Definition 35.15.1. 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

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

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