Lemma 66.5.2. Let $S$ be a scheme. Let $\mathcal{P}$ be one of the properties $(\alpha )$, $(\beta )$, $(\gamma )$, $(\delta )$, $(\epsilon )$, $(\zeta )$, or $(\theta )$ of algebraic spaces listed in Lemma 66.5.1. Then if $X$ is an algebraic space over $S$, and $X = \bigcup X_ i$ is a Zariski open covering such that each $X_ i$ has $\mathcal{P}$, then $X$ has $\mathcal{P}$.

Proof. Let $X$ be an algebraic space over $S$, and let $X = \bigcup X_ i$ is a Zariski open covering such that each $X_ i$ has $\mathcal{P}$.

The case $\mathcal{P} = (\alpha )$. The condition $(\alpha )$ for $X_ i$ means that for every $x \in |X_ i|$ and every affine scheme $U$, and étale morphism $\varphi : U \to X_ i$ the fibre of $\varphi : |U| \to |X_ i|$ over $x$ is finite. Consider $x \in X$, an affine scheme $U$ and an étale morphism $U \to X$. Since $X = \bigcup X_ i$ is a Zariski open covering there exits a finite affine open covering $U = U_1 \cup \ldots \cup U_ n$ such that each $U_ j \to X$ factors through some $X_{i_ j}$. By assumption the fibres of $|U_ j | \to |X_{i_ j}|$ over $x$ are finite for $j = 1, \ldots , n$. Clearly this means that the fibre of $|U| \to |X|$ over $x$ is finite. This proves the result for $(\alpha )$.

The case $\mathcal{P} = (\beta )$. The condition $(\beta )$ for $X_ i$ means that every $x \in |X_ i|$ is represented by a monomorphism from the spectrum of a field towards $X_ i$. Hence the same follows for $X$ as $X_ i \to X$ is a monomorphism and $X = \bigcup X_ i$.

The case $\mathcal{P} = (\gamma )$. Note that $(\gamma ) = (\alpha ) + (\beta )$ by Lemma 66.5.1 hence the lemma for $(\gamma )$ follows from the cases treated above.

The case $\mathcal{P} = (\delta )$. The condition $(\delta )$ for $X_ i$ means there exist schemes $U_{ij}$ and étale morphisms $U_{ij} \to X_ i$ with universally bounded fibres which cover $X_ i$. These schemes also give an étale surjective morphism $\coprod U_{ij} \to X$ and $U_{ij} \to X$ still has universally bounded fibres.

The case $\mathcal{P} = (\epsilon )$. The condition $(\epsilon )$ for $X_ i$ means we can find a set $J_ i$ and morphisms $\varphi _{ij} : U_{ij} \to X_ i$ such that each $\varphi _{ij}$ is étale, both projections $U_{ij} \times _{X_ i} U_{ij} \to U_{ij}$ are quasi-compact, and $\coprod _{j \in J_ i} U_{ij} \to X_ i$ is surjective. In this case the compositions $U_{ij} \to X_ i \to X$ are étale (combine Morphisms, Lemmas 29.35.3 and 29.35.9 and Spaces, Lemmas 63.5.4 and 63.5.3 ). Since $X_ i \subset X$ is a subspace we see that $U_{ij} \times _{X_ i} U_{ij} = U_{ij} \times _ X U_{ij}$, and hence the condition on fibre products is preserved. And clearly $\coprod _{i, j} U_{ij} \to X$ is surjective. Hence $X$ satisfies $(\epsilon )$.

The case $\mathcal{P} = (\zeta )$. The condition $(\zeta )$ for $X_ i$ means that $X_ i$ is Zariski locally quasi-separated. It is immediately clear that this means $X$ is Zariski locally quasi-separated.

For $(\theta )$, see Properties of Spaces, Lemma 64.13.1. $\square$

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