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?
  • 1
    $\begingroup$ Why do you assume that the statement was removed because of a mistake? $\endgroup$ Apr 12, 2022 at 8:14
  • $\begingroup$ I’m not sure about that yet, but I think the statement need not be removed if there is no mistake. $\endgroup$
    – Stsn Y
    Apr 13, 2022 at 12:35

1 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.


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