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

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

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 (

Comments (3)

Comment #6377 by Peng DU on

I find in some other books, the map 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 is an idempotent operator on F, and hence condition (2) is just the condition that it is an isomorphism." I think if we assume 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 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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