Definition 77.10.1. Let $B \to S$ as in Section 77.3. Let $(G, m)$ be a group algebraic space over $B$, and let $a : G \times _ B X \to X$ be an action of $G$ on the algebraic space $X$ over $B$. An $G$-equivariant quasi-coherent $\mathcal{O}_ X$-module, or simply a 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 $\mathcal{O}_{G \times _ B X}$-module map

$\alpha : a^*\mathcal{F} \longrightarrow \text{pr}_1^*\mathcal{F}$

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

1. the diagram

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

is a commutative in the category of $\mathcal{O}_{G \times _ B G \times _ B X}$-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 (77.8.1.1).

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