Lemma 68.5.3. 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 68.5.1. 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 68.4.5 (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 68.4.5.
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 68.3.3 the base change $V \times _ Y X \to X$ also has universally bounded fibres. Hence the first part of Lemma 68.4.6 applies and we see that $Y$ also has property $(\delta )$.
The case $\mathcal{P} = (\epsilon )$. We will repeatedly use Spaces, Lemma 65.5.5. Let $V_ i \to Y$ be as in Lemma 68.4.7 (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 68.4.7 (2).
The case $\mathcal{P} = (\theta )$. In this case the result is Categories, Lemma 4.8.3. $\square$
Post a comment
Your email address will not be published. Required fields are marked.
In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).
All contributions are licensed under the GNU Free Documentation License.
Comments (0)