About Multiplicities and Applications to Bezout Numbers

Abstract

Let $(A,\mathfrak{m}, \mathbb{k})$ denote a local Noetherian ring and $\mathfrak{q}$ an ideal such that $\ell_{A}(M/\mathfrak{q}M) <\infty$ for a finitely generated $A$-module $M$. Let $\underline{a} = a_{1},\ldots,a_{d}$ denote a system of parameters of $M$ such that $a_{i} \in \mathfrak{q}^{c_{i}}\setminus \mathfrak{q}^{c_{i}+1}$ for $i = 1,\ldots,d$. It follows that $\chi:= e_{0}(\underline{a};M) - c \cdot e_{0}(\mathfrak{q};M) \geq 0$, where $c = c_{1} \cdot \ldots \cdot c_{d}$. The main results of the report are a discussion when $\chi = 0$ resp. to describe the value of $\chi$ in some particular cases. Applications concern results on the multiplicity $e_{0}(\underline{a};M)$ and applications to Bezout numbers.