Section 59.18 (03OK): Čech cohomology

59.18 Čech cohomology

Our next goal is to use descent theory to show that $H^ i(\mathcal{C}, \mathcal{F}^ a) = H_{Zar}^ i(S, \mathcal{F})$ for all quasi-coherent sheaves $\mathcal{F}$ on $S$ , and any site $\mathcal{C}$ as in Theorem 59.17.4. To this end, we introduce Čech cohomology on sites. See [ArtinTopologies] and Cohomology on Sites, Sections 21.8, 21.9 and 21.10 for more details.

Definition 59.18.1. Let $\mathcal{C}$ be a category. Let $\mathcal{U} = \{ U_ i \to U\} _{i \in I}$ be a family of morphisms of $\mathcal{C}$ with fixed target. Assume that all the fibre products $U_{i_0} \times _ U \ldots \times _ U U_{i_ p}$ exists in $\mathcal{C}$ . Let $\mathcal{F} \in \textit{PAb}(\mathcal{C})$ be an abelian presheaf. We define the Čech complex $\check{\mathcal{C}}^\bullet (\mathcal{U}, \mathcal{F})$ by

$\prod _{i_0 \in I} \mathcal{F}(U_{i_0}) \to \prod _{i_0, i_1 \in I} \mathcal{F}(U_{i_0} \times _ U U_{i_1}) \to \prod _{i_0, i_1, i_2 \in I} \mathcal{F}(U_{i_0} \times _ U U_{i_1} \times _ U U_{i_2}) \to \ldots$

where the first term is in degree 0 and the maps are the usual ones. The Čech cohomology groups are defined by

$\check{H}^ p(\mathcal{U}, \mathcal{F}) = H^ p(\check{\mathcal{C}}^\bullet (\mathcal{U}, \mathcal{F})).$

In the above definition, it is essential to allow the indices $(i_0, \ldots , i_ p)$ to have repetitions.

Lemma 59.18.2. Notation and assumptions as in Definition 59.18.1. The functor $\check{\mathcal{C}}^\bullet (\mathcal{U}, -)$ is exact on the category $\textit{PAb}(\mathcal{C})$ .

In other words, if $0\to \mathcal{F}_1\to \mathcal{F}_2\to \mathcal{F}_3\to 0$ is a short exact sequence of presheaves of abelian groups, then

$0 \to \check{\mathcal{C}}^\bullet \left(\mathcal{U}, \mathcal{F}_1\right) \to \check{\mathcal{C}}^\bullet (\mathcal{U}, \mathcal{F}_2) \to \check{\mathcal{C}}^\bullet (\mathcal{U}, \mathcal{F}_3)\to 0$

is a short exact sequence of complexes.

Proof. This follows at once from the definition of a short exact sequence of presheaves. Namely, as the category of abelian presheaves is the category of functors on some category with values in $\textit{Ab}$ , it is automatically an abelian category: a sequence $\mathcal{F}_1\to \mathcal{F}_2\to \mathcal{F}_3$ is exact in $\textit{PAb}$ if and only if for all $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ , the sequence $\mathcal{F}_1(U) \to \mathcal{F}_2(U) \to \mathcal{F}_3(U)$ is exact in $\textit{Ab}$ . So the complex above is merely a product of short exact sequences in each degree. See also Cohomology on Sites, Lemma 21.9.1. $\square$

This shows that $\check{H}^\bullet (\mathcal{U}, -)$ is a $\delta$ -functor. We now proceed to show that it is a universal $\delta$ -functor. We thus need to show that it is an effaceable functor. We start by recalling the Yoneda lemma.

Lemma 59.18.3 (Yoneda Lemma). For any presheaf $\mathcal{F}$ on a category $\mathcal{C}$ and $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ there is a functorial isomorphism

$\mathop{\mathrm{Hom}}\nolimits _{\textit{PSh}(\mathcal{C})}(h_ U, \mathcal{F}) = \mathcal{F}(U).$

Proof. See Categories, Lemma 4.3.5. $\square$

Given a set $E$ we denote (in this section) $\mathbf{Z}[E]$ the free abelian group on $E$ . In a formula $\mathbf{Z}[E] = \bigoplus _{e \in E} \mathbf{Z}$ , i.e., $\mathbf{Z}[E]$ is a free $\mathbf{Z}$ -module having a basis consisting of the elements of $E$ . Using this notation we introduce the free abelian presheaf on a presheaf of sets.

