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Tag 01EH

20.11. Čech cohomology as a functor on presheaves

Warning: In this section we work almost exclusively with presheaves and categories of presheaves and the results are completely wrong in the setting of sheaves and categories of sheaves!

Let $X$ be a ringed space. Let $\mathcal{U} : U = \bigcup_{i \in I} U_i$ be an open covering. Let $\mathcal{F}$ be a presheaf of $\mathcal{O}_X$-modules. We have the Čech complex $\check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{F})$ of $\mathcal{F}$ just by thinking of $\mathcal{F}$ as a presheaf of abelian groups. However, each term $\check{\mathcal{C}}^p(\mathcal{U}, \mathcal{F})$ has a natural structure of a $\mathcal{O}_X(U)$-module and the differential is given by $\mathcal{O}_X(U)$-module maps. Moreover, it is clear that the construction $$ \mathcal{F} \longmapsto \check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{F}) $$ is functorial in $\mathcal{F}$. In fact, it is a functor \begin{equation} \tag{20.11.0.1} \check{\mathcal{C}}^\bullet(\mathcal{U}, -) : \textit{PMod}(\mathcal{O}_X) \longrightarrow \text{Comp}^{+}(\text{Mod}_{\mathcal{O}_X(U)}) \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 12.12.9.

Lemma 20.11.1. The functor given by Equation (20.11.0.1) is an exact functor (see Homology, Lemma 12.7.2).

Proof. For any open $W \subset U$ the functor $\mathcal{F} \mapsto \mathcal{F}(W)$ is an additive exact functor from $\textit{PMod}(\mathcal{O}_X)$ to $\text{Mod}_{\mathcal{O}_X(U)}$. 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 20.11.2. Let $X$ be a ringed space. Let $\mathcal{U} : U = \bigcup_{i \in I} U_i$ be an open covering. The functors $\mathcal{F} \mapsto \check{H}^n(\mathcal{U}, \mathcal{F})$ form a $\delta$-functor from the abelian category of presheaves of $\mathcal{O}_X$-modules to the category of $\mathcal{O}_X(U)$-modules (see Homology, Definition 12.11.1).

Proof. By Lemma 20.11.1 a short exact sequence of presheaves of $\mathcal{O}_X$-modules $0 \to \mathcal{F}_1 \to \mathcal{F}_2 \to \mathcal{F}_3 \to 0$ is turned into a short exact sequence of complexes of $\mathcal{O}_X(U)$-modules. Hence we can use Homology, Lemma 12.12.12 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. $\square$

In the formulation of the following lemma we use the functor $j_{p!}$ of extension by $0$ for presheaves of modules relative to an open immersion $j : U \to X$. See Sheaves, Section 6.31. For any open $W \subset X$ and any presheaf $\mathcal{G}$ of $\mathcal{O}_X|_U$-modules we have $$ (j_{p!}\mathcal{G})(W) = \left\{ \begin{matrix} \mathcal{G}(W) & \text{if } W \subset U \\ 0 & \text{else.} \end{matrix} \right. $$ Moreover, the functor $j_{p!}$ is a left adjoint to the restriction functor see Sheaves, Lemma 6.31.8. In particular we have the following formula $$ \mathop{\mathrm{Hom}}\nolimits_{\mathcal{O}_X}(j_{p!}\mathcal{O}_U, \mathcal{F}) = \mathop{\mathrm{Hom}}\nolimits_{\mathcal{O}_U}(\mathcal{O}_U, \mathcal{F}|_U) = \mathcal{F}(U). $$ Since the functor $\mathcal{F} \mapsto \mathcal{F}(U)$ is an exact functor on the category of presheaves we conclude that the presheaf $j_{p!}\mathcal{O}_U$ is a projective object in the category $\textit{PMod}(\mathcal{O}_X)$, see Homology, Lemma 12.25.2.

