Let $R=\oplus_{n \ge 0} R_n$ be a commutative graded ring. For a prime ideal $P$ of $R$, let $ht P$ denote the usual height of a prime ideal $P$ of $R$. Now let $P$ be a prime ideal of $R$ which is also a homogeneous ideal. Define the homogeneous height of such a homogeneous prime ideal $P$ to be $ht_h P:=\sup\{n \in \mathbb N : P_0 \subset ...\subset P_n=P, $ where $P_i$ s are homogeneous prime ideals of $R\}$ . Now if $R$ is a Noetherian ring, then we know that $ht P <\infty$ for every prime ideal $P$, hence in particular, for every homogeneous prime ideal $P$ of $R$, $ht_h P \le ht P < \infty$ . My question is : For a Noetherian ring , is there any equality kind of relation between $ht P$ and $ht_h P$ for homogeneous prime ideals $P$ ? And what is the relation between usual Krull dimension $\dim R$ and say homogeneous Krull-dimension $\dim_h R :=\sup \{ht_h P : P$ is homogeneous prime ideal of $R\}$ ? I am partiularly interested in the case when $R=k[x_0,...,x_n]$ , graded by the usual homogeneous polynomials, for an algebraically closed field $k$ (so the idea of "homogeneous" height relates to usual Krull-dimension of irreducible projective algebraic sets ) .
Is there any reference in literature for this "homogeneous" height or dimension kind of thing ?