About multiplicities and applications to Bezout numbers

Let $(A,\mathfrak{m},\mathbb{k})$ denote a local Noetherian ring and $\mathfrak{q}$ an ideal such that ${\mathrm{\ell }}_A(M\mathrm{/}\mathfrak{q}M)<\mathrm{\infty }$ for a finitely generated $A$-module $M$. Let $\text{au}=a_1,\dots ,a_d$ denote a system of parameters of $M$ such that $a_i \in \mathfrak{q}^{c_i} \setminus \mathfrak{q}^{c_i+1}$for$i=1,\ldots,d$\mathrm{.}Itfollowsthat$ \chi := e_0(\au;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 $e_0(\text{au};M)$ and applications to Bezout numbers.