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The Stacks project

Lemma 76.14.2. Let S be a scheme. Let f : X \to Y be a morphism of algebraic spaces over S. The following are equivalent:

  1. f is formally unramified,

  2. for every diagram

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

    where U and V are schemes and the vertical arrows are étale the morphism of schemes \psi is formally unramified (as in More on Morphisms, Definition 37.6.1), and

  3. for one such diagram with surjective vertical arrows the morphism \psi is formally unramified.

Proof. Assume f is formally unramified. By Lemma 76.13.5 the morphisms U \to X and V \to Y are formally unramified. Thus by Lemma 76.13.3 the composition U \to Y is formally unramified. Then it follows from Lemma 76.13.8 that U \to V is formally unramified. Thus (1) implies (2). And (2) implies (3) trivially

Assume given a diagram as in (3). By Lemma 76.13.5 the morphism V \to Y is formally unramified. Thus by Lemma 76.13.3 the composition U \to Y is formally unramified. Then it follows from Lemma 76.13.6 that X \to Y is formally unramified, i.e., (1) holds. \square


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