Definition 35.26.1. Let $\mathcal{P}$ be a property of morphisms of schemes. Let $\tau \in \{ Zariski, \linebreak[0] fpqc, \linebreak[0] fppf, \linebreak[0] {\acute{e}tale}, \linebreak[0] smooth, \linebreak[0] syntomic\} $. We say $\mathcal{P}$ is $\tau $ local on the source, or local on the source for the $\tau $-topology if for any morphism of schemes $f : X \to Y$ over $S$, and any $\tau $-covering $\{ X_ i \to X\} _{i \in I}$ we have
\[ f \text{ has }\mathcal{P} \Leftrightarrow \text{each }X_ i \to Y\text{ has }\mathcal{P}. \]
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