Note that if we are given open subsets $U \subset V \subset X$ with associated open immersions $j_U, j_V$, then we have a canonical map $(j_U)_{p!}\mathcal{O}_U \to (j_V)_{p!}\mathcal{O}_V$. It is the identity on sections over any open $W \subset U$ and $0$ else. In terms of the identification $\mathop{\mathrm{Hom}}\nolimits_{\mathcal{O}_X}((j_U)_{p!}\mathcal{O}_U, (j_V)_{p!}\mathcal{O}_V) = (j_V)_{p!}\mathcal{O}_V(U) = \mathcal{O}_V(U)$ it corresponds to the element $1 \in \mathcal{O}_V(U)$.

Lemma 20.11.3. Let $X$ be a ringed space. Let $\mathcal{U} : U = \bigcup_{i \in I} U_i$ be a covering. Denote $j_{i_0\ldots i_p} : U_{i_0 \ldots i_p} \to X$ the open immersion. Consider the chain complex $K(\mathcal{U})_\bullet$ of presheaves of $\mathcal{O}_X$-modules $$ \ldots \to \bigoplus_{i_0i_1i_2} (j_{i_0i_1i_2})_{p!}\mathcal{O}_{U_{i_0i_1i_2}} \to \bigoplus_{i_0i_1} (j_{i_0i_1})_{p!}\mathcal{O}_{U_{i_0i_1}} \to \bigoplus_{i_0} (j_{i_0})_{p!}\mathcal{O}_{U_{i_0}} \to 0 \to \ldots $$ where the last nonzero term is placed in degree $0$ and where the map $$ (j_{i_0\ldots i_{p + 1}})_{p!}\mathcal{O}_{U_{i_0\ldots i_{p + 1}}} \longrightarrow (j_{i_0\ldots \hat i_j \ldots i_{p + 1}})_{p!} \mathcal{O}_{U_{i_0\ldots \hat i_j \ldots i_{p + 1}}} $$ is given by $(-1)^j$ times the canonical map. Then there is an isomorphism $$ \mathop{\mathrm{Hom}}\nolimits_{\mathcal{O}_X}(K(\mathcal{U})_\bullet, \mathcal{F}) = \check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{F}) $$ functorial in $\mathcal{F} \in \mathop{\mathrm{Ob}}\nolimits(\textit{PMod}(\mathcal{O}_X))$.

Proof. We saw in the discussion just above the lemma that $$ \mathop{\mathrm{Hom}}\nolimits_{\mathcal{O}_X}( (j_{i_0\ldots i_p})_{p!}\mathcal{O}_{U_{i_0\ldots i_p}}, \mathcal{F}) = \mathcal{F}(U_{i_0\ldots i_p}). $$ Hence we see that it is indeed the case that the direct sum $$ \bigoplus\nolimits_{i_0 \ldots i_p} (j_{i_0 \ldots i_p})_{p!}\mathcal{O}_{U_{i_0 \ldots i_p}} $$ represents the functor $$ \mathcal{F} \longmapsto \prod\nolimits_{i_0\ldots i_p} \mathcal{F}(U_{i_0\ldots i_p}). $$ Hence by Categories, Yoneda Lemma 4.3.5 we see that there is a complex $K(\mathcal{U})_\bullet$ with terms as given. It is a simple matter to see that the maps are as given in the lemma. $\square$

Lemma 20.11.4. Let $X$ be a ringed space. Let $\mathcal{U} : U = \bigcup_{i \in I} U_i$ be a covering. Let $\mathcal{O}_\mathcal{U} \subset \mathcal{O}_X$ be the image presheaf of the map $\bigoplus j_{p!}\mathcal{O}_{U_i} \to \mathcal{O}_X$. The chain complex $K(\mathcal{U})_\bullet$ of presheaves of Lemma 20.11.3 above has homology presheaves $$ H_i(K(\mathcal{U})_\bullet) = \left\{ \begin{matrix} 0 & \text{if} & i \not = 0 \\ \mathcal{O}_\mathcal{U} & \text{if} & i = 0 \end{matrix} \right. $$

Proof. Consider the extended complex $K^{ext}_\bullet$ one gets by putting $\mathcal{O}_\mathcal{U}$ in degree $-1$ with the obvious map $K(\mathcal{U})_0 = \bigoplus_{i_0} (j_{i_0})_{p!}\mathcal{O}_{U_{i_0}} \to \mathcal{O}_\mathcal{U}$. It suffices to show that taking sections of this extended complex over any open $W \subset X$ leads to an acyclic complex. In fact, we claim that for every $W \subset X$ the complex $K^{ext}_\bullet(W)$ is homotopy equivalent to the zero complex. Write $I = I_1 \amalg I_2$ where $W \subset U_i$ if and only if $i \in I_1$.

