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

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 $U$.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.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 $S$.

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