The Stacks project

Lemma 100.36.8. Let $\mathcal{X} \to \mathcal{Y} \to \mathcal{Z}$ be morphisms of algebraic stacks. If $\mathcal{X} \to \mathcal{Z}$ is unramified and $\mathcal{Y} \to \mathcal{Z}$ is DM, then $\mathcal{X} \to \mathcal{Y}$ is unramified.

Proof. Assume $\mathcal{X} \to \mathcal{Z}$ is unramified. By Lemma 100.4.12 the morphism $\mathcal{X} \to \mathcal{Y}$ is DM. Choose a commutative diagram

\[ \xymatrix{ U \ar[d] \ar[r] & V \ar[d] \ar[r] & W \ar[d] \\ \mathcal{X} \ar[r] & \mathcal{Y} \ar[r] & \mathcal{Z} } \]

with $U, V, W$ algebraic spaces, with $W \to \mathcal{Z}$ surjective smooth, $V \to \mathcal{Y} \times _\mathcal {Z} W$ surjective étale, and $U \to \mathcal{X} \times _\mathcal {Y} V$ surjective étale (see Lemma 100.21.7). Then also $U \to \mathcal{X} \times _\mathcal {Z} W$ is surjective and étale. Hence we know that $U \to W$ is unramified and we have to show that $U \to V$ is unramified. This follows from Morphisms of Spaces, Lemma 66.38.11. $\square$

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