If $I_1 = \emptyset$, then the complex $K^{ext}_\bullet(W) = 0$ so there is nothing to prove.

If $I_1 \not = \emptyset$, then $\mathcal{O}_\mathcal{U}(W) = \mathcal{O}_X(W)$ and $$ K^{ext}_p(W) = \bigoplus\nolimits_{i_0 \ldots i_p \in I_1} \mathcal{O}_X(W). $$ This is true because of the simple description of the presheaves $(j_{i_0 \ldots i_p})_{p!}\mathcal{O}_{U_{i_0 \ldots i_p}}$. Moreover, the differential of the complex $K^{ext}_\bullet(W)$ is given by $$ d(s)_{i_0 \ldots i_p} = \sum\nolimits_{j = 0, \ldots, p + 1} \sum\nolimits_{i \in I_1} (-1)^j s_{i_0 \ldots i_{j - 1} i i_j \ldots i_p}. $$ The sum is finite as the element $s$ has finite support. Fix an element $i_{\text{fix}} \in I_1$. Define a map $$ h : K^{ext}_p(W) \longrightarrow K^{ext}_{p + 1}(W) $$ by the rule $$ h(s)_{i_0 \ldots i_{p + 1}} = \left\{ \begin{matrix} 0 & \text{if} & i_0 \not = i \\ s_{i_1 \ldots i_{p + 1}} & \text{if} & i_0 = i_{\text{fix}} \end{matrix} \right. $$ We will use the shorthand $h(s)_{i_0 \ldots i_{p + 1}} = (i_0 = i_{\text{fix}}) s_{i_1 \ldots i_p}$ for this. Then we compute \begin{eqnarray*} & & (dh + hd)(s)_{i_0 \ldots i_p} \\ & = & \sum_j \sum_{i \in I_1} (-1)^j h(s)_{i_0 \ldots i_{j - 1} i i_j \ldots i_p} + (i = i_0) d(s)_{i_1 \ldots i_p} \\ & = & s_{i_0 \ldots i_p} + \sum_{j \geq 1}\sum_{i \in I_1} (-1)^j (i_0 = i_{\text{fix}}) s_{i_1 \ldots i_{j - 1} i i_j \ldots i_p} + (i_0 = i_{\text{fix}}) d(s)_{i_1 \ldots i_p} \end{eqnarray*} which is equal to $s_{i_0 \ldots i_p}$ as desired. $\square$

Lemma 20.11.5. Let $X$ be a ringed space. Let $\mathcal{U} : U = \bigcup_{i \in I} U_i$ be an open covering of $U \subset X$. The Čech cohomology functors $\check{H}^p(\mathcal{U}, -)$ are canonically isomorphic as a $\delta$-functor to the right derived functors of the functor $$ \check{H}^0(\mathcal{U}, -) : \textit{PMod}(\mathcal{O}_X) \longrightarrow \text{Mod}_{\mathcal{O}_X(U)}. $$ Moreover, there is a functorial quasi-isomorphism $$ \check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{F}) \longrightarrow R\check{H}^0(\mathcal{U}, \mathcal{F}) $$ where the right hand side indicates the right derived functor $$ R\check{H}^0(\mathcal{U}, -) : D^{+}(\textit{PMod}(\mathcal{O}_X)) \longrightarrow D^{+}(\mathcal{O}_X(U)) $$ of the left exact functor $\check{H}^0(\mathcal{U}, -)$.

