Lemma 21.9.1. The functor given by Equation (21.9.0.1) is an exact functor (see Homology, Lemma 12.7.2).
21.9 Čech cohomology as a functor on presheaves
Warning: In this section we work exclusively with abelian presheaves on a category. The results are completely wrong in the setting of sheaves and categories of sheaves!
Let $\mathcal{C}$ be a category. Let $\mathcal{U} = \{ U_ i \to U\} _{i \in I}$ be a family of morphisms with fixed target such that all fibre products $U_{i_0} \times _ U \ldots \times _ U U_{i_ p}$ exist in $\mathcal{C}$. Let $\mathcal{F}$ be an abelian presheaf on $\mathcal{C}$. The construction
\[ \mathcal{F} \longmapsto \check{\mathcal{C}}^\bullet (\mathcal{U}, \mathcal{F}) \]
is functorial in $\mathcal{F}$. In fact, it is a functor
21.9.0.1
\begin{equation} \label{sites-cohomology-equation-cech-functor} \check{\mathcal{C}}^\bullet (\mathcal{U}, -) : \textit{PAb}(\mathcal{C}) \longrightarrow \text{Comp}^{+}(\textit{Ab}) \end{equation}
see Derived Categories, Definition 13.8.1 for notation. Recall that the category of bounded below complexes in an abelian category is an abelian category, see Homology, Lemma 12.13.9.
Proof. For any object $W$ of $\mathcal{C}$ the functor $\mathcal{F} \mapsto \mathcal{F}(W)$ is an additive exact functor from $\textit{PAb}(\mathcal{C})$ to $\textit{Ab}$. The terms $\check{\mathcal{C}}^ p(\mathcal{U}, \mathcal{F})$ of the complex are products of these exact functors and hence exact. Moreover a sequence of complexes is exact if and only if the sequence of terms in a given degree is exact. Hence the lemma follows. $\square$
Lemma 21.9.2. Let $\mathcal{C}$ be a category. Let $\mathcal{U} = \{ U_ i \to U\} _{i \in I}$ be a family of morphisms with fixed target such that all fibre products $U_{i_0} \times _ U \ldots \times _ U U_{i_ p}$ exist in $\mathcal{C}$. The functors $\mathcal{F} \mapsto \check{H}^ n(\mathcal{U}, \mathcal{F})$ form a $\delta $-functor from the abelian category $\textit{PAb}(\mathcal{C})$ to the category of $\mathbf{Z}$-modules (see Homology, Definition 12.12.1).
Proof. By Lemma 21.9.1 a short exact sequence of abelian presheaves $0 \to \mathcal{F}_1 \to \mathcal{F}_2 \to \mathcal{F}_3 \to 0$ is turned into a short exact sequence of complexes of $\mathbf{Z}$-modules. Hence we can use Homology, Lemma 12.13.12 to get the boundary maps $\delta _{\mathcal{F}_1 \to \mathcal{F}_2 \to \mathcal{F}_3} : \check{H}^ n(\mathcal{U}, \mathcal{F}_3) \to \check{H}^{n + 1}(\mathcal{U}, \mathcal{F}_1)$ and a corresponding long exact sequence. We omit the verification that these maps are compatible with maps between short exact sequences of presheaves. $\square$
Lemma 21.9.3. Let $\mathcal{C}$ be a category. Let $\mathcal{U} = \{ U_ i \to U\} _{i \in I}$ be a family of morphisms with fixed target such that all fibre products $U_{i_0} \times _ U \ldots \times _ U U_{i_ p}$ exist in $\mathcal{C}$. Consider the chain complex $\mathbf{Z}_{\mathcal{U}, \bullet }$ of abelian presheaves where the last nonzero term is placed in degree $0$ and where the map is given by $(-1)^ j$ times the canonical map. Then there is an isomorphism functorial in $\mathcal{F} \in \mathop{\mathrm{Ob}}\nolimits (\textit{PAb}(\mathcal{C}))$.
