Lemma of the day

Let P be a property of morphisms of algebraic spaces. Assume

  1. P is smooth local on the source,
  2. P is smooth local on the target, and
  3. P is stable under postcomposing with smooth morphisms: if f : X —> Y has P and Y —> Z is a smooth morphism then X —> Z has P.

Then P is smooth local on the source-and-target. See Tag 06FB.