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



systems of parameters vs. minimal "exhausting" systems in a Noetherian local ring 
Suppose $M=A=k[[y,z]]$, $x\_1=yz,\ x\_2=y(y+z),\ x\_3=z(y+z)$. This is a counterexample.


