The Stacks project

Lemma 34.4.3. Let $T$ be a scheme.

1. If $T' \to T$ is an isomorphism then $\{ T' \to T\}$ is an étale covering of $T$.

2. If $\{ T_ i \to T\} _{i\in I}$ is an étale covering and for each $i$ we have an étale covering $\{ T_{ij} \to T_ i\} _{j\in J_ i}$, then $\{ T_{ij} \to T\} _{i \in I, j\in J_ i}$ is an étale covering.

3. If $\{ T_ i \to T\} _{i\in I}$ is an étale covering and $T' \to T$ is a morphism of schemes then $\{ T' \times _ T T_ i \to T'\} _{i\in I}$ is an étale covering.