## 78.6 Properties of groupoids

This section is the analogue of More on Groupoids, Section 40.6. The reader is strongly encouraged to read that section first.

The following lemma is the analogue of More on Groupoids, Lemma 40.6.4.

Lemma 78.6.1. Let $B \to S$ be as in Section 78.2. Let $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $B$. Let $\tau \in \{ fppf, \linebreak[0] {\acute{e}tale}, \linebreak[0] smooth, \linebreak[0] syntomic\}$. Let $\mathcal{P}$ be a property of morphisms of algebraic spaces which is $\tau$-local on the target (Descent on Spaces, Definition 73.10.1). Assume $\{ s : R \to U\}$ and $\{ t : R \to U\}$ are coverings for the $\tau$-topology. Let $W \subset U$ be the maximal open subspace such that $s^{-1}(W) \to W$ has property $\mathcal{P}$. Then $W$ is $R$-invariant (Groupoids in Spaces, Definition 77.18.1).

Proof. The existence and properties of the open $W \subset U$ are described in Descent on Spaces, Lemma 73.10.3. In Diagram (78.3.0.1) let $W_1 \subset R$ be the maximal open subscheme over which the morphism $\text{pr}_1 : R \times _{s, U, t} R \to R$ has property $\mathcal{P}$. It follows from the aforementioned Descent on Spaces, Lemma 73.10.3 and the assumption that $\{ s : R \to U\}$ and $\{ t : R \to U\}$ are coverings for the $\tau$-topology that $t^{-1}(W) = W_1 = s^{-1}(W)$ as desired. $\square$

Lemma 78.6.2. Let $B \to S$ be as in Section 78.2. Let $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $B$. Let $G \to U$ be its stabilizer group algebraic space. Let $\tau \in \{ fppf, \linebreak[0] {\acute{e}tale}, \linebreak[0] smooth, \linebreak[0] syntomic\}$. Let $\mathcal{P}$ be a property of morphisms of algebraic spaces which is $\tau$-local on the target. Assume $\{ s : R \to U\}$ and $\{ t : R \to U\}$ are coverings for the $\tau$-topology. Let $W \subset U$ be the maximal open subspace such that $G_ W \to W$ has property $\mathcal{P}$. Then $W$ is $R$-invariant (see Groupoids in Spaces, Definition 77.18.1).

Proof. The existence and properties of the open $W \subset U$ are described in Descent on Spaces, Lemma 73.10.3. The morphism

$G \times _{U, t} R \longrightarrow R \times _{s, U} G, \quad (g, r) \longmapsto (r, r^{-1} \circ g \circ r)$

is an isomorphism of algebraic spaces over $R$ (where $\circ$ denotes composition in the groupoid). Hence $s^{-1}(W) = t^{-1}(W)$ by the properties of $W$ proved in the aforementioned Descent on Spaces, Lemma 73.10.3. $\square$

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