Let be a local Noetherian ring and let be a system of parameters, i.e. . Then
I know just a few basic facts about dimension theory. I think I can prove the inequality via Krull's Hauptidealsatz in this way: the maximal ideal of is
So it must be .
But how to prove the other inequality? I think I should do it by induction, but I cannot understand how to begin. So, if what I said so far is right, my question is: how can I prove that