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 σ : G × S X → X {\displaystyle \sigma :G\times _{S}X\to X} such that (associativity) σ ∘ ( 1 G × σ ) = σ ∘ ( m × 1 X ) {\displaystyle \sigma \circ (1_{G}\times \sigma )=\sigma \circ (m\times 1_{X})} , where m : G × S G → G {\displaystyle m:G\times _{S}G\to G} is the group law, (unitality) σ ∘ ( e × 1 X ) = 1 X {\displaystyle \sigma \circ (e\times 1_{X})=1_{X}} , where e : S → G {\displaystyle e:S\to G} is the identity section of G. A right action of G on X is defined analogously.