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 \input{preamble} % OK, start here. % \begin{document} \title{Cohomology of Sheaves} \maketitle \phantomsection \label{section-phantom} \tableofcontents \section{Introduction} \label{section-introduction} \noindent In this document we work out some topics on cohomology of sheaves on topological spaces. We mostly work in the generality of modules over a sheaf of rings and we work with morphisms of ringed spaces. To see what happens for sheaves on sites take a look at the chapter Cohomology on Sites, Section \ref{sites-cohomology-section-introduction}. Basic references are \cite{Godement} and \cite{Iversen}. \section{Topics} \label{section-topics} \noindent Here are some topics that should be discussed in this chapter, and have not yet been written. \begin{enumerate} \item Ext-groups. \item Ext sheaves. \item Tor functors. \item Derived pullback for morphisms between ringed spaces. \item Cup-product. \item Etc, etc, etc. \end{enumerate} \section{Cohomology of sheaves} \label{section-cohomology-sheaves} \noindent Let $X$ be a topological space. Let $\mathcal{F}$ be a abelian sheaf. We know that the category of abelian sheaves on $X$ has enough injectives, see Injectives, Lemma \ref{injectives-lemma-abelian-sheaves-space}. Hence we can choose an injective resolution $\mathcal{F}[0] \to \mathcal{I}^\bullet$. As is customary we define \begin{equation} \label{equation-cohomology} H^i(X, \mathcal{F}) = H^i(\Gamma(X, \mathcal{I}^\bullet)) \end{equation} to be the {\it $i$th cohomology group of the abelian sheaf $\mathcal{F}$}. The family of functors $H^i((X, -)$ forms a universal $\delta$-functor from $\textit{Ab}(X) \to \textit{Ab}$. \medskip\noindent Let $f : X \to Y$ be a continuous map of topological spaces. With $\mathcal{F}[0] \to \mathcal{I}^\bullet$ as above we define \begin{equation} \label{equation-higher-direct-image} R^if_*\mathcal{F} = H^i(f_*\mathcal{I}^\bullet) \end{equation} to be the {\it $i$th higher direct image of $\mathcal{F}$}. The family of functors $R^if_*$ forms a universal $\delta$-functor from $\textit{Ab}(X) \to \textit{Ab}(Y)$. \medskip\noindent Let $(X, \mathcal{O}_X)$ be a ringed space. Let $\mathcal{F}$ be an $\mathcal{O}_X$-module. We know that the category of $\mathcal{O}_X$-modules on $X$ has enough injectives, see Injectives, Lemma \ref{injectives-lemma-sheaves-modules-space}. Hence we can choose an injective resolution $\mathcal{F}[0] \to \mathcal{I}^\bullet$. As is customary we define \begin{equation} \label{equation-cohomology-modules} H^i(X, \mathcal{F}) = H^i(\Gamma(X, \mathcal{I}^\bullet)) \end{equation} to be the {\it $i$th cohomology group of $\mathcal{F}$}. The family of functors $H^i((X, -)$ forms a universal $\delta$-functor from $\textit{Mod}(\mathcal{O}_X) \to \text{Mod}_{\mathcal{O}_X(X)}$. \medskip\noindent Let $f : (X, \mathcal{O}_X) \to (Y, \mathcal{O}_Y)$ be a morphism of ringed spaces. With $\mathcal{F}[0] \to \mathcal{I}^\bullet$ as above we define \begin{equation} \label{equation-higher-direct-image-modules} R^if_*\mathcal{F} = H^i(f_*\mathcal{I}^\bullet) \end{equation} to be the {\it $i$th higher direct image of $\mathcal{F}$}. The family of functors $R^if_*$ forms a universal $\delta$-functor from $\textit{Mod}(\mathcal{O}_X) \to \textit{Mod}(\mathcal{O}_Y)$. \section{Derived functors} \label{section-derived-functors} \noindent We briefly explain an approach to right derived functors using resolution functors. Let $(X, \mathcal{O}_X)$ be a ringed space. The category $\textit{Mod}(\mathcal{O}_X)$ is abelian, see Modules, Lemma \ref{modules-lemma-abelian}. In this chapter we will write $$K(X) = K(\mathcal{O}_X) = K(\textit{Mod}(\mathcal{O}_X)) \quad \text{and} \quad D(X) = D(\mathcal{O}_X) = D(\textit{Mod}(\mathcal{O}_X)).$$ and similarly for the bounded versions for the triangulated categories introduced in Derived Categories, Definition \ref{derived-definition-complexes-notation} and Definition \ref{derived-definition-unbounded-derived-category}. By Derived Categories, Remark \ref{derived-remark-big-abelian-category} there exists a resolution functor $$j = j_X : K^{+}(\textit{Mod}(\mathcal{O}_X)) \longrightarrow K^{+}(\mathcal{I})$$ where $\mathcal{I}$ is the strictly full additive subcategory of $\textit{Mod}(\mathcal{O}_X)$ consisting of injective sheaves. For any left exact functor $F : \textit{Mod}(\mathcal{O}_X) \to \mathcal{B}$ into any abelian category $\mathcal{B}$ we will denote $RF$ the right derived functor described in Derived Categories, Section \ref{derived-section-right-derived-functor} and constructed using the resolution functor $j_X$ just described: \begin{equation} \label{equation-RF} RF = F \circ j_X' : D^{+}(X) \longrightarrow D^{+}(\mathcal{B}) \end{equation} see Derived Categories, Lemma \ref{derived-lemma-right-derived-functor} for notation. Note that we may think of $RF$ as defined on $\textit{Mod}(\mathcal{O}_X)$, $\text{Comp}^{+}(\textit{Mod}(\mathcal{O}_X))$, $K^{+}(X)$, or $D^{+}(X)$ depending on the situation. According to Derived Categories, Definition \ref{derived-definition-higher-derived-functors} we obtain the $i$th right derived functor \begin{equation} \label{equation-RFi} R^iF = H^i \circ RF : \textit{Mod}(\mathcal{O}_X) \longrightarrow \mathcal{B} \end{equation} so that $R^0F = F$ and $\{R^iF, \delta\}_{i \geq 0}$ is universal $\delta$-functor, see Derived Categories, Lemma \ref{derived-lemma-higher-derived-functors}. \medskip\noindent Here are two special cases of this construction. Given a ring $R$ we write $K(R) = K(\text{Mod}_R)$ and $D(R) = D(\text{Mod}_R)$ and similarly for bounded versions. For any open $U \subset X$ we have a left exact functor $\Gamma(U, -) : \textit{Mod}(\mathcal{O}_X) \longrightarrow \text{Mod}_{\mathcal{O}_X(U)}$ which gives rise to \begin{equation} \label{equation-total-derived-cohomology} R\Gamma(U, -) : D^{+}(X) \longrightarrow D^{+}(\mathcal{O}_X(U)) \end{equation} by the discussion above. We set $H^i(U, -) = R^i\Gamma(U, -)$. If $U = X$ we recover (\ref{equation-cohomology-modules}). If $f : X \to Y$ is a morphism of ringed spaces, then we have the left exact functor $f_* : \textit{Mod}(\mathcal{O}_X) \longrightarrow \textit{Mod}(\mathcal{O}_Y)$ which gives rise to the {\it derived pushforward} \begin{equation} \label{equation-total-derived-direct-image} Rf_* : D^{+}(X) \longrightarrow D^{+}(Y) \end{equation} The $i$th cohomology sheaf of $Rf_*\mathcal{F}^\bullet$ is denoted $R^if_*\mathcal{F}^\bullet$ and called the $i$th {\it higher direct image} in accordance with (\ref{equation-higher-direct-image-modules}). The two displayed functors above are exact functors of derived categories. \medskip\noindent {\bf Abuse of notation:} When the functor $Rf_*$, or any other derived functor, is applied to a sheaf $\mathcal{F}$ on $X$ or a complex of sheaves it is understood that $\mathcal{F}$ has been replaced by a suitable resolution of $\mathcal{F}$. To facilitate this kind of operation we will say, given an object $\mathcal{F}^\bullet \in D(X)$, that a bounded below complex $\mathcal{I}^\bullet$ of injectives of $\textit{Mod}(\mathcal{O}_X)$ {\it represents $\mathcal{F}^\bullet$ in the derived category} if there exists a quasi-isomorphism $\mathcal{F}^\bullet \to \mathcal{I}^\bullet$. In the same vein the phrase let $\alpha : \mathcal{F}^\bullet \to \mathcal{G}^\bullet$ be a morphism of $D(X)$'' does not mean that $\alpha$ is represented by a morphism of complexes. If we have an actual morphism of complexes we will say so. \section{First cohomology and torsors} \label{section-h1-torsors} \begin{definition} \label{definition-torsor} Let $X$ be a topological space. Let $\mathcal{G}$ be a sheaf of (possibly non-commutative) groups on $X$. A {\it torsor}, or more precisely a {\it $\mathcal{G}$-torsor}, is a sheaf of sets $\mathcal{F}$ on $X$ endowed with an action $\mathcal{G} \times \mathcal{F} \to \mathcal{F}$ such that \begin{enumerate} \item whenever $\mathcal{F}(U)$ is nonempty the action $\mathcal{G}(U) \times \mathcal{F}(U) \to \mathcal{F}(U)$ is simply transitive, and \item for every $x \in X$ the stalk $\mathcal{F}_x$ is nonempty. \end{enumerate} A {\it morphism of $\mathcal{G}$-torsors} $\mathcal{F} \to \mathcal{F}'$ is simply a morphism of sheaves of sets compatible with the $\mathcal{G}$-actions. The {\it trivial $\mathcal{G}$-torsor} is the sheaf $\mathcal{G}$ endowed with the obvious left $\mathcal{G}$-action. \end{definition} \noindent It is clear that a morphism of torsors is automatically an isomorphism. \begin{lemma} \label{lemma-trivial-torsor} Let $X$ be a topological space. Let $\mathcal{G}$ be a sheaf of (possibly non-commutative) groups on $X$. A $\mathcal{G}$-torsor $\mathcal{F}$ is trivial if and only if $\mathcal{F}(X) \not = \emptyset$. \end{lemma} \begin{proof} Omitted. \end{proof} \begin{lemma} \label{lemma-torsors-h1} Let $X$ be a topological space. Let $\mathcal{H}$ be an abelian sheaf on $X$. There is a canonical bijection between the set of isomorphism classes of $\mathcal{H}$-torsors and $H^1(X, \mathcal{H})$. \end{lemma} \begin{proof} Let $\mathcal{F}$ be a $\mathcal{H}$-torsor. Consider the free abelian sheaf $\mathbf{Z}[\mathcal{F}]$ on $\mathcal{F}$. It is the sheafification of the rule which associates to $U \subset X$ open the collection of finite formal sums $\sum n_i[s_i]$ with $n_i \in \mathbf{Z}$ and $s_i \in \mathcal{F}(U)$. There is a natural map $$\sigma : \mathbf{Z}[\mathcal{F}] \longrightarrow \underline{\mathbf{Z}}$$ which to a local section $\sum n_i[s_i]$ associates $\sum n_i$. The kernel of $\sigma$ is generated by the local section of the form $[s] - [s']$. There is a canonical map $a : \Ker(\sigma) \to \mathcal{H}$ which maps $[s] - [s'] \mapsto h$ where $h$ is the local section of $\mathcal{H}$ such that $h \cdot s = s'$. Consider the pushout diagram $$\xymatrix{ 0 \ar[r] & \Ker(\sigma) \ar[r] \ar[d]^a & \mathbf{Z}[\mathcal{F}] \ar[r] \ar[d] & \underline{\mathbf{Z}} \ar[r] \ar[d] & 0 \\ 0 \ar[r] & \mathcal{H} \ar[r] & \mathcal{E} \ar[r] & \underline{\mathbf{Z}} \ar[r] & 0 }$$ Here $\mathcal{E}$ is the extension obtained by pushout. From the long exact cohomology sequence associated to the lower short exact sequence we obtain an element $\xi = \xi_\mathcal{F} \in H^1(X, \mathcal{H})$ by applying the boundary operator to $1 \in H^0(X, \underline{\mathbf{Z}})$. \medskip\noindent Conversely, given $\xi \in H^1(X, \mathcal{H})$ we can associate to $\xi$ a torsor as follows. Choose an embedding $\mathcal{H} \to \mathcal{I}$ of $\mathcal{H}$ into an injective abelian sheaf $\mathcal{I}$. We set $\mathcal{Q} = \mathcal{I}/\mathcal{H}$ so that we have a short exact sequence $$\xymatrix{ 0 \ar[r] & \mathcal{H} \ar[r] & \mathcal{I} \ar[r] & \mathcal{Q} \ar[r] & 0 }$$ The element $\xi$ is the image of a global section $q \in H^0(X, \mathcal{Q})$ because $H^1(X, \mathcal{I}) = 0$ (see Derived Categories, Lemma \ref{derived-lemma-higher-derived-functors}). Let $\mathcal{F} \subset \mathcal{I}$ be the subsheaf (of sets) of sections that map to $q$ in the sheaf $\mathcal{Q}$. It is easy to verify that $\mathcal{F}$ is a torsor. \medskip\noindent We omit the verification that the two constructions given above are mutually inverse. \end{proof} \section{First cohomology and extensions} \label{section-h1-extensions} \begin{lemma} \label{lemma-h1-extensions} Let $(X, \mathcal{O}_X)$ be a ringed space. Let $\mathcal{F}$ be a sheaf of $\mathcal{O}_X$-modules. There is a canonical bijection $$\text{Ext}^1_{\textit{Mod}(\mathcal{O}_X)}(\mathcal{O}_X, \mathcal{F}) \longrightarrow H^1(X, \mathcal{F})$$ which associates to the extension $$0 \to \mathcal{F} \to \mathcal{E} \to \mathcal{O}_X \to 0$$ the image of $1 \in \Gamma(X, \mathcal{O}_X)$ in $H^1(X, \mathcal{F})$. \end{lemma} \begin{proof} Let us construct the inverse of the map given in the lemma. Let $\xi \in H^1(X, \mathcal{F})$. Choose an injection $\mathcal{F} \subset \mathcal{I}$ with $\mathcal{I}$ injective in $\textit{Mod}(\mathcal{O}_X)$. Set $\mathcal{Q} = \mathcal{I}/\mathcal{F}$. By the long exact sequence of cohomology, we see that $\xi$ is the image of of a section $\tilde \xi \in \Gamma(X, \mathcal{Q}) = \Hom_{\mathcal{O}_X}(\mathcal{O}_X, \mathcal{Q})$. Now, we just form the pullback $$\xymatrix{ 0 \ar[r] & \mathcal{F} \ar[r] \ar@{=}[d] & \mathcal{E} \ar[r] \ar[d] & \mathcal{O}_X \ar[r] \ar[d]^{\tilde \xi} & 0 \\ 0 \ar[r] & \mathcal{F} \ar[r] & \mathcal{I} \ar[r] & \mathcal{Q} \ar[r] & 0 }$$ see Homology, Section \ref{homology-section-extensions}. \end{proof} \section{First cohomology and invertible sheaves} \label{section-invertible-sheaves} \noindent The Picard group of a ringed space is defined in Modules, Section \ref{modules-section-invertible}. \begin{lemma} \label{lemma-h1-invertible} Let $(X, \mathcal{O}_X)$ be a locally ringed space. There is a canonical isomorphism $$H^1(X, \mathcal{O}_X^*) = \text{Pic}(X).