79.8 Restricting groupoids
In this section we collect a bunch of lemmas on properties of groupoids which are inherited by restrictions. Most of these lemmas can be proved by contemplating the defining diagram
79.8.0.1
\begin{equation} \label{spaces-more-groupoids-equation-restriction} \vcenter { \xymatrix{ R' \ar[d] \ar[r] \ar@/_3pc/[dd]_{t'} \ar@/^1pc/[rr]^{s'}& R \times _{s, U} U' \ar[r] \ar[d] & U' \ar[d]^ g \\ U' \times _{U, t} R \ar[d] \ar[r] & R \ar[r]^ s \ar[d]_ t & U \\ U' \ar[r]^ g & U } } \end{equation}
of a restriction. See Groupoids in Spaces, Lemma 78.17.1.
Lemma 79.8.1. 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 $g : U' \to U$ be a morphism of algebraic spaces over $B$. Let $(U', R', s', t', c')$ be the restriction of $(U, R, s, t, c)$ via $g$.
If $s, t$ are locally of finite type and $g$ is locally of finite type, then $s', t'$ are locally of finite type.
If $s, t$ are locally of finite presentation and $g$ is locally of finite presentation, then $s', t'$ are locally of finite presentation.
If $s, t$ are flat and $g$ is flat, then $s', t'$ are flat.
Add more here.
Proof.
The property of being locally of finite type is stable under composition and arbitrary base change, see Morphisms of Spaces, Lemmas 67.23.2 and 67.23.3. Hence (1) is clear from Diagram (79.8.0.1). For the other cases, see Morphisms of Spaces, Lemmas 67.28.2, 67.28.3, 67.30.3, and 67.30.4.
$\square$
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