Definition 35.12.1. Let $\mathcal{P}$ be a property of schemes. Let $\tau \in \{ fpqc, \linebreak[0] fppf, \linebreak[0] syntomic, \linebreak[0] smooth, \linebreak[0] {\acute{e}tale}, \linebreak[0] Zariski\}$. We say $\mathcal{P}$ is local in the $\tau$-topology if for any $\tau$-covering $\{ S_ i \to S\} _{i \in I}$ (see Topologies, Section 34.2) we have

$S \text{ has }\mathcal{P} \Leftrightarrow \text{each }S_ i \text{ has }\mathcal{P}.$

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).