Lemma 76.7.2. In Situation 76.7.1. 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 72.9.3. 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 66.30). 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. $\square$

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 083H. Beware of the difference between the letter 'O' and the digit '0'.