The Stacks project

Lemma 74.7.2. In Situation 74.7.1. Each of the functors $F_{iso}$, $F_{inj}$, $F_{surj}$, $F_{zero}$ satisfies the sheaf property for the fpqc topology.

Proof. Let $\{ T_ i \to T\} _{i \in I}$ be an fpqc covering of schemes over $B$. Set $X_ i = X_{T_ i} = X \times _ S T_ i$ and $u_ i = u_{T_ i}$. Note that $\{ X_ i \to X_ T\} _{i \in I}$ is an fpqc covering of $X_ T$, see Topologies on Spaces, Lemma 70.9.3. In particular, for every $x \in |X_ T|$ there exists an $i \in I$ and an $x_ i \in |X_ i|$ mapping to $x$. Since $\mathcal{O}_{X_ T, \overline{x}} \to \mathcal{O}_{X_ i, \overline{x_ i}}$ is flat, hence faithfully flat (see Morphisms of Spaces, Section 64.30). we conclude that $(u_ i)_{x_ i}$ is injective, surjective, bijective, or zero if and only if $(u_ T)_ x$ is injective, surjective, bijective, or zero. The lemma follows. $\square$


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