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
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
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
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
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