Definition 35.22.1. Let $\mathcal{P}$ be a property of morphisms of schemes over a base. Let $\tau \in \{ fpqc, fppf, syntomic, smooth, {\acute{e}tale}, Zariski\}$. We say $\mathcal{P}$ is $\tau$ local on the base, or $\tau$ local on the target, or local on the base for the $\tau$-topology if for any $\tau$-covering $\{ Y_ i \to Y\} _{i \in I}$ (see Topologies, Section 34.2) and any morphism of schemes $f : X \to Y$ over $S$ we have

$f \text{ has }\mathcal{P} \Leftrightarrow \text{each }Y_ i \times _ Y X \to Y_ i\text{ has }\mathcal{P}.$

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).