The removed statement from the original edition (Matsumura's "Commutative Ring Theory")

Question

From Theorem 13.7 ii) in the original edition (written in Japanese) of Matsumura's "Commutative Ring Theory," the following statement is got rid of:

Let $A = \bigoplus_{n \geq 0}A_n$ be a noetherian graded ring. If $P \subset A$ is a homogeneous prime ideal of height $r$, then there exists an ideal $I = (b_1, \ldots, b_r)$ generated by $r$ homogeneous elements $b_i$ such that $P$ is a minimal prime divisor of $I$.

The proof is written as follows:

The case $r = 0$ (hence $I = (0)$) is no problem. Assume $r > 0$. By Theorem 13.6, we can choose $a_1, \ldots, a_r \in P$ such that $P$ is a minimal prime divisor of $(a_1, \ldots, a_r)$. Put $J = (a_1, \ldots, a_r)$, and let $a_{ij}$ be the homogeneous term of $a_i$ of degree $j$. Since $JA_P$ is generated by $\{a_{ij}\}_{i, j}$, we can take a minimal basis from $\{a_{ij}\}$. Hence there exist $r$ homogeneuous elements $b_1, \ldots, b_r \in P$ with $JA_P = (b_1, \ldots, b_r)A_P$. Then $P$ is a minimal prime divisor of $(b_1, \ldots, b_r)A$.

Here is my question.

1. Where are the mistakes in the proof above?
2. What is a counterexample against the removed statement?

Answer 1

I think this statement is wrong, because, e.g., a general $\mathbb{N}_0$-graded Noetherian graded local ring $A$ (some texts say $*$-local) does not admit a full homogeneous system of parameters necessarily, unless $A_0$ is a field. In the case where $A_0$ is field, the existence of a full homogeneous system of parameters follows from the graded version of the prime avoidance (see, e.g. [1, Proposition 5.2.]), which it is apparent from the statement that this prime avoidance needs the assumption that the ideal is generated in positive degrees. A general example is many Rees algebras $R[\mathfrak{a}t]$, where $\mathfrak{a}$ is an ideal of a Noetherian local ring $R$, which do not admit a full homogeneous system of parameters. For a general full system of parameters of a Rees algebra, which necessarily is inhomogeneous, see [2, Proposition 3.1].

(By a full homogeneous system of parameters for an $\mathbb{N}_0$-graded ring with a homogeneous maximal ideal $\mathfrak{m}$ I mean a sequence of homogeneous elements of length $\text{height}(\mathfrak{m})$ whose generated ideal is $\mathfrak{m}$-primary).

[1]: T. Marely, GRADED RINGS AND MODULES, https://math.unl.edu/faculty/Marley/905notes.pdf.

[2]: W. Vasconcelos, Integral Closure Rees Algebras, Multiplicities, algorithms, https://link.springer.com/book/10.1007/b137713.