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.

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