Lemma 87.34.6. Let $S$ be a scheme. Let $X$ be a formal algebraic space over $S$. Then $X_{spaces, {\acute{e}tale}}$ is equivalent to the category whose objects are morphisms $\varphi : U \to X$ of formal algebraic spaces such that $\varphi $ is representable by algebraic spaces and étale.
Proof. Denote $\mathcal{C}$ the category introduced in the lemma. Recall that $X_{spaces, {\acute{e}tale}} = X_{red, spaces, {\acute{e}tale}}$. Hence we can define a functor
\[ \mathcal{C} \longrightarrow X_{spaces, {\acute{e}tale}},\quad (U \to X) \longmapsto U \times _ X X_{red} \]
because $U \times _ X X_{red}$ is an algebraic space étale over $X_{red}$.
To finish the proof we will construct a quasi-inverse. Choose an object $\psi : V \to X_{red}$ of $X_{red, spaces, {\acute{e}tale}}$. Consider the functor $U_{V, \psi } : (\mathit{Sch}/S)_{fppf} \to \textit{Sets}$ given by
\[ U_{V, \psi }(T) = \{ (a, b) \mid a : T \to X, \ b : T \times _{a, X} X_{red} \to V, \ \psi \circ b = a|_{T \times _{a, X} X_{red}}\} \]
We claim that the transformation $U_{V, \psi } \to X$, $(a, b) \mapsto a$ defines an object of the category $\mathcal{C}$. First, let's prove that $U_{V, \psi }$ is a formal algebraic space. Observe that $U_{V, \psi }$ is a sheaf for the fppf topology (some details omitted). Next, suppose that $X_ i \to X$ is an étale covering by affine formal algebraic spaces as in Definition 87.11.1. Set $V_ i = V \times _{X_{red}} X_{i, red}$ and denote $\psi _ i : V_ i \to X_{i, red}$ the projection. Then we have
\[ U_{V, \psi } \times _ X X_ i = U_{V_ i, \psi _ i} \]
by a formal argument because $X_{i, red} = X_ i \times _ X X_{red}$ (as $X_ i \to X$ is representable by algebraic spaces and étale). Hence it suffices to show that $U_{V_ i, \psi _ i}$ is an affine formal algebraic space, because then we will have a covering $U_{V_ i, \psi _ i} \to U_{V, \psi }$ as in Definition 87.11.1. On the other hand, we have seen in the proof of Lemma 87.34.3 that $\psi _ i : V_ i \to X_ i$ is the base change of a representable and étale morphism $U_ i \to X_ i$ of affine formal algebraic spaces. Then it is not hard to see that $U_ i = U_{V_ i, \psi _ i}$ as desired.
We omit the verification that $U_{V, \psi } \to X$ is representable by algebraic spaces and étale. Thus we obtain our functor $(V, \psi ) \mapsto (U_{V, \psi } \to X)$ in the other direction. We omit the verification that the constructions are mutually inverse to each other. $\square$
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Your email address will not be published.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)