The Stacks project

Lemma 68.13.3 (Artin-Rees). Let $S$ be a scheme. Let $X$ be a Noetherian algebraic space over $S$. 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. Choose an affine scheme $U$ and a surjective ├ętale morphism $U \to X$ (see Properties of Spaces, Lemma 65.6.3). Then $U$ is a Noetherian scheme (by Morphisms of Spaces, Lemma 66.23.5). The equality of the lemma holds if and only if it holds after restricting to $U$. Hence the result follows from the case of schemes, see Cohomology of Schemes, Lemma 30.10.3. $\square$


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