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

Abstract

In this article, we prove that the Buchsbaum-Rim function ${\ell }_A({\S }_{\nu +1}(F)/N^{\nu +1})$ of a parameter module $N$ in $F$ is bounded above by $e(F/N)\left(\genfrac{}{}{0ex}{}{\nu +d+r-1}{d+r-1}\right)$ 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 $e_1(F/N)$ of a parameter module $N$ is always non-positive.

11 pages