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

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

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).

Comment #6377 by Peng DU on

I find in some other books, the map $\alpha$ is assumed to be an isomorphism, don't we need this assumption here, or will it be a consequence in some way?

And I noted that after this Definition you wrote "Note that the commutativity of the first diagram guarantees that $(e\times 1_X)^*\alpha$ is an idempotent operator on F, and hence condition (2) is just the condition that it is an isomorphism." I think if we assume $\alpha$ is an isomorphism, then (2) will be automatic.

The notion of an equivariant quasi-coherent module is always mysterious for me.

Comment #6378 by Peng DU on

Sorry that I just noted my above issue is already treated in other Comments to this section.

Comment #6379 by on

Yes, you are right (both times). As I said on the other page, given condition (1) condition (2) is equivalent to the condition that $\alpha$ is an isomorphism and this is how usually the definition is made.

There are also:

• 12 comment(s) on Section 39.12: Equivariant quasi-coherent sheaves

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