The Stacks project

Definition 35.34.3. Let $S$ be a scheme. Let $\{ X_ i \to S\} _{i \in I}$ be a family of morphisms with target $S$.

  1. A descent datum $(V_ i, \varphi _{ij})$ relative to the family $\{ X_ i \to S\} $ is given by a scheme $V_ i$ over $X_ i$ for each $i \in I$, an isomorphism $\varphi _{ij} : V_ i \times _ S X_ j \to X_ i \times _ S V_ j$ of schemes over $X_ i \times _ S X_ j$ for each pair $(i, j) \in I^2$ such that for every triple of indices $(i, j, k) \in I^3$ the diagram

    \[ \xymatrix{ V_ i \times _ S X_ j \times _ S X_ k \ar[rd]^{\text{pr}_{01}^*\varphi _{ij}} \ar[rr]_{\text{pr}_{02}^*\varphi _{ik}} & & X_ i \times _ S X_ j \times _ S V_ k\\ & X_ i \times _ S V_ j \times _ S X_ k \ar[ru]^{\text{pr}_{12}^*\varphi _{jk}} } \]

    of schemes over $X_ i \times _ S X_ j \times _ S X_ k$ commutes (with obvious notation).

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

    \[ \xymatrix{ V_ i \times _ S X_ j \ar[r]_{\varphi _{ij}} \ar[d]_{\psi _ i \times \text{id}} & X_ i \times _ S V_ j \ar[d]^{\text{id} \times \psi _ j} \\ V'_ i \times _ S X_ j \ar[r]^{\varphi '_{ij}} & X_ i \times _ S V'_ j } \]


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