The Stacks project

Lemma 83.4.1. 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 65.19.9 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 68.18.1. 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 √Čtale Cohomology, Lemma 59.102.1 (in fact it is in some sense equivalent to this lemma). $\square$

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