Gruppenschema/Gruppenoperation/en/Definition

Group scheme action

Let ${}(G,\alpha ,n)$ denote a group scheme over a scheme ${}X$ and let

$T\longrightarrow X$

denote a scheme over ${}X$. A morphism

$\beta \colon G\times _{X}T\longrightarrow T$

is called a group scheme action of ${}G$ on ${}T$, if the diagram

${\begin{matrix}G\times _{X}G\times _{X}T&{\stackrel {\operatorname {Id} _{G}\times \beta }{\longrightarrow }}&G\times _{X}T&\\\!\!\!\!\!\alpha \times \operatorname {Id} _{T}\downarrow &&\downarrow \beta \!\!\!\!\!&\\G\times _{X}T&{\stackrel {\beta }{\longrightarrow }}&T&\!\!\!\!\!\\\end{matrix}}$

commutes and if the composition

$T{\stackrel {\cong }{\longrightarrow }}X\times _{X}T\,{\stackrel {n\times \operatorname {Id} _{T}}{\longrightarrow }}\,G\times _{X}T{\stackrel {\beta }{\longrightarrow }}T$

is the identity on ${}T$.