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The Stacks project

Lemma 70.14.5. Let S be a scheme. Let X be an algebraic space over S. Let T \subset |X| be a closed subset and let U \subset X be the open subspace such that T \amalg |U| = |X|. Let \mathcal{F} be a quasi-coherent \mathcal{O}_ X-module. Consider the sheaf of \mathcal{O}_ X-modules \mathcal{F}' which associates to every object \varphi : W \to X of X_{\acute{e}tale} the module

\mathcal{F}'(W) = \{ s \in \mathcal{F}(W) \mid \text{the support of }s\text{ is contained in }|\varphi |^{-1}(T)\}

If U \to X is quasi-compact, then

  1. for W affine there exist a finitely generated ideal I \subset \mathcal{O}_ X(W) such that |\varphi |^{-1}(T) = V(I),

  2. for W and I as in (1) we have \mathcal{F}'(W) = \{ x \in \mathcal{F}(W) \mid I^ nx = 0 \text{ for some } n\} ,

  3. \mathcal{F}' is a quasi-coherent sheaf of \mathcal{O}_ X-modules.

Proof. It is clear that the rule defining \mathcal{F}' gives a subsheaf of \mathcal{F}. Hence we may work étale locally on X to verify the other statements. Thus the lemma reduces to the case of schemes which is Properties, Lemma 28.24.5. \square


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