The Stacks project

Proof. Let $U$ be a scheme and $U \to X$ a surjective étale morphism. Then $Z \times _ X U \to U$ is an immersion of schemes, hence gives a homeomorphism of $|Z \times _ X U|$ with a locally closed subset $T'$ of $|U|$. By Lemma 66.4.3 the subset $T'$ is the inverse image of the image $T$ of $|Z| \to |X|$. The map $|Z| \to |X|$ is injective because the transformation of functors $Z \to X$ is injective, see Spaces, Section 65.12. By Topology, Lemma 5.6.4 we see that $T$ is locally closed in $|X|$. Moreover, the continuous map $|Z| \to T$ is a homeomorphism as the map $|Z \times _ X U| \to T'$ is a homeomorphism and $|Z \times _ Y U| \to |Z|$ is submersive. $\square$

Proof. Denote $\varphi : U \to X$ the canonical map. The bijection sends $Z \to X$ to $Z' = Z \times _ X U \to U$. It is immediate from the definition that $Z' \to U$ is an immersion, resp. closed immersion, resp. open immersion if $Z \to X$ is so. It is also clear that $Z'$ is $R$-invariant (see Groupoids, Definition 39.19.1).

Conversely, assume that $Z' \to U$ is an immersion which is $R$-invariant. Let $R'$ be the restriction of $R$ to $Z'$, see Groupoids, Definition 39.18.2. Since $R' = R \times _{s, U} Z' = Z' \times _{U, t} R$ in this case we see that $R'$ is an étale equivalence relation on $Z'$. By Spaces, Theorem 65.10.5 we see $Z = Z'/R'$ is an algebraic space. By construction we have $U \times _ X Z = Z'$, so $U \times _ X Z \to Z$ is an immersion. Note that the property “immersion” is preserved under base change and fppf local on the base (see Spaces, Section 65.4). Moreover, immersions are separated and locally quasi-finite (see Schemes, Lemma 26.23.8 and Morphisms, Lemma 29.20.16). Hence by More on Morphisms, Lemma 37.57.1 immersions satisfy descent for fppf covering. This means all the hypotheses of Spaces, Lemma 65.11.5 are satisfied for $Z \to X$, $\mathcal{P}=$“immersion”, and the étale surjective morphism $U \to X$. We conclude that $Z \to X$ is representable and an immersion, which is the definition of a subspace (see Spaces, Definition 65.12.1).

It is clear that these constructions are inverse to each other and we win. $\square$

Proof. Let $U \to X$ be a surjective étale morphism, where $U$ is a scheme. Set $R = U \times _ X U$, so that $X = U/R$, see Spaces, Lemma 65.9.1. As usual we denote $s, t : R \to U$ the two projection morphisms. By Lemma 66.4.5 we see that $T$ corresponds to a closed subset $T' \subset |U|$ such that $s^{-1}(T') = t^{-1}(T')$. Let $Z' \subset U$ be the reduced induced scheme structure on $T'$. In this case the fibre products $Z' \times _{U, t} R$ and $Z' \times _{U, s} R$ are closed subschemes of $R$ (Schemes, Lemma 26.18.2) which are étale over $Z'$ (Morphisms, Lemma 29.36.4), and hence reduced (because being reduced is local in the étale topology, see Remark 66.7.3). Since they have the same underlying topological space (see above) we conclude that $Z' \times _{U, t} R = Z' \times _{U, s} R$. Thus we can apply Lemma 66.12.2 to obtain a closed subspace $Z \subset X$ whose pullback to $U$ is $Z'$. By construction $|Z| = T$ and $Z$ is reduced. This proves existence. We omit the proof of uniqueness. $\square$

Proof. Assume $f(|Y|) \subset |Z|$. Choose a diagram

where $U$, $V$ are schemes, and the vertical arrows are surjective and étale. The scheme $V$ is reduced, see Lemma 66.7.1. Hence $h$ factors through $a^{-1}(Z)$ by Schemes, Lemma 26.12.7. So $a \circ h$ factors through $Z$. As $Z \subset X$ is a subsheaf, and $V \to Y$ is a surjection of sheaves on $(\mathit{Sch}/S)_{fppf}$ we conclude that $X \to Y$ factors through $Z$. $\square$