The Stacks project

Definition 74.3.1. Let $S$ be a scheme. Let $\{ f_ i : X_ i \to X\} _{i \in I}$ be a family of morphisms of algebraic spaces over $S$ with fixed target $X$.

  1. A descent datum $(\mathcal{F}_ i, \varphi _{ij})$ for quasi-coherent sheaves with respect to the given family is given by a quasi-coherent sheaf $\mathcal{F}_ i$ on $X_ i$ for each $i \in I$, an isomorphism of quasi-coherent $\mathcal{O}_{X_ i \times _ X X_ j}$-modules $\varphi _{ij} : \text{pr}_0^*\mathcal{F}_ i \to \text{pr}_1^*\mathcal{F}_ j$ for each pair $(i, j) \in I^2$ such that for every triple of indices $(i, j, k) \in I^3$ the diagram

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

    of $\mathcal{O}_{X_ i \times _ X X_ j \times _ X X_ k}$-modules commutes. This is called the cocycle condition.

  2. A morphism $\psi : (\mathcal{F}_ i, \varphi _{ij}) \to (\mathcal{F}'_ i, \varphi '_{ij})$ of descent data is given by a family $\psi = (\psi _ i)_{i\in I}$ of morphisms of $\mathcal{O}_{X_ i}$-modules $\psi _ i : \mathcal{F}_ i \to \mathcal{F}'_ i$ such that all the diagrams

    \[ \xymatrix{ \text{pr}_0^*\mathcal{F}_ i \ar[r]_{\varphi _{ij}} \ar[d]_{\text{pr}_0^*\psi _ i} & \text{pr}_1^*\mathcal{F}_ j \ar[d]^{\text{pr}_1^*\psi _ j} \\ \text{pr}_0^*\mathcal{F}'_ i \ar[r]^{\varphi '_{ij}} & \text{pr}_1^*\mathcal{F}'_ j \\ } \]


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