The Stacks project

Lemma 39.17.5. Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid scheme over $S$. Let $p \in U \times _ S U$ be a point. Denote $R_ p$ the scheme theoretic fibre of $j = (t, s) : R \to U \times _ S U$. If $R_ p \not= \emptyset $, then the action

\[ G_{0, \kappa (p)} \times _{\kappa (p)} R_ p \longrightarrow R_ p \]

(see Lemma 39.17.4) which turns $R_ p$ into a $G_{\kappa (p)}$-torsor over $\kappa (p)$.

Proof. The action is a pseudo-torsor by the lemma cited in the statement. And if $R_ p$ is not the empty scheme, then $\{ R_ p \to p\} $ is an fpqc covering which trivializes the pseudo-torsor. $\square$

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