Processing math: 100%

The Stacks project

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


Comments (0)


Your email address will not be published.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.