Proof. Note that the category of presheaves of $\mathcal{O}_X$-modules has enough injectives, see Injectives, Proposition 19.8.5. Note that $\check{H}^0(\mathcal{U}, -)$ is a left exact functor from the category of presheaves of $\mathcal{O}_X$-modules to the category of $\mathcal{O}_X(U)$-modules. Hence the derived functor and the right derived functor exist, see Derived Categories, Section 13.20.

Let $\mathcal{I}$ be a injective presheaf of $\mathcal{O}_X$-modules. In this case the functor $\mathop{\mathrm{Hom}}\nolimits_{\mathcal{O}_X}(-, \mathcal{I})$ is exact on $\textit{PMod}(\mathcal{O}_X)$. By Lemma 20.11.3 we have $$ \mathop{\mathrm{Hom}}\nolimits_{\mathcal{O}_X}(K(\mathcal{U})_\bullet, \mathcal{I}) = \check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{I}). $$ By Lemma 20.11.4 we have that $K(\mathcal{U})_\bullet$ is quasi-isomorphic to $\mathcal{O}_\mathcal{U}[0]$. 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 20.11.2) satisfies the assumptions of Homology, Lemma 12.11.4, 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 12.11.5 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 presheaf of $\mathcal{O}_X$-modules. Choose an injective resolution $\mathcal{F} \to \mathcal{I}^\bullet$ in the category $\textit{PMod}(\mathcal{O}_X)$. Consider the double complex $A^{\bullet, \bullet}$ with terms $$ A^{p, q} = \check{\mathcal{C}}^p(\mathcal{U}, \mathcal{I}^q). $$ Consider the simple complex $sA^\bullet$ associated to this double complex. There is a map of complexes $$ \check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{F}) \longrightarrow sA^\bullet $$ coming from the maps $\check{\mathcal{C}}^p(\mathcal{U}, \mathcal{F}) \to A^{p, 0} = \check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{I}^0)$ and there is a map of complexes $$ \check{H}^0(\mathcal{U}, \mathcal{I}^\bullet) \longrightarrow sA^\bullet $$ coming from the maps $\check{H}^0(\mathcal{U}, \mathcal{I}^q) \to A^{0, q} = \check{\mathcal{C}}^0(\mathcal{U}, \mathcal{I}^q)$. Both of these maps are quasi-isomorphisms by an application of Homology, Lemma 12.22.7. Namely, the columns of the double complex are exact in positive degrees because the Čech complex as a functor is exact (Lemma 20.11.1) 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 $D^{+}(\mathcal{O}_X(U))$ this gives the last displayed morphism of the lemma. We omit the verification that this morphism is functorial. $\square$

    The code snippet corresponding to this tag is a part of the file cohomology.tex and is located in lines 955–1347 (see updates for more information).

    \section{{\v C}ech cohomology as a functor on presheaves}
    \label{section-cech-functor}
    
    \noindent
    Warning: In this section we work almost exclusively with presheaves and
    categories of presheaves and the results are completely wrong in the
    setting of sheaves and categories of sheaves!
    
    \medskip\noindent
    Let $X$ be a ringed space.
    Let $\mathcal{U} : U = \bigcup_{i \in I} U_i$ be an open covering.
    Let $\mathcal{F}$ be a presheaf of $\mathcal{O}_X$-modules.
    We have the {\v C}ech complex
    $\check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{F})$
    of $\mathcal{F}$ just by thinking of $\mathcal{F}$
    as a presheaf of abelian groups. However, each term
    $\check{\mathcal{C}}^p(\mathcal{U}, \mathcal{F})$ has a natural
    structure of a $\mathcal{O}_X(U)$-module and the differential is given by
    $\mathcal{O}_X(U)$-module maps. Moreover, it is clear that the
    construction
    $$
    \mathcal{F} \longmapsto \check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{F})
    $$
    is functorial in $\mathcal{F}$. In fact, it is a functor
    \begin{equation}
    \label{equation-cech-functor}
    \check{\mathcal{C}}^\bullet(\mathcal{U}, -) :
    \textit{PMod}(\mathcal{O}_X)
    \longrightarrow
    \text{Comp}^{+}(\text{Mod}_{\mathcal{O}_X(U)})
    \end{equation}
    see
    Derived Categories, Definition \ref{derived-definition-complexes-notation}
    for notation. Recall that the category of bounded below complexes
    in an abelian category is an abelian category, see
    Homology, Lemma \ref{homology-lemma-cat-cochain-abelian}.
    
