The Stacks project

Lemma 61.14.1. Let $X$ be a scheme.

  1. For every object $U$ of $X_{pro\text{-}\acute{e}tale}$ there exists a hypercovering $K$ of $U$ in $X_{pro\text{-}\acute{e}tale}$ such that each term $K_ n$ consists of a single weakly contractible object of $X_{pro\text{-}\acute{e}tale}$ covering $U$.

  2. For every quasi-compact and quasi-separated object $U$ of $X_{pro\text{-}\acute{e}tale}$ there exists a hypercovering $K$ of $U$ in $X_{pro\text{-}\acute{e}tale}$ such that each term $K_ n$ consists of a single affine and weakly contractible object of $X_{pro\text{-}\acute{e}tale}$ covering $U$.

Proof. Let $\mathcal{B} \subset \mathop{\mathrm{Ob}}\nolimits (X_{pro\text{-}\acute{e}tale})$ be the set of weakly contractible objects of $X_{pro\text{-}\acute{e}tale}$. Every object $T$ of $X_{pro\text{-}\acute{e}tale}$ has a covering $\{ T_ i \to T\} _{i \in I}$ with $I$ finite and $T_ i \in \mathcal{B}$ by Lemma 61.13.5. By Hypercoverings, Lemma 25.12.6 we get a hypercovering $K$ of $U$ such that $K_ n = \{ U_{n, i}\} _{i \in I_ n}$ with $I_ n$ finite and $U_{n, i}$ weakly contractible. Then we can replace $K$ by the hypercovering of $U$ given by $\{ U_ n\} $ in degree $n$ where $U_ n = \coprod _{i \in I_ n} U_{n, i}$ This is allowed by Hypercoverings, Remark 25.12.9.

Let $X_{qcqs, {pro\text{-}\acute{e}tale}} \subset X_{pro\text{-}\acute{e}tale}$ be the full subcategory consisting of quasi-compact and quasi-separated objects. A covering of $X_{qcqs, {pro\text{-}\acute{e}tale}}$ will be a finite pro-├ętale covering. Then $X_{qcqs, {pro\text{-}\acute{e}tale}}$ is a site, has fibre products, and the inclusion functor $X_{qcqs, {pro\text{-}\acute{e}tale}} \to X_{pro\text{-}\acute{e}tale}$ is continuous and commutes with fibre products. In particular, if $K$ is a hypercovering of an object $U$ in $X_{qcqs, {pro\text{-}\acute{e}tale}}$ then $K$ is a hypercovering of $U$ in $X_{pro\text{-}\acute{e}tale}$ by Hypercoverings, Lemma 25.12.5. Let $\mathcal{B} \subset \mathop{\mathrm{Ob}}\nolimits (X_{qcqs, {pro\text{-}\acute{e}tale}})$ be the set of affine and weakly contractible objects. By Lemma 61.13.3 and the fact that finite unions of affines are affine, for every object $U$ of $X_{qcqs, {pro\text{-}\acute{e}tale}}$ there exists a covering $\{ V \to U\} $ of $X_{qcqs, {pro\text{-}\acute{e}tale}}$ with $V \in \mathcal{B}$. By Hypercoverings, Lemma 25.12.6 we get a hypercovering $K$ of $U$ such that $K_ n = \{ U_{n, i}\} _{i \in I_ n}$ with $I_ n$ finite and $U_{n, i}$ affine and weakly contractible. Then we can replace $K$ by the hypercovering of $U$ given by $\{ U_ n\} $ in degree $n$ where $U_ n = \coprod _{i \in I_ n} U_{n, i}$. This is allowed by Hypercoverings, Remark 25.12.9. $\square$


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