A Note on the Buchsbaum-Rim function of a parameter module

In this article, we prove that the Buchsbaum-Rim function ${\mathrm{\ell }}_A({\mathrm{\S }}_{\nu +1}(F)\mathrm{/}{N}^{\nu +1})$ of a parameter module $N$ in $F$ is bounded above by $e(F\mathrm{/}N)\left(\genfrac{}{}{0px}{}{\nu +d+r-1}{d+r-1}\right)$ for every integer $\nu \ge 0$. Moreover, it turns out that the base ring $A$ is Cohen-Macaulay once the equality holds for some integer $\nu$. As a direct consequence, we observe that the first Buchsbaum-Rim coefficient $e_1(F\mathrm{/}N)$ of a parameter module $N$ is always non-positive.