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 $\au = a_1,\ldots,a_d$ denote a system of parameters of $M$ such that $a_i\in {\mathfrak{q}}^{c_i}\setminus {\mathfrak{q}}^{c_i+1}$ for $i=1,\dots ,d$. It follows that $ χ:= e_0(\au;M) - c \cdot e_0(\mathfrak{q};M) \geq 0$, where $c=c_1\cdot \dots \cdot c_d$. 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 $e_0(\au;M)$ and applications to Bezout numbers.

11 pages, to appear Springer INdAM-Series, Vol. 20 (2017)