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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