The Stacks project

Proof. We will prove (1) and (2) and omit the proof of (3) and (4). Let $h : X' \to X$ be a morphism of schemes. If (1) holds for $X$ and $X'$, then $X' \times _ X X^{awn} \to X'$ is a universal homeomorphism and hence we get a unique morphism $(X')^{awn} \to X' \times _ X X^{awn}$ over $X'$ by the universal property of $(X')^{awn} \to X'$. Composed with the projection $X' \times _ X X^{awn} \to X^{awn}$ we obtain $h^{awn}$. If in addition (2) holds for $X$ and $X'$ and $h$ is an open immersion, then $X' \times _ X X^{awn}$ is absolutely weakly normal (Lemma 29.47.4) and we deduce that $(X')^{awn} \to X' \times _ X X^{awn}$ is an isomorphism.

Recall that any universal homeomorphism is affine, see Lemma 29.45.4. Thus if $X$ is affine then (1) and (2) follow immediately from Lemma 29.47.6. Let $X$ be a scheme and let $\mathcal{B}$ be the set of affine opens of $X$. For each $U \in \mathcal{B}$ we obtain $U^{awn} \to U$ and for $V \subset U$, $V, U \in \mathcal{B}$ we obtain a canonical isomorphism $\rho _{V, U} : V^{awn} \to V \times _ U U^{awn}$ by the discussion in the previous paragraph. Thus by relative glueing (Constructions, Lemma 27.2.1) we obtain a morphism $X^{awn} \to X$ which restricts to $U^{awn}$ over $U$ compatibly with the $\rho _{V, U}$. Next, let $Y \to X$ be a universal homeomorphism. Then $U \times _ X Y \to U$ is a universal homeomorphism for $U \in \mathcal{B}$ and we obtain a unique morphism $g_ U : U^{awn} \to U \times _ X Y$ over $U$. These $g_ U$ are compatible with the morphisms $\rho _{V, U}$; details omitted. Hence there is a unique morphism $g : X^{awn} \to Y$ over $X$ agreeing with $g_ U$ over $U$, see Constructions, Remark 27.2.3. This proves (1) for $X$. Part (2) follows because it holds affine locally. $\square$