Let $\mathfrak{ q}$ denote an ideal in a Noetherian local ring $(A,\mathfrak{ m})$. Let $\underline{ a}=a_1,\ldots,a_d \subset \mathfrak{ q}$ denote a system of parameters in a finitely generated $A$-module $M$. This note investigate an improvement of the inequality $c_1\cdot \ldots \cdot c_d \cdot e_0(\mathfrak{ q};M) \leq \ell_A(M/\underline{ a}\,M)$, where $c_i$ denote the initial degrees of $a_i$ in the form ring $G_A(\mathfrak{ q})$. To this end, there is an investigation of regular sequences in the form module $G_M(\mathfrak{ q})$ by homology of a factor complex of the Koszul complex. In a particular case, there is a discussion of classical local B\'ezout inequality in the affine $d$-space $\mathbb{ A}^d_k$.
Categories: math.AC