The Stacks project

Lemma 75.14.2. 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 ├ętale 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 75.13.5 the morphisms $U \to X$ and $V \to Y$ are formally unramified. Thus by Lemma 75.13.3 the composition $U \to Y$ is formally unramified. Then it follows from Lemma 75.13.8 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 75.13.5 the morphism $V \to Y$ is formally unramified. Thus by Lemma 75.13.3 the composition $U \to Y$ is formally unramified. Then it follows from Lemma 75.13.6 that $X \to Y$ is formally unramified, i.e., (1) holds. $\square$


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