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

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 $$\le \dim R -1$$ ?

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 $$\dim (\mathcal{R}/I^m\mathcal{R}) = \dim R$$.