The Stacks project

Lemma 61.13.3. Let $\mathit{Sch}_{pro\text{-}\acute{e}tale}$ be a big pro-étale site as in Definition 61.12.7. For every object $T$ of $\mathit{Sch}_{pro\text{-}\acute{e}tale}$ there exists a covering $\{ T_ i \to T\} $ in $\mathit{Sch}_{pro\text{-}\acute{e}tale}$ with each $T_ i$ affine and the spectrum of a w-contractible ring. In particular, $T_ i$ is weakly contractible in $\mathit{Sch}_{pro\text{-}\acute{e}tale}$.

Proof. For those readers who do not care about set-theoretical issues this lemma is a trivial consequence of Lemma 61.13.2 and Proposition 61.11.3. Here are the details. Choose an affine open covering $T = \bigcup U_ i$. Write $U_ i = \mathop{\mathrm{Spec}}(A_ i)$. Choose faithfully flat, ind-étale ring maps $A_ i \to D_ i$ such that $D_ i$ is w-contractible as in Proposition 61.11.3. The family of morphisms $\{ \mathop{\mathrm{Spec}}(D_ i) \to T\} $ is a pro-étale covering. If we can show that $\mathop{\mathrm{Spec}}(D_ i)$ is isomorphic to an object, say $T_ i$, of $\mathit{Sch}_{pro\text{-}\acute{e}tale}$, then $\{ T_ i \to T\} $ will be combinatorially equivalent to a covering of $\mathit{Sch}_{pro\text{-}\acute{e}tale}$ by the construction of $\mathit{Sch}_{pro\text{-}\acute{e}tale}$ in Definition 61.12.7 and more precisely the application of Sets, Lemma 3.11.1 in the last step. To prove $\mathop{\mathrm{Spec}}(D_ i)$ is isomorphic to an object of $\mathit{Sch}_{pro\text{-}\acute{e}tale}$, it suffices to prove that $|D_ i| \leq Bound(\text{size}(T))$ by the construction of $\mathit{Sch}_{pro\text{-}\acute{e}tale}$ in Definition 61.12.7 and more precisely the application of Sets, Lemma 3.9.2 in step (3). Since $|A_ i| \leq \text{size}(U_ i) \leq \text{size}(T)$ by Sets, Lemmas 3.9.4 and 3.9.7 we get $|D_ i| \leq \kappa ^{2^{2^{2^\kappa }}}$ where $\kappa = \text{size}(T)$ by Remark 61.11.4. Thus by our choice of the function $Bound$ in Definition 61.12.7 we win. $\square$

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