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I've just read the definition of a closed subscheme in Hartshorne's recently and I collected here and there (notes that people put online) the following statement.

Claim. Suppose that $(X,\mathcal{O}_X)$ is a scheme, $Y \subset X$ is a closed subset. Then $Y$ can be equipped with a sheaf $\mathcal{O}_Y$ such that $(Y,\mathcal{O}_Y)$ is a reduced closed subscheme of $X$ with the following universal property: for any reduced scheme $Z$ together with a morphism $f: Z \rightarrow X$ such that $f(Z) \subset X$, $f$ factors uniquely through $i: Y \hookrightarrow X$ ($i^{\#}: \mathcal{O}_X \rightarrow i_{\ast}\mathcal{O}_Y$).

Here are my questions (I expect an answer to either (1) or (2) and an answer to (3). Thanks!

1) Can you provide me some good references for the statement of the result, for a proof or a construction of this result?

2) Since I haven't found a satisfactory description of the above fact in popular texts yet, I came up with my own try. I define an ideal sheaf $\mathcal{I}_Y(U) = \{s \in \mathcal{O}_X(U) \mid s_p \in \mathfrak{m}_{X,p} \text{ for all } p \in U \cap Y\}$ on $X$, and I define $\mathcal{O}_Y = i^{-1}(\mathcal{O}_X/\mathcal{I}_Y)$. I expect $\mathcal{O}_Y$ to be the desired sheaf on $Y$. So far, I have managed to show that $(Y,\mathcal{O}_Y)$ is a reduced subscheme of $X$, but I haven't had any idea how to prove the universal property yet. Do you know whether or not the above construction gives the right sheaf in the claim? If yes, could you provide me some idea how to prove the universal property?

3) Suppose that $\mathcal{F}$ is another sheaf on $Y$ that makes $(Y,\mathcal{F})$ a reduced subscheme of $X$, does this imply that $(Y,\mathcal{F})$ also has the universal property stated in the claim?

mr.bigproblem
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1 Answers1

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The best way to address your question might be to split the answer in two parts:

1) Given a closed subset $Y$ of a scheme $X$ (or more precisely of its underlying topological space $|X|$), there is a unique way to endow it with the structure of reduced scheme and with a closed embedding $i: Y \hookrightarrow X$ whose underlying set-theoretic map is the inclusion $|Y| \hookrightarrow |X|$ .
Note that unicity implies that necessarily $\mathcal F=\mathcal O_Y$ in your question 3), whose answer is thus "yes".

2) There is a functorial way to associate to a completely arbitrary morphism of schemes $f: Y\to X$ a morphism of the corresponding reduced schemes $f_{red}: Y_{red}\to X_{red}$ compatible with the two immersions $Y_{red}\hookrightarrow Y$ and $X_{red}\hookrightarrow X$.
It follows that any completely arbitrary morphism $f:Y\to X$ from a reduced scheme $Y$ (not related in any way to $X$) to $X$ can be factored uniquely as a composite $Y\to X_{red}\hookrightarrow X$.
This is a generalization of your claim : the fact that for you $Y$ is a subscheme of $X$ turns out to actually be irrelevant.

These results are proved in EGA I (surprise, surprise!), (5.2.1) and (5.1.5) respectively.

And to end on a pleasant note: your construction of $\mathcal O_Y$ is absolutely correct. Bravo!

Georges Elencwajg
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  • Thanks a lot for the detailed and helpful answer! – mr.bigproblem Apr 28 '13 at 16:23
  • Dear mr.bigproblem, it is my pleasure. – Georges Elencwajg Apr 28 '13 at 17:31
  • @GeorgesElencwajg I was wondering if this follows from your answer 1): Let $X$ be an affine scheme, and $Y$ a closed subset with the reduced induced structure. Then there is a unique closed embedding $Y \to X$ such that the set-theoretic map is the inclusion? [I may not be understanding the sentence in 1) correctly, and I was wondering] – Johnny T. Nov 09 '19 at 09:54
  • Dear @Johnny T. Yes, this is correct. – Georges Elencwajg Nov 10 '19 at 09:10
  • @GeorgesElencwajg Ok, thank you! – Johnny T. Nov 10 '19 at 09:38
  • $\def\n{\operatorname{Nil}} \def\O{\mathcal{O}}$I would like to point out that the construction 2) extends to all ringed spaces. A ringed space is said to be reduced if all stalks of its structure sheaf are reduced (equivalently, all rings of sections over all open sets are reduced). The category of reduced ringed spaces is coreflective in ringed spaces. That is, for a ringed space $X$ there is a ringed space morphism $X_\mathrm{red}\to X$ satisfying the expected universal property. This morphism is a homeomorphism on spaces and the structure sheaf of $X_\mathrm{red}$ is $\O_X/\n\O_X$. – Elías Guisado Villalgordo Oct 27 '23 at 16:34