61.14 Weakly contractible hypercoverings

The results of Section 61.13 leads to the existence of hypercoverings made up out weakly contractible objects.

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-étale 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. $\square$

In the following lemma we use the Čech complex $s(\mathcal{F}(K))$ associated to a hypercovering $K$ in a site. See Hypercoverings, Section 25.5. If $K$ is a hypercovering of $U$ and $K_ n = \{ U_ n \to U\}$, then the Čech complex looks like this:

$s(\mathcal{F}(K)) = \left( \mathcal{F}(U_0) \to \mathcal{F}(U_1) \to \mathcal{F}(U_2) \to \ldots \right)$

where $s(\mathcal{F}(U_ n))$ is placed in cohomological degree $n$.

Lemma 61.14.2. Let $X$ be a scheme. Let $E \in D^+(X_{pro\text{-}\acute{e}tale})$ be represented by a bounded below complex $\mathcal{E}^\bullet$ of abelian sheaves. Let $K$ be a hypercovering of $U \in \mathop{\mathrm{Ob}}\nolimits (X_{pro\text{-}\acute{e}tale})$ with $K_ n = \{ U_ n \to U\}$ where $U_ n$ is a weakly contractible object of $X_{pro\text{-}\acute{e}tale}$. Then

$R\Gamma (U, E) = \text{Tot}(s(\mathcal{E}^\bullet (K)))$

in $D(\textit{Ab})$.

Proof. If $\mathcal{E}$ is an abelian sheaf on $X_{pro\text{-}\acute{e}tale}$, then the spectral sequence of Hypercoverings, Lemma 25.5.3 implies that

$R\Gamma (X_{pro\text{-}\acute{e}tale}, \mathcal{E}) = s(\mathcal{E}(K))$

because the higher cohomology groups of any sheaf over $U_ n$ vanish, see Cohomology on Sites, Lemma 21.51.1.

If $\mathcal{E}^\bullet$ is bounded below, then we can choose an injective resolution $\mathcal{E}^\bullet \to \mathcal{I}^\bullet$ and consider the map of complexes

$\text{Tot}(s(\mathcal{E}^\bullet (K))) \longrightarrow \text{Tot}(s(\mathcal{I}^\bullet (K)))$

For every $n$ the map $\mathcal{E}^\bullet (U_ n) \to \mathcal{I}^\bullet (U_ n)$ is a quasi-isomorphism because taking sections over $U_ n$ is exact. Hence the displayed map is a quasi-isomorphism by one of the spectral sequences of Homology, Lemma 12.25.3. Using the result of the first paragraph we see that for every $p$ the complex $s(\mathcal{I}^ p(K))$ is acyclic in degrees $n > 0$ and computes $\mathcal{I}^ p(U)$ in degree $0$. Thus the other spectral sequence of Homology, Lemma 12.25.3 shows $\text{Tot}(s(\mathcal{I}^\bullet (K)))$ computes $R\Gamma (U, E) = \mathcal{I}^\bullet (U)$. $\square$

Lemma 61.14.3. Let $X$ be a quasi-compact and quasi-separated scheme. The functor $R\Gamma (X, -) : D^+(X_{pro\text{-}\acute{e}tale}) \to D(\textit{Ab})$ commutes with direct sums and homotopy colimits.

Proof. The statement means the following: Suppose we have a family of objects $E_ i$ of $D^+(X_{pro\text{-}\acute{e}tale})$ such that $\bigoplus E_ i$ is an object of $D^+(X_{pro\text{-}\acute{e}tale})$. Then $R\Gamma (X, \bigoplus E_ i) = \bigoplus R\Gamma (X, E_ i)$. To see this choose a hypercovering $K$ of $X$ with $K_ n = \{ U_ n \to X\}$ where $U_ n$ is an affine and weakly contractible scheme, see Lemma 61.14.1. Let $N$ be an integer such that $H^ p(E_ i) = 0$ for $p < N$. Choose a complex of abelian sheaves $\mathcal{E}_ i^\bullet$ representing $E_ i$ with $\mathcal{E}_ i^ p = 0$ for $p < N$. The termwise direct sum $\bigoplus \mathcal{E}_ i^\bullet$ represents $\bigoplus E_ i$ in $D(X_{pro\text{-}\acute{e}tale})$, see Injectives, Lemma 19.13.4. By Lemma 61.14.2 we have

$R\Gamma (X, \bigoplus E_ i) = \text{Tot}(s((\bigoplus \mathcal{E}^\bullet _ i)(K)))$

and

$R\Gamma (X, E_ i) = \text{Tot}(s(\mathcal{E}^\bullet _ i(K)))$

Since each $U_ n$ is quasi-compact we see that

$\text{Tot}(s((\bigoplus \mathcal{E}^\bullet _ i)(K))) = \bigoplus \text{Tot}(s(\mathcal{E}^\bullet _ i(K)))$

by Modules on Sites, Lemma 18.30.3. The statement on homotopy colimits is a formal consequence of the fact that $R\Gamma$ is an exact functor of triangulated categories and the fact (just proved) that it commutes with direct sums. $\square$

Remark 61.14.4. Let $X$ be a scheme. Because $X_{pro\text{-}\acute{e}tale}$ has enough weakly contractible objects for all $K$ in $D(X_{pro\text{-}\acute{e}tale})$ we have $K = R\mathop{\mathrm{lim}}\nolimits \tau _{\geq -n}K$ by Cohomology on Sites, Proposition 21.51.2. Since $R\Gamma$ commutes with $R\mathop{\mathrm{lim}}\nolimits$ by Injectives, Lemma 19.13.6 we see that

$R\Gamma (X, K) = R\mathop{\mathrm{lim}}\nolimits R\Gamma (X, \tau _{\geq -n}K)$

in $D(\textit{Ab})$. This will sometimes allow us to extend results from bounded below complexes to all complexes.

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