The Stacks project

Remark 35.32.12. Lemma 35.32.11 says that morphisms of schemes satisfy fpqc descent. In other words, given a scheme $S$ and schemes $X$, $Y$ over $S$ the functor

\[ (\mathit{Sch}/S)^{opp} \longrightarrow \textit{Sets}, \quad T \longmapsto \mathop{Mor}\nolimits _ T(X_ T, Y_ T) \]

satisfies the sheaf condition for the fpqc topology. The simplest case of this is the following. Suppose that $T \to S$ is a surjective flat morphism of affines. Let $\psi _0 : X_ T \to Y_ T$ be a morphism of schemes over $T$ which is compatible with the canonical descent data. Then there exists a unique morphism $\psi : X \to Y$ whose base change to $T$ is $\psi _0$. In fact this special case follows in a straightforward manner from Lemma 35.32.4. And, in turn, that lemma is a formal consequence of the following two facts: (a) the base change functor by a faithfully flat morphism is faithful, see Lemma 35.32.2 and (b) a scheme satisfies the sheaf condition for the fpqc topology, see Lemma 35.10.7.


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