Let $k$ be a field, and consider the polynomial ring $k[x_1,...,x_n]$. My question is this:
Given a homogeneous ideal $I\subseteq (x_1,...,x_n)$, does every maximal chain of homogeneous primes $I\subseteq p_0\subseteq\cdots\subseteq (x_1,...,x_n)$ have the same length?
I know the statement to be true if we consider all chains of primes between $I$ and $(x_1,...,x_n)$ and not only homogeneous ones, since the ring is catenary. But I'm not sure what happens in my case. Any help would be appreciated.