The Stacks project

76.15 The stabilizer group algebraic space

Please compare with Groupoids, Section 39.17. Given a groupoid in algebraic spaces we get a group algebraic space as follows.

Lemma 76.15.1. Let $B \to S$ as in Section 76.3. Let $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $B$. The algebraic space $G$ defined by the cartesian square

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

is a group algebraic space 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 a monomorphism we see that $G = j^{-1}(\Delta _{U/B})$ is a subsheaf of $R$. Thinking of it in this way, the structure morphism $G = j^{-1}(\Delta _{U/B}) \to U$ is induced by either $s$ or $t$ (it is the same), and $m$ is induced by $c$.

Definition 76.15.2. Let $B \to S$ as in Section 76.3. Let $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $B$. The group algebraic space $j^{-1}(\Delta _{U/B}) \to U$ is called the stabilizer of the groupoid in algebraic spaces $(U, R, s, t, c)$.

In the literature the stabilizer group algebraic space is often denoted $S$ (because the word stabilizer starts with an ā€œsā€ presumably); we cannot do this since we have already used $S$ for the base scheme.

Lemma 76.15.3. Let $B \to S$ as in Section 76.3. Let $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $B$, and let $G/U$ be its stabilizer. Denote $R_ t/U$ the algebraic space $R$ seen as an algebraic space 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 $T/B$ we define $a(g, r) = c(g, r)$. $\square$


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