## 39.17 The stabilizer group scheme

Given a groupoid scheme we get a group scheme as follows.

Lemma 39.17.1. Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid over $S$. The scheme $G$ defined by the cartesian square

$\xymatrix{ G \ar[r] \ar[d] & R \ar[d]^{j = (t, s)} \\ U \ar[r]^-{\Delta } & U \times _ S U }$

is a group scheme over $U$ with composition law $m$ induced by the composition law $c$.

Proof. This is true because in a groupoid category the set of self maps of any object forms a group. $\square$

Since $\Delta$ is an immersion we see that $G = j^{-1}(\Delta _{U/S})$ is a locally closed subscheme of $R$. Thinking of it in this way, the structure morphism $j^{-1}(\Delta _{U/S}) \to U$ is induced by either $s$ or $t$ (it is the same), and $m$ is induced by $c$.

Definition 39.17.2. Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid over $S$. The group scheme $j^{-1}(\Delta _{U/S})\to U$ is called the stabilizer of the groupoid scheme $(U, R, s, t, c)$.

In the literature the stabilizer group scheme is often denoted $S$ (because the word stabilizer starts with an “s” presumably); we cannot do this since we have already used $S$ for the base scheme.

Lemma 39.17.3. Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid over $S$, and let $G/U$ be its stabilizer. Denote $R_ t/U$ the scheme $R$ seen as a scheme over $U$ via the morphism $t : R \to U$. There is a canonical left action

$a : G \times _ U R_ t \longrightarrow R_ t$

induced by the composition law $c$.

Proof. In terms of points over $T/S$ we define $a(g, r) = c(g, r)$. $\square$

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 $\mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(y, y)$. $\square$

Lemma 39.17.5. Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid scheme over $S$. Let $p \in U \times _ S U$ be a point. Denote $R_ p$ the scheme theoretic fibre of $j = (t, s) : R \to U \times _ S U$. If $R_ p \not= \emptyset$, then the action

$G_{0, \kappa (p)} \times _{\kappa (p)} R_ p \longrightarrow R_ p$

(see Lemma 39.17.4) which turns $R_ p$ into a $G_{\kappa (p)}$-torsor over $\kappa (p)$.

Proof. The action is a pseudo-torsor by the lemma cited in the statement. And if $R_ p$ is not the empty scheme, then $\{ R_ p \to p\}$ is an fpqc covering which trivializes the pseudo-torsor. $\square$

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