TITLE: The removed statement from the original edition (Matsumura's "Commutative Ring Theory")
QUESTION [8 upvotes]: 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.

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

REPLY [2 votes]: 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.