The Stacks project

Lemma 61.13.2. Let $\mathit{Sch}_{pro\text{-}\acute{e}tale}$ be a big pro-étale site as in Definition 61.12.7. Let $T = \mathop{\mathrm{Spec}}(A)$ be an affine object of $\mathit{Sch}_{pro\text{-}\acute{e}tale}$. The following are equivalent

  1. $A$ is w-contractible,

  2. $T$ is a weakly contractible (Sites, Definition 7.40.2) object of $\mathit{Sch}_{pro\text{-}\acute{e}tale}$, and

  3. every pro-étale covering of $T$ can be refined by a Zariski covering of the form $T = \coprod _{i = 1, \ldots , n} U_ i$.

Proof. We have seen the equivalence of (1) and (3) in Lemma 61.13.1.

Assume (3) and let $\mathcal{F} \to \mathcal{G}$ be a surjection of sheaves on $\mathit{Sch}_{pro\text{-}\acute{e}tale}$. Let $s \in \mathcal{G}(T)$. To prove (2) we will show that $s$ is in the image of $\mathcal{F}(T) \to \mathcal{G}(T)$. We can find a covering $\{ T_ i \to T\} $ of $\mathit{Sch}_{pro\text{-}\acute{e}tale}$ such that $s$ lifts to a section of $\mathcal{F}$ over $T_ i$ (Sites, Definition 7.11.1). By (3) we may assume we have a finite covering $T = \coprod _{j = 1, \ldots , m} U_ j$ by open and closed subsets and we have $t_ j \in \mathcal{F}(U_ j)$ mapping to $s|_{U_ j}$. Since Zariski coverings are coverings in $\mathit{Sch}_{pro\text{-}\acute{e}tale}$ (Lemma 61.12.3) we conclude that $\mathcal{F}(T) = \prod \mathcal{F}(U_ j)$. Thus $t = (t_1, \ldots , t_ m) \in \mathcal{F}(T)$ is a section mapping to $s$.

Assume (2). Let $A \to D$ be as in Proposition 61.11.3. Then $\{ V \to T\} $ is a covering of $\mathit{Sch}_{pro\text{-}\acute{e}tale}$. (Note that $V = \mathop{\mathrm{Spec}}(D)$ is an object of $\mathit{Sch}_{pro\text{-}\acute{e}tale}$ by Remark 61.11.4 combined with our choice of the function $Bound$ in Definition 61.12.7 and the computation of the size of affine schemes in Sets, Lemma 3.9.5.) Since the topology on $\mathit{Sch}_{pro\text{-}\acute{e}tale}$ is subcanonical (Lemma 61.12.22) we see that $h_ V \to h_ T$ is a surjective map of sheaves (Sites, Lemma 7.12.4). Since $T$ is assumed weakly contractible, we see that there is an element $f \in h_ V(T) = \mathop{\mathrm{Mor}}\nolimits (T, V)$ whose image in $h_ T(T)$ is $\text{id}_ T$. Thus $A \to D$ has a section $\sigma : D \to A$. Now if $A \to B$ is faithfully flat and weakly étale, then $D \to D \otimes _ A B$ has the same properties, hence there is a section $D \otimes _ A B \to D$ and combined with $\sigma $ we get a section $B \to D \otimes _ A B \to D \to A$ of $A \to B$. Thus $A$ is w-contractible and (1) holds. $\square$


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