Definition 26.12.5 (01J4)—The Stacks project

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Table of contents
Part 2: Schemes
Chapter 26: Schemes
Section 26.12: Reduced schemes
Definition 26.12.5 (cite)

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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.

Comments (2)

Comment #2616 by Harry on July 04, 2017 at 15:00

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 on July 07, 2017 at 12:59

Thanks Harry, fixed here.

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