Lemma 40.10.13 (04RB)—The Stacks project

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Table of contents
Part 2: Schemes
Chapter 40: More on Groupoid Schemes
Section 40.10: Properties of groupoids on fields
Lemma 40.10.13 (cite)

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

$U = \mathop{\mathrm{Spec}}(k)$ with $k$ a field,
$s, t$ are locally of finite type,
$R$ is reduced, and
$k$ 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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