    \begin{lemma}
    \label{lemma-cech-exact-presheaves}
    The functor given by Equation (\ref{equation-cech-functor})
    is an exact functor (see Homology, Lemma \ref{homology-lemma-exact-functor}).
    \end{lemma}
    
    \begin{proof}
    For any open $W \subset U$ the functor
    $\mathcal{F} \mapsto \mathcal{F}(W)$ is an additive exact functor
    from $\textit{PMod}(\mathcal{O}_X)$ to $\text{Mod}_{\mathcal{O}_X(U)}$.
    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.
    \end{proof}
    
    \begin{lemma}
    \label{lemma-cech-cohomology-delta-functor-presheaves}
    Let $X$ be a ringed space.
    Let $\mathcal{U} : U = \bigcup_{i \in I} U_i$ be an open covering.
    The functors $\mathcal{F} \mapsto \check{H}^n(\mathcal{U}, \mathcal{F})$
    form a $\delta$-functor from the abelian category of
    presheaves of $\mathcal{O}_X$-modules to the category
    of $\mathcal{O}_X(U)$-modules (see
    Homology, Definition \ref{homology-definition-cohomological-delta-functor}).
    \end{lemma}
    
    \begin{proof}
    By
    Lemma \ref{lemma-cech-exact-presheaves}
    a short exact sequence of presheaves of
    $\mathcal{O}_X$-modules
    $0 \to \mathcal{F}_1 \to \mathcal{F}_2 \to \mathcal{F}_3 \to 0$
    is turned into a short exact sequence of complexes of
    $\mathcal{O}_X(U)$-modules. Hence we can use
    Homology, Lemma \ref{homology-lemma-long-exact-sequence-cochain}
    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.
    \end{proof}
    
    
    \noindent
    In the formulation of the following lemma we use the functor $j_{p!}$ of
    extension by $0$ for presheaves of modules
    relative to an open immersion $j : U \to X$.
    See Sheaves, Section \ref{sheaves-section-open-immersions}. For any open
    $W \subset X$ and any presheaf $\mathcal{G}$ of $\mathcal{O}_X|_U$-modules
    we have
    $$
    (j_{p!}\mathcal{G})(W) =
    \left\{
    \begin{matrix}
    \mathcal{G}(W) & \text{if } W \subset U \\
    0 & \text{else.}
    \end{matrix}
    \right.
    $$
    Moreover, the functor $j_{p!}$ is a left adjoint to the restriction functor
    see Sheaves, Lemma \ref{sheaves-lemma-j-shriek-modules}.
    In particular we have the following formula
    $$
    \Hom_{\mathcal{O}_X}(j_{p!}\mathcal{O}_U, \mathcal{F})
    =
    \Hom_{\mathcal{O}_U}(\mathcal{O}_U, \mathcal{F}|_U)
    =
    \mathcal{F}(U).
    $$
    Since the functor $\mathcal{F} \mapsto \mathcal{F}(U)$ is an exact functor
    on the category of presheaves we conclude that the presheaf
    $j_{p!}\mathcal{O}_U$ is a projective object in the category
    $\textit{PMod}(\mathcal{O}_X)$, see
    Homology, Lemma \ref{homology-lemma-characterize-projectives}.
    