\[ \ldots \to \bigoplus _{i_0i_1i_2} \mathbf{Z}_{U_{i_0} \times _ U U_{i_1} \times _ U U_{i_2}} \to \bigoplus _{i_0i_1} \mathbf{Z}_{U_{i_0} \times _ U U_{i_1}} \to \bigoplus _{i_0} \mathbf{Z}_{U_{i_0}} \to 0 \to \ldots \]
\[ \mathbf{Z}_{U_{i_0} \times _ U \ldots \times _ u U_{i_{p + 1}}} \longrightarrow \mathbf{Z}_{U_{i_0} \times _ U \ldots \widehat{U_{i_ j}} \ldots \times _ U U_{i_{p + 1}}} \]
\[ \mathop{\mathrm{Hom}}\nolimits _{\textit{PAb}(\mathcal{C})}(\mathbf{Z}_{\mathcal{U}, \bullet }, \mathcal{F}) = \check{\mathcal{C}}^\bullet (\mathcal{U}, \mathcal{F}) \]
Proof. This is a tautology based on the fact that
\begin{align*} \mathop{\mathrm{Hom}}\nolimits _{\textit{PAb}(\mathcal{C})}( \bigoplus _{i_0 \ldots i_ p} \mathbf{Z}_{U_{i_0} \times _ U \ldots \times _ U U_{i_ p}}, \mathcal{F}) & = \prod _{i_0 \ldots i_ p} \mathop{\mathrm{Hom}}\nolimits _{\textit{PAb}(\mathcal{C})}( \mathbf{Z}_{U_{i_0} \times _ U \ldots \times _ U U_{i_ p}}, \mathcal{F}) \\ & = \prod _{i_0 \ldots i_ p} \mathcal{F}(U_{i_0} \times _ U \ldots \times _ U U_{i_ p}) \end{align*}
see Modules on Sites, Lemma 18.4.2. $\square$
Lemma 21.9.4. Let $\mathcal{C}$ be a category. Let $\mathcal{U} = \{ f_ i : U_ i \to U\} _{i \in I}$ be a family of morphisms with fixed target such that all fibre products $U_{i_0} \times _ U \ldots \times _ U U_{i_ p}$ exist in $\mathcal{C}$. The chain complex $\mathbf{Z}_{\mathcal{U}, \bullet }$ of presheaves of Lemma 21.9.3 above is exact in positive degrees, i.e., the homology presheaves $H_ i(\mathbf{Z}_{\mathcal{U}, \bullet })$ are zero for $i > 0$.
Proof. Let $V$ be an object of $\mathcal{C}$. We have to show that the chain complex of abelian groups $\mathbf{Z}_{\mathcal{U}, \bullet }(V)$ is exact in degrees $> 0$. This is the complex
\[ \xymatrix{ \ldots \ar[d] \\ \bigoplus _{i_0i_1i_2} \mathbf{Z}[ \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(V, U_{i_0} \times _ U U_{i_1} \times _ U U_{i_2}) ] \ar[d] \\ \bigoplus _{i_0i_1} \mathbf{Z}[ \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(V, U_{i_0} \times _ U U_{i_1}) ] \ar[d] \\ \bigoplus _{i_0} \mathbf{Z}[ \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(V, U_{i_0}) ] \ar[d] \\ 0 } \]
For any morphism $\varphi : V \to U$ denote $\mathop{\mathrm{Mor}}\nolimits _\varphi (V, U_ i) = \{ \varphi _ i : V \to U_ i \mid f_ i \circ \varphi _ i = \varphi \} $. We will use a similar notation for $\mathop{\mathrm{Mor}}\nolimits _\varphi (V, U_{i_0} \times _ U \ldots \times _ U U_{i_ p})$. Note that composing with the various projection maps between the fibred products $U_{i_0} \times _ U \ldots \times _ U U_{i_ p}$ preserves these morphism sets. Hence we see that the complex above is the same as the complex
\[ \xymatrix{ \ldots \ar[d] \\ \bigoplus _\varphi \bigoplus _{i_0i_1i_2} \mathbf{Z}[ \mathop{\mathrm{Mor}}\nolimits _\varphi (V, U_{i_0} \times _ U U_{i_1} \times _ U U_{i_2}) ] \ar[d] \\ \bigoplus _\varphi \bigoplus _{i_0i_1} \mathbf{Z}[ \mathop{\mathrm{Mor}}\nolimits _\varphi (V, U_{i_0} \times _ U U_{i_1}) ] \ar[d] \\ \bigoplus _\varphi \bigoplus _{i_0} \mathbf{Z}[ \mathop{\mathrm{Mor}}\nolimits _\varphi (V, U_{i_0}) ] \ar[d] \\ 0 } \]
Next, we make the remark that we have
\[ \mathop{\mathrm{Mor}}\nolimits _\varphi (V, U_{i_0} \times _ U \ldots \times _ U U_{i_ p}) = \mathop{\mathrm{Mor}}\nolimits _\varphi (V, U_{i_0}) \times \ldots \times \mathop{\mathrm{Mor}}\nolimits _\varphi (V, U_{i_ p}) \]
