85.25 Proper hypercoverings in topology
Let's work in the category $\textit{LC}$ of Hausdorff and locally quasi-compact topological spaces and continuous maps, see Cohomology on Sites, Section 21.31. Let $X$ be an object of $\textit{LC}$ and let $U$ be a simplicial object of $\textit{LC}$. Assume we have an augmentation
\[ a : U \to X \]
We say that $U$ is a proper hypercovering of $X$ if
$U_0 \to X$ is a proper surjective map,
$U_1 \to U_0 \times _ X U_0$ is a proper surjective map,
$U_{n + 1} \to (\text{cosk}_ n\text{sk}_ n U)_{n + 1}$ is a proper surjective map for $n \geq 1$.
The category $\textit{LC}$ has all finite limits, hence the coskeleta used in the formulation above exist.
\[ \fbox{Principle: Proper hypercoverings can be used to compute cohomology.} \]
A key idea behind the proof of the principle is to find a topology on $\textit{LC}$ which is stronger than the usual one such that (a) a surjective proper map defines a covering, and (b) cohomology of usual sheaves with respect to this stronger topology agrees with the usual cohomology. Properties (a) and (b) hold for the qc topology, see Cohomology on Sites, Section 21.31. Once we have (a) and (b) we deduce the principle via the earlier work done in this chapter.
Lemma 85.25.1. Let $U$ be a simplicial object of $\textit{LC}$ and let $a : U \to X$ be an augmentation. There is a commutative diagram
\[ \xymatrix{ \mathop{\mathit{Sh}}\nolimits ((\textit{LC}_{qc}/U)_{total}) \ar[r]_-h \ar[d]_{a_{qc}} & \mathop{\mathit{Sh}}\nolimits (U_{Zar}) \ar[d]^ a \\ \mathop{\mathit{Sh}}\nolimits (\textit{LC}_{qc}/X) \ar[r]^-{h_{-1}} & \mathop{\mathit{Sh}}\nolimits (X) } \]
where the left vertical arrow is defined in Section 85.21 and the right vertical arrow is defined in Lemma 85.2.8.
Proof.
Write $\mathop{\mathit{Sh}}\nolimits (X) = \mathop{\mathit{Sh}}\nolimits (X_{Zar})$. Observe that both $(\textit{LC}_{qc}/U)_{total}$ and $U_{Zar}$ fall into case A of Situation 85.3.3. This is immediate from the construction of $U_{Zar}$ in Section 85.2 and it follows from Lemma 85.21.5 for $(\textit{LC}_{qc}/U)_{total}$. Next, consider the functors $U_{n, Zar} \to \textit{LC}_{qc}/U_ n$, $U \mapsto U/U_ n$ and $X_{Zar} \to \textit{LC}_{qc}/X$, $U \mapsto U/X$. We have seen that these define morphisms of sites in Cohomology on Sites, Section 21.31. 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 85.25.2. Let $U$ be a simplicial object of $\textit{LC}$ and let $a : U \to X$ be an augmentation. If $a : U \to X$ gives a proper hypercovering of $X$, then
\[ a^{-1} : \mathop{\mathit{Sh}}\nolimits (X) \to \mathop{\mathit{Sh}}\nolimits (U_{Zar}) \quad \text{and}\quad a^{-1} : \textit{Ab}(X) \to \textit{Ab}(U_{Zar}) \]
are fully faithful with essential image the cartesian sheaves and quasi-inverse given by $a_*$. Here $a : \mathop{\mathit{Sh}}\nolimits (U_{Zar}) \to \mathop{\mathit{Sh}}\nolimits (X)$ is as in Lemma 85.2.8.