$$ of abelian groups. \end{lemma} \begin{proof} Let $\mathcal{L}$ be an invertible $\mathcal{O}_X$-module. Consider the presheaf $\mathcal{L}^*$ defined by the rule $$U \longmapsto \{s \in \mathcal{L}(U) \text{ such that } \mathcal{O}_U \xrightarrow{s \cdot -} \mathcal{L}_U \text{ is an isomorphism}\}$$ This presheaf satisfies the sheaf condition. Moreover, if $f \in \mathcal{O}_X^*(U)$ and $s \in \mathcal{L}^*(U)$, then clearly $fs \in \mathcal{L}^*(U)$. By the same token, if $s, s' \in \mathcal{L}^*(U)$ then there exists a unique $f \in \mathcal{O}_X^*(U)$ such that $fs = s'$. Moreover, the sheaf $\mathcal{L}^*$ has sections locally by Modules, Lemma \ref{modules-lemma-invertible-is-locally-free-rank-1}. In other words we see that $\mathcal{L}^*$ is a $\mathcal{O}_X^*$-torsor. Thus we get a map $$\begin{matrix} \text{invertible sheaves on }(X, \mathcal{O}_X) \\ \text{ up to isomorphism} \end{matrix} \longrightarrow \begin{matrix} \mathcal{O}_X^*\text{-torsors} \\ \text{ up to isomorphism} \end{matrix}$$ We omit the verification that this is a homomorphism of abelian groups. By Lemma \ref{lemma-torsors-h1} the right hand side is canonically bijective to $H^1(X, \mathcal{O}_X^*)$. Thus we have to show this map is injective and surjective. \medskip\noindent Injective. If the torsor $\mathcal{L}^*$ is trivial, this means by Lemma \ref{lemma-trivial-torsor} that $\mathcal{L}^*$ has a global section. Hence this means exactly that $\mathcal{L} \cong \mathcal{O}_X$ is the neutral element in $\text{Pic}(X)$. \medskip\noindent Surjective. Let $\mathcal{F}$ be an $\mathcal{O}_X^*$-torsor. Consider the presheaf of sets $$\mathcal{L}_1 : U \longmapsto (\mathcal{F}(U) \times \mathcal{O}_X(U))/\mathcal{O}_X^*(U)$$ where the action of $f \in \mathcal{O}_X^*(U)$ on $(s, g)$ is $(fs, f^{-1}g)$. Then $\mathcal{L}_1$ is a presheaf of $\mathcal{O}_X$-modules by setting $(s, g) + (s', g') = (s, g + (s'/s)g')$ where $s'/s$ is the local section $f$ of $\mathcal{O}_X^*$ such that $fs = s'$, and $h(s, g) = (s, hg)$ for $h$ a local section of $\mathcal{O}_X$. We omit the verification that the sheafification $\mathcal{L} = \mathcal{L}_1^\#$ is an invertible $\mathcal{O}_X$-module whose associated $\mathcal{O}_X^*$-torsor $\mathcal{L}^*$ is isomorphic to $\mathcal{F}$. \end{proof} \section{Locality of cohomology} \label{section-locality} \noindent The following lemma says there is no ambiguity in defining the cohomology of a sheaf $\mathcal{F}$ over an open. \begin{lemma} \label{lemma-cohomology-of-open} Let $X$ be a ringed space. Let $U \subset X$ be an open subspace. \begin{enumerate} \item If $\mathcal{I}$ is an injective $\mathcal{O}_X$-module then $\mathcal{I}|_U$ is an injective $\mathcal{O}_U$-module. \item For any sheaf of $\mathcal{O}_X$-modules $\mathcal{F}$ we have $H^p(U, \mathcal{F}) = H^p(U, \mathcal{F}|_U)$. \end{enumerate} \end{lemma} \begin{proof} Denote $j : U \to X$ the open immersion. Recall that the functor $j^{-1}$ of restriction to $U$ is a right adjoint to the functor $j_!$ of extension by $0$, see Sheaves, Lemma \ref{sheaves-lemma-j-shriek-modules}. Moreover, $j_!$ is exact. Hence (1) follows from Homology, Lemma \ref{homology-lemma-adjoint-preserve-injectives}. \medskip\noindent By definition $H^p(U, \mathcal{F}) = H^p(\Gamma(U, \mathcal{I}^\bullet))$ where $\mathcal{F} \to \mathcal{I}^\bullet$ is an injective resolution in $\textit{Mod}(\mathcal{O}_X)$. By the above we see that $\mathcal{F}|_U \to \mathcal{I}^\bullet|_U$ is an injective resolution in $\textit{Mod}(\mathcal{O}_U)$. Hence $H^p(U, \mathcal{F}|_U)$ is equal to $H^p(\Gamma(U, \mathcal{I}^\bullet|_U))$. Of course $\Gamma(U, \mathcal{F}) = \Gamma(U, \mathcal{F}|_U)$ for any sheaf $\mathcal{F}$ on $X$. Hence the equality in (2). \end{proof} \noindent Let $X$ be a ringed space. Let $\mathcal{F}$ be a sheaf of $\mathcal{O}_X$-modules. Let $U \subset V \subset X$ be open subsets. Then there is a canonical {\it restriction mapping} \begin{equation} \label{equation-restriction-mapping} H^n(V, \mathcal{F}) \longrightarrow H^n(U, \mathcal{F}), \quad \xi \longmapsto \xi|_U \end{equation} functorial in $\mathcal{F}$. Namely, choose any injective resolution $\mathcal{F} \to \mathcal{I}^\bullet$. The restriction mappings of the sheaves $\mathcal{I}^p$ give a morphism of complexes $$\Gamma(V, \mathcal{I}^\bullet) \longrightarrow \Gamma(U, \mathcal{I}^\bullet)$$ The LHS is a complex representing $R\Gamma(V, \mathcal{F})$ and the RHS is a complex representing $R\Gamma(U, \mathcal{F})$. We get the map on cohomology groups by applying the functor $H^n$. As indicated we will use the notation $\xi \mapsto \xi|_U$ to denote this map. Thus the rule $U \mapsto H^n(U, \mathcal{F})$ is a presheaf of $\mathcal{O}_X$-modules. This presheaf is customarily denoted $\underline{H}^n(\mathcal{F})$. We will give another interpretation of this presheaf in Lemma \ref{lemma-include}. \begin{lemma} \label{lemma-kill-cohomology-class-on-covering} Let $X$ be a ringed space. Let $\mathcal{F}$ be a sheaf of $\mathcal{O}_X$-modules. Let $U \subset X$ be an open subspace. Let $n > 0$ and let $\xi \in H^n(U, \mathcal{F})$. Then there exists an open covering $U = \bigcup_{i\in I} U_i$ such that $\xi|_{U_i} = 0$ for all $i \in I$. \end{lemma} \begin{proof} Let $\mathcal{F} \to \mathcal{I}^\bullet$ be an injective resolution. Then $$H^n(U, \mathcal{F}) = \frac{\Ker(\mathcal{I}^n(U) \to \mathcal{I}^{n + 1}(U))} {\Im(\mathcal{I}^{n - 1}(U) \to \mathcal{I}^n(U))}.$$ Pick an element $\tilde \xi \in \mathcal{I}^n(U)$ representing the cohomology class in the presentation above. Since $\mathcal{I}^\bullet$ is an injective resolution of $\mathcal{F}$ and $n > 0$ we see that the complex $\mathcal{I}^\bullet$ is exact in degree $n$. Hence $\Im(\mathcal{I}^{n - 1} \to \mathcal{I}^n) = \Ker(\mathcal{I}^n \to \mathcal{I}^{n + 1})$ as sheaves. Since $\tilde \xi$ is a section of the kernel sheaf over $U$ we conclude there exists an open covering $U = \bigcup_{i \in I} U_i$ such that $\tilde \xi|_{U_i}$ is the image under $d$ of a section $\xi_i \in \mathcal{I}^{n - 1}(U_i)$. By our definition of the restriction $\xi|_{U_i}$ as corresponding to the class of $\tilde \xi|_{U_i}$ we conclude. \end{proof} \begin{lemma} \label{lemma-describe-higher-direct-images} Let $f : X \to Y$ be a morphism of ringed spaces. Let $\mathcal{F}$ be a $\mathcal{O}_X$-module. The sheaves $R^if_*\mathcal{F}$ are the sheaves associated to the presheaves $$V \longmapsto H^i(f^{-1}(V), \mathcal{F})$$ with restriction mappings as in Equation (\ref{equation-restriction-mapping}). There is a similar statement for $R^if_*$ applied to a bounded below complex $\mathcal{F}^\bullet$. \end{lemma} \begin{proof} Let $\mathcal{F} \to \mathcal{I}^\bullet$ be an injective resolution. Then $R^if_*\mathcal{F}$ is by definition the $i$th cohomology sheaf of the complex $$f_*\mathcal{I}^0 \to f_*\mathcal{I}^1 \to f_*\mathcal{I}^2 \to \ldots$$ By definition of the abelian category structure on $\mathcal{O}_Y$-modules this cohomology sheaf is the sheaf associated to the presheaf $$V \longmapsto \frac{\Ker(f_*\mathcal{I}^i(V) \to f_*\mathcal{I}^{i + 1}(V))} {\Im(f_*\mathcal{I}^{i - 1}(V) \to f_*\mathcal{I}^i(V))}$$ and this is obviously equal to $$\frac{\Ker(\mathcal{I}^i(f^{-1}(V)) \to \mathcal{I}^{i + 1}(f^{-1}(V)))} {\Im(\mathcal{I}^{i - 1}(f^{-1}(V)) \to \mathcal{I}^i(f^{-1}(V)))}$$ which is equal to $H^i(f^{-1}(V), \mathcal{F})$ and we win. \end{proof} \begin{lemma} \label{lemma-localize-higher-direct-images} Let $f : X \to Y$ be a morphism of ringed spaces. Let $\mathcal{F}$ be an $\mathcal{O}_X$-module. Let $V \subset Y$ be an open subspace. Denote $g : f^{-1}(V) \to V$ the restriction of $f$. Then we have $$R^pg_*(\mathcal{F}|_{f^{-1}(V)}) = (R^pf_*\mathcal{F})|_V$$ There is a similar statement for the derived image $Rf_*\mathcal{F}^\bullet$ where $\mathcal{F}^\bullet$ is a bounded below complex of $\mathcal{O}_X$-modules. \end{lemma} \begin{proof} First proof. Apply Lemmas \ref{lemma-describe-higher-direct-images} and \ref{lemma-cohomology-of-open} to see the displayed equality. Second proof. Choose an injective resolution $\mathcal{F} \to \mathcal{I}^\bullet$ and use that $\mathcal{F}|_{f^{-1}(V)} \to \mathcal{I}^\bullet|_{f^{-1}(V)}$ is an injective resolution also. \end{proof} \begin{remark} \label{remark-daniel} Here is a different approach to the proofs of Lemmas \ref{lemma-kill-cohomology-class-on-covering} and \ref{lemma-describe-higher-direct-images} above. Let $(X, \mathcal{O}_X)$ be a ringed space. Let $i_X : \textit{Mod}(\mathcal{O}_X) \to \textit{PMod}(\mathcal{O}_X)$ be the inclusion functor and let $\#$ be the sheafification functor. Recall that $i_X$ is left exact and $\#$ is exact. \begin{enumerate} \item First prove Lemma \ref{lemma-include} below which says that the right derived functors of $i_X$ are given by $R^pi_X\mathcal{F} = \underline{H}^p(\mathcal{F})$. Here is another proof: The equality is clear for $p = 0$. Both $(R^pi_X)_{p \geq 0}$ and $(\underline{H}^p)_{p \geq 0}$ are delta functors vanishing on injectives, hence both are universal, hence they are isomorphic. See Homology, Section \ref{homology-section-cohomological-delta-functor}. \item A restatement of Lemma \ref{lemma-kill-cohomology-class-on-covering} is that $(\underline{H}^p(\mathcal{F}))^\# = 0$, $p > 0$ for any sheaf of $\mathcal{O}_X$-modules $\mathcal{F}$. To see this is true, use that ${}^\#$ is exact so $$(\underline{H}^p(\mathcal{F}))^\# = (R^pi_X\mathcal{F})^\# = R^p(\# \circ i_X)(\mathcal{F}) = 0$$ because $\# \circ i_X$ is the identity functor. \item Let $f : X \to Y$ be a morphism of ringed spaces. Let $\mathcal{F}$ be an $\mathcal{O}_X$-module. The presheaf $V \mapsto H^p(f^{-1}V, \mathcal{F})$ is equal to $R^p (i_Y \circ f_*)\mathcal{F}$. You can prove this by noticing that both give universal delta functors as in the argument of (1) above. Hence Lemma \ref{lemma-describe-higher-direct-images} says that $R^p f_* \mathcal{F}= (R^p (i_Y \circ f_*)\mathcal{F})^\#$. Again using that $\#$ is exact a that $\# \circ i_Y$ is the identity functor we see that $$R^p f_* \mathcal{F} = R^p(\# \circ i_Y \circ f_*)\mathcal{F} = (R^p (i_Y \circ f_*)\mathcal{F})^\#$$ as desired. \end{enumerate} \end{remark} \section{Mayer-Vietoris} \label{section-mayer-vietoris} \noindent Below will construct the {\v C}ech-to-cohomology spectral sequence, see Lemma \ref{lemma-cech-spectral-sequence}. A special case of that spectral sequence is the Mayer-Vietoris long exact sequence. Since it is such a basic, useful and easy to understand variant of the spectral sequence we treat it here separately. \begin{lemma} \label{lemma-injective-restriction-surjective} Let $X$ be a ringed space. Let $U' \subset U \subset X$ be open subspaces. For any injective $\mathcal{O}_X$-module $\mathcal{I}$ the restriction mapping $\mathcal{I}(U) \to \mathcal{I}(U')$ is surjective. \end{lemma} \begin{proof} Let $j : U \to X$ and $j' : U' \to X$ be the open immersions. Recall that $j_!\mathcal{O}_U$ is the extension by zero of $\mathcal{O}_U = \mathcal{O}_X|_U$, see Sheaves, Section \ref{sheaves-section-open-immersions}. Since $j_!$ is a left adjoint to restriction we see that for any sheaf $\mathcal{F}$ of $\mathcal{O}_X$-modules $$\Hom_{\mathcal{O}_X}(j_!\mathcal{O}_U, \mathcal{F}) = \Hom_{\mathcal{O}_U}(\mathcal{O}_U, \mathcal{F}|_U) = \mathcal{F}(U)$$ see Sheaves, Lemma \ref{sheaves-lemma-j-shriek-modules}. Similarly, the sheaf $j'_!\mathcal{O}_{U'}$ represents the functor $\mathcal{F} \mapsto \mathcal{F}(U')$. Moreover there is an obvious canonical map of $\mathcal{O}_X$-modules $$j'_!\mathcal{O}_{U'} \longrightarrow j_!\mathcal{O}_U$$ which corresponds to the restriction mapping $\mathcal{F}(U) \to \mathcal{F}(U')$ via Yoneda's lemma (Categories, Lemma \ref{categories-lemma-yoneda}). By the description of the stalks of the sheaves $j'_!\mathcal{O}_{U'}$, $j_!\mathcal{O}_U$ we see that the displayed map above is injective (see lemma cited above). Hence if $\mathcal{I}$ is an injective $\mathcal{O}_X$-module, then the map $$\Hom_{\mathcal{O}_X}(j_!\mathcal{O}_U, \mathcal{I}) \longrightarrow \Hom_{\mathcal{O}_X}(j'_!\mathcal{O}_{U'}, \mathcal{I})$$ is surjective, see Homology, Lemma \ref{homology-lemma-characterize-injectives}. Putting everything together we obtain the lemma. \end{proof} \begin{lemma}[Mayer-Vietoris] \label{lemma-mayer-vietoris} Let $X$ be a ringed space. Suppose that $X = U \cup V$ is a union of two open subsets. For every $\mathcal{O}_X$-module $\mathcal{F}$ there exists a long exact cohomology sequence $$0 \to H^0(X, \mathcal{F}) \to H^0(U, \mathcal{F}) \oplus H^0(V, \mathcal{F}) \to H^0(U \cap V, \mathcal{F}) \to H^1(X, \mathcal{F}) \to \ldots$$ This long exact sequence is functorial in $\mathcal{F}$. \end{lemma} \begin{proof} The sheaf condition says that the kernel of $(1, -1) : \mathcal{F}(U) \oplus \mathcal{F}(V) \to \mathcal{F}(U \cap V)$ is equal to the image of $\mathcal{F}(X)$ by the first map for any abelian sheaf $\mathcal{F}$. Lemma \ref{lemma-injective-restriction-surjective} above implies that the map $(1, -1) : \mathcal{I}(U) \oplus \mathcal{I}(V) \to \mathcal{I}(U \cap V)$ is surjective whenever $\mathcal{I}$ is an injective $\mathcal{O}_X$-module. Hence if $\mathcal{F} \to \mathcal{I}^\bullet$ is an injective resolution of $\mathcal{F}$, then we get a short exact sequence of complexes $$0 \to \mathcal{I}^\bullet(X) \to \mathcal{I}^\bullet(U) \oplus \mathcal{I}^\bullet(V) \to \mathcal{I}^\bullet(U \cap V) \to 0.$$ Taking cohomology gives the result (use Homology, Lemma \ref{homology-lemma-long-exact-sequence-cochain}). We omit the proof of the functoriality of the sequence. \end{proof} \begin{lemma}[Relative Mayer-Vietoris] \label{lemma-relative-mayer-vietoris} Let $f : X \to Y$ be a morphism of ringed spaces. Suppose that $X = U \cup V$ is a union of two open subsets. Denote $a = f|_U : U \to Y$, $b = f|_V : V \to Y$, and $c = f|_{U \cap V} : U \cap V \to Y$. For every $\mathcal{O}_X$-module $\mathcal{F}$ there exists a long exact sequence $$0 \to f_*\mathcal{F} \to a_*(\mathcal{F}|_U) \oplus b_*(\mathcal{F}|_V) \to c_*(\mathcal{F}|_{U \cap V}) \to R^1f_*\mathcal{F} \to \ldots$$ This long exact sequence is functorial in $\mathcal{F}$. \end{lemma} \begin{proof} Let $\mathcal{F} \to \mathcal{I}^\bullet$ be an injective resolution of $\mathcal{F}$. We claim that we get a short exact sequence of complexes $$0 \to f_*\mathcal{I}^\bullet \to a_*\mathcal{I}^\bullet|_U \oplus b_*\mathcal{I}^\bullet|_V \to c_*\mathcal{I}^\bullet|_{U \cap V} \to 0.