The Stacks project

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

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

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

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

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