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

$f \text{ has }\mathcal{P} \Leftrightarrow \text{each }X_ i \to Y\text{ has }\mathcal{P}.$

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).