The Stacks project

Definition 74.3.3. Let $S$ be a scheme. Let $\{ U_ i \to U\} _{i \in I}$ be a family of morphisms of algebraic spaces over $S$ with fixed target.

  1. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ U$-module. We call the unique descent on $\mathcal{F}$ datum with respect to the covering $\{ U \to U\} $ the trivial descent datum.

  2. The pullback of the trivial descent datum to $\{ U_ i \to U\} $ is called the canonical descent datum. Notation: $(\mathcal{F}|_{U_ i}, can)$.

  3. A descent datum $(\mathcal{F}_ i, \varphi _{ij})$ for quasi-coherent sheaves with respect to the given family is said to be effective if there exists a quasi-coherent sheaf $\mathcal{F}$ on $U$ such that $(\mathcal{F}_ i, \varphi _{ij})$ is isomorphic to $(\mathcal{F}|_{U_ i}, can)$.

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