The Stacks project

Lemma 66.18.8. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$.

  1. The continuous functor

    \[ Y_{spaces, {\acute{e}tale}} \longrightarrow X_{spaces, {\acute{e}tale}}, \quad V \longmapsto X \times _ Y V \]

    induces a morphism of sites

    \[ f_{spaces, {\acute{e}tale}} : X_{spaces, {\acute{e}tale}} \to Y_{spaces, {\acute{e}tale}}. \]

  2. The rule $f \mapsto f_{spaces, {\acute{e}tale}}$ is compatible with compositions, in other words $(f \circ g)_{spaces, {\acute{e}tale}} = f_{spaces, {\acute{e}tale}} \circ g_{spaces, {\acute{e}tale}}$ (see Sites, Definition 7.14.5).

  3. The morphism of topoi associated to $f_{spaces, {\acute{e}tale}}$ induces, via Lemma 66.18.3, a morphism of topoi $f_{small} : \mathop{\mathit{Sh}}\nolimits (X_{\acute{e}tale}) \to \mathop{\mathit{Sh}}\nolimits (Y_{\acute{e}tale})$ whose construction is compatible with compositions.

  4. If $f$ is a representable morphism of algebraic spaces, then $f_{small}$ comes from a morphism of sites $X_{\acute{e}tale}\to Y_{\acute{e}tale}$, corresponding to the continuous functor $V \mapsto X \times _ Y V$.

Proof. Let us show that the functor described in (1) satisfies the assumptions of Sites, Proposition 7.14.7. Thus we have to show that $Y_{spaces, {\acute{e}tale}}$ has a final object (namely $Y$) and that the functor transforms this into a final object in $X_{spaces, {\acute{e}tale}}$ (namely $X$). This is clear as $X \times _ Y Y = X$ in any category. Next, we have to show that $Y_{spaces, {\acute{e}tale}}$ has fibre products. This is true since the category of algebraic spaces has fibre products, and since $V \times _ Y V'$ is étale over $Y$ if $V$ and $V'$ are étale over $Y$ (see Lemmas 66.16.4 and 66.16.5 above). OK, so the proposition applies and we see that we get a morphism of sites as described in (1).

Part (2) you get by unwinding the definitions. Part (3) is clear by using the equivalences for $X$ and $Y$ from Lemma 66.18.3 above. Part (4) follows, because if $f$ is representable, then the functors above fit into a commutative diagram

\[ \xymatrix{ X_{\acute{e}tale}\ar[r] & X_{spaces, {\acute{e}tale}} \\ Y_{\acute{e}tale}\ar[r] \ar[u] & Y_{spaces, {\acute{e}tale}} \ar[u] } \]

of categories. $\square$

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