Group-scheme action

In algebraic geometry, an action of a group scheme is a generalization of a group action to a group scheme. Precisely, given a group S-scheme G, a left action of G on an S-scheme X is an S-morphism

${\displaystyle \sigma :G\times _{S}X\to X}$

such that

• (associativity) ${\displaystyle \sigma \circ (1_{G}\times \sigma )=\sigma \circ (m\times 1_{X})}$, where ${\displaystyle m:G\times _{S}G\to G}$ is the group law,
• (unitality) ${\displaystyle \sigma \circ (e\times 1_{X})=1_{X}}$, where ${\displaystyle e:S\to G}$ is the identity section of G.

A right action of G on X is defined analogously. A scheme equipped with a left or right action of a group scheme G is called a G-scheme. An equivariant morphism between G-schemes is a morphism of schemes that intertwines the respective G-actions.

More generally, one can also consider (at least some special case of) an action of a group functor: viewing G as a functor, an action is given as a natural transformation satisfying the conditions analogous to the above. Alternatively, some authors study group action in the language of a groupoid; a group-scheme action is then an example of a groupoid scheme.