Lemma 39.17.4. Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid scheme over $S$. Let $G$ be the stabilizer group scheme of $R$. Let

$G_0 = G \times _{U, \text{pr}_0} (U \times _ S U) = G \times _ S U$

as a group scheme over $U \times _ S U$. The action of $G$ on $R$ of Lemma 39.17.3 induces an action of $G_0$ on $R$ over $U \times _ S U$ which turns $R$ into a pseudo $G_0$-torsor over $U \times _ S U$.

Proof. This is true because in a groupoid category $\mathcal{C}$ the set $\mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(x, y)$ is a principal homogeneous set under the group $\mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(y, y)$. $\square$

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