The Stacks project

Definition 74.22.10. Let $S$ be a scheme. Let $f : Y \to X$ be a morphism of algebraic spaces over $S$.

  1. Given an algebraic space $U$ over $X$ we have the trivial descent datum of $U$ relative to $\text{id} : X \to X$, namely the identity morphism on $U$.

  2. By Lemma 74.22.6 we get a canonical descent datum on $Y \times _ X U$ relative to $Y \to X$ by pulling back the trivial descent datum via $f$. We often denote $(Y \times _ X U, can)$ this descent datum.

  3. A descent datum $(V, \varphi )$ relative to $Y/X$ is called effective if $(V, \varphi )$ is isomorphic to the canonical descent datum $(Y \times _ X U, can)$ for some algebraic space $U$ over $X$.

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