The Stacks project

Lemma 86.8.1. 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 86.5.1. 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 86.5.3 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 65.12.5, and Properties of Spaces, Lemma 65.12.4).

Let $X$ and $\{ X_ i \to X\} _{i \in I}$ be as in Definition 86.7.1. 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 86.5.11). Hence $X_{i, red} \times _ X X_ j$ is reduced (see Descent, Lemma 35.17.1). 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 64.10.5. 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 79.5.2 (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 64.5.6 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 65.7). 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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