The Stacks project

Lemma 66.11.1. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Let $U \to X$ be a surjective ├ętale morphism from a scheme to $X$. Set $R = U \times _ X U$ and denote $t, s : R \to U$ the projection morphisms as usual. Denote $a : U \to Y$ and $b : R \to Y$ the induced morphisms. For any object $\mathcal{F}$ of $\textit{Mod}(\mathcal{O}_ X)$ there exists an exact sequence

\[ 0 \to f_*\mathcal{F} \to a_*(\mathcal{F}|_ U) \to b_*(\mathcal{F}|_ R) \]

where the second arrow is the difference $t^* - s^*$.

Proof. We denote $\mathcal{F}$ also its extension to a sheaf of modules on $X_{spaces, {\acute{e}tale}}$, see Properties of Spaces, Remark 65.18.4. Let $V \to Y$ be an object of $Y_{\acute{e}tale}$. Then $V \times _ Y X$ is an object of $X_{spaces, {\acute{e}tale}}$, and by definition $f_*\mathcal{F}(V) = \mathcal{F}(V \times _ Y X)$. Since $U \to X$ is surjective ├ętale, we see that $\{ V \times _ Y U \to V \times _ Y X\} $ is a covering. Also, we have $(V \times _ Y U) \times _ X (V \times _ Y U) = V \times _ Y R$. Hence, by the sheaf condition of $\mathcal{F}$ on $X_{spaces, {\acute{e}tale}}$ we have a short exact sequence

\[ 0 \to \mathcal{F}(V \times _ Y X) \to \mathcal{F}(V \times _ Y U) \to \mathcal{F}(V \times _ Y R) \]

where the second arrow is the difference of restricting via $t$ or $s$. This exact sequence is functorial in $V$ and hence we obtain the lemma. $\square$


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