The Stacks project

Definition 39.14.1. Let $S$ be a scheme, let $(U, R, s, t, c)$ be a groupoid scheme over $S$. A quasi-coherent module on $(U, R, s, t, c)$ is a pair $(\mathcal{F}, \alpha )$, where $\mathcal{F}$ is a quasi-coherent $\mathcal{O}_ U$-module, and $\alpha $ is a $\mathcal{O}_ R$-module map

\[ \alpha : t^*\mathcal{F} \longrightarrow s^*\mathcal{F} \]

such that

  1. the diagram

    \[ \xymatrix{ & \text{pr}_1^*t^*\mathcal{F} \ar[r]_-{\text{pr}_1^*\alpha } & \text{pr}_1^*s^*\mathcal{F} \ar@{=}[rd] & \\ \text{pr}_0^*s^*\mathcal{F} \ar@{=}[ru] & & & c^*s^*\mathcal{F} \\ & \text{pr}_0^*t^*\mathcal{F} \ar[lu]^{\text{pr}_0^*\alpha } \ar@{=}[r] & c^*t^*\mathcal{F} \ar[ru]_{c^*\alpha } } \]

    is a commutative in the category of $\mathcal{O}_{R \times _{s, U, t} R}$-modules, and

  2. the pullback

    \[ e^*\alpha : \mathcal{F} \longrightarrow \mathcal{F} \]

    is the identity map.

Compare with the commutative diagrams of Lemma 39.13.4.

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