The Stacks project

Lemma 61.31.1. Let $\mathit{Sch}_{pro\text{-}\acute{e}tale}$ be a big pro-étale site as in Definition 61.12.7. Let $T \in \mathop{\mathrm{Ob}}\nolimits (\mathit{Sch}_{pro\text{-}\acute{e}tale})$. Let $\{ T_ i \to T\} _{i \in I}$ be an arbitrary pro-étale covering of $T$. There exists a covering $\{ U_ j \to T\} _{j \in J}$ of $T$ in the site $\mathit{Sch}_{pro\text{-}\acute{e}tale}$ which refines $\{ T_ i \to T\} _{i \in I}$.

Proof. Namely, we first let $\{ V_ k \to T\} $ be a covering as in Lemma 61.13.3. Then the pro-étale coverings $\{ T_ i \times _ T V_ k \to V_ k\} $ can be refined by a finite disjoint open covering $V_ k = V_{k, 1} \amalg \ldots \amalg V_{k, n_ k}$, see Lemma 61.13.1. Then $\{ V_{k, i} \to T\} $ is a covering of $\mathit{Sch}_{pro\text{-}\acute{e}tale}$ which refines $\{ T_ i \to T\} _{i \in I}$. $\square$

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