The Stacks project

Definition 79.15.1. Let $S$ be a scheme. Let $B$ be an algebraic space over $S$ Let $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $B$. Let $u \in |U|$ be a point.

1. We say $R$ is strongly split over $u$ if there exists an open subspace $P \subset R$ such that

   1. $(U, P, s|_ P, t|_ P, c|_{P \times _{s, U, t} P})$ is a groupoid in algebraic spaces over $B$,

   2. $s|_ P$, $t|_ P$ are finite, and

   3. $\{ r \in |R| : s(r) = u, t(r) = u\} \subset |P|$.

   The choice of such a $P$ will be called a strong splitting of $R$ over $u$.

2. We say $R$ is split over $u$ if there exists an open subspace $P \subset R$ such that

   1. $(U, P, s|_ P, t|_ P, c|_{P \times _{s, U, t} P})$ is a groupoid in algebraic spaces over $B$,

   2. $s|_ P$, $t|_ P$ are finite, and

   3. $\{ g \in |G| : g\text{ maps to }u\} \subset |P|$ where $G \to U$ is the stabilizer.

   The choice of such a $P$ will be called a splitting of $R$ over $u$.

3. We say $R$ is quasi-split over $u$ if there exists an open subspace $P \subset R$ such that

   1. $(U, P, s|_ P, t|_ P, c|_{P \times _{s, U, t} P})$ is a groupoid in algebraic spaces over $B$,

   2. $s|_ P$, $t|_ P$ are finite, and

   3. $e(u) \in |P|$¹.

   The choice of such a $P$ will be called a quasi-splitting of $R$ over $u$.