
Lemma 54.70.8. Let $X$ be a quasi-compact and quasi-separated scheme.

1. Let $\mathcal{F} \to \mathcal{G}$ be a map of constructible sheaves of sets on $X_{\acute{e}tale}$. Then the set of points $x \in X$ where $\mathcal{F}_{\overline{x}} \to \mathcal{G}_{\overline{x}}$ is surjective, resp. injective, resp. is isomorphic to a given map of sets, is constructible in $X$.

2. Let $\mathcal{F}$ be a constructible abelian sheaf on $X_{\acute{e}tale}$. The support of $\mathcal{F}$ is constructible.

3. Let $\Lambda$ be a Noetherian ring. Let $\mathcal{F}$ be a constructible sheaf of $\Lambda$-modules on $X_{\acute{e}tale}$. The support of $\mathcal{F}$ is constructible.

Proof. Proof of (1). Let $X = \coprod X_ i$ be a partition of $X$ by locally closed constructible subschemes such that both $\mathcal{F}$ and $\mathcal{G}$ are finite locally constant over the parts (use Lemma 54.70.2 for both $\mathcal{F}$ and $\mathcal{G}$ and choose a common refinement). Then apply Lemma 54.63.5 to the restriction of the map to each part.

The proof of (2) and (3) is omitted. $\square$

Comment #2357 by Simon Pepin Lehalleur on

Suggested slogan: Notions of constructibility for étale sheaves and for subsets of schemes are compatible

Comment #3387 by Dario on

Typo in (1): the map on stalks should also go from F to G

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