## 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 $\textit{Mod}(\mathcal{O}_ X) \to \textit{Mod}(\mathcal{O}_ Y)$.

## Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like `$\pi$`

). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

## Comments (4)

Comment #350 by Fan on

Comment #353 by Johan on

Comment #4526 by Théo de Oliveira Santos on

Comment #4742 by Johan on