Lemma 66.17.5. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Let $\omega \in \{ \beta , decent, reasonable\}$. Suppose that $Y$ has property $(\omega )$ and $f : X \to Y$ has $(\omega )$. Then $X$ has $(\omega )$.

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

$\xymatrix{ \mathop{\mathrm{Spec}}(k') \ar[r]_ x \ar[d] & X \ar[d]^ f \\ \mathop{\mathrm{Spec}}(k) \ar[r]^ y & Y }$

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 65.8.4 and 65.8.5). 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 66.3.3 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 66.3.2 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. $\square$

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