In this section we discuss the notion of a smooth covering of algebraic spaces, and we define the big smooth site of an algebraic space. Please compare with Topologies, Section 34.5.
Definition 73.5.1. Let $S$ be a scheme, and let $X$ be an algebraic space over $S$. A smooth covering of $X$ is a family of morphisms $\{ f_ i : X_ i \to X\} _{i \in I}$ of algebraic spaces over $S$ such that each $f_ i$ is smooth and such that i.e., the morphisms are jointly surjective.
\[ |X| = \bigcup \nolimits _{i \in I} |f_ i|(|X_ i|), \]
This is exactly the same as Topologies, Definition 34.5.1. In particular, if $X$ and all the $X_ i$ are schemes, then we recover the usual notion of a smooth covering of schemes.
Lemma 73.5.2. Any étale covering is a smooth covering, and a fortiori, any Zariski covering is a smooth covering.
Proof. This is clear from the definitions, the fact that an étale morphism is smooth (Morphisms of Spaces, Lemma 67.39.6), and Lemma 73.4.2. $\square$
Lemma 73.5.3. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.
If $X' \to X$ is an isomorphism then $\{ X' \to X\} $ is a smooth covering of $X$.
If $\{ X_ i \to X\} _{i\in I}$ is a smooth covering and for each $i$ we have a smooth covering $\{ X_{ij} \to X_ i\} _{j\in J_ i}$, then $\{ X_{ij} \to X\} _{i \in I, j\in J_ i}$ is a smooth covering.
If $\{ X_ i \to X\} _{i\in I}$ is a smooth covering and $X' \to X$ is a morphism of algebraic spaces then $\{ X' \times _ X X_ i \to X'\} _{i\in I}$ is a smooth covering.
Proof. Omitted. $\square$
To be continued...
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