## 85.36 Proper hypercoverings of algebraic spaces

This section is the analogue of Section 85.25 for the case of algebraic spaces. The reader who wishes to do so, can replace “algebraic space” everywhere with “scheme” and get equally valid results. This has the advantage of replacing the references to More on Cohomology of Spaces, Section 84.8 with references to Étale Cohomology, Section 59.102.

We fix a base scheme $S$. Let $X$ be an algebraic space over $S$ and let $U$ be a simplicial algebraic space over $S$. Assume we have an augmentation

$a : U \to X$

See Section 85.32. We say that $U$ is a proper hypercovering of $X$ if

1. $U_0 \to X$ is proper and surjective,

2. $U_1 \to U_0 \times _ X U_0$ is proper and surjective,

3. $U_{n + 1} \to (\text{cosk}_ n\text{sk}_ n U)_{n + 1}$ is proper and surjective for $n \geq 1$.

The category of algebraic spaces over $S$ has all finite limits, hence the coskeleta used in the formulation above exist.

$\fbox{Principle: Proper hypercoverings can be used to compute étale cohomology.}$

The key idea behind the proof of the principle is to compare the ph and étale topologies on the category $\textit{Spaces}/S$. Namely, the ph topology is stronger than the étale topology and we have (a) a proper surjective map defines a ph covering, and (b) ph cohomology of sheaves pulled back from the small étale site agrees with étale cohomology as we have seen in More on Cohomology of Spaces, Section 84.8.

All results in this section generalize to the case where $U \to X$ is merely a “ph hypercovering”, meaning a hypercovering of $X$ in the site $(\textit{Spaces}/S)_{ph}$ as defined in Section 85.21. If we ever need this, we will precisely formulate and prove this here.

Lemma 85.36.1. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $U$ be a simplicial algebraic space over $S$. Let $a : U \to X$ be an augmentation. There is a commutative diagram

$\xymatrix{ \mathop{\mathit{Sh}}\nolimits ((\textit{Spaces}/U)_{ph, total}) \ar[r]_-h \ar[d]_{a_{ph}} & \mathop{\mathit{Sh}}\nolimits (U_{\acute{e}tale}) \ar[d]^ a \\ \mathop{\mathit{Sh}}\nolimits ((\textit{Spaces}/X)_{ph}) \ar[r]^-{h_{-1}} & \mathop{\mathit{Sh}}\nolimits (X_{\acute{e}tale}) }$

where the left vertical arrow is defined in Section 85.21 and the right vertical arrow is defined in Section 85.32.

Proof. The notation $(\textit{Spaces}/U)_{ph, total}$ indicates that we are using the construction of Section 85.21 for the site $(\textit{Spaces}/S)_{ph}$ and the simplicial object $U$ of this site1. We will use the sites $X_{spaces, {\acute{e}tale}}$ and $U_{spaces, {\acute{e}tale}}$ for the topoi on the right hand side; this is permissible see discussion in Section 85.32.

Observe that both $(\textit{Spaces}/U)_{ph, total}$ and $U_{spaces, {\acute{e}tale}}$ fall into case A of Situation 85.3.3. This is immediate from the construction of $U_{\acute{e}tale}$ in Section 85.32 and it follows from Lemma 85.21.5 for $(\textit{Spaces}/U)_{ph, total}$. Next, consider the functors $U_{n, spaces, {\acute{e}tale}} \to (\textit{Spaces}/U_ n)_{ph}$, $U \mapsto U/U_ n$ and $X_{spaces, {\acute{e}tale}} \to (\textit{Spaces}/X)_{ph}$, $U \mapsto U/X$. We have seen that these define morphisms of sites in More on Cohomology of Spaces, Section 84.8 where these were denoted $a_{U_ n} = \epsilon _{U_ n} \circ \pi _{u_ n}$ and $a_ X = \epsilon _ X \circ \pi _ X$. Thus we obtain a morphism of simplicial sites compatible with augmentations as in Remark 85.5.4 and we may apply Lemma 85.5.5 to conclude. $\square$

Lemma 85.36.2. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $U$ be a simplicial algebraic space over $S$. Let $a : U \to X$ be an augmentation. If $a : U \to X$ is a proper hypercovering of $X$, then

$a^{-1} : \mathop{\mathit{Sh}}\nolimits (X_{\acute{e}tale}) \to \mathop{\mathit{Sh}}\nolimits (U_{\acute{e}tale}) \quad \text{and}\quad a^{-1} : \textit{Ab}(X_{\acute{e}tale}) \to \textit{Ab}(U_{\acute{e}tale})$

are fully faithful with essential image the cartesian sheaves and quasi-inverse given by $a_*$. Here $a : \mathop{\mathit{Sh}}\nolimits (U_{\acute{e}tale}) \to \mathop{\mathit{Sh}}\nolimits (X_{\acute{e}tale})$ is as in Section 85.32.

