The Stacks project

Definition 39.12.1. Let $S$ be a scheme, let $(G, m)$ be a group scheme over $S$, and let $a : G \times _ S X \to X$ be an action of the group scheme $G$ on $X/S$. A $G$-equivariant quasi-coherent $\mathcal{O}_ X$-module, or simply an equivariant quasi-coherent $\mathcal{O}_ X$-module, is a pair $(\mathcal{F}, \alpha )$, where $\mathcal{F}$ is a quasi-coherent $\mathcal{O}_ X$-module, and $\alpha $ is a $\mathcal{O}_{G \times _ S X}$-module map

where $\text{pr}_1 : G \times _ S X \to X$ is the projection such that

1. the diagram

   \[ \xymatrix{ (1_ G \times a)^*\text{pr}_1^*\mathcal{F} \ar[r]_-{\text{pr}_{12}^*\alpha } & \text{pr}_2^*\mathcal{F} \\ (1_ G \times a)^*a^*\mathcal{F} \ar[u]^{(1_ G \times a)^*\alpha } \ar@{=}[r] & (m \times 1_ X)^*a^*\mathcal{F} \ar[u]_{(m \times 1_ X)^*\alpha } } \]

   is a commutative in the category of $\mathcal{O}_{G \times _ S G \times _ S X}$-modules, and

2. the pullback

   \[ (e \times 1_ X)^*\alpha : \mathcal{F} \longrightarrow \mathcal{F} \]

   is the identity map.

For explanation compare with the relevant diagrams of Equation (39.10.1.1).