Group-scheme action

In algebraic geometry, a 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

such that

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.

Given a group-scheme action ${\displaystyle \sigma }$, for each morphism ${\displaystyle T\to S}$, ${\displaystyle \sigma }$ determines a group action ${\displaystyle G(T)\times X(T)\to X(T)}$; i.e., the group ${\displaystyle G(T)}$ acts on the set of T-points ${\displaystyle X(T)}$. Conversely, if for each ${\displaystyle T\to S}$, there is a group action ${\displaystyle \sigma _{T}:G(T)\times X(T)\to X(T)}$ and if those actions are compatible; i.e., they form a natural transformation, then, by the Yoneda lemma, they determine a group-scheme action ${\displaystyle \sigma :G\times _{S}X\to X}$.

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