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.