The Stacks project

Situation 40.12.1. Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid scheme over $S$. Let $g : U' \to U$ be a morphism of schemes. Let $u \in U$ be a point, and let $u' \in U'$ be a point such that $g(u') = u$. Given these data, denote $(U', R', s', t', c')$ the restriction of $(U, R, s, t, c)$ via the morphism $g$. Denote $G \to U$ the stabilizer group scheme of $R$, which is a locally closed subscheme of $R$. Denote $h$ the composition

\[ h = s \circ \text{pr}_1 : U' \times _{g, U, t} R \longrightarrow U. \]

Denote $F_ u = s^{-1}(u)$ (scheme theoretic fibre), and $G_ u$ the scheme theoretic fibre of $G$ over $u$. Similarly for $R'$ we denote $F'_{u'} = (s')^{-1}(u')$. Because $g(u') = u$ we have

\[ F'_{u'} = h^{-1}(u) \times _{\mathop{\mathrm{Spec}}(\kappa (u))} \mathop{\mathrm{Spec}}(\kappa (u')). \]

The point $e(u) \in R$ may be viewed as a point on $G_ u$ and $F_ u$ also, and $e'(u')$ is a point of $R'$ (resp. $G'_{u'}$, resp. $F'_{u'}$) which maps to $e(u)$ in $R$ (resp. $G_ u$, resp. $F_ u$).

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