The Stacks project

Definition 35.34.10. Let $S$ be a scheme. Let $f : X \to S$ be a morphism of schemes.

1. Given a scheme $U$ over $S$ we have the trivial descent datum of $U$ relative to $\text{id} : S \to S$, namely the identity morphism on $U$.

2. By Lemma 35.34.6 we get a canonical descent datum on $X \times _ S U$ relative to $X \to S$ by pulling back the trivial descent datum via $f$. We often denote $(X \times _ S U, can)$ this descent datum.

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