How do maximal homogeneous ideals look like in polynomial ring?

Question

I would like to show that the homogeneous ideal of any point $x$ in projective space $\mathbb{P}^n(k)$, where $k$ is an algebraic closed field, is a maximal homogeneous ideal in the polynomial ring $k[X_0, \dots, X_n]$.

By maximal homogeneous ideal I mean a homogeneous ideal in the polynomial ring that is properly included in the irrelevant ideal $(X_0, \dots, X_n)$, and that is maximal among all homogeneous ideals that also are properly included in the irrelevant ideal.

I have already found, that if $x=(x_0: \dots: x_n)$ with $x_i = 1$, I can write its homogeneous ideal as $(X_0 - x_0 X_i, \dots, X_n - x_n X_i)$. I think the next step should be to show that this is indeed a maximal homogeneous ideal, but I just don't know how to do this, and I would be grateful for some help. Is it possible to check it directly by showing that any homogeneous ideal containing properly $(X_0 - x_0 X_i, \dots, X_n - x_n X_i)$ has to be the the irrelevant ideal $(X_0, \dots, X_n)$? Is there some general criterion for checking this kind of maximality, similar to the maximal ideals ($R/I$ is a field $\Leftrightarrow $ I is a maximal ideal in the ring $R$)?

Also, I was wondering whether, given any homogeneous ideal properly included in the irrelevant ideal, there always exists a maximal homogeneous ideal containing it. Since we have this with ideals, I was thinking it was in this situation also true, and started proving this using Zorn's Lemma, but got stuck at some point because of some technical problem. Is this statement true?

Thanks very much in a advance!

Answer 1

The maximal homogeneous relevant ideals of $k[X_0, \dots, X_n]$ are exactly the ideals of the form $$\mathfrak m_a=\langle a_iX_j-a_jX_i\vert i,j=0\cdots n\rangle $$ where $a=[a_0:a_1:\cdots:a_n]\in \mathbb P^n(k) $.
The correspondence $a \leftrightarrow \mathfrak m_a$ is the link between the classical points of projective space $\mathbb P^n(k)$ and the closed points of projective space seen as the scheme $\mathbb P^n_k=\operatorname {Proj} (k[X_0, \dots, X_n])$.

Answer 2

Thanks very much for your help and your answers! However I am still confused, because it seems to me that the ideal $m_a = (a_iX_j - a_jX_i \vert i,j=0, \dots,n)$ is the same as $(X_0 - \tilde{a}_0 X_i, \dots, X_n - \tilde{a}_n X_i)$, supposing $$a = (a_0:\dots:a_n) = (\tilde{a}_0:\dots:\tilde{a}_{i-1}:1:\tilde{a}_{i+1}:\dots:\tilde{a}_n) \in \mathbb{P}^n(k),\ \tilde{a}_i=1.$$ On the other hand we have $k[X_0, \dots, X_n]/(X_0 - \tilde{a}_0 X_i, \dots, X_n - \tilde{a}_n X_i) \cong k[X_i]$, and so by the remark of user26857, $(X_0 - \tilde{a}_0 X_i, \dots, X_n - \tilde{a}_n X_i)=m_a$ cannot be a maximal homogeneous ideal. Am I missing something? Maybe we don't use the same definition of maximal homogeneous ideal? In any case this correspondence is what I was looking for. I wanted to obtain a subset of $\mathrm{Proj} \, k[X_0, \dots, X_n]$, that is isomorphic to $\mathbb{P}^n(k)$ as a locally ringed space. Because in the affine case this subset is the set $\mathrm{Spm} \,k[X_1, \dots, X_n]$ of all maximal ideals of $k[X_1, \dots, X_n]$, I thought it would be similar in the projective case if I adapted the definition of maximality, hence if I considered the subset of all maximal homogeneous ideals in $k[X_0, \dots, X_n]$.