On the Degree of Hilbert Polynomials of Derived Functors

Given a $d$-dimensional Cohen–Macaulay local ring $(R,\mathfrak m)$, let $I$ be an $\mathfrak{m}$-primary ideal, and let $J$ be a minimal reduction ideal of $I$. If $M$ is a maximal Cohen–Macaulay $R$-module, then, for $n$ large enough and $1\le i\le d$, the lengths of the modules $\operatorname{Ext}^i_R(R/J,M/I^nM)$ and $\operatorname{Tor}_i^R(R/J,M/I^nM)$ are polynomials of degree $d-1$. It is also shown that
$$ \operatorname{deg}\beta_i^R(M/I^nM) =\operatorname{deg}\mu^i_R(M/I^nM)=d-1, $$
where $\beta_i^R(\,\cdot\,)$ and $\mu^i_R(\,\cdot\,)$ are the $i$th Betti number and the $i$th Bass number, respectively.