The Stacks project

Lemma 59.71.6. Let $X$ be a scheme.

  1. The category of constructible sheaves of sets is closed under finite limits and colimits inside $\mathop{\mathit{Sh}}\nolimits (X_{\acute{e}tale})$.

  2. The category of constructible abelian sheaves is a weak Serre subcategory of $\textit{Ab}(X_{\acute{e}tale})$.

  3. Let $\Lambda $ be a Noetherian ring. The category of constructible sheaves of $\Lambda $-modules on $X_{\acute{e}tale}$ is a weak Serre subcategory of $\textit{Mod}(X_{\acute{e}tale}, \Lambda )$.

Proof. We prove (3). We will use the criterion of Homology, Lemma 12.10.3. Suppose that $\varphi : \mathcal{F} \to \mathcal{G}$ is a map of constructible sheaves of $\Lambda $-modules. We have to show that $\mathcal{K} = \mathop{\mathrm{Ker}}(\varphi )$ and $\mathcal{Q} = \mathop{\mathrm{Coker}}(\varphi )$ are constructible. Similarly, suppose that $0 \to \mathcal{F} \to \mathcal{E} \to \mathcal{G} \to 0$ is a short exact sequence of sheaves of $\Lambda $-modules with $\mathcal{F}$, $\mathcal{G}$ constructible. We have to show that $\mathcal{E}$ is constructible. In both cases we can replace $X$ with the members of an affine open covering. Hence we may assume $X$ is affine. Then we may further replace $X$ by the members of a finite partition of $X$ by constructible locally closed subschemes on which $\mathcal{F}$ and $\mathcal{G}$ are of finite type and locally constant. Thus we may apply Lemma 59.64.6 to conclude.

The proofs of (1) and (2) are very similar and are omitted. $\square$

Comments (2)

Comment #5422 by Alexander Schmidt on

Typo: The --> Then at the beginning of the third-last sentence.

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