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The Stacks project

Lemma 84.4.1. Let S be a scheme. Let f : Y \to X be a surjective proper morphism of algebraic spaces over S. Let \mathcal{F} be a sheaf on X_{\acute{e}tale}. Then \mathcal{F} \to f_*f^{-1}\mathcal{F} is injective with image the equalizer of the two maps f_*f^{-1}\mathcal{F} \to g_*g^{-1}\mathcal{F} where g is the structure morphism g : Y \times _ X Y \to X.

Proof. For any surjective morphism f : Y \to X of algebraic spaces over S, the map \mathcal{F} \to f_*f^{-1}\mathcal{F} is injective. Namely, if \overline{x} is a geometric point of X, then we choose a geometric point \overline{y} of Y lying over \overline{x} and we consider

\mathcal{F}_{\overline{x}} \to (f_*f^{-1}\mathcal{F})_{\overline{x}} \to (f^{-1}\mathcal{F})_{\overline{y}} = \mathcal{F}_{\overline{x}}

See Properties of Spaces, Lemma 66.19.9 for the last equality.

The second statement is local on X in the étale topology, hence we may and do assume Y is an affine scheme.

Choose a surjective proper morphism Z \to Y where Z is a scheme, see Cohomology of Spaces, Lemma 69.18.1. The result for Z \to X implies the result for Y \to X. Since Z \to X is a surjective proper morphism of schemes and hence a ph covering (Topologies, Lemma 34.8.6) the result for Z \to X follows from Étale Cohomology, Lemma 59.102.1 (in fact it is in some sense equivalent to this lemma). \square


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