The Stacks project

Lemma 73.14.7. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. The following are equivalent:

  1. The morphism $f$ is unramified,

  2. the morphism $f$ is locally of finite type and $\Omega _{X/Y} = 0$, and

  3. the morphism $f$ is locally of finite type and formally unramified.

Proof. Choose a 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 and surjective. Then we see

\begin{align*} f\text{ unramified} & \Leftrightarrow \psi \text{ unramified} \\ & \Leftrightarrow \psi \text{ locally finite type and }\Omega _{U/V} = 0 \\ & \Leftrightarrow f\text{ locally finite type and }\Omega _{X/Y} = 0 \\ & \Leftrightarrow f\text{ locally finite type and formally unramified} \end{align*}

Here we have used Morphisms, Lemma 29.33.2 and Lemma 73.14.6. $\square$


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