Lemma 97.9.2. Let $Z \to U$ be a finite locally free morphism of schemes. Let $W$ be an algebraic space and let $W \to Z$ be an étale morphism. Then the functor

defined by the rule

is an algebraic space and the morphism $F \to U$ is étale.

Lemma 97.9.2. Let $Z \to U$ be a finite locally free morphism of schemes. Let $W$ be an algebraic space and let $W \to Z$ be an étale morphism. Then the functor

\[ F : (\mathit{Sch}/U)_{fppf}^{opp} \longrightarrow \textit{Sets}, \]

defined by the rule

\[ U' \longmapsto F(U') = \{ \sigma : Z_{U'} \to W_{U'}\text{ section of }W_{U'} \to Z_{U'}\} \]

is an algebraic space and the morphism $F \to U$ is étale.

**Proof.**
Assume first that $W \to Z$ is also separated. Let $U'$ be a scheme over $U$ and let $\sigma \in F(U')$. By Morphisms of Spaces, Lemma 67.4.7 the morphism $\sigma $ is a closed immersion. Moreover, $\sigma $ is étale by Properties of Spaces, Lemma 66.16.6. Hence $\sigma $ is also an open immersion, see Morphisms of Spaces, Lemma 67.51.2. In other words, $Z_\sigma = \sigma (Z_{U'}) \subset W_{U'}$ is an open subspace such that the morphism $Z_\sigma \to Z_{U'}$ is an isomorphism. In particular, the morphism $Z_\sigma \to U'$ is finite. Hence we obtain a transformation of functors

\[ F \longrightarrow (W/U)_{fin}, \quad \sigma \longmapsto (U' \to U, Z_\sigma ) \]

where $(W/U)_{fin}$ is the finite part of the morphism $W \to U$ introduced in More on Groupoids in Spaces, Section 79.12. It is clear that this transformation of functors is injective (since we can recover $\sigma $ from $Z_\sigma $ as the inverse of the isomorphism $Z_\sigma \to Z_{U'}$). By More on Groupoids in Spaces, Proposition 79.12.11 we know that $(W/U)_{fin}$ is an algebraic space étale over $U$. Hence to finish the proof in this case it suffices to show that $F \to (W/U)_{fin}$ is representable and an open immersion. To see this suppose that we are given a morphism of schemes $U' \to U$ and an open subspace $Z' \subset W_{U'}$ such that $Z' \to U'$ is finite. Then it suffices to show that there exists an open subscheme $U'' \subset U'$ such that a morphism $T \to U'$ factors through $U''$ if and only if $Z' \times _{U'} T$ maps isomorphically to $Z \times _{U'} T$. This follows from More on Morphisms of Spaces, Lemma 76.49.6 (here we use that $Z \to B$ is flat and locally of finite presentation as well as finite). Hence we have proved the lemma in case $W \to Z$ is separated as well as étale.

In the general case we choose a separated scheme $W'$ and a surjective étale morphism $W' \to W$. Note that the morphisms $W' \to W$ and $W \to Z$ are separated as their source is separated. Denote $F'$ the functor associated to $W' \to Z \to U$ as in the lemma. In the first paragraph of the proof we showed that $F'$ is representable by an algebraic space étale over $U$. By Lemma 97.9.1 the map of functors $F' \to F$ is surjective for the étale topology on $\mathit{Sch}/U$. Moreover, if $U'$ and $\sigma : Z_{U'} \to W_{U'}$ define a point $\xi \in F(U')$, then the fibre product

\[ F'' = F' \times _{F, \xi } U' \]

is the functor on $\mathit{Sch}/U'$ associated to the morphisms

\[ W'_{U'} \times _{W_{U'}, \sigma } Z_{U'} \to Z_{U'} \to U'. \]

Since the first morphism is separated as a base change of a separated morphism, we see that $F''$ is an algebraic space étale over $U'$ by the result of the first paragraph. It follows that $F' \to F$ is a surjective étale transformation of functors, which is representable by algebraic spaces. Hence $F$ is an algebraic space by Bootstrap, Theorem 80.10.1. Since $F' \to F$ is an étale surjective morphism of algebraic spaces it follows that $F \to U$ is étale because $F' \to U$ is étale. $\square$

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).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

## Comments (4)

Comment #4924 by Robot0079 on

Comment #5192 by Johan on

Comment #6313 by Robot0079 on

Comment #6315 by Johan on