Proof. We will prove the statement for sheaves of sets. It will be an almost formal consequence of results already established. Consider the diagram of Lemma 85.36.1. In the proof of this lemma we have seen that $h_{-1}$ is the morphism $a_ X$ of More on Cohomology of Spaces, Section 84.8. Thus it follows from More on Cohomology of Spaces, Lemma 84.8.1 that $(h_{-1})^{-1}$ is fully faithful with quasi-inverse $h_{-1, *}$. The same holds true for the components $h_ n$ of $h$. By the description of the functors $h^{-1}$ and $h_*$ of Lemma 85.5.2 we conclude that $h^{-1}$ is fully faithful with quasi-inverse $h_*$. Observe that $U$ is a hypercovering of $X$ in $(\textit{Spaces}/S)_{ph}$ as defined in Section 85.21 since a surjective proper morphism gives a ph covering by Topologies on Spaces, Lemma 73.8.3. By Lemma 85.21.1 we see that $a_{ph}^{-1}$ is fully faithful with quasi-inverse $a_{ph, *}$ and with essential image the cartesian sheaves on $(\textit{Spaces}/U)_{ph, total}$. A formal argument (chasing around the diagram) now shows that $a^{-1}$ is fully faithful.

Finally, suppose that $\mathcal{G}$ is a cartesian sheaf on $U_{\acute{e}tale}$. Then $h^{-1}\mathcal{G}$ is a cartesian sheaf on $(\textit{Spaces}/U)_{ph, total}$. Hence $h^{-1}\mathcal{G} = a_{ph}^{-1}\mathcal{H}$ for some sheaf $\mathcal{H}$ on $(\textit{Spaces}/X)_{ph}$. We compute using somewhat pedantic notation

\begin{align*} (h_{-1})^{-1}(a_*\mathcal{G}) & = (h_{-1})^{-1} \text{Eq}( \xymatrix{ a_{0, small, *}\mathcal{G}_0 \ar@<1ex>[r] \ar@<-1ex>[r] & a_{1, small, *}\mathcal{G}_1 } ) \\ & = \text{Eq}( \xymatrix{ (h_{-1})^{-1}a_{0, small, *}\mathcal{G}_0 \ar@<1ex>[r] \ar@<-1ex>[r] & (h_{-1})^{-1}a_{1, small, *}\mathcal{G}_1 } ) \\ & = \text{Eq}( \xymatrix{ a_{0, big, ph, *}h_0^{-1}\mathcal{G}_0 \ar@<1ex>[r] \ar@<-1ex>[r] & a_{1, big, ph, *}h_1^{-1}\mathcal{G}_1 } ) \\ & = \text{Eq}( \xymatrix{ a_{0, big, ph, *}(a_{0, big, ph})^{-1}\mathcal{H} \ar@<1ex>[r] \ar@<-1ex>[r] & a_{1, big, ph, *}(a_{1, big, ph})^{-1}\mathcal{H} } ) \\ & = a_{ph, *}a_{ph}^{-1}\mathcal{H} \\ & = \mathcal{H} \end{align*}

Here the first equality follows from Lemma 85.4.2, the second equality follows as $(h_{-1})^{-1}$ is an exact functor, the third equality follows from More on Cohomology of Spaces, Lemma 84.8.5 (here we use that $a_0 : U_0 \to X$ and $a_1: U_1 \to X$ are proper), the fourth follows from $a_{ph}^{-1}\mathcal{H} = h^{-1}\mathcal{G}$, the fifth from Lemma 85.4.2, and the sixth we've seen above. Since $a_{ph}^{-1}\mathcal{H} = h^{-1}\mathcal{G}$ we deduce that $h^{-1}\mathcal{G} \cong h^{-1}a^{-1}a_*\mathcal{G}$ which ends the proof by fully faithfulness of $h^{-1}$. $\square$

Lemma 85.36.3. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $U$ be a simplicial algebraic space over $S$. Let $a : U \to X$ be an augmentation. If $a : U \to X$ is a proper hypercovering of $X$, then for $K \in D^+(X_{\acute{e}tale})$

$K \to Ra_*(a^{-1}K)$

is an isomorphism. Here $a : \mathop{\mathit{Sh}}\nolimits (U_{\acute{e}tale}) \to \mathop{\mathit{Sh}}\nolimits (X_{\acute{e}tale})$ is as in Section 85.32.

