Lemma 75.19.11. The property $\mathcal{P}(f) =$“$f$ is formally smooth” is fpqc local on the base.

**Proof.**
Let $f : X \to Y$ be a morphism of algebraic spaces over a scheme $S$. Choose an index set $I$ and diagrams

with étale vertical arrows and $U_ i$, $V_ i$ affine schemes. Moreover, assume that $\coprod U_ i \to X$ and $\coprod V_ i \to Y$ are surjective, see Properties of Spaces, Lemma 65.6.1. By Lemma 75.19.10 we see that $f$ is formally smooth if and only if each of the morphisms $\psi _ i$ are formally smooth. Hence we reduce to the case of a morphism of affine schemes. In this case the result follows from Algebra, Lemma 10.138.16. Some details omitted. $\square$

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