Lemma 76.19.11. 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 66.6.1. By Lemma 76.19.10 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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