The Stacks project

77.8 Actions of group algebraic spaces

Please refer to Groupoids, Section 39.10 for notation.

Definition 77.8.1. Let $B \to S$ as in Section 77.3. Let $(G, m)$ be a group algebraic space over $B$. Let $X$ be an algebraic space over $B$.

  1. An action of $G$ on the algebraic space $X/B$ is a morphism $a : G \times _ B X \to X$ over $B$ such that for every scheme $T$ over $B$ the map $a : G(T) \times X(T) \to X(T)$ defines the structure of a $G(T)$-set on $X(T)$.

  2. Suppose that $X$, $Y$ are algebraic spaces over $B$ each endowed with an action of $G$. An equivariant or more precisely a $G$-equivariant morphism $\psi : X \to Y$ is a morphism of algebraic spaces over $B$ such that for every $T$ over $B$ the map $\psi : X(T) \to Y(T)$ is a morphism of $G(T)$-sets.

In situation (1) this means that the diagrams

77.8.1.1
\begin{equation} \label{spaces-groupoids-equation-action} \xymatrix{ G \times _ B G \times _ B X \ar[r]_-{1_ G \times a} \ar[d]_{m \times 1_ X} & G \times _ B X \ar[d]^ a \\ G \times _ B X \ar[r]^ a & X } \quad \xymatrix{ G \times _ B X \ar[r]_-a & X \\ X\ar[u]^{e \times 1_ X} \ar[ru]_{1_ X} } \end{equation}

are commutative. In situation (2) this just means that the diagram

\[ \xymatrix{ G \times _ B X \ar[r]_-{\text{id} \times f} \ar[d]_ a & G \times _ B Y \ar[d]^ a \\ X \ar[r]^ f & Y } \]

commutes.

Definition 77.8.2. Let $B \to S$, $G \to B$, and $X \to B$ as in Definition 77.8.1. Let $a : G \times _ B X \to X$ be an action of $G$ on $X/B$. We say the action is free if for every scheme $T$ over $B$ the action $a : G(T) \times X(T) \to X(T)$ is a free action of the group $G(T)$ on the set $X(T)$.

Lemma 77.8.3. Situation as in Definition 77.8.2, The action $a$ is free if and only if

\[ G \times _ B X \to X \times _ B X, \quad (g, x) \mapsto (a(g, x), x) \]

is a monomorphism of algebraic spaces.

Proof. Immediate from the definitions. $\square$


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