Lemma 87.35.1. Every formal algebraic space has a structure sheaf.
Proof. Let $S$ be a scheme. Let $X$ be a formal algebraic space over $S$. By (87.34.1.1) it suffices to construct $\mathcal{O}_ X$ as a sheaf of topological rings on $X_{affine, {\acute{e}tale}}$. Denote $\mathcal{C}$ the category whose objects are morphisms $\varphi : U \to X$ of formal algebraic spaces such that $U$ is an affine formal algebraic space and $\varphi $ is representable by algebraic spaces and étale. By Lemma 87.34.7 the functor $U \mapsto U_{red}$ is an equivalence of categories $\mathcal{C} \to X_{affine, {\acute{e}tale}}$. Hence by the rule given above the lemma, we already have $\mathcal{O}_ X$ as a presheaf of topological rings on $X_{affine, {\acute{e}tale}}$. Thus it suffices to check the sheaf condition.
By definition of $X_{affine, {\acute{e}tale}}$ a covering corresponds to a finite family $\{ g_ i : U_ i \to U\} _{i = 1, \ldots , n}$ of morphisms of $\mathcal{C}$ such that $\{ U_{i, red} \to U_{red}\} $ is an étale covering. The morphisms $g_ i$ are representably by algebraic spaces (Lemma 87.19.3) hence affine (Lemma 87.19.7). Then $g_ i$ is étale (follows formally from Properties of Spaces, Lemma 66.16.6 as $U_ i$ and $U$ are étale over $X$ in the sense of Bootstrap, Section 80.4). Finally, write $U = \mathop{\mathrm{colim}}\nolimits U_\lambda $ as in Definition 87.9.1.
With these preparations out of the way, we can prove the sheaf property as follows. For each $\lambda $ we set $U_{i, \lambda } = U_ i \times _ U U_\lambda $ and $U_{ij, \lambda } = (U_ i \times _ U U_ j) \times _ U U_\lambda $. By the above, these are affine schemes, $\{ U_{i, \lambda } \to U_\lambda \} $ is an étale covering, and $U_{ij, \lambda } = U_{i, \lambda } \times _{U_\lambda } U_{j, \lambda }$. Also we have $U_ i = \mathop{\mathrm{colim}}\nolimits U_{i, \lambda }$ and $U_ i \times _ U U_ j = \mathop{\mathrm{colim}}\nolimits U_{ij, \lambda }$. For each $\lambda $ we have an exact sequence
\[ 0 \to \Gamma (U_\lambda , \mathcal{O}_{U_\lambda }) \to \prod \nolimits _ i \Gamma (U_{i, \lambda }, \mathcal{O}_{U_{i, \lambda }}) \to \prod \nolimits _{i, j} \Gamma (U_{ij, \lambda }, \mathcal{O}_{U_{ij, \lambda }}) \]
as we have the sheaf condition for the structure sheaf on $U_\lambda $ and the étale topology (see Étale Cohomology, Proposition 59.17.1). Since limits commute with limits, the inverse limit of these exact sequences is an exact sequence
\[ 0 \to \mathop{\mathrm{lim}}\nolimits \Gamma (U_\lambda , \mathcal{O}_{U_\lambda }) \to \prod \nolimits _ i \mathop{\mathrm{lim}}\nolimits \Gamma (U_{i, \lambda }, \mathcal{O}_{U_{i, \lambda }}) \to \prod \nolimits _{i, j} \mathop{\mathrm{lim}}\nolimits \Gamma (U_{ij, \lambda }, \mathcal{O}_{U_{ij, \lambda }}) \]
which exactly means that
\[ 0 \to \mathcal{O}_ X(U_{red}) \to \prod \nolimits _ i \mathcal{O}_ X(U_{i, red}) \to \prod \nolimits _{i, j} \mathcal{O}_ X((U_ i \times _ U U_ j)_{red}) \]
is exact and hence the sheaf property holds as desired. $\square$
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