Definition 59.18.4. Let $\mathcal{C}$ be a category. Given a presheaf of sets $\mathcal{G}$ , we define the free abelian presheaf on $\mathcal{G}$ , denoted $\mathbf{Z}_\mathcal {G}$ , by the rule

$\mathbf{Z}_\mathcal {G}(U) = \mathbf{Z}[\mathcal{G}(U)]$

for $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ with restriction maps induced by the restriction maps of $\mathcal{G}$ . In the special case $\mathcal{G} = h_ U$ we write simply $\mathbf{Z}_ U = \mathbf{Z}_{h_ U}$ .

The functor $\mathcal{G} \mapsto \mathbf{Z}_\mathcal {G}$ is left adjoint to the forgetful functor $\textit{PAb}(\mathcal{C}) \to \textit{PSh}(\mathcal{C})$ . Thus, for any presheaf $\mathcal{F}$ , there is a canonical isomorphism

$\mathop{\mathrm{Hom}}\nolimits _{\textit{PAb}(\mathcal{C})}(\mathbf{Z}_ U, \mathcal{F}) = \mathop{\mathrm{Hom}}\nolimits _{\textit{PSh}(\mathcal{C})}(h_ U, \mathcal{F}) = \mathcal{F}(U)$

the last equality by the Yoneda lemma. In particular, we have the following result.

Lemma 59.18.5. Notation and assumptions as in Definition 59.18.1. The Čech complex $\check{\mathcal{C}}^\bullet (\mathcal{U}, \mathcal{F})$ can be described explicitly as follows

$\begin{eqnarray*} \check{\mathcal{C}}^\bullet (\mathcal{U}, \mathcal{F}) & = & \left( \prod _{i_0 \in I} \mathop{\mathrm{Hom}}\nolimits _{\textit{PAb}(\mathcal{C})}(\mathbf{Z}_{U_{i_0}}, \mathcal{F}) \to \prod _{i_0, i_1 \in I} \mathop{\mathrm{Hom}}\nolimits _{\textit{PAb}(\mathcal{C})}( \mathbf{Z}_{U_{i_0} \times _ U U_{i_1}}, \mathcal{F}) \to \ldots \right) \\ & = & \mathop{\mathrm{Hom}}\nolimits _{\textit{PAb}(\mathcal{C})}\left( \left( \bigoplus _{i_0 \in I} \mathbf{Z}_{U_{i_0}} \leftarrow \bigoplus _{i_0, i_1 \in I} \mathbf{Z}_{U_{i_0} \times _ U U_{i_1}} \leftarrow \ldots \right), \mathcal{F}\right) \end{eqnarray*}$

Proof. This follows from the formula above. See Cohomology on Sites, Lemma 21.9.3. $\square$

This reduces us to studying only the complex in the first argument of the last $\mathop{\mathrm{Hom}}\nolimits$ .

Lemma 59.18.6. Notation and assumptions as in Definition 59.18.1. The complex of abelian presheaves

$\begin{align*} \mathbf{Z}_\mathcal {U}^\bullet \quad : \quad \bigoplus _{i_0 \in I} \mathbf{Z}_{U_{i_0}} \leftarrow \bigoplus _{i_0, i_1 \in I} \mathbf{Z}_{U_{i_0} \times _ U U_{i_1}} \leftarrow \bigoplus _{i_0, i_1, i_2 \in I} \mathbf{Z}_{U_{i_0} \times _ U U_{i_1} \times _ U U_{i_2}} \leftarrow \ldots \end{align*}$

is exact in all degrees except $0$ in $\textit{PAb}(\mathcal{C})$ .

