynomial type of degree$=\dim M-1 \le \dim A -1$ (Bruns&Herzog, Cohen-Macaulay rings, Theorem 4.1.3).

Now let $I$ be an ideal of a Noetherian local ring $(R, \mathfrak m)$ of dimension $d>0$. Assume $I$ is $\mathfrak m$-primary. Consider the Rees Algebra $\mathcal R_I:=\oplus_{n=0}^\infty I^n$ which satisfies the conditions of a graded ring as defined in the first paragraph. Note that the degree zero piece of $\mathcal R_I$ is $R$ and $\dim \mathcal R_I=d+1$. Now let $E=\oplus_{n=0}^\infty E_n$ be a finitely generated graded $\mathcal R_I$-module such that $\lambda_R(E_n)<\infty, \forall n\ge 0$. Then by the discussions in the first paragraph, it follows that $n \mapsto \lambda_R(E_n)$ is a function eventually of polynomial type of degree $=\dim E -1\le \dim \mathcal R_I-1=d=\dim R$.

My question is: Can we actually show that the function $n \mapsto \lambda_R(E_n)$ is eventually of polynomial type of degree $\le \dim R -1 $ ?

Answer 1

The map $(\mathcal{R_I})_1 \times E_n \to E_{n+1}$ is $R$-linear for all $n$ and surjective for all $n \gg 0$. Since $E_n$ is a finite-length $R$-module for every $n$, it follows that there exist $m$ such that $I^mE_n = 0$ for each $n$. Therefore $E$ is a finitely generated $\mathcal{R}/I^m\mathcal{R}$-module. Note that $\dim (\mathcal{R}/I^m\mathcal{R}) = \dim R$.