I'm needing help in this question.
Let $k$ be a field. Consider the $k$-algebra $R:=k[x,y,z,w]/(z+w,xy+xw)$ and define the ring $A$ the localization of $R$ in its maximal ideal $\mathfrak{m} = (\overline{x},\overline{y},\overline{z},\overline{w})$. Prove that a sequence $a_1,\dots,a_n$ of elements of $\mathfrak{m}$ is system of parameters if and only if $a_1,\dots,a_n$ is a regular sequence maximal.
Well, we know that every regular sequence is part of a system of parameters, so if $a_1,\dots,a_n$ is maximal regular sequence, then $a_1,\dots,a_n$ is part of system of parameters, but I'm not getting ace in the hole to solve this problem.
I think the correct approach to prove it is to prove that $A$ is Cohen-Macaulay. Can someone give me some hint?
First of all $$R\simeq k[x,y,z]/(x(y-z)).$$ This is clearly a Cohen-Macaulay ring, and $k[x,y,z]_{(x,y,z)}/(x(y-z))$ is a Cohen-Macaulay local ring. But in such a ring a system of parameters is a regular sequence.