Question

Hello,

Probably this is a very easy question.

Fix a Noetherian local ring $A$, and an $A$-module of finite type $M$. Lets call a system $ x_1 , \ldots , x_m \in \mathfrak{m}$ $M$-exhausting, if $M / (x_1 M + \ldots + x_m M)$ is of finite length.

Definition: a system of parameters for $M$ is an exhausting system with $dim(M)$ members.

Note that $dim(M)$ itself can be defined as the minimal possible number of elements in an exhausting system.

My question is: Given an exhausting system, such that no subsystem is exhausting, how do we show that it is a system of parameters for $M$. Or maybe there is some subtlety which I don't see and this is not true.

Thank you,

Sasha

Answer 1

Suppose $M=A=k[[y,z]]$, $x_1=yz,\ x_2=y(y+z),\ x_3=z(y+z)$. This is a counterexample.