Section 39.10 (022Y): Actions of group schemes

39.10 Actions of group schemes

Let $(G, m)$ be a group and let $V$ be a set. Recall that a (left) action of $G$ on $V$ is given by a map $a : G \times V \to V$ such that

(associativity) $a(m(g, g'), v) = a(g, a(g', v))$ for all $g, g' \in G$ and $v \in V$, and
(identity) $a(e, v) = v$ for all $v \in V$.

We also say that $V$ is a $G$-set (this usually means we drop the $a$ from the notation – which is abuse of notation). A map of $G$-sets $\psi : V \to V'$ is any set map such that $\psi (a(g, v)) = a(g, \psi (v))$ for all $v \in V$.

Definition 39.10.1. Let $S$ be a scheme. Let $(G, m)$ be a group scheme over $S$.

An action of $G$ on the scheme $X/S$ is a morphism $a : G \times _ S X \to X$ over $S$ such that for every $T/S$ the map $a : G(T) \times X(T) \to X(T)$ defines the structure of a $G(T)$-set on $X(T)$.
Suppose that $X$, $Y$ are schemes over $S$ each endowed with an action of $G$. An equivariant or more precisely a $G$-equivariant morphism $\psi : X \to Y$ is a morphism of schemes over $S$ such that for every $T/S$ the map $\psi : X(T) \to Y(T)$ is a morphism of $G(T)$-sets.

In situation (1) this means that the diagrams

39.10.1.1

\begin{equation} \label{groupoids-equation-action} \vcenter { \xymatrix{ G \times _ S G \times _ S X \ar[r]_-{1_ G \times a} \ar[d]_{m \times 1_ X} & G \times _ S X \ar[d]^ a \\ G \times _ S X \ar[r]^ a & X } } \quad \quad \vcenter { \xymatrix{ G \times _ S 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

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

commutes.

Definition 39.10.2. Let $S$, $G \to S$, and $X \to S$ as in Definition 39.10.1. Let $a : G \times _ S X \to X$ be an action of $G$ on $X/S$. We say the action is free if for every scheme $T$ over $S$ 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 39.10.3. Situation as in Definition 39.10.2, The action $a$ is free if and only if

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

is a monomorphism.

Proof. Immediate from the definitions. $\square$

Comments (4)

Comment #5569 by Lucas das Dores on November 04, 2020 at 14:06

Typo: In the diagram describing situation (2) after definition 022Z, the horizontal arrows $id \times f$ and $f$ should read $id \times \psi$ and $\psi$ respectively.

Comment #5750 by Johan on November 16, 2020 at 17:14

Thanks and fixed here.

Comment #11587 by Elías Guisado on June 18, 2026 at 06:48

Maybe the following is worth to mention as a comment/remark/lemma. Let $G$ be a group. Let $S$ be a scheme. Let $X$ be an $S$ -scheme. There is a bijective correspondence between $G_S$ -actions on $X$ and group homomorphisms $G\to\operatorname{Aut}_S(X)$ . Namely, since $G_S\cong\coprod_{g\in G}S$ , giving a morphism $G_S\times_SX\to X$ is the same as giving a morphism $\coprod_{g\in G}X\to X$ over $S$ , and the latter is the same as giving a map $G\to\operatorname{Hom}_S(X,X)$ . It is not difficult to see that commutativity of the diagrams 39.10.1.1 amounts to the induced map $G\to\operatorname{Hom}_S(X,X)$ defining a group homomorphism $G\to\operatorname{Aut}_S(X)\subset\operatorname{Hom}_S(X,X)$ .

Comment #11588 by Elías Guisado on June 18, 2026 at 08:19

Another tentative addition to this section:

Lemma. Let $G$ be a group. Let $X$ be an $S$ -scheme equipped with a $G_S$ -action $a:G_S\times_SX\to X$ . Then the action is free if and only if the following condition holds:

(*) For any point $x \in X$ and $g\in G$ , if $g(x)=x$ and $g$ induces the trivial automorphism on $\kappa(x)$ , then $g=e$ is the identity.

Proof. We will use the identification of $a$ with $G\to\operatorname{Aut}_S(X)$ explained in #11587.

( $\Leftarrow$ ). It's done in the proof of Algebraic Spaces, Lemma 65.14.3.

( $\Rightarrow$ ). Let $x$ and $g$ be as in the statement and consider the morphisms $\operatorname{Spec}(\kappa(x))\overset{g}{\underset{e}{\rightrightarrows}} G_S\times_SX\cong_S\coprod_{g'\in G}X$ given by the canonical morphism $\operatorname{Spec}(\kappa(x))\to X$ followed by the insertion into the $g$ -th and $e$ -th components. These two morphisms become equal after postcomposition by $G_S\times_SX\to X\times_SX$ . But the latter morphism is monic (Lemma 39.10.3), hence $g=e$ . $\square$

The Stacks project

39.10 Actions of group schemes

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