The Stacks project

Lemma 64.3.5. Let $\mathcal{F}$ be a sheaf on $X_{\acute{e}tale}$. Then there are canonical isomorphisms $\pi _ X^{-1} \mathcal{F} \cong \mathcal{F}$ and $\mathcal{F} \cong {\pi _ X}_*\mathcal{F}$.

Proof. Let $\varphi : U \to X$ be ├ętale. Recall that ${\pi _ X}_* \mathcal{F} (U) = \mathcal{F} (U \times _{\varphi , X, \pi _ X} X)$. Since $\pi _ X = F_ X^ f$, it follows from the proof of Theorem 64.3.3 that there is a functorial isomorphism

\[ \xymatrix{ U \ar[rd]_{\varphi } \ar[rr]_-{\gamma _ U} & & U \times _{\varphi , X, \pi _ X} X \ar[ld]^{\text{pr}_2} \\ & X } \]

where $\gamma _ U = (\varphi , F_ U^ f)$. Now we define an isomorphism

\[ \mathcal{F} (U) \longrightarrow {\pi _ X}_* \mathcal{F} (U) = \mathcal{F} (U \times _{\varphi , X, \pi _ X} X) \]

by taking the restriction map of $\mathcal{F}$ along $\gamma _ U^{-1}$. The other isomorphism is analogous. $\square$

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