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

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

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

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

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

Comment #350 by Fan on

Two lines below (20.3.0.2): is $R^i f_*$ a functor from $Ab(X) \to Ab(Y)$ or $Ab(X) \to Ab$?

Comment #353 by on

This is correct as is. Here $H^i(f_*\mathcal{I}^\bullet)$ means taking the $i$th cohomology sheaf of the object $Rf_*\mathcal{F}$ of the derived category of $Y$. Thus it is an object of $Ab(Y)$.

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