    \medskip\noindent
    Note that if we are given open subsets $U \subset V \subset X$
    with associated open immersions $j_U, j_V$, then we have a canonical
    map $(j_U)_{p!}\mathcal{O}_U \to (j_V)_{p!}\mathcal{O}_V$. It is the
    identity on sections over any open $W \subset U$ and $0$ else.
    In terms of the identification
    $\Hom_{\mathcal{O}_X}((j_U)_{p!}\mathcal{O}_U, (j_V)_{p!}\mathcal{O}_V) =
    (j_V)_{p!}\mathcal{O}_V(U) = \mathcal{O}_V(U)$ it corresponds to
    the element $1 \in \mathcal{O}_V(U)$.
    
    \begin{lemma}
    \label{lemma-cech-map-into}
    Let $X$ be a ringed space.
    Let $\mathcal{U} : U = \bigcup_{i \in I} U_i$ be a covering.
    Denote $j_{i_0\ldots i_p} : U_{i_0 \ldots i_p} \to X$ the open immersion.
    Consider the chain complex $K(\mathcal{U})_\bullet$
    of presheaves of $\mathcal{O}_X$-modules
    $$
    \ldots
    \to
    \bigoplus_{i_0i_1i_2} (j_{i_0i_1i_2})_{p!}\mathcal{O}_{U_{i_0i_1i_2}}
    \to
    \bigoplus_{i_0i_1} (j_{i_0i_1})_{p!}\mathcal{O}_{U_{i_0i_1}}
    \to
    \bigoplus_{i_0} (j_{i_0})_{p!}\mathcal{O}_{U_{i_0}}
    \to 0 \to \ldots
    $$
    where the last nonzero term is placed in degree $0$
    and where the map
    $$
    (j_{i_0\ldots i_{p + 1}})_{p!}\mathcal{O}_{U_{i_0\ldots i_{p + 1}}}
    \longrightarrow
    (j_{i_0\ldots \hat i_j \ldots i_{p + 1}})_{p!}
    \mathcal{O}_{U_{i_0\ldots \hat i_j \ldots i_{p + 1}}}
    $$
    is given by $(-1)^j$ times the canonical map.
    Then there is an isomorphism
    $$
    \Hom_{\mathcal{O}_X}(K(\mathcal{U})_\bullet, \mathcal{F})
    =
    \check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{F})
    $$
    functorial in $\mathcal{F} \in \Ob(\textit{PMod}(\mathcal{O}_X))$.
    \end{lemma}
    
    \begin{proof}
    We saw in the discussion just above the lemma that
    $$
    \Hom_{\mathcal{O}_X}(
    (j_{i_0\ldots i_p})_{p!}\mathcal{O}_{U_{i_0\ldots i_p}},
    \mathcal{F})
    =
    \mathcal{F}(U_{i_0\ldots i_p}).
    $$
    Hence we see that it is indeed the case that the direct sum
    $$
    \bigoplus\nolimits_{i_0 \ldots i_p}
    (j_{i_0 \ldots i_p})_{p!}\mathcal{O}_{U_{i_0 \ldots i_p}}
    $$
    represents the functor
    $$
    \mathcal{F}
    \longmapsto
    \prod\nolimits_{i_0\ldots i_p} \mathcal{F}(U_{i_0\ldots i_p}).
    $$
    Hence by Categories, Yoneda Lemma \ref{categories-lemma-yoneda}
    we see that there is a complex $K(\mathcal{U})_\bullet$ with terms
    as given. It is a simple matter to see that the maps are as given
    in the lemma.
    \end{proof}
    
    \begin{lemma}
    \label{lemma-homology-complex}
    Let $X$ be a ringed space.
    Let $\mathcal{U} : U = \bigcup_{i \in I} U_i$ be a covering.
    Let $\mathcal{O}_\mathcal{U} \subset \mathcal{O}_X$
    be the image presheaf of the map
    $\bigoplus j_{p!}\mathcal{O}_{U_i} \to \mathcal{O}_X$.
    The chain complex $K(\mathcal{U})_\bullet$ of presheaves
    of Lemma \ref{lemma-cech-map-into} above has homology presheaves
    $$
    H_i(K(\mathcal{U})_\bullet) =
    \left\{
    \begin{matrix}
    0 & \text{if} & i \not = 0 \\
    \mathcal{O}_\mathcal{U} & \text{if} & i = 0
    \end{matrix}
    \right.
    $$
    \end{lemma}
    
    \begin{proof}
    Consider the extended complex $K^{ext}_\bullet$ one gets by putting
    $\mathcal{O}_\mathcal{U}$ in degree $-1$ with the obvious map
    $K(\mathcal{U})_0 =
    \bigoplus_{i_0} (j_{i_0})_{p!}\mathcal{O}_{U_{i_0}} \to
    \mathcal{O}_\mathcal{U}$.
    It suffices to show that taking sections of this extended complex over
    any open $W \subset X$ leads to an acyclic complex.
    In fact, we claim that for every $W \subset X$ the complex
    $K^{ext}_\bullet(W)$ is homotopy equivalent to the zero complex.
    Write $I = I_1 \amalg I_2$ where $W \subset U_i$ if and only
    if $i \in I_1$.
    
