I'm studying these notes, in particular page 17 theorem 1. I have some problems in the implication iii $\Rightarrow$ ii:
Let $A$ be a local ring of dimension $n$ and maximal ideal $m$
ii) $A$ is Cohen-Macaulay and there exists a system of parameter generating an irreducible ideal (irreducible means I cannot see it as an intersection of 2 different ideals)
iii) every system of parameters generates an irreducible ideal.
Here it is how the proof goes:
Induction on $\mathrm{dim}\;A$. If it is 0 we are ok. Suppose $\mathrm{dim}\;A=n>0$, we want to prove $\mathrm{depth}\;A\geq1$. Suppose $\underline{f}=(f_1,\ldots,f_n)$ is a system of parameters. Define $\underline{f}_r=(f_1^r,\ldots,f_n^r)$, it is another system of parameters. We have
$\mathrm{Hom}_A(k,\underline{f}_r/\underline{f}_{r+1})\neq0$ Why?
then we have:
$\mathrm{Hom}_A(k,A/\underline{f}_{r+1})$ is a $k$-vectorspace of dimension 1, Why?
this should imply
$\underline{f}_{r+1}:m\subset\underline{f}_r$ for every $r$, why?
And so $0:m\subset\bigcap_r(\underline{f}_{r+1}:m)\subset\bigcap_r\underline{f}_r=0$.
And now we can easily conclude.
where did we use the hypothesis that the $\underline{f}_r$ are irreducible?
Could you help me in solving these 4 doubts that I have?