25.6 Hypercoverings a la Verdier
The astute reader will have noticed that all we need in order to get the Čech to cohomology spectral sequence for a hypercovering of an object $X$, is the conclusion of Lemma 25.3.9. Therefore the following definition makes sense.
Definition 25.6.1. Let $\mathcal{C}$ be a site. Assume $\mathcal{C}$ has equalizers and fibre products. Let $\mathcal{G}$ be a presheaf of sets. A hypercovering of $\mathcal{G}$ is a simplicial object $K$ of $\text{SR}(\mathcal{C})$ endowed with an augmentation $F(K) \to \mathcal{G}$ such that
$F(K_0) \to \mathcal{G}$ becomes surjective after sheafification,
$F(K_1) \to F(K_0) \times _\mathcal {G} F(K_0)$ becomes surjective after sheafification, and
$F(K_{n + 1}) \longrightarrow F((\text{cosk}_ n \text{sk}_ n K)_{n + 1})$ for $n \geq 1$ becomes surjective after sheafification.
We say that a simplicial object $K$ of $\text{SR}(\mathcal{C})$ is a hypercovering if $K$ is a hypercovering of the final object $*$ of $\textit{PSh}(\mathcal{C})$.
The assumption that $\mathcal{C}$ has fibre products and equalizers guarantees that $\text{SR}(\mathcal{C})$ has fibre products and equalizers and $F$ commutes with these (Lemma 25.2.3) which suffices to define the coskeleton functors used (see Simplicial, Remark 14.19.11 and Categories, Lemma 4.18.2). If $\mathcal{C}$ is general, we can replace the condition (3) by the condition that $F(K_{n + 1}) \longrightarrow ((\text{cosk}_ n \text{sk}_ n F(K))_{n + 1})$ for $n \geq 1$ becomes surjective after sheafification and the results of this section remain valid.
Let $\mathcal{F}$ be an abelian sheaf on $\mathcal{C}$. In the previous section, we defined the Čech complex of $\mathcal{F}$ with respect to a simplicial object $K$ of $\text{SR}(\mathcal{C})$. Next, given a presheaf $\mathcal{G}$ we set
\[ H^0(\mathcal{G}, \mathcal{F}) = \mathop{\mathrm{Mor}}\nolimits _{\textit{PSh}(\mathcal{C})}(\mathcal{G}, \mathcal{F}) = \mathop{\mathrm{Mor}}\nolimits _{\mathop{\mathit{Sh}}\nolimits (\mathcal{C})}(\mathcal{G}^\# , \mathcal{F}) = H^0(\mathcal{G}^\# , \mathcal{F}) \]
with notation as in Cohomology on Sites, Section 21.13. This is a left exact functor and its higher derived functors (briefly studied in Cohomology on Sites, Section 21.13) are denoted $H^ i(\mathcal{G}, \mathcal{F})$. We will show that given a hypercovering $K$ of $\mathcal{G}$, there is a Čech to cohomology spectral sequence converging to the cohomology $H^ i(\mathcal{G}, \mathcal{F})$. Note that if $\mathcal{G} = *$, then $H^ i(*, \mathcal{F}) = H^ i(\mathcal{C}, \mathcal{F})$ recovers the cohomology of $\mathcal{F}$ on the site $\mathcal{C}$.
Lemma 25.6.2. Let $\mathcal{C}$ be a site with equalizers and fibre products. Let $\mathcal{G}$ be a presheaf on $\mathcal{C}$. Let $K$ be a hypercovering of $\mathcal{G}$. Let $\mathcal{F}$ be a sheaf of abelian groups on $\mathcal{C}$. Then $\check{H}^0(K, \mathcal{F}) = H^0(\mathcal{G}, \mathcal{F})$.
Proof.
This follows from the definition of $H^0(\mathcal{G}, \mathcal{F})$ and the fact that
\[ \xymatrix{ F(K_1) \ar@<1ex>[r] \ar@<-1ex>[r] & F(K_0) \ar[r] & \mathcal{G} } \]
becomes an coequalizer diagram after sheafification.
$\square$
Lemma 25.6.3. Let $\mathcal{C}$ be a site with equalizers and fibre products. Let $\mathcal{G}$ be a presheaf on $\mathcal{C}$. Let $K$ be a hypercovering of $\mathcal{G}$. Let $\mathcal{I}$ be an injective sheaf of abelian groups on $\mathcal{C}$. Then
\[ \check{H}^ p(K, \mathcal{I}) = \left\{ \begin{matrix} H^0(\mathcal{G}, \mathcal{I})
& \text{if}
& p = 0
\\ 0
& \text{if}
& p > 0
\end{matrix} \right. \]
Proof.