Using this and the fact that $\mathbf{Z}[A] \oplus \mathbf{Z}[B] = \mathbf{Z}[A \amalg B]$ we see that the complex becomes
\[ \xymatrix{ \ldots \ar[d] \\ \bigoplus _\varphi \mathbf{Z}\left[ \coprod _{i_0i_1i_2} \mathop{\mathrm{Mor}}\nolimits _\varphi (V, U_{i_0}) \times \mathop{\mathrm{Mor}}\nolimits _\varphi (V, U_{i_1}) \times \mathop{\mathrm{Mor}}\nolimits _\varphi (V, U_{i_2}) \right] \ar[d] \\ \bigoplus _\varphi \mathbf{Z}\left[ \coprod _{i_0i_1} \mathop{\mathrm{Mor}}\nolimits _\varphi (V, U_{i_0}) \times \mathop{\mathrm{Mor}}\nolimits _\varphi (V, U_{i_1}) \right] \ar[d] \\ \bigoplus _\varphi \mathbf{Z}\left[ \coprod _{i_0} \mathop{\mathrm{Mor}}\nolimits _\varphi (V, U_{i_0}) \right] \ar[d] \\ 0 } \]
Finally, on setting $S_\varphi = \coprod _{i \in I} \mathop{\mathrm{Mor}}\nolimits _\varphi (V, U_ i)$ we see that we get
\[ \bigoplus \nolimits _\varphi \left(\ldots \to \mathbf{Z}[S_\varphi \times S_\varphi \times S_\varphi ] \to \mathbf{Z}[S_\varphi \times S_\varphi ] \to \mathbf{Z}[S_\varphi ] \to 0 \to \ldots \right) \]
Thus we have simplified our task. Namely, it suffices to show that for any nonempty set $S$ the (extended) complex of free abelian groups
\[ \ldots \to \mathbf{Z}[S \times S \times S] \to \mathbf{Z}[S \times S] \to \mathbf{Z}[S] \xrightarrow {\Sigma } \mathbf{Z} \to 0 \to \ldots \]
is exact in all degrees. To see this fix an element $s \in S$, and use the homotopy
\[ n_{(s_0, \ldots , s_ p)} \longmapsto n_{(s, s_0, \ldots , s_ p)} \]
with obvious notations. $\square$
Lemma 21.9.5. Let $\mathcal{C}$ be a category. Let $\mathcal{U} = \{ f_ i : U_ i \to U\} _{i \in I}$ be a family of morphisms with fixed target such that all fibre products $U_{i_0} \times _ U \ldots \times _ U U_{i_ p}$ exist in $\mathcal{C}$. Let $\mathcal{O}$ be a presheaf of rings on $\mathcal{C}$. The chain complex is exact in positive degrees. Here $\mathbf{Z}_{\mathcal{U}, \bullet }$ is the chain complex of Lemma 21.9.3, and the tensor product is over the constant presheaf of rings with value $\mathbf{Z}$.
\[ \mathbf{Z}_{\mathcal{U}, \bullet } \otimes _{p, \mathbf{Z}} \mathcal{O} \]
Proof. Let $V$ be an object of $\mathcal{C}$. In the proof of Lemma 21.9.4 we saw that $\mathbf{Z}_{\mathcal{U}, \bullet }(V)$ is isomorphic as a complex to a direct sum of complexes which are homotopic to $\mathbf{Z}$ placed in degree zero. Hence also $\mathbf{Z}_{\mathcal{U}, \bullet }(V) \otimes _\mathbf {Z} \mathcal{O}(V)$ is isomorphic as a complex to a direct sum of complexes which are homotopic to $\mathcal{O}(V)$ placed in degree zero. Or you can use Modules on Sites, Lemma 18.28.11, which applies since the presheaves $\mathbf{Z}_{\mathcal{U}, i}$ are flat, and the proof of Lemma 21.9.4 shows that $H_0(\mathbf{Z}_{\mathcal{U}, \bullet })$ is a flat presheaf also. $\square$
Lemma 21.9.6. Let $\mathcal{C}$ be a category. Let $\mathcal{U} = \{ f_ i : U_ i \to U\} _{i \in I}$ be a family of morphisms with fixed target such that all fibre products $U_{i_0} \times _ U \ldots \times _ U U_{i_ p}$ exist in $\mathcal{C}$. The Čech cohomology functors $\check{H}^ p(\mathcal{U}, -)$ are canonically isomorphic as a $\delta $-functor to the right derived functors of the functor Moreover, there is a functorial quasi-isomorphism where the right hand side indicates the derived functor of the left exact functor $\check{H}^0(\mathcal{U}, -)$.
