The Stacks project

Definition 74.22.11. Let $S$ be a scheme. Let $\{ X_ i \to X\} $ be a family of morphisms of algebraic spaces over $S$ with fixed target $X$.

  1. Given an algebraic space $U$ over $X$ we have a canonical descent datum on the family of algebraic spaces $X_ i \times _ X U$ by pulling back the trivial descent datum for $U$ relative to $\{ \text{id} : S \to S\} $. We denote this descent datum $(X_ i \times _ X U, can)$.

  2. A descent datum $(V_ i, \varphi _{ij})$ relative to $\{ X_ i \to S\} $ is called effective if there exists an algebraic space $U$ over $X$ such that $(V_ i, \varphi _{ij})$ is isomorphic to $(X_ i \times _ X U, can)$.

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