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80.5 Descending étale morphisms of algebraic spaces

In this section we combine the glueing results for étale sheaves given in Section 80.4 with the flexibility of algebraic spaces to get some descent statements for étale morphisms of algebraic spaces.

Lemma 80.5.1. Let $S$ be a scheme. Let $f : X \to Y$ be a proper surjective morphism of algebraic spaces over $S$. Any descent datum $(U/X, \varphi )$ relative to $f$ (Descent on Spaces, Definition 73.22.1) with $U$ étale over $X$ is effective (Descent on Spaces, Definition 73.22.10). More precisely, there exists an étale morphism $V \to Y$ of algebraic spaces whose corresponding canonical descent datum is isomorphic to $(U/X, \varphi )$.

Proof. Recall that $U$ gives rise to a representable sheaf $\mathcal{F} = h_ U$ in $\mathop{\mathit{Sh}}\nolimits (X_{spaces, {\acute{e}tale}}) = \mathop{\mathit{Sh}}\nolimits (X_{\acute{e}tale})$, see Properties of Spaces, Section 65.18. The descent datum on $U$ relative to $f$ exactly gives a descent datum $(\mathcal{F}, \varphi )$ for étale sheaves with respect to $\{ X \to Y\} $. By Lemma 80.4.5 this descent datum is effective. Let $\mathcal{G}$ be the corresponding sheaf on $Y_{\acute{e}tale}$. By Properties of Spaces, Lemma 65.27.3 we obtain an étale morphism $V \to Y$ of algebraic spaces corresponding to $\mathcal{G}$; we omit the verification of the set theoretic condition1. The given isomorphism $\mathcal{F} \to f_{small}^{-1}\mathcal{G}$ corresponds to an isomorphism $U \to V \times _ Y X$ compatible with the descent datum. $\square$

Lemma 80.5.2. Let $S$ be a scheme. Let $f : Y' \to Y$ be a proper morphism of algebraic spaces over $S$. Let $i : Z \to Y$ be a closed immersion. Set $E = Z \times _ Y Y'$. Picture

\[ \xymatrix{ E \ar[d]_ g \ar[r]_ j & Y' \ar[d]^ f \\ Z \ar[r]^ i & Y } \]

If $f$ is an isomorphism over $Y \setminus Z$, then the functor

\[ Y_{spaces, {\acute{e}tale}} \longrightarrow Y'_{spaces, {\acute{e}tale}} \times _{E_{spaces, {\acute{e}tale}}} Z_{spaces, {\acute{e}tale}} \]

is an equivalence of categories.

Proof. Let $(V' \to Y', W \to Z, \alpha )$ be an object of the right hand side. Recall that $V'$, resp. $W$ gives rise to a representable sheaf $\mathcal{G}' = h_{V'}$ in $\mathop{\mathit{Sh}}\nolimits (Y'_{spaces, {\acute{e}tale}}) = \mathop{\mathit{Sh}}\nolimits (Y'_{\acute{e}tale})$, resp. $\mathcal{G} = h_ W$ in $\mathop{\mathit{Sh}}\nolimits (Z_{spaces, {\acute{e}tale}}) = \mathop{\mathit{Sh}}\nolimits (Z_{\acute{e}tale})$, see Properties of Spaces, Section 65.18. The isomorphism $\alpha : V' \times _{Y'} E \to W \times _ Z E$ determines an isomorphism $j_{small}^{-1}\mathcal{G}' \to g_{small}^{-1}\mathcal{G}$ of sheaves on $E$. By Lemma 80.4.7 we obtain a unique sheaf $\mathcal{F}$ on $Y$ pulling pack to $\mathcal{G}'$ and $\mathcal{G}$ compatibly with the isomorphism. By Properties of Spaces, Lemma 65.27.3 we obtain an étale morphism $V \to Y$ of algebraic spaces corresponding to $\mathcal{F}$; we omit the verification of the set theoretic condition2. The given isomorphism $\mathcal{G}' \to f_{small}^{-1}\mathcal{F}$ and $\mathcal{G} \to i_{small}^{-1}\mathcal{F}$ corresponds to isomorphisms $V' \to V \times _ Y Y'$ and $W \to V \times _ Y Z$ compatible with $\alpha $ as desired. $\square$

[1] It follows from the fact that $\mathcal{F}$ satisfies the corresponding condition.
[2] It follows from the fact that $\mathcal{G}$ and $\mathcal{G}'$ satisfies the corresponding condition.

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