The Stacks project

58.18 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 58.17.4. 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 58.18.1. Let $\mathcal{C}$ be a category, $\mathcal{U} = \{ U_ i \to U\} _{i \in I}$ a family of morphisms of $\mathcal{C}$ with fixed target, and $\mathcal{F} \in \textit{PAb}(\mathcal{C})$ 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. Again, it is essential to allow the case $i_0 = i_1$ etc. The Čech cohomology groups are defined by

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

Lemma 58.18.2. 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 58.18.3 (Yoneda Lemma). For any presheaf $\mathcal{F}$ on a category $\mathcal{C}$ 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 58.18.4. 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 58.18.5. 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 58.18.6. 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

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

where

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

Set $S_\varphi = \coprod _{i\in I} \mathop{Mor}\nolimits _\varphi (V, U_ i)$, 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 58.18.7. Let $\mathcal{C}$ be a category. If $\mathcal{I}$ is an injective object of $\textit{PAb}(\mathcal{C})$ and $\mathcal{U}$ is a family of morphisms with fixed target in $\mathcal{C}$, 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 58.18.8. 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 58.18.7, 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 58.18.9. Observe that all of the preceding statements are about presheaves so we haven't made use of the topology yet.


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