The Stacks project

\begin{equation*} \DeclareMathOperator\Coim{Coim} \DeclareMathOperator\Coker{Coker} \DeclareMathOperator\Ext{Ext} \DeclareMathOperator\Hom{Hom} \DeclareMathOperator\Im{Im} \DeclareMathOperator\Ker{Ker} \DeclareMathOperator\Mor{Mor} \DeclareMathOperator\Ob{Ob} \DeclareMathOperator\Sh{Sh} \DeclareMathOperator\SheafExt{\mathcal{E}\mathit{xt}} \DeclareMathOperator\SheafHom{\mathcal{H}\mathit{om}} \DeclareMathOperator\Spec{Spec} \newcommand\colim{\mathop{\mathrm{colim}}\nolimits} \newcommand\lim{\mathop{\mathrm{lim}}\nolimits} \newcommand\Qcoh{\mathit{Qcoh}} \newcommand\Sch{\mathit{Sch}} \newcommand\QCohstack{\mathcal{QC}\!\mathit{oh}} \newcommand\Cohstack{\mathcal{C}\!\mathit{oh}} \newcommand\Spacesstack{\mathcal{S}\!\mathit{paces}} \newcommand\Quotfunctor{\mathrm{Quot}} \newcommand\Hilbfunctor{\mathrm{Hilb}} \newcommand\Curvesstack{\mathcal{C}\!\mathit{urves}} \newcommand\Polarizedstack{\mathcal{P}\!\mathit{olarized}} \newcommand\Complexesstack{\mathcal{C}\!\mathit{omplexes}} \newcommand\Pic{\mathop{\mathrm{Pic}}\nolimits} \newcommand\Picardstack{\mathcal{P}\!\mathit{ic}} \newcommand\Picardfunctor{\mathrm{Pic}} \newcommand\Deformationcategory{\mathcal{D}\!\mathit{ef}} \end{equation*}

20.3 Cohomology of sheaves

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 19.4.1. Hence we can choose an injective resolution $\mathcal{F}[0] \to \mathcal{I}^\bullet $. As is customary we define

20.3.0.1
\begin{equation} \label{cohomology-equation-cohomology} H^ i(X, \mathcal{F}) = H^ i(\Gamma (X, \mathcal{I}^\bullet )) \end{equation}

to be the $i$th cohomology group of the abelian sheaf $\mathcal{F}$. The family of functors $H^ i((X, -)$ forms a universal $\delta $-functor from $\textit{Ab}(X) \to \textit{Ab}$.

Let $f : X \to Y$ be a continuous map of topological spaces. With $\mathcal{F}[0] \to \mathcal{I}^\bullet $ as above we define

20.3.0.2
\begin{equation} \label{cohomology-equation-higher-direct-image} R^ if_*\mathcal{F} = H^ i(f_*\mathcal{I}^\bullet ) \end{equation}

to be the $i$th higher direct image of $\mathcal{F}$. The family of functors $R^ if_*$ forms a universal $\delta $-functor from $\textit{Ab}(X) \to \textit{Ab}(Y)$.

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 19.5.1. Hence we can choose an injective resolution $\mathcal{F}[0] \to \mathcal{I}^\bullet $. As is customary we define

20.3.0.3
\begin{equation} \label{cohomology-equation-cohomology-modules} H^ i(X, \mathcal{F}) = H^ i(\Gamma (X, \mathcal{I}^\bullet )) \end{equation}

to be the $i$th cohomology group of $\mathcal{F}$. The family of functors $H^ i((X, -)$ forms a universal $\delta $-functor from $\textit{Mod}(\mathcal{O}_ X) \to \text{Mod}_{\mathcal{O}_ X(X)}$.

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

20.3.0.4
\begin{equation} \label{cohomology-equation-higher-direct-image-modules} R^ if_*\mathcal{F} = H^ i(f_*\mathcal{I}^\bullet ) \end{equation}

to be the $i$th higher direct image of $\mathcal{F}$. The family of functors $R^ if_*$ forms a universal $\delta $-functor from $\textit{Mod}(\mathcal{O}_ X) \to \textit{Mod}(\mathcal{O}_ Y)$.


Comments (2)

Comment #350 by Fan on

Two lines below (20.3.0.2): is a functor from or ?

Comment #353 by on

This is correct as is. Here means taking the th cohomology sheaf of the object of the derived category of . Thus it is an object of .


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