---
tag: 990YQ65S1D
kind: definition
name: Group action on a variety
msc: [14L30]
defines: RW85C34GQW
uses: [GH81SYAPPN, HV6YBS0YDH, CDS1VWB7A3, 0WCY42EV9V]
sources:
  - url: https://en.wikipedia.org/wiki/Group_action
---

## Definition

Let $k$ be a field, and let $G$ be an [[algebraic group|GH81SYAPPN]] over $k$ with multiplication $m$ and unit $e$.
Let $X$ be a [[variety|HV6YBS0YDH]] over $k$.
An *action* of $G$ on $X$ is a [[morphism|CDS1VWB7A3]] $\sigma \colon G \times_k X \to X$ such that:

1. $\sigma \circ (e \times \id_X)$ is the identity of $X$, where $\operatorname{Spec} k \times_k X$ is identified with $X$;
2. $\sigma \circ (m \times \id_X) = \sigma \circ (\id_G \times \sigma)$ as morphisms $G \times_k G \times_k X \to X$.

## Remarks

On points, with $g \cdot x = \sigma(g, x)$, the clauses say $e \cdot x = x$ and $(gh) \cdot x = g \cdot (h \cdot x)$.
So for each $k$-algebra $R$ one gets a [[group action|0WCY42EV9V]] of $G(R)$ on $X(R)$.

Example: $\mathrm{GL}_n$ acts on $\mathbb{A}^n$ by $\sigma(g, v) = g v$.
The coordinates of $g v$ are polynomials in the entries of $g$ and $v$, the identity matrix fixes every $v$, and $(gh) v = g (h v)$.

Non-example: the morphism $\sigma \colon \mathbb{G}_m \times \mathbb{A}^1 \to \mathbb{A}^1$, $\sigma(t, x) = x + t - 1$, satisfies clause 1.
It fails clause 2, since $\sigma(st, x) = x + st - 1$ while $\sigma(s, \sigma(t, x)) = x + s + t - 2$.