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The Stacks project

Lemma 34.6.7. Let \mathit{Sch}_{syntomic} be a big syntomic site as in Definition 34.6.6. Let T \in \mathop{\mathrm{Ob}}\nolimits (\mathit{Sch}_{syntomic}). Let \{ T_ i \to T\} _{i \in I} be an arbitrary syntomic covering of T.

  1. There exists a covering \{ U_ j \to T\} _{j \in J} of T in the site \mathit{Sch}_{syntomic} which refines \{ T_ i \to T\} _{i \in I}.

  2. If \{ T_ i \to T\} _{i \in I} is a standard syntomic covering, then it is tautologically equivalent to a covering in \mathit{Sch}_{syntomic}.

  3. If \{ T_ i \to T\} _{i \in I} is a Zariski covering, then it is tautologically equivalent to a covering in \mathit{Sch}_{syntomic}.

Proof. For each i choose an affine open covering T_ i = \bigcup _{j \in J_ i} T_{ij} such that each T_{ij} maps into an affine open subscheme of T. By Lemma 34.6.3 the refinement \{ T_{ij} \to T\} _{i \in I, j \in J_ i} is a syntomic covering of T as well. Hence we may assume each T_ i is affine, and maps into an affine open W_ i of T. Applying Sets, Lemma 3.9.9 we see that W_ i is isomorphic to an object of \mathit{Sch}_{syntomic}. But then T_ i as a finite type scheme over W_ i is isomorphic to an object V_ i of \mathit{Sch}_{syntomic} by a second application of Sets, Lemma 3.9.9. The covering \{ V_ i \to T\} _{i \in I} refines \{ T_ i \to T\} _{i \in I} (because they are isomorphic). Moreover, \{ V_ i \to T\} _{i \in I} is combinatorially equivalent to a covering \{ U_ j \to T\} _{j \in J} of T in the site \mathit{Sch}_{syntomic} by Sets, Lemma 3.9.9. The covering \{ U_ j \to T\} _{j \in J} is a covering as in (1). In the situation of (2), (3) each of the schemes T_ i is isomorphic to an object of \mathit{Sch}_{syntomic} by Sets, Lemma 3.9.9, and another application of Sets, Lemma 3.11.1 gives what we want. \square


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