On Regular Sequences in the Form Module with Applications to Local B\'ezout Inequalities

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 $\mathbb{A}^d_k$.