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Suppose $X$ is a scheme, and $T$ its closet subset. I want to ask that is there a canonical way to obtain a quasi-coherent $\mathscr O_X$-ideal $I$ such that $T=supp \mathscr O_X/I$? This is the inverse question of the construction of closed subschemes. Could you give a counterexample or a proof?

Recently I'm reading algebraic geometry but I don't know why the inverse image of a closed subscheme is also a closed subscheme. So I thought that I may need to solve the above question. Or could you give another way to show "the inverse image of a closed subscheme is also a closed subscheme?" Thank you.

Richard
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The canonical way to obtain a quasi-coherent sheaf of ideals $I$ with $V(I)=T$ is to put the reduced induced subscheme structure on $T$ and then let $I$ be the kernel of $\mathcal{O}_X\to\mathcal{O}_T$. In general, though, there may be many different subscheme structures on $T$, so the collection of quasi-coherent sheaves of ideals $I$ which have $V(I)=T$ as sets can be large.

This is not really relevant to your second paragraph, though - the way we define the preimage of a closed subscheme $T\subset X$ under a map $Y\to X$ is to take the fiber product $Y\times_X T$ (this is the definition as subschemes - if you were only interested in the definition as subsets, you can just use the fact that the map on topological spaces associated to a morphism of schemes is continuous by definition). The proof that the morphism of schemes $Y\times_X T\to Y$ is a closed immersion boils down to checking the affine case where you can see that if $X=\operatorname{Spec} A$, $Y=\operatorname{Spec} B$, and $T=\operatorname{Spec} A/I$, then $Y\times_X T=\operatorname{Spec} B\otimes_A A/I \cong \operatorname{Spec} B/IB$.

KReiser
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  • Thank you very much! The book I read (Iitaka's Algebraic Geometry) doesn't state the inverse image precisely...(actually lots of properties and details are not stated clearly on that book...) As a beginner, I may need to change a book to read. Could you suggest some "clear" books for me? Thanks! – Richard Apr 04 '21 at 01:33
  • I grew up on Hartshorne, though I would not describe it as totally clear. OTOH, since it's so widely used, if you run in to an issue it is likely that someone else has had this problem before and asked about it on the internet. There are other options: see here for some discussion. I think Vakil's text is pretty good, though I haven't taught out of it and I already had a fair bit of AG under my belt when I first read it. – KReiser Apr 04 '21 at 02:03
  • Thank you for your help! I will try it! – Richard Apr 04 '21 at 05:11