$$ Namely, for any open $W \subset Y$, and for any $n \geq 0$ the corresponding sequence of groups of sections over $W$ $$0 \to \mathcal{I}^n(f^{-1}(W)) \to \mathcal{I}^n(U \cap f^{-1}(W)) \oplus \mathcal{I}^n(V \cap f^{-1}(W)) \to \mathcal{I}^n(U \cap V \cap f^{-1}(W)) \to 0$$ was shown to be short exact in the proof of Lemma \ref{lemma-mayer-vietoris}. The lemma follows by taking cohomology sheaves and using the fact that $\mathcal{I}^\bullet|_U$ is an injective resolution of $\mathcal{F}|_U$ and similarly for $\mathcal{I}^\bullet|_V$, $\mathcal{I}^\bullet|_{U \cap V}$ see Lemma \ref{lemma-cohomology-of-open}. \end{proof} \section{The {\v C}ech complex and {\v C}ech cohomology} \label{section-cech} \noindent Let $X$ be a topological space. Let $\mathcal{U} : U = \bigcup_{i \in I} U_i$ be an open covering, see Topology, Basic notion (\ref{topology-item-covering}). As is customary we denote $U_{i_0\ldots i_p} = U_{i_0} \cap \ldots \cap U_{i_p}$ for the $(p + 1)$-fold intersection of members of $\mathcal{U}$. Let $\mathcal{F}$ be an abelian presheaf on $X$. Set $$\check{\mathcal{C}}^p(\mathcal{U}, \mathcal{F}) = \prod\nolimits_{(i_0, \ldots, i_p) \in I^{p + 1}} \mathcal{F}(U_{i_0\ldots i_p}).$$ This is an abelian group. For $s \in \check{\mathcal{C}}^p(\mathcal{U}, \mathcal{F})$ we denote $s_{i_0\ldots i_p}$ its value in $\mathcal{F}(U_{i_0\ldots i_p})$. Note that if $s \in \check{\mathcal{C}}^1(\mathcal{U}, \mathcal{F})$ and $i, j \in I$ then $s_{ij}$ and $s_{ji}$ are both elements of $\mathcal{F}(U_i \cap U_j)$ but there is no imposed relation between $s_{ij}$ and $s_{ji}$. In other words, we are {\it not} working with alternating cochains (these will be defined in Section \ref{section-alternating-cech}). We define $$d : \check{\mathcal{C}}^p(\mathcal{U}, \mathcal{F}) \longrightarrow \check{\mathcal{C}}^{p + 1}(\mathcal{U}, \mathcal{F})$$ by the formula \begin{equation} \label{equation-d-cech} d(s)_{i_0\ldots i_{p + 1}} = \sum\nolimits_{j = 0}^{p + 1} (-1)^j s_{i_0\ldots \hat i_j \ldots i_{p + 1}}|_{U_{i_0\ldots i_{p + 1}}} \end{equation} It is straightforward to see that $d \circ d = 0$. In other words $\check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{F})$ is a complex. \begin{definition} \label{definition-cech-complex} Let $X$ be a topological space. Let $\mathcal{U} : U = \bigcup_{i \in I} U_i$ be an open covering. Let $\mathcal{F}$ be an abelian presheaf on $X$. The complex $\check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{F})$ is the {\it {\v C}ech complex} associated to $\mathcal{F}$ and the open covering $\mathcal{U}$. Its cohomology groups $H^i(\check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{F}))$ are called the {\it {\v C}ech cohomology groups} associated to $\mathcal{F}$ and the covering $\mathcal{U}$. They are denoted $\check H^i(\mathcal{U}, \mathcal{F})$. \end{definition} \begin{lemma} \label{lemma-cech-h0} Let $X$ be a topological space. Let $\mathcal{F}$ be an abelian presheaf on $X$. The following are equivalent \begin{enumerate} \item $\mathcal{F}$ is an abelian sheaf and \item for every open covering $\mathcal{U} : U = \bigcup_{i \in I} U_i$ the natural map $$\mathcal{F}(U) \to \check{H}^0(\mathcal{U}, \mathcal{F})$$ is bijective. \end{enumerate} \end{lemma} \begin{proof} This is true since the sheaf condition is exactly that $\mathcal{F}(U) \to \check{H}^0(\mathcal{U}, \mathcal{F})$ is bijective for every open covering. \end{proof} \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} \section{{\v C}ech cohomology and cohomology} \label{section-cech-cohomology-cohomology} \begin{lemma} \label{lemma-injective-trivial-cech} Let $X$ be a ringed space. Let $\mathcal{U} : U = \bigcup_{i \in I} U_i$ be a covering. Let $\mathcal{I}$ be an injective $\mathcal{O}_X$-module. Then $$\check{H}^p(\mathcal{U}, \mathcal{I}) = \left\{ \begin{matrix} \mathcal{I}(U) & \text{if} & p = 0 \\ 0 & \text{if} & p > 0 \end{matrix} \right.$$ \end{lemma} \begin{proof} An injective $\mathcal{O}_X$-module is also injective as an object in the category $\textit{PMod}(\mathcal{O}_X)$ (for example since sheafification is an exact left adjoint to the inclusion functor, using Homology, Lemma \ref{homology-lemma-adjoint-preserve-injectives}). Hence we can apply Lemma \ref{lemma-cech-cohomology-derived-presheaves} (or its proof) to see the result. \end{proof} \begin{lemma} \label{lemma-cech-cohomology} Let $X$ be a ringed space. Let $\mathcal{U} : U = \bigcup_{i \in I} U_i$ be a covering. There is a transformation $$\check{\mathcal{C}}^\bullet(\mathcal{U}, -) \longrightarrow R\Gamma(U, -)$$ of functors $\textit{Mod}(\mathcal{O}_X) \to D^{+}(\mathcal{O}_X(U))$. In particular this provides canonical maps $\check{H}^p(\mathcal{U}, \mathcal{F}) \to H^p(U, \mathcal{F})$ for $\mathcal{F}$ ranging over $\textit{Mod}(\mathcal{O}_X)$. \end{lemma} \begin{proof} Let $\mathcal{F}$ be an $\mathcal{O}_X$-module. Choose an injective resolution $\mathcal{F} \to \mathcal{I}^\bullet$. Consider the double complex $\check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{I}^\bullet)$ with terms $\check{\mathcal{C}}^p(\mathcal{U}, \mathcal{I}^q)$. There is a map of complexes $$\alpha : \Gamma(U, \mathcal{I}^\bullet) \longrightarrow \text{Tot}(\check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{I}^\bullet))$$ coming from the maps $\mathcal{I}^q(U) \to \check{H}^0(\mathcal{U}, \mathcal{I}^q)$ and a map of complexes $$\beta : \check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{F}) \longrightarrow \text{Tot}(\check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{I}^\bullet))$$ coming from the map $\mathcal{F} \to \mathcal{I}^0$. We can apply Homology, Lemma \ref{homology-lemma-double-complex-gives-resolution} to see that $\alpha$ is a quasi-isomorphism. Namely, Lemma \ref{lemma-injective-trivial-cech} implies that the $q$th row of the double complex $\check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{I}^\bullet)$ is a resolution of $\Gamma(U, \mathcal{I}^q)$. Hence $\alpha$ becomes invertible in $D^{+}(\mathcal{O}_X(U))$ and the transformation of the lemma is the composition of $\beta$ followed by the inverse of $\alpha$. We omit the verification that this is functorial. \end{proof} \begin{lemma} \label{lemma-cech-h1} Let $X$ be a topological space. Let $\mathcal{H}$ be an abelian sheaf on $X$. Let $\mathcal{U} : X = \bigcup_{i \in I} U_i$ be an open covering. The map $$\check{H}^1(\mathcal{U}, \mathcal{H}) \longrightarrow H^1(X, \mathcal{H})$$ is injective and identifies $\check{H}^1(\mathcal{U}, \mathcal{H})$ via the bijection of Lemma \ref{lemma-torsors-h1} with the set of isomorphism classes of $\mathcal{H}$-torsors which restrict to trivial torsors over each $U_i$. \end{lemma} \begin{proof} To see this we construct an inverse map. Namely, let $\mathcal{F}$ be a $\mathcal{H}$-torsor whose restriction to $U_i$ is trivial. By Lemma \ref{lemma-trivial-torsor} this means there exists a section $s_i \in \mathcal{F}(U_i)$. On $U_{i_0} \cap U_{i_1}$ there is a unique section $s_{i_0i_1}$ of $\mathcal{H}$ such that $s_{i_0i_1} \cdot s_{i_0}|_{U_{i_0} \cap U_{i_1}} = s_{i_1}|_{U_{i_0} \cap U_{i_1}}$. A computation shows that $s_{i_0i_1}$ is a {\v C}ech cocycle and that its class is well defined (i.e., does not depend on the choice of the sections $s_i$). The inverse maps the isomorphism class of $\mathcal{F}$ to the cohomology class of the cocycle $(s_{i_0i_1})$. We omit the verification that this map is indeed an inverse. \end{proof} \begin{lemma} \label{lemma-include} Let $X$ be a ringed space. Consider the functor $i : \textit{Mod}(\mathcal{O}_X) \to \textit{PMod}(\mathcal{O}_X)$. It is a left exact functor with right derived functors given by $$R^pi(\mathcal{F}) = \underline{H}^p(\mathcal{F}) : U \longmapsto H^p(U, \mathcal{F})$$ see discussion in Section \ref{section-locality}. \end{lemma} \begin{proof} It is clear that $i$ is left exact. Choose an injective resolution $\mathcal{F} \to \mathcal{I}^\bullet$. By definition $R^pi$ is the $p$th cohomology {\it presheaf} of the complex $\mathcal{I}^\bullet$. In other words, the sections of $R^pi(\mathcal{F})$ over an open $U$ are given by $$\frac{\Ker(\mathcal{I}^n(U) \to \mathcal{I}^{n + 1}(U))} {\Im(\mathcal{I}^{n - 1}(U) \to \mathcal{I}^n(U))}.$$ which is the definition of $H^p(U, \mathcal{F})$. \end{proof} \begin{lemma} \label{lemma-cech-spectral-sequence} Let $X$ be a ringed space. Let $\mathcal{U} : U = \bigcup_{i \in I} U_i$ be a covering. For any sheaf of $\mathcal{O}_X$-modules $\mathcal{F}$ there is a spectral sequence $(E_r, d_r)_{r \geq 0}$ with $$E_2^{p, q} = \check{H}^p(\mathcal{U}, \underline{H}^q(\mathcal{F}))$$ converging to $H^{p + q}(U, \mathcal{F})$. This spectral sequence is functorial in $\mathcal{F}$. \end{lemma} \begin{proof} This is a Grothendieck spectral sequence (see Derived Categories, Lemma \ref{derived-lemma-grothendieck-spectral-sequence}) for the functors $$i : \textit{Mod}(\mathcal{O}_X) \to \textit{PMod}(\mathcal{O}_X) \quad\text{and}\quad \check{H}^0(\mathcal{U}, - ) : \textit{PMod}(\mathcal{O}_X) \to \text{Mod}_{\mathcal{O}_X(U)}.$$ Namely, we have $\check{H}^0(\mathcal{U}, i(\mathcal{F})) = \mathcal{F}(U)$ by Lemma \ref{lemma-cech-h0}. We have that $i(\mathcal{I})$ is {\v C}ech acyclic by Lemma \ref{lemma-injective-trivial-cech}. And we have that $\check{H}^p(\mathcal{U}, -) = R^p\check{H}^0(\mathcal{U}, -)$ as functors on $\textit{PMod}(\mathcal{O}_X)$ by Lemma \ref{lemma-cech-cohomology-derived-presheaves}. Putting everything together gives the lemma. \end{proof} \begin{lemma} \label{lemma-cech-spectral-sequence-application} Let $X$ be a ringed space. Let $\mathcal{U} : U = \bigcup_{i \in I} U_i$ be a covering. Let $\mathcal{F}$ be an $\mathcal{O}_X$-module. Assume that $H^i(U_{i_0 \ldots i_p}, \mathcal{F}) = 0$ for all $i > 0$, all $p \geq 0$ and all $i_0, \ldots, i_p \in I$. Then $\check{H}^p(\mathcal{U}, \mathcal{F}) = H^p(U, \mathcal{F})$ as $\mathcal{O}_X(U)$-modules. \end{lemma} \begin{proof} We will use the spectral sequence of Lemma \ref{lemma-cech-spectral-sequence}. The assumptions mean that $E_2^{p, q} = 0$ for all $(p, q)$ with $q \not = 0$. Hence the spectral sequence degenerates at $E_2$ and the result follows. \end{proof} \begin{lemma} \label{lemma-ses-cech-h1} Let $X$ be a ringed space. Let $$0 \to \mathcal{F} \to \mathcal{G} \to \mathcal{H} \to 0$$ be a short exact sequence of $\mathcal{O}_X$-modules. Let $U \subset X$ be an open subset. If there exists a cofinal system of open coverings $\mathcal{U}$ of $U$ such that $\check{H}^1(\mathcal{U}, \mathcal{F}) = 0$, then the map $\mathcal{G}(U) \to \mathcal{H}(U)$ is surjective. \end{lemma} \begin{proof} Take an element $s \in \mathcal{H}(U)$. Choose an open covering $\mathcal{U} : U = \bigcup_{i \in I} U_i$ such that (a) $\check{H}^1(\mathcal{U}, \mathcal{F}) = 0$ and (b) $s|_{U_i}$ is the image of a section $s_i \in \mathcal{G}(U_i)$. Since we can certainly find a covering such that (b) holds it follows from the assumptions of the lemma that we can find a covering such that (a) and (b) both hold. Consider the sections $$s_{i_0i_1} = s_{i_1}|_{U_{i_0i_1}} - s_{i_0}|_{U_{i_0i_1}}.$$ Since $s_i$ lifts $s$ we see that $s_{i_0i_1} \in \mathcal{F}(U_{i_0i_1})$. By the vanishing of $\check{H}^1(\mathcal{U}, \mathcal{F})$ we can find sections $t_i \in \mathcal{F}(U_i)$ such that $$s_{i_0i_1} = t_{i_1}|_{U_{i_0i_1}} - t_{i_0}|_{U_{i_0i_1}}.$$ Then clearly the sections $s_i - t_i$ satisfy the sheaf condition and glue to a section of $\mathcal{G}$ over $U$ which maps to $s$. Hence we win. \end{proof} \begin{lemma} \label{lemma-cech-vanish} \begin{slogan} If higher {\v C}ech cohomology of an abelian sheaf vanishes for all open covers, then higher cohomology vanishes. \end{slogan} Let $X$ be a ringed space. Let $\mathcal{F}$ be an $\mathcal{O}_X$-module such that $$\check{H}^p(\mathcal{U}, \mathcal{F}) = 0$$ for all $p > 0$ and any open covering $\mathcal{U} : U = \bigcup_{i \in I} U_i$ of an open of $X$. Then $H^p(U, \mathcal{F}) = 0$ for all $p > 0$ and any open $U \subset X$. \end{lemma} \begin{proof} Let $\mathcal{F}$ be a sheaf satisfying the assumption of the lemma. We will indicate this by saying $\mathcal{F}$ has vanishing higher {\v C}ech cohomology for any open covering''. Choose an embedding $\mathcal{F} \to \mathcal{I}$ into an injective $\mathcal{O}_X$-module. By Lemma \ref{lemma-injective-trivial-cech} $\mathcal{I}$ has vanishing higher {\v C}ech cohomology for any open covering. Let $\mathcal{Q} = \mathcal{I}/\mathcal{F}$ so that we have a short exact sequence $$0 \to \mathcal{F} \to \mathcal{I} \to \mathcal{Q} \to 0.