Definition 35.19.1. 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

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

## Post a comment

Your email address will not be published. Required fields are marked.

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).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

## Comments (0)