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

## TL;DR

This paper explores regular sequences in form modules of Noetherian local rings to improve inequalities relating multiplicities and lengths, with applications to classical local Bézout inequalities in algebraic geometry.

## Contribution

It introduces new conditions for regular sequences in form modules to refine existing inequalities involving multiplicities and lengths.

## Key findings

- Improved inequality relating multiplicities and lengths in Noetherian local rings.
- Characterization of regular sequences in form modules via Koszul complex homology.
- Application to classical local Bézout inequality in affine space.

## Abstract

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 $\mathbb{A}^d_k$.