Lemma 81.5.1. 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 74.22.1) with $U$ étale over $X$ is effective (Descent on Spaces, Definition 74.22.10). More precisely, there exists an étale morphism $V \to Y$ of algebraic spaces whose corresponding canonical descent datum is isomorphic to $(U/X, \varphi )$.
81.5 Descending étale morphisms of algebraic spaces
In this section we combine the glueing results for étale sheaves given in Section 81.4 with the flexibility of algebraic spaces to get some descent statements for étale morphisms of algebraic spaces.
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 66.18. 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 81.4.5 this descent datum is effective. Let $\mathcal{G}$ be the corresponding sheaf on $Y_{\acute{e}tale}$. By Properties of Spaces, Lemma 66.27.3 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 81.5.2. 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 If $f$ is an isomorphism over $Y \setminus Z$, then the functor is an equivalence of categories.
\[ \xymatrix{ E \ar[d]_ g \ar[r]_ j & Y' \ar[d]^ f \\ Z \ar[r]^ i & Y } \]
\[ Y_{spaces, {\acute{e}tale}} \longrightarrow Y'_{spaces, {\acute{e}tale}} \times _{E_{spaces, {\acute{e}tale}}} Z_{spaces, {\acute{e}tale}} \]
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 66.18. 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 81.4.7 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 66.27.3 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.
Post a comment
Your email address will not be published. Required fields are marked.
In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).
All contributions are licensed under the GNU Free Documentation License.
Comments (0)