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If $ I $ is a homogeneous ideal of the ring of homogeneous polynomials $ \mathbb {C} [X_0, \dots, X_n] $ , under which conditions on the homogeneous ideal $ I $, and particularly on $ I_m $, the $m$ -th graded piece of $I$ for every $m$, the quotient ring $ \mathbb{C} [X_0, \dots, X_n]/I $ is a regular ring?

According to the following wiki link: https://fr.wikipedia.org/wiki/Anneau_local_r%C3%A9gulier , we find that:

If $ A $ is a regular Noetherian local ring, and if: $ I $ is an ideal of $ A $. Then $ A/I $ is regular if and only if $ I $ is generated by a part of a regular parameter system of $ A $.

But here, for our case, $ \mathbb {C} [X_0, \dots, X_n] $ is not a local ring, which implies that we can not apply this proposition to our case of the quotient ring: $ \mathbb{C} [X_0, \dots, X_n] / I $. What is the solution ? How to answer to my questions in this case?

Thanks in advance for your help.

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$\mathbb{C}[x_1,\dotsc,x_n]/I$ is regular if and only if the affine variety $V(I)$ is smooth. When $I$ is homogeneous,a $V(I)$ is a cone (with vertex at the origin). The only way for a cone to be smooth is if it's a linear subspace. So, for homogeneous $I$, the ring is regular if and only if $I$ is generated by linear forms.

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  • $\begingroup$ Please, is $I$ homogeneous, generated by linear forms if and only if : $I = I^{ \mathrm{lin} } = \{ \ \text{ the ideal generated by the linear terms } f^{ \mathrm{lin} } \text{ of all } f \in I \ \} $ such that : $ f^{ \mathrm{lin} } $ is the linear term of $f$ as the degree one homogeneous polynomial in its expression as a sum of homogeneous polynomials in the variables $x_i$ 's ? Thank you . $\endgroup$ – YoYo Sep 9 '18 at 20:33
  • $\begingroup$ @YoYo: Yes. If $I$ is generated by linear forms $\ell_i$ then generators of $I^{\text{lin}}$ include $\ell_i^{\text{lin}} = \ell_i$. Conversely $I^{\text{lin}}$ is generated by linear forms by definition, so if $I = I^{\text{lin}}$ then $I$ is generated by linear forms, too. $\endgroup$ – Zach Teitler Oct 23 '18 at 7:47

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