Definition 26.12.5. 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 26.12.4. The reduction $X_{red}$ of $X$ is the reduced induced scheme structure on $X$ itself.

Comment #2616 by Harry on

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.

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