Definition 39.10.1. Let $S$ be a scheme. Let $(G, m)$ be a group scheme over $S$.
An action of $G$ on the scheme $X/S$ is a morphism $a : G \times _ S X \to X$ over $S$ such that for every $T/S$ the map $a : G(T) \times X(T) \to X(T)$ defines the structure of a $G(T)$-set on $X(T)$.
Suppose that $X$, $Y$ are schemes over $S$ each endowed with an action of $G$. An equivariant or more precisely a $G$-equivariant morphism $\psi : X \to Y$ is a morphism of schemes over $S$ such that for every $T/S$ the map $\psi : X(T) \to Y(T)$ is a morphism of $G(T)$-sets.