## 21.2 Cohomology of sheaves

Let $\mathcal{C}$ be a site, see Sites, Definition 7.6.2. Let $\mathcal{F}$ be an abelian sheaf on $\mathcal{C}$. We know that the category of abelian sheaves on $\mathcal{C}$ has enough injectives, see Injectives, Theorem 19.7.4. Hence we can choose an injective resolution $\mathcal{F} \to \mathcal{I}^\bullet$. For any object $U$ of the site $\mathcal{C}$ we define

21.2.0.1
\begin{equation} \label{sites-cohomology-equation-cohomology-object-site} H^ i(U, \mathcal{F}) = H^ i(\Gamma (U, \mathcal{I}^\bullet )) \end{equation}

to be the $i$th cohomology group of the abelian sheaf $\mathcal{F}$ over the object $U$. In other words, these are the right derived functors of the functor $\mathcal{F} \mapsto \mathcal{F}(U)$. The family of functors $H^ i(U, -)$ forms a universal $\delta$-functor $\textit{Ab}(\mathcal{C}) \to \textit{Ab}$.

It sometimes happens that the site $\mathcal{C}$ does not have a final object. In this case we define the global sections of a presheaf of sets $\mathcal{F}$ over $\mathcal{C}$ to be the set

21.2.0.2
\begin{equation} \label{sites-cohomology-equation-global-sections} \Gamma (\mathcal{C}, \mathcal{F}) = \mathop{\mathrm{Mor}}\nolimits _{\textit{PSh}(\mathcal{C})}(e, \mathcal{F}) \end{equation}

where $e$ is a final object in the category of presheaves on $\mathcal{C}$. In this case, given an abelian sheaf $\mathcal{F}$ on $\mathcal{C}$, we define the $i$th cohomology group of $\mathcal{F}$ on $\mathcal{C}$ as follows

21.2.0.3
\begin{equation} \label{sites-cohomology-equation-cohomology} H^ i(\mathcal{C}, \mathcal{F}) = H^ i(\Gamma (\mathcal{C}, \mathcal{I}^\bullet )) \end{equation}

in other words, it is the $i$th right derived functor of the global sections functor. The family of functors $H^ i(\mathcal{C}, -)$ forms a universal $\delta$-functor $\textit{Ab}(\mathcal{C}) \to \textit{Ab}$.

Let $f : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{D})$ be a morphism of topoi, see Sites, Definition 7.15.1. With $\mathcal{F} \to \mathcal{I}^\bullet$ as above we define

21.2.0.4
\begin{equation} \label{sites-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 $\mathcal{F}$. These are the right derived functors of $f_*$. The family of functors $R^ if_*$ forms a universal $\delta$-functor from $\textit{Ab}(\mathcal{C}) \to \textit{Ab}(\mathcal{D})$.

Let $(\mathcal{C}, \mathcal{O})$ be a ringed site, see Modules on Sites, Definition 18.6.1. Let $\mathcal{F}$ be an $\mathcal{O}$-module. We know that the category of $\mathcal{O}$-modules has enough injectives, see Injectives, Theorem 19.8.4. Hence we can choose an injective resolution $\mathcal{F} \to \mathcal{I}^\bullet$. For any object $U$ of the site $\mathcal{C}$ we define

21.2.0.5
\begin{equation} \label{sites-cohomology-equation-cohomology-object-site-modules} H^ i(U, \mathcal{F}) = H^ i(\Gamma (U, \mathcal{I}^\bullet )) \end{equation}

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

21.2.0.6
\begin{equation} \label{sites-cohomology-equation-cohomology-modules} H^ i(\mathcal{C}, \mathcal{F}) = H^ i(\Gamma (\mathcal{C}, \mathcal{I}^\bullet )) \end{equation}

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

Let $f : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}), \mathcal{O}')$ be a morphism of ringed topoi, see Modules on Sites, Definition 18.7.1. With $\mathcal{F} \to \mathcal{I}^\bullet$ as above we define

21.2.0.7
\begin{equation} \label{sites-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 $\mathcal{F}$. These are the right derived functors of $f_*$. The family of functors $R^ if_*$ forms a universal $\delta$-functor from $\textit{Mod}(\mathcal{O}) \to \textit{Mod}(\mathcal{O}')$.

Comment #3238 by Dario Weißmann on

In the second sentence fo the introduction: a abelian

Comment #5118 by Weixiao Lu on

If $f: Sh(\mathcal C) \to Sh(\mathcal D)$ is a morphism of topoi, then $f_\ast(\mathcal F)$ might just be a sheaf of sets, not a sheaf of abelian groups, even if $\mathcal F$ is. Then how do we define $Rf_\ast(\mathcal F)$?

Comment #5325 by on

@#5118: No, the pushforward of an abelian sheaf is always considered as an abelian sheaf. This is discussed (a bit) in Section 7.44 and a lot more in Chapter 18.

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