On the Buchsbaum-Rim function of a parameter module

In this article, we prove that the Buchsbaum-Rim function ℓA(Sν+1(F)/Nν+1) of a parameter module N in F is bounded above by e(F/N)(ν+d+r−1d+r−1) for every integer ν⩾0. Moreover, it turns out that the base ring A is Cohen-Macaulay once the equality holds for some integer ν. As a direct consequence, we observe that the first Buchsbaum-Rim coefficient e1(F/N) of a parameter module N is always non-positive.