Chapter 1, Exercise 2.1: A homogeneous form of the Nullstellensatz

Prove the "homogeneous Nullstellensatz," which says that if $\mathfrak a\subseteq S$ is a homogeneous ideal, and if $f\in S$ is a homogeneous polynomial with $\deg f>0$, such that $f(P)=0$ for all $P\in Z(\mathfrak a)$ in $\mathbb P^n$, then $f^q\in\mathfrak a$ for some $q>0$. [Hint: Interpret the problem in terms of the affine $(n+1)$-space whose affine coordinate ring is $S$, and use the usual Nullstellensatz, (1.3A).]

The first case of this problem (a trivial case) is when $Z(\mathfrak a)=\emptyset$ in $\mathbb P^n$ and the condition that $f$ vanishes on $Z(\mathfrak a)$ is vacuous. In this case, in $\mathbb A^{n+1}$ either $Z(\mathfrak a)=\emptyset$ or $Z(\mathfrak a)=\{(0,\dots,0)\}$. In the first case $f$ vanishes on $Z(\mathfrak a)$ vacuously and thus $f^q\in\mathfrak a$ for some $q$. In the second case, since $f$ is homogeneous with $\deg f>0$ we have $f(0,\dots,0)=0$, hence the same holds and we are done.

The nontrivial case occurs when $Z(\mathfrak a)\neq\emptyset$ in $\mathbb P^n$. Interpreting this in $\mathbb A^{n+1}$, if $\mathfrak a$ is homogeneous then we can interpret $Z(\mathfrak a)$ as an affine cone: a union of the lines through the origin representing points through the origin. To prove this, let $P=(a_0,\dots,a_n)\in Z(\mathfrak a)$ be a point. The ideal $\mathfrak a$ has generators that are homogeneous, and thus it suffices to show that any homogeneous polynomial vanishing at $P$ vanishes on such a line. Indeed, since it is homogeneous it vanishes on any multiple $\lambda P$, and thus on the entire line. Hence each line is contained in $Z(\mathfrak a)$. The other inclusion is trivial.

Now, since $f$ vanishes on $Z(\mathfrak a)$ and $f$ is homogeneous, it also vanishes on any affine lines through the origin and thus on the cone $Z(\mathfrak a)\subseteq\mathbb A^{n+1}$. By the ordinary Nullstellensatz, $f^q\in\mathfrak a$ for some $q>0$.