The Stacks project

Lemma 66.5.3. Let $S$ be a scheme. Let $\mathcal{P}$ be one of the properties $(\beta )$, $(\gamma )$, $(\delta )$, $(\epsilon )$, or $(\theta )$ of algebraic spaces listed in Lemma 66.5.1. Let $X$, $Y$ be algebraic spaces over $S$. Let $X \to Y$ be a representable morphism. If $Y$ has property $\mathcal{P}$, so does $X$.

Proof. Assume $f : X \to Y$ is a representable morphism of algebraic spaces, and assume that $Y$ has $\mathcal{P}$. Let $x \in |X|$, and set $y = f(x) \in |Y|$.

The case $\mathcal{P} = (\beta )$. Condition $(\beta )$ for $Y$ means there exists a monomorphism $\mathop{\mathrm{Spec}}(k) \to Y$ representing $y$. The fibre product $X_ y = \mathop{\mathrm{Spec}}(k) \times _ Y X$ is a scheme, and $x$ corresponds to a point of $X_ y$, i.e., to a monomorphism $\mathop{\mathrm{Spec}}(k') \to X_ y$. As $X_ y \to X$ is a monomorphism also we see that $x$ is represented by the monomorphism $\mathop{\mathrm{Spec}}(k') \to X_ y \to X$. In other words $(\beta )$ holds for $X$.

The case $\mathcal{P} = (\gamma )$. Since $(\gamma ) \Rightarrow (\beta )$ we have seen in the preceding paragraph that $y$ and $x$ can be represented by monomorphisms as in the following diagram

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

Also, by definition of property $(\gamma )$ via Lemma 66.4.5 (2) there exist schemes $V_ i$ and étale morphisms $V_ i \to Y$ such that $\coprod V_ i \to Y$ is surjective and for each $i$, setting $R_ i = V_ i \times _ Y V_ i$ the fibres of both

\[ |V_ i| \longrightarrow |Y| \quad \text{and}\quad |R_ i| \longrightarrow |Y| \]

over $y$ are finite. This means that the schemes $(V_ i)_ y$ and $(R_ i)_ y$ are finite schemes over $y = \mathop{\mathrm{Spec}}(k)$. As $X \to Y$ is representable, the fibre products $U_ i = V_ i \times _ Y X$ are schemes. The morphisms $U_ i \to X$ are étale, and $\coprod U_ i \to X$ is surjective. Finally, for each $i$ we have

\[ (U_ i)_ x = (V_ i \times _ Y X)_ x = (V_ i)_ y \times _{\mathop{\mathrm{Spec}}(k)} \mathop{\mathrm{Spec}}(k') \]

and

\[ (U_ i \times _ X U_ i)_ x = \left((V_ i \times _ Y X) \times _ X (V_ i \times _ Y X)\right)_ x = (R_ i)_ y \times _{\mathop{\mathrm{Spec}}(k)} \mathop{\mathrm{Spec}}(k') \]

hence these are finite over $k'$ as base changes of the finite schemes $(V_ i)_ y$ and $(R_ i)_ y$. This implies that $(\gamma )$ holds for $X$, again via the second condition of Lemma 66.4.5.

The case $\mathcal{P} = (\delta )$. Let $V \to Y$ be an étale morphism with $V$ an affine scheme. Since $Y$ has property $(\delta )$ this morphism has universally bounded fibres. By Lemma 66.3.3 the base change $V \times _ Y X \to X$ also has universally bounded fibres. Hence the first part of Lemma 66.4.6 applies and we see that $Y$ also has property $(\delta )$.

The case $\mathcal{P} = (\epsilon )$. We will repeatedly use Spaces, Lemma 63.5.5. Let $V_ i \to Y$ be as in Lemma 66.4.7 (2). Set $U_ i = X \times _ Y V_ i$. The morphisms $U_ i \to X$ are étale, and $\coprod U_ i \to X$ is surjective. Because $U_ i \times _ X U_ i = X \times _ Y (V_ i \times _ Y V_ i)$ we see that the projections $U_ i \times _ Y U_ i \to U_ i$ are base changes of the projections $V_ i \times _ Y V_ i \to V_ i$, and so quasi-compact as well. Hence $X$ satisfies Lemma 66.4.7 (2).

The case $\mathcal{P} = (\theta )$. In this case the result is Categories, Lemma 4.8.3. $\square$


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