Definition 59.20.1. (See Topologies, Definitions 34.7.1, 34.6.1, 34.5.1, 34.4.1, and 34.3.1.) Let $\tau \in \{ fppf, syntomic, smooth, {\acute{e}tale}, Zariski\}$. A family of morphisms of schemes $\{ f_ i : T_ i \to T\} _{i \in I}$ with fixed target is called a $\tau$-covering if and only if each $f_ i$ is flat of finite presentation, syntomic, smooth, étale, resp. an open immersion, and we have $\bigcup f_ i(T_ i) = T$.

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