Lemma 30.10.3 (01YA): Artin-Rees

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Table of contents
Part 2: Schemes
Chapter 30: Cohomology of Schemes
Section 30.10: Coherent sheaves on Noetherian schemes
Lemma 30.10.3: Artin-Rees (cite)

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Lemma 30.10.3 (Artin-Rees). Let $X$ be a Noetherian scheme. Let $\mathcal{F}$ be a coherent sheaf on $X$. Let $\mathcal{G} \subset \mathcal{F}$ be a quasi-coherent subsheaf. Let $\mathcal{I} \subset \mathcal{O}_ X$ be a quasi-coherent sheaf of ideals. Then there exists a $c \geq 0$ such that for all $n \geq c$ we have

\[ \mathcal{I}^{n - c}(\mathcal{I}^ c\mathcal{F} \cap \mathcal{G}) = \mathcal{I}^ n\mathcal{F} \cap \mathcal{G} \]

Proof. This follows immediately from Algebra, Lemma 10.51.2 because $X$ has a finite covering by spectra of Noetherian rings. $\square$

Comments (1)

Comment #943 by correction_bot on August 23, 2014 at 02:31

In the statement of the lemma, it should be $\mathcal{I}^n\mathcal{F} \cap \mathcal{G}$ not $\mathcal{I}^n\mathcal{F}$ .

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