    \medskip\noindent
    If $I_1 = \emptyset$, then the complex $K^{ext}_\bullet(W) = 0$ so there is
    nothing to prove.
    
    \medskip\noindent
    If $I_1 \not = \emptyset$, then
    $\mathcal{O}_\mathcal{U}(W) = \mathcal{O}_X(W)$
    and
    $$
    K^{ext}_p(W) =
    \bigoplus\nolimits_{i_0 \ldots i_p \in I_1} \mathcal{O}_X(W).
    $$
    This is true because of the simple description of the presheaves
    $(j_{i_0 \ldots i_p})_{p!}\mathcal{O}_{U_{i_0 \ldots i_p}}$.
    Moreover, the differential of the complex $K^{ext}_\bullet(W)$
    is given by
    $$
    d(s)_{i_0 \ldots i_p} =
    \sum\nolimits_{j = 0, \ldots, p + 1} \sum\nolimits_{i \in I_1}
    (-1)^j s_{i_0 \ldots i_{j - 1} i i_j \ldots i_p}.
    $$
    The sum is finite as the element $s$ has finite support.
    Fix an element $i_{\text{fix}} \in I_1$. Define a map
    $$
    h : K^{ext}_p(W) \longrightarrow K^{ext}_{p + 1}(W)
    $$
    by the rule
    $$
    h(s)_{i_0 \ldots i_{p + 1}} =
    \left\{
    \begin{matrix}
    0 & \text{if} & i_0 \not = i \\
    s_{i_1 \ldots i_{p + 1}} & \text{if} & i_0 = i_{\text{fix}}
    \end{matrix}
    \right.
    $$
    We will use the shorthand
    $h(s)_{i_0 \ldots i_{p + 1}} = (i_0 = i_{\text{fix}}) s_{i_1 \ldots i_p}$
    for this. Then we compute
    \begin{eqnarray*}
    & & (dh + hd)(s)_{i_0 \ldots i_p} \\
    & = &
    \sum_j \sum_{i \in I_1} (-1)^j h(s)_{i_0 \ldots i_{j - 1} i i_j \ldots i_p}
    +
    (i = i_0) d(s)_{i_1 \ldots i_p} \\
    & = &
    s_{i_0 \ldots i_p} +
    \sum_{j \geq 1}\sum_{i \in I_1}
    (-1)^j (i_0 = i_{\text{fix}}) s_{i_1 \ldots i_{j - 1} i i_j \ldots i_p}
    +
    (i_0 = i_{\text{fix}}) d(s)_{i_1 \ldots i_p}
    \end{eqnarray*}
    which is equal to $s_{i_0 \ldots i_p}$ as desired.
    \end{proof}
    
    \begin{lemma}
    \label{lemma-cech-cohomology-derived-presheaves}
    Let $X$ be a ringed space.
    Let $\mathcal{U} : U = \bigcup_{i \in I} U_i$
    be an open covering of $U \subset X$.
    The {\v C}ech cohomology functors $\check{H}^p(\mathcal{U}, -)$
    are canonically isomorphic as a $\delta$-functor to
    the right derived functors of the functor
    $$
    \check{H}^0(\mathcal{U}, -) :
    \textit{PMod}(\mathcal{O}_X)
    \longrightarrow
    \text{Mod}_{\mathcal{O}_X(U)}.
    $$
    Moreover, there is a functorial quasi-isomorphism
    $$
    \check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{F})
    \longrightarrow
    R\check{H}^0(\mathcal{U}, \mathcal{F})
    $$
    where the right hand side indicates the right derived functor
    $$
    R\check{H}^0(\mathcal{U}, -) :
    D^{+}(\textit{PMod}(\mathcal{O}_X))
    \longrightarrow
    D^{+}(\mathcal{O}_X(U))
    $$
    of the left exact functor $\check{H}^0(\mathcal{U}, -)$.
    \end{lemma}
    