Proof. For any $V\in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ the complex of abelian groups $\mathbf{Z}_\mathcal {U}^\bullet (V)$ is

$\begin{matrix} \mathbf{Z}\left[ \coprod _{i_0\in I} \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(V, U_{i_0})\right] \leftarrow \mathbf{Z}\left[ \coprod _{i_0, i_1 \in I} \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(V, U_{i_0} \times _ U U_{i_1})\right] \leftarrow \ldots = \\ \bigoplus _{\varphi : V \to U} \left( \mathbf{Z}\left[\coprod _{i_0 \in I} \mathop{\mathrm{Mor}}\nolimits _\varphi (V, U_{i_0})\right] \leftarrow \mathbf{Z}\left[\coprod _{i_0, i_1\in I} \mathop{\mathrm{Mor}}\nolimits _\varphi (V, U_{i_0}) \times \mathop{\mathrm{Mor}}\nolimits _\varphi (V, U_{i_1})\right] \leftarrow \ldots \right) \end{matrix}$

where

$\mathop{\mathrm{Mor}}\nolimits _{\varphi }(V, U_ i) = \{ V \to U_ i \text{ such that } V \to U_ i \to U \text{ equals } \varphi \} .$

Set $S_\varphi = \coprod _{i\in I} \mathop{\mathrm{Mor}}\nolimits _\varphi (V, U_ i)$ , so that

$\mathbf{Z}_\mathcal {U}^\bullet (V) = \bigoplus _{\varphi : V \to U} \left( \mathbf{Z}[S_\varphi ] \leftarrow \mathbf{Z}[S_\varphi \times S_\varphi ] \leftarrow \mathbf{Z}[S_\varphi \times S_\varphi \times S_\varphi ] \leftarrow \ldots \right).$

Thus it suffices to show that for each $S = S_\varphi$ , the complex

$\begin{align*} \mathbf{Z}[S] \leftarrow \mathbf{Z}[S \times S] \leftarrow \mathbf{Z}[S \times S \times S] \leftarrow \ldots \end{align*}$

is exact in negative degrees. To see this, we can give an explicit homotopy. Fix $s\in S$ and define $K: n_{(s_0, \ldots , s_ p)} \mapsto n_{(s, s_0, \ldots , s_ p)}.$ One easily checks that $K$ is a nullhomotopy for the operator

$\delta : \eta _{(s_0, \ldots , s_ p)} \mapsto \sum \nolimits _{i = 0}^ p (-1)^ p \eta _{(s_0, \ldots , \hat s_ i, \ldots , s_ p)}.$

See Cohomology on Sites, Lemma 21.9.4 for more details. $\square$

Lemma 59.18.7. Notation and assumptions as in Definition 59.18.1. If $\mathcal{I}$ is an injective object of $\textit{PAb}(\mathcal{C})$ , then $\check H^ p(\mathcal{U}, \mathcal{I}) = 0$ for all $p > 0$ .

Proof. The Čech complex is the result of applying the functor $\mathop{\mathrm{Hom}}\nolimits _{\textit{PAb}(\mathcal{C})}(-, \mathcal{I})$ to the complex $\mathbf{Z}^\bullet _\mathcal {U}$ , i.e.,

$\check H^ p(\mathcal{U}, \mathcal{I}) = H^ p (\mathop{\mathrm{Hom}}\nolimits _{\textit{PAb}(\mathcal{C})} (\mathbf{Z}^\bullet _\mathcal {U}, \mathcal{I})).$

But we have just seen that $\mathbf{Z}^\bullet _\mathcal {U}$ is exact in negative degrees, and the functor $\mathop{\mathrm{Hom}}\nolimits _{\textit{PAb}(\mathcal{C})}(-, \mathcal{I})$ is exact, hence $\mathop{\mathrm{Hom}}\nolimits _{\textit{PAb}(\mathcal{C})} (\mathbf{Z}^\bullet _\mathcal {U}, \mathcal{I})$ is exact in positive degrees. $\square$

Theorem 59.18.8. Notation and assumptions as in Definition 59.18.1. On $\textit{PAb}(\mathcal{C})$ the functors $\check{H}^ p(\mathcal{U}, -)$ are the right derived functors of $\check{H}^0(\mathcal{U}, -)$ .

Proof. By the Lemma 59.18.7, the functors $\check H^ p(\mathcal{U}, -)$ are universal $\delta$ -functors since they are effaceable. So are the right derived functors of $\check H^0(\mathcal{U}, -)$ . Since they agree in degree $0$ , they agree by the universal property of universal $\delta$ -functors. For more details see Cohomology on Sites, Lemma 21.9.6. $\square$

Remark 59.18.9. Observe that all of the preceding statements are about presheaves so we haven't made use of the topology yet.

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59.18 Čech cohomology

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