The Stacks project

Lemma 59.73.1. Let $j : U \to X$ be an étale morphism of quasi-compact and quasi-separated schemes.

  1. The sheaf $h_ U$ is a constructible sheaf of sets.

  2. The sheaf $j_!\underline{M}$ is a constructible abelian sheaf for a finite abelian group $M$.

  3. If $\Lambda $ is a Noetherian ring and $M$ is a finite $\Lambda $-module, then $j_!\underline{M}$ is a constructible sheaf of $\Lambda $-modules on $X_{\acute{e}tale}$.

Proof. By Lemma 59.72.1 there is a partition $\coprod _ i X_ i$ such that $\pi _ i : j^{-1}(X_ i) \to X_ i$ is finite étale. The restriction of $h_ U$ to $X_ i$ is $h_{j^{-1}(X_ i)}$ which is finite locally constant by Lemma 59.64.4. For cases (2) and (3) we note that

\[ j_!(\underline{M})|_{X_ i} = \pi _{i!}(\underline{M}) = \pi _{i*}(\underline{M}) \]

by Lemmas 59.70.5 and 59.70.7. Thus it suffices to show the lemma for $\pi : Y \to X$ finite étale. This is Lemma 59.64.3. $\square$

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