The Stacks project

Lemma 61.13.5. Let $S$ be a scheme. The pro-étale sites $\mathit{Sch}_{pro\text{-}\acute{e}tale}$, $S_{pro\text{-}\acute{e}tale}$, $(\mathit{Sch}/S)_{pro\text{-}\acute{e}tale}$ have the following property: for any object $U$ there exists a covering $\{ V \to U\} $ with $V$ a weakly contractible object. If $U$ is quasi-compact, then we may choose $V$ affine and weakly contractible.

Proof. Suppose that $V = \coprod _{j \in J} V_ j$ is an object of $(\mathit{Sch}/S)_{pro\text{-}\acute{e}tale}$ which is the disjoint union of weakly contractible objects $V_ j$. Since a disjoint union decomposition is a pro-étale covering we see that $\mathcal{F}(V) = \prod _{j \in J} \mathcal{F}(V_ j)$ for any pro-étale sheaf $\mathcal{F}$. Let $\mathcal{F} \to \mathcal{G}$ be a surjective map of sheaves of sets. Since $V_ j$ is weakly contractible, the map $\mathcal{F}(V_ j) \to \mathcal{G}(V_ j)$ is surjective, see Sites, Definition 7.40.2. Thus $\mathcal{F}(V) \to \mathcal{G}(V)$ is surjective as a product of surjective maps of sets and we conclude that $V$ is weakly contractible.

Choose a covering $\{ U_ i \to U\} _{i \in I}$ with $U_ i$ affine and weakly contractible as in Lemma 61.13.3. Take $V = \coprod _{i \in I} U_ i$ (there is a set theoretic issue here which we will address below). Then $\{ V \to U\} $ is the desired pro-étale covering by a weakly contractible object (to check it is a covering use Lemma 61.12.2). If $U$ is quasi-compact, then it follows immediately from Lemma 61.12.2 that we can choose a finite subset $I' \subset I$ such that $\{ U_ i \to U\} _{i \in I'}$ is still a covering and then $\{ \coprod _{i \in I'} U_ i \to U\} $ is the desired covering by an affine and weakly contractible object.

In this paragraph, which we urge the reader to skip, we address set theoretic problems. In order to know that the disjoint union lies in our partial universe, we need to bound the cardinality of the index set $I$. It is seen immediately from the construction of the covering $\{ U_ i \to U\} _{i \in I}$ in the proof of Lemma 61.13.3 that $|I| \leq \text{size}(U)$ where the size of a scheme is as defined in Sets, Section 3.9. Moreover, for each $i$ we have $\text{size}(U_ i) \leq Bound(\text{size}(U))$; this follows for the bound of the cardinality of $\Gamma (U_ i, \mathcal{O}_{U_ i})$ in the proof of Lemma 61.13.3 and Sets, Lemma 3.9.4. Thus $\text{size}(\coprod _{i \in I} U_ i)) \leq Bound(\text{size}(U))$ by Sets, Lemma 3.9.5. Hence by construction of the big pro-étale site through Sets, Lemma 3.9.2 we see that $\coprod _{i \in I} U_ i$ is isomorphic to an object of our site and the proof is complete. $\square$

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