Lemma 57.73.5. Let $X$ be a Noetherian scheme. Let $E \subset X$ be a subset closed under specialization.

Let $\mathcal{F}$ be a torsion abelian sheaf on $X_{\acute{e}tale}$ whose support is contained in $E$. 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 subset contained in $E$.

Let $\Lambda $ be a Noetherian ring and $\mathcal{F}$ a sheaf of $\Lambda $-modules on $X_{\acute{e}tale}$ whose support is contained in $E$. 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 subset contained in $E$.

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