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Let $(X,\mathcal O_X)$ be a Noetherian scheme. Let $Z$ be a closed subset of $X$ .

Can we always induce $Z$ with a structure sheaf $\mathcal O_Z$ such that $(Z,\mathcal O_Z)$ becomes a closed subscheme of $(X,\mathcal O_X)$ where the closed immersion is the natural inclusion $i:Z \to X$ ?

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  • $\begingroup$ For any scheme $X$ and a closed subset $Z \subset X$ you can equip $Z$ with the structure of a reduced subscheme. On an affine subset $U$ of $X$ you just take the sheaf $\mathcal{O}_U / \mathcal{I}_{U \cap Z}$, where $\mathcal{I}_{U \cap Z}$ is the radical ideal associated to $Z \cap U$. You can then glue these sheafs together. $\endgroup$ Jul 3, 2019 at 22:15

2 Answers 2

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For any scheme (you don't need noetherian) $X$ and a closed subset $i: Z \hookrightarrow X$ you can equip $Z$ with the structure of a reduced subscheme.

On an affine subset $\DeclareMathOperator{Spec}{Spec} U = \Spec A$ of $X$ you just take the sheaf $\mathcal{O}_{\Spec A/I_{U \cap Z}}$, where $I_{U \cap Z}$ is the radical ideal associated to $Z \cap U$. This makes $U \cap Z$ a closed subscheme of $U$.

You can then glue these sheafs together: If $U$ and $U' = \Spec A'$ are affine open subsets of $X$, cover their intersection by sets that are distinguished open in both $U$ and $U'$. Let $W = \Spec B$ be one of these. Then $\mathcal{O}_{\Spec A/I_{U \cap Z}}\vert_W$ and $\mathcal{O}_{\Spec A'/I_{U' \cap Z}}\vert_W$ both are isomorphic to $\mathcal{O}_{\Spec B/I_{W \cap Z}}$.

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    $\begingroup$ This is known as the reduced induced subscheme structure, if you're looking to search for more information on it. $\endgroup$
    – KReiser
    Jul 3, 2019 at 23:27
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In general you can define several sheaves such that this works (already for affine schemes where you can work very explicitly). If you have a closed subset $Z \subset X = (X, \mathcal{O}_X)$ can define closed subscheme structures by ideal sheaves $\mathcal{I} \subset \mathcal{O}_X$ such that the quotient sheaf $\mathcal{O}_X/\mathcal{I}$ is supported on the image of $Z$ under the inclusion $i \colon Z \rightarrow X$. One restricts the quotient sheaf and gets the structure sheaf on $Z$. A very natural one would maybe the the reduced subscheme structure. As this is a rather quick summary, I would suggest that you just read about that in some textbook.

You can for example try to compute all closed subscheme structures of $Z = 0 \subset \mathbb{A}_2$ such that $h^0(Z, \mathcal{O}_Z) = 2$. You will find three types then if I recall correctly.

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