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The Stacks project

Lemma 67.11.1. 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 étale 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 66.18.4. 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 étale, 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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