The Stacks project

Lemma 59.64.4. Let $X$ be a scheme and $\mathcal{F}$ a sheaf of sets on $X_{\acute{e}tale}$. Then the following are equivalent

  1. $\mathcal{F}$ is finite locally constant, and

  2. $\mathcal{F} = h_ U$ for some finite étale morphism $U \to X$.

Proof. A finite étale morphism is locally isomorphic to a disjoint union of isomorphisms, see Étale Morphisms, Lemma 41.18.3. Thus (2) implies (1). Conversely, if $\mathcal{F}$ is finite locally constant, then there exists an étale covering $\{ X_ i \to X\} $ such that $\mathcal{F}|_{X_ i}$ is representable by $U_ i \to X_ i$ finite étale. Arguing exactly as in the proof of Descent, Lemma 35.39.1 we obtain a descent datum for schemes $(U_ i, \varphi _{ij})$ relative to $\{ X_ i \to X\} $ (details omitted). This descent datum is effective for example by Descent, Lemma 35.37.1 and the resulting morphism of schemes $U \to X$ is finite étale by Descent, Lemmas 35.23.23 and 35.23.29. $\square$

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