The Stacks project

Lemma 34.3.6. Let $\mathit{Sch}_{Zar}$ be a big Zariski site as in Definition 34.3.5. Let $T \in \mathop{\mathrm{Ob}}\nolimits (\mathit{Sch}_{Zar})$. Let $\{ T_ i \to T\} _{i \in I}$ be an arbitrary Zariski covering of $T$. There exists a covering $\{ U_ j \to T\} _{j \in J}$ of $T$ in the site $\mathit{Sch}_{Zar}$ which is tautologically equivalent (see Sites, Definition 7.8.2) to $\{ T_ i \to T\} _{i \in I}$.

Proof. Since each $T_ i \to T$ is an open immersion, we see by Sets, Lemma 3.9.9 that each $T_ i$ is isomorphic to an object $V_ i$ of $\mathit{Sch}_{Zar}$. The covering $\{ V_ i \to T\} _{i \in I}$ is tautologically equivalent to $\{ T_ i \to T\} _{i \in I}$ (using the identity map on $I$ both ways). 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 $\mathit{Sch}_{Zar}$ by Sets, Lemma 3.11.1. $\square$

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