The Stacks project

Lemma 73.5.3. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.

  1. If $X' \to X$ is an isomorphism then $\{ X' \to X\} $ is a smooth covering of $X$.

  2. 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.

  3. 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. $\square$

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