By (25.5.2.1) we have
\[ s(\mathcal{F}(K)) = \mathop{\mathrm{Hom}}\nolimits _{\textit{Ab}(\mathcal{C})}(s(\mathbf{Z}_{F(K)}^\# ), \mathcal{F}) \]
The complex $s(\mathbf{Z}_{F(K)}^\# )$ is quasi-isomorphic to $\mathbf{Z}_\mathcal {G}^\# $, see Lemma 25.4.4. We conclude that if $\mathcal{I}$ is an injective abelian sheaf, then the complex $s(\mathcal{I}(K))$ is acyclic except possibly in degree $0$. In other words, we have $\check{H}^ i(K, \mathcal{I}) = 0$ for $i > 0$. Combined with Lemma 25.6.2 the lemma is proved.
$\square$
Lemma 25.6.4. Let $\mathcal{C}$ be a site with equalizers and fibre products. Let $\mathcal{G}$ be a presheaf on $\mathcal{C}$. Let $K$ be a hypercovering of $\mathcal{G}$. Let $\mathcal{F}$ be a sheaf of abelian groups on $\mathcal{C}$. There is a map
\[ s(\mathcal{F}(K)) \longrightarrow R\Gamma (\mathcal{G}, \mathcal{F}) \]
in $D^{+}(\textit{Ab})$ functorial in $\mathcal{F}$, which induces a natural transformation
\[ \check{H}^ i(K, -) \longrightarrow H^ i(\mathcal{G}, -) \]
of functors $\textit{Ab}(\mathcal{C}) \to \textit{Ab}$. Moreover, there is a spectral sequence $(E_ r, d_ r)_{r \geq 0}$ with
\[ E_2^{p, q} = \check{H}^ p(K, \underline{H}^ q(\mathcal{F})) \]
converging to $H^{p + q}(\mathcal{G}, \mathcal{F})$. This spectral sequence is functorial in $\mathcal{F}$ and in the hypercovering $K$.
Proof.
Choose an injective resolution $\mathcal{F} \to \mathcal{I}^\bullet $ in the category of abelian sheaves on $\mathcal{C}$. Consider the double complex $A^{\bullet , \bullet }$ with terms
\[ A^{p, q} = \mathcal{I}^ q(K_ p) \]
where the differential $d_1^{p, q} : A^{p, q} \to A^{p + 1, q}$ is the one coming from the differential $\mathcal{I}^ p \to \mathcal{I}^{p + 1}$ and the differential $d_2^{p, q} : A^{p, q} \to A^{p, q + 1}$ is the one coming from the differential on the complex $s(\mathcal{I}^ p(K))$ associated to the cosimplicial abelian group $\mathcal{I}^ p(K)$ as explained above. We will use the two spectral sequences $({}'E_ r, {}'d_ r)$ and $({}''E_ r, {}''d_ r)$ associated to this double complex, see Homology, Section 12.25.
By Lemma 25.6.3 the complexes $s(\mathcal{I}^ p(K))$ are acyclic in positive degrees and have $H^0$ equal to $H^0(\mathcal{G}, \mathcal{I}^ p)$. Hence by Homology, Lemma 12.25.4 and its proof the spectral sequence $({}'E_ r, {}'d_ r)$ degenerates, and the natural map
\[ H^0(\mathcal{G}, \mathcal{I}^\bullet ) \longrightarrow \text{Tot}(A^{\bullet , \bullet }) \]
is a quasi-isomorphism of complexes of abelian groups. The map $s(\mathcal{F}(K)) \longrightarrow R\Gamma (\mathcal{G}, \mathcal{F})$ of the lemma is the composition of the natural map $s(\mathcal{F}(K)) \to \text{Tot}(A^{\bullet , \bullet })$ followed by the inverse of the displayed quasi-isomorphism above. This works because $H^0(\mathcal{G}, \mathcal{I}^\bullet )$ is a representative of $R\Gamma (\mathcal{G}, \mathcal{F})$.
Consider the spectral sequence $({}''E_ r, {}''d_ r)_{r \geq 0}$. By Homology, Lemma 12.25.1 we see that
\[ {}''E_2^{p, q} = H^ p_{II}(H^ q_ I(A^{\bullet , \bullet })) \]
In other words, we first take cohomology with respect to $d_1$ which gives the groups ${}''E_1^{p, q} = \underline{H}^ p(\mathcal{F})(K_ q)$. Hence it is indeed the case (by the description of the differential ${}''d_1$) that ${}''E_2^{p, q} = \check{H}^ p(K, \underline{H}^ q(\mathcal{F}))$. Since this spectral sequence converges to the cohomology of $\text{Tot}(A^{\bullet , \bullet })$ the proof is finished.
$\square$
Lemma 25.6.5. Let $\mathcal{C}$ be a site with equalizers and fibre products. Let $K$ be a hypercovering. Let $\mathcal{F}$ be an abelian sheaf. There is a spectral sequence $(E_ r, d_ r)_{r \geq 0}$ with
\[ E_2^{p, q} = \check{H}^ p(K, \underline{H}^ q(\mathcal{F})) \]
converging to the global cohomology groups $H^{p + q}(\mathcal{F})$.
Proof.
This is a special case of Lemma 25.6.4.
$\square$
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