Lemma 87.12.1. Let S be a scheme. Let X be a formal algebraic space over S. There exists a reduced algebraic space X_{red} and a representable morphism X_{red} \to X which is a thickening. A morphism U \to X with U a reduced algebraic space factors uniquely through X_{red}.
Proof. First assume that X is an affine formal algebraic space. Say X = \mathop{\mathrm{colim}}\nolimits X_\lambda as in Definition 87.9.1. Since the transition morphisms are thickenings, the affine schemes X_\lambda all have isomorphic reductions X_{red}. The morphism X_{red} \to X is representable and a thickening by Lemma 87.9.3 and the fact that compositions of thickenings are thickenings. We omit the verification of the universal property (use Schemes, Definition 26.12.5, Schemes, Lemma 26.12.7, Properties of Spaces, Definition 66.12.5, and Properties of Spaces, Lemma 66.12.4).
Let X and \{ X_ i \to X\} _{i \in I} be as in Definition 87.11.1. For each i let X_{i, red} \to X_ i be the reduction as constructed above. For i, j \in I the projection X_{i, red} \times _ X X_ j \to X_{i, red} is an étale (by assumption) morphism of schemes (by Lemma 87.9.11). Hence X_{i, red} \times _ X X_ j is reduced (see Descent, Lemma 35.18.1). Thus the projection X_{i, red} \times _ X X_ j \to X_ j factors through X_{j, red} by the universal property. We conclude that
R_{ij} = X_{i, red} \times _ X X_ j = X_{i, red} \times _ X X_{j, red} = X_ i \times _ X X_{j, red}
because the morphisms X_{i, red} \to X_ i are injections of sheaves. Set U = \coprod X_{i, red}, set R = \coprod R_{ij}, and denote s, t : R \to U the two projections. As a sheaf R = U \times _ X U and s and t are étale. Then (t, s) : R \to U defines an étale equivalence relation by our observations above. Thus X_{red} = U/R is an algebraic space by Spaces, Theorem 65.10.5. By construction the diagram
\xymatrix{ \coprod X_{i, red} \ar[r] \ar[d] & \coprod X_ i \ar[d] \\ X_{red} \ar[r] & X }
is cartesian. Since the right vertical arrow is étale surjective and the top horizontal arrow is representable and a thickening we conclude that X_{red} \to X is representable by Bootstrap, Lemma 80.5.2 (to verify the assumptions of the lemma use that a surjective étale morphism is surjective, flat, and locally of finite presentation and use that thickenings are separated and locally quasi-finite). Then we can use Spaces, Lemma 65.5.6 to conclude that X_{red} \to X is a thickening (use that being a thickening is equivalent to being a surjective closed immersion).
Finally, suppose that U \to X is a morphism with U a reduced algebraic space over S. Then each X_ i \times _ X U is étale over U and therefore reduced (by our definition of reduced algebraic spaces in Properties of Spaces, Section 66.7). Then X_ i \times _ X U \to X_ i factors through X_{i, red}. Hence U \to X factors through X_{red} because \{ X_ i \times _ X U \to U\} is an étale covering. \square
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