Definition 72.13.1. Let $S$ be a scheme. Let $\mathcal{P}$ be a property of morphisms of algebraic spaces over $S$. Let $\tau \in \{ fpqc, \linebreak[0] fppf, \linebreak[0] syntomic, \linebreak[0] smooth, \linebreak[0] {\acute{e}tale}\} $. We say $\mathcal{P}$ is *$\tau $ local on the source*, or *local on the source for the $\tau $-topology* if for any morphism $f : X \to Y$ of algebraic spaces over $S$, and any $\tau $-covering $\{ X_ i \to X\} _{i \in I}$ of algebraic spaces we have

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

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