## Tag `01J4`

Chapter 25: Schemes > Section 25.12: Reduced schemes

Definition 25.12.5. Let $X$ be a scheme. Let $Z \subset X$ be a closed subset. A

scheme structure on $Z$is given by a closed subscheme $Z'$ of $X$ whose underlying set is equal to $Z$. We often say ''let $(Z, \mathcal{O}_Z)$ be a scheme structure on $Z$'' to indicate this. Thereduced induced scheme structureon $Z$ is the one constructed in Lemma 25.12.4. Thereduction $X_{red}$ of $X$is the reduced induced scheme structure on $X$ itself.

The code snippet corresponding to this tag is a part of the file `schemes.tex` and is located in lines 2095–2105 (see updates for more information).

```
\begin{definition}
\label{definition-reduced-induced-scheme}
Let $X$ be a scheme. Let $Z \subset X$ be a closed subset.
A {\it scheme structure on $Z$} is given by a closed subscheme $Z'$ of
$X$ whose underlying set is equal to $Z$. We often say
``let $(Z, \mathcal{O}_Z)$ be a scheme structure on $Z$'' to
indicate this. The {\it reduced induced scheme structure}
on $Z$ is the one constructed in Lemma \ref{lemma-reduced-closed-subscheme}.
The {\it reduction $X_{red}$ of $X$} is the reduced induced scheme
structure on $X$ itself.
\end{definition}
```

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