$$ By Lemma \ref{lemma-ses-cech-h1} and our assumptions this sequence is actually exact as a sequence of presheaves! In particular we have a long exact sequence of {\v C}ech cohomology groups for any open covering $\mathcal{U}$, see Lemma \ref{lemma-cech-cohomology-delta-functor-presheaves} for example. This implies that $\mathcal{Q}$ is also an $\mathcal{O}_X$-module with vanishing higher {\v C}ech cohomology for all open coverings. \medskip\noindent Next, we look at the long exact cohomology sequence $$\xymatrix{ 0 \ar[r] & H^0(U, \mathcal{F}) \ar[r] & H^0(U, \mathcal{I}) \ar[r] & H^0(U, \mathcal{Q}) \ar[lld] \\ & H^1(U, \mathcal{F}) \ar[r] & H^1(U, \mathcal{I}) \ar[r] & H^1(U, \mathcal{Q}) \ar[lld] \\ & \ldots & \ldots & \ldots \\ }$$ for any open $U \subset X$. Since $\mathcal{I}$ is injective we have $H^n(U, \mathcal{I}) = 0$ for $n > 0$ (see Derived Categories, Lemma \ref{derived-lemma-higher-derived-functors}). By the above we see that $H^0(U, \mathcal{I}) \to H^0(U, \mathcal{Q})$ is surjective and hence $H^1(U, \mathcal{F}) = 0$. Since $\mathcal{F}$ was an arbitrary $\mathcal{O}_X$-module with vanishing higher {\v C}ech cohomology we conclude that also $H^1(U, \mathcal{Q}) = 0$ since $\mathcal{Q}$ is another of these sheaves (see above). By the long exact sequence this in turn implies that $H^2(U, \mathcal{F}) = 0$. And so on and so forth. \end{proof} \begin{lemma} \label{lemma-cech-vanish-basis} (Variant of Lemma \ref{lemma-cech-vanish}.) Let $X$ be a ringed space. Let $\mathcal{B}$ be a basis for the topology on $X$. Let $\mathcal{F}$ be an $\mathcal{O}_X$-module. Assume there exists a set of open coverings $\text{Cov}$ with the following properties: \begin{enumerate} \item For every $\mathcal{U} \in \text{Cov}$ with $\mathcal{U} : U = \bigcup_{i \in I} U_i$ we have $U, U_i \in \mathcal{B}$ and every $U_{i_0 \ldots i_p} \in \mathcal{B}$. \item For every $U \in \mathcal{B}$ the open coverings of $U$ occurring in $\text{Cov}$ is a cofinal system of open coverings of $U$. \item For every $\mathcal{U} \in \text{Cov}$ we have $\check{H}^p(\mathcal{U}, \mathcal{F}) = 0$ for all $p > 0$. \end{enumerate} Then $H^p(U, \mathcal{F}) = 0$ for all $p > 0$ and any $U \in \mathcal{B}$. \end{lemma} \begin{proof} Let $\mathcal{F}$ and $\text{Cov}$ be as in the lemma. We will indicate this by saying $\mathcal{F}$ has vanishing higher {\v C}ech cohomology for any $\mathcal{U} \in \text{Cov}$''. Choose an embedding $\mathcal{F} \to \mathcal{I}$ into an injective $\mathcal{O}_X$-module. By Lemma \ref{lemma-injective-trivial-cech} $\mathcal{I}$ has vanishing higher {\v C}ech cohomology for any $\mathcal{U} \in \text{Cov}$. Let $\mathcal{Q} = \mathcal{I}/\mathcal{F}$ so that we have a short exact sequence $$0 \to \mathcal{F} \to \mathcal{I} \to \mathcal{Q} \to 0.$$ By Lemma \ref{lemma-ses-cech-h1} and our assumption (2) this sequence gives rise to an exact sequence $$0 \to \mathcal{F}(U) \to \mathcal{I}(U) \to \mathcal{Q}(U) \to 0.$$ for every $U \in \mathcal{B}$. Hence for any $\mathcal{U} \in \text{Cov}$ we get a short exact sequence of {\v C}ech complexes $$0 \to \check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{F}) \to \check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{I}) \to \check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{Q}) \to 0$$ since each term in the {\v C}ech complex is made up out of a product of values over elements of $\mathcal{B}$ by assumption (1). In particular we have a long exact sequence of {\v C}ech cohomology groups for any open covering $\mathcal{U} \in \text{Cov}$. This implies that $\mathcal{Q}$ is also an $\mathcal{O}_X$-module with vanishing higher {\v C}ech cohomology for all $\mathcal{U} \in \text{Cov}$. \medskip\noindent Next, we look at the long exact cohomology sequence $$\xymatrix{ 0 \ar[r] & H^0(U, \mathcal{F}) \ar[r] & H^0(U, \mathcal{I}) \ar[r] & H^0(U, \mathcal{Q}) \ar[lld] \\ & H^1(U, \mathcal{F}) \ar[r] & H^1(U, \mathcal{I}) \ar[r] & H^1(U, \mathcal{Q}) \ar[lld] \\ & \ldots & \ldots & \ldots \\ }$$ for any $U \in \mathcal{B}$. Since $\mathcal{I}$ is injective we have $H^n(U, \mathcal{I}) = 0$ for $n > 0$ (see Derived Categories, Lemma \ref{derived-lemma-higher-derived-functors}). By the above we see that $H^0(U, \mathcal{I}) \to H^0(U, \mathcal{Q})$ is surjective and hence $H^1(U, \mathcal{F}) = 0$. Since $\mathcal{F}$ was an arbitrary $\mathcal{O}_X$-module with vanishing higher {\v C}ech cohomology for all $\mathcal{U} \in \text{Cov}$ we conclude that also $H^1(U, \mathcal{Q}) = 0$ since $\mathcal{Q}$ is another of these sheaves (see above). By the long exact sequence this in turn implies that $H^2(U, \mathcal{F}) = 0$. And so on and so forth. \end{proof} \begin{lemma} \label{lemma-pushforward-injective} Let $f : X \to Y$ be a morphism of ringed spaces. Let $\mathcal{I}$ be an injective $\mathcal{O}_X$-module. Then \begin{enumerate} \item $\check{H}^p(\mathcal{V}, f_*\mathcal{I}) = 0$ for all $p > 0$ and any open covering $\mathcal{V} : V = \bigcup_{j \in J} V_j$ of $Y$. \item $H^p(V, f_*\mathcal{I}) = 0$ for all $p > 0$ and every open $V \subset Y$. \end{enumerate} In other words, $f_*\mathcal{I}$ is right acyclic for $\Gamma(V, -)$ (see Derived Categories, Definition \ref{derived-definition-derived-functor}) for any $V \subset Y$ open. \end{lemma} \begin{proof} Set $\mathcal{U} : f^{-1}(V) = \bigcup_{j \in J} f^{-1}(V_j)$. It is an open covering of $X$ and $$\check{\mathcal{C}}^\bullet(\mathcal{V}, f_*\mathcal{I}) = \check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{I}).$$ This is true because $$f_*\mathcal{I}(V_{j_0 \ldots j_p}) = \mathcal{I}(f^{-1}(V_{j_0 \ldots j_p})) = \mathcal{I}(f^{-1}(V_{j_0}) \cap \ldots \cap f^{-1}(V_{j_p})) = \mathcal{I}(U_{j_0 \ldots j_p}).$$ Thus the first statement of the lemma follows from Lemma \ref{lemma-injective-trivial-cech}. The second statement follows from the first and Lemma \ref{lemma-cech-vanish}. \end{proof} \noindent The following lemma implies in particular that $f_* : \textit{Ab}(X) \to \textit{Ab}(Y)$ transforms injective abelian sheaves into injective abelian sheaves. \begin{lemma} \label{lemma-pushforward-injective-flat} Let $f : X \to Y$ be a morphism of ringed spaces. Assume $f$ is flat. Then $f_*\mathcal{I}$ is an injective $\mathcal{O}_Y$-module for any injective $\mathcal{O}_X$-module $\mathcal{I}$. \end{lemma} \begin{proof} In this case the functor $f^*$ transforms injections into injections (Modules, Lemma \ref{modules-lemma-pullback-flat}). Hence the result follows from Homology, Lemma \ref{homology-lemma-adjoint-preserve-injectives}. \end{proof} \begin{lemma} \label{lemma-cohomology-products} Let $(X, \mathcal{O}_X)$ be a ringed space. Let $I$ be a set. For $i \in I$ let $\mathcal{F}_i$ be an $\mathcal{O}_X$-module. Let $U \subset X$ be open. The canonical map $$H^p(U, \prod\nolimits_{i \in I} \mathcal{F}_i) \longrightarrow \prod\nolimits_{i \in I} H^p(U, \mathcal{F}_i)$$ is an isomorphism for $p = 0$ and injective for $p = 1$. \end{lemma} \begin{proof} The statement for $p = 0$ is true because the product of sheaves is equal to the product of the underlying presheaves, see Sheaves, Section \ref{sheaves-section-limits-sheaves}. Proof for $p = 1$. Set $\mathcal{F} = \prod \mathcal{F}_i$. Let $\xi \in H^1(U, \mathcal{F})$ map to zero in $\prod H^1(U, \mathcal{F}_i)$. By locality of cohomology, see Lemma \ref{lemma-kill-cohomology-class-on-covering}, there exists an open covering $\mathcal{U} : U = \bigcup U_j$ such that $\xi|_{U_j} = 0$ for all $j$. By Lemma \ref{lemma-cech-h1} this means $\xi$ comes from an element $\check \xi \in \check H^1(\mathcal{U}, \mathcal{F})$. Since the maps $\check H^1(\mathcal{U}, \mathcal{F}_i) \to H^1(U, \mathcal{F}_i)$ are injective for all $i$ (by Lemma \ref{lemma-cech-h1}), and since the image of $\xi$ is zero in $\prod H^1(U, \mathcal{F}_i)$ we see that the image $\check \xi_i = 0$ in $\check H^1(\mathcal{U}, \mathcal{F}_i)$. However, since $\mathcal{F} = \prod \mathcal{F}_i$ we see that $\check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{F})$ is the product of the complexes $\check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{F}_i)$, hence by Homology, Lemma \ref{homology-lemma-product-abelian-groups-exact} we conclude that $\check \xi = 0$ as desired. \end{proof} \section{Flasque sheaves} \label{section-flasque} \noindent Here is the definition. \begin{definition} \label{definition-flasque} Let $X$ be a topological space. We say a presheaf of sets $\mathcal{F}$ is {\it flasque} or {\it flabby} if for every $U \subset V$ open in $X$ the restriction map $\mathcal{F}(V) \to \mathcal{F}(U)$ is surjective. \end{definition} \noindent We will use this terminology also for abelian sheaves and sheaves of modules if $X$ is a ringed space. Clearly it suffices to assume the restriction maps $\mathcal{F}(X) \to \mathcal{F}(U)$ is surjective for every open $U \subset X$. \begin{lemma} \label{lemma-injective-flasque} Let $(X, \mathcal{O}_X)$ be a ringed space. Then any injective $\mathcal{O}_X$-module is flasque. \end{lemma} \begin{proof} This is a reformulation of Lemma \ref{lemma-injective-restriction-surjective}. \end{proof} \begin{lemma} \label{lemma-flasque-acyclic} Let $(X, \mathcal{O}_X)$ be a ringed space. Any flasque $\mathcal{O}_X$-module is acyclic for $R\Gamma(X, -)$ as well as $R\Gamma(U, -)$ for any open $U$ of $X$. \end{lemma} \begin{proof} We will prove this using Derived Categories, Lemma \ref{derived-lemma-subcategory-right-acyclics}. Since every injective module is flasque we see that we can embed every $\mathcal{O}_X$-module into a flasque module, see Injectives, Lemma \ref{injectives-lemma-abelian-sheaves-space}. Thus it suffices to show that given a short exact sequence $$0 \to \mathcal{F} \to \mathcal{G} \to \mathcal{H} \to 0$$ with $\mathcal{F}$, $\mathcal{G}$ flasque, then $\mathcal{H}$ is flasque and the sequence remains short exact after taking sections on any open of $X$. In fact, the second statement implies the first. Thus, let $U \subset X$ be an open subspace. Let $s \in \mathcal{H}(U)$. We will show that we can lift $s$ to a sequence of $\mathcal{G}$ over $U$. To do this consider the set $T$ of pairs $(V, t)$ where $V \subset U$ is open and $t \in \mathcal{G}(V)$ is a section mapping to $s|_V$ in $\mathcal{H}$. We put a partial ordering on $T$ by setting $(V, t) \leq (V', t')$ if and only if $V \subset V'$ and $t'|_V = t$. If $(V_\alpha, t_\alpha)$, $\alpha \in A$ is a totally ordered subset of $T$, then $V = \bigcup V_\alpha$ is open and there is a unique section $t \in \mathcal{G}(V)$ restricting to $t_\alpha$ over $V_\alpha$ by the sheaf condition on $\mathcal{G}$. Thus by Zorn's lemma there exists a maximal element $(V, t)$ in $T$. We will show that $V = U$ thereby finishing the proof. Namely, pick any $x \in U$. We can find a small open neighbourhood $W \subset U$ of $x$ and $t' \in \mathcal{G}(W)$ mapping to $s|_W$ in $\mathcal{H}$. Then $t'|_{W \cap V} - t|_{W \cap V}$ maps to zero in $\mathcal{H}$, hence comes from some section $r' \in \mathcal{F}(W \cap V)$. Using that $\mathcal{F}$ is flasque we find a section $r \in \mathcal{F}(W)$ restricting to $r'$ over $W \cap V$. Modifying $t'$ by the image of $r$ we may assume that $t$ and $t'$ restrict to the same section over $W \cap V$. By the sheaf condition of $\mathcal{G}$ we can find a section $\tilde t$ of $\mathcal{G}$ over $W \cup V$ restricting to $t$ and $t'$. By maximality of $(V, t)$ we see that $V \cap W = V$. Thus $x \in V$ and we are done. \end{proof} \noindent The following lemma does not hold for flasque presheaves. \begin{lemma} \label{lemma-flasque-acyclic-cech} Let $(X, \mathcal{O}_X)$ be a ringed space. Let $\mathcal{F}$ be a sheaf of $\mathcal{O}_X$-modules. Let $\mathcal{U} : U = \bigcup U_i$ be an open covering. If $\mathcal{F}$ is flasque, then $\check{H}^p(\mathcal{U}, \mathcal{F}) = 0$ for $p > 0$. \end{lemma} \begin{proof} The presheaves $\underline{H}^q(\mathcal{F})$ used in the statement of Lemma \ref{lemma-cech-spectral-sequence} are zero by Lemma \ref{lemma-flasque-acyclic}. Hence $\check{H}^p(U, \mathcal{F}) = H^p(U, \mathcal{F}) = 0$ by Lemma \ref{lemma-flasque-acyclic} again. \end{proof} \begin{lemma} \label{lemma-flasque-acyclic-pushforward} Let $(X, \mathcal{O}_X) \to (Y, \mathcal{O}_Y)$ be a morphism of ringed spaces. Let $\mathcal{F}$ be a sheaf of $\mathcal{O}_X$-modules. If $\mathcal{F}$ is flasque, then $R^pf_*\mathcal{F} = 0$ for $p > 0$. \end{lemma} \begin{proof} Immediate from Lemma \ref{lemma-describe-higher-direct-images} and Lemma \ref{lemma-flasque-acyclic}. \end{proof} \noindent The following lemma can be proved by an elementary induction argument for finite coverings, compare with the discussion of {\v C}ech cohomology in \cite{FOAG}. \begin{lemma} \label{lemma-vanishing-ravi} Let $X$ be a topological space. Let $\mathcal{F}$ be an abelian sheaf on $X$. Let $\mathcal{U} : U = \bigcup_{i \in I} U_i$ be an open covering. Assume the restriction mappings $\mathcal{F}(U) \to \mathcal{F}(U')$ are surjective for $U'$ an arbitrary union of opens of the form $U_{i_0 \ldots i_p}$. Then $\check{H}^p(\mathcal{U}, \mathcal{F})$ vanishes for $p > 0$. \end{lemma} \begin{proof} Let $Y$ be the set of nonempty subsets of $I$. We will use the letters $A, B, C, \ldots$ to denote elements of $Y$, i.e., nonempty subsets of $I$. For a finite nonempty subset $J \subset I$ let $$V_J = \{A \in Y \mid J \subset A\}$$ This means that $V_{\{i\}} = \{A \in Y \mid i \in A\}$ and $V_J = \bigcap_{j \in J} V_{\{j\}}$. Then $V_J \subset V_K$ if and only if $J \supset K$. There is a unique topology on $Y$ such that the collection of subsets $V_J$ is a basis for the topology on $Y$. Any open is of the form $$V = \bigcup\nolimits_{t \in T} V_{J_t}$$ for some family of finite subsets $J_t$. If $J_t \subset J_{t'}$ then we may remove $J_{t'}$ from the family without changing $V$. Thus we may assume there are no