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
\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

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


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. $\square$

Comments (0)

Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 043P. Beware of the difference between the letter 'O' and the digit '0'.