    \begin{proof}
    Note that the category of presheaves of $\mathcal{O}_X$-modules
    has enough injectives, see
    Injectives, Proposition \ref{injectives-proposition-presheaves-modules}.
    Note that $\check{H}^0(\mathcal{U}, -)$ is a left exact functor
    from the category of presheaves of $\mathcal{O}_X$-modules
    to the category of $\mathcal{O}_X(U)$-modules.
    Hence the derived functor and the right derived functor exist, see
    Derived Categories, Section \ref{derived-section-right-derived-functor}.
    
    \medskip\noindent
    Let $\mathcal{I}$ be a injective presheaf of $\mathcal{O}_X$-modules.
    In this case the functor $\Hom_{\mathcal{O}_X}(-, \mathcal{I})$
    is exact on $\textit{PMod}(\mathcal{O}_X)$. By
    Lemma \ref{lemma-cech-map-into} we have
    $$
    \Hom_{\mathcal{O}_X}(K(\mathcal{U})_\bullet, \mathcal{I})
    =
    \check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{I}).
    $$
    By Lemma \ref{lemma-homology-complex} we have that $K(\mathcal{U})_\bullet$ is
    quasi-isomorphic to $\mathcal{O}_\mathcal{U}[0]$. 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 \ref{lemma-cech-cohomology-delta-functor-presheaves})
    satisfies the assumptions of
    Homology, Lemma \ref{homology-lemma-efface-implies-universal},
    and hence is a universal $\delta$-functor.
    
    \medskip\noindent
    By
    Derived Categories, Lemma \ref{derived-lemma-higher-derived-functors}
    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 \ref{homology-lemma-uniqueness-universal-delta-functor}
    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.
    
    \medskip\noindent
    Let $\mathcal{F}$ be any presheaf of $\mathcal{O}_X$-modules.
    Choose an injective resolution $\mathcal{F} \to \mathcal{I}^\bullet$
    in the category $\textit{PMod}(\mathcal{O}_X)$.
    Consider the double complex $A^{\bullet, \bullet}$ with terms
    $$
    A^{p, q} =
    \check{\mathcal{C}}^p(\mathcal{U}, \mathcal{I}^q).
    $$
    Consider the simple complex $sA^\bullet$ associated to this double
    complex. There is a map of complexes
    $$
    \check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{F})
    \longrightarrow
    sA^\bullet
    $$
    coming from the maps
    $\check{\mathcal{C}}^p(\mathcal{U}, \mathcal{F})
    \to A^{p, 0} = \check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{I}^0)$
    and there is a map of complexes
    $$
    \check{H}^0(\mathcal{U}, \mathcal{I}^\bullet)
    \longrightarrow
    sA^\bullet
    $$
    coming from the maps
    $\check{H}^0(\mathcal{U}, \mathcal{I}^q) \to
    A^{0, q} = \check{\mathcal{C}}^0(\mathcal{U}, \mathcal{I}^q)$.
    Both of these maps are quasi-isomorphisms by an application of
    Homology, Lemma \ref{homology-lemma-double-complex-gives-resolution}.
    Namely, the columns of the double complex are exact in positive degrees
    because the {\v C}ech complex as a functor is exact
    (Lemma \ref{lemma-cech-exact-presheaves})
    and the rows of the double complex are exact in positive degrees
    since as we just saw the higher {\v C}ech cohomology groups of the injective
    presheaves $\mathcal{I}^q$ are zero.
    Since quasi-isomorphisms become invertible
    in $D^{+}(\mathcal{O}_X(U))$ this gives the last displayed morphism
    of the lemma. We omit the verification that this morphism is
    functorial.
    \end{proof}

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