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This is a follow up to my previous question: Zero subscheme of a section: Making computations.

I want to show that every closed subscheme of $\mathbb{P}^n_R$ when $R$ is Noetherian is of the form $V(s)$ (i.e. the vanishing scheme of a section of a vector bundle). The description $\mathbb{P}^n_R$ we have in our algebraic geomtry course is through explicit gluing, so I have no tools regarding graded ideals and graded modules.

Here is what I tried ( I think it is wrong) : Let $Z$ denote the closed the closed subscheme and $\mathcal{O}_X \to i_{*}\mathcal{O}_Z$ the associated surjection. Since $i$ is a closed embedding $i_{*}\mathcal{O}_Z$ is a quasi-coherent sheaf on $X$. Therefore its kernel is also quasi-coherent. Since $R$ is noetherian this kernel is of finite type. Let us denote the kernel by $\mathscr{I}$.

I know that there exists some integer such that $\mathscr{I}(m)$ is generated by global sections and those global sections can be identified with some $f_i \in \mathcal{O}_{\mathbb{P}^n_R}(m)(\mathbb{P}^n_R)$ homogeneous polynomials of degree $m$. I would like to say that the vanishing subscheme of that family $\{f_i\}_{i=0}^k$ is exactly $Z$.

If that were true every closed subscheme would be cut out by polynomials of the same degree and that does not make sense to me at all. How can I solve this? Furthermore, how can I show that once I get this right that closed subscheme is going to satisfy the pullback property I stated in the other question?

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Abellan
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  • Any closed subscheme is defined by finitely many homogeneous polynomials and further we may assume they have the same degree. So, if $F_i, 1\leq i\leq m$ are these polynomials, the same scheme id defined as the zeroes of the single polynomial $\sum F_i^2$ over the reals. – Mohan Dec 04 '16 at 20:59
  • So are the polynomials that I found those defining the ideal? Because in that case i can take the section to be $(f_1,\dots,f_k) \in \mathcal{O}^k(m)(\mathbb{P}^n_R)$ and take that vanishing subscheme right? – Abellan Dec 04 '16 at 21:05
  • To clarify your confusion, the scheme defined by $f=0$, a homogeneous polynomial and that defined by $x_if=0, 0\leq i\leq n$ are exactly the same. – Mohan Dec 05 '16 at 01:14

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