# Does every closed subset of the underlying space of a Noetherian scheme admits a natural closed subscheme structure?

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 $$i:Z \to X$$ ?

• 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. – Carlos Esparza Jul 3 '19 at 22:15

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 $$\mathcal{O}_{\Spec B/I_{W \cap Z}}$$.
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.