inclusions among the $J_t$. In this case the minimal elements of $V$ are the sets $A = J_t$. Hence we can read off the family $(J_t)_{t \in T}$ from the open $V$. \medskip\noindent We can completely understand open coverings in $Y$. First, because the elements $A \in Y$ are nonempty subsets of $I$ we have $$Y = \bigcup\nolimits_{i \in I} V_{\{i\}}$$ To understand other coverings, let $V$ be as above and let $V_s \subset Y$ be an open corresponding to the family $(J_{s, t})_{t \in T_s}$. Then $$V = \bigcup\nolimits_{s \in S} V_s$$ if and only if for each $t \in T$ there exists an $s \in S$ and $t_s \in T_s$ such that $J_t = J_{s, t_s}$. Namely, as the family $(J_t)_{t \in T}$ is minimal, the minimal element $A = J_t$ has to be in $V_s$ for some $s$, hence $A \in V_{J_{t_s}}$ for some $t_s \in T_s$. But since $A$ is also minimal in $V_s$ we conclude that $J_{t_s} = J_t$. \medskip\noindent Next we map the set of opens of $Y$ to opens of $X$. Namely, we send $Y$ to $U$, we use the rule $$V_J \mapsto U_J = \bigcap\nolimits_{i \in J} U_i$$ on the opens $V_J$, and we extend it to arbitrary opens $V$ by the rule $$V = \bigcup\nolimits_{t \in T} V_{J_t} \mapsto \bigcup\nolimits_{t \in T} U_{J_t}$$ The classification of open coverings of $Y$ given above shows that this rule transforms open coverings into open coverings. Thus we obtain an abelian sheaf $\mathcal{G}$ on $Y$ by setting $\mathcal{G}(Y) = \mathcal{F}(U)$ and for $V = \bigcup\nolimits_{t \in T} V_{J_t}$ setting $$\mathcal{G}(V) = \mathcal{F}\left(\bigcup\nolimits_{t \in T} U_{J_t}\right)$$ and using the restriction maps of $\mathcal{F}$. \medskip\noindent With these preliminaries out of the way we can prove our lemma as follows. We have an open covering $\mathcal{V} : Y = \bigcup_{i \in I} V_{\{i\}}$ of $Y$. By construction we have an equality $$\check{C}^\bullet(\mathcal{V}, \mathcal{G}) = \check{C}^\bullet(\mathcal{U}, \mathcal{F})$$ of {\v C}ech complexes. Since the sheaf $\mathcal{G}$ is flasque on $Y$ (by our assumption on $\mathcal{F}$ in the statement of the lemma) the vanishing follows from Lemma \ref{lemma-flasque-acyclic-cech}. \end{proof} \section{The Leray spectral sequence} \label{section-Leray} \begin{lemma} \label{lemma-before-Leray} Let $f : X \to Y$ be a morphism of ringed spaces. There is a commutative diagram $$\xymatrix{ D^{+}(X) \ar[rr]_-{R\Gamma(X, -)} \ar[d]_{Rf_*} & & D^{+}(\mathcal{O}_X(X)) \ar[d]^{\text{restriction}} \\ D^{+}(Y) \ar[rr]^-{R\Gamma(Y, -)} & & D^{+}(\mathcal{O}_Y(Y)) }$$ More generally for any $V \subset Y$ open and $U = f^{-1}(V)$ there is a commutative diagram $$\xymatrix{ D^{+}(X) \ar[rr]_-{R\Gamma(U, -)} \ar[d]_{Rf_*} & & D^{+}(\mathcal{O}_X(U)) \ar[d]^{\text{restriction}} \\ D^{+}(Y) \ar[rr]^-{R\Gamma(V, -)} & & D^{+}(\mathcal{O}_Y(V)) }$$ See also Remark \ref{remark-elucidate-lemma} for more explanation. \end{lemma} \begin{proof} Let $\Gamma_{res} : \textit{Mod}(\mathcal{O}_X) \to \text{Mod}_{\mathcal{O}_Y(Y)}$ be the functor which associates to an $\mathcal{O}_X$-module $\mathcal{F}$ the global sections of $\mathcal{F}$ viewed as a $\mathcal{O}_Y(Y)$-module via the map $f^\sharp : \mathcal{O}_Y(Y) \to \mathcal{O}_X(X)$. Let $restriction : \text{Mod}_{\mathcal{O}_X(X)} \to \text{Mod}_{\mathcal{O}_Y(Y)}$ be the restriction functor induced by $f^\sharp : \mathcal{O}_Y(Y) \to \mathcal{O}_X(X)$. Note that $restriction$ is exact so that its right derived functor is computed by simply applying the restriction functor, see Derived Categories, Lemma \ref{derived-lemma-right-derived-exact-functor}. It is clear that $$\Gamma_{res} = restriction \circ \Gamma(X, -) = \Gamma(Y, -) \circ f_*$$ We claim that Derived Categories, Lemma \ref{derived-lemma-compose-derived-functors} applies to both compositions. For the first this is clear by our remarks above. For the second, it follows from Lemma \ref{lemma-pushforward-injective} which implies that injective $\mathcal{O}_X$-modules are mapped to $\Gamma(Y, -)$-acyclic sheaves on $Y$. \end{proof} \begin{remark} \label{remark-elucidate-lemma} Here is a down-to-earth explanation of the meaning of Lemma \ref{lemma-before-Leray}. It says that given $f : X \to Y$ and $\mathcal{F} \in \textit{Mod}(\mathcal{O}_X)$ and given an injective resolution $\mathcal{F} \to \mathcal{I}^\bullet$ we have $$\begin{matrix} R\Gamma(X, \mathcal{F}) & \text{is represented by} & \Gamma(X, \mathcal{I}^\bullet) \\ Rf_*\mathcal{F} & \text{is represented by} & f_*\mathcal{I}^\bullet \\ R\Gamma(Y, Rf_*\mathcal{F}) & \text{is represented by} & \Gamma(Y, f_*\mathcal{I}^\bullet) \end{matrix}$$ the last fact coming from Leray's acyclicity lemma (Derived Categories, Lemma \ref{derived-lemma-leray-acyclicity}) and Lemma \ref{lemma-pushforward-injective}. Finally, it combines this with the trivial observation that $$\Gamma(X, \mathcal{I}^\bullet) = \Gamma(Y, f_*\mathcal{I}^\bullet).$$ to arrive at the commutativity of the diagram of the lemma. \end{remark} \begin{lemma} \label{lemma-modules-abelian} Let $X$ be a ringed space. Let $\mathcal{F}$ be an $\mathcal{O}_X$-module. \begin{enumerate} \item The cohomology groups $H^i(U, \mathcal{F})$ for $U \subset X$ open of $\mathcal{F}$ computed as an $\mathcal{O}_X$-module, or computed as an abelian sheaf are identical. \item Let $f : X \to Y$ be a morphism of ringed spaces. The higher direct images $R^if_*\mathcal{F}$ of $\mathcal{F}$ computed as an $\mathcal{O}_X$-module, or computed as an abelian sheaf are identical. \end{enumerate} There are similar statements in the case of bounded below complexes of $\mathcal{O}_X$-modules. \end{lemma} \begin{proof} Consider the morphism of ringed spaces $(X, \mathcal{O}_X) \to (X, \underline{\mathbf{Z}}_X)$ given by the identity on the underlying topological space and by the unique map of sheaves of rings $\underline{\mathbf{Z}}_X \to \mathcal{O}_X$. Let $\mathcal{F}$ be an $\mathcal{O}_X$-module. Denote $\mathcal{F}_{ab}$ the same sheaf seen as an $\underline{\mathbf{Z}}_X$-module, i.e., seen as a sheaf of abelian groups. Let $\mathcal{F} \to \mathcal{I}^\bullet$ be an injective resolution. By Remark \ref{remark-elucidate-lemma} we see that $\Gamma(X, \mathcal{I}^\bullet)$ computes both $R\Gamma(X, \mathcal{F})$ and $R\Gamma(X, \mathcal{F}_{ab})$. This proves (1). \medskip\noindent To prove (2) we use (1) and Lemma \ref{lemma-describe-higher-direct-images}. The result follows immediately. \end{proof} \begin{lemma}[Leray spectral sequence] \label{lemma-Leray} Let $f : X \to Y$ be a morphism of ringed spaces. Let $\mathcal{F}^\bullet$ be a bounded below complex of $\mathcal{O}_X$-modules. There is a spectral sequence $$E_2^{p, q} = H^p(Y, R^qf_*(\mathcal{F}^\bullet))$$ converging to $H^{p + q}(X, \mathcal{F}^\bullet)$. \end{lemma} \begin{proof} This is just the Grothendieck spectral sequence Derived Categories, Lemma \ref{derived-lemma-grothendieck-spectral-sequence} coming from the composition of functors $\Gamma_{res} = \Gamma(Y, -) \circ f_*$ where $\Gamma_{res}$ is as in the proof of Lemma \ref{lemma-before-Leray}. To see that the assumptions of Derived Categories, Lemma \ref{derived-lemma-grothendieck-spectral-sequence} are satisfied, see the proof of Lemma \ref{lemma-before-Leray} or Remark \ref{remark-elucidate-lemma}. \end{proof} \begin{remark} \label{remark-Leray-ss-more-structure} The Leray spectral sequence, the way we proved it in Lemma \ref{lemma-Leray} is a spectral sequence of $\Gamma(Y, \mathcal{O}_Y)$-modules. However, it is quite easy to see that it is in fact a spectral sequence of $\Gamma(X, \mathcal{O}_X)$-modules. For example $f$ gives rise to a morphism of ringed spaces $f' : (X, \mathcal{O}_X) \to (Y, f_*\mathcal{O}_X)$. By Lemma \ref{lemma-modules-abelian} the terms $E_r^{p, q}$ of the Leray spectral sequence for an $\mathcal{O}_X$-module $\mathcal{F}$ and $f$ are identical with those for $\mathcal{F}$ and $f'$ at least for $r \geq 2$. Namely, they both agree with the terms of the Leray spectral sequence for $\mathcal{F}$ as an abelian sheaf. And since $(f_*\mathcal{O}_X)(Y) = \mathcal{O}_X(X)$ we see the result. It is often the case that the Leray spectral sequence carries additional structure. \end{remark} \begin{lemma} \label{lemma-apply-Leray} Let $f : X \to Y$ be a morphism of ringed spaces. Let $\mathcal{F}$ be an $\mathcal{O}_X$-module. \begin{enumerate} \item If $R^qf_*\mathcal{F} = 0$ for $q > 0$, then $H^p(X, \mathcal{F}) = H^p(Y, f_*\mathcal{F})$ for all $p$. \item If $H^p(Y, R^qf_*\mathcal{F}) = 0$ for all $q$ and $p > 0$, then $H^q(X, \mathcal{F}) = H^0(Y, R^qf_*\mathcal{F})$ for all $q$. \end{enumerate} \end{lemma} \begin{proof} These are two simple conditions that force the Leray spectral sequence to degenerate at $E_2$. You can also prove these facts directly (without using the spectral sequence) which is a good exercise in cohomology of sheaves. \end{proof} \begin{lemma} \label{lemma-higher-direct-images-compose} \begin{slogan} The total derived functor of a composition is the composition of the total derived functors. \end{slogan} Let $f : X \to Y$ and $g : Y \to Z$ be morphisms of ringed spaces. In this case $Rg_* \circ Rf_* = R(g \circ f)_*$ as functors from $D^{+}(X) \to D^{+}(Z)$. \end{lemma} \begin{proof} We are going to apply Derived Categories, Lemma \ref{derived-lemma-compose-derived-functors}. It is clear that $g_* \circ f_* = (g \circ f)_*$, see Sheaves, Lemma \ref{sheaves-lemma-pushforward-composition}. It remains to show that $f_*\mathcal{I}$ is $g_*$-acyclic. This follows from Lemma \ref{lemma-pushforward-injective} and the description of the higher direct images $R^ig_*$ in Lemma \ref{lemma-describe-higher-direct-images}. \end{proof} \begin{lemma}[Relative Leray spectral sequence] \label{lemma-relative-Leray} Let $f : X \to Y$ and $g : Y \to Z$ be morphisms of ringed spaces. Let $\mathcal{F}$ be an $\mathcal{O}_X$-module. There is a spectral sequence with $$E_2^{p, q} = R^pg_*(R^qf_*\mathcal{F})$$ converging to $R^{p + q}(g \circ f)_*\mathcal{F}$. This spectral sequence is functorial in $\mathcal{F}$, and there is a version for bounded below complexes of $\mathcal{O}_X$-modules. \end{lemma} \begin{proof} This is a Grothendieck spectral sequence for composition of functors and follows from Lemma \ref{lemma-higher-direct-images-compose} and Derived Categories, Lemma \ref{derived-lemma-grothendieck-spectral-sequence}. \end{proof} \section{Functoriality of cohomology} \label{section-functoriality} \begin{lemma} \label{lemma-functoriality} Let $f : X \to Y$ be a morphism of ringed spaces. Let $\mathcal{G}^\bullet$, resp.\ $\mathcal{F}^\bullet$ be a bounded below complex of $\mathcal{O}_Y$-modules, resp.\ $\mathcal{O}_X$-modules. Let $\varphi : \mathcal{G}^\bullet \to f_*\mathcal{F}^\bullet$ be a morphism of complexes. There is a canonical morphism $$\mathcal{G}^\bullet \longrightarrow Rf_*(\mathcal{F}^\bullet)$$ in $D^{+}(Y)$. Moreover this construction is functorial in the triple $(\mathcal{G}^\bullet, \mathcal{F}^\bullet, \varphi)$. \end{lemma} \begin{proof} Choose an injective resolution $\mathcal{F}^\bullet \to \mathcal{I}^\bullet$. By definition $Rf_*(\mathcal{F}^\bullet)$ is represented by $f_*\mathcal{I}^\bullet$ in $K^{+}(\mathcal{O}_Y)$. The composition $$\mathcal{G}^\bullet \to f_*\mathcal{F}^\bullet \to f_*\mathcal{I}^\bullet$$ is a morphism in $K^{+}(Y)$ which turns into the morphism of the lemma upon applying the localization functor $j_Y : K^{+}(Y) \to D^{+}(Y)$. \end{proof} \noindent Let $f : X \to Y$ be a morphism of ringed spaces. Let $\mathcal{G}$ be an $\mathcal{O}_Y$-module and let $\mathcal{F}$ be an $\mathcal{O}_X$-module. Recall that an $f$-map $\varphi$ from $\mathcal{G}$ to $\mathcal{F}$ is a map $\varphi : \mathcal{G} \to f_*\mathcal{F}$, or what is the same thing, a map $\varphi : f^*\mathcal{G} \to \mathcal{F}$. See Sheaves, Definition \ref{sheaves-definition-f-map}. Such an $f$-map gives rise to a morphism of complexes \begin{equation} \label{equation-functorial-derived} \varphi : R\Gamma(Y, \mathcal{G}) \longrightarrow R\Gamma(X, \mathcal{F}) \end{equation} in $D^{+}(\mathcal{O}_Y(Y))$. Namely, we use the morphism $\mathcal{G} \to Rf_*\mathcal{F}$ in $D^{+}(Y)$ of Lemma \ref{lemma-functoriality}, and we apply $R\Gamma(Y, -)$. By Lemma \ref{lemma-before-Leray} we see that $R\Gamma(X, \mathcal{F}) = R\Gamma(Y, Rf_*\mathcal{F})$ and we get the displayed arrow. We spell this out completely in Remark \ref{remark-explain-arrow} below. In particular it gives rise to maps on cohomology \begin{equation} \label{equation-functorial} \varphi : H^i(Y, \mathcal{G}) \longrightarrow H^i(X, \mathcal{F}). \end{equation} \begin{remark} \label{remark-explain-arrow} Let $f : X \to Y$ be a morphism of ringed spaces. Let $\mathcal{G}$ be an $\mathcal{O}_Y$-module. Let $\mathcal{F}$ be an $\mathcal{O}_X$-module. Let $\varphi$ be an $f$-map from $\mathcal{G}$ to $\mathcal{F}$. Choose a resolution $\mathcal{F} \to \mathcal{I}^\bullet$ by a complex of injective $\mathcal{O}_X$-modules. Choose resolutions $\mathcal{G} \to \mathcal{J}^\bullet$ and $f_*\mathcal{I}^\bullet \to (\mathcal{J}')^\bullet$ by complexes of injective $\mathcal{O}_Y$-modules. By Derived Categories, Lemma \ref{derived-lemma-morphisms-lift} there exists a map of complexes $\beta$ such that the diagram \begin{equation} \label{equation-choice} \xymatrix{ \mathcal{G} \ar[d] \ar[r] & f_*\mathcal{F} \ar[r] & f_*\mathcal{I}^\bullet \ar[d] \\ \mathcal{J}^\bullet \ar[rr]^\beta & & (\mathcal{J}')^\bullet } \end{equation} commutes. Applying global section functors we see that