Definition 74.10.1. Let $S$ be a scheme. Let $\mathcal{P}$ be a property of morphisms of algebraic spaces over $S$. Let $\tau \in \{ fpqc, fppf, syntomic, smooth, {\acute{e}tale}\} $. 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 $\{ Y_ i \to Y\} _{i \in I}$ of algebraic spaces and any morphism of algebraic spaces $f : X \to Y$ we have

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

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