# Exercise 4.5.H in Vakil

Suppose $$I$$ is any homogeneous ideal of $$S_•$$ contained in $$S_+$$, and $$f$$ is a homogeneous element of positive degree. Show that $$f$$ vanishes on $$V(I)$$ (i.e., $$V(I) ⊂ V(f)$$) if and only if $$f^n ∈ I$$ for some $$n$$. (Hint: Mimic the affine case; see Exercise 3.4.J.)
It seems like one cannot exactly mimic the affine case without knowing something like "for a homogenus ideal $$I$$, $$\sqrt{I}$$ is the intersection of all homogeneous primes that contain it." (EDIT: Maybe a better way of saying this is that minimal primes of a homogeneous ideal are homogeneous. )But this statement has not been mentioned so far in Vakil, so I am wondering if there is an easier way to do it.
• Whatever you are saying is in fact true. If you have a homogeneous ideal $I$, then $I$ is the intersection of homogeneous prime ideals containing $I$. To see this, show that $p\in \text{Spec }S$, then $p^h=\langle x\in p | x \text{ is homogeneous} \rangle$ is in fact a homogeneous prime ideal. – Ignorant Mathematician May 23 at 23:31