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In the polynomial ring $\mathbb{C}[x,y,z]$, I should prove that the ideal $$I=(\underbrace{x^4-y^3}_{=:\,p_1},\underbrace{x^5-z^3}_{=:\,p_2},\underbrace{y^5-z^4}_{=:\,p_3})$$ can not be generated by two elements.

I showed that $I\neq (p_1,p_2)$, $I\neq (p_1,p_3)$, $I\neq (p_2,p_3)$. Is it enough to conclude? If I was working with linear subspaces it would be, but I don't know if things works in the same way with ideals.

qwertyuio
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1 Answers1

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Consider $R=\mathbb C[X,Y,Z]$ as a graded ring by assigning to the indeterminates the following degrees: $\deg X=3$, $\deg Y=4$, $\deg Z=5$. (This way the ideal $I$ becomes homogeneous.) If $\mathfrak m=(X,Y,Z)$, then the minimal number of generators of $I$ is $\dim_{\mathbb C}I/\mathfrak mI$. Can you show that the residue classes of $p_1,p_2,p_3$ modulo $\mathfrak mI$ are linearly independent over $\mathbb C$?

  • Thank you for your answer. I don't know how to compute those residue classes. And why one needs to work with a homogeneous ideal? – qwertyuio Sep 04 '13 at 08:23
  • If the ideal isn't homogeneous I don't think this claim is necessarily true: "the minimal number of generators of $I$ is $\dim_{\mathbb C}I/\mathfrak mI$" since in this case $\mathfrak m$ is a maximal ideal among others, but when we introduce a grading then $\mathfrak m$ becomes the maximal homogeneous ideal of $R$. (You don't need to compute some residue classes, only to prove that them are linearly independent which is easy enough if take into account the grading.) –  Sep 04 '13 at 15:20