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20.2 Cohomology of sheaves

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

20.2.0.1
\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

20.2.0.2
\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

20.2.0.3
\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

20.2.0.4
\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 (4)

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 .

Comment #4526 by Théo de Oliveira Santos on

A few trivial typos: * Let F be a abelian sheaf * Extra parenthesis: The family of functors forms [...] from * Extra parenthesis: The family of functors forms [...] from


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