The Stacks project

Lemma 59.73.12. Let $X$ be an irreducible scheme with generic point $\eta $.

  1. Let $S' \subset S$ be an inclusion of sets. If we have $\underline{S'} \subset \mathcal{G} \subset \underline{S}$ in $\mathop{\mathit{Sh}}\nolimits (X_{\acute{e}tale})$ and $S' = \mathcal{G}_{\overline{\eta }}$, then $\mathcal{G} = \underline{S'}$.

  2. Let $A' \subset A$ be an inclusion of abelian groups. If we have $\underline{A'} \subset \mathcal{G} \subset \underline{A}$ in $\textit{Ab}(X_{\acute{e}tale})$ and $A' = \mathcal{G}_{\overline{\eta }}$, then $\mathcal{G} = \underline{A'}$.

  3. Let $M' \subset M$ be an inclusion of modules over a ring $\Lambda $. If we have $\underline{M'} \subset \mathcal{G} \subset \underline{M}$ in $\textit{Mod}(X_{\acute{e}tale}, \underline{\Lambda })$ and $M' = \mathcal{G}_{\overline{\eta }}$, then $\mathcal{G} = \underline{M'}$.

Proof. This is true because for every étale morphism $U \to X$ with $U \not= \emptyset $ the point $\eta $ is in the image. $\square$

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