Proof. Consider the diagram of Lemma 85.36.1. Observe that $Rh_{n, *}h_ n^{-1}$ is the identity functor on $D^+(U_{n, {\acute{e}tale}})$ by More on Cohomology of Spaces, Lemma 84.8.2. Hence $Rh_*h^{-1}$ is the identity functor on $D^+(U_{\acute{e}tale})$ by Lemma 85.5.3. We have

\begin{align*} Ra_*(a^{-1}K) & = Ra_*Rh_*h^{-1}a^{-1}K \\ & = Rh_{-1, *}Ra_{ph, *}a_{ph}^{-1}(h_{-1})^{-1}K \\ & = Rh_{-1, *}(h_{-1})^{-1}K \\ & = K \end{align*}

The first equality by the discussion above, the second equality because of the commutativity of the diagram in Lemma 85.25.1, the third equality by Lemma 85.21.2 as $U$ is a hypercovering of $X$ in $(\textit{Spaces}/S)_{ph}$ by Topologies on Spaces, Lemma 73.8.3, and the last equality by the already used More on Cohomology of Spaces, Lemma 84.8.2. $\square$

Lemma 85.36.4. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $U$ be a simplicial algebraic space over $S$. Let $a : U \to X$ be an augmentation. If $a : U \to X$ is a proper hypercovering of $X$, then

$R\Gamma (X_{\acute{e}tale}, K) = R\Gamma (U_{\acute{e}tale}, a^{-1}K)$

for $K \in D^+(X_{\acute{e}tale})$. Here $a : \mathop{\mathit{Sh}}\nolimits (U_{\acute{e}tale}) \to \mathop{\mathit{Sh}}\nolimits (X_{\acute{e}tale})$ is as in Section 85.32.

Proof. This follows from Lemma 85.36.3 because $R\Gamma (U_{\acute{e}tale}, -) = R\Gamma (X_{\acute{e}tale}, -) \circ Ra_*$ by Cohomology on Sites, Remark 21.14.4. $\square$

Lemma 85.36.5. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $U$ be a simplicial algebraic space over $S$. Let $a : U \to X$ be an augmentation. Let $\mathcal{A} \subset \textit{Ab}(U_{\acute{e}tale})$ denote the weak Serre subcategory of cartesian abelian sheaves. If $U$ is a proper hypercovering of $X$, then the functor $a^{-1}$ defines an equivalence

$D^+(X_{\acute{e}tale}) \longrightarrow D_\mathcal {A}^+(U_{\acute{e}tale})$

with quasi-inverse $Ra_*$. Here $a : \mathop{\mathit{Sh}}\nolimits (U_{\acute{e}tale}) \to \mathop{\mathit{Sh}}\nolimits (X_{\acute{e}tale})$ is as in Section 85.32.

Proof. Observe that $\mathcal{A}$ is a weak Serre subcategory by Lemma 85.12.6. The equivalence is a formal consequence of the results obtained so far. Use Lemmas 85.36.2 and 85.36.3 and Cohomology on Sites, Lemma 21.28.5. $\square$

Lemma 85.36.6. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $U$ be a simplicial algebraic space over $S$. Let $a : U \to X$ be an augmentation. Let $\mathcal{F}$ be an abelian sheaf on $X_{\acute{e}tale}$. Let $\mathcal{F}_ n$ be the pullback to $U_{n, {\acute{e}tale}}$. If $U$ is a ph hypercovering of $X$, then there exists a canonical spectral sequence

$E_1^{p, q} = H^ q_{\acute{e}tale}(U_ p, \mathcal{F}_ p)$

converging to $H^{p + q}_{\acute{e}tale}(X, \mathcal{F})$.

Proof. Immediate consequence of Lemmas 85.36.4 and 85.8.3. $\square$

[1] To distinguish from $(\textit{Spaces}/U)_{fppf, total}$ defined using the fppf topology in Section 85.33.

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