Cohomology of Sheaves

en · official · sha256:992dfb482cc74c054eb720fd53a56dd386827983f1cb4214bf3e7230a56ea9d3

\section{Cohomology of sheaves} \label{section-cohomology-sheaves} \noindent Let $X$ be a topological space. Let $\mathcal{F}$ be an 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 $\textit{Mod}(\mathcal{O}_X) \to \textit{Mod}(\mathcal{O}_Y)$.

Referencing norms: CohomologyIsDerivedGlobalSections, CohomologyUniversalDeltaFunctor, CohomologyVanishesForAcyclic, LongExactCohomologySequence, abelian_groups_category

en · official · sha256:95b540b6925fdab8fa3a53def4a39d102181a21ef47fcce25d076b6a97bc0b1a

\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}

Referencing norms: CechH0Order, CechH1Dimension, CechH1Order, CompatiblePair

en · official · sha256:4c5b22ab222601aaae95952557fde5950661f40d81749b9bb69b4a8f61ff6108

\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}

Referencing norms: CechH0MatchesSections

en · official · sha256:4dbf509c03629779d780d60d6502aae3001874670f84eb6bc4cd9e0dab533882

\begin{definition} \label{definition-ordered-cech-complex} Let $X$ be a topological space. Let $\mathcal{U} : U = \bigcup_{i \in I} U_i$ be an open covering. Assume given a total ordering on $I$. Let $\mathcal{F}$ be an abelian presheaf on $X$. The complex $\check{\mathcal{C}}_{ord}^\bullet(\mathcal{U}, \mathcal{F})$ is the {\it ordered {\v C}ech complex} associated to $\mathcal{F}$, the open covering $\mathcal{U}$ and the given total ordering on $I$. \end{definition}

Referencing norms: CechC0Order, CechC1Order, CechH0Order, CechH1Order, CompatiblePair, first_member, second_member

en · official · sha256:783d414a41461bccfb07b242a527657ee8fda1055c33a6ef41abf92ee925f966

\begin{lemma} \label{lemma-alternating-usual} Let $X$ be a topological space. Let $\mathcal{U} : U = \bigcup_{i \in I} U_i$ be an open covering. Assume $I$ comes equipped with a total ordering. The map $c \circ \pi$ is homotopic to the identity on $\check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{F})$. In particular the inclusion map $\check{\mathcal{C}}_{alt}^\bullet(\mathcal{U}, \mathcal{F}) \to \check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{F})$ is a homotopy equivalence. \end{lemma}

Referencing norms: CechH1Order

en · official · sha256:f61344b89b5447a2349b548070137ca4e6f491c90033bf637527fbd9f371c876

\section{Derived functors} \label{section-derived-functors} \noindent We briefly explain how to get right derived functors using resolution functors. For the unbounded derived functors, please see Section \ref{section-unbounded}. \medskip\noindent 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 $$ K(\mathcal{O}_X) = K(\textit{Mod}(\mathcal{O}_X)) \quad \text{and} \quad D(\mathcal{O}_X) = D(\textit{Mod}(\mathcal{O}_X)). $$ 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 $\mathcal{F}^\bullet \in D(\mathcal{O}_X)$, 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 $D(\mathcal{O}_X)$'' does not mean that $\alpha$ is represented by a morphism of complexes. If we have an actual morphism of complexes we will say so.

Referencing norms: CohomologyIsDerivedGlobalSections, GlobalSectionsBetween, GlobalSectionsLeftExact, H0IsGlobalSections, global_sections_functor

en · official · sha256:730a7c4f799967f2799cc62798491a402948504907086636e81743211b35b8b0

\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}

Referencing norms: FlasqueByRestrictions, restrictions_surjective

en · official · sha256:d2662eefd4a3574b5c328b30b8abcb92218c774fd5994f253a2e2e7e4820d036

\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}

Referencing norms: InjectiveSheafIsFlasque

en · official · sha256:707ef368c24b80d3d736e07759dff795af4faae92c0ece2f17999e27898ff0d2

\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}

Referencing norms: CohomologyVanishesForAcyclic, FlasqueComplexIsAcyclicTermwise, FlasqueIsAcyclic, terms_flasque

en · official · sha256:8b044713268f8366ca6cc0212d88985ff0517eb2417f464d218f454e93c283fe

\begin{lemma} \label{lemma-godement-resolution} Let $(X, \mathcal{O}_X)$ be a ringed space. For every sheaf of $\mathcal{O}_X$-modules $\mathcal{F}$ there is a resolution $$ 0 \to \mathcal{F} \to f_*f^*\mathcal{F} \to f_*f^*f_*f^*\mathcal{F} \to f_*f^*f_*f^*f_*f^*\mathcal{F} \to \ldots $$ functorial in $\mathcal{F}$ such that each term $f_*f^* \ldots f_*f^*\mathcal{F}$ is a flasque $\mathcal{O}_X$-module and such that for all $x \in X$ the map $$ \mathcal{F}_x[0] \to \Big( (f_*f^*\mathcal{F})_x \to (f_*f^*f_*f^*\mathcal{F})_x \to (f_*f^*f_*f^*f_*f^*\mathcal{F})_x \to \ldots \Big) $$ is a homotopy equivalence in the category of complexes of $\mathcal{O}_{X, x}$-modules. \end{lemma}

Referencing norms: FlasqueComplexIsAcyclicTermwise, GodementResolutionExists, terms_flasque