The Stacks project

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24.5 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 20.10, 20.11 and 20.12.

Lemma 24.5.1. 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 $\check{H}^0(K, \mathcal{F}) = \mathcal{F}(X)$.

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 24.5.2 in this case is the following.

Lemma 24.5.2. 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{Mor}\nolimits _{\textit{PSh}(\mathcal{C})}(h_{U_ i}, \mathcal{F})\\ & = & \mathop{Mor}\nolimits _{\textit{PSh}(\mathcal{C})}(F(Z), \mathcal{F})\\ & = & \mathop{Mor}\nolimits _{\textit{PAb}(\mathcal{C})}(\mathbf{Z}_{F(Z)}, \mathcal{F}) \\ & = & \mathop{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
\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 24.4.1 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 24.4.5. 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 24.5.1 the lemma is proved. $\square$

Next we come to the analogue of Cohomology on Sites, Lemma 21.11.6. 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 24.5.3. 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

\[ s(\mathcal{F}(K)) \longrightarrow R\Gamma (X, \mathcal{F}) \]

in $D^{+}(\textit{Ab})$ functorial in $\mathcal{F}$, which induces natural transformations

\[ \check{H}^ i(K, -) \longrightarrow H^ i(X, -) \]

as 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 $H^{p + q}(X, \mathcal{F})$. This spectral sequence is functorial in $\mathcal{F}$ and in the hypercovering $K$.

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 $\mathcal{I}^ p \to \mathcal{I}^{p + 1}$ 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. As usual we denote $sA^\bullet $ the simple complex associated to the double complex $A^{\bullet , \bullet }$. We will use the two spectral sequences $({}'E_ r, {}'d_ r)$ and $({}''E_ r, {}''d_ r)$ associated to this double complex, see Homology, Section 12.22.

By Lemma 24.5.2 the complexes $s(\mathcal{I}^ p(K))$ are acyclic in positive degrees and have $H^0$ equal to $\mathcal{I}^ p(X)$. Hence by Homology, Lemma 12.22.7 and its proof the spectral sequence $({}'E_ r, {}'d_ r)$ degenerates, and the natural map

\[ \mathcal{I}^\bullet (X) \longrightarrow sA^\bullet \]

is a quasi-isomorphism of complexes of abelian groups. In particular we conclude that $H^ n(sA^\bullet ) = H^ n(X, \mathcal{F})$.

The map $s(\mathcal{F}(K)) \longrightarrow R\Gamma (X, \mathcal{F})$ of the lemma is the composition of the natural map $s(\mathcal{F}(K)) \to sA^\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 12.22.4 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 ${}''E_2^{p, q} = \check{H}^ p(K, \underline{H}^ q(\mathcal{F}))$. And by the other spectral sequence above we see that this one 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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