we get a diagram $$\xymatrix{ & & \Gamma(Y, f_*\mathcal{I}^\bullet) \ar[d]_{qis} \ar@{=}[r] & \Gamma(X, \mathcal{I}^\bullet) \\ \Gamma(Y, \mathcal{J}^\bullet) \ar[rr]^\beta & & \Gamma(Y, (\mathcal{J}')^\bullet) & }$$ The complex on the bottom left represents $R\Gamma(Y, \mathcal{G})$ and the complex on the top right represents $R\Gamma(X, \mathcal{F})$. The vertical arrow is a quasi-isomorphism by Lemma \ref{lemma-before-Leray} which becomes invertible after applying the localization functor $K^{+}(\mathcal{O}_Y(Y)) \to D^{+}(\mathcal{O}_Y(Y))$. The arrow (\ref{equation-functorial-derived}) is given by the composition of the horizontal map by the inverse of the vertical map. \end{remark} \section{Refinements and {\v C}ech cohomology} \label{section-refinements-cech} \noindent Let $(X, \mathcal{O}_X)$ be a ringed space. Let $\mathcal{U} : X = \bigcup_{i \in I} U_i$ and $\mathcal{V} : X = \bigcup_{j \in J} V_j$ be open coverings. Assume that $\mathcal{U}$ is a refinement of $\mathcal{V}$. Choose a map $c : I \to J$ such that $U_i \subset V_{c(i)}$ for all $i \in I$. This induces a map of {\v C}ech complexes $$\gamma : \check{\mathcal{C}}^\bullet(\mathcal{V}, \mathcal{F}) \longrightarrow \check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{F}), \quad (\xi_{j_0 \ldots j_p}) \longmapsto (\xi_{c(i_0) \ldots c(i_p)}|_{U_{i_0 \ldots i_p}})$$ functorial in the sheaf of $\mathcal{O}_X$-modules $\mathcal{F}$. Suppose that $c' : I \to J$ is a second map such that $U_i \subset V_{c'(i)}$ for all $i \in I$. Then the corresponding maps $\gamma$ and $\gamma'$ are homotopic. Namely, $\gamma - \gamma' = \text{d} \circ h + h \circ \text{d}$ with $h : \check{\mathcal{C}}^{p + 1}(\mathcal{V}, \mathcal{F}) \to \check{\mathcal{C}}^p(\mathcal{U}, \mathcal{F})$ given by the rule $$h(\xi)_{i_0 \ldots i_p} = \sum\nolimits_{a = 0}^{p} (-1)^a \alpha_{c(i_0)\ldots c(i_a) c'(i_a) \ldots c'(i_p)}$$ We omit the computation showing this works; please see the discussion following (\ref{equation-transformation}) for the proof in a more general case. In particular, the map on {\v C}ech cohomology groups is independent of the choice of $c$. Moreover, it is clear that if $\mathcal{W} : X = \bigcup_{k \in K} W_k$ is a third open covering and $\mathcal{V}$ is a refinement of $\mathcal{W}$, then the composition of the maps $$\check{\mathcal{C}}^\bullet(\mathcal{W}, \mathcal{F}) \longrightarrow \check{\mathcal{C}}^\bullet(\mathcal{V}, \mathcal{F}) \longrightarrow \check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{F})$$ associated to maps $I \to J$ and $J \to K$ is the map associated to the composition $I \to K$. In particular, we can define the {\v C}ech cohomology groups $$\check{H}^p(X, \mathcal{F}) = \colim_\mathcal{U} \check{H}^p(\mathcal{U}, \mathcal{F})$$ where the colimit is over all open coverings of $X$ preordered by refinement. \medskip\noindent It turns out that the maps $\gamma$ defined above are compatible with the map to cohomology, in other words, the composition $$\check{H}^p(\mathcal{V}, \mathcal{F}) \to \check{H}^p(\mathcal{U}, \mathcal{F}) \xrightarrow{\text{Lemma \ref{lemma-cech-cohomology}}} H^p(X, \mathcal{F})$$ is the canonical map from the first group to cohomology of Lemma \ref{lemma-cech-cohomology}. In the lemma below we will prove this in a slightly more general setting. A consequence is that we obtain a well defined map \begin{equation} \label{equation-cech-to-cohomology} \check{H}^p(X, \mathcal{F}) = \colim_\mathcal{U} \check{H}^p(\mathcal{U}, \mathcal{F}) \longrightarrow H^p(X, \mathcal{F}) \end{equation} from {\v C}ech cohomology to cohomology. \begin{lemma} \label{lemma-functoriality-cech} Let $f : X \to Y$ be a morphism of ringed spaces. Let $\varphi : f^*\mathcal{G} \to \mathcal{F}$ be an $f$-map from an $\mathcal{O}_Y$-module $\mathcal{G}$ to an $\mathcal{O}_X$-module $\mathcal{F}$. Let $\mathcal{U} : X = \bigcup_{i \in I} U_i$ and $\mathcal{V} : Y = \bigcup_{j \in J} V_j$ be open coverings. Assume that $\mathcal{U}$ is a refinement of $f^{-1}\mathcal{V} : X = \bigcup_{j \in J} f^{-1}(V_j)$. In this case there exists a commutative diagram $$\xymatrix{ \check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{F}) \ar[r] & R\Gamma(X, \mathcal{F}) \\ \check{\mathcal{C}}^\bullet(\mathcal{V}, \mathcal{G}) \ar[r] \ar[u]^\gamma & R\Gamma(Y, \mathcal{G}) \ar[u] }$$ in $D^{+}(\mathcal{O}_X(X))$ with horizontal arrows given by Lemma \ref{lemma-cech-cohomology} and right vertical arrow by (\ref{equation-functorial-derived}). In particular we get commutative diagrams of cohomology groups $$\xymatrix{ \check{H}^p(\mathcal{U}, \mathcal{F}) \ar[r] & H^p(X, \mathcal{F}) \\ \check{H}^p(\mathcal{V}, \mathcal{G}) \ar[r] \ar[u]^\gamma & H^p(Y, \mathcal{G}) \ar[u] }$$ where the right vertical arrow is (\ref{equation-functorial}) \end{lemma} \begin{proof} We first define the left vertical arrow. Namely, choose a map $c : I \to J$ such that $U_i \subset f^{-1}(V_{c(i)})$ for all $i \in I$. In degree $p$ we define the map by the rule $$\gamma(s)_{i_0 \ldots i_p} = \varphi(s)_{c(i_0) \ldots c(i_p)}$$ This makes sense because $\varphi$ does indeed induce maps $\mathcal{G}(V_{c(i_0) \ldots c(i_p)}) \to \mathcal{F}(U_{i_0 \ldots i_p})$ by assumption. It is also clear that this defines a morphism of complexes. Choose injective resolutions $\mathcal{F} \to \mathcal{I}^\bullet$ on $X$ and $\mathcal{G} \to J^\bullet$ on $Y$. According to the proof of Lemma \ref{lemma-cech-cohomology} we introduce the double complexes $A^{\bullet, \bullet}$ and $B^{\bullet, \bullet}$ with terms $$B^{p, q} = \check{\mathcal{C}}^p(\mathcal{V}, \mathcal{J}^q) \quad \text{and} \quad A^{p, q} = \check{\mathcal{C}}^p(\mathcal{U}, \mathcal{I}^q).$$ As in Remark \ref{remark-explain-arrow} above we also choose an injective resolution $f_*\mathcal{I} \to (\mathcal{J}')^\bullet$ on $Y$ and a morphism of complexes $\beta : \mathcal{J} \to (\mathcal{J}')^\bullet$ making (\ref{equation-choice}) commutes. We introduce some more double complexes, namely $(B')^{\bullet, \bullet}$ and $(B''){\bullet, \bullet}$ with $$(B')^{p, q} = \check{\mathcal{C}}^p(\mathcal{V}, (\mathcal{J}')^q) \quad \text{and} \quad (B'')^{p, q} = \check{\mathcal{C}}^p(\mathcal{V}, f_*\mathcal{I}^q).$$ Note that there is an $f$-map of complexes from $f_*\mathcal{I}^\bullet$ to $\mathcal{I}^\bullet$. Hence it is clear that the same rule as above defines a morphism of double complexes $$\gamma : (B'')^{\bullet, \bullet} \longrightarrow A^{\bullet, \bullet}.$$ Consider the diagram of complexes $$\xymatrix{ \check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{F}) \ar[r] & sA^\bullet & & & \Gamma(X, \mathcal{I}^\bullet) \ar[lll]^{qis} \ar@{=}[ddl]\\ \check{\mathcal{C}}^\bullet(\mathcal{V}, \mathcal{G}) \ar[r] \ar[u]^\gamma & sB^\bullet \ar[r]^\beta & s(B')^\bullet & s(B'')^\bullet \ar[l] \ar[llu]_{s\gamma} \\ & \Gamma(Y, \mathcal{J}^\bullet) \ar[u]^{qis} \ar[r]^\beta & \Gamma(Y, (\mathcal{J}')^\bullet) \ar[u] & \Gamma(Y, f_*\mathcal{I}^\bullet) \ar[u] \ar[l]_{qis} }$$ The two horizontal arrows with targets $sA^\bullet$ and $sB^\bullet$ are the ones explained in Lemma \ref{lemma-cech-cohomology}. The left upper shape (a pentagon) is commutative simply because (\ref{equation-choice}) is commutative. The two lower squares are trivially commutative. It is also immediate from the definitions that the right upper shape (a square) is commutative. The result of the lemma now follows from the definitions and the fact that going around the diagram on the outer sides from $\check{\mathcal{C}}^\bullet(\mathcal{V}, \mathcal{G})$ to $\Gamma(X, \mathcal{I}^\bullet)$ either on top or on bottom is the same (where you have to invert any quasi-isomorphisms along the way). \end{proof} \section{Cohomology on Hausdorff quasi-compact spaces} \label{section-cohomology-LC} \noindent For such a space {\v C}ech cohomology agrees with cohomology. \begin{lemma} \label{lemma-cech-always} Let $X$ be a topological space. Let $\mathcal{F}$ be an abelian sheaf. Then the map $\check{H}^1(X, \mathcal{F}) \to H^1(X, \mathcal{F})$ defined in (\ref{equation-cech-to-cohomology}) is an isomorphism. \end{lemma} \begin{proof} Let $\mathcal{U}$ be an open covering of $X$. By Lemma \ref{lemma-cech-spectral-sequence} there is an exact sequence $$0 \to \check{H}^1(\mathcal{U}, \mathcal{F}) \to H^1(X, \mathcal{F}) \to \check{H}^0(\mathcal{U}, \underline{H}^1(\mathcal{F}))$$ Thus the map is injective. To show surjectivity it suffices to show that any element of $\check{H}^0(\mathcal{U}, \underline{H}^1(\mathcal{F}))$ maps to zero after replacing $\mathcal{U}$ by a refinement. This is immediate from the definitions and the fact that $\underline{H}^1(\mathcal{F})$ is a presheaf of abelian groups whose sheafification is zero by locality of cohomology, see Lemma \ref{lemma-kill-cohomology-class-on-covering}. \end{proof} \begin{lemma} \label{lemma-cech-Hausdorff-quasi-compact} Let $X$ be a Hausdorff and quasi-compact topological space. Let $\mathcal{F}$ be an abelian sheaf on $X$. Then the map $\check{H}^n(X, \mathcal{F}) \to H^n(X, \mathcal{F})$ defined in (\ref{equation-cech-to-cohomology}) is an isomorphism for all $n$. \end{lemma} \begin{proof} We already know that $\check{H}^n(X, -) \to H^p(X, -)$ is an isomorphism of functors for $n = 0, 1$, see Lemma \ref{lemma-cech-always}. The functors $H^n(X, -)$ form a universal $\delta$-functor, see Derived Categories, Lemma \ref{derived-lemma-higher-derived-functors}. If we show that $\check{H}^n(X, -)$ forms a universal $\delta$-functor and that $\check{H}^n(X, -) \to H^n(X, -)$ is compatible with boundary maps, then the map will automatically be an isomorphism by uniqueness of universal $\delta$-functors, see Homology, Lemma \ref{homology-lemma-uniqueness-universal-delta-functor}. \medskip\noindent Let $0 \to \mathcal{F} \to \mathcal{G} \to \mathcal{H} \to 0$ be a short exact sequence of abelian sheaves on $X$. Let $\mathcal{U} : X = \bigcup_{i \in I} U_i$ be an open covering. This gives a complex of complexes $$0 \to \check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{F}) \to \check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{G}) \to \check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{H}) \to 0$$ which is in general not exact on the right. The sequence defines the maps $$\check{H}^n(\mathcal{U}, \mathcal{F}) \to \check{H}^n(\mathcal{U}, \mathcal{G}) \to \check{H}^n(\mathcal{U}, \mathcal{H})$$ but isn't good enough to define a boundary operator $\delta : \check{H}^n(\mathcal{U}, \mathcal{H}) \to \check{H}^{n + 1}(\mathcal{U}, \mathcal{F})$. Indeed such a thing will not exist in general. However, given an element $\overline{h} \in \check{H}^n(\mathcal{U}, \mathcal{H})$ which is the cohomology class of a cocycle $h = (h_{i_0 \ldots i_n})$ we can choose open coverings $$U_{i_0 \ldots i_n} = \bigcup W_{i_0 \ldots i_n, k}$$ such that $h_{i_0 \ldots i_n}|_{W_{i_0 \ldots i_n, k}}$ lifts to a section of $\mathcal{G}$ over $W_{i_0 \ldots i_n, k}$. By Topology, Lemma \ref{topology-lemma-refine-covering} we can choose an open covering $\mathcal{V} : X = \bigcup_{j \in J} V_j$ and $\alpha : J \to I$ such that $V_j \subset U_{\alpha(j)}$ (it is a refinement) and such that for all $j_0, \ldots, j_n \in J$ there is a $k$ such that $V_{j_0 \ldots j_n} \subset W_{\alpha(j_0) \ldots \alpha(j_n), k}$. We obtain maps of complexes $$\xymatrix{ 0 \ar[r] & \check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{F}) \ar[d] \ar[r] & \check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{G}) \ar[d] \ar[r] & \check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{H}) \ar[d] \ar[r] & 0 \\ 0 \ar[r] & \check{\mathcal{C}}^\bullet(\mathcal{V}, \mathcal{F}) \ar[r] & \check{\mathcal{C}}^\bullet(\mathcal{V}, \mathcal{G}) \ar[r] & \check{\mathcal{C}}^\bullet(\mathcal{V}, \mathcal{H}) \ar[r] & 0 }$$ In fact, the vertical arrows are the maps of complexes used to define the transition maps between the {\v C}ech cohomology groups. Our choice of refinement shows that we may choose $$g_{j_0 \ldots j_n} \in \mathcal{G}(V_{j_0 \ldots j_n}),\quad g_{j_0 \ldots j_n} \longmapsto h_{\alpha(j_0) \ldots \alpha(j_n)}|_{V_{j_0 \ldots j_n}}$$ The cochain $g = (g_{j_0 \ldots j_n})$ is not a cocycle in general but we know that its {\v C}ech boundary $\text{d}(g)$ maps to zero in $\check{\mathcal{C}}^{n + 1}(\mathcal{V}, \mathcal{H})$ (by the commutative diagram above and the fact that $h$ is a cocycle). Hence $\text{d}(g)$ is a cocycle in $\check{\mathcal{C}}^\bullet(\mathcal{V}, \mathcal{F})$. This allows us to define $$\delta(\overline{h}) = \text{class of }\text{d}(g)\text{ in } \check{H}^{n + 1}(\mathcal{V}, \mathcal{F})$$ Now, given an element $\xi \in \check{H}^n(X, \mathcal{G})$ we choose an open covering $\mathcal{U}$ and an element $\overline{h} \in \check{H}^n(\mathcal{U}, \mathcal{G})$ mapping to $\xi$ in the colimit defining {\v C}ech cohomology. Then we choose $\mathcal{V}$ and $g$ as above and set $\delta(\xi)$ equal to the image of $\delta(\overline{h})$ in $\check{H}^n(X, \mathcal{F})$. At this point a lot of properties have to be checked, all of which are straightforward. For example, we need to check that our construction is independent of the choice of $\mathcal{U}, \overline{h}, \mathcal{V}, \alpha : J \to I, g$. The class of $\text{d}(g)$ is independent of the choice of the lifts $g_{i_0 \ldots i_n}$ because the difference will be a coboundary. Independence of $\alpha$ holds\footnote{This is an important check because the nonuniqueness of $\alpha$ is the only thing preventing us from taking the colimit of {\v C}ech complexes over all open coverings of $X$ to get a short exact sequence of complexes computing {\v C}ech cohomology.