Vakil's 4.5.H reads as follows

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.

A similar question has been asked here, but that seemed to use a lot of results that had not been developed in Vakil (yet).

  • 1
    $\begingroup$ One minor typographical thing: your quoted section from Vakil seems to have lost its math formatting. Secondly, in Vakil's introduction, he says that "[t]he reader should be familiar with some basic notions in commutative ring theory, in particular the notion of ideals ... and localization." I think the answer to the linked question is that it exactly uses these ideas, which Vakil says one should already be somewhat familiar with (and thus he doesn't develop these ideas in the text). If this is unsatisfying, what level of technology would suffice for "easier"? $\endgroup$ – KReiser May 23 at 22:09
  • $\begingroup$ 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. $\endgroup$ – Ignorant Mathematician May 23 at 23:31

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.