Lemma 31.4.7 (02OM)—The Stacks project

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Table of contents
Part 2: Schemes
Chapter 31: Divisors
Section 31.4: Embedded points
Lemma 31.4.7 (cite)

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Lemma 31.4.7. Let $X$ be a locally Noetherian scheme. Let $\mathcal{F}$ be a coherent $\mathcal{O}_ X$ -module without embedded associated points. Set

$\mathcal{I} = \mathop{\mathrm{Ker}}(\mathcal{O}_ X \longrightarrow \mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(\mathcal{F}, \mathcal{F})).$

This is a coherent sheaf of ideals which defines a closed subscheme $Z \subset X$ without embedded points. Moreover there exists a coherent sheaf $\mathcal{G}$ on $Z$ such that (a) $\mathcal{F} = (Z \to X)_*\mathcal{G}$ , (b) $\mathcal{G}$ has no associated embedded points, and (c) $\text{Supp}(\mathcal{G}) = Z$ (as sets).

Proof. Some of the statements we have seen in the proof of Cohomology of Schemes, Lemma 30.9.7. The others follow from Algebra, Lemma 10.67.4. $\square$

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