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The Stacks project

Lemma 59.73.14. Let X be a quasi-compact and quasi-separated scheme. Let \eta \in X be a generic point of an irreducible component of X.

  1. Let \mathcal{F} be a torsion abelian sheaf on X_{\acute{e}tale} whose stalk \mathcal{F}_{\overline{\eta }} is zero. Then \mathcal{F} = \mathop{\mathrm{colim}}\nolimits \mathcal{F}_ i is a filtered colimit of constructible abelian sheaves \mathcal{F}_ i such that for each i the support of \mathcal{F}_ i is contained in a closed subscheme not containing \eta .

  2. Let \Lambda be a Noetherian ring and \mathcal{F} a sheaf of \Lambda -modules on X_{\acute{e}tale} whose stalk \mathcal{F}_{\overline{\eta }} is zero. Then \mathcal{F} = \mathop{\mathrm{colim}}\nolimits \mathcal{F}_ i is a filtered colimit of constructible sheaves of \Lambda -modules \mathcal{F}_ i such that for each i the support of \mathcal{F}_ i is contained in a closed subscheme not containing \eta .

Proof. Proof of (1). We can write \mathcal{F} = \mathop{\mathrm{colim}}\nolimits _{i \in I} \mathcal{F}_ i with \mathcal{F}_ i constructible abelian by Lemma 59.73.2. Choose i \in I. Since \mathcal{F}|_\eta is zero by assumption, we see that there exists an i'(i) \geq i such that \mathcal{F}_ i|_\eta \to \mathcal{F}_{i'(i)}|_\eta is zero, see Lemma 59.71.8. Then \mathcal{G}_ i = \mathop{\mathrm{Im}}(\mathcal{F}_ i \to \mathcal{F}_{i'(i)}) is a constructible abelian sheaf (Lemma 59.71.6) whose stalk at \eta is zero. Hence the support E_ i of \mathcal{G}_ i is a constructible subset of X not containing \eta . Since \eta is a generic point of an irreducible component of X, we see that \eta \not\in Z_ i = \overline{E_ i} by Topology, Lemma 5.15.15. Define a new directed set I' by using the set I with ordering defined by the rule i_1 is bigger or equal to i_2 if and only if i_1 \geq i'(i_2). Then the sheaves \mathcal{G}_ i form a system over I' with colimit \mathcal{F} and the proof is complete.

The proof in case (2) is exactly the same and we omit it. \square


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