Equidimensionality of stalks of $\operatorname{Proj} S$ when $S$ is equidimensional.

Question

I would like to know a reference of the following statement (or counter example).

Let $S$ be a (commutative) Noetherian standard graded ring over a local ring, i.e., $S = S_0[S_1]$, where $S_0$ is Noetherian local and $S_1$ is finitely generated over $S_0$. If $S$ is an integral domain, then $X = \operatorname{Proj} S$ is a irreducible and reduced. Hence $\mathcal{O}_{X,x}$ is an integral domain for any $x \in X$. But what about equidimensionality? That is

Let $S$ be a Noetherian standard graded equidimensional ring over a Noetherian local ring. Then is the local ring $\mathcal{O}_{X,x}$ equidimensional for all $x \in X$?

A ring $R$ is called $\textit{equidimensional}$ if $\dim R = \dim R/p$ for any minimal prime $p$ of $R$.

Answer 1

Here is a counter example. Set $S_0 = k[x]$ where $k$ is a field. Let $S = S_0[Y, Z, W]/(xY, xZ)$ with $Y$, $Z$, $W$ in degree $1$. Then $X = \text{Proj}(S)$ is the union of a copy of $\text{Spec}(S_0) \cong \mathbf{A}^1_k$ and a copy of $\mathbf{P}^2_k$ glued in a point $x$. Hence $\mathcal{O}_{X, x}$ is not equidimensional.