# Question about the “nullstellensatz” for projective schemes

Yeasterday I asked a question on math stackexchange which simplifies to the following:

Assume that $G$ is a graded ring and $A \subseteq G$ is a homogeneous radical ideal. Is it true that $IV(A) = A$? Here $I(X)$ is the homogeneous ideal generated by homogeneous elements that vanish at every point of $X$ and $V(S)$ is the set of relevent primes that contain $S$.

It is still unanswered. As explained in the math stackexchange question, it suffices to prove the following: If $A \subseteq \mathfrak{p}$ is a prime, then there exists a relevent homogeneous prime $\mathfrak{q}$ such that $A \subseteq \mathfrak{q} \subseteq \mathfrak{p}$. I understand how to deal with the case when $\mathfrak{p}$ does not contain the irrelvent ideal, but I am stuck with the case when $\mathfrak{p}$ does contain the irrelevent ideal. Even in the special case $G = \mathbb{Z}[X,Y]$ I am not sure how to proceed. The primes containing $(X,Y)$ all look like $(p,X,Y)$ for some prime $p$, but I don't know how to "choose a line" which passes through this point like you can in the case of $\mathbb{C}[X,Y]$. I debated whether to ask this question here, but I feel like somthing interesting is going and I just can't see it. I am looking forward to reading some enightened responses!

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The Nullstellensatz is a theorem about ideals in a polynomial ring over an algebraically closed field: The statement you are looking for is not true, even for $G=\mathbb{R}[x,y]$ and $A=(x^2+y^2)$. – J.C. Ottem May 19 '13 at 22:14
@J.C Ottem: I don't understand your example. $A$ is a relevent homogeneous maximal ideal. This means that $V(A) = \{ A \}$ so $IV(A) = A$ – Daniel Barter May 19 '13 at 22:37
I meant that the equality $I(Z(A))=\sqrt{A}$, which is the usual statement of the Nullstellensatz does not hold in this case. – J.C. Ottem May 19 '13 at 22:53
@Ottem: Oh i see what you mean. I definitely what to include the "non line" points in Z(A) – Daniel Barter May 19 '13 at 22:56
You can find "Projective Nullstellensatz" in the wiki entry "Hilbert's Nullstellensatz". Hopefully it will answer your question. – Xiaobo Zhuang May 20 '13 at 15:08