Remark 34.11.1. Take any category $\mathit{Sch}_\alpha $ constructed as in Sets, Lemma 3.9.2 starting with the set of schemes $\{ X, Y, S\} $. Choose any set of coverings $\text{Cov}_{fppf}$ on $\mathit{Sch}_\alpha $ as in Sets, Lemma 3.11.1 starting with the category $\mathit{Sch}_\alpha $ and the class of fppf coverings. Let $\mathit{Sch}_{fppf}$ denote the big fppf site so obtained. Next, for $\tau \in \{ Zariski, {\acute{e}tale}, smooth, syntomic\} $ let $\mathit{Sch}_\tau $ have the same underlying category as $\mathit{Sch}_{fppf}$ with coverings $\text{Cov}_\tau \subset \text{Cov}_{fppf}$ simply the subset of $\tau $-coverings. It is straightforward to check that this gives rise to a big site $\mathit{Sch}_\tau $.
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