The Stacks project

Lemma 34.7.7. Let $\mathit{Sch}_{fppf}$ be a big fppf site as in Definition 34.7.6. Let $T \in \mathop{\mathrm{Ob}}\nolimits (\mathit{Sch}_{fppf})$. Let $\{ T_ i \to T\} _{i \in I}$ be an arbitrary fppf covering of $T$.

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

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

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

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 34.7.3 the refinement $\{ T_{ij} \to T\} _{i \in I, j \in J_ i}$ is an fppf 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 $\mathit{Sch}_{fppf}$. But then $T_ i$ as a finite type scheme over $W_ i$ is isomorphic to an object $V_ i$ of $\mathit{Sch}_{fppf}$ 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 $\mathit{Sch}_{fppf}$ by Sets, Lemma 3.9.9. The covering $\{ U_ j \to T\} _{j \in J}$ is a refinement as in (1). In the situation of (2), (3) each of the schemes $T_ i$ is isomorphic to an object of $\mathit{Sch}_{fppf}$ by Sets, Lemma 3.9.9, and another application of Sets, Lemma 3.11.1 gives what we want. $\square$


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