The Stacks project

Lemma 34.5.3. Let $T$ be a scheme.

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

  2. 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.

  3. 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

Typo in (1): "an smooth covering". Missing spaces (not sure!) in "If " and "of ".

Comment #3662 by on

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

There are also:

  • 3 comment(s) on Section 34.5: The smooth topology

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