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 $X$.

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 $.

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 $.

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