Is every minimal set of generators for a homogeneous ideal composed by homogeneous elements?

Question

An ideal $\mathfrak a$ of a graded ring $A$ is said to be homogeneous if I can find a set of homogeneous generators for $\mathfrak a$. Is it true that every minimal set of generators for a homogeneous ideal $\mathfrak a$ is composed by homogeneous elements?

Thanks in advance

As you stated the problem, I believe the answer is no. Suppose you have an ideal minimally generated by an element $a$ of weight $2$ and an element $b$ of weight $3$. Then it is also minimally generated by $a+ b$ and $b$. — Andreas Caranti, May 13 '14 at 13:27

score 7 · Accepted Answer · answered May 13 '14 at 13:26

7

The answer is no. Consider the following homogenous ideal of $\mathbb{C}[x,y,z]$ :

$$ I = (x^2+y^2+z^2, xyz, z^5).$$

This is also equal to $ I = (x^2+y^2+z^2, xyz+z^5, xyz- z^5).$

answered May 13 '14 at 13:26

Ragib Zaman

36,065

What if we enforce that $I$ is homogeneous and radical, must every minimal set of generators be homogeneous? – Ben Jun 01 '20 at 13:01
@doremifasolatido I would recommend asking a new question for that. – Ragib Zaman Jun 01 '20 at 16:08

Is every minimal set of generators for a homogeneous ideal composed by homogeneous elements?

1 Answers1

Linked