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 85.25.1. By Cohomology on Sites, Lemma 21.31.6 the functor $(h_{-1})^{-1}$ 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 $\textit{LC}_{qc}$ (as defined in Section 85.21) by Cohomology on Sites, Lemma 21.31.4. By Lemma 85.21.1 we see that $a_{qc}^{-1}$ is fully faithful with quasi-inverse $a_{qc, *}$ and with essential image the cartesian sheaves on $(\textit{LC}_{qc}/U)_{total}$. 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 $U_{Zar}$. Then $h^{-1}\mathcal{G}$ is a cartesian sheaf on $\textit{LC}_{qc}/U$. Hence $h^{-1}\mathcal{G} = a_{qc}^{-1}\mathcal{H}$ for some sheaf $\mathcal{H}$ on $\textit{LC}_{qc}/X$. We compute
\begin{align*} (h_{-1})^{-1}(a_*\mathcal{G}) & = (h_{-1})^{-1} \text{Eq}( \xymatrix{ a_{0, *}\mathcal{G}_0 \ar@<1ex>[r] \ar@<-1ex>[r] & a_{1, *}\mathcal{G}_1 } ) \\ & = \text{Eq}( \xymatrix{ (h_{-1})^{-1}a_{0, *}\mathcal{G}_0 \ar@<1ex>[r] \ar@<-1ex>[r] & (h_{-1})^{-1}a_{1, *}\mathcal{G}_1 } ) \\ & = \text{Eq}( \xymatrix{ a_{qc, 0, *}h_0^{-1}\mathcal{G}_0 \ar@<1ex>[r] \ar@<-1ex>[r] & a_{qc, 1, *}h_1^{-1}\mathcal{G}_1 } ) \\ & = \text{Eq}( \xymatrix{ a_{qc, 0, *}a_{qc, 0}^{-1}\mathcal{H} \ar@<1ex>[r] \ar@<-1ex>[r] & a_{qc, 1, *}a_{qc, 1}^{-1}\mathcal{H} } ) \\ & = a_{qc, *}a_{qc}^{-1}\mathcal{H} \\ & = \mathcal{H} \end{align*}
Here the first equality follows from Lemma 85.2.8, the second equality follows as $(h_{-1})^{-1}$ is an exact functor, the third equality follows from Cohomology on Sites, Lemma 21.31.8 (here we use that $a_0 : U_0 \to X$ and $a_1: U_1 \to X$ are proper), the fourth follows from $a_{qc}^{-1}\mathcal{H} = h^{-1}\mathcal{G}$, the fifth from Lemma 85.4.2, and the sixth we've seen above. Since $a_{qc}^{-1}\mathcal{H} = h^{-1}\mathcal{G}$ 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 85.25.3. Let $U$ be a simplicial object of $\textit{LC}$ and let $a : U \to X$ be an augmentation. If $a : U \to X$ gives a proper hypercovering of $X$, then for $K \in D^+(X)$
\[ K \to Ra_*(a^{-1}K) \]
is an isomorphism where $a : \mathop{\mathit{Sh}}\nolimits (U_{Zar}) \to \mathop{\mathit{Sh}}\nolimits (X)$ is as in Lemma 85.2.8.
Proof.
Consider the diagram of Lemma 85.25.1. Observe that $Rh_{n, *}h_ n^{-1}$ is the identity functor on $D^+(U_ n)$ by Cohomology on Sites, Lemma 21.31.11. Hence $Rh_*h^{-1}$ is the identity functor on $D^+(U_{Zar})$ by Lemma 85.5.3. We have
\begin{align*} Ra_*(a^{-1}K) & = Ra_*Rh_*h^{-1}a^{-1}K \\ & = Rh_{-1, *}Ra_{qc, *}a_{qc}^{-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 ($U$ is a hypercovering of $X$ in $\textit{LC}_{qc}$ by Cohomology on Sites, Lemma 21.31.4), and the last equality by the already used Cohomology on Sites, Lemma 21.31.11.
$\square$
Lemma 85.25.4. Let $U$ be a simplicial object of $\textit{LC}$ and let $a : U \to X$ be an augmentation. If $U$ is a proper hypercovering of $X$, then
\[ R\Gamma (X, K) = R\Gamma (U_{Zar}, a^{-1}K) \]
for $K \in D^+(X)$ where $a : \mathop{\mathit{Sh}}\nolimits (U_{Zar}) \to \mathop{\mathit{Sh}}\nolimits (X)$ is as in Lemma 85.2.8.
Proof.
This follows from Lemma 85.25.3 because $R\Gamma (U_{Zar}, -) = R\Gamma (X, -) \circ Ra_*$ by Cohomology on Sites, Remark 21.14.4.
$\square$
Lemma 85.25.5. Let $U$ be a simplicial object of $\textit{LC}$ and let $a : U \to X$ be an augmentation. Let $\mathcal{A} \subset \textit{Ab}(U_{Zar})$ 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
\[ D^+(X) \longrightarrow D_\mathcal {A}^+(U_{Zar}) \]
with quasi-inverse $Ra_*$ where $a : \mathop{\mathit{Sh}}\nolimits (U_{Zar}) \to \mathop{\mathit{Sh}}\nolimits (X)$ is as in Lemma 85.2.8.
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 85.25.2 and 85.25.3 and Cohomology on Sites, Lemma 21.28.5.
$\square$
Lemma 85.25.6. Let $U$ be a simplicial object of $\textit{LC}$ and let $a : U \to X$ be an augmentation. Let $\mathcal{F}$ be an abelian sheaf on $X$. Let $\mathcal{F}_ n$ be the pullback to $U_ n$. If $U$ is a proper hypercovering of $X$, then there exists a canonical spectral sequence
\[ E_1^{p, q} = H^ q(U_ p, \mathcal{F}_ p) \]
converging to $H^{p + q}(X, \mathcal{F})$.
Proof.
Immediate consequence of Lemmas 85.25.4 and 85.2.10.
$\square$
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