The Stacks project

Remark 66.18.4. Let us explain the meaning of Lemma 66.18.3. Let $S$ be a scheme, and let $X$ be an algebraic space over $S$. Let $\mathcal{F}$ be a sheaf on the small étale site $X_{\acute{e}tale}$ of $X$. The lemma says that there exists a unique sheaf $\mathcal{F}'$ on $X_{spaces, {\acute{e}tale}}$ which restricts back to $\mathcal{F}$ on the subcategory $X_{\acute{e}tale}$. If $U \to X$ is an étale morphism of algebraic spaces, then how do we compute $\mathcal{F}'(U)$? Well, by definition of an algebraic space there exists a scheme $U'$ and a surjective étale morphism $U' \to U$. Then $\{ U' \to U\} $ is a covering in $X_{spaces, {\acute{e}tale}}$ and hence we get an equalizer diagram

\[ \xymatrix{ \mathcal{F}'(U) \ar[r] & \mathcal{F}(U') \ar@<1ex>[r] \ar@<-1ex>[r] & \mathcal{F}(U' \times _ U U'). } \]

Note that $U' \times _ U U'$ is a scheme, and hence we may write $\mathcal{F}$ and not $\mathcal{F}'$. Thus we see how to compute $\mathcal{F}'$ when given the sheaf $\mathcal{F}$.

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