I have been wondering about the following (and allready posted a similar question, see Dimension of ring completion wrt to a decreasing chain of ideals):

Let $R$ be the ring of formal power series in $n$ indeterminates over $\mathbb{C}$, and let $(I_{k})_{k\in \mathbb{N}}$ be a strictly decreasing chain of unmixed radical ideals, which **all** have the same height $s$. Further assume that $\bigcap I_{n} =: \mathfrak{p}$ is prime and that $I_{1}$ is prime.

Then obviously $ht(\mathfrak{p}) \le s$.

Is it true that $ht(\mathfrak{p}) = s$?