} because a different choice of $\alpha$ determines homotopic vertical maps of complexes in the diagram above, see Section \ref{section-refinements-cech}. For the other choices we use that given a finite collection of coverings of $X$ we can always find a covering refining all of them. We also need to check additivity which is shown in the same manner. Finally, we need to check that the maps $\check{H}^n(X, -) \to H^n(X, -)$ are compatible with boundary maps. To do this we choose injective resolutions $$\xymatrix{ 0 \ar[r] & \mathcal{F} \ar[r] \ar[d] & \mathcal{G} \ar[r] \ar[d] & \mathcal{H} \ar[r] \ar[d] & 0 \\ 0 \ar[r] & \mathcal{I}_1^\bullet \ar[r] & \mathcal{I}_2^\bullet \ar[r] & \mathcal{I}_3^\bullet \ar[r] & 0 }$$ as in Derived Categories, Lemma \ref{derived-lemma-injective-resolution-ses}. This will give a commutative diagram $$\xymatrix{ 0 \ar[r] & \check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{F}) \ar[r] \ar[d] & \check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{F}) \ar[r] \ar[d] & \check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{F}) \ar[r] \ar[d] & 0 \\ 0 \ar[r] & \text{Tot}(\check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{I}_1^\bullet)) \ar[r] & \text{Tot}(\check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{I}_2^\bullet)) \ar[r] & \text{Tot}(\check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{I}_3^\bullet)) \ar[r] & 0 }$$ Here $\mathcal{U}$ is an open covering as above and the vertical maps are those used to define the maps $\check{H}^n(\mathcal{U}, -) \to H^n(X, -)$, see Lemma \ref{lemma-cech-cohomology}. The bottom complex is exact as the sequence of complexes of injectives is termwise split exact. Hence the boundary map in cohomology is computed by the usual procedure for this lower exact sequence, see Homology, Lemma \ref{homology-lemma-long-exact-sequence-cochain}. The same will be true after passing to the refinement $\mathcal{V}$ where the boundary map for {\v C}ech cohomology was defined. Hence the boundary maps agree because they use the same construction (whenever the first one is defined on an element in {\v C}ech cohomology on a given covering). This finishes our discussion of the construction of the structure of a $\delta$-functor on {\v C}ech cohomology and why this structure is compatible with the given $\delta$-functor structure on usual cohomology. \medskip\noindent Finally, we may apply Lemma \ref{lemma-injective-trivial-cech} to see that higher {\v C}ech cohomology is trivial on injective sheaves. Hence we see that {\v C}ech cohomology is a universal $\delta$-functor by Homology, Lemma \ref{homology-lemma-efface-implies-universal}. \end{proof} \begin{lemma} \label{lemma-cohomology-of-closed} \begin{reference} \cite[Expose V bis, 4.1.3]{SGA4} \end{reference} Let $X$ be a topological space. Let $Z \subset X$ be a quasi-compact subset such that any two points of $Z$ have disjoint open neighbourhoods in $X$. For every abelian sheaf $\mathcal{F}$ on $X$ the canonical map $$\colim H^p(U, \mathcal{F}) \longrightarrow H^p(Z, \mathcal{F}|_Z)$$ where the colimit is over open neighbourhoods $U$ of $Z$ in $X$ is an isomorphism. \end{lemma} \begin{proof} We first prove this for $p = 0$. Injectivity follows from the definition of $\mathcal{F}|_Z$ and holds in general (for any subset of any topological space $X$). Next, suppose that $s \in H^0(Z, \mathcal{F}|_Z)$. Then we can find opens $U_i \subset X$ such that $Z \subset \bigcup U_i$ and such that $s|_{Z \cap U_i}$ comes from $s_i \in \mathcal{F}(U_i)$. It follows that there exist opens $W_{ij} \subset U_i \cap U_j$ with $W_{ij} \cap Z = U_i \cap U_j \cap Z$ such that $s_i|_{W_{ij}} = s_j|_{W_{ij}}$. Applying Topology, Lemma \ref{topology-lemma-lift-covering-of-quasi-compact-hausdorff-subset} we find opens $V_i$ of $X$ such that $V_i \subset U_i$ and such that $V_i \cap V_j \subset W_{ij}$. Hence we see that $s_i|_{V_i}$ glue to a section of $\mathcal{F}$ over the open neighbourhood $\bigcup V_i$ of $Z$. \medskip\noindent To finish the proof, it suffices to show that if $\mathcal{I}$ is an injective abelian sheaf on $X$, then $H^p(Z, \mathcal{I}|_Z) = 0$ for $p > 0$. This follows using short exact sequences and dimension shifting; details omitted. Thus, suppose $\overline{\xi}$ is an element of $H^p(Z, \mathcal{I}|_Z)$ for some $p > 0$. By Lemma \ref{lemma-cech-Hausdorff-quasi-compact} the element $\overline{\xi}$ comes from $\check{H}^p(\mathcal{V}, \mathcal{I}|_Z)$ for some open covering $\mathcal{V} : Z = \bigcup V_i$ of $Z$. Say $\overline{\xi}$ is the image of the class of a cocycle $\xi = (\xi_{i_0 \ldots i_p})$ in $\check{\mathcal{C}}^p(\mathcal{V}, \mathcal{I}|_Z)$. \medskip\noindent Let $\mathcal{I}' \subset \mathcal{I}|_Z$ be the subpresheaf defined by the rule $$\mathcal{I}'(V) = \{s \in \mathcal{I}|_Z(V) \mid \exists (U, t),\ U \subset X\text{ open}, \ t \in \mathcal{I}(U),\ V = Z \cap U,\ s = t|_{Z \cap U} \}$$ Then $\mathcal{I}|_Z$ is the sheafification of $\mathcal{I}'$. Thus for every $(p + 1)$-tuple $i_0 \ldots i_p$ we can find an open covering $V_{i_0 \ldots i_p} = \bigcup W_{i_0 \ldots i_p, k}$ such that $\xi_{i_0 \ldots i_p}|_{W_{i_0 \ldots i_p, k}}$ is a section of $\mathcal{I}'$. Applying Topology, Lemma \ref{topology-lemma-refine-covering} we may after refining $\mathcal{V}$ assume that each $\xi_{i_0 \ldots i_p}$ is a section of the presheaf $\mathcal{I}'$. \medskip\noindent Write $V_i = Z \cap U_i$ for some opens $U_i \subset X$. Since $\mathcal{I}$ is flasque (Lemma \ref{lemma-injective-flasque}) and since $\xi_{i_0 \ldots i_p}$ is a section of $\mathcal{I}'$ for every $(p + 1)$-tuple $i_0 \ldots i_p$ we can choose a section $s_{i_0 \ldots i_p} \in \mathcal{I}(U_{i_0 \ldots i_p})$ which restricts to $\xi_{i_0 \ldots i_p}$ on $V_{i_0 \ldots i_p} = Z \cap U_{i_0 \ldots i_p}$. (This appeal to injectives being flasque can be avoided by an additional application of Topology, Lemma \ref{topology-lemma-lift-covering-of-quasi-compact-hausdorff-subset}.) Let $s = (s_{i_0 \ldots i_p})$ be the corresponding cochain for the open covering $U = \bigcup U_i$. Since $\text{d}(\xi) = 0$ we see that the sections $\text{d}(s)_{i_0 \ldots i_{p + 1}}$ restrict to zero on $Z \cap U_{i_0 \ldots i_{p + 1}}$. Hence, by the initial remarks of the proof, there exists open subsets $W_{i_0 \ldots i_{p + 1}} \subset U_{i_0 \ldots i_{p + 1}}$ with $Z \cap W_{i_0 \ldots i_{p + 1}} = Z \cap U_{i_0 \ldots i_{p + 1}}$ such that $\text{d}(s)_{i_0 \ldots i_{p + 1}}|_{W_{i_0 \ldots i_{p + 1}}} = 0$. By Topology, Lemma \ref{topology-lemma-lift-covering-of-quasi-compact-hausdorff-subset} we can find $U'_i \subset U_i$ such that $Z \subset \bigcup U'_i$ and such that $U'_{i_0 \ldots i_{p + 1}} \subset W_{i_0 \ldots i_{p + 1}}$. Then $s' = (s'_{i_0 \ldots i_p})$ with $s'_{i_0 \ldots i_p} = s_{i_0 \ldots i_p}|_{U'_{i_0 \ldots i_p}}$ is a cocycle for $\mathcal{I}$ for the open covering $U' = \bigcup U'_i$ of an open neighbourhood of $Z$. Since $\mathcal{I}$ has trivial higher {\v C}ech cohomology groups (Lemma \ref{lemma-injective-trivial-cech}) we conclude that $s'$ is a coboundary. It follows that the image of $\xi$ in the {\v C}ech complex for the open covering $Z = \bigcup Z \cap U'_i$ is a coboundary and we are done. \end{proof} \section{The base change map} \label{section-base-change-map} \noindent We will need to know how to construct the base change map in some cases. Since we have not yet discussed derived pullback we only discuss this in the case of a base change by a flat morphism of ringed spaces. Before we state the result, let us discuss flat pullback on the derived category. Namely, suppose that $g : X \to Y$ is a flat morphism of ringed spaces. By Modules, Lemma \ref{modules-lemma-pullback-flat} the functor $g^* : \textit{Mod}(\mathcal{O}_Y) \to \textit{Mod}(\mathcal{O}_X)$ is exact. Hence it has a derived functor $$g^* : D^{+}(Y) \to D^{+}(X)$$ which is computed by simply pulling back an representative of a given object in $D^{+}(Y)$, see Derived Categories, Lemma \ref{derived-lemma-right-derived-exact-functor}. Hence as indicated we indicate this functor by $g^*$ rather than $Lg^*$. \begin{lemma} \label{lemma-base-change-map-flat-case} Let $$\xymatrix{ X' \ar[r]_{g'} \ar[d]_{f'} & X \ar[d]^f \\ S' \ar[r]^g & S }$$ be a commutative diagram of ringed spaces. Let $\mathcal{F}^\bullet$ be a bounded below complex of $\mathcal{O}_X$-modules. Assume both $g$ and $g'$ are flat. Then there exists a canonical base change map $$g^*Rf_*\mathcal{F}^\bullet \longrightarrow R(f')_*(g')^*\mathcal{F}^\bullet$$ in $D^{+}(S')$. \end{lemma} \begin{proof} Choose injective resolutions $\mathcal{F}^\bullet \to \mathcal{I}^\bullet$ and $(g')^*\mathcal{F}^\bullet \to \mathcal{J}^\bullet$. By Lemma \ref{lemma-pushforward-injective-flat} we see that $(g')_*\mathcal{J}^\bullet$ is a complex of injectives representing $R(g')_*(g')^*\mathcal{F}^\bullet$. Hence by Derived Categories, Lemmas \ref{derived-lemma-morphisms-lift} and \ref{derived-lemma-morphisms-equal-up-to-homotopy} the arrow $\beta$ in the diagram $$\xymatrix{ (g')_*(g')^*\mathcal{F}^\bullet \ar[r] & (g')_*\mathcal{J}^\bullet \\ \mathcal{F}^\bullet \ar[u]^{adjunction} \ar[r] & \mathcal{I}^\bullet \ar[u]_\beta }$$ exists and is unique up to homotopy. Pushing down to $S$ we get $$f_*\beta : f_*\mathcal{I}^\bullet \longrightarrow f_*(g')_*\mathcal{J}^\bullet = g_*(f')_*\mathcal{J}^\bullet$$ By adjunction of $g^*$ and $g_*$ we get a map of complexes $g^*f_*\mathcal{I}^\bullet \to (f')_*\mathcal{J}^\bullet$. Note that this map is unique up to homotopy since the only choice in the whole process was the choice of the map $\beta$ and everything was done on the level of complexes. \end{proof} \begin{remark} \label{remark-correct-version-base-change-map} The correct'' version of the base change map is map $$Lg^* Rf_* \mathcal{F}^\bullet \longrightarrow R(f')_* L(g')^*\mathcal{F}^\bullet.$$ The construction of this map involves unbounded complexes, see Remark \ref{remark-base-change}. \end{remark} \section{Proper base change in topology} \label{section-proper-base-change} \noindent In this section we prove a very general version of the proper base change theorem in topology. It tells us that the stalks of the higher direct images $R^pf_*$ can be computed on the fibre. \begin{lemma} \label{lemma-proper-base-change} Let $f : (X, \mathcal{O}_X) \to (Y, \mathcal{O}_Y)$ be a morphism of ringed spaces. Let $y \in Y$. Assume that \begin{enumerate} \item $f$ is closed, \item $f$ is separated, and \item $f^{-1}(y)$ is quasi-compact. \end{enumerate} Then for $E$ in $D^+(\mathcal{O}_X)$ we have $(Rf_*E)_y = R\Gamma(f^{-1}(y), E|_{f^{-1}(y)})$ in $D^+(\mathcal{O}_{Y, y})$. \end{lemma} \begin{proof} The base change map of Lemma \ref{lemma-base-change-map-flat-case} gives a canonical map $(Rf_*E)_y \to R\Gamma(f^{-1}(y), E|_{f^{-1}(y)})$. To prove this map is an isomorphism, we represent $E$ by a bounded below complex of injectives $\mathcal{I}^\bullet$. Set $Z = f^{-1}(\{y\})$. The assumptions of Lemma \ref{lemma-cohomology-of-closed} are satisfied, see Topology, Lemma \ref{topology-lemma-separated}. Hence the restrictions $\mathcal{I}^n|_Z$ are acyclic for $\Gamma(Z, -)$. Thus $R\Gamma(Z, E|_Z)$ is represented by the complex $\Gamma(Z, \mathcal{I}^\bullet|_Z)$, see Derived Categories, Lemma \ref{derived-lemma-leray-acyclicity}. In other words, we have to show the map $$\colim_V \mathcal{I}^\bullet(f^{-1}(V)) \longrightarrow \Gamma(Z, \mathcal{I}^\bullet|_Z)$$ is an isomorphism. Using Lemma \ref{lemma-cohomology-of-closed} we see that it suffices to show that the collection of open neighbourhoods $f^{-1}(V)$ of $Z = f^{-1}(\{y\})$ is cofinal in the system of all open neighbourhoods. If $f^{-1}(\{y\}) \subset U$ is an open neighbourhood, then as $f$ is closed the set $V = Y \setminus f(X \setminus U)$ is an open neighbourhood of $y$ with $f^{-1}(V) \subset U$. This proves the lemma. \end{proof} \begin{theorem}[Proper base change] \label{theorem-proper-base-change} \begin{reference} \cite[Expose V bis, 4.1.1]{SGA4} \end{reference} Consider a cartesian square of topological spaces $$\xymatrix{ X' = Y' \times_Y X \ar[d]_{f'} \ar[r]_-{g'} & X \ar[d]^f \\ Y' \ar[r]^g & Y }$$ Assume that $f$ is proper and separated. Let $E$ be an object of $D^+(X)$. Then the base change map $$g^{-1}Rf_*E \longrightarrow Rf'_*(g')^{-1}E$$ of Lemma \ref{lemma-base-change-map-flat-case} is an isomorphism in $D^+(Y')$. \end{theorem} \begin{proof} Let $y' \in Y'$ be a point with image $y \in Y$. It suffices to show that the base change map induces an isomorphism on stalks at $y'$. As $f$ is proper it follows that $f'$ is proper, the fibres of $f$ and $f'$ are quasi-compact and $f$ and $f'$ are closed, see Topology, Theorem \ref{topology-theorem-characterize-proper}. Moreover $f'$ is separated by Topology, Lemma \ref{topology-lemma-base-change-separated}. Thus we can apply Lemma \ref{lemma-proper-base-change} twice to see that $$(Rf'_*(g')^{-1}E)_{y'} = R\Gamma((f')^{-1}(y'), (g')^{-1}E|_{(f')^{-1}(y')})$$ and $$(Rf_*E)_y = R\Gamma(f^{-1}(y), E|_{f^{-1}(y)})$$ The induced map of fibres $(f')^{-1}(y') \to f^{-1}(y)$ is a homeomorphism of topological spaces and the pull back of $E|_{f^{-1}(y)}$ is $(g')^{-1}E|_{(f')^{-1}(y')}$. The desired result