The Stacks project

Lemma 35.32.8. Let $f : X \to X'$ be a morphism of schemes over a base scheme $S$. Assume $\{ X \to S\} $ is an fpqc covering (for example if $f$ is surjective, flat and quasi-compact). Then the pullback functor of Lemma 35.31.6 is a fully faithful functor from the category of descent data relative to $X'/S$ to the category of descent data relative to $X/S$.

Proof. We may factor $X \to X'$ as $X \to X \times _ S X' \to X'$. The first morphism has a section, hence induces an equivalence of categories of descent data by Lemma 35.32.6. The second morphism is an fpqc covering hence induces a fully faithful functor by Lemma 35.32.4. $\square$


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