The Stacks project

Proof. Let us prove the lemma in case $\omega = \beta $. In this case we have to show that any $x \in |X|$ is represented by a monomorphism from the spectrum of a field into $X$. Let $y = f(x) \in |Y|$. By assumption there exists a field $k$ and a monomorphism $\mathop{\mathrm{Spec}}(k) \to Y$ representing $y$. Then $x$ corresponds to a point $x'$ of $\mathop{\mathrm{Spec}}(k) \times _ Y X$. By assumption $x'$ is represented by a monomorphism $\mathop{\mathrm{Spec}}(k') \to \mathop{\mathrm{Spec}}(k) \times _ Y X$. Clearly the composition $\mathop{\mathrm{Spec}}(k') \to X$ is a monomorphism representing $x$.

Let us prove the lemma in case $\omega = decent$. Let $x \in |X|$ and $y = f(x) \in |Y|$. By the result of the preceding paragraph we can choose a diagram

whose horizontal arrows monomorphisms. As $Y$ is decent the morphism $y$ is quasi-compact. As $f$ is decent the algebraic space $\mathop{\mathrm{Spec}}(k) \times _ Y X$ is decent. Hence the monomorphism $\mathop{\mathrm{Spec}}(k') \to \mathop{\mathrm{Spec}}(k) \times _ Y X$ is quasi-compact. Then the monomorphism $x : \mathop{\mathrm{Spec}}(k') \to X$ is quasi-compact as a composition of quasi-compact morphisms (use Morphisms of Spaces, Lemmas 67.8.4 and 67.8.5). As the point $x$ was arbitrary this implies $X$ is decent.

Let us prove the lemma in case $\omega = reasonable$. Choose $V \to Y$ étale with $V$ an affine scheme. Choose $U \to V \times _ Y X$ étale with $U$ an affine scheme. By assumption $V \to Y$ has universally bounded fibres. By Lemma 68.3.3 the morphism $V \times _ Y X \to X$ has universally bounded fibres. By assumption on $f$ we see that $U \to V \times _ Y X$ has universally bounded fibres. By Lemma 68.3.2 the composition $U \to X$ has universally bounded fibres. Hence there exists sufficiently many étale morphisms $U \to X$ from schemes with universally bounded fibres, and we conclude that $X$ is reasonable. $\square$