Definition 59.20.4. (See Topologies, Definitions 34.7.5, 34.6.5, 34.5.5, 34.4.5, and 34.3.4.) Let $\tau \in \{ fppf, syntomic, smooth, {\acute{e}tale}, Zariski\} $. Let $T$ be an affine scheme. A *standard $\tau $-covering* of $T$ is a family $\{ f_ j : U_ j \to T\} _{j = 1, \ldots , m}$ with each $U_ j$ is affine, and each $f_ j$ flat and of finite presentation, standard syntomic, standard smooth, étale, resp. the immersion of a standard principal open in $T$ and $T = \bigcup f_ j(U_ j)$.

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