The Stacks project

Lemma 65.28.1. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. The morphism of ringed topoi $(f_{small}, f^\sharp )$ associated to $f$ is a morphism of locally ringed topoi, see Modules on Sites, Definition 18.40.9.

Proof. Note that the assertion makes sense since we have seen that $(X_{\acute{e}tale}, \mathcal{O}_{X_{\acute{e}tale}})$ and $(Y_{\acute{e}tale}, \mathcal{O}_{Y_{\acute{e}tale}})$ are locally ringed sites, see Lemma 65.22.3. Moreover, we know that $X_{\acute{e}tale}$ has enough points, see Theorem 65.19.12. Hence it suffices to prove that $(f_{small}, f^\sharp )$ satisfies condition (3) of Modules on Sites, Lemma 18.40.8. To see this take a point $p$ of $X_{\acute{e}tale}$. By Lemma 65.19.13 $p$ corresponds to a geometric point $\overline{x}$ of $X$. By Lemma 65.19.9 the point $q = f_{small} \circ p$ corresponds to the geometric point $\overline{y} = f \circ \overline{x}$ of $Y$. Hence the assertion we have to prove is that the induced map of étale local rings

\[ \mathcal{O}_{Y, \overline{y}} \longrightarrow \mathcal{O}_{X, \overline{x}} \]

is a local ring map. You can prove this directly, but instead we deduce it from the corresponding result for schemes. To do this choose a commutative diagram

\[ \xymatrix{ U \ar[d] \ar[r]_\psi & V \ar[d] \\ X \ar[r] & Y } \]

where $U$ and $V$ are schemes, and the vertical arrows are surjective étale (see Spaces, Lemma 64.11.6). Choose a lift $\overline{u} : \overline{x} \to U$ (possible by Lemma 65.19.5). Set $\overline{v} = \psi \circ \overline{u}$. We obtain a commutative diagram of étale local rings

\[ \xymatrix{ \mathcal{O}_{U, \overline{u}} & \mathcal{O}_{V, \overline{v}} \ar[l] \\ \mathcal{O}_{X, \overline{x}} \ar[u] & \mathcal{O}_{Y, \overline{y}}. \ar[l] \ar[u] } \]

By Étale Cohomology, Lemma 59.40.1 the top horizontal arrow is a local ring map. Finally by Lemma 65.22.1 the vertical arrows are isomorphisms. Hence we win. $\square$


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