The Stacks project

Definition 58.16.1. Let $\mathcal{U} = \{ t_ i : T_ i \to T\} _{i \in I}$ be a family of morphisms of schemes with fixed target. A descent datum for quasi-coherent sheaves with respect to $\mathcal{U}$ is a family $(\mathcal{F}_ i, \varphi _{ij})_{i, j\in I}$ where

  1. for all $i$, $\mathcal{F}_ i$ is a quasi-coherent sheaf on $T_ i$, and

  2. for all $i, j \in I$ the map $\varphi _{ij} : \text{pr}_0^* \mathcal{F}_ i \cong \text{pr}_1^* \mathcal{F}_ j$ is an isomorphism on $T_ i \times _ T T_ j$ such that the diagrams

    \[ \xymatrix{ \text{pr}_0^*\mathcal{F}_ i \ar[dr]_{\text{pr}_{02}^*\varphi _{ik}} \ar[rr]^{\text{pr}_{01}^*\varphi _{ij}} & & \text{pr}_1^*\mathcal{F}_ j \ar[dl]^{\text{pr}_{12}^*\varphi _{jk}} \\ & \text{pr}_2^*\mathcal{F}_ k } \]

    commute on $T_ i \times _ T T_ j \times _ T T_ k$.

This descent datum is called effective if there exist a quasi-coherent sheaf $\mathcal{F}$ over $T$ and $\mathcal{O}_{T_ i}$-module isomorphisms $\varphi _ i : t_ i^* \mathcal{F} \cong \mathcal{F}_ i$ satisfying the cocycle condition, namely

\[ \varphi _{ij} = \text{pr}_1^* (\varphi _ j) \circ \text{pr}_0^* (\varphi _ i)^{-1}. \]

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