The Stacks project

Lemma 74.19.11. 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

\[ \xymatrix{ U_ i \ar[d] \ar[r]_{\psi _ i} & V_ i \ar[d] \\ X \ar[r]^ f & Y } \]

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 64.6.1. By Lemma 74.19.10 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 10.137.16. Some details omitted. $\square$


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