Lemma 69.11.3. Let $f : X \to Y$ be a quasi-compact, separated, étale morphism of algebraic spaces. Then for any quasi-coherent $\mathcal{O}_ X$-module $\mathcal{F}$ the map $f^*f_*\mathcal{F} \to \mathcal{F}$ is split.
Proof. Consider the cartesian diagram
\[ \xymatrix{ X \times _ Y X \ar[r]_-p \ar[d]_ q & X \ar[d]^ f \\ X \ar[r]^ f & Y } \]
By Lemma 69.11.1 we have $f^*f_*\mathcal{F} = q_*p^*\mathcal{F}$. The morphism $\Delta : X \to X \times _ Y X$ is open (as a morphism between algebraic spaces étale over $Y$) and closed as $f$ is separated. Thus we see that $p^*\mathcal{F}$ is a direct sum of $\Delta _*\mathcal{F}$ and a quasi-coherent module supported on the (open and closed) complement of $\Delta (X)$. Tracing the maps the reader verifies that
\[ \mathcal{F} = q_*\Delta _*\mathcal{F} \to q_*p^*\mathcal{F} = f^*f_*\mathcal{F} \to \mathcal{F} \]
is the identity map. Details omitted. $\square$
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