Proposition 59.74.1. Let $X$ be a Noetherian scheme. Let $\Lambda$ be a Noetherian ring.

1. Any sub or quotient sheaf of a constructible sheaf of sets is constructible.

2. The category of constructible abelian sheaves on $X_{\acute{e}tale}$ is a (strong) Serre subcategory of $\textit{Ab}(X_{\acute{e}tale})$. In particular, every sub and quotient sheaf of a constructible abelian sheaf on $X_{\acute{e}tale}$ is constructible.

3. The category of constructible sheaves of $\Lambda$-modules on $X_{\acute{e}tale}$ is a (strong) Serre subcategory of $\textit{Mod}(X_{\acute{e}tale}, \Lambda )$. In particular, every submodule and quotient module of a constructible sheaf of $\Lambda$-modules on $X_{\acute{e}tale}$ is constructible.

Proof. Proof of (1). Let $\mathcal{G} \subset \mathcal{F}$ with $\mathcal{F}$ a constructible sheaf of sets on $X_{\acute{e}tale}$. Let $\eta \in X$ be a generic point of an irreducible component of $X$. By Noetherian induction it suffices to find an open neighbourhood $U$ of $\eta$ such that $\mathcal{G}|_ U$ is locally constant. To do this we may replace $X$ by an étale neighbourhood of $\eta$. Hence we may assume $\mathcal{F}$ is constant and $X$ is irreducible.

Say $\mathcal{F} = \underline{S}$ for some finite set $S$. Then $S' = \mathcal{G}_{\overline{\eta }} \subset S$ say $S' = \{ s_1, \ldots , s_ t\}$. Pick an étale neighbourhood $(U, \overline{u})$ of $\overline{\eta }$ and sections $\sigma _1, \ldots , \sigma _ t \in \mathcal{G}(U)$ which map to $s_ i$ in $\mathcal{G}_{\overline{\eta }} \subset S$. Since $\sigma _ i$ maps to an element $s_ i \in S' \subset S = \Gamma (X, \mathcal{F})$ we see that the two pullbacks of $\sigma _ i$ to $U \times _ X U$ are the same as sections of $\mathcal{G}$. By the sheaf condition for $\mathcal{G}$ we find that $\sigma _ i$ comes from a section of $\mathcal{G}$ over the open $\mathop{\mathrm{Im}}(U \to X)$ of $X$. Shrinking $X$ we may assume $\underline{S'} \subset \mathcal{G} \subset \underline{S}$. Then we see that $\underline{S'} = \mathcal{G}$ by Lemma 59.73.12.

Let $\mathcal{F} \to \mathcal{Q}$ be a surjection with $\mathcal{F}$ a constructible sheaf of sets on $X_{\acute{e}tale}$. Then set $\mathcal{G} = \mathcal{F} \times _\mathcal {Q} \mathcal{F}$. By the first part of the proof we see that $\mathcal{G}$ is constructible as a subsheaf of $\mathcal{F} \times \mathcal{F}$. This in turn implies that $\mathcal{Q}$ is constructible, see Lemma 59.71.6.

Proof of (3). we already know that constructible sheaves of modules form a weak Serre subcategory, see Lemma 59.71.6. Thus it suffices to show the statement on submodules.

Let $\mathcal{G} \subset \mathcal{F}$ be a submodule of a constructible sheaf of $\Lambda$-modules on $X_{\acute{e}tale}$. Let $\eta \in X$ be a generic point of an irreducible component of $X$. By Noetherian induction it suffices to find an open neighbourhood $U$ of $\eta$ such that $\mathcal{G}|_ U$ is locally constant. To do this we may replace $X$ by an étale neighbourhood of $\eta$. Hence we may assume $\mathcal{F}$ is constant and $X$ is irreducible.

Say $\mathcal{F} = \underline{M}$ for some finite $\Lambda$-module $M$. Then $M' = \mathcal{G}_{\overline{\eta }} \subset M$. Pick finitely many elements $s_1, \ldots , s_ t$ generating $M'$ as a $\Lambda$-module. (This is possible as $\Lambda$ is Noetherian and $M$ is finite.) Pick an étale neighbourhood $(U, \overline{u})$ of $\overline{\eta }$ and sections $\sigma _1, \ldots , \sigma _ t \in \mathcal{G}(U)$ which map to $s_ i$ in $\mathcal{G}_{\overline{\eta }} \subset M$. Since $\sigma _ i$ maps to an element $s_ i \in M' \subset M = \Gamma (X, \mathcal{F})$ we see that the two pullbacks of $\sigma _ i$ to $U \times _ X U$ are the same as sections of $\mathcal{G}$. By the sheaf condition for $\mathcal{G}$ we find that $\sigma _ i$ comes from a section of $\mathcal{G}$ over the open $\mathop{\mathrm{Im}}(U \to X)$ of $X$. Shrinking $X$ we may assume $\underline{M'} \subset \mathcal{G} \subset \underline{M}$. Then we see that $\underline{M'} = \mathcal{G}$ by Lemma 59.73.12.

Proof of (2). This follows in the usual manner from (3). Details omitted. $\square$

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