The Stacks project

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. $\square$


Comments (0)


Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 09BH. Beware of the difference between the letter 'O' and the digit '0'.