follows. \end{proof} \begin{lemma}[Proper base change for sheaves of sets] \label{lemma-proper-base-change-sheaves-of-sets} Consider a cartesian square of topological spaces $$\xymatrix{ X' \ar[d]_{f'} \ar[r]_-{g'} & X \ar[d]^f \\ Y' \ar[r]^g & Y }$$ Assume that $f$ is proper and separated. Then $g^{-1}f_*\mathcal{F} = f'_*(g')^{-1}\mathcal{F}$ for any sheaf of sets $\mathcal{F}$ on $X$. \end{lemma} \begin{proof} We argue exactly as in the proof of Theorem \ref{theorem-proper-base-change} and we find it suffices to show $(f_*\mathcal{F})_y = \Gamma(X_y, \mathcal{F}|_{X_y})$. Then we argue as in Lemma \ref{lemma-proper-base-change} to reduce this to the $p = 0$ case of Lemma \ref{lemma-cohomology-of-closed} for sheaves of sets. The first part of the proof of Lemma \ref{lemma-cohomology-of-closed} works for sheaves of sets and this finishes the proof. Some details omitted. \end{proof} \section{Cohomology and colimits} \label{section-limits} \noindent Let $X$ be a ringed space. Let $(\mathcal{F}_i, \varphi_{ii'})$ be a system of sheaves of $\mathcal{O}_X$-modules over the directed set $I$, see Categories, Section \ref{categories-section-posets-limits}. Since for each $i$ there is a canonical map $\mathcal{F}_i \to \colim_i \mathcal{F}_i$ we get a canonical map $$\colim_i H^p(X, \mathcal{F}_i) \longrightarrow H^p(X, \colim_i \mathcal{F}_i)$$ for every $p \geq 0$. Of course there is a similar map for every open $U \subset X$. These maps are in general not isomorphisms, even for $p = 0$. In this section we generalize the results of Sheaves, Lemma \ref{sheaves-lemma-directed-colimits-sections}. See also Modules, Lemma \ref{modules-lemma-finite-presentation-quasi-compact-colimit} (in the special case $\mathcal{G} = \mathcal{O}_X$). \begin{lemma} \label{lemma-quasi-separated-cohomology-colimit} Let $X$ be a ringed space. Assume that the underlying topological space of $X$ has the following properties: \begin{enumerate} \item there exists a basis of quasi-compact open subsets, and \item the intersection of any two quasi-compact opens is quasi-compact. \end{enumerate} Then for any directed system $(\mathcal{F}_i, \varphi_{ii'})$ of sheaves of $\mathcal{O}_X$-modules and for any quasi-compact open $U \subset X$ the canonical map $$\colim_i H^q(U, \mathcal{F}_i) \longrightarrow H^q(U, \colim_i \mathcal{F}_i)$$ is an isomorphism for every $q \geq 0$. \end{lemma} \begin{proof} It is important in this proof to argue for all quasi-compact opens $U \subset X$ at the same time. The result is true for $i = 0$ and any quasi-compact open $U \subset X$ by Sheaves, Lemma \ref{sheaves-lemma-directed-colimits-sections} (combined with Topology, Lemma \ref{topology-lemma-topology-quasi-separated-scheme}). Assume that we have proved the result for all $q \leq q_0$ and let us prove the result for $q = q_0 + 1$. \medskip\noindent By our conventions on directed systems the index set $I$ is directed, and any system of $\mathcal{O}_X$-modules $(\mathcal{F}_i, \varphi_{ii'})$ over $I$ is directed. By Injectives, Lemma \ref{injectives-lemma-sheaves-modules-space} the category of $\mathcal{O}_X$-modules has functorial injective embeddings. Thus for any system $(\mathcal{F}_i, \varphi_{ii'})$ there exists a system $(\mathcal{I}_i, \varphi_{ii'})$ with each $\mathcal{I}_i$ an injective $\mathcal{O}_X$-module and a morphism of systems given by injective $\mathcal{O}_X$-module maps $\mathcal{F}_i \to \mathcal{I}_i$. Denote $\mathcal{Q}_i$ the cokernel so that we have short exact sequences $$0 \to \mathcal{F}_i \to \mathcal{I}_i \to \mathcal{Q}_i \to 0.$$ We claim that the sequence $$0 \to \colim_i \mathcal{F}_i \to \colim_i \mathcal{I}_i \to \colim_i \mathcal{Q}_i \to 0.$$ is also a short exact sequence of $\mathcal{O}_X$-modules. We may check this on stalks. By Sheaves, Sections \ref{sheaves-section-limits-presheaves} and \ref{sheaves-section-limits-sheaves} taking stalks commutes with colimits. Since a directed colimit of short exact sequences of abelian groups is short exact (see Algebra, Lemma \ref{algebra-lemma-directed-colimit-exact}) we deduce the result. We claim that $H^q(U, \colim_i \mathcal{I}_i) = 0$ for all quasi-compact open $U \subset X$ and all $q \geq 1$. Accepting this claim for the moment consider the diagram $$\xymatrix{ \colim_i H^{q_0}(U, \mathcal{I}_i) \ar[d] \ar[r] & \colim_i H^{q_0}(U, \mathcal{Q}_i) \ar[d] \ar[r] & \colim_i H^{q_0 + 1}(U, \mathcal{F}_i) \ar[d] \ar[r] & 0 \ar[d] \\ H^{q_0}(U, \colim_i \mathcal{I}_i) \ar[r] & H^{q_0}(U, \colim_i \mathcal{Q}_i) \ar[r] & H^{q_0 + 1}(U, \colim_i \mathcal{F}_i) \ar[r] & 0 }$$ The zero at the lower right corner comes from the claim and the zero at the upper right corner comes from the fact that the sheaves $\mathcal{I}_i$ are injective. The top row is exact by an application of Algebra, Lemma \ref{algebra-lemma-directed-colimit-exact}. Hence by the snake lemma we deduce the result for $q = q_0 + 1$. \medskip\noindent It remains to show that the claim is true. We will use Lemma \ref{lemma-cech-vanish-basis}. Let $\mathcal{B}$ be the collection of all quasi-compact open subsets of $X$. This is a basis for the topology on $X$ by assumption. Let $\text{Cov}$ be the collection of finite open coverings $\mathcal{U} : U = \bigcup_{j = 1, \ldots, m} U_j$ with each of $U$, $U_j$ quasi-compact open in $X$. By the result for $q = 0$ we see that for $\mathcal{U} \in \text{Cov}$ we have $$\check{\mathcal{C}}^\bullet(\mathcal{U}, \colim_i \mathcal{I}_i) = \colim_i \check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{I}_i)$$ because all the multiple intersections $U_{j_0 \ldots j_p}$ are quasi-compact. By Lemma \ref{lemma-injective-trivial-cech} each of the complexes in the colimit of {\v C}ech complexes is acyclic in degree $\geq 1$. Hence by Algebra, Lemma \ref{algebra-lemma-directed-colimit-exact} we see that also the {\v C}ech complex $\check{\mathcal{C}}^\bullet(\mathcal{U}, \colim_i \mathcal{I}_i)$ is acyclic in degrees $\geq 1$. In other words we see that $\check{H}^p(\mathcal{U}, \colim_i \mathcal{I}_i) = 0$ for all $p \geq 1$. Thus the assumptions of Lemma \ref{lemma-cech-vanish-basis} are satisfied and the claim follows. \end{proof} \noindent Next we formulate the analogy of Sheaves, Lemma \ref{sheaves-lemma-descend-opens} for cohomology. Let $X$ be a spectral space which is written as a cofiltered limit of spectral spaces $X_i$ for a diagram with spectral transition morphisms as in Topology, Lemma \ref{topology-lemma-directed-inverse-limit-spectral-spaces}. Assume given \begin{enumerate} \item an abelian sheaf $\mathcal{F}_i$ on $X_i$ for all $i \in \Ob(\mathcal{I})$, \item for $a : j \to i$ an $f_a$-map $\varphi_a : \mathcal{F}_i \to \mathcal{F}_j$ of abelian sheaves (see Sheaves, Definition \ref{sheaves-definition-f-map}) \end{enumerate} such that $\varphi_c = \varphi_b \circ \varphi_a$ whenever $c = a \circ b$. Set $\mathcal{F} = \colim p_i^{-1}\mathcal{F}_i$ on $X$. \begin{lemma} \label{lemma-colimit} In the situation discussed above. Let $i \in \Ob(\mathcal{I})$ and let $U_i \subset X_i$ be quasi-compact open. Then $$\colim_{a : j \to i} H^p(f_a^{-1}(U_i), \mathcal{F}_j) = H^p(p_i^{-1}(U_i), \mathcal{F})$$ for all $p \geq 0$. In particular we have $H^p(X, \mathcal{F}) = \colim H^p(X_i, \mathcal{F}_i)$. \end{lemma} \begin{proof} The case $p = 0$ is Sheaves, Lemma \ref{sheaves-lemma-descend-opens}. \medskip\noindent In this paragraph we show that we can find a map of systems $(\gamma_i) : (\mathcal{F}_i, \varphi_a) \to (\mathcal{G}_i, \psi_a)$ with $\mathcal{G}_i$ an injective abelian sheaf and $\gamma_i$ injective. For each $i$ we pick an injection $\mathcal{F}_i \to \mathcal{I}_i$ where $\mathcal{I}_i$ is an injective abelian sheaf on $X_i$. Then we can consider the family of maps $$\gamma_i : \mathcal{F}_i \longrightarrow \prod\nolimits_{b : k \to i} f_{b, *}\mathcal{I}_k = \mathcal{G}_i$$ where the component maps are the maps adjoint to the maps $f_b^{-1}\mathcal{F}_i \to \mathcal{F}_k \to \mathcal{I}_k$. For $a : j \to i$ in $\mathcal{I}$ there is a canonical map $$\psi_a : f_a^{-1}\mathcal{G}_i \to \mathcal{G}_j$$ whose components are the canonical maps $f_b^{-1}f_{a \circ b, *}\mathcal{I}_k \to f_{b, *}\mathcal{I}_k$ for $b : k \to j$. Thus we find an injection $\{\gamma_i\} : \{\mathcal{F}_i, \varphi_a) \to (\mathcal{G}_i, \psi_a)$ of systems of abelian sheaves. Note that $\mathcal{G}_i$ is an injective sheaf of abelian groups on $\mathcal{C}_i$, see Lemma \ref{lemma-pushforward-injective-flat} and Homology, Lemma \ref{homology-lemma-product-injectives}. This finishes the construction. \medskip\noindent Arguing exactly as in the proof of Lemma \ref{lemma-quasi-separated-cohomology-colimit} we see that it suffices to prove that $H^p(X, \colim f_i^{-1}\mathcal{G}_i) = 0$ for $p > 0$. \medskip\noindent Set $\mathcal{G} = \colim f_i^{-1}\mathcal{G}_i$. To show vanishing of cohomology of $\mathcal{G}$ on every quasi-compact open of $X$, it suffices to show that the {\v C}ech cohomology of $\mathcal{G}$ for any covering $\mathcal{U}$ of a quasi-compact open of $X$ by finitely many quasi-compact opens is zero, see Lemma \ref{lemma-cech-vanish-basis}. Such a covering is the inverse by $p_i$ of such a covering $\mathcal{U}_i$ on the space $X_i$ for some $i$ by Topology, Lemma \ref{topology-lemma-descend-opens}. We have $$\check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{G}) = \colim_{a : j \to i} \check{\mathcal{C}}^\bullet(f_a^{-1}(\mathcal{U}_i), \mathcal{G}_j)$$ by the case $p = 0$. The right hand side is a filtered colimit of complexes each of which is acyclic in positive degrees by Lemma \ref{lemma-injective-trivial-cech}. Thus we conclude by Algebra, Lemma \ref{algebra-lemma-directed-colimit-exact}. \end{proof} \section{Vanishing on Noetherian topological spaces} \label{section-vanishing-Noetherian} \noindent The aim is to prove a theorem of Grothendieck namely Proposition \ref{proposition-vanishing-Noetherian}. See \cite{Tohoku}. \begin{lemma} \label{lemma-cohomology-and-closed-immersions} Let $i : Z \to X$ be a closed immersion of topological spaces. For any abelian sheaf $\mathcal{F}$ on $Z$ we have $H^p(Z, \mathcal{F}) = H^p(X, i_*\mathcal{F})$. \end{lemma} \begin{proof} This is true because $i_*$ is exact (see Modules, Lemma \ref{modules-lemma-i-star-exact}), and hence $R^pi_* = 0$ as a functor (Derived Categories, Lemma \ref{derived-lemma-right-derived-exact-functor}). Thus we may apply Lemma \ref{lemma-apply-Leray}. \end{proof} \begin{lemma} \label{lemma-irreducible-constant-cohomology-zero} Let $X$ be an irreducible topological space. Then $H^p(X, \underline{A}) = 0$ for all $p > 0$ and any abelian group $A$. \end{lemma} \begin{proof} Recall that $\underline{A}$ is the constant sheaf as defined in Sheaves, Definition \ref{sheaves-definition-constant-sheaf}. It is clear that for any nonempty open $U \subset X$ we have $\underline{A}(U) = A$ as $X$ is irreducible (and hence $U$ is connected). We will show that the higher {\v C}ech cohomology groups $\check{H}^p(\mathcal{U}, \underline{A})$ are zero for any open covering $\mathcal{U} : U = \bigcup_{i\in I} U_i$ of an open $U \subset X$. Then the lemma will follow from Lemma \ref{lemma-cech-vanish}. \medskip\noindent Recall that the value of an abelian sheaf on the empty open set is $0$. Hence we may clearly assume $U_i \not = \emptyset$ for all $i \in I$. In this case we see that $U_i \cap U_{i'} \not = \emptyset$ for all $i, i' \in I$. Hence we see that the {\v C}ech complex is simply the complex $$\prod_{i_0 \in I} A \to \prod_{(i_0, i_1) \in I^2} A \to \prod_{(i_0, i_1, i_2) \in I^3} A \to \ldots$$ We have to see this has trivial higher cohomology groups. We can see this for example because this is the {\v C}ech complex for the covering of a $1$-point space and {\v C}ech cohomology agrees with cohomology on such a space. (You can also directly verify it by writing an explicit homotopy.) \end{proof} \begin{lemma} \label{lemma-subsheaf-of-constant-sheaf} \begin{reference} \cite[Page 168]{Tohoku}. \end{reference} Let $X$ be a topological space such that the intersection of any two quasi-compact opens is quasi-compact. Let $\mathcal{F} \subset \underline{\mathbf{Z}}$ be a subsheaf generated by finitely many sections over quasi-compact opens. Then there exists a finite filtration $$(0) = \mathcal{F}_0 \subset \mathcal{F}_1 \subset \ldots \subset \mathcal{F}_n = \mathcal{F}$$ by abelian subsheaves such that for each $0 < i \leq n$ there exists a short exact sequence $$0 \to j'_!\underline{\mathbf{Z}}_V \to j_!\underline{\mathbf{Z}}_U \to \mathcal{F}_i/\mathcal{F}_{i - 1} \to 0$$ with $j : U \to X$ and $j' : V \to X$ the inclusion of quasi-compact opens into $X$. \end{lemma} \begin{proof} Say $\mathcal{F}$ is generated by the sections $s_1, \ldots, s_t$ over the quasi-compact opens $U_1, \ldots, U_t$. Since $U_i$ is quasi-compact and $s_i$ a locally constant function to $\mathbf{Z}$ we may assume, after possibly replacing $U_i$ by the parts of a finite decomposition into open and closed subsets, that $s_i$ is a constant section. Say $s_i = n_i$ with $n_i \in \mathbf{Z}$. Of course we can remove $(U_i, n_i)$ from the list if $n_i = 0$. Flipping signs if necessary we may also assume $n_i > 0$. Next, for any subset $I \subset \{1, \ldots, t\}$ we may add $\bigcup_{i \in I} U_i$ and $\gcd(n_i, i \in I)$ to the list. After doing this we see that our list $(U_1, n_1), \ldots, (U_t, n_t)$ satisfies the following property: For $x \in X$ set \$I_x = \{i \in \{1, \ldots, t\} \mid x