\[ \check{H}^0(\mathcal{U}, -) : \textit{PAb}(\mathcal{C}) \longrightarrow \textit{Ab}. \]
\[ \check{\mathcal{C}}^\bullet (\mathcal{U}, \mathcal{F}) \longrightarrow R\check{H}^0(\mathcal{U}, \mathcal{F}) \]
\[ R\check{H}^0(\mathcal{U}, -) : D^{+}(\textit{PAb}(\mathcal{C})) \longrightarrow D^{+}(\mathbf{Z}) \]
Proof. Note that the category of abelian presheaves has enough injectives, see Injectives, Proposition 19.6.1. Note that $\check{H}^0(\mathcal{U}, -)$ is a left exact functor from the category of abelian presheaves to the category of $\mathbf{Z}$-modules. Hence the derived functor and the right derived functor exist, see Derived Categories, Section 13.20.
Let $\mathcal{I}$ be a injective abelian presheaf. In this case the functor $\mathop{\mathrm{Hom}}\nolimits _{\textit{PAb}(\mathcal{C})}(-, \mathcal{I})$ is exact on $\textit{PAb}(\mathcal{C})$. By Lemma 21.9.3 we have
\[ \mathop{\mathrm{Hom}}\nolimits _{\textit{PAb}(\mathcal{C})}( \mathbf{Z}_{\mathcal{U}, \bullet }, \mathcal{I}) = \check{\mathcal{C}}^\bullet (\mathcal{U}, \mathcal{I}). \]
By Lemma 21.9.4 we have that $\mathbf{Z}_{\mathcal{U}, \bullet }$ is exact in positive degrees. Hence by the exactness of Hom into $\mathcal{I}$ mentioned above we see that $\check{H}^ i(\mathcal{U}, \mathcal{I}) = 0$ for all $i > 0$. Thus the $\delta $-functor $(\check{H}^ n, \delta )$ (see Lemma 21.9.2) satisfies the assumptions of Homology, Lemma 12.12.4, and hence is a universal $\delta $-functor.
By Derived Categories, Lemma 13.20.4 also the sequence $R^ i\check{H}^0(\mathcal{U}, -)$ forms a universal $\delta $-functor. By the uniqueness of universal $\delta $-functors, see Homology, Lemma 12.12.5 we conclude that $R^ i\check{H}^0(\mathcal{U}, -) = \check{H}^ i(\mathcal{U}, -)$. This is enough for most applications and the reader is suggested to skip the rest of the proof.
Let $\mathcal{F}$ be any abelian presheaf on $\mathcal{C}$. Choose an injective resolution $\mathcal{F} \to \mathcal{I}^\bullet $ in the category $\textit{PAb}(\mathcal{C})$. Consider the double complex $\check{\mathcal{C}}^\bullet (\mathcal{U}, \mathcal{I}^\bullet )$ with terms $\check{\mathcal{C}}^ p(\mathcal{U}, \mathcal{I}^ q)$. Next, consider the total complex $\text{Tot}(\check{\mathcal{C}}^\bullet (\mathcal{U}, \mathcal{I}^\bullet ))$ associated to this double complex, see Homology, Section 12.18. There is a map of complexes
\[ \check{\mathcal{C}}^\bullet (\mathcal{U}, \mathcal{F}) \longrightarrow \text{Tot}(\check{\mathcal{C}}^\bullet (\mathcal{U}, \mathcal{I}^\bullet )) \]
coming from the maps $\check{\mathcal{C}}^ p(\mathcal{U}, \mathcal{F}) \to \check{\mathcal{C}}^ p(\mathcal{U}, \mathcal{I}^0)$ and there is a map of complexes
\[ \check{H}^0(\mathcal{U}, \mathcal{I}^\bullet ) \longrightarrow \text{Tot}(\check{\mathcal{C}}^\bullet (\mathcal{U}, \mathcal{I}^\bullet )) \]
coming from the maps $\check{H}^0(\mathcal{U}, \mathcal{I}^ q) \to \check{\mathcal{C}}^0(\mathcal{U}, \mathcal{I}^ q)$. Both of these maps are quasi-isomorphisms by an application of Homology, Lemma 12.25.4. Namely, the columns of the double complex are exact in positive degrees because the Čech complex as a functor is exact (Lemma 21.9.1) and the rows of the double complex are exact in positive degrees since as we just saw the higher Čech cohomology groups of the injective presheaves $\mathcal{I}^ q$ are zero. Since quasi-isomorphisms become invertible in $D^{+}(\mathbf{Z})$ this gives the last displayed morphism of the lemma. We omit the verification that this morphism is functorial. $\square$
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