Lemma 59.104.6 (0GF3)—The Stacks project

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Part 3: Topics in Scheme Theory
Chapter 59: Étale Cohomology
Section 59.104: Descending étale sheaves
Lemma 59.104.6 (cite)

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Lemma 59.104.6. Let $\{ f_ i : X_ i \to X\} $ be an fppf covering of schemes. The functor

\[ \mathop{\mathit{Sh}}\nolimits (X_{\acute{e}tale}) \longrightarrow \text{descent data for étale sheaves wrt }\{ f_ i : X_ i \to X\} \]

is an equivalence of categories.

Proof. We have Lemma 59.104.5 for the morphism $f : \coprod X_ i \to X$. Then a formal argument shows that descent data for $f$ are the same thing as descent data for the covering, compare with Descent, Lemma 35.34.5. Details omitted. $\square$

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