# The Stacks Project

## Tag 01J4

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. The reduced induced scheme structure on $Z$ is the one constructed in Lemma 25.12.4. The reduction $X_{red}$ 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 2096–2106 (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}

Comment #2616 by Harry on July 4, 2017 a 3:00 pm UTC

I guess "underlying closed" should be "underlying set". Another cosmetic suggestion: Change the second sentence to "Let $Z\subset X$ be a closed subset.", which is the form that's used in the subsequent paragraphs.

Comment #2636 by Johan (site) on July 7, 2017 a 12:59 pm UTC

Thanks Harry, fixed here.

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