The Stacks project

Lemma 40.10.13. Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid scheme over $S$. Assume

  1. $U = \mathop{\mathrm{Spec}}(k)$ with $k$ a field,

  2. $s, t$ are locally of finite type,

  3. $R$ is reduced, and

  4. $k$ is perfect.

Then $s, t : R \to U$ are smooth.

Proof. By Lemma 40.4.1 the sheaf $\Omega _{R/U}$ is free. Hence the lemma follows from Varieties, Lemma 33.25.2. $\square$

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