The Stacks project

Lemma 59.74.2. Let $X$ be a Noetherian scheme. Let $\Lambda $ be a Noetherian ring. Consider inclusions

\[ \mathcal{F}_1 \subset \mathcal{F}_2 \subset \mathcal{F}_3 \subset \ldots \subset \mathcal{F} \]

in the category of sheaves of sets, abelian groups, or $\Lambda $-modules. If $\mathcal{F}$ is constructible, then for some $n$ we have $\mathcal{F}_ n = \mathcal{F}_{n + 1} = \mathcal{F}_{n + 2} = \ldots $.

Proof. By Proposition 59.74.1 we see that $\mathcal{F}_ i$ and $\mathop{\mathrm{colim}}\nolimits \mathcal{F}_ i$ are constructible. Then the lemma follows from Lemma 59.71.8. $\square$

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