Lemma 76.19.5 (061H)—The Stacks project

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Table of contents
Part 4: Algebraic Spaces
Chapter 76: More on Morphisms of Spaces
Section 76.19: Formally smooth morphisms
Lemma 76.19.5 (cite)

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Lemma 76.19.5. Let $S$ be a scheme. Let

\[ \xymatrix{ U \ar[d] \ar[r]_\psi & V \ar[d] \\ X \ar[r]^ f & Y } \]

be a commutative diagram of morphisms of algebraic spaces over $S$. If the vertical arrows are étale and $f$ is formally smooth, then $\psi $ is formally smooth.

Proof. By Lemma 76.13.5 the morphisms $U \to X$ and $V \to Y$ are formally étale. By Lemma 76.13.3 the composition $U \to Y$ is formally smooth. By Lemma 76.13.8 we see $\psi : U \to V$ is formally smooth. $\square$

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