Lemma 25.5.1. Let $\mathcal{C}$ be a site with fibre products. Let $X$ be an object of $\mathcal{C}$. Let $K$ be a hypercovering of $X$. Let $\mathcal{F}$ be a sheaf of abelian groups on $\mathcal{C}$. Then $\check{H}^0(K, \mathcal{F}) = \mathcal{F}(X)$.
25.5 Čech cohomology and hypercoverings
Let $\mathcal{C}$ be a site. Consider a presheaf of abelian groups $\mathcal{F}$ on the site $\mathcal{C}$. It defines a functor
\begin{eqnarray*} \mathcal{F} : \text{SR}(\mathcal{C})^{opp} & \longrightarrow & \textit{Ab} \\ \{ U_ i\} _{i \in I} & \longmapsto & \prod \nolimits _{i \in I} \mathcal{F}(U_ i) \end{eqnarray*}
Thus a simplicial object $K$ of $\text{SR}(\mathcal{C})$ is turned into a cosimplicial object $\mathcal{F}(K)$ of $\textit{Ab}$. The cochain complex $s(\mathcal{F}(K))$ associated to $\mathcal{F}(K)$ (Simplicial, Section 14.25) is called the Čech complex of $\mathcal{F}$ with respect to the simplicial object $K$. We set
\[ \check{H}^ i(K, \mathcal{F}) = H^ i(s(\mathcal{F}(K))). \]
and we call it the $i$th Čech cohomology group of $\mathcal{F}$ with respect to $K$. In this section we prove analogues of some of the results for Čech cohomology of open coverings proved in Cohomology, Sections 20.9, 20.10 and 20.11.
Proof. We have
\[ \check{H}^0(K, \mathcal{F}) = \mathop{\mathrm{Ker}}(\mathcal{F}(K_0) \longrightarrow \mathcal{F}(K_1)) \]
Write $K_0 = \{ U_ i \to X\} $. It is a covering in the site $\mathcal{C}$. As well, we have that $K_1 \to K_0 \times K_0$ is a covering in $\text{SR}(\mathcal{C}, X)$. Hence we may write $K_1 = \amalg _{i_0, i_1 \in I} \{ V_{i_0i_1j} \to X\} $ so that the morphism $K_1 \to K_0 \times K_0$ is given by coverings $\{ V_{i_0i_1j} \to U_{i_0} \times _ X U_{i_1}\} $ of the site $\mathcal{C}$. Thus we can further identify
\[ \check{H}^0(K, \mathcal{F}) = \mathop{\mathrm{Ker}}( \prod \nolimits _ i \mathcal{F}(U_ i) \longrightarrow \prod \nolimits _{i_0i_1 j} \mathcal{F}(V_{i_0i_1j}) ) \]
with obvious map. The sheaf property of $\mathcal{F}$ implies that $\check{H}^0(K, \mathcal{F}) = H^0(X, \mathcal{F})$. $\square$
In fact this property characterizes the abelian sheaves among all abelian presheaves on $\mathcal{C}$ of course. The analogue of Cohomology, Lemma 25.5.2 in this case is the following.
Lemma 25.5.2. Let $\mathcal{C}$ be a site with fibre products. Let $X$ be an object of $\mathcal{C}$. Let $K$ be a hypercovering of $X$. Let $\mathcal{I}$ be an injective sheaf of abelian groups on $\mathcal{C}$. Then
\[ \check{H}^ p(K, \mathcal{I}) = \left\{ \begin{matrix} \mathcal{I}(X)
& \text{if}
& p = 0
\\ 0
& \text{if}
& p > 0
\end{matrix} \right. \]
Proof. Observe that for any object $Z = \{ U_ i \to X\} $ of $\text{SR}(\mathcal{C}, X)$ and any abelian sheaf $\mathcal{F}$ on $\mathcal{C}$ we have
\begin{eqnarray*} \mathcal{F}(Z) & = & \prod \mathcal{F}(U_ i) \\ & = & \prod \mathop{\mathrm{Mor}}\nolimits _{\textit{PSh}(\mathcal{C})}(h_{U_ i}, \mathcal{F})\\ & = & \mathop{\mathrm{Mor}}\nolimits _{\textit{PSh}(\mathcal{C})}(F(Z), \mathcal{F})\\ & = & \mathop{\mathrm{Mor}}\nolimits _{\textit{PAb}(\mathcal{C})}(\mathbf{Z}_{F(Z)}, \mathcal{F}) \\ & = & \mathop{\mathrm{Mor}}\nolimits _{\textit{Ab}(\mathcal{C})}(\mathbf{Z}_{F(Z)}^\# , \mathcal{F}) \end{eqnarray*}
Thus we see, for any simplicial object $K$ of $\text{SR}(\mathcal{C}, X)$ that we have
25.5.2.1
\begin{equation} \label{hypercovering-equation-identify-cech} s(\mathcal{F}(K)) = \mathop{\mathrm{Hom}}\nolimits _{\textit{Ab}(\mathcal{C})}(s(\mathbf{Z}_{F(K)}^\# ), \mathcal{F}) \end{equation}
see Definition 25.4.1 for notation. The complex of sheaves $s(\mathbf{Z}_{F(K)}^\# )$ is quasi-isomorphic to $\mathbf{Z}_ X^\# $ if $K$ is a hypercovering, see Lemma 25.4.5. We conclude that if $\mathcal{I}$ is an injective abelian sheaf, and $K$ a hypercovering, 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 25.5.1 the lemma is proved. $\square$
Next we come to the analogue of Cohomology on Sites, Lemma 21.10.6. Let $\mathcal{C}$ be a site. Let $\mathcal{F}$ be a sheaf of abelian groups on $\mathcal{C}$. Recall that $\underline{H}^ i(\mathcal{F})$ indicates the presheaf of abelian groups on $\mathcal{C}$ which is defined by the rule $\underline{H}^ i(\mathcal{F}) : U \longmapsto H^ i(U, \mathcal{F})$. We extend this to $\text{SR}(\mathcal{C})$ as in the introduction to this section.
Lemma 25.5.3. Let $\mathcal{C}$ be a site with fibre products. Let $X$ be an object of $\mathcal{C}$. Let $K$ be a hypercovering of $X$. Let $\mathcal{F}$ be a sheaf of abelian groups on $\mathcal{C}$. There is a map in $D^{+}(\textit{Ab})$ functorial in $\mathcal{F}$, which induces natural transformations as functors $\textit{Ab}(\mathcal{C}) \to \textit{Ab}$. Moreover, there is a spectral sequence $(E_ r, d_ r)_{r \geq 0}$ with converging to $H^{p + q}(X, \mathcal{F})$. This spectral sequence is functorial in $\mathcal{F}$ and in the hypercovering $K$.
\[ s(\mathcal{F}(K)) \longrightarrow R\Gamma (X, \mathcal{F}) \]
\[ \check{H}^ i(K, -) \longrightarrow H^ i(X, -) \]
\[ E_2^{p, q} = \check{H}^ p(K, \underline{H}^ q(\mathcal{F})) \]
Proof. We could prove this by the same method as employed in the corresponding lemma in the chapter on cohomology. Instead let us prove this by a double complex argument.
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 on the complex $s(\mathcal{I}^ q(K))$ associated to the cosimplicial abelian group $\mathcal{I}^ p(K)$ and the differential $d_2^{p, q} : A^{p, q} \to A^{p, q + 1}$ is the one coming from the differential $\mathcal{I}^ q \to \mathcal{I}^{q + 1}$. Denote $\text{Tot}(A^{\bullet , \bullet })$ the total complex associated to the double complex $A^{\bullet , \bullet }$, see Homology, Section 12.18. 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 25.5.2 the complexes $s(\mathcal{I}^ q(K))$ are acyclic in positive degrees and have $H^0$ equal to $\mathcal{I}^ q(X)$. Hence by Homology, Lemma 12.25.4 the natural map
\[ \mathcal{I}^\bullet (X) \longrightarrow \text{Tot}(A^{\bullet , \bullet }) \]
is a quasi-isomorphism of complexes of abelian groups. In particular we conclude that $H^ n(\text{Tot}(A^{\bullet , \bullet })) = H^ n(X, \mathcal{F})$.
The map $s(\mathcal{F}(K)) \longrightarrow R\Gamma (X, \mathcal{F})$ of the lemma is the composition of the map $s(\mathcal{F}(K)) \to \text{Tot}(A^{\bullet , \bullet })$ followed by the inverse of the displayed quasi-isomorphism above. This works because $\mathcal{I}^\bullet (X)$ is a representative of $R\Gamma (X, \mathcal{F})$.
Consider the spectral sequence $({}'E_ r, {}'d_ r)_{r \geq 0}$. By Homology, Lemma 12.25.1 we see that
\[ {}'E_2^{p, q} = H^ p_ I(H^ q_{II}(A^{\bullet , \bullet })) \]
In other words, we first take cohomology with respect to $d_2$ which gives the groups ${}'E_1^{p, q} = \underline{H}^ q(\mathcal{F})(K_ p)$. Hence it is indeed the case (by the description of the differential ${}'d_1$) that ${}'E_2^{p, q} = \check{H}^ p(K, \underline{H}^ q(\mathcal{F}))$. By the above and Homology, Lemma 12.25.3 we see that this converges to $H^ n(X, \mathcal{F})$ as desired.
We omit the proof of the statements regarding the functoriality of the above constructions in the abelian sheaf $\mathcal{F}$ and the hypercovering $K$. $\square$
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