Lemma 34.5.3 (0221)—The Stacks project

Lemma 34.5.3. Let $T$ be a scheme.

If $T' \to T$ is an isomorphism then $\{ T' \to T\} $ is a smooth covering of $T$.
If $\{ T_ i \to T\} _{i\in I}$ is a smooth covering and for each $i$ we have a smooth covering $\{ T_{ij} \to T_ i\} _{j\in J_ i}$, then $\{ T_{ij} \to T\} _{i \in I, j\in J_ i}$ is a smooth covering.
If $\{ T_ i \to T\} _{i\in I}$ is a smooth covering and $T' \to T$ is a morphism of schemes then $\{ T' \times _ T T_ i \to T'\} _{i\in I}$ is a smooth covering.

Proof. Omitted. $\square$

Comments (2)

Comment #3522 by Laurent Moret-Bailly on August 18, 2018 at 12:09

Typo in (1): "an smooth covering". Missing spaces (not sure!) in "If $T'\to T$ " and "of $T$ ".

Comment #3662 by Johan on October 22, 2018 at 09:30

Somehow this already got fixed earlier. The spaces are there in the latex.

There are also: