The Stacks project

77.10 Equivariant quasi-coherent sheaves

Please compare with Groupoids, Section 39.12.

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).

Note that the commutativity of the first diagram guarantees that $(e \times 1_ X)^*\alpha $ is an idempotent operator on $\mathcal{F}$, and hence condition (2) is just the condition that it is an isomorphism.

Lemma 77.10.2. Let $B \to S$ as in Section 77.3. Let $G$ be a group algebraic space over $B$. Let $f : X \to Y$ be a $G$-equivariant morphism between algebraic spaces over $B$ endowed with $G$-actions. Then pullback $f^*$ given by $(\mathcal{F}, \alpha ) \mapsto (f^*\mathcal{F}, (1_ G \times f)^*\alpha )$ defines a functor from the category of quasi-coherent $G$-equivariant sheaves on $Y$ to the category of quasi-coherent $G$-equivariant sheaves on $X$.

Proof. Omitted. $\square$


Comments (0)


Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 043S. Beware of the difference between the letter 'O' and the digit '0'.