The Stacks project

Lemma 38.34.12. Let $\mathit{Sch}_ h$ be a big h site as in Definition 38.34.10. Let $T \in \mathop{\mathrm{Ob}}\nolimits (\mathit{Sch}_ h)$. Let $\{ T_ i \to T\} _{i \in I}$ be an